--- /srv/rebuilderd/tmp/rebuilderdQDImcZ/inputs/macaulay2-common_1.26.05+ds-2_all.deb +++ /srv/rebuilderd/tmp/rebuilderdQDImcZ/out/macaulay2-common_1.26.05+ds-2_all.deb ├── file list │ @@ -1,3 +1,3 @@ │ -rw-r--r-- 0 0 0 4 2026-05-18 11:29:46.000000 debian-binary │ --rw-r--r-- 0 0 0 559032 2026-05-18 11:29:46.000000 control.tar.xz │ --rw-r--r-- 0 0 0 32686888 2026-05-18 11:29:46.000000 data.tar.xz │ +-rw-r--r-- 0 0 0 559156 2026-05-18 11:29:46.000000 control.tar.xz │ +-rw-r--r-- 0 0 0 32687740 2026-05-18 11:29:46.000000 data.tar.xz ├── control.tar.xz │ ├── control.tar │ │ ├── ./control │ │ │ @@ -1,13 +1,13 @@ │ │ │ Package: macaulay2-common │ │ │ Source: macaulay2 │ │ │ Version: 1.26.05+ds-2 │ │ │ Architecture: all │ │ │ Maintainer: Debian Math Team │ │ │ -Installed-Size: 318738 │ │ │ +Installed-Size: 318743 │ │ │ Depends: fonts-katex (>= 0.16.10+~cs6.1.0), libjs-bootsidemenu (>= 2.2.2), libjs-bootstrap5 (>= 5.3.8+dfsg), libjs-d3 (>= 3.5.17), libjs-jquery (>= 3.7.1+dfsg+~3.5.33), libjs-katex (>= 0.16.10+~cs6.1.0), libjs-nouislider (>= 15.8.1+ds), libjs-three (>= 111+dfsg1), node-clipboard (>= 2.0.11+ds+ds+~cs7.6.4), node-fortawesome-fontawesome-free (>= 7.2.0+ds1) │ │ │ Section: math │ │ │ Priority: optional │ │ │ Multi-Arch: foreign │ │ │ Homepage: http://macaulay2.com │ │ │ Description: Software system for algebraic geometry research (common files) │ │ │ Macaulay 2 is a software system for algebraic geometry research, written by │ │ ├── ./md5sums │ │ │ ├── ./md5sums │ │ │ │┄ Files differ ├── data.tar.xz │ ├── data.tar │ │ ├── file list │ │ │ @@ -3550,25 +3550,25 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37097 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1918 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1748 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ -rw-r--r-- 0 root (0) root (0) 558 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_expected__Dimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 558 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_linear__System__On__Rational__Surface.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1889 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_parametrization.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1888 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_parametrization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1272 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_rational__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1596 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_slow__Adjunction__Calculation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2944 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_special__Families__Of__Sommese__Vande__Ven.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 23 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 9420 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11531 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6822 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_expected__Dimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7460 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_linear__System__On__Rational__Surface.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9782 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9781 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9943 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_rational__Surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9521 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_slow__Adjunction__Calculation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12686 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_special__Families__Of__Sommese__Vande__Ven.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14004 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8478 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4639 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/AlgebraicSplines/ │ │ │ @@ -3804,18 +3804,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 77440 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4428 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3111 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Benchmark/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2927 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 423 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 433 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Benchmark/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 29 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Benchmark/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 5776 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5786 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5435 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Benchmark/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4444 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Benchmark/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3114 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Benchmark/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/BernsteinSato/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 289778 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/BernsteinSato/example-output/ │ │ │ @@ -4415,22 +4415,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5493 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Browse/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4576 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Browse/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3259 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Browse/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 19549 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 4571 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4572 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1717 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2553 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_elementary.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1652 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_evans__Griffith.out │ │ │ -rw-r--r-- 0 root (0) root (0) 544 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_is__Syzygy.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 49 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 14242 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14243 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9033 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/html/_bruns__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10670 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/html/_elementary.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8302 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/html/_evans__Griffith.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6745 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/html/_is__Syzygy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9793 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7011 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4277 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Bruns/html/toc.html │ │ │ @@ -4457,28 +4457,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 963 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cells.out │ │ │ -rw-r--r-- 0 root (0) root (0) 574 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cells_lp__Z__Z_cm__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1427 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_complex_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 539 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_describe_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 266 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_dim_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 291 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_dim_lp__Cell_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 790 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_face__Poset_lp__Cell__Complex_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 964 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_hull__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 963 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_hull__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 446 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Cycle.out │ │ │ -rw-r--r-- 0 root (0) root (0) 257 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Free_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 422 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Minimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Simplex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 505 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Well__Defined_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 595 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Well__Defined_lp__Cell_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 363 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_max__Cells.out │ │ │ -rw-r--r-- 0 root (0) root (0) 451 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_net_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 275 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_new__Cell.out │ │ │ -rw-r--r-- 0 root (0) root (0) 228 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_new__Simplex__Cell.out │ │ │ --rw-r--r-- 0 root (0) root (0) 733 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_relabel__Cell__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 734 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_relabel__Cell__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_ring_lp__Cell__Complex_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 517 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_scarf__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 519 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_scarf__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2473 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_skeleton_lp__Z__Z_cm__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 678 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_subcomplex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 821 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_taylor__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 524 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_tex__Math_lp__Cell__Complex_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 39 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 9064 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/___Cell.html │ │ │ @@ -4501,28 +4501,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8003 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_cells.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7185 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_cells_lp__Z__Z_cm__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13019 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_complex_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7286 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_describe_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6264 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_dim_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6590 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_dim_lp__Cell_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7354 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_face__Poset_lp__Cell__Complex_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8628 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_hull__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8627 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_hull__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6838 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Cycle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6980 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Free_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7601 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Minimal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5895 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Simplex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8389 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Well__Defined_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9176 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Well__Defined_lp__Cell_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6658 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_max__Cells.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7480 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_net_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8662 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_new__Cell.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7560 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_new__Simplex__Cell.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8807 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_relabel__Cell__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8808 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_relabel__Cell__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6076 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_ring_lp__Cell__Complex_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7040 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_scarf__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7042 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_scarf__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8632 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_skeleton_lp__Z__Z_cm__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8981 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_subcomplex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6797 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_taylor__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7326 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_tex__Math_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 28785 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21213 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10808 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/toc.html │ │ │ @@ -4540,15 +4540,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1448 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Quasi__Isomorphism.out │ │ │ -rw-r--r-- 0 root (0) root (0) 771 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Quasi__Isomorphism_lp..._cm__Length__Limit_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 278 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_koszul__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1957 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 694 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_nonzero__Max.out │ │ │ -rw-r--r-- 0 root (0) root (0) 684 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_prepend__Zero__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 899 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_remove__Zero__Trailing__Terms.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3451 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3448 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 541 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2570 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_scarf__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 537 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_substitute_lp__Chain__Complex_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 672 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_taylor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1333 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_taylor__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1351 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_trivial__Homological__Truncation.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/ │ │ │ @@ -4571,15 +4571,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5675 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_koszul__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7422 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_koszul__Complex_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10293 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6814 2026-05-18 11:29:46.000000 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./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1503 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1502 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3419 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Toric__Variety__Valid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3442 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chern.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2404 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 775 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Toric__Chow__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2016 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4380 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2015 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4377 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ -rw-r--r-- 0 root (0) root (0) 342 2026-05-18 11:29:46.000000 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./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Output.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3327 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Segre.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1666 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Toric__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_is__Multi__Homogeneous.out │ │ │ -rw-r--r-- 0 root (0) root (0) 802 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_probabilistic_spalgorithm.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 665 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 55 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 23897 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6579 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6578 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11441 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Toric__Variety__Valid.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18381 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Chern.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9779 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6611 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root (0) root (0) 6105 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6213 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6215 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6666 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8579 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Multi__Proj__Coord__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16600 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16912 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Segre.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8892 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Toric__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5112 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_bertini__Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5330 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_configuring_sp__Bertini.html │ │ │ @@ -5006,15 +5006,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 231378 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 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root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Complexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Complexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 740220 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Complexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Complexes/example-output/ │ │ │ @@ -5463,29 +5463,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 21522 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16348 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9720 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50933 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 36624 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 36623 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Gauss_sq_sphypergeometric_spfunction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3314 2026-05-18 11:29:46.000000 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(0) 46681 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8052 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Gauss_sq_sphypergeometric_spfunction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9415 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5813 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_base__Fraction__Field.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7036 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_connection__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6391 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_connection__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7739 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_gauge__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7468 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_gauge__Transform.html │ │ │ @@ -5707,136 +5707,136 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 38722 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CpMackeyFunctors/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35476 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CpMackeyFunctors/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16782 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/CpMackeyFunctors/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cremona/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cremona/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 239171 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cremona/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) 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./usr/share/doc/Macaulay2/Cremona/html/_target_lp__Rational__Map_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7173 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cremona/html/_to__External__String_lp__Rational__Map_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7172 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cremona/html/_to__External__String_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13197 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cremona/html/_to__Map.html │ │ │ --rw-r--r-- 0 root (0) root (0) 66339 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cremona/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 66331 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cremona/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37303 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cremona/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16389 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cremona/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cyclotomic/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cyclotomic/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 6101 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cyclotomic/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cyclotomic/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 455 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cyclotomic/example-output/_cyclotomic__Field.out │ │ │ @@ -5854,15 +5854,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3784 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Cyclotomic/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 881301 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2851 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Base_spchange_spand_sptensor_spwith_spnon-__D__G_sptypes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9363 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.out │ │ │ --rw-r--r-- 0 root (0) root (0) 7201 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebras.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 7200 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebras.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2706 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Module_sp__Maps.out │ │ │ -rw-r--r-- 0 root (0) root (0) 644 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Building_sp__D__G_spalgebras_spfrom_spexisting_sp__D__G_spalgebras.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1278 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Building_sp__D__G_spmodules_cm_spsubmodules_cm_spand_spquotients.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1245 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Computing_spmodule_spdifferentials_spand_spvisualizing_sp__D__G_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1841 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Algebra__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 813 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Algebra__Map_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 884 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Algebra_sp_sl_sp__D__G__Ideal.out │ │ │ @@ -5886,15 +5886,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1264 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Quotient__Module_sp_st_st_sp__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1225 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Quotient__Module_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 938 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Submodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1726 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Submodule_sp_eq_eq_sp__D__G__Submodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1428 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Submodule_sp_pl_sp__D__G__Submodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1210 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Submodule_sp_st_st_sp__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1220 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Submodule_sp_st_st_sp__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 601 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 600 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1463 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1883 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1510 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Module__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1849 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Quotient__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 403 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_us__Z__Z_sp__D__G__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2029 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_us__Z__Z_sp__D__G__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1966 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_us__Z__Z_sp__D__G__Quotient__Module.out │ │ │ @@ -5902,15 +5902,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1900 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Image_cm_spkernel_cm_spand_spcokernel_spof_sp__D__G_spmodule_spmaps.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1758 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Module-like_spoperations_spon_sp__D__G_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1289 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Operations_spon_sp__D__G_sp__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1926 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Operations_spon_sp__D__G_sp__Submodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1201 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Pruning_sp__D__G_spmodules_cm_spsubmodules_cm_spquotients_cm_spand_spmaps.out │ │ │ -rw-r--r-- 0 root (0) root (0) 567 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Ring__Element_sp_pc_sp__D__G__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1644 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Semifree_spresolutions_spof_sp__D__G_spmodules.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5183 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 5182 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1320 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Well-definedness_cm_spacyclicity_cm_spand_spquasi-isomorphism.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2814 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_acyclic__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1894 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_acyclic__Closure_lp..._cm__End__Degree_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2317 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_acyclic__Closure_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1131 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_adjoin__Generators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2499 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_adjoin__Variables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1396 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_ambient.out │ │ │ @@ -5944,18 +5944,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 830 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_generator__Degrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1151 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_generator__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 683 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Basis.out │ │ │ -rw-r--r-- 0 root (0) root (0) 728 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Basis_lp__Z__Z_cm__D__G__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1392 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Boundary__Preimage.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1429 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Boundary__Preimage_lp__D__G__Module_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2072 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Boundary__Preimage_lp__D__G__Module_cm__Vector_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 861 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Deg__N__Module.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 860 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Deg__N__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1535 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Generators.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4462 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Algebra.out │ │ │ --rw-r--r-- 0 root (0) root (0) 852 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4458 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Algebra.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 853 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1583 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class_lp__D__G__Module_cm__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2000 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3481 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology_lp__D__G__Module__Map_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_identity__D__G__Algebra__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2324 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_identity__D__G__Module__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1439 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_image_lp__D__G__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 946 2026-05-18 11:29:46.000000 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./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6660 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_reorder__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24985 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FastMinors/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26906 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FastMinors/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7735 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FastMinors/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 25936 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1848 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1849 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 334 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_affine__Part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 694 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_co1__Fitting.out │ │ │ -rw-r--r-- 0 root (0) root (0) 848 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_gauss__Col.out │ │ │ -rw-r--r-- 0 root (0) root (0) 365 2026-05-18 11:29:46.000000 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6355 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_affine__Part.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7650 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_co1__Fitting.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7579 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_gauss__Col.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8388 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_gotzmann__Test.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7448 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_next__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6051 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_quot__Scheme.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9740 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/index.html │ │ │ @@ -7482,16 +7482,16 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 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--rw-r--r-- 0 root (0) root (0) 20018 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 20015 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7223 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_projective__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7393 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pullback_lp__Equivariant__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7948 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pushforward.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7166 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_set__Indicator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5918 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_trivial__K__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5739 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_underlying__Graph_lp__Moment__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27022 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/index.html │ │ │ @@ -8684,42 +8684,42 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 71461 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Graphs/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 57648 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Graphs/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27600 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Graphs/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 56295 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 33163 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 33164 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1505 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_find__Weight__Constraints.out │ │ │ -rw-r--r-- 0 root (0) root (0) 598 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_find__Weight__Vector.out │ │ │ -rw-r--r-- 0 root (0) root (0) 30368 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_groebner__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2709 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_groebner__Stratum.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_linear__Part.out │ │ │ --rw-r--r-- 0 root (0) root (0) 13202 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out │ │ │ --rw-r--r-- 0 root (0) root (0) 14750 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13718 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 14718 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6751 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 675 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_smaller__Monomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 960 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_standard__Monomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1344 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_tail__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6232 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/___All__Standard.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5179 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/___Minimalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10013 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_find__Weight__Constraints.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8666 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_find__Weight__Vector.html │ │ │ -rw-r--r-- 0 root (0) root (0) 42335 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_groebner__Family.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10545 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_groebner__Stratum.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6202 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_linear__Part.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23437 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23765 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23953 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23733 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15197 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7491 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_smaller__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7862 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_standard__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8696 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_tail__Monomials.html │ │ │ --rw-r--r-- 0 root (0) root (0) 50436 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 50437 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11481 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5901 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 18071 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 705 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/___Groebner__Walk.out │ │ │ @@ -9093,15 +9093,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 289 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__P__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_idealizer.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Keep_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Limit_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 28069 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 370 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Variable_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2401 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3676 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3674 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2595 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ring_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1221 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 173 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_is__Normal_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2065 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_make__S2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 496 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ring__From__Fractions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2043 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_test__Huneke__Question.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/ │ │ │ @@ -9128,15 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-rw-r--r-- 0 root (0) root (0) 10264 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14492 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14490 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14920 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ring_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12936 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6289 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_is__Normal_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9850 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_make__S2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6675 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_make__S2_lp..._cm__Variable_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8000 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_ring__From__Fractions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9442 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_test__Huneke__Question.html │ │ │ @@ -9151,44 +9151,44 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 690 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_action__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 415 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_ambient_lp__Ring__Of__Invariants_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 419 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_cyclic__Factors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 557 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_defining__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 617 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_degrees__Ring_lp__Diagonal__Action_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1384 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_diagonal__Action.out │ │ │ -rw-r--r-- 0 root (0) root (0) 300 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_dim_lp__Group__Action_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1735 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_equivariant__Hilbert.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1736 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_equivariant__Hilbert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3512 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_equivariant__Hilbert__Series.out │ │ │ -rw-r--r-- 0 root (0) root (0) 668 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_finite__Action.out │ │ │ -rw-r--r-- 0 root (0) root (0) 677 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_generators_lp__Finite__Group__Action_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 555 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_generators_lp__Ring__Of__Invariants_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 656 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_group.out │ │ │ -rw-r--r-- 0 root (0) root (0) 636 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_group__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1302 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hilbert__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 606 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hilbert__Series_lp__Ring__Of__Invariants_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 313 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hironaka__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9643 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hsop_spalgorithms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 586 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariant__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1321 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1320 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1180 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 513 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Use__Coefficient__Ring_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1487 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Diagonal__Action_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 560 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Finite__Group__Action_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2661 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Finite__Group__Action_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 773 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Linearly__Reductive__Action_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 798 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Linearly__Reductive__Action_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1070 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_is__Abelian.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2617 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_is__Invariant.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1075 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_linearly__Reductive__Action.out │ │ │ -rw-r--r-- 0 root (0) root (0) 515 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_molien__Series.out │ │ │ -rw-r--r-- 0 root (0) root (0) 397 2026-05-18 11:29:46.000000 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./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 269 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6823 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_correspondence__Scroll.out │ │ │ -rw-r--r-- 0 root (0) root (0) 804 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_cox__Matrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1634 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6876 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6874 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2293 2026-05-18 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root (0) 2223 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out │ │ │ -rw-r--r-- 0 root (0) root (0) 129 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Scroll.out │ │ │ -rw-r--r-- 0 root (0) root (0) 758 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_scheme__In__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3037 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_schreyer__Name.out │ │ │ -rw-r--r-- 0 root (0) root (0) 813 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_small__Diagonal.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 30 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5115 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/___Fine__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4691 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/___Scrolls.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8129 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_all__Gradings.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9736 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6767 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Carpet.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13501 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Homotopies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12195 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9990 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9993 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10823 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7050 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6938 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7052 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6939 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18391 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_correspondence__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7594 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_cox__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9630 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15104 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15102 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5308 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_gorenstein__Double.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8680 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_hankel__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8313 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_homotopy__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6379 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_irrelevant__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8222 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_product__Of__Projective__Spaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7320 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Equations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7406 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8546 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution__Twists.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8521 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8525 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6053 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7725 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_scheme__In__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11056 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_schreyer__Name.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6586 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_small__Diagonal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 32113 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22554 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8983 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/toc.html │ │ │ @@ -9607,27 +9607,27 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1849 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LAYOUT.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 33504 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2858 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2411 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Change__Matrix_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1435 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1436 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 476 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/_gcd__L__L__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 678 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/_is__L__L__L.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 28 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4290 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___B__K__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4236 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Cohen__Engine.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4350 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Cohen__Top__Level.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4153 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Givens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3646 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Hermite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11043 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10067 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Change__Matrix_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 22486 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 22487 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4198 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___N__T__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3999 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__F__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4019 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__Q__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4201 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__Q__P1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4035 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__R__R.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4068 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__X__D.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4672 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Threshold.html │ │ │ @@ -9643,15 +9643,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 70564 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/___Working_spwith_splattice_sppolytopes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_adjoint__Polytope.out │ │ │ -rw-r--r-- 0 root (0) root (0) 329 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_ambient__Halfspaces.out │ │ │ --rw-r--r-- 0 root (0) root (0) 596 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 595 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 684 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_cayley.out │ │ │ -rw-r--r-- 0 root (0) root (0) 85 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_codegree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 281 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_degree__Of__Jet__Separation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 312 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_epsilon__Bounds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gauss__Fiber.out │ │ │ -rw-r--r-- 0 root (0) root (0) 373 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gauss__Image.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gaussk__Fiber.out │ │ │ @@ -9668,15 +9668,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_toric__Div.out │ │ │ -rw-r--r-- 0 root (0) root (0) 167 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_torus__Embedding.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 17 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5207 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/___Working_spwith_splattice_sppolytopes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5861 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_adjoint__Polytope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5951 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_ambient__Halfspaces.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7914 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7913 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9664 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_cayley.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5159 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_codegree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7109 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_degree__Of__Jet__Separation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6789 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_epsilon__Bounds.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6932 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gauss__Fiber.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6966 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gauss__Image.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7234 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gaussk__Fiber.html │ │ │ @@ -10183,15 +10183,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1999 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Expression.out │ │ │ -rw-r--r-- 0 root (0) root (0) 788 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ext^__Z__Z_lp__Matrix_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1449 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ext^__Z__Z_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 548 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Flat__Monoid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 340 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_at_at_sp__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 836 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_us_sp__Thing.out │ │ │ --rw-r--r-- 0 root (0) root (0) 414 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 415 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 611 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__F.out │ │ │ -rw-r--r-- 0 root (0) root (0) 194 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Rev__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Gamma.out │ │ │ -rw-r--r-- 0 root (0) root (0) 230 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Assign__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 371 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Release__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6869 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Grassmannian.out │ │ │ @@ -10258,15 +10258,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2458 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 587 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Monomial__Ideal_sp-_sp__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Monomial__Order.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4407 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Multigraded__Betti__Tally.out │ │ │ -rw-r--r-- 0 root (0) root (0) 218 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutable__Hash__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 795 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutable__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 298 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutable__Matrix_sp_us_sp__Sequence_sp_eq_sp__Thing.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1960 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1979 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 699 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Net__File.out │ │ │ -rw-r--r-- 0 root (0) root (0) 185 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Net_sp^_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 205 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Net_sp_vb_sp__Net.out │ │ │ -rw-r--r-- 0 root (0) root (0) 193 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Net_sp_vb_vb_sp__Net.out │ │ │ -rw-r--r-- 0 root (0) root (0) 339 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Numbered__Vertical__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 430 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Option.out │ │ │ -rw-r--r-- 0 root (0) root (0) 571 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Option__Table_sp_gt_gt_sp__Function.out │ │ │ @@ -10437,15 +10437,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apply_lp__Z__Z_cm__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1347 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apropos.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ascii.out │ │ │ -rw-r--r-- 0 root (0) root (0) 112 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asin.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asinh.out │ │ │ -rw-r--r-- 0 root (0) 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(0) 773 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_augmented_spassignment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1258 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_autoload.out │ │ │ -rw-r--r-- 0 root (0) root (0) 137 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Name.out │ │ │ @@ -10468,15 +10468,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8939 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_capture.out │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ceiling_lp__Number_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_center__String.out │ │ │ 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2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_clean.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2022 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 164 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Monomial__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 267 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Quotient__Ring_rp.out │ │ │ @@ -10492,22 +10492,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 288 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_column__Swap.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_columnate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_combine.out │ │ │ -rw-r--r-- 0 root (0) root (0) 198 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_command__Interpreter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 149 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comments.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_common__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 465 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_commonest.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1535 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1547 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 225 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 372 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compact__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 251 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compare__Exchange.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1085 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compose.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2477 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compositions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compress.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4287 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4284 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ -rw-r--r-- 0 root (0) root (0) 640 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_spsyzygies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_concatenate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1767 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_concatenating_spmatrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 392 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conditional_spexecution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 239 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conjugate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 478 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conjugate_lp__Partition_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_constructing_spmaps_spbetween_spmodules.out │ │ │ @@ -10533,15 +10533,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 130 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Directory.out │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__File__Directory.out │ │ │ -rw-r--r-- 0 root (0) root (0) 90 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__File__Name.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2981 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Layout.out │ │ │ -rw-r--r-- 0 root (0) root (0) 193 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 81 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Row__Number.out │ │ │ --rw-r--r-- 0 root (0) root (0) 329 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Time.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 330 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 318 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_debug_lp__Package_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 965 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_debug_lp__String_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 691 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_deep__Splice.out │ │ │ -rw-r--r-- 0 root (0) root (0) 201 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_default.out │ │ │ -rw-r--r-- 0 root (0) root (0) 406 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_default__Precision.out │ │ │ -rw-r--r-- 0 root (0) root (0) 443 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_definition_spof_spproduct_sp_lpblock_rp_sporders.out │ │ │ -rw-r--r-- 0 root (0) root (0) 735 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_degree__Group.out │ │ │ @@ -10588,15 +10588,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1720 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eagon__Northcott.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvalues.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvectors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eint.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 162 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 366 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elements.out │ │ │ --rw-r--r-- 0 root (0) root (0) 21212 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 21209 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 782 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3675 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 487 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries.out │ │ │ -rw-r--r-- 0 root (0) root (0) 185 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_equality_spand_spcontainment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erfc.out │ │ │ @@ -10737,15 +10737,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1260 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1039 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inheritance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1187 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inputting_spa_spmatrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 592 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_insert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 686 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spassignment_spmethods.out │ │ │ -rw-r--r-- 0 root (0) root (0) 936 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spaugmented_spassignment_spmethods.out │ │ │ --rw-r--r-- 0 root (0) root (0) 932 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 933 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integers_spmodulo_spa_spprime.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integrate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1162 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Ideal_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 246 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__R__Ri_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 189 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Set_cm__Set_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 190 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersection_spof_spideals.out │ │ │ @@ -10884,15 +10884,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 769 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Mutable__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 422 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring_cm__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 228 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 560 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max.out │ │ │ --rw-r--r-- 0 root (0) root (0) 82 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 83 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ -rw-r--r-- 0 root (0) root (0) 83 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 311 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1637 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1730 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_merge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2673 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 928 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method__Options_lp__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6785 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods.out │ │ │ @@ -10900,15 +10900,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 867 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_midpoint.out │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 310 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 925 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingens_lp__Groebner__Basis_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2181 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingens_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 737 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingle.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1804 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1805 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 951 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 607 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 643 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1229 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 160 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimize__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 504 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minors_lp__Z__Z_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 318 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mkdir.out │ │ │ @@ -10984,16 +10984,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 681 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pack.out │ │ │ -rw-r--r-- 0 root (0) root (0) 144 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_packages.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_packing_spmonomials_spfor_spefficiency.out │ │ │ -rw-r--r-- 0 root (0) root (0) 131 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pad.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1059 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pairs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 144 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel__Apply.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1650 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8671 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1652 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8673 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 358 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parse.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parsing_spprecedence_cm_spin_spdetail.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3030 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1297 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1097 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partitions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 652 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_path.out │ │ │ @@ -11002,15 +11002,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_peek_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 267 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_permanents.out │ │ │ -rw-r--r-- 0 root (0) root (0) 874 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_permutations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 495 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pfaffian.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2202 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pfaffians.out │ │ │ -rw-r--r-- 0 root (0) root (0) 301 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pi.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pivots_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 13709 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13710 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2331 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polarize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polylog.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4055 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polynomial_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 624 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 475 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_positions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 274 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_powermod.out │ │ │ -rw-r--r-- 0 root (0) root (0) 173 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_precision.out │ │ │ @@ -11046,15 +11046,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 796 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 825 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3975 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_lp__Matrix_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1476 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1400 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 94 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__Element.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1250 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1251 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out │ │ │ -rw-r--r-- 0 root (0) root (0) 669 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__Mutable__Matrix_lp__Z__Z_cm__Z__Z_cm__R__R_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__Subset.out │ │ │ -rw-r--r-- 0 root (0) root (0) 843 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__List_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1109 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Q__Q_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Type_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1087 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Z__Z_cm__Ideal_rp.out │ │ │ @@ -11142,15 +11142,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_shuffle.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sign.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2452 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_simple_sp__Groebner_spbasis_spcomputations_spover_spvarious_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sin.out │ │ │ -rw-r--r-- 0 root (0) root (0) 933 2026-05-18 11:29:46.000000 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sort_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 582 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sort_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 278 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_source_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_source_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_span.out │ │ │ @@ -11217,15 +11217,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 687 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_terms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 730 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_tests.out │ │ │ -rw-r--r-- 0 root (0) root (0) 214 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./usr/share/doc/Macaulay2/MinimalPrimes/html/_minimal_spprimes_spof_span_spideal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11494 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8779 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical__Containment.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11493 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8780 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical__Containment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5225 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical_spof_span_spideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10088 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8701 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4222 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Miura/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Miura/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 15700 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Miura/dump/rawdocumentation.dump │ │ │ @@ -14108,58 +14108,58 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 31134 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 260 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_hom__Ideal__Polytope.out │ │ │ -rw-r--r-- 0 root (0) root (0) 302 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_hom__Ideal__Polytope_lp..._cm__Coefficient__Ring_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 535 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_m__Mixed__Volume.out │ │ │ -rw-r--r-- 0 root (0) root (0) 896 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_mixed__Multiplicity.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2839 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2838 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 612 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal_lp..._cm__Variable__Base__Name_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 229 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_sec__Milnor__Numbers.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 648 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 30 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 7812 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_hom__Ideal__Polytope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7295 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_hom__Ideal__Polytope_lp..._cm__Coefficient__Ring_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8519 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_m__Mixed__Volume.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10487 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_mixed__Multiplicity.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12591 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12590 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9072 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal_lp..._cm__Variable__Base__Name_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7865 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_sec__Milnor__Numbers.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13547 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8418 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4908 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 20012 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 709 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/_deform__M__C__M__Module_lp__Module_cm__Ring__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2985 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/_deform__M__C__M__Module_lp__Module_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2987 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/_deform__M__C__M__Module_lp__Module_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 607 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 53 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6617 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8079 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp..._cm__Degree__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9384 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_cm__Ring__Map_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12121 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12123 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3769 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_xi.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9749 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5436 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4093 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 133490 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 447 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/___Monodromy__Solver.out │ │ │ -rw-r--r-- 0 root (0) root (0) 985 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/___Monodromy__Solver__Options.out │ │ │ -rw-r--r-- 0 root (0) root (0) 333 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_complete__Graph__Augment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 243 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_complete__Graph__Init.out │ │ │ -rw-r--r-- 0 root (0) root (0) 418 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_compute__Mixed__Volume.out │ │ │ -rw-r--r-- 0 root (0) root (0) 407 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_create__Seed__Pair.out │ │ │ --rw-r--r-- 0 root (0) root (0) 943 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 944 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_flower__Graph__Augment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 242 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_flower__Graph__Init.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9570 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Group.out │ │ │ -rw-r--r-- 0 root (0) root (0) 427 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Solve_lp__System_cm__Abstract__Point_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1395 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Solve_lp__System_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 956 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_potential__E.out │ │ │ -rw-r--r-- 0 root (0) root (0) 442 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_potential__Lower__Bound.out │ │ │ @@ -14174,15 +14174,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6071 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/___Point__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4702 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_append__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4754 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_append__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5077 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_complete__Graph__Augment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4899 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_complete__Graph__Init.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5392 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_compute__Mixed__Volume.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8016 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_create__Seed__Pair.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7827 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7828 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5211 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_flower__Graph__Augment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4872 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_flower__Graph__Init.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5091 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_get__Track__Time.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6401 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_homotopy__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4808 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_indices_lp__Point__Array_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5049 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_is__Member_lp__Abstract__Point_cm__Point__Array_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4938 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_length_lp__Point__Array_rp.html │ │ │ @@ -14420,28 +14420,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 19311 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13051 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7631 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 41382 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2242 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2239 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 719 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_compute__Component.out │ │ │ -rw-r--r-- 0 root (0) root (0) 723 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_interpolate__Component.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1000 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_max__Grading.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_trim__Basis__In__Degree.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5673 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Coefficient__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4437 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4693 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Previous__Gens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4952 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Return__Target__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5214 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Use__Interpolation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4849 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Use__Matroid.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12058 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12055 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6804 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9426 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_compute__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10300 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_interpolate__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8210 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_max__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9672 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_trim__Basis__In__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14253 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13335 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/master.html │ │ │ @@ -14449,27 +14449,27 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 35822 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 369 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/___N__P.out │ │ │ -rw-r--r-- 0 root (0) root (0) 656 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_gr__Gr.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1249 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_hilbert__Sequence.out │ │ │ --rw-r--r-- 0 root (0) root (0) 413 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 414 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 262 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out │ │ │ -rw-r--r-- 0 root (0) root (0) 573 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Reduction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 596 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_monj__Mult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 805 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_multiplicity__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1239 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_print__Hilbert__Sequence.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 594 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 47 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6392 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/___N__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6908 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_gr__Gr.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9087 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_hilbert__Sequence.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6242 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6243 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5908 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7033 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Reduction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6680 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9787 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_multiplicity__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7123 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_print__Hilbert__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15467 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9778 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/master.html │ │ │ @@ -14538,16 +14538,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 728 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_bs_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 507 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_eq_eq_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 390 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_pc_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_pl_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 447 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_st_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 905 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_st_sp__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1340 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1276 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1341 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1273 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1200 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_lt_lt_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 693 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_st_sp__Multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1848 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 772 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 803 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_sp__Multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 763 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_vb_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 744 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_vb_sp__Multirational__Map.out │ │ │ @@ -14566,42 +14566,42 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_coefficient__Ring_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3115 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_cone__Of__Lines.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_conormal__Variety_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 575 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_cycle__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 671 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_decompose_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 324 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 158 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./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2265 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2264 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 155 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_dim_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 482 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_dual_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 809 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_entries_lp__Multirational__Map_rp.out │ │ │ 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(0) root (0) 932 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Isomorphism_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1017 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_image_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 802 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse2.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1407 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 929 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Isomorphism_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 386 2026-05-18 11:29:46.000000 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9285 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_shortcuts.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8654 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_show_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5942 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_singular__Locus_lp__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5751 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_source_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6910 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_super_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6862 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_support_lp__Multiprojective__Variety_rp.html │ │ │ @@ -15124,15 +15124,15 @@ │ │ │ -rw-r--r-- 0 root 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./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1405 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1437 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_add__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 240 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2137 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_build__Graph__Filter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 165 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_count__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 363 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_filter__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2806 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Bipartite__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3579 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Graphs.out │ │ │ @@ -15153,15 +15153,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_string__To__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 245 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_string__To__Graph.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 575 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 18 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6190 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5916 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8722 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8723 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10518 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/html/_add__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7193 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14397 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/html/_build__Graph__Filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8007 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/html/_count__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8305 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/html/_filter__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13917 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Bipartite__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14290 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Graphs.html │ │ │ @@ -15193,15 +15193,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1406 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 826 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_add__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 388 2026-05-18 11:29:46.000000 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(0) root (0) 7935 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7934 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9116 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__To__String.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6131 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_is__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7565 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_neighborhood__Complements.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6592 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_new__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6503 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_only__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7157 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_relabel__Bipartite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9231 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_relabel__Graph.html │ │ │ @@ -15255,15 +15255,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179759 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 431 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Dependent__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 791 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Diff__Op.out │ │ │ -rw-r--r-- 0 root (0) root (0) 553 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Diff__Op_sp__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 289 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Hybrid_dq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Macaulay__Matrix_dq.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2349 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2350 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_amult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 931 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_coordinate__Change__Ops.out │ │ │ -rw-r--r-- 0 root (0) 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11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Dependent__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9234 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Diff__Op.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7518 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Diff__Op_sp__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5228 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Sampler.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7481 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Hybrid_dq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8007 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Macaulay__Matrix_dq.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8016 2026-05-18 11:29:46.000000 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./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2228 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_nef__Generators_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1002 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3171 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__List_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1693 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1964 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ -rw-r--r-- 0 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11158 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_nef__Generators_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10638 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19357 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__List_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14017 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13755 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9994 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12286 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_orbits_lp__Normal__Toric__Variety_cm__Z__Z_rp.html │ │ │ @@ -15884,46 +15884,46 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 40691 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24606 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6456 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 146200 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root 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./usr/share/doc/Macaulay2/NumericalImplicitization/html/_pseudo__Witness__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11102 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_pseudo__Witness__Set_lp..._cm__Software_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9003 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_real__Point.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9004 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_real__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 34897 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 34669 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6367 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalLinearAlgebra/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalLinearAlgebra/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 18725 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalLinearAlgebra/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalLinearAlgebra/example-output/ │ │ │ @@ -15999,15 +15999,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 25255 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16770 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9775 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 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./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_semigroup__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 336 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_socle.out │ │ │ -rw-r--r-- 0 root (0) root (0) 439 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_sums.out │ │ │ -rw-r--r-- 0 root (0) root (0) 73 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_type.out │ │ │ -rw-r--r-- 0 root (0) root (0) 329 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_weight.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalSemigroups/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 66 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalSemigroups/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 11840 2026-05-18 11:29:46.000000 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./usr/share/doc/Macaulay2/NumericalSemigroups/html/_buchweitz__Criterion.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7884 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_buchweitz__Semigroups.html │ │ │ @@ -16100,25 +16100,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 691 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Degree__Shifts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1610 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1610 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module__Map_sp__Vector__In__Width.out │ │ │ -rw-r--r-- 0 root (0) root (0) 395 2026-05-18 11:29:46.000000 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./usr/share/doc/Macaulay2/OIGroebnerBases/html/_to__String_lp__Free__O__I__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5625 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_to__String_lp__Polynomial__O__I__Algebra_rp.html │ │ │ @@ -16255,15 +16255,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 703 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Ring_sp^_sp__Betti__Tally.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1814 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp..._cm__Minimize_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp__Graded__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 518 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 358 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 832 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_complete_lp__Chain__Complex_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1283 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_computing_spresolutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1282 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_computing_spresolutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1250 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_cone_lp__Chain__Complex__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 621 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_dd.out │ │ │ -rw-r--r-- 0 root (0) root (0) 536 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_dual_lp__Chain__Complex__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1335 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_extend_lp__Chain__Complex_cm__Chain__Complex_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1130 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_extracting_spinformation_spfrom_spchain_spcomplexes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_free_spresolutions_spof_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_graded__Module__Map.out │ │ │ @@ -16338,15 +16338,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5199 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Graded__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6787 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6067 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6430 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5793 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_complete_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4492 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_complete_lp__Graded__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4416 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_components_lp__Chain__Complex_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7676 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7675 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7353 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_cone_lp__Chain__Complex__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5668 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dd.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5987 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dual_lp__Chain__Complex__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4043 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dual_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8439 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_extend_lp__Chain__Complex_cm__Chain__Complex_cm__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7732 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_extracting_spinformation_spfrom_spchain_spcomplexes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6102 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_free_spresolutions_spof_spmodules.html │ │ │ @@ -16416,49 +16416,49 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6562 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OpenMath/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5201 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OpenMath/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3345 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/OpenMath/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 84707 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 17044 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 17050 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1080 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.1_co_spunique_spgraph_spon_sp8_spvertices_spwith_spexotic_spsolutions_spand_spno_spinduced_spcycle_spof_splength_spat_spleast_sp5.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2543 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2081 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.3_co_spexamples_spof_spgluing_sptwo_spcycles_spalong_span_spedge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5401 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.4_co_sp__The_spsquare_spwithin_spa_spsquare.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Generation_spof_spall_sp__S__C__T_sp_lpsimple_cm_spconnected_cm_sp2-connected_rp_spgraphs_spon_spsmall_spnumbers_spof_spvertices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 16187 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Oscillators.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3813 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3812 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1682 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_all__Unique__Principal__Minors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10260 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_find__Real__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 151 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Angles.out │ │ │ --rw-r--r-- 0 root (0) root (0) 300 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Linearly__Stable__Solutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 301 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Linearly__Stable__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2330 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_identify__Stability.out │ │ │ -rw-r--r-- 0 root (0) root (0) 459 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_is__Stable__Solution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10680 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__Jacobian.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1227 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__Quadrics.out │ │ │ -rw-r--r-- 0 root (0) root (0) 673 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 23276 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__System.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1924 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_show__Exotic__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_standard__Sols.out │ │ │ -rw-r--r-- 0 root (0) root (0) 252 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_vertex__Spanning__Polynomial.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 33 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 25605 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25611 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6436 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.1_co_spunique_spgraph_spon_sp8_spvertices_spwith_spexotic_spsolutions_spand_spno_spinduced_spcycle_spof_splength_spat_spleast_sp5.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7938 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7846 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.3_co_spexamples_spof_spgluing_sptwo_spcycles_spalong_span_spedge.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10847 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.4_co_sp__The_spsquare_spwithin_spa_spsquare.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6893 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Generation_spof_spall_sp__S__C__T_sp_lpsimple_cm_spconnected_cm_sp2-connected_rp_spgraphs_spon_spsmall_spnumbers_spof_spvertices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3765 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Harrington-__Schenck-__Stillman.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8394 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8393 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9009 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_all__Unique__Principal__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19912 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_find__Real__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7066 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_get__Angles.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6811 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6812 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10775 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_identify__Stability.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7695 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_is__Stable__Solution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20221 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_osc__Jacobian.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8065 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_osc__Quadrics.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8985 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_osc__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 33005 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_osc__System.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10010 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_show__Exotic__Solutions.html │ │ │ @@ -16846,15 +16846,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 221 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_qq__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 962 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_quadratic__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2452 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_rand__Nice__Pencil.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2319 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_random__Extension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 423 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_random__Isotropic__Subspace.out │ │ │ -rw-r--r-- 0 root (0) root (0) 917 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_random__Line__Bundle.out │ │ │ -rw-r--r-- 0 root (0) root (0) 750 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_random__Nice__Pencil.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1672 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_search__Ulrich.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1673 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_search__Ulrich.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2177 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_sym__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 734 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_symmetric__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1764 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_tensor__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 613 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_translate__Isotropic__Subspace.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1547 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_vector__Bundle__On__E.out │ │ │ -rw-r--r-- 0 root (0) root (0) 513 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_y__Action.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/ │ │ │ @@ -16888,15 +16888,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6353 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_qq__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7744 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_quadratic__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10142 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_rand__Nice__Pencil.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10967 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_random__Extension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7855 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_random__Isotropic__Subspace.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9237 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_random__Line__Bundle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10694 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_random__Nice__Pencil.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11232 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_search__Ulrich.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11233 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_search__Ulrich.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8751 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_sym__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6998 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_symmetric__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11810 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_tensor__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8426 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_translate__Isotropic__Subspace.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10831 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_vector__Bundle__On__E.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7268 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_y__Action.html │ │ │ -rw-r--r-- 0 root (0) root (0) 29502 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/index.html │ │ │ @@ -17971,15 +17971,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 411 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_irreducible__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 280 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_is__Primary.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2223 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 627 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_localize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1734 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7407 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary__Decomposition_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 765 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary_spdecomposition.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1389 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1390 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 423 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_remove__Lowest__Dimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2249 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_strategies_spfor_spcomputing_spprimary_spdecomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 242 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_top__Components.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 17892 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated__Primes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4872 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated_spprimes.html │ │ │ @@ -17987,15 +17987,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8434 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_is__Primary.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8935 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10374 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_localize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8411 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12015 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21273 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Decomposition_lp__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5937 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary_spdecomposition.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10087 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10088 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7523 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_remove__Lowest__Dimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10692 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_strategies_spfor_spcomputing_spprimary_spdecomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7563 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_top__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21043 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13443 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5682 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Probability/ │ │ │ @@ -18391,15 +18391,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1962 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2633 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300c].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2039 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spall_sppossible_spbetti_sptables_spfor_spquadratic_spcomponent_spof_spinverse_spsystem_spfor_spquartics_spin_sp4_spvariables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9657 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sp16_spbetti_sptables_sppossible_spfor_spquartic_spforms_spin_sp4_spvariables_cm_spand_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 13630 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sp__Betti_spstratum_spof_spa_spgiven_spquartic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3258 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sppossible_spbetti_sptables_spfor_sppoints_spin_sp__P^3_spwith_spgiven_spgeometry.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1713 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Half_spcanonical_spdegree_sp20.out │ │ │ --rw-r--r-- 0 root (0) root (0) 22975 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 28490 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2741 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Noether-__Lefschetz_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7051 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Pfaffians_spon_spquadrics.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Singularities_spof_splifting_spof_sptype_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 4091 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[000]_cm_sp__C__Y_spof_spdegree_sp20.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2380 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[210]_cm_sp__C__Y_spof_spdegree_sp18_spvia_splinkage.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1645 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[310]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2176 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[331]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.out │ │ │ @@ -18433,15 +18433,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9263 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 10050 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300c].html │ │ │ -rw-r--r-- 0 root (0) root (0) 7030 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spall_sppossible_spbetti_sptables_spfor_spquadratic_spcomponent_spof_spinverse_spsystem_spfor_spquartics_spin_sp4_spvariables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16374 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sp16_spbetti_sptables_sppossible_spfor_spquartic_spforms_spin_sp4_spvariables_cm_spand_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20058 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sp__Betti_spstratum_spof_spa_spgiven_spquartic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9470 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sppossible_spbetti_sptables_spfor_sppoints_spin_sp__P^3_spwith_spgiven_spgeometry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8895 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Half_spcanonical_spdegree_sp20.html │ │ │ --rw-r--r-- 0 root (0) root (0) 48823 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 54338 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11186 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Noether-__Lefschetz_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4325 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12642 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Pfaffians_spon_spquadrics.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8788 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Singularities_spof_splifting_spof_sptype_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 14894 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[000]_cm_sp__C__Y_spof_spdegree_sp20.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12432 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[210]_cm_sp__C__Y_spof_spdegree_sp18_spvia_splinkage.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9486 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[310]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.html │ │ │ @@ -18705,27 +18705,27 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2643 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_disturb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1358 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_histogram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1300 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_maximal__Entry.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1502 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_normalize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5291 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 435 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Simplicial__Complex.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1161 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1158 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4139 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Discrete.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4497 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___With__L__L__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4353 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Zero__Mean.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10112 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_disturb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7443 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_histogram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7418 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_maximal__Entry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7083 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14749 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6768 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Simplicial__Complex.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8274 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8271 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11532 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9695 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4992 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1330 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/ │ │ │ @@ -18772,33 +18772,33 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6521 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4368 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 85866 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Finding_sp__Extreme_sp__Examples.out │ │ │ --rw-r--r-- 0 root (0) root (0) 448 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 430 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 481 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__Chain__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_is__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 250 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Addition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 906 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 564 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 479 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Elements__From__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 461 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 289 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 285 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 384 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 359 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Pure__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 733 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal__Chain.out │ │ │ -rw-r--r-- 0 root (0) root (0) 752 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 426 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Sparse__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8938 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8838 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1627 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Toric__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 233 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_reg__Seq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_square__Free.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 39 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5297 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/___Alexander__Probability.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10502 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/___Finding_sp__Extreme_sp__Examples.html │ │ │ @@ -18807,28 +18807,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5991 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_is__Shelling.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7976 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Addition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7500 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7902 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) 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./usr/share/doc/Macaulay2/RandomIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 151393 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/example-output/___C__M__Stats.out │ │ │ @@ -18956,15 +18956,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 96387 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 356 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/___Max__Coordinates__To__Replace.out │ │ │ -rw-r--r-- 0 root (0) root (0) 268 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/___Point__Check__Attempts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2427 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/___Replacement.out │ │ │ --rw-r--r-- 0 root (0) root (0) 292 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_dim__Via__Bezout.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 291 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_dim__Via__Bezout.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1455 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_extend__Ideal__By__Non__Zero__Minor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 379 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_find__A__Non__Zero__Minor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1379 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_generic__Projection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 841 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_get__Random__Linear__Forms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2154 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_projection__To__Hypersurface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 805 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Coordinate__Change.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1227 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Points.out │ │ │ @@ -18974,15 +18974,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4950 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Decomposition__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6061 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Dimension__Function.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6203 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Extend__Field.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5945 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Max__Coordinates__To__Replace.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5105 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Num__Threads__To__Use.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6223 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Point__Check__Attempts.html │ │ │ -rw-r--r-- 0 root (0) root 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./usr/share/doc/Macaulay2/Resultants/example-output/_tangential__Chow__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1268 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_veronese.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 41 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6151 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Affine__Chart__Grass.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5450 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Affine__Chart__Proj.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5368 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Assume__Ordinary.html │ │ │ -rw-r--r-- 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11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25301 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10852 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14552 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25299 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8175 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/html/_conormal__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11980 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/html/_conormal__Variety_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12278 2026-05-18 11:29:46.000000 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./usr/share/doc/Macaulay2/Resultants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7538 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Resultants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37681 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 415 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5991 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 5995 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 61 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5468 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/___Keep__Statistics__Command.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5729 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/___Pre__Run__Script.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5231 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_is__External__M2__Child.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5297 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_is__External__M2__Parent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7803 2026-05-18 11:29:46.000000 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(0) 7692 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5080 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SCMAlgebras/ │ │ │ @@ -19967,15 +19967,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1688 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Generation_spof_spformulas.out │ │ │ -rw-r--r-- 0 root (0) root (0) 902 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Grassmannian_spof_splines_spin_sp__P3.out │ │ │ -rw-r--r-- 0 root (0) root (0) 639 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Hilbert_sppolynomial_spand_sp__Todd_spclass_spof_spprojective_sp3-space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 803 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Lines_spon_spa_spquintic_spthreefold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 511 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Riemann-__Roch_spformulas.out │ │ │ -rw-r--r-- 0 root (0) root (0) 569 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__The_spnumber_spof_spelliptic_spcubics_spon_spa_spsextic_sp4-fold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 247 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Hom_lp__Abstract__Sheaf_cm__Abstract__Sheaf_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1585 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1583 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ -rw-r--r-- 0 root (0) root (0) 256 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___O__O_sp_us_sp__Abstract__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1332 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___O__O_sp_us_sp__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 913 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Quotient__Bundles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spon_spa_spcurve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1741 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spon_spa_spsurface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7382 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spwithout_spdenominators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Ring_sp_us_sp__Chern__Class__Variable.out │ │ │ @@ -20087,15 +20087,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5807 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Examples_spfrom_sp__Schubert_cm_sptranslated.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9185 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Flag__Bundle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6572 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Hom_lp__Abstract__Sheaf_cm__Abstract__Sheaf_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5753 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Ideas_spfor_spfuture_spdevelopment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6553 2026-05-18 11:29:46.000000 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│ -rw-r--r-- 0 root (0) root (0) 13950 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7012 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44566 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/example-output/ │ │ │ @@ -20800,15 +20800,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10013 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_coimage_lp__Simplicial__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root 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│ │ │ -rw-r--r-- 0 root (0) root (0) 209 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_num__Columns_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 209 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_num__Rows_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 445 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_permutation__Matrix_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 389 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_permutation__Sign.out │ │ │ -rw-r--r-- 0 root (0) root (0) 556 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_permute__Polynomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 297 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_power__Sum__Symmetric__Polynomials_lp__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 274 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_reading__Word_lp__Young__Tableau_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1936 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_representation__Multiplicity.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1935 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_representation__Multiplicity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 410 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_row__Permutation__Tableaux_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 335 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_row__Stabilizer_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 563 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_schur__Polynomial_lp__List_cm__Partition_cm__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 45652 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1508 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_semistandard__Tableaux_lp__Partition_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 206 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_size_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 603 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_sort__Columns__Tableau_lp__Specht__Module__Element_rp.out │ │ │ @@ -21317,15 +21317,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5590 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_cycle__Decomposition_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6030 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_elementary__Symmetric__Polynomials_lp__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5260 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_entries_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6553 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_first__Row__Descent_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8285 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_garnir__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6960 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_generalized__Vandermonde__Matrix_lp__List_cm__List_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6542 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_generate__Permutation__Group_lp__List_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13751 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13753 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21069 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6357 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_hook__Length__Formula_lp__Partition_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7437 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_index__Monomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6226 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_index__Tableau_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6948 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_inner__Product_lp__Z__Z_cm__Mutable__Matrix_cm__Mutable__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5471 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_list__To__Tableau_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8312 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_matrix__Representation.html │ │ │ @@ -21333,15 +21333,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5236 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_num__Columns_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5237 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_num__Rows_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5602 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_permutation__Matrix_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6963 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_permutation__Sign.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8878 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_permute__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6039 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_power__Sum__Symmetric__Polynomials_lp__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5794 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_reading__Word_lp__Young__Tableau_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11265 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_representation__Multiplicity.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11264 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_representation__Multiplicity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6299 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_row__Permutation__Tableaux_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5985 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_row__Stabilizer_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7629 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_schur__Polynomial_lp__List_cm__Partition_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 53747 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7438 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_semistandard__Tableaux_lp__Partition_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5385 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_size_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6705 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_sort__Columns__Tableau_lp__Specht__Module__Element_rp.html │ │ │ @@ -21373,45 +21373,45 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3084 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Cubic__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3473 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Gushel__Mukai__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_beauville__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 351 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_clean_lp__Hodge__Special__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6082 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_cubic__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_cubic__Fourfold_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 386 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_cubic__Fourfold_lp__String_cm__Ring_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1165 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Cubic__Fourfold_cm__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1166 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Cubic__Fourfold_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1544 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Cubic__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 221 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Gushel__Mukai__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 672 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_fano__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 593 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_from__Ordinary__To__Gushel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1210 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_gushel__Mukai__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 733 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_gushel__Mukai__Fourfold_lp__Array_cm__Array_cm__String_cm__Thing_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 462 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_gushel__Mukai__Fourfold_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 508 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_gushel__Mukai__Fourfold_lp__String_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_is__Admissible.out │ │ │ -rw-r--r-- 0 root (0) root (0) 164 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_is__Admissible__G__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 708 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_map_lp__Congruence__Of__Curves_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1356 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_mirror__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_normal__Sheaf.out │ │ │ --rw-r--r-- 0 root (0) root (0) 811 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out │ │ │ --rw-r--r-- 0 root (0) root (0) 779 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Cubic__Fourfold_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 814 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 778 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Cubic__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 966 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Gushel__Mukai__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 652 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 653 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1966 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize_lp__Hodge__Special__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 13809 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_polarized__K3surface.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13810 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./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_swap.out │ │ │ -rw-r--r-- 0 root (0) root (0) 744 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__External__String_lp__Hodge__Special__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5171 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4352 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass_lp__Embedded__Projective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 525 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4354 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass_lp__Embedded__Projective__Variety_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 524 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 610 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 23 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6578 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Congruence__Of__Curves.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7185 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Congruence__Of__Curves_sp__Embedded__Projective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10059 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Cubic__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11435 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Doubly__Special__Cubic__Fourfold.html │ │ │ @@ -21429,15 +21429,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7029 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_beauville__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6103 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_check_lp__Z__Z_cm__Congruence__Of__Curves_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5739 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_clean_lp__Hodge__Special__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15111 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_cubic__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6977 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_cubic__Fourfold_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7088 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_cubic__Fourfold_lp__String_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6230 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8781 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Cubic__Fourfold_cm__Z__Z_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8782 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Cubic__Fourfold_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9646 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Gushel__Mukai__Fourfold_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6678 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6828 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6812 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_fano__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6067 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_from__Ordinary__To__Gushel.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11646 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_gushel__Mukai__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11531 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_gushel__Mukai__Fourfold_lp__Array_cm__Array_cm__String_cm__Thing_rp.html │ │ │ @@ -21447,32 +21447,32 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5928 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_is__Admissible__G__M.html │ │ │ 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2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8057 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Cubic__Fourfold_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8575 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8056 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8283 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Gushel__Mukai__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8411 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8412 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8929 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize_lp__Hodge__Special__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25242 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_polarized__K3surface.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25243 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_polarized__K3surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10548 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6743 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold_lp__String_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7524 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7913 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9419 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5853 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surfaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6632 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_swap.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6196 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__External__String_lp__Hodge__Special__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11960 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10973 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10975 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5606 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_trisecant__Flop.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7165 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7164 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 40647 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 36048 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15507 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpectralSequences/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpectralSequences/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 229836 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpectralSequences/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SpectralSequences/example-output/ │ │ │ @@ -21984,15 +21984,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 212 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_no__Packed__All__Subs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_no__Packed__Sub.out │ │ │ -rw-r--r-- 0 root (0) root (0) 174 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_squarefree__Gens.out │ │ │ -rw-r--r-- 0 root (0) root (0) 169 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_squarefree__In__Codim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 651 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symb__Power__Prime__Pos__Char.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Defect.out │ │ │ -rw-r--r-- 0 root (0) root (0) 193 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Polyhedron.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1209 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1211 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power__Join.out │ │ │ -rw-r--r-- 0 root (0) root (0) 320 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_waldschmidt.out │ │ │ -rw-r--r-- 0 root (0) root (0) 200 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_waldschmidt_lp..._cm__Sample__Size_eq_gt..._rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 593 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 15 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4818 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/___A_spquick_spintroduction_spto_spthis_sppackage.html │ │ │ @@ -22025,15 +22025,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6168 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_no__Packed__All__Subs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5913 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_no__Packed__Sub.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5905 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_squarefree__Gens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6053 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_squarefree__In__Codim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6692 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symb__Power__Prime__Pos__Char.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6735 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Defect.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6216 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Polyhedron.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9872 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9874 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6010 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power__Join.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8128 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_waldschmidt.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7220 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_waldschmidt_lp..._cm__Sample__Size_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21914 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20086 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11292 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/SymmetricPolynomials/ │ │ │ @@ -22463,22 +22463,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_floor__Log.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1518 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2026-05-18 11:29:46.000000 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2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Rational.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1492 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1493 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_multiplicative__Order.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_parameter__Test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 216 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Element.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1514 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1513 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2756 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Module.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 590 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4528 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Ascent__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4587 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__C__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5332 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__Domain.html │ │ │ @@ -22514,22 +22514,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5427 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_floor__Log.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10813 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12610 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5439 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Preimage.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16946 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8756 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Trace__On__Canonical__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8417 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16015 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16013 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10548 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Pure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9863 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Rational.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16594 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16595 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6026 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_multiplicative__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8092 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_parameter__Test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6504 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Element.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15196 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15195 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17910 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37779 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35016 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12686 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Text/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Text/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 137217 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Text/dump/rawdocumentation.dump │ │ │ @@ -22730,30 +22730,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 38833 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27904 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11715 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 24254 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1517 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1392 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6776 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Threaded__G__B.out │ │ │ --rw-r--r-- 0 root (0) root (0) 610 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1159 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1155 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4263 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 572 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1036 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1030 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2663 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb_lp..._cm__Verbose_eq_gt..._rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6465 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Lineage__Table.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7228 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6834 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7206 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7275 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14998 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7103 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6796 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7083 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7150 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13398 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7599 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21563 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6806 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4705 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Topcom/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Topcom/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 66598 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Topcom/dump/rawdocumentation.dump │ │ │ @@ -22856,25 +22856,25 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 20320 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 944 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_cm__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 567 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_cm__Volumes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_dual__Deg__Codim.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1537 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1536 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1032 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_polar__Degrees.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 86 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4590 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Force__Amat.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4416 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4408 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Text__Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9049 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_cm__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7712 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_cm__Volumes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7673 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_dual__Deg__Codim.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9020 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9019 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9051 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_polar__Degrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10176 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7391 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4869 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricTopology/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricTopology/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 38537 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/ToricTopology/dump/rawdocumentation.dump │ │ │ @@ -23082,15 +23082,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 17582 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12090 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7419 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 31886 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2273 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2275 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 18048 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_all__Triangulations_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 75242 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 714 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_is__Regular__Triangulation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1654 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_triangulation.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4662 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/html/___Chirotope.html │ │ │ @@ -23114,15 +23114,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4009 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_neighbors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4152 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Fine__Star__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4172 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Fine__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4346 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Triangulation__Weights.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9921 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3929 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_vectors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4039 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_volume__Vector.html │ │ │ --rw-r--r-- 0 root (0) root (0) 24899 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24901 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10543 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6525 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triangulations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triplets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triplets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 49146 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triplets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triplets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/Triplets/example-output/___Betti1_lp__Triplet_rp.out │ │ │ @@ -23906,15 +23906,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11316 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VectorGraphics/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9379 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VectorGraphics/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 226382 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1050 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Def__Param.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1213 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1214 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_check__Comparison__Theorem.out │ │ │ -rw-r--r-- 0 root (0) root (0) 406 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_check__Tangent__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 862 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_correct__Deformation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 433 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_cotangent__Cohomology1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 774 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_cotangent__Cohomology2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 562 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_ext__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3316 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_families.out │ │ │ @@ -23943,15 +23943,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5915 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Correction__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7910 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Def__Param.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5805 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Degree__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5135 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Highest__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9665 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Nested__Deformation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5260 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Polynomial__Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6269 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Sanity__Check.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8283 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8284 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7160 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Verbose.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4647 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Versal__Deformation__Results.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8052 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_check__Comparison__Theorem.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7843 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_check__Tangent__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10158 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_correct__Deformation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5637 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_correction__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10095 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_cotangent__Cohomology1.html │ │ │ @@ -24075,15 +24075,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 620 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ceiling_lp__R__Weil__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 264 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_clean__Support.out │ │ │ -rw-r--r-- 0 root (0) root (0) 470 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_clear__Cache.out │ │ │ -rw-r--r-- 0 root (0) root (0) 254 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_coefficient_lp__Basic__List_cm__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 335 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_coefficient_lp__Ideal_cm__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 728 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_coefficients_lp__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2714 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_divisor.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1719 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_dualize.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1714 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_dualize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1174 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_embed__As__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 500 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_find__Element__Of__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_gbs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 831 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Linear__Diophantine__Solution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 501 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Prime__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Prime__Divisors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 336 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ideal__Power.out │ │ │ @@ -24107,16 +24107,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 441 2026-05-18 11:29:46.000000 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./usr/share/doc/Macaulay2/WeilDivisors/html/_to__Weil__Divisor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7028 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_torsion__Submodule.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6378 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_trim_lp__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5126 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_zero__Divisor.html │ │ │ @@ -24517,15 +24517,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 37102 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WeylGroups/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WhitneyStratifications/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/doc/Macaulay2/WhitneyStratifications/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 62178 2026-05-18 11:29:46.000000 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root (0) 14769 2026-05-18 11:29:46.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 14770 2026-05-18 11:29:46.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19752 2026-05-18 11:29:46.000000 ./usr/share/info/WittVectors.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8866 2026-05-18 11:29:46.000000 ./usr/share/info/XML.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 49215 2026-05-18 11:29:46.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 49214 2026-05-18 11:29:46.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/lintian/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/lintian/overrides/ │ │ │ -rw-r--r-- 0 root (0) root (0) 10748 2026-05-18 11:29:46.000000 ./usr/share/lintian/overrides/macaulay2-common │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/Macaulay2/Style/katex/contrib/auto-render.min.js -> ../../../../javascript/katex/contrib/auto-render.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/Macaulay2/Style/katex/contrib/copy-tex.min.js -> ../../../../javascript/katex/contrib/copy-tex.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/Macaulay2/Style/katex/contrib/render-a11y-string.min.js -> ../../../../javascript/katex/contrib/render-a11y-string.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2026-05-18 11:29:46.000000 ./usr/share/Macaulay2/Style/katex/fonts/KaTeX_AMS-Regular.ttf -> ../../../../fonts/truetype/katex/KaTeX_AMS-Regular.ttf │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ @@ -10,25 +10,25 @@ │ │ │ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ i3 : elapsedTime burkeResolution(M, 7, Check => false) │ │ │ - -- 1.86369s elapsed │ │ │ + -- 1.61629s elapsed │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ o3 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ o3 : Complex │ │ │ │ │ │ i4 : elapsedTime burkeResolution(M, 7, Check => true) │ │ │ - -- 2.26837s elapsed │ │ │ + -- 2.03621s elapsed │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ o4 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ @@ -95,28 +95,28 @@ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i3 : elapsedTime burkeResolution(M, 7, Check => false)
│ │ │ - -- 1.86369s elapsed
│ │ │ + -- 1.61629s elapsed
│ │ │  
│ │ │        1      3      9      27      81      243      729      2187
│ │ │  o3 = R  <-- R  <-- R  <-- R   <-- R   <-- R    <-- R    <-- R
│ │ │                                                               
│ │ │       0      1      2      3       4       5        6        7
│ │ │  
│ │ │  o3 : Complex
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i4 : elapsedTime burkeResolution(M, 7, Check => true)
│ │ │ - -- 2.26837s elapsed
│ │ │ + -- 2.03621s elapsed
│ │ │  
│ │ │        1      3      9      27      81      243      729      2187
│ │ │  o4 = R  <-- R  <-- R  <-- R   <-- R   <-- R    <-- R    <-- R
│ │ │                                                               
│ │ │       0      1      2      3       4       5        6        7
│ │ │  
│ │ │  o4 : Complex
│ │ │ ├── html2text {} │ │ │ │ @@ -23,24 +23,24 @@ │ │ │ │ i2 : M = coker vars R │ │ │ │ │ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ │ │ 1 │ │ │ │ o2 : R-module, quotient of R │ │ │ │ i3 : elapsedTime burkeResolution(M, 7, Check => false) │ │ │ │ - -- 1.86369s elapsed │ │ │ │ + -- 1.61629s elapsed │ │ │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ │ o3 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ │ │ o3 : Complex │ │ │ │ i4 : elapsedTime burkeResolution(M, 7, Check => true) │ │ │ │ - -- 2.26837s elapsed │ │ │ │ + -- 2.03621s elapsed │ │ │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ │ o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ │ │ o4 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AbstractToricVarieties/dump/rawdocumentation.dump │ │ │ @@ -1,8 +1,8 @@ │ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon May 18 11:29:47 2026 │ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon May 18 11:29:46 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=22 │ │ │ QWJzdHJhY3RUb3JpY1ZhcmlldGllcw== │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ @@ -49,15 +49,15 @@ │ │ │ o8 : BettiTally │ │ │ │ │ │ i9 : c=codim I │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ i10 : elapsedTime fI=res I │ │ │ - -- .0363244s elapsed │ │ │ + -- .0501221s elapsed │ │ │ │ │ │ 1 14 33 28 8 │ │ │ o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o10 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ @@ -87,30 +87,30 @@ │ │ │ o13 : BettiTally │ │ │ │ │ │ i14 : phi=map(P2,Pn,H); │ │ │ │ │ │ o14 : RingMap P2 <-- Pn │ │ │ │ │ │ i15 : elapsedTime betti(I'=trim ker phi) │ │ │ - -- .671672s elapsed │ │ │ + -- .581181s elapsed │ │ │ │ │ │ 0 1 │ │ │ o15 = total: 1 11 │ │ │ 0: 1 . │ │ │ 1: . 3 │ │ │ 2: . 8 │ │ │ │ │ │ o15 : BettiTally │ │ │ │ │ │ i16 : I'== I │ │ │ │ │ │ o16 = true │ │ │ │ │ │ i17 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 6.43471s elapsed │ │ │ + -- 5.31875s elapsed │ │ │ │ │ │ i18 : tally apply(basePts,c->(dim c, degree c, betti c)) │ │ │ │ │ │ 0 1 │ │ │ o18 = Tally{(1, 1, total: 1 2) => 5} │ │ │ 0: 1 2 │ │ │ 0 1 │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_parametrization.out │ │ │ @@ -79,40 +79,40 @@ │ │ │ 1: . . │ │ │ 2: . . │ │ │ 3: . 8 │ │ │ │ │ │ o13 : BettiTally │ │ │ │ │ │ i14 : elapsedTime sub(I,H) │ │ │ - -- .0123755s elapsed │ │ │ + -- .0141104s elapsed │ │ │ │ │ │ o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) │ │ │ │ │ │ o14 : Ideal of P2 │ │ │ │ │ │ i15 : phi=map(P2,Pn,H); │ │ │ │ │ │ o15 : RingMap P2 <-- Pn │ │ │ │ │ │ i16 : elapsedTime betti(I'=trim ker phi) │ │ │ - -- .0544883s elapsed │ │ │ + -- .0645586s elapsed │ │ │ │ │ │ 0 1 │ │ │ o16 = total: 1 12 │ │ │ 0: 1 . │ │ │ 1: . 12 │ │ │ │ │ │ o16 : BettiTally │ │ │ │ │ │ i17 : I'== I │ │ │ │ │ │ o17 = true │ │ │ │ │ │ i18 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 1.94343s elapsed │ │ │ + -- 1.6258s elapsed │ │ │ │ │ │ i19 : tally apply(basePts,c->(dim c, degree c, betti c)) │ │ │ │ │ │ 0 1 │ │ │ o19 = Tally{(0, 34, total: 1 15) => 1} │ │ │ 0: 1 . │ │ │ 1: . . │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ @@ -154,15 +154,15 @@ │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i10 : elapsedTime fI=res I
│ │ │ - -- .0363244s elapsed
│ │ │ + -- .0501221s elapsed
│ │ │  
│ │ │          1       14       33       28       8
│ │ │  o10 = Pn  <-- Pn   <-- Pn   <-- Pn   <-- Pn
│ │ │                                            
│ │ │        0       1        2        3        4
│ │ │  
│ │ │  o10 : Complex
│ │ │ ├── html2text {} │ │ │ │ @@ -54,15 +54,15 @@ │ │ │ │ 2: . 12 │ │ │ │ │ │ │ │ o8 : BettiTally │ │ │ │ i9 : c=codim I │ │ │ │ │ │ │ │ o9 = 4 │ │ │ │ i10 : elapsedTime fI=res I │ │ │ │ - -- .0363244s elapsed │ │ │ │ + -- .0501221s elapsed │ │ │ │ │ │ │ │ 1 14 33 28 8 │ │ │ │ o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn │ │ │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ │ │ o10 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ @@ -222,15 +222,15 @@ │ │ │ │ │ │ o14 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .671672s elapsed
│ │ │ + -- .581181s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o15 = total: 1 11
│ │ │            0: 1  .
│ │ │            1: .  3
│ │ │            2: .  8
│ │ │  
│ │ │ @@ -243,15 +243,15 @@
│ │ │  
│ │ │  o16 = true
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 6.43471s elapsed
│ │ │ + -- 5.31875s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i18 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │  
│ │ │                            0 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -110,28 +110,28 @@
│ │ │ │            6: . 7
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : phi=map(P2,Pn,H);
│ │ │ │  
│ │ │ │  o14 : RingMap P2 <-- Pn
│ │ │ │  i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ │ - -- .671672s elapsed
│ │ │ │ + -- .581181s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o15 = total: 1 11
│ │ │ │            0: 1  .
│ │ │ │            1: .  3
│ │ │ │            2: .  8
│ │ │ │  
│ │ │ │  o15 : BettiTally
│ │ │ │  i16 : I'== I
│ │ │ │  
│ │ │ │  o16 = true
│ │ │ │  i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 6.43471s elapsed
│ │ │ │ + -- 5.31875s elapsed
│ │ │ │  i18 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │ │  
│ │ │ │                            0 1
│ │ │ │  o18 = Tally{(1, 1, total: 1 2) => 5}
│ │ │ │                         0: 1 2
│ │ │ │                            0 1
│ │ │ │              (1, 3, total: 1 3) => 8
│ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html
│ │ │ @@ -198,15 +198,15 @@
│ │ │  
│ │ │  o13 : BettiTally
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : elapsedTime sub(I,H)
│ │ │ - -- .0123755s elapsed
│ │ │ + -- .0141104s elapsed
│ │ │  
│ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o14 : Ideal of P2
│ │ │ │ │ │ │ │ │ │ │ │ @@ -215,15 +215,15 @@ │ │ │ │ │ │ o15 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .0544883s elapsed
│ │ │ + -- .0645586s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o16 = total: 1 12
│ │ │            0: 1  .
│ │ │            1: . 12
│ │ │  
│ │ │  o16 : BettiTally
│ │ │ @@ -235,15 +235,15 @@ │ │ │ │ │ │ o17 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 1.94343s elapsed
│ │ │ + -- 1.6258s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i19 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │  
│ │ │                             0  1
│ │ │ ├── html2text {}
│ │ │ │ @@ -82,36 +82,36 @@
│ │ │ │            0: 1 .
│ │ │ │            1: . .
│ │ │ │            2: . .
│ │ │ │            3: . 8
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : elapsedTime sub(I,H)
│ │ │ │ - -- .0123755s elapsed
│ │ │ │ + -- .0141104s elapsed
│ │ │ │  
│ │ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │ │  
│ │ │ │  o14 : Ideal of P2
│ │ │ │  i15 : phi=map(P2,Pn,H);
│ │ │ │  
│ │ │ │  o15 : RingMap P2 <-- Pn
│ │ │ │  i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ │ - -- .0544883s elapsed
│ │ │ │ + -- .0645586s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o16 = total: 1 12
│ │ │ │            0: 1  .
│ │ │ │            1: . 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ │ │  i17 : I'== I
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 1.94343s elapsed
│ │ │ │ + -- 1.6258s elapsed
│ │ │ │  i19 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │ │  
│ │ │ │                             0  1
│ │ │ │  o19 = Tally{(0, 34, total: 1 15) => 1}
│ │ │ │                          0: 1  .
│ │ │ │                          1: .  .
│ │ │ │                          2: .  .
│ │ ├── ./usr/share/doc/Macaulay2/BGG/example-output/_pure__Resolution.out
│ │ │ @@ -114,26 +114,26 @@
│ │ │        | 19a+19b  -38a-16b -18a-13b 16a+22b  |
│ │ │        | -10a-29b 39a+21b  -43a-15b 45a-34b  |
│ │ │  
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │  
│ │ │  i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.449866s (cpu); 0.387314s (thread); 0s (gc)
│ │ │ + -- used 0.493959s (cpu); 0.414208s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o14 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │  
│ │ │  o14 : BettiTally
│ │ │  
│ │ │  i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.493859s (cpu); 0.409637s (thread); 0s (gc)
│ │ │ + -- used 0.585825s (cpu); 0.498057s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o15 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/BGG/html/_pure__Resolution.html
│ │ │ @@ -258,15 +258,15 @@
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.449866s (cpu); 0.387314s (thread); 0s (gc)
│ │ │ + -- used 0.493959s (cpu); 0.414208s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o14 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │  
│ │ │ @@ -277,15 +277,15 @@
│ │ │          

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

│ │ │          

│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -192,27 +192,27 @@
│ │ │ │  o18 : ActionOnComplex
│ │ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │ │  
│ │ │ │  o19 = Complex with 6 actors
│ │ │ │  
│ │ │ │  o19 : ActionOnComplex
│ │ │ │  i20 : elapsedTime a1 = character A1
│ │ │ │ - -- .772423s elapsed
│ │ │ │ + -- .666105s elapsed
│ │ │ │  
│ │ │ │  o20 = Character over kk
│ │ │ │  
│ │ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │ │                   |                4    2         4    2
│ │ │ │         (1, {8})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ │        (2, {11})  |  1   1  1  1            1                  1
│ │ │ │        (2, {13})  |  1   1  1  1            1                  1
│ │ │ │  
│ │ │ │  o20 : Character
│ │ │ │  i21 : elapsedTime a2 = character A2
│ │ │ │ - -- 32.3092s elapsed
│ │ │ │ + -- 26.2348s elapsed
│ │ │ │  
│ │ │ │  o21 = Character over kk
│ │ │ │  
│ │ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │ │        (1, {16})  |  6   2  0  0           -1                 -1
│ │ │ │                   |                4    2         4    2
│ │ │ │        (2, {19})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ │ @@ -319,15 +319,15 @@
│ │ │ │  i30 : M = Is2 / I2;
│ │ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │ │  
│ │ │ │  o31 = Module with 6 actors
│ │ │ │  
│ │ │ │  o31 : ActionOnGradedModule
│ │ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ │ - -- 15.1894s elapsed
│ │ │ │ + -- 12.0247s elapsed
│ │ │ │  
│ │ │ │  o32 = Character over kk
│ │ │ │  
│ │ │ │        (0, {21})  |  1  1  1  1  1  1
│ │ │ │  
│ │ │ │  o32 : Character
│ │ │ │  i33 : b/T
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns.out
│ │ │ @@ -230,15 +230,15 @@
│ │ │            0: 1 . . . .
│ │ │            1: . 4 2 . .
│ │ │            2: . 1 6 5 1
│ │ │  
│ │ │  o22 : BettiTally
│ │ │  
│ │ │  i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.268195s (cpu); 0.19787s (thread); 0s (gc)
│ │ │ + -- used 0.311946s (cpu); 0.245962s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of S
│ │ │  
│ │ │  i24 : betti res j
│ │ │  
│ │ │               0 1 2 3 4
│ │ │  o24 = total: 1 3 6 5 1
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html
│ │ │ @@ -385,15 +385,15 @@
│ │ │  
│ │ │  o22 : BettiTally
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │              
│ │ │            
│ │ │          

│ │ │                
i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.493859s (cpu); 0.409637s (thread); 0s (gc)
│ │ │ + -- used 0.585825s (cpu); 0.498057s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o15 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │ ├── html2text {}
│ │ │ │ @@ -161,30 +161,30 @@
│ │ │ │        | -30a-29b -29a-24b -47a-39b 38a+2b   |
│ │ │ │        | 19a+19b  -38a-16b -18a-13b 16a+22b  |
│ │ │ │        | -10a-29b 39a+21b  -43a-15b 45a-34b  |
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o13 : Matrix A  <-- A
│ │ │ │  i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ │ - -- used 0.449866s (cpu); 0.387314s (thread); 0s (gc)
│ │ │ │ + -- used 0.493959s (cpu); 0.414208s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0 1 2
│ │ │ │  o14 = total: 3 6 3
│ │ │ │            0: 3 . .
│ │ │ │            1: . 6 .
│ │ │ │            2: . . 3
│ │ │ │  
│ │ │ │  o14 : BettiTally
│ │ │ │  With the form pureResolution(p,q,D) we can directly create the situation of
│ │ │ │  pureResolution(M,D) where M is generic product(m_i+1) x #D-1+sum(m_i) matrix of
│ │ │ │  linear forms defined over a ring with product(m_i+1) * #D-1+sum(m_i) variables
│ │ │ │  of characteristic p, created by the script. For a given number of variables in
│ │ │ │  A this runs much faster than taking a random matrix M.
│ │ │ │  i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ │ - -- used 0.493859s (cpu); 0.409637s (thread); 0s (gc)
│ │ │ │ + -- used 0.585825s (cpu); 0.498057s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0 1 2
│ │ │ │  o15 = total: 3 6 3
│ │ │ │            0: 3 . .
│ │ │ │            1: . 6 .
│ │ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out
│ │ │ @@ -1,10 +1,10 @@
│ │ │  -- -*- M2-comint -*- hash: 1330545576567
│ │ │  
│ │ │  i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Mon May 18 12:38:41 UTC 2026
│ │ │ --- Linux sbuild 6.12.88+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.88-1 (2026-05-15) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250  
│ │ │ +-- beginning computation Wed May 20 17:27:15 UTC 2026
│ │ │ +-- Linux sbuild 6.12.88+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.88-1 (2026-05-15) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- Macaulay2 1.26.05, compiled with gcc 15.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .131161 seconds
│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .178585 seconds
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html
│ │ │ @@ -80,19 +80,19 @@
│ │ │          

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

│ │ │          

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

│ │ │                
i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Mon May 18 12:38:41 UTC 2026
│ │ │ --- Linux sbuild 6.12.88+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.88-1 (2026-05-15) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250  
│ │ │ +-- beginning computation Wed May 20 17:27:15 UTC 2026
│ │ │ +-- Linux sbuild 6.12.88+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.88-1 (2026-05-15) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- Macaulay2 1.26.05, compiled with gcc 15.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .131161 seconds

│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .178585 seconds
│ │ │              

│ │ │        
│ │ │        

│ │ │          

│ │ │            
For the programmer

│ │ │ ├── html2text {}
│ │ │ │ @@ -23,18 +23,18 @@
│ │ │ │  "gb3445" -- gb of an ideal with elements of degree 3,4,4,5 in 8 variables
│ │ │ │  "gbB148" -- gb of Bayesian graph ideal #148
│ │ │ │  "res39" -- res of a generic 3 by 9 matrix over ZZ/101
│ │ │ │  "resG25" -- res of the coordinate ring of Grassmannian(2,5)
│ │ │ │  "yang-gb1" -- an example of Yang-Hui He arising in string theory
│ │ │ │  "yang-subring" -- an example of Yang-Hui He
│ │ │ │  i1 : runBenchmarks "res39"
│ │ │ │ --- beginning computation Mon May 18 12:38:41 UTC 2026
│ │ │ │ --- Linux sbuild 6.12.88+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.88-1
│ │ │ │ -(2026-05-15) x86_64 GNU/Linux
│ │ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250
│ │ │ │ +-- beginning computation Wed May 20 17:27:15 UTC 2026
│ │ │ │ +-- Linux sbuild 6.12.88+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +6.12.88-1 (2026-05-15) x86_64 GNU/Linux
│ │ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998
│ │ │ │  -- Macaulay2 1.26.05, compiled with gcc 15.2.0
│ │ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .131161 seconds
│ │ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .178585 seconds
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_u_n_B_e_n_c_h_m_a_r_k_s is a _c_o_m_m_a_n_d.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.05+ds/M2/Macaulay2/packages/Benchmark.m2:297:0.
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump
│ │ │ @@ -515,15 +515,15 @@
│ │ │  Pi4uLikiLCJCZXJ0aW5pIn0sIlJhbmRvbUNvbXBsZXgifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwi
│ │ │  LCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwie30ifSwiLCAiLFNQQU57fX0sU1BBTntUTzJ7bmV3
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│ │ │  aW5pVXNlckhvbW90b3B5KC4uLixSYW5kb21SZWFsPT4uLi4pIiwiQmVydGluaSJ9LCJSYW5kb21S
│ │ │  ZWFsIn0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZhdWx0IHZhbHVlICIsInt9
│ │ │  In0sIiwgIixTUEFOe319LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHsiVG9wRGlyZWN0
│ │ │  b3J5IiwiVG9wRGlyZWN0b3J5IiwiQmVydGluaSJ9LCJUb3BEaXJlY3RvcnkifSxUVHsiID0+ICJ9
│ │ │ -LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiXCIvdG1wL00yLTIzNDUwLTAv
│ │ │ +LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiXCIvdG1wL00yLTI5OTA1LTAv
│ │ │  MFwiIn0sIiwgIixTUEFOeyJPcHRpb24gdG8gY2hhbmdlIGRpcmVjdG9yeSBmb3IgZmlsZSBzdG9y
│ │ │  YWdlLiJ9fSxTUEFOe1RPMntuZXcgRG9jdW1lbnRUYWcgZnJvbSB7W2JlcnRpbmlVc2VySG9tb3Rv
│ │ │  cHksVmVyYm9zZV0sImJlcnRpbmlVc2VySG9tb3RvcHkoLi4uLFZlcmJvc2U9Pi4uLikiLCJCZXJ0
│ │ │  aW5pIn0sIlZlcmJvc2UifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQg
│ │ │  dmFsdWUgIiwiZmFsc2UifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBzaWxlbmNlIGFkZGl0aW9uYWwg
│ │ │  b3V0cHV0In19fSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsi
│ │ │  YmVydGluaVVzZXJIb21vdG9weSIsImJlcnRpbmlVc2VySG9tb3RvcHkiLCJCZXJ0aW5pIn0sIEtl
│ │ │ @@ -1100,15 +1100,15 @@
│ │ │  ZXJ0aW5pUGFyYW1ldGVySG9tb3RvcHkoLi4uLFJhbmRvbVJlYWw9Pi4uLikiLCJCZXJ0aW5pIn0s
│ │ │  IlJhbmRvbVJlYWwifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFs
│ │ │  dWUgIiwie30ifSwiLCAiLFNQQU57ImFuIG9wdGlvbiB3aGljaCBkZXNpZ25hdGVzIHN5bWJvbHMv
│ │ │  c3RyaW5ncy92YXJpYWJsZXMgdGhhdCB3aWxsIGJlIHNldCB0byBiZSBhIHJhbmRvbSByZWFsIG51
│ │ │  bWJlciBvciByYW5kb20gY29tcGxleCBudW1iZXIifX0sU1BBTntUTzJ7bmV3IERvY3VtZW50VGFn
│ │ │  IGZyb20geyJUb3BEaXJlY3RvcnkiLCJUb3BEaXJlY3RvcnkiLCJCZXJ0aW5pIn0sIlRvcERpcmVj
│ │ │  dG9yeSJ9LFRUeyIgPT4gIn0sVFR7Ii4uLiJ9LCIsICIsU1BBTnsiZGVmYXVsdCB2YWx1ZSAiLCJc
│ │ │ -Ii90bXAvTTItMjM0NTAtMC8wXCIifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBjaGFuZ2UgZGlyZWN0
│ │ │ +Ii90bXAvTTItMjk5MDUtMC8wXCIifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBjaGFuZ2UgZGlyZWN0
│ │ │  b3J5IGZvciBmaWxlIHN0b3JhZ2UuIn19LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHtb
│ │ │  YmVydGluaVBhcmFtZXRlckhvbW90b3B5LFZlcmJvc2VdLCJiZXJ0aW5pUGFyYW1ldGVySG9tb3Rv
│ │ │  cHkoLi4uLFZlcmJvc2U9Pi4uLikiLCJCZXJ0aW5pIn0sIlZlcmJvc2UifSxUVHsiID0+ICJ9LFRU
│ │ │  eyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiZmFsc2UifSwiLCAiLFNQQU57Ik9w
│ │ │  dGlvbiB0byBzaWxlbmNlIGFkZGl0aW9uYWwgb3V0cHV0In19fSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiYmVydGluaVBhcmFtZXRlckhvbW90b3B5IiwiYmVy
│ │ │  dGluaVBhcmFtZXRlckhvbW90b3B5IiwiQmVydGluaSJ9LCBLZXkgPT4gYmVydGluaVBhcmFtZXRl
│ │ │ @@ -2449,15 +2449,15 @@
│ │ │  YWw9Pi4uLikiLCJCZXJ0aW5pIn0sIlJhbmRvbVJlYWwifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwi
│ │ │  LCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwie30ifSwiLCAiLFNQQU57ImFuIG9wdGlvbiB3aGlj
│ │ │  aCBkZXNpZ25hdGVzIHN5bWJvbHMvc3RyaW5ncy92YXJpYWJsZXMgdGhhdCB3aWxsIGJlIHNldCB0
│ │ │  byBiZSBhIHJhbmRvbSByZWFsIG51bWJlciBvciByYW5kb20gY29tcGxleCBudW1iZXIifX0sU1BB
│ │ │  TntUTzJ7bmV3IERvY3VtZW50VGFnIGZyb20ge1tiZXJ0aW5pWmVyb0RpbVNvbHZlLFRvcERpcmVj
│ │ │  dG9yeV0sImJlcnRpbmlaZXJvRGltU29sdmUoLi4uLFRvcERpcmVjdG9yeT0+Li4uKSIsIkJlcnRp
│ │ │  bmkifSwiVG9wRGlyZWN0b3J5In0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZh
│ │ │ -dWx0IHZhbHVlICIsIlwiL3RtcC9NMi0yMzQ1MC0wLzBcIiJ9LCIsICIsU1BBTnsiT3B0aW9uIHRv
│ │ │ +dWx0IHZhbHVlICIsIlwiL3RtcC9NMi0yOTkwNS0wLzBcIiJ9LCIsICIsU1BBTnsiT3B0aW9uIHRv
│ │ │  IGNoYW5nZSBkaXJlY3RvcnkgZm9yIGZpbGUgc3RvcmFnZS4ifX0sU1BBTntUTzJ7bmV3IERvY3Vt
│ │ │  ZW50VGFnIGZyb20ge1tiZXJ0aW5pWmVyb0RpbVNvbHZlLFVzZVJlZ2VuZXJhdGlvbl0sImJlcnRp
│ │ │  bmlaZXJvRGltU29sdmUoLi4uLFVzZVJlZ2VuZXJhdGlvbj0+Li4uKSIsIkJlcnRpbmkifSwiVXNl
│ │ │  UmVnZW5lcmF0aW9uIn0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZhdWx0IHZh
│ │ │  bHVlICIsIi0xIn0sIiwgIixTUEFOe319LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHtb
│ │ │  YmVydGluaVplcm9EaW1Tb2x2ZSxWZXJib3NlXSwiYmVydGluaVplcm9EaW1Tb2x2ZSguLi4sVmVy
│ │ │  Ym9zZT0+Li4uKSIsIkJlcnRpbmkifSwiVmVyYm9zZSJ9LFRUeyIgPT4gIn0sVFR7Ii4uLiJ9LCIs
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Parameter__Homotopy.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │              
  • HomVariableGroup => ..., default value {}, an option to group variables and use multihomogeneous homotopies
    • 
│ │ │              
  • M2Precision (missing documentation)
│ │ │   => ..., default value 53, 
    • 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │              
  • RandomReal => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-23450-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-29905-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S, a list, a list whose entries are lists of solutions for each target system
      • 
│ │ │            
    
│ │ │          
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │              "OutPoints",
│ │ │ │            o _R_a_n_d_o_m_C_o_m_p_l_e_x => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │            o _R_a_n_d_o_m_R_e_a_l => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-23450-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-29905-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S, a _l_i_s_t, a list whose entries are lists of solutions for each
│ │ │ │              target system
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__User__Homotopy.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │   => ..., default value 53, 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │              
  • RandomReal (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-23450-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-29905-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S0, a list, a list of solutions to the target system
      • 
│ │ │            
    
│ │ │          
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │              value {},
│ │ │ │            o HomVariableGroup (missing documentation) => ..., default value {},
│ │ │ │            o M2Precision (missing documentation) => ..., default value 53,
│ │ │ │            o OutputStyle (missing documentation) => ..., default value
│ │ │ │              "OutPoints",
│ │ │ │            o RandomComplex (missing documentation) => ..., default value {},
│ │ │ │            o RandomReal (missing documentation) => ..., default value {},
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-23450-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-29905-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S0, a _l_i_s_t, a list of solutions to the target system
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This method calls Bertini to track a user-defined homotopy. The user needs to
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Zero__Dim__Solve.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │   => ..., default value "main_data", 
│ │ │              
  • NameSolutionsFile (missing documentation)
│ │ │   => ..., default value "raw_solutions", 
    • 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │              
  • RandomReal => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-23450-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-29905-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • UseRegeneration (missing documentation)
│ │ │   => ..., default value -1, 
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S, a list, a list of points that are contained in the variety of F
      • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,15 +32,15 @@
│ │ │ │              "OutPoints",
│ │ │ │            o _R_a_n_d_o_m_C_o_m_p_l_e_x => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │            o _R_a_n_d_o_m_R_e_a_l => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-23450-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-29905-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o UseRegeneration (missing documentation) => ..., default value -1,
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S, a _l_i_s_t, a list of points that are contained in the variety of F
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp1.out
│ │ │ @@ -76,15 +76,15 @@
│ │ │  i8 : A = action(RI,S7)
│ │ │  
│ │ │  o8 = Complex with 15 actors
│ │ │  
│ │ │  o8 : ActionOnComplex
│ │ │  
│ │ │  i9 : elapsedTime c = character A
│ │ │ - -- .383699s elapsed
│ │ │ + -- .308328s elapsed
│ │ │  
│ │ │  o9 = Character over QQ
│ │ │        
│ │ │       (0, {0})  |  1   1   1   1   1  1   1   1   1   1   1   1   1   1   1
│ │ │       (1, {2})  |  0  -1   1  -1   0  0   0  -1   2   0   2   2   2   6  14
│ │ │       (2, {3})  |  0   1   0   0  -1  1  -1  -1  -1  -1  -1   1  -1   5  35
│ │ │       (3, {4})  |  0  -1   0   0   1  1   1  -1  -1   1  -1  -1  -1  -5  35
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp2.out
│ │ │ @@ -100,15 +100,15 @@
│ │ │  i6 : A=action(RI,S6)
│ │ │  
│ │ │  o6 = Complex with 11 actors
│ │ │  
│ │ │  o6 : ActionOnComplex
│ │ │  
│ │ │  i7 : elapsedTime c=character A
│ │ │ - -- .842169s elapsed
│ │ │ + -- .462508s elapsed
│ │ │  
│ │ │  o7 = Character over QQ
│ │ │        
│ │ │        (0, {0})  |   1  1   1   1   1   1  1   1   1   1    1
│ │ │        (1, {5})  |   0  1   0   2   0   1  3   0   2   4    6
│ │ │        (1, {7})  |   0  0   0   0   0   1  3   0   4  16   60
│ │ │        (1, {9})  |   0  0   0   0   2   2  2   0   4   8   20
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out
│ │ │ @@ -187,28 +187,28 @@
│ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │  
│ │ │  o19 = Complex with 6 actors
│ │ │  
│ │ │  o19 : ActionOnComplex
│ │ │  
│ │ │  i20 : elapsedTime a1 = character A1
│ │ │ - -- .772423s elapsed
│ │ │ + -- .666105s elapsed
│ │ │  
│ │ │  o20 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │                   |                4    2         4    2
│ │ │         (1, {8})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │        (2, {11})  |  1   1  1  1            1                  1
│ │ │        (2, {13})  |  1   1  1  1            1                  1
│ │ │  
│ │ │  o20 : Character
│ │ │  
│ │ │  i21 : elapsedTime a2 = character A2
│ │ │ - -- 32.3092s elapsed
│ │ │ + -- 26.2348s elapsed
│ │ │  
│ │ │  o21 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │        (1, {16})  |  6   2  0  0           -1                 -1
│ │ │                   |                4    2         4    2
│ │ │        (2, {19})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ @@ -308,15 +308,15 @@
│ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │  
│ │ │  o31 = Module with 6 actors
│ │ │  
│ │ │  o31 : ActionOnGradedModule
│ │ │  
│ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ - -- 15.1894s elapsed
│ │ │ + -- 12.0247s elapsed
│ │ │  
│ │ │  o32 = Character over kk
│ │ │         
│ │ │        (0, {21})  |  1  1  1  1  1  1
│ │ │  
│ │ │  o32 : Character
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp1.html
│ │ │ @@ -167,15 +167,15 @@
│ │ │  
│ │ │  o8 : ActionOnComplex
    • 
│ │ │              
    
│ │ │                
    i9 : elapsedTime c = character A
│ │ │ - -- .383699s elapsed
│ │ │ + -- .308328s elapsed
│ │ │  
│ │ │  o9 = Character over QQ
│ │ │        
│ │ │       (0, {0})  |  1   1   1   1   1  1   1   1   1   1   1   1   1   1   1
│ │ │       (1, {2})  |  0  -1   1  -1   0  0   0  -1   2   0   2   2   2   6  14
│ │ │       (2, {3})  |  0   1   0   0  -1  1  -1  -1  -1  -1  -1   1  -1   5  35
│ │ │       (3, {4})  |  0  -1   0   0   1  1   1  -1  -1   1  -1  -1  -1  -5  35
│ │ │ ├── html2text {}
│ │ │ │ @@ -91,15 +91,15 @@
│ │ │ │  o7 : List
│ │ │ │  i8 : A = action(RI,S7)
│ │ │ │  
│ │ │ │  o8 = Complex with 15 actors
│ │ │ │  
│ │ │ │  o8 : ActionOnComplex
│ │ │ │  i9 : elapsedTime c = character A
│ │ │ │ - -- .383699s elapsed
│ │ │ │ + -- .308328s elapsed
│ │ │ │  
│ │ │ │  o9 = Character over QQ
│ │ │ │  
│ │ │ │       (0, {0})  |  1   1   1   1   1  1   1   1   1   1   1   1   1   1   1
│ │ │ │       (1, {2})  |  0  -1   1  -1   0  0   0  -1   2   0   2   2   2   6  14
│ │ │ │       (2, {3})  |  0   1   0   0  -1  1  -1  -1  -1  -1  -1   1  -1   5  35
│ │ │ │       (3, {4})  |  0  -1   0   0   1  1   1  -1  -1   1  -1  -1  -1  -5  35
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp2.html
│ │ │ @@ -185,15 +185,15 @@
│ │ │  
│ │ │  o6 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime c=character A
│ │ │ - -- .842169s elapsed
│ │ │ + -- .462508s elapsed
│ │ │  
│ │ │  o7 = Character over QQ
│ │ │        
│ │ │        (0, {0})  |   1  1   1   1   1   1  1   1   1   1    1
│ │ │        (1, {5})  |   0  1   0   2   0   1  3   0   2   4    6
│ │ │        (1, {7})  |   0  0   0   0   0   1  3   0   4  16   60
│ │ │        (1, {9})  |   0  0   0   0   2   2  2   0   4   8   20
│ │ │ ├── html2text {}
│ │ │ │ @@ -113,15 +113,15 @@
│ │ │ │  o5 : List
│ │ │ │  i6 : A=action(RI,S6)
│ │ │ │  
│ │ │ │  o6 = Complex with 11 actors
│ │ │ │  
│ │ │ │  o6 : ActionOnComplex
│ │ │ │  i7 : elapsedTime c=character A
│ │ │ │ - -- .842169s elapsed
│ │ │ │ + -- .462508s elapsed
│ │ │ │  
│ │ │ │  o7 = Character over QQ
│ │ │ │  
│ │ │ │        (0, {0})  |   1  1   1   1   1   1  1   1   1   1    1
│ │ │ │        (1, {5})  |   0  1   0   2   0   1  3   0   2   4    6
│ │ │ │        (1, {7})  |   0  0   0   0   0   1  3   0   4  16   60
│ │ │ │        (1, {9})  |   0  0   0   0   2   2  2   0   4   8   20
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html
│ │ │ @@ -315,15 +315,15 @@
│ │ │  
│ │ │  o19 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i20 : elapsedTime a1 = character A1
│ │ │ - -- .772423s elapsed
│ │ │ + -- .666105s elapsed
│ │ │  
│ │ │  o20 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │                   |                4    2         4    2
│ │ │         (1, {8})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │        (2, {11})  |  1   1  1  1            1                  1
│ │ │ @@ -331,15 +331,15 @@
│ │ │  
│ │ │  o20 : Character
    
│ │ │              
    
│ │ │                
    i21 : elapsedTime a2 = character A2
│ │ │ - -- 32.3092s elapsed
│ │ │ + -- 26.2348s elapsed
│ │ │  
│ │ │  o21 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │        (1, {16})  |  6   2  0  0           -1                 -1
│ │ │                   |                4    2         4    2
│ │ │        (2, {19})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ @@ -483,15 +483,15 @@
│ │ │  
│ │ │  o31 : ActionOnGradedModule
    
│ │ │              
    
│ │ │                
    i32 : elapsedTime b = character(B,21)
│ │ │ - -- 15.1894s elapsed
│ │ │ + -- 12.0247s elapsed
│ │ │  
│ │ │  o32 = Character over kk
│ │ │         
│ │ │        (0, {21})  |  1  1  1  1  1  1
│ │ │  
│ │ │  o32 : Character
    
│ │ │              
    
│ │ │                
    i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.268195s (cpu); 0.19787s (thread); 0s (gc)
│ │ │ + -- used 0.311946s (cpu); 0.245962s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of S
    
│ │ │              
    
│ │ │                
    i24 : betti res j
│ │ │ ├── html2text {}
│ │ │ │ @@ -230,15 +230,15 @@
│ │ │ │  o22 = total: 1 5 8 5 1
│ │ │ │            0: 1 . . . .
│ │ │ │            1: . 4 2 . .
│ │ │ │            2: . 1 6 5 1
│ │ │ │  
│ │ │ │  o22 : BettiTally
│ │ │ │  i23 : time j=bruns F.dd_3;
│ │ │ │ - -- used 0.268195s (cpu); 0.19787s (thread); 0s (gc)
│ │ │ │ + -- used 0.311946s (cpu); 0.245962s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 : Ideal of S
│ │ │ │  i24 : betti res j
│ │ │ │  
│ │ │ │               0 1 2 3 4
│ │ │ │  o24 = total: 1 3 6 5 1
│ │ │ │            0: 1 . . . .
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/___Ring__Map_sp_st_st_sp__Cell__Complex.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  
│ │ │  i8 : e23 = newCell({v2,v3});
│ │ │  
│ │ │  i9 : C = cellComplex(S,{e12,e23});
│ │ │  
│ │ │  i10 : cells(1,C)/cellLabel
│ │ │  
│ │ │ -o10 = {y*z, x*y}
│ │ │ +o10 = {x*y, y*z}
│ │ │  
│ │ │  o10 : List
│ │ │  
│ │ │  i11 : D = f ** C;
│ │ │  
│ │ │  i12 : cells(1,D)/cellLabel
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_boundary.out
│ │ │ @@ -34,14 +34,14 @@
│ │ │  
│ │ │  i12 : f = (cells(2,C))#0;
│ │ │  
│ │ │  i13 : boundary(f)
│ │ │  
│ │ │  o13 = {(Cell of dimension 1 with label 1, 1), (Cell of dimension 1 with label
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 1), (Cell of dimension 1 with label 1, -1), (Cell of dimension 1
│ │ │ +      1, -1), (Cell of dimension 1 with label 1, -1), (Cell of dimension 1
│ │ │        -----------------------------------------------------------------------
│ │ │ -      with label 1, -1)}
│ │ │ +      with label 1, 1)}
│ │ │  
│ │ │  o13 : List
│ │ │  
│ │ │  i14 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_boundary__Map_lp__Z__Z_cm__Cell__Complex_rp.out
│ │ │ @@ -16,24 +16,24 @@
│ │ │  
│ │ │  i8 : f = newSimplexCell {exy,exz,eyz};
│ │ │  
│ │ │  i9 : C = cellComplex(R,{f});
│ │ │  
│ │ │  i10 : d1 = boundaryMap_1 C
│ │ │  
│ │ │ -o10 = {1} | -x 0  -y |
│ │ │ -      {1} | z  y  0  |
│ │ │ -      {1} | 0  -x z  |
│ │ │ +o10 = {1} | -y -x 0  |
│ │ │ +      {1} | 0  z  y  |
│ │ │ +      {1} | z  0  -x |
│ │ │  
│ │ │  o10 : Matrix
│ │ │  
│ │ │  i11 : d2 = boundaryMap_2 C
│ │ │  
│ │ │ -o11 = {2} | -y |
│ │ │ +o11 = {2} | x  |
│ │ │ +      {2} | -y |
│ │ │        {2} | z  |
│ │ │ -      {2} | x  |
│ │ │  
│ │ │  o11 : Matrix
│ │ │  
│ │ │  i12 : assert(d1*d2==0)
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex_lp__Ring_cm__Polyhedral__Complex_rp.out
│ │ │ @@ -12,27 +12,27 @@
│ │ │  
│ │ │  i6 : F = polyhedralComplex {P1,P2,P3,P4};
│ │ │  
│ │ │  i7 : C = cellComplex(R,F);
│ │ │  
│ │ │  i8 : facePoset C
│ │ │  
│ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 1 1 1 0 0 0 1 0 1 1 1 1 |
│ │ │ -                      | 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 |
│ │ │ -                      | 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 |
│ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 |
│ │ │ +                      | 0 0 0 0 1 0 0 0 1 0 1 1 1 1 1 1 1 |
│ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o8 : Poset
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex_lp__Ring_cm__Simplicial__Complex_rp.out
│ │ │ @@ -24,15 +24,15 @@
│ │ │  
│ │ │  o7 : CellComplex
│ │ │  
│ │ │  i8 : applyValues(cells C, l -> apply(l,cellLabel))
│ │ │  
│ │ │                        5   4    3 2   2 3     4   5
│ │ │  o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }                                       }
│ │ │ -                      5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3   5 4   5    5 2
│ │ │ -               1 => {x y , x y , x y , x y , x y, x y , x y , x y , x y , x y , x y, x y }
│ │ │ -                      5 3   5 4   5 2   5 4   5 3   5 4   5 2   5 4
│ │ │ +                      4 4   5    3 3   5 2   2 4   5 3   5 4   5    5 2   5 3   5 4   4 2
│ │ │ +               1 => {x y , x y, x y , x y , x y , x y , x y , x y, x y , x y , x y , x y }
│ │ │ +                      5 2   5 4   5 3   5 4   5 2   5 4   5 3   5 4
│ │ │                 2 => {x y , x y , x y , x y , x y , x y , x y , x y }
│ │ │  
│ │ │  o8 : HashTable
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cells.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  
│ │ │  i5 : exy = newSimplexCell {vx,vy};
│ │ │  
│ │ │  i6 : C = cellComplex(R,{exy,vz});
│ │ │  
│ │ │  i7 : cells(C)
│ │ │  
│ │ │ -o7 = HashTable{0 => {Cell of dimension 0 with label z, Cell of dimension 0 with label x, Cell of dimension 0 with label y}}
│ │ │ +o7 = HashTable{0 => {Cell of dimension 0 with label y, Cell of dimension 0 with label x, Cell of dimension 0 with label z}}
│ │ │                 1 => {Cell of dimension 1 with label x*y}
│ │ │  
│ │ │  o7 : HashTable
│ │ │  
│ │ │  i8 : R = QQ;
│ │ │  
│ │ │  i9 : P = convexHull matrix {{1,1,-1,-1},{1,-1,1,-1}};
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cells_lp__Z__Z_cm__Cell__Complex_rp.out
│ │ │ @@ -10,17 +10,17 @@
│ │ │  
│ │ │  i5 : exy = newSimplexCell {vx,vy};
│ │ │  
│ │ │  i6 : C = cellComplex(R,{exy,vz});
│ │ │  
│ │ │  i7 : cells(0,C)
│ │ │  
│ │ │ -o7 = {Cell of dimension 0 with label z, Cell of dimension 0 with label x,
│ │ │ +o7 = {Cell of dimension 0 with label y, Cell of dimension 0 with label x,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Cell of dimension 0 with label y}
│ │ │ +     Cell of dimension 0 with label z}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : cells(1,C)
│ │ │  
│ │ │  o8 = {Cell of dimension 1 with label x*y}
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_face__Poset_lp__Cell__Complex_rp.out
│ │ │ @@ -20,18 +20,18 @@
│ │ │  
│ │ │  i10 : f = newCell({e12,e23,e34,e41});
│ │ │  
│ │ │  i11 : C = cellComplex(R,{f});
│ │ │  
│ │ │  i12 : facePoset C
│ │ │  
│ │ │ -o12 = Relation Matrix: | 1 0 0 0 1 0 1 0 1 |
│ │ │ -                       | 0 1 0 0 0 0 1 1 1 |
│ │ │ -                       | 0 0 1 0 0 1 0 1 1 |
│ │ │ -                       | 0 0 0 1 1 1 0 0 1 |
│ │ │ +o12 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 |
│ │ │ +                       | 0 1 0 0 0 1 0 1 1 |
│ │ │ +                       | 0 0 1 0 1 0 0 1 1 |
│ │ │ +                       | 0 0 0 1 1 0 1 0 1 |
│ │ │                         | 0 0 0 0 1 0 0 0 1 |
│ │ │                         | 0 0 0 0 0 1 0 0 1 |
│ │ │                         | 0 0 0 0 0 0 1 0 1 |
│ │ │                         | 0 0 0 0 0 0 0 1 1 |
│ │ │                         | 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o12 : Poset
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_hull__Complex.out
│ │ │ @@ -19,16 +19,16 @@
│ │ │                             
│ │ │       -1     0      1      2
│ │ │  
│ │ │  o4 : Complex
│ │ │  
│ │ │  i5 : cells(1,H)/cellLabel
│ │ │  
│ │ │ -       3 5    4 5   2        2    4 4    5 3    5 4
│ │ │ -o5 = {x y z, x y , x y*z, x*y z, x y z, x y z, x y }
│ │ │ +       4 4    5 3    5 4   3 5    4 5   2        2
│ │ │ +o5 = {x y z, x y z, x y , x y z, x y , x y*z, x*y z}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : cells(2,H)/cellLabel
│ │ │  
│ │ │         5 4    4 5
│ │ │  o6 = {x y z, x y z}
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_relabel__Cell__Complex.out
│ │ │ @@ -29,13 +29,13 @@
│ │ │          2      2   2
│ │ │  o13 = {a b, b*c , b , a*c}
│ │ │  
│ │ │  o13 : List
│ │ │  
│ │ │  i14 : for c in cells(1,relabeledC) list cellLabel(c)
│ │ │  
│ │ │ -        2 2       2   2 2   2   2     2
│ │ │ -o14 = {a b , a*b*c , b c , a b*c , a*b c}
│ │ │ +            2     2    2 2   2 2   2   2
│ │ │ +o14 = {a*b*c , a*b c, a b , b c , a b*c }
│ │ │  
│ │ │  o14 : List
│ │ │  
│ │ │  i15 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_scarf__Complex.out
│ │ │ @@ -19,16 +19,16 @@
│ │ │                      
│ │ │       -1     0      1
│ │ │  
│ │ │  o4 : Complex
│ │ │  
│ │ │  i5 : cells(1,C)/cellLabel
│ │ │  
│ │ │ -       4 5     2    5 4   2
│ │ │ -o5 = {x y , x*y z, x y , x y*z}
│ │ │ +       2      4 5   5 4     2
│ │ │ +o5 = {x y*z, x y , x y , x*y z}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : cells(2,C)/cellLabel
│ │ │  
│ │ │  o6 = {}
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_taylor__Complex.out
│ │ │ @@ -25,20 +25,20 @@
│ │ │  
│ │ │             1                       3
│ │ │  o5 = -1 : S  <------------------- S  : 0
│ │ │                  | -x2 -y2 -z2 |
│ │ │  
│ │ │            3                           3
│ │ │       0 : S  <----------------------- S  : 1
│ │ │ -               {2} | -y2 0   -z2 |
│ │ │ -               {2} | x2  -z2 0   |
│ │ │ -               {2} | 0   y2  x2  |
│ │ │ +               {2} | -z2 -y2 0   |
│ │ │ +               {2} | 0   x2  -z2 |
│ │ │ +               {2} | x2  0   y2  |
│ │ │  
│ │ │            3                   1
│ │ │       1 : S  <--------------- S  : 2
│ │ │ +               {4} | y2  |
│ │ │                 {4} | -z2 |
│ │ │                 {4} | -x2 |
│ │ │ -               {4} | y2  |
│ │ │  
│ │ │  o5 : ComplexMap
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/___Ring__Map_sp_st_st_sp__Cell__Complex.html
│ │ │ @@ -125,15 +125,15 @@
│ │ │                
    i9 : C = cellComplex(S,{e12,e23});
    
│ │ │              
    
│ │ │                
    i10 : cells(1,C)/cellLabel
│ │ │  
│ │ │ -o10 = {y*z, x*y}
│ │ │ +o10 = {x*y, y*z}
│ │ │  
│ │ │  o10 : List
    
│ │ │              
    
│ │ │                
    i11 : D = f ** C;
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  i5 : v2 = newCell({},y);
│ │ │ │  i6 : v3 = newCell({},z);
│ │ │ │  i7 : e12 = newCell({v1,v2});
│ │ │ │  i8 : e23 = newCell({v2,v3});
│ │ │ │  i9 : C = cellComplex(S,{e12,e23});
│ │ │ │  i10 : cells(1,C)/cellLabel
│ │ │ │  
│ │ │ │ -o10 = {y*z, x*y}
│ │ │ │ +o10 = {x*y, y*z}
│ │ │ │  
│ │ │ │  o10 : List
│ │ │ │  i11 : D = f ** C;
│ │ │ │  i12 : cells(1,D)/cellLabel
│ │ │ │  
│ │ │ │                 2
│ │ │ │  o12 = {a*b, b*c }
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_boundary.html
│ │ │ @@ -150,17 +150,17 @@
│ │ │            
    
│ │ │                
    i13 : boundary(f)
│ │ │  
│ │ │  o13 = {(Cell of dimension 1 with label 1, 1), (Cell of dimension 1 with label
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 1), (Cell of dimension 1 with label 1, -1), (Cell of dimension 1
│ │ │ +      1, -1), (Cell of dimension 1 with label 1, -1), (Cell of dimension 1
│ │ │        -----------------------------------------------------------------------
│ │ │ -      with label 1, -1)}
│ │ │ +      with label 1, 1)}
│ │ │  
│ │ │  o13 : List
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,17 +42,17 @@
│ │ │ │  i10 : P = convexHull matrix {{1,1,-1,-1},{1,-1,1,-1}};
│ │ │ │  i11 : C = cellComplex(R,P);
│ │ │ │  i12 : f = (cells(2,C))#0;
│ │ │ │  i13 : boundary(f)
│ │ │ │  
│ │ │ │  o13 = {(Cell of dimension 1 with label 1, 1), (Cell of dimension 1 with label
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      1, 1), (Cell of dimension 1 with label 1, -1), (Cell of dimension 1
│ │ │ │ +      1, -1), (Cell of dimension 1 with label 1, -1), (Cell of dimension 1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      with label 1, -1)}
│ │ │ │ +      with label 1, 1)}
│ │ │ │  
│ │ │ │  o13 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _b_o_u_n_d_a_r_y_C_e_l_l_s_(_C_e_l_l_) -- returns the boundary cells of the given cell
│ │ │ │  ********** WWaayyss ttoo uussee bboouunnddaarryy:: **********
│ │ │ │      * boundary(Cell)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_boundary__Map_lp__Z__Z_cm__Cell__Complex_rp.html
│ │ │ @@ -130,28 +130,28 @@
│ │ │                
    i9 : C = cellComplex(R,{f});
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : d1 = boundaryMap_1 C
│ │ │  
│ │ │ -o10 = {1} | -x 0  -y |
│ │ │ -      {1} | z  y  0  |
│ │ │ -      {1} | 0  -x z  |
│ │ │ +o10 = {1} | -y -x 0  |
│ │ │ +      {1} | 0  z  y  |
│ │ │ +      {1} | z  0  -x |
│ │ │  
│ │ │  o10 : Matrix
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : d2 = boundaryMap_2 C
│ │ │  
│ │ │ -o11 = {2} | -y |
│ │ │ +o11 = {2} | x  |
│ │ │ +      {2} | -y |
│ │ │        {2} | z  |
│ │ │ -      {2} | x  |
│ │ │  
│ │ │  o11 : Matrix
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : assert(d1*d2==0)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,24 +28,24 @@
│ │ │ │  i5 : exy = newSimplexCell {vx,vy};
│ │ │ │  i6 : exz = newSimplexCell {vx,vz};
│ │ │ │  i7 : eyz = newSimplexCell {vy,vz};
│ │ │ │  i8 : f = newSimplexCell {exy,exz,eyz};
│ │ │ │  i9 : C = cellComplex(R,{f});
│ │ │ │  i10 : d1 = boundaryMap_1 C
│ │ │ │  
│ │ │ │ -o10 = {1} | -x 0  -y |
│ │ │ │ -      {1} | z  y  0  |
│ │ │ │ -      {1} | 0  -x z  |
│ │ │ │ +o10 = {1} | -y -x 0  |
│ │ │ │ +      {1} | 0  z  y  |
│ │ │ │ +      {1} | z  0  -x |
│ │ │ │  
│ │ │ │  o10 : Matrix
│ │ │ │  i11 : d2 = boundaryMap_2 C
│ │ │ │  
│ │ │ │ -o11 = {2} | -y |
│ │ │ │ +o11 = {2} | x  |
│ │ │ │ +      {2} | -y |
│ │ │ │        {2} | z  |
│ │ │ │ -      {2} | x  |
│ │ │ │  
│ │ │ │  o11 : Matrix
│ │ │ │  i12 : assert(d1*d2==0)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_m_p_l_e_x_(_C_e_l_l_C_o_m_p_l_e_x_) -- compute the cellular chain complex for a cell
│ │ │ │        complex
│ │ │ │      * _c_o_m_p_l_e_x_(_S_i_m_p_l_i_c_i_a_l_C_o_m_p_l_e_x_) -- create the chain complex associated to a
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex_lp__Ring_cm__Polyhedral__Complex_rp.html
│ │ │ @@ -117,27 +117,27 @@
│ │ │                
    i7 : C = cellComplex(R,F);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : facePoset C
│ │ │  
│ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 1 1 1 0 0 0 1 0 1 1 1 1 |
│ │ │ -                      | 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 |
│ │ │ -                      | 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 |
│ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 |
│ │ │ +                      | 0 0 0 0 1 0 0 0 1 0 1 1 1 1 1 1 1 |
│ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o8 : Poset
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,27 +26,27 @@
│ │ │ │  i3 : P2 = convexHull matrix {{2,-2,0},{1,1,0}};
│ │ │ │  i4 : P3 = convexHull matrix {{-2,-2,0},{1,-1,0}};
│ │ │ │  i5 : P4 = convexHull matrix {{-2,2,0},{-1,-1,0}};
│ │ │ │  i6 : F = polyhedralComplex {P1,P2,P3,P4};
│ │ │ │  i7 : C = cellComplex(R,F);
│ │ │ │  i8 : facePoset C
│ │ │ │  
│ │ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 1 1 1 0 0 0 1 0 1 1 1 1 |
│ │ │ │ -                      | 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 |
│ │ │ │ -                      | 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 |
│ │ │ │ -                      | 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 |
│ │ │ │ -                      | 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 1 |
│ │ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 |
│ │ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 |
│ │ │ │ +                      | 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 |
│ │ │ │ +                      | 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 |
│ │ │ │ +                      | 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 |
│ │ │ │ +                      | 0 0 0 0 1 0 0 0 1 0 1 1 1 1 1 1 1 |
│ │ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │ │  
│ │ │ │  o8 : Poset
│ │ │ │  The labels on the vertices can be controlled via the optional parameter Labels
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex_lp__Ring_cm__Simplicial__Complex_rp.html
│ │ │ @@ -134,17 +134,17 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : applyValues(cells C, l -> apply(l,cellLabel))
│ │ │  
│ │ │                        5   4    3 2   2 3     4   5
│ │ │  o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }                                       }
│ │ │ -                      5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3   5 4   5    5 2
│ │ │ -               1 => {x y , x y , x y , x y , x y, x y , x y , x y , x y , x y , x y, x y }
│ │ │ -                      5 3   5 4   5 2   5 4   5 3   5 4   5 2   5 4
│ │ │ +                      4 4   5    3 3   5 2   2 4   5 3   5 4   5    5 2   5 3   5 4   4 2
│ │ │ +               1 => {x y , x y, x y , x y , x y , x y , x y , x y, x y , x y , x y , x y }
│ │ │ +                      5 2   5 4   5 3   5 4   5 2   5 4   5 3   5 4
│ │ │                 2 => {x y , x y , x y , x y , x y , x y , x y , x y }
│ │ │  
│ │ │  o8 : HashTable
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,19 +41,19 @@
│ │ │ │  
│ │ │ │  o7 : CellComplex
│ │ │ │  i8 : applyValues(cells C, l -> apply(l,cellLabel))
│ │ │ │  
│ │ │ │                        5   4    3 2   2 3     4   5
│ │ │ │  o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }
│ │ │ │  }
│ │ │ │ -                      5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3   5 4
│ │ │ │ -5    5 2
│ │ │ │ -               1 => {x y , x y , x y , x y , x y, x y , x y , x y , x y , x y ,
│ │ │ │ -x y, x y }
│ │ │ │ -                      5 3   5 4   5 2   5 4   5 3   5 4   5 2   5 4
│ │ │ │ +                      4 4   5    3 3   5 2   2 4   5 3   5 4   5    5 2   5 3
│ │ │ │ +5 4   4 2
│ │ │ │ +               1 => {x y , x y, x y , x y , x y , x y , x y , x y, x y , x y ,
│ │ │ │ +x y , x y }
│ │ │ │ +                      5 2   5 4   5 3   5 4   5 2   5 4   5 3   5 4
│ │ │ │                 2 => {x y , x y , x y , x y , x y , x y , x y , x y }
│ │ │ │  
│ │ │ │  o8 : HashTable
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_e_l_l_C_o_m_p_l_e_x -- create a cell complex
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _c_e_l_l_C_o_m_p_l_e_x_(_R_i_n_g_,_S_i_m_p_l_i_c_i_a_l_C_o_m_p_l_e_x_) -- Creates a cell complex from a
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cells.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │                
    i6 : C = cellComplex(R,{exy,vz});
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : cells(C)
│ │ │  
│ │ │ -o7 = HashTable{0 => {Cell of dimension 0 with label z, Cell of dimension 0 with label x, Cell of dimension 0 with label y}}
│ │ │ +o7 = HashTable{0 => {Cell of dimension 0 with label y, Cell of dimension 0 with label x, Cell of dimension 0 with label z}}
│ │ │                 1 => {Cell of dimension 1 with label x*y}
│ │ │  
│ │ │  o7 : HashTable
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,16 +19,16 @@
│ │ │ │  i2 : vx = newSimplexCell({},x);
│ │ │ │  i3 : vy = newSimplexCell({},y);
│ │ │ │  i4 : vz = newSimplexCell({},z);
│ │ │ │  i5 : exy = newSimplexCell {vx,vy};
│ │ │ │  i6 : C = cellComplex(R,{exy,vz});
│ │ │ │  i7 : cells(C)
│ │ │ │  
│ │ │ │ -o7 = HashTable{0 => {Cell of dimension 0 with label z, Cell of dimension 0 with
│ │ │ │ -label x, Cell of dimension 0 with label y}}
│ │ │ │ +o7 = HashTable{0 => {Cell of dimension 0 with label y, Cell of dimension 0 with
│ │ │ │ +label x, Cell of dimension 0 with label z}}
│ │ │ │                 1 => {Cell of dimension 1 with label x*y}
│ │ │ │  
│ │ │ │  o7 : HashTable
│ │ │ │  i8 : R = QQ;
│ │ │ │  i9 : P = convexHull matrix {{1,1,-1,-1},{1,-1,1,-1}};
│ │ │ │  i10 : C = cellComplex(R,P);
│ │ │ │  i11 : cells C
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cells_lp__Z__Z_cm__Cell__Complex_rp.html
│ │ │ @@ -108,17 +108,17 @@
│ │ │                
    i6 : C = cellComplex(R,{exy,vz});
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : cells(0,C)
│ │ │  
│ │ │ -o7 = {Cell of dimension 0 with label z, Cell of dimension 0 with label x,
│ │ │ +o7 = {Cell of dimension 0 with label y, Cell of dimension 0 with label x,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Cell of dimension 0 with label y}
│ │ │ +     Cell of dimension 0 with label z}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │                
    i8 : cells(1,C)
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,17 +19,17 @@
│ │ │ │  i2 : vx = newSimplexCell({},x);
│ │ │ │  i3 : vy = newSimplexCell({},y);
│ │ │ │  i4 : vz = newSimplexCell({},z);
│ │ │ │  i5 : exy = newSimplexCell {vx,vy};
│ │ │ │  i6 : C = cellComplex(R,{exy,vz});
│ │ │ │  i7 : cells(0,C)
│ │ │ │  
│ │ │ │ -o7 = {Cell of dimension 0 with label z, Cell of dimension 0 with label x,
│ │ │ │ +o7 = {Cell of dimension 0 with label y, Cell of dimension 0 with label x,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Cell of dimension 0 with label y}
│ │ │ │ +     Cell of dimension 0 with label z}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : cells(1,C)
│ │ │ │  
│ │ │ │  o8 = {Cell of dimension 1 with label x*y}
│ │ │ │  
│ │ │ │  o8 : List
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_face__Poset_lp__Cell__Complex_rp.html
│ │ │ @@ -132,18 +132,18 @@
│ │ │                
    i11 : C = cellComplex(R,{f});
    
│ │ │              
    
│ │ │                
    i12 : facePoset C
│ │ │  
│ │ │ -o12 = Relation Matrix: | 1 0 0 0 1 0 1 0 1 |
│ │ │ -                       | 0 1 0 0 0 0 1 1 1 |
│ │ │ -                       | 0 0 1 0 0 1 0 1 1 |
│ │ │ -                       | 0 0 0 1 1 1 0 0 1 |
│ │ │ +o12 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 |
│ │ │ +                       | 0 1 0 0 0 1 0 1 1 |
│ │ │ +                       | 0 0 1 0 1 0 0 1 1 |
│ │ │ +                       | 0 0 0 1 1 0 1 0 1 |
│ │ │                         | 0 0 0 0 1 0 0 0 1 |
│ │ │                         | 0 0 0 0 0 1 0 0 1 |
│ │ │                         | 0 0 0 0 0 0 1 0 1 |
│ │ │                         | 0 0 0 0 0 0 0 1 1 |
│ │ │                         | 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o12 : Poset
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,18 +25,18 @@
│ │ │ │  i7 : e23 = newCell({v2,v3});
│ │ │ │  i8 : e34 = newCell({v3,v4});
│ │ │ │  i9 : e41 = newCell({v4,v1});
│ │ │ │  i10 : f = newCell({e12,e23,e34,e41});
│ │ │ │  i11 : C = cellComplex(R,{f});
│ │ │ │  i12 : facePoset C
│ │ │ │  
│ │ │ │ -o12 = Relation Matrix: | 1 0 0 0 1 0 1 0 1 |
│ │ │ │ -                       | 0 1 0 0 0 0 1 1 1 |
│ │ │ │ -                       | 0 0 1 0 0 1 0 1 1 |
│ │ │ │ -                       | 0 0 0 1 1 1 0 0 1 |
│ │ │ │ +o12 = Relation Matrix: | 1 0 0 0 0 1 1 0 1 |
│ │ │ │ +                       | 0 1 0 0 0 1 0 1 1 |
│ │ │ │ +                       | 0 0 1 0 1 0 0 1 1 |
│ │ │ │ +                       | 0 0 0 1 1 0 1 0 1 |
│ │ │ │                         | 0 0 0 0 1 0 0 0 1 |
│ │ │ │                         | 0 0 0 0 0 1 0 0 1 |
│ │ │ │                         | 0 0 0 0 0 0 1 0 1 |
│ │ │ │                         | 0 0 0 0 0 0 0 1 1 |
│ │ │ │                         | 0 0 0 0 0 0 0 0 1 |
│ │ │ │  
│ │ │ │  o12 : Poset
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_hull__Complex.html
│ │ │ @@ -114,16 +114,16 @@
│ │ │  o4 : Complex
│ │ │              
    
│ │ │                
    i5 : cells(1,H)/cellLabel
│ │ │  
│ │ │ -       3 5    4 5   2        2    4 4    5 3    5 4
│ │ │ -o5 = {x y z, x y , x y*z, x*y z, x y z, x y z, x y }
│ │ │ +       4 4    5 3    5 4   3 5    4 5   2        2
│ │ │ +o5 = {x y z, x y z, x y , x y z, x y , x y*z, x*y z}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : cells(2,H)/cellLabel
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,16 +39,16 @@
│ │ │ │  o4 = S  <-- S  <-- S  <-- S
│ │ │ │  
│ │ │ │       -1     0      1      2
│ │ │ │  
│ │ │ │  o4 : Complex
│ │ │ │  i5 : cells(1,H)/cellLabel
│ │ │ │  
│ │ │ │ -       3 5    4 5   2        2    4 4    5 3    5 4
│ │ │ │ -o5 = {x y z, x y , x y*z, x*y z, x y z, x y z, x y }
│ │ │ │ +       4 4    5 3    5 4   3 5    4 5   2        2
│ │ │ │ +o5 = {x y z, x y z, x y , x y z, x y , x y*z, x*y z}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : cells(2,H)/cellLabel
│ │ │ │  
│ │ │ │         5 4    4 5
│ │ │ │  o6 = {x y z, x y z}
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_relabel__Cell__Complex.html
│ │ │ @@ -151,16 +151,16 @@
│ │ │  o13 : List
    
│ │ │              
    
│ │ │                
    i14 : for c in cells(1,relabeledC) list cellLabel(c)
│ │ │  
│ │ │ -        2 2       2   2 2   2   2     2
│ │ │ -o14 = {a b , a*b*c , b c , a b*c , a*b c}
│ │ │ +            2     2    2 2   2 2   2   2
│ │ │ +o14 = {a*b*c , a*b c, a b , b c , a b*c }
│ │ │  
│ │ │  o14 : List
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,16 +37,16 @@
│ │ │ │  
│ │ │ │          2      2   2
│ │ │ │  o13 = {a b, b*c , b , a*c}
│ │ │ │  
│ │ │ │  o13 : List
│ │ │ │  i14 : for c in cells(1,relabeledC) list cellLabel(c)
│ │ │ │  
│ │ │ │ -        2 2       2   2 2   2   2     2
│ │ │ │ -o14 = {a b , a*b*c , b c , a b*c , a*b c}
│ │ │ │ +            2     2    2 2   2 2   2   2
│ │ │ │ +o14 = {a*b*c , a*b c, a b , b c , a b*c }
│ │ │ │  
│ │ │ │  o14 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_e_l_l_L_a_b_e_l -- return the label of a cell
│ │ │ │      * _R_i_n_g_M_a_p_ _*_*_ _C_e_l_l_C_o_m_p_l_e_x -- tensors labels via a ring map
│ │ │ │  ********** WWaayyss ttoo uussee rreellaabbeellCCeellllCCoommpplleexx:: **********
│ │ │ │      * relabelCellComplex(CellComplex,HashTable)
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_scarf__Complex.html
│ │ │ @@ -109,16 +109,16 @@
│ │ │  o4 : Complex
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : cells(1,C)/cellLabel
│ │ │  
│ │ │ -       4 5     2    5 4   2
│ │ │ -o5 = {x y , x*y z, x y , x y*z}
│ │ │ +       2      4 5   5 4     2
│ │ │ +o5 = {x y*z, x y , x y , x*y z}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : cells(2,C)/cellLabel
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,16 +29,16 @@
│ │ │ │  o4 = S  <-- S  <-- S
│ │ │ │  
│ │ │ │       -1     0      1
│ │ │ │  
│ │ │ │  o4 : Complex
│ │ │ │  i5 : cells(1,C)/cellLabel
│ │ │ │  
│ │ │ │ -       4 5     2    5 4   2
│ │ │ │ -o5 = {x y , x*y z, x y , x y*z}
│ │ │ │ +       2      4 5   5 4     2
│ │ │ │ +o5 = {x y*z, x y , x y , x*y z}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : cells(2,C)/cellLabel
│ │ │ │  
│ │ │ │  o6 = {}
│ │ │ │  
│ │ │ │  o6 : List
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_taylor__Complex.html
│ │ │ @@ -115,23 +115,23 @@
│ │ │  
│ │ │             1                       3
│ │ │  o5 = -1 : S  <------------------- S  : 0
│ │ │                  | -x2 -y2 -z2 |
│ │ │  
│ │ │            3                           3
│ │ │       0 : S  <----------------------- S  : 1
│ │ │ -               {2} | -y2 0   -z2 |
│ │ │ -               {2} | x2  -z2 0   |
│ │ │ -               {2} | 0   y2  x2  |
│ │ │ +               {2} | -z2 -y2 0   |
│ │ │ +               {2} | 0   x2  -z2 |
│ │ │ +               {2} | x2  0   y2  |
│ │ │  
│ │ │            3                   1
│ │ │       1 : S  <--------------- S  : 2
│ │ │ +               {4} | y2  |
│ │ │                 {4} | -z2 |
│ │ │                 {4} | -x2 |
│ │ │ -               {4} | y2  |
│ │ │  
│ │ │  o5 : ComplexMap
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -35,23 +35,23 @@ │ │ │ │ │ │ │ │ 1 3 │ │ │ │ o5 = -1 : S <------------------- S : 0 │ │ │ │ | -x2 -y2 -z2 | │ │ │ │ │ │ │ │ 3 3 │ │ │ │ 0 : S <----------------------- S : 1 │ │ │ │ - {2} | -y2 0 -z2 | │ │ │ │ - {2} | x2 -z2 0 | │ │ │ │ - {2} | 0 y2 x2 | │ │ │ │ + {2} | -z2 -y2 0 | │ │ │ │ + {2} | 0 x2 -z2 | │ │ │ │ + {2} | x2 0 y2 | │ │ │ │ │ │ │ │ 3 1 │ │ │ │ 1 : S <--------------- S : 2 │ │ │ │ + {4} | y2 | │ │ │ │ {4} | -z2 | │ │ │ │ {4} | -x2 | │ │ │ │ - {4} | y2 | │ │ │ │ │ │ │ │ o5 : ComplexMap │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _h_u_l_l_C_o_m_p_l_e_x_(_M_o_n_o_m_i_a_l_I_d_e_a_l_) -- gives the hull complex of a monomial ideal │ │ │ │ ********** WWaayyss ttoo uussee ttaayylloorrCCoommpplleexx:: ********** │ │ │ │ * taylorComplex(MonomialIdeal) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize_lp__Chain__Complex_rp.out │ │ │ @@ -63,15 +63,15 @@ │ │ │ o11 : ChainComplex │ │ │ │ │ │ i12 : isMinimalChainComplex E │ │ │ │ │ │ o12 = false │ │ │ │ │ │ i13 : time m = minimize (E[1]); │ │ │ - -- used 0.305932s (cpu); 0.225305s (thread); 0s (gc) │ │ │ + -- used 0.232816s (cpu); 0.232815s (thread); 0s (gc) │ │ │ │ │ │ i14 : isQuasiIsomorphism m │ │ │ │ │ │ o14 = true │ │ │ │ │ │ i15 : E[1] == source m │ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ @@ -27,18 +27,18 @@ │ │ │ i5 : C = res(R^1/(ideal vars R))**(R^1/(ideal vars R)^5); │ │ │ │ │ │ i6 : mods = for i from 0 to max C list pushForward(f, C_i); │ │ │ │ │ │ i7 : C = chainComplex for i from min C+1 to max C list map(mods_(i-1),mods_i,substitute(matrix C.dd_i,S)); │ │ │ │ │ │ i8 : time m = resolutionOfChainComplex C; │ │ │ - -- used 0.09451s (cpu); 0.0949109s (thread); 0s (gc) │ │ │ + -- used 0.119998s (cpu); 0.117279s (thread); 0s (gc) │ │ │ │ │ │ i9 : time n = cartanEilenbergResolution C; │ │ │ - -- used 0.118192s (cpu); 0.118576s (thread); 0s (gc) │ │ │ + -- used 0.151893s (cpu); 0.153s (thread); 0s (gc) │ │ │ │ │ │ i10 : betti source m │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ o10 = total: 1 19 80 181 312 484 447 156 │ │ │ 0: 1 3 3 1 . . . . │ │ │ 1: . . 1 3 3 . . . │ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize_lp__Chain__Complex_rp.html │ │ │ @@ -186,15 +186,15 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -306,15 +306,15 @@ │ │ │ │ │ │ o21 : A │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -160,15 +160,15 @@ │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o14 = ideal (x x - x x x , x x ) │ │ │ │ 0 3 1 2 4 2 5 │ │ │ │ │ │ │ │ o14 : Ideal of R │ │ │ │ i15 : time csmK=CSM(A,K) │ │ │ │ - -- used 0.392129s (cpu); 0.284143s (thread); 0s (gc) │ │ │ │ + -- used 0.988269s (cpu); 0.415782s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ │ o15 = 7h h + 5h h + 4h h + h + 3h h + h │ │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ │ │ o15 : A │ │ │ │ i16 : csmKHash= CSM(A,K,Output=>HashForm) │ │ │ │ @@ -199,15 +199,15 @@ │ │ │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ │ o21 = 9h h + 9h h + 9h h + 3h + 7h h + 3h + 3h + 2h │ │ │ │ 1 2 1 2 1 2 1 1 2 2 1 2 │ │ │ │ │ │ │ │ o21 : A │ │ │ │ i22 : time CSM(A,K,m) │ │ │ │ - -- used 0.0601472s (cpu); 0.0572141s (thread); 0s (gc) │ │ │ │ + -- used 0.0968978s (cpu); 0.0771196s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ │ o22 = 7h h + 5h h + 4h h + h + 3h h + h │ │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ │ │ o22 : A │ │ │ │ In the case where the ambient space is a toric variety which is not a product │ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html │ │ │ @@ -77,15 +77,15 @@ │ │ │ │ │ │ o2 : NormalToricVariety │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i13 : time m = minimize (E[1]);
│ │ │ - -- used 0.305932s (cpu); 0.225305s (thread); 0s (gc)
    │ │ │ + -- used 0.232816s (cpu); 0.232815s (thread); 0s (gc) │ │ │
    │ │ │ 
    i14 : isQuasiIsomorphism m
│ │ │  
│ │ │  o14 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -81,15 +81,15 @@ │ │ │ │ o11 : ChainComplex │ │ │ │ i12 : isMinimalChainComplex E │ │ │ │ │ │ │ │ o12 = false │ │ │ │ Now we minimize the result. The free summand we added to the end maps to zero, │ │ │ │ and thus is part of the minimization. │ │ │ │ i13 : time m = minimize (E[1]); │ │ │ │ - -- used 0.305932s (cpu); 0.225305s (thread); 0s (gc) │ │ │ │ + -- used 0.232816s (cpu); 0.232815s (thread); 0s (gc) │ │ │ │ i14 : isQuasiIsomorphism m │ │ │ │ │ │ │ │ o14 = true │ │ │ │ i15 : E[1] == source m │ │ │ │ │ │ │ │ o15 = true │ │ │ │ i16 : E' = target m │ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html │ │ │ @@ -134,21 +134,21 @@ │ │ │     		│ │ │ 
    i7 : C = chainComplex for i from min C+1 to max C list map(mods_(i-1),mods_i,substitute(matrix C.dd_i,S));
    │ │ │
    │ │ │ 
    i8 : time m = resolutionOfChainComplex C;
│ │ │ - -- used 0.09451s (cpu); 0.0949109s (thread); 0s (gc)
    │ │ │ + -- used 0.119998s (cpu); 0.117279s (thread); 0s (gc) │ │ │
    │ │ │ 
    i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.118192s (cpu); 0.118576s (thread); 0s (gc)
    │ │ │ + -- used 0.151893s (cpu); 0.153s (thread); 0s (gc) │ │ │
    │ │ │ 
    i10 : betti source m
│ │ │  
│ │ │               0  1  2   3   4   5   6   7
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,17 +49,17 @@
│ │ │ │  
│ │ │ │  o4 : RingMap R <-- S
│ │ │ │  i5 : C = res(R^1/(ideal vars R))**(R^1/(ideal vars R)^5);
│ │ │ │  i6 : mods = for i from 0 to max C list pushForward(f, C_i);
│ │ │ │  i7 : C = chainComplex for i from min C+1 to max C list map(mods_(i-
│ │ │ │  1),mods_i,substitute(matrix C.dd_i,S));
│ │ │ │  i8 : time m = resolutionOfChainComplex C;
│ │ │ │ - -- used 0.09451s (cpu); 0.0949109s (thread); 0s (gc)
│ │ │ │ + -- used 0.119998s (cpu); 0.117279s (thread); 0s (gc)
│ │ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ │ - -- used 0.118192s (cpu); 0.118576s (thread); 0s (gc)
│ │ │ │ + -- used 0.151893s (cpu); 0.153s (thread); 0s (gc)
│ │ │ │  i10 : betti source m
│ │ │ │  
│ │ │ │               0  1  2   3   4   5   6   7
│ │ │ │  o10 = total: 1 19 80 181 312 484 447 156
│ │ │ │            0: 1  3  3   1   .   .   .   .
│ │ │ │            1: .  .  1   3   3   .   .   .
│ │ │ │            2: .  1  3   3   2   .   .   .
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out
│ │ │ @@ -83,15 +83,15 @@
│ │ │                2              2
│ │ │  o14 = ideal (x x  - x x x , x x )
│ │ │                0 3    1 2 4   2 5
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : time csmK=CSM(A,K)
│ │ │ - -- used 0.392129s (cpu); 0.284143s (thread); 0s (gc)
│ │ │ + -- used 0.988269s (cpu); 0.415782s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o15 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o15 : A
│ │ │  
│ │ │ @@ -124,15 +124,15 @@
│ │ │          2 2     2         2     2             2
│ │ │  o21 = 9h h  + 9h h  + 9h h  + 3h  + 7h h  + 3h  + 3h  + 2h
│ │ │          1 2     1 2     1 2     1     1 2     2     1     2
│ │ │  
│ │ │  o21 : A
│ │ │  
│ │ │  i22 : time CSM(A,K,m)
│ │ │ - -- used 0.0601472s (cpu); 0.0572141s (thread); 0s (gc)
│ │ │ + -- used 0.0968978s (cpu); 0.0771196s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o22 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o22 : A
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out
│ │ │ @@ -9,28 +9,28 @@
│ │ │  i2 : U = toricProjectiveSpace 7
│ │ │  
│ │ │  o2 = U
│ │ │  
│ │ │  o2 : NormalToricVariety
│ │ │  
│ │ │  i3 : time CSM U
│ │ │ - -- used 0.266893s (cpu); 0.194827s (thread); 0s (gc)
│ │ │ + -- used 0.282992s (cpu); 0.192923s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │  o3 : -----------------------------------------------------------------------------------------------
│ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x )
│ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5     0    6     0    7
│ │ │  
│ │ │  i4 : time CSM(U,CheckSmooth=>false)
│ │ │ - -- used 0.370147s (cpu); 0.289154s (thread); 0s (gc)
│ │ │ + -- used 0.40081s (cpu); 0.327775s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out
│ │ │ @@ -18,29 +18,29 @@
│ │ │  i3 : R=ZZ/32749[v_0..v_5];
│ │ │  
│ │ │  i4 : I=ideal(4*v_3*v_1*v_2-8*v_1*v_3^2,v_5*(v_0*v_1*v_4-v_2^3));
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.52357s (cpu); 0.322636s (thread); 0s (gc)
│ │ │ + -- used 1.00843s (cpu); 0.411282s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │         1      1      1      1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o5 : ------
│ │ │          6
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ - -- used 2.53559s (cpu); 2.22953s (thread); 0s (gc)
│ │ │ + -- used 2.23707s (cpu); 2.05437s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          6
│ │ │ @@ -53,29 +53,29 @@
│ │ │  i8 : S=QQ[s_0..s_3];
│ │ │  
│ │ │  i9 : K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));
│ │ │  
│ │ │  o9 : Ideal of S
│ │ │  
│ │ │  i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.287927s (cpu); 0.199448s (thread); 0s (gc)
│ │ │ + -- used 0.316646s (cpu); 0.227054s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │  o10 : ------
│ │ │           4
│ │ │          h
│ │ │           1
│ │ │  
│ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0818983s (cpu); 0.0819088s (thread); 0s (gc)
│ │ │ + -- used 0.096974s (cpu); 0.0968747s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o11 = 3H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │           4
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out
│ │ │ @@ -21,20 +21,20 @@
│ │ │               2                                                        2
│ │ │       - 14254x  - 11226x x  + 2653x x  + 12365x x  - 10226x x  - 12696x )
│ │ │               3         0 4        1 4         2 4         3 4         4
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0335711s (cpu); 0.0326314s (thread); 0s (gc)
│ │ │ + -- used 0.0743873s (cpu); 0.0417517s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : time Euler I
│ │ │ - -- used 0.237054s (cpu); 0.141798s (thread); 0s (gc)
│ │ │ + -- used 0.280241s (cpu); 0.167637s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 4
│ │ │  
│ │ │  i6 : EulerIHash=Euler(I,Output=>HashForm);
│ │ │  
│ │ │  i7 : A=ring EulerIHash#"CSM"
│ │ │  
│ │ │ @@ -62,20 +62,20 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x )
│ │ │          0 3
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : time Euler(J,Method=>DirectCompleteInt)
│ │ │ - -- used 0.0727122s (cpu); 0.0711736s (thread); 0s (gc)
│ │ │ + -- used 0.17632s (cpu); 0.0973445s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
│ │ │  
│ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.182232s (cpu); 0.103237s (thread); 0s (gc)
│ │ │ + -- used 0.223838s (cpu); 0.10167s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = 2
│ │ │  
│ │ │  i12 : R=MultiProjCoordRing({2,2})
│ │ │  
│ │ │  o12 = R
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler__Affine.out
│ │ │ @@ -13,12 +13,12 @@
│ │ │              2    2    2
│ │ │  o3 = ideal(x  + x  + x  - 1)
│ │ │              1    2    3
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time EulerAffine I
│ │ │ - -- used 0.0476101s (cpu); 0.0472749s (thread); 0s (gc)
│ │ │ + -- used 0.0999706s (cpu); 0.0707451s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out
│ │ │ @@ -7,29 +7,29 @@
│ │ │  o1 : PolynomialRing
│ │ │  
│ │ │  i2 : I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 1.44931s (cpu); 1.05344s (thread); 0s (gc)
│ │ │ + -- used 4.45599s (cpu); 1.26987s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │  o3 : ----------
│ │ │          3   3
│ │ │        (h , h )
│ │ │          1   2
│ │ │  
│ │ │  i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 1.55498s (cpu); 1.16715s (thread); 0s (gc)
│ │ │ + -- used 5.28056s (cpu); 1.37429s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out
│ │ │ @@ -3,43 +3,43 @@
│ │ │  i1 : R = ZZ/32749[x_0..x_4];
│ │ │  
│ │ │  i2 : I=ideal(random(2,R),random(2,R),random(1,R));
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM I
│ │ │ - -- used 0.579858s (cpu); 0.398931s (thread); 0s (gc)
│ │ │ + -- used 0.941589s (cpu); 0.494475s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0315102s (cpu); 0.03088s (thread); 0s (gc)
│ │ │ + -- used 0.0762452s (cpu); 0.0459221s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o4 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i5 : time Chern I
│ │ │ - -- used 0.0313383s (cpu); 0.0307065s (thread); 0s (gc)
│ │ │ + -- used 0.0675067s (cpu); 0.0398229s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o5 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out
│ │ │ @@ -7,29 +7,29 @@
│ │ │  o1 : PolynomialRing
│ │ │  
│ │ │  i2 : I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM I
│ │ │ - -- used 1.15414s (cpu); 0.859181s (thread); 0s (gc)
│ │ │ + -- used 2.58895s (cpu); 1.09767s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o3 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          7
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ - -- used 0.305858s (cpu); 0.2109s (thread); 0s (gc)
│ │ │ + -- used 0.562209s (cpu); 0.25065s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o4 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html
│ │ │ @@ -239,15 +239,15 @@
│ │ │  
│ │ │  o14 : Ideal of R
    │ │ │
    │ │ │ 
    i15 : time csmK=CSM(A,K)
│ │ │ - -- used 0.392129s (cpu); 0.284143s (thread); 0s (gc)
│ │ │ + -- used 0.988269s (cpu); 0.415782s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o15 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o15 : A
    │ │ │
    │ │ │ 
    i22 : time CSM(A,K,m)
│ │ │ - -- used 0.0601472s (cpu); 0.0572141s (thread); 0s (gc)
│ │ │ + -- used 0.0968978s (cpu); 0.0771196s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o22 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o22 : A
    │ │ │
    │ │ │ 
    i3 : time CSM U
│ │ │ - -- used 0.266893s (cpu); 0.194827s (thread); 0s (gc)
│ │ │ + -- used 0.282992s (cpu); 0.192923s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │ @@ -93,15 +93,15 @@
│ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x )
│ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5     0    6     0    7
    │ │ │
    │ │ │ 
    i4 : time CSM(U,CheckSmooth=>false)
│ │ │ - -- used 0.370147s (cpu); 0.289154s (thread); 0s (gc)
│ │ │ + -- used 0.40081s (cpu); 0.327775s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,30 +16,30 @@
│ │ │ │  o1 : Package
│ │ │ │  i2 : U = toricProjectiveSpace 7
│ │ │ │  
│ │ │ │  o2 = U
│ │ │ │  
│ │ │ │  o2 : NormalToricVariety
│ │ │ │  i3 : time CSM U
│ │ │ │ - -- used 0.266893s (cpu); 0.194827s (thread); 0s (gc)
│ │ │ │ + -- used 0.282992s (cpu); 0.192923s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         7      6      5      4      3      2
│ │ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │ │         7      7      7      7      7      7     7
│ │ │ │  
│ │ │ │                                                  ZZ[x ..x ]
│ │ │ │                                                      0   7
│ │ │ │  o3 : --------------------------------------------------------------------------
│ │ │ │  ---------------------
│ │ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x ,
│ │ │ │  - x  + x , - x  + x )
│ │ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5
│ │ │ │  0    6     0    7
│ │ │ │  i4 : time CSM(U,CheckSmooth=>false)
│ │ │ │ - -- used 0.370147s (cpu); 0.289154s (thread); 0s (gc)
│ │ │ │ + -- used 0.40081s (cpu); 0.327775s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         7      6      5      4      3      2
│ │ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │ │         7      7      7      7      7      7     7
│ │ │ │  
│ │ │ │                                                  ZZ[x ..x ]
│ │ │ │                                                      0   7
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    │ │ │
    │ │ │ 
    i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.52357s (cpu); 0.322636s (thread); 0s (gc)
│ │ │ + -- used 1.00843s (cpu); 0.411282s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │         1      1      1      1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -114,15 +114,15 @@
│ │ │         h
│ │ │          1
    │ │ │
    │ │ │ 
    i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ - -- used 2.53559s (cpu); 2.22953s (thread); 0s (gc)
│ │ │ + -- used 2.23707s (cpu); 2.05437s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          6
│ │ │ @@ -147,15 +147,15 @@
│ │ │  
│ │ │  o9 : Ideal of S
    │ │ │
    │ │ │ 
    i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.287927s (cpu); 0.199448s (thread); 0s (gc)
│ │ │ + -- used 0.316646s (cpu); 0.227054s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │ @@ -164,15 +164,15 @@
│ │ │          h
│ │ │           1
    │ │ │
    │ │ │ 
    i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0818983s (cpu); 0.0819088s (thread); 0s (gc)
│ │ │ + -- used 0.096974s (cpu); 0.0968747s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o11 = 3H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │           4
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,28 +32,28 @@
│ │ │ │  using the regenerative cascade implemented in Bertini. This is done by choosing
│ │ │ │  the option bertini, provided Bertini is _i_n_s_t_a_l_l_e_d_ _a_n_d_ _c_o_n_f_i_g_u_r_e_d.
│ │ │ │  i3 : R=ZZ/32749[v_0..v_5];
│ │ │ │  i4 : I=ideal(4*v_3*v_1*v_2-8*v_1*v_3^2,v_5*(v_0*v_1*v_4-v_2^3));
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ │ - -- used 0.52357s (cpu); 0.322636s (thread); 0s (gc)
│ │ │ │ + -- used 1.00843s (cpu); 0.411282s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         5      4      3      2
│ │ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │ │         1      1      1      1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o5 : ------
│ │ │ │          6
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ │ - -- used 2.53559s (cpu); 2.22953s (thread); 0s (gc)
│ │ │ │ + -- used 2.23707s (cpu); 2.05437s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         5      4      3      2
│ │ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o6 : -----
│ │ │ │          6
│ │ │ │ @@ -62,28 +62,28 @@
│ │ │ │  
│ │ │ │  o7 = 2
│ │ │ │  i8 : S=QQ[s_0..s_3];
│ │ │ │  i9 : K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));
│ │ │ │  
│ │ │ │  o9 : Ideal of S
│ │ │ │  i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ │ - -- used 0.287927s (cpu); 0.199448s (thread); 0s (gc)
│ │ │ │ + -- used 0.316646s (cpu); 0.227054s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o10 = 3h  + 5h
│ │ │ │          1     1
│ │ │ │  
│ │ │ │        ZZ[h ]
│ │ │ │            1
│ │ │ │  o10 : ------
│ │ │ │           4
│ │ │ │          h
│ │ │ │           1
│ │ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ │ - -- used 0.0818983s (cpu); 0.0819088s (thread); 0s (gc)
│ │ │ │ + -- used 0.096974s (cpu); 0.0968747s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o11 = 3H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │           4
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html
│ │ │ @@ -130,23 +130,23 @@
│ │ │  
│ │ │  o3 : Ideal of R
    │ │ │
    │ │ │ 
    i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0335711s (cpu); 0.0326314s (thread); 0s (gc)
│ │ │ + -- used 0.0743873s (cpu); 0.0417517s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
    │ │ │
    │ │ │ 
    i5 : time Euler I
│ │ │ - -- used 0.237054s (cpu); 0.141798s (thread); 0s (gc)
│ │ │ + -- used 0.280241s (cpu); 0.167637s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 4
    │ │ │
    │ │ │ 
    i6 : EulerIHash=Euler(I,Output=>HashForm);
    │ │ │ @@ -194,23 +194,23 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i10 : time Euler(J,Method=>DirectCompleteInt)
│ │ │ - -- used 0.0727122s (cpu); 0.0711736s (thread); 0s (gc)
│ │ │ + -- used 0.17632s (cpu); 0.0973445s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
    │ │ │
    │ │ │ 
    i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.182232s (cpu); 0.103237s (thread); 0s (gc)
│ │ │ + -- used 0.223838s (cpu); 0.10167s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = 2
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │ ├── html2text {} │ │ │ │ @@ -74,19 +74,19 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 │ │ │ │ - 14254x - 11226x x + 2653x x + 12365x x - 10226x x - 12696x ) │ │ │ │ 3 0 4 1 4 2 4 3 4 4 │ │ │ │ │ │ │ │ o3 : Ideal of R │ │ │ │ i4 : time Euler(I,InputIsSmooth=>true) │ │ │ │ - -- used 0.0335711s (cpu); 0.0326314s (thread); 0s (gc) │ │ │ │ + -- used 0.0743873s (cpu); 0.0417517s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 4 │ │ │ │ i5 : time Euler I │ │ │ │ - -- used 0.237054s (cpu); 0.141798s (thread); 0s (gc) │ │ │ │ + -- used 0.280241s (cpu); 0.167637s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 4 │ │ │ │ i6 : EulerIHash=Euler(I,Output=>HashForm); │ │ │ │ i7 : A=ring EulerIHash#"CSM" │ │ │ │ │ │ │ │ o7 = A │ │ │ │ │ │ │ │ @@ -114,19 +114,19 @@ │ │ │ │ o9 : Ideal of R │ │ │ │ Note that the ideal J above is a complete intersection, thus we may change the │ │ │ │ method option which may speed computation in some cases. We may also note that │ │ │ │ the ideal generated by the first 2 generators of I defines a smooth scheme and │ │ │ │ input this information into the method. This may also improve computation │ │ │ │ speed. │ │ │ │ i10 : time Euler(J,Method=>DirectCompleteInt) │ │ │ │ - -- used 0.0727122s (cpu); 0.0711736s (thread); 0s (gc) │ │ │ │ + -- used 0.17632s (cpu); 0.0973445s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 = 2 │ │ │ │ i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1}) │ │ │ │ - -- used 0.182232s (cpu); 0.103237s (thread); 0s (gc) │ │ │ │ + -- used 0.223838s (cpu); 0.10167s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 = 2 │ │ │ │ Now consider an example in \PP^2 \times \PP^2. │ │ │ │ i12 : R=MultiProjCoordRing({2,2}) │ │ │ │ │ │ │ │ o12 = R │ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler__Affine.html │ │ │ @@ -100,15 +100,15 @@ │ │ │ │ │ │ o3 : Ideal of R │ │ │
    │ │ │ 
    i4 : time EulerAffine I
│ │ │ - -- used 0.0476101s (cpu); 0.0472749s (thread); 0s (gc)
│ │ │ + -- used 0.0999706s (cpu); 0.0707451s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │ ├── html2text {} │ │ │ │ @@ -23,15 +23,15 @@ │ │ │ │ │ │ │ │ 2 2 2 │ │ │ │ o3 = ideal(x + x + x - 1) │ │ │ │ 1 2 3 │ │ │ │ │ │ │ │ o3 : Ideal of R │ │ │ │ i4 : time EulerAffine I │ │ │ │ - -- used 0.0476101s (cpu); 0.0472749s (thread); 0s (gc) │ │ │ │ + -- used 0.0999706s (cpu); 0.0707451s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ Observe that the algorithm is a probabilistic algorithm and may give a wrong │ │ │ │ answer with a small but nonzero probability. Read more under _p_r_o_b_a_b_i_l_i_s_t_i_c │ │ │ │ _a_l_g_o_r_i_t_h_m. │ │ │ │ ********** WWaayyss ttoo uussee EEuulleerrAAffffiinnee:: ********** │ │ │ │ * EulerAffine(Ideal) │ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ @@ -75,15 +75,15 @@ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 1.44931s (cpu); 1.05344s (thread); 0s (gc)
│ │ │ + -- used 4.45599s (cpu); 1.26987s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │ @@ -92,15 +92,15 @@
│ │ │        (h , h )
│ │ │          1   2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 1.55498s (cpu); 1.16715s (thread); 0s (gc)
│ │ │ + -- used 5.28056s (cpu); 1.37429s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,28 +16,28 @@
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ │ - -- used 1.44931s (cpu); 1.05344s (thread); 0s (gc)
│ │ │ │ + -- used 4.45599s (cpu); 1.26987s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2 2     2         2
│ │ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │ │         1 2     1 2     1 2
│ │ │ │  
│ │ │ │       ZZ[h ..h ]
│ │ │ │           1   2
│ │ │ │  o3 : ----------
│ │ │ │          3   3
│ │ │ │        (h , h )
│ │ │ │          1   2
│ │ │ │  i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ │ - -- used 1.55498s (cpu); 1.16715s (thread); 0s (gc)
│ │ │ │ + -- used 5.28056s (cpu); 1.37429s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2 2     2         2
│ │ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │ │         1 2     1 2     1 2
│ │ │ │  
│ │ │ │       ZZ[h ..h ]
│ │ │ │           1   2
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html
│ │ │ @@ -71,15 +71,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time CSM I
│ │ │ - -- used 0.579858s (cpu); 0.398931s (thread); 0s (gc)
│ │ │ + -- used 0.941589s (cpu); 0.494475s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -88,15 +88,15 @@
│ │ │         h
│ │ │          1
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0315102s (cpu); 0.03088s (thread); 0s (gc)
│ │ │ + -- used 0.0762452s (cpu); 0.0459221s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -110,15 +110,15 @@
│ │ │          
    
│ │ │            
    Note that one could, equivalently, use the command Chern instead in this case.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -438,26 +438,26 @@
│ │ │  
│ │ │  o26 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -182,15 +182,15 @@
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {0, -1, 0, 0, 0, -1}, {0, 0, -1, 0, 0, -1}, {0, 0, 0, -1, 0, -1}, {0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        0, 0, 0, -1, -1}}
│ │ │ │  
│ │ │ │  o19 : List
│ │ │ │  i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ │ - -- 3.44748s elapsed
│ │ │ │ + -- 3.3735s elapsed
│ │ │ │  
│ │ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -262,42 +262,42 @@
│ │ │ │                         {2, 2, 3, 1, -4, -6} => {{0, 1, 0, 0, 0}, {{1, 1x1*x2}}}
│ │ │ │  i22 : degree(X_3 + X_7 + X_8)
│ │ │ │  
│ │ │ │  o22 = {0, 0, 1, 2, 0, -1}
│ │ │ │  
│ │ │ │  o22 : List
│ │ │ │  i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ │ - -- .360827s elapsed
│ │ │ │ + -- .515359s elapsed
│ │ │ │  
│ │ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o23 : List
│ │ │ │  i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X
│ │ │ │  (0,0,1,2,0,-1))
│ │ │ │ - -- 10.6243s elapsed
│ │ │ │ + -- 10.0765s elapsed
│ │ │ │  
│ │ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o24 : List
│ │ │ │  i25 : assert(cohomvec1 == cohomvec2)
│ │ │ │  i26 : degree(X_3 + X_7 - X_8)
│ │ │ │  
│ │ │ │  o26 = {0, 0, 1, 2, -2, -1}
│ │ │ │  
│ │ │ │  o26 : List
│ │ │ │  i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ │ - -- .331804s elapsed
│ │ │ │ + -- .544485s elapsed
│ │ │ │  
│ │ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j
│ │ │ │  (X, OO_X(0,0,1,2,-2,-1))
│ │ │ │ - -- .553928s elapsed
│ │ │ │ - -- .55396s elapsed
│ │ │ │ + -- .445109s elapsed
│ │ │ │ + -- .445139s elapsed
│ │ │ │  
│ │ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o28 : List
│ │ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ │ │  _c_o_h_o_m_C_a_l_g computes cohomology vectors by calling CohomCalg. It also stashes
│ │ │ │  it's results in the toric variety's cache table, so computations need not be
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : len = 10
│ │ │  
│ │ │  o6 = 10
│ │ │  
│ │ │  i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 6.31114s (cpu); 4.81547s (thread); 0s (gc)
│ │ │ + -- used 7.57796s (cpu); 5.98248s (thread); 0s (gc)
│ │ │  
│ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S   \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /    S   \68     /    S   \76
│ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|
│ │ │       |  2   3 |      |  2   3 |      |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |
│ │ │       |(x , x )|      |(x , x )|      |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|
│ │ │       \  0   1 /      \  0   1 /      \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /
│ │ │                                                                                                                                                                                
│ │ │ @@ -140,37 +140,37 @@
│ │ │  i19 : R1 = R/ideal ff
│ │ │  
│ │ │  o19 = R1
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │  
│ │ │  i20 : FF = time Shamash(R1,F,4)
│ │ │ - -- used 0.0677985s (cpu); 0.0677563s (thread); 0s (gc)
│ │ │ + -- used 0.101019s (cpu); 0.101018s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        0       1       2        3        4
│ │ │  
│ │ │  o20 : Complex
│ │ │  
│ │ │  i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ - -- used 1.06441s (cpu); 0.797691s (thread); 0s (gc)
│ │ │ + -- used 1.25759s (cpu); 0.940751s (thread); 0s (gc)
│ │ │  
│ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │                                                   
│ │ │        0         1         2          3          4
│ │ │  
│ │ │  o21 : Complex
│ │ │  
│ │ │  i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 0.915388s (cpu); 0.724566s (thread); 0s (gc)
│ │ │ + -- used 1.15963s (cpu); 0.924254s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        -2      -1      0        1        2
│ │ │  
│ │ │  o22 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out
│ │ │ @@ -2,21 +2,21 @@
│ │ │  
│ │ │  i1 : setRandomSeed 0
│ │ │   -- setting random seed to 0
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 1.20315s (cpu); 0.562216s (thread); 0s (gc)
│ │ │ + -- used 1.27474s (cpu); 0.559822s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.40989s (cpu); 0.176145s (thread); 0s (gc)
│ │ │ + -- used 0.198487s (cpu); 0.139064s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 4.068e-06s (cpu); 3.406e-06s (thread); 0s (gc)
│ │ │ + -- used 3.834e-06s (cpu); 3.256e-06s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out
│ │ │ @@ -2,23 +2,23 @@
│ │ │  
│ │ │  i1 : setRandomSeed 0
│ │ │   -- setting random seed to 0
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : twoMonomials(2,3)
│ │ │ - -- used 0.718914s (cpu); 0.551814s (thread); 0s (gc)
│ │ │ + -- used 1.24606s (cpu); 0.744577s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.50418s (cpu); 0.37207s (thread); 0s (gc)
│ │ │ + -- used 0.731217s (cpu); 0.44469s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.0982566s (cpu); 0.0978823s (thread); 0s (gc)
│ │ │ + -- used 0.132642s (cpu); 0.120555s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html
│ │ │ @@ -136,15 +136,15 @@
│ │ │  
│ │ │  o6 = 10
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time Chern I
│ │ │ - -- used 0.0313383s (cpu); 0.0307065s (thread); 0s (gc)
│ │ │ + -- used 0.0675067s (cpu); 0.0398229s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o5 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,42 +9,42 @@
│ │ │ │  input ideal is known to define a smooth subscheme setting this option to true
│ │ │ │  will speed up computations (it is set to false by default).
│ │ │ │  i1 : R = ZZ/32749[x_0..x_4];
│ │ │ │  i2 : I=ideal(random(2,R),random(2,R),random(1,R));
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM I
│ │ │ │ - -- used 0.579858s (cpu); 0.398931s (thread); 0s (gc)
│ │ │ │ + -- used 0.941589s (cpu); 0.494475s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o3 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ │ - -- used 0.0315102s (cpu); 0.03088s (thread); 0s (gc)
│ │ │ │ + -- used 0.0762452s (cpu); 0.0459221s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o4 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o4 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  Note that one could, equivalently, use the command _C_h_e_r_n instead in this case.
│ │ │ │  i5 : time Chern I
│ │ │ │ - -- used 0.0313383s (cpu); 0.0307065s (thread); 0s (gc)
│ │ │ │ + -- used 0.0675067s (cpu); 0.0398229s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o5 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : time CSM I
│ │ │ - -- used 1.15414s (cpu); 0.859181s (thread); 0s (gc)
│ │ │ + -- used 2.58895s (cpu); 1.09767s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o3 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -92,15 +92,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ - -- used 0.305858s (cpu); 0.2109s (thread); 0s (gc)
│ │ │ + -- used 0.562209s (cpu); 0.25065s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o4 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,28 +18,28 @@
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM I
│ │ │ │ - -- used 1.15414s (cpu); 0.859181s (thread); 0s (gc)
│ │ │ │ + -- used 2.58895s (cpu); 1.09767s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o3 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          7
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ │ - -- used 0.305858s (cpu); 0.2109s (thread); 0s (gc)
│ │ │ │ + -- used 0.562209s (cpu); 0.25065s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o4 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out
│ │ │ @@ -184,15 +184,15 @@
│ │ │        {0, -1, 0, 0, 0, -1}, {0, 0, -1, 0, 0, -1}, {0, 0, 0, -1, 0, -1}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, -1, -1}}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 3.44748s elapsed
│ │ │ + -- 3.3735s elapsed
│ │ │  
│ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │        -----------------------------------------------------------------------
│ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -265,45 +265,45 @@
│ │ │  i22 : degree(X_3 + X_7 + X_8)
│ │ │  
│ │ │  o22 = {0, 0, 1, 2, 0, -1}
│ │ │  
│ │ │  o22 : List
│ │ │  
│ │ │  i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ - -- .360827s elapsed
│ │ │ + -- .515359s elapsed
│ │ │  
│ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o23 : List
│ │ │  
│ │ │  i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,0,-1))
│ │ │ - -- 10.6243s elapsed
│ │ │ + -- 10.0765s elapsed
│ │ │  
│ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o24 : List
│ │ │  
│ │ │  i25 : assert(cohomvec1 == cohomvec2)
│ │ │  
│ │ │  i26 : degree(X_3 + X_7 - X_8)
│ │ │  
│ │ │  o26 = {0, 0, 1, 2, -2, -1}
│ │ │  
│ │ │  o26 : List
│ │ │  
│ │ │  i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ - -- .331804s elapsed
│ │ │ + -- .544485s elapsed
│ │ │  
│ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,-2,-1))
│ │ │ - -- .553928s elapsed
│ │ │ - -- .55396s elapsed
│ │ │ + -- .445109s elapsed
│ │ │ + -- .445139s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
│ │ │  
│ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/html/index.html
│ │ │ @@ -314,15 +314,15 @@
│ │ │  
│ │ │  o19 : List
    
│ │ │              
    
│ │ │                
    i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 3.44748s elapsed
│ │ │ + -- 3.3735s elapsed
│ │ │  
│ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │        -----------------------------------------------------------------------
│ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -404,25 +404,25 @@
│ │ │  
│ │ │  o22 : List
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ - -- .360827s elapsed
│ │ │ + -- .515359s elapsed
│ │ │  
│ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o23 : List
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,0,-1))
│ │ │ - -- 10.6243s elapsed
│ │ │ + -- 10.0765s elapsed
│ │ │  
│ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o24 : List
    
│ │ │              
    
│ │ │                
    i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ - -- .331804s elapsed
│ │ │ + -- .544485s elapsed
│ │ │  
│ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o27 : List
    
│ │ │              
    
│ │ │                
    i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,-2,-1))
│ │ │ - -- .553928s elapsed
│ │ │ - -- .55396s elapsed
│ │ │ + -- .445109s elapsed
│ │ │ + -- .445139s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
    
│ │ │              
    
│ │ │                
    i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 6.31114s (cpu); 4.81547s (thread); 0s (gc)
│ │ │ + -- used 7.57796s (cpu); 5.98248s (thread); 0s (gc)
│ │ │  
│ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S   \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /    S   \68     /    S   \76
│ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|
│ │ │       |  2   3 |      |  2   3 |      |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |
│ │ │       |(x , x )|      |(x , x )|      |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|
│ │ │       \  0   1 /      \  0   1 /      \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /
│ │ │                                                                                                                                                                                
│ │ │ @@ -300,28 +300,28 @@
│ │ │  
│ │ │  o19 : QuotientRing
    
│ │ │              
    
│ │ │                
    i20 : FF = time Shamash(R1,F,4)
│ │ │ - -- used 0.0677985s (cpu); 0.0677563s (thread); 0s (gc)
│ │ │ + -- used 0.101019s (cpu); 0.101018s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        0       1       2        3        4
│ │ │  
│ │ │  o20 : Complex
    
│ │ │              
    
│ │ │                
    i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ - -- used 1.06441s (cpu); 0.797691s (thread); 0s (gc)
│ │ │ + -- used 1.25759s (cpu); 0.940751s (thread); 0s (gc)
│ │ │  
│ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │                                                   
│ │ │        0         1         2          3          4
│ │ │ @@ -333,15 +333,15 @@
│ │ │          
    
│ │ │            
    The function also deals correctly with complexes F where min F is not 0:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 0.915388s (cpu); 0.724566s (thread); 0s (gc)
│ │ │ + -- used 1.15963s (cpu); 0.924254s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        -2      -1      0        1        2
│ │ │  
│ │ │  o22 : Complex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │  o5 = R
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : len = 10
│ │ │ │  
│ │ │ │  o6 = 10
│ │ │ │  i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ │ - -- used 6.31114s (cpu); 4.81547s (thread); 0s (gc)
│ │ │ │ + -- used 7.57796s (cpu); 5.98248s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S
│ │ │ │  \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /
│ │ │ │  S   \68     /    S   \76
│ │ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |-------
│ │ │ │  -|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |-
│ │ │ │  -------|   <-- |--------|
│ │ │ │ @@ -165,36 +165,36 @@
│ │ │ │  o18 : Matrix R  <-- R
│ │ │ │  i19 : R1 = R/ideal ff
│ │ │ │  
│ │ │ │  o19 = R1
│ │ │ │  
│ │ │ │  o19 : QuotientRing
│ │ │ │  i20 : FF = time Shamash(R1,F,4)
│ │ │ │ - -- used 0.0677985s (cpu); 0.0677563s (thread); 0s (gc)
│ │ │ │ + -- used 0.101019s (cpu); 0.101018s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          1       6       18       38       66
│ │ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │ │  
│ │ │ │        0       1       2        3        4
│ │ │ │  
│ │ │ │  o20 : Complex
│ │ │ │  i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ │ - -- used 1.06441s (cpu); 0.797691s (thread); 0s (gc)
│ │ │ │ + -- used 1.25759s (cpu); 0.940751s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │ │  
│ │ │ │        0         1         2          3          4
│ │ │ │  
│ │ │ │  o21 : Complex
│ │ │ │  The function also deals correctly with complexes F where min F is not 0:
│ │ │ │  i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ │ - -- used 0.915388s (cpu); 0.724566s (thread); 0s (gc)
│ │ │ │ + -- used 1.15963s (cpu); 0.924254s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          1       6       18       38       66
│ │ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │ │  
│ │ │ │        -2      -1      0        1        2
│ │ │ │  
│ │ │ │  o22 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html
│ │ │ @@ -84,23 +84,23 @@
│ │ │  
│ │ │  o1 = 0
│ │ │              
    
│ │ │                
    i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 1.20315s (cpu); 0.562216s (thread); 0s (gc)
│ │ │ + -- used 1.27474s (cpu); 0.559822s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.40989s (cpu); 0.176145s (thread); 0s (gc)
│ │ │ + -- used 0.198487s (cpu); 0.139064s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 4.068e-06s (cpu); 3.406e-06s (thread); 0s (gc)
│ │ │ + -- used 3.834e-06s (cpu); 3.256e-06s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{}
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,23 +18,23 @@
│ │ │ │  appropriate syzygy M of M0 = R/(m1+m2) where m1 and m2 are monomials of the
│ │ │ │  same degree.
│ │ │ │  i1 : setRandomSeed 0
│ │ │ │   -- setting random seed to 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : sumTwoMonomials(2,3)
│ │ │ │ - -- used 1.20315s (cpu); 0.562216s (thread); 0s (gc)
│ │ │ │ + -- used 1.27474s (cpu); 0.559822s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │ │  
│ │ │ │ - -- used 0.40989s (cpu); 0.176145s (thread); 0s (gc)
│ │ │ │ + -- used 0.198487s (cpu); 0.139064s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │ │  
│ │ │ │ - -- used 4.068e-06s (cpu); 3.406e-06s (thread); 0s (gc)
│ │ │ │ + -- used 3.834e-06s (cpu); 3.256e-06s (thread); 0s (gc)
│ │ │ │  4
│ │ │ │  Tally{}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_w_o_M_o_n_o_m_i_a_l_s -- tally the sequences of BRanks for certain examples
│ │ │ │  ********** WWaayyss ttoo uussee ssuummTTwwooMMoonnoommiiaallss:: **********
│ │ │ │      * sumTwoMonomials(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html
│ │ │ @@ -88,25 +88,25 @@
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │                
    i2 : twoMonomials(2,3)
│ │ │ - -- used 0.718914s (cpu); 0.551814s (thread); 0s (gc)
│ │ │ + -- used 1.24606s (cpu); 0.744577s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.50418s (cpu); 0.37207s (thread); 0s (gc)
│ │ │ + -- used 0.731217s (cpu); 0.44469s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.0982566s (cpu); 0.0978823s (thread); 0s (gc)
│ │ │ + -- used 0.132642s (cpu); 0.120555s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
    
│ │ │              
    
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -20,25 +20,25 @@ │ │ │ │ that is, for an appropriate syzygy M of M0 = R/(m1, m2) where m1 and m2 are │ │ │ │ monomials of the same degree. │ │ │ │ i1 : setRandomSeed 0 │ │ │ │ -- setting random seed to 0 │ │ │ │ │ │ │ │ o1 = 0 │ │ │ │ i2 : twoMonomials(2,3) │ │ │ │ - -- used 0.718914s (cpu); 0.551814s (thread); 0s (gc) │ │ │ │ + -- used 1.24606s (cpu); 0.744577s (thread); 0s (gc) │ │ │ │ 2 │ │ │ │ Tally{{{1, 1}} => 2 } │ │ │ │ {{2, 2}, {1, 2}} => 4 │ │ │ │ │ │ │ │ - -- used 0.50418s (cpu); 0.37207s (thread); 0s (gc) │ │ │ │ + -- used 0.731217s (cpu); 0.44469s (thread); 0s (gc) │ │ │ │ 3 │ │ │ │ Tally{{{2, 2}, {1, 2}} => 2} │ │ │ │ {{3, 3}, {2, 3}} => 1 │ │ │ │ │ │ │ │ - -- used 0.0982566s (cpu); 0.0978823s (thread); 0s (gc) │ │ │ │ + -- used 0.132642s (cpu); 0.120555s (thread); 0s (gc) │ │ │ │ 4 │ │ │ │ Tally{{{2, 2}, {1, 2}} => 1} │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_w_o_M_o_n_o_m_i_a_l_s -- tally the sequences of BRanks for certain examples │ │ │ │ ********** WWaayyss ttoo uussee ttwwooMMoonnoommiiaallss:: ********** │ │ │ │ * twoMonomials(ZZ,ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.out │ │ │ @@ -27,18 +27,18 @@ │ │ │ - ϵ*z*dy + 2ϵ - ϵ, x*dx + y*dy + z*dz - 2ϵ) │ │ │ │ │ │ o7 : Ideal of D │ │ │ │ │ │ i8 : assert(holonomicRank I == 4) │ │ │ │ │ │ i9 : elapsedTime A = connectionMatrices I; │ │ │ - -- 3.01135s elapsed │ │ │ + -- 2.51128s elapsed │ │ │ │ │ │ i10 : elapsedTime assert isIntegrable A │ │ │ - -- 5.57359s elapsed │ │ │ + -- 4.10474s elapsed │ │ │ │ │ │ i11 : netList A │ │ │ │ │ │ +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ o11 = || 2ϵ/x -y/x -z/x 0 | | │ │ │ || (4x2y2ϵ^2+4xy2zϵ^2-2x2z2ϵ^2-2y2z2ϵ^2-4xz3ϵ^2+x3zϵ-3xy2zϵ+2xz3ϵ)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3) (2x3y2ϵ-2x2y3ϵ+2x3yzϵ-2xy3zϵ-x3z2ϵ+x2yz2ϵ-xy2z2ϵ+y3z2ϵ-2x3yz+2xy3z)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3) (-2x2y2zϵ-x3z2ϵ-3xy2z2ϵ+x2z3ϵ+y2z3ϵ+4xz4ϵ+2xy2z2-2xz4)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3) (-xyz+xz2+yz2-z3)/(2x2y+2xy2-x2z-2xyz-y2z) | | │ │ │ || (-2xyz2ϵ^2-2y2z2ϵ^2-4yz3ϵ^2+2x2y2ϵ+x2yzϵ+xy2zϵ+2y2z2ϵ+2yz3ϵ)/(2x3y2z+x3yz2+x2y2z2-x3z3-xy2z3-x2z4-xyz4) (x2yz2ϵ+2xy2z2ϵ+y3z2ϵ+2xyz3ϵ+2y2z3ϵ-2x2y3-x2y2z-xy3z-x2yz2-y3z2-xyz3-y2z3)/(2x3y2z+x3yz2+x2y2z2-x3z3-xy2z3-x2z4-xyz4) (2x2y2ϵ+x2yzϵ+xy2zϵ-2x2z2ϵ+xyz2ϵ+y2z2ϵ-2xz3ϵ+2yz3ϵ-2x2y2-x2yz-xy2z+x2z2-y2z2+xz3-yz3)/(2x3y2+x3yz+x2y2z-x3z2-xy2z2-x2z3-xyz3) (-yz+z2)/(2xy-xz-yz) | | │ │ │ @@ -56,24 +56,24 @@ │ │ │ +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ │ │ │ i12 : F = baseFractionField D; │ │ │ │ │ │ i13 : B = {1_D,dx,dy,dx*dy}; │ │ │ │ │ │ i14 : elapsedTime g = gaugeMatrix(I, B); │ │ │ - -- .666811s elapsed │ │ │ + -- .512682s elapsed │ │ │ │ │ │ 4 4 │ │ │ o14 : Matrix F <-- F │ │ │ │ │ │ i15 : elapsedTime A1 = gaugeTransform(g, A); │ │ │ - -- 1.61881s elapsed │ │ │ + -- 1.0998s elapsed │ │ │ │ │ │ i16 : elapsedTime assert isIntegrable A1 │ │ │ - -- 1.04705s elapsed │ │ │ + -- .862916s elapsed │ │ │ │ │ │ i17 : netList A1 │ │ │ │ │ │ +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ o17 = || 0 1 0 0 | | │ │ │ || (-2ϵ^2+ϵ)/(x2-z2) (3xϵ+zϵ-2x)/(x2-z2) (yϵ+zϵ)/(x2-z2) (-y-z)/(x-z) | | │ │ │ || 0 0 0 1 | | │ │ │ @@ -96,18 +96,18 @@ │ │ │ {0, 0, ϵ*(y^2-z^2), ϵ*(x+y)*(y+z)}, │ │ │ {0, 0, 0, -(x+y)*(x+z)*(y+z)}}); │ │ │ │ │ │ 4 4 │ │ │ o18 : Matrix F <-- F │ │ │ │ │ │ i19 : elapsedTime A2 = gaugeTransform(changeEps, A1); │ │ │ - -- .546223s elapsed │ │ │ + -- .416064s elapsed │ │ │ │ │ │ i20 : elapsedTime assert isIntegrable A2 │ │ │ - -- .724237s elapsed │ │ │ + -- .662088s elapsed │ │ │ │ │ │ i21 : netList A2 │ │ │ │ │ │ +-------------------------------------------------------------------------------------------+ │ │ │ o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0 0 | | │ │ │ || 0 ϵ/(x-z) 0 ϵ/(x+y) | | │ │ │ || 0 0 ϵ/(x+z) (-yϵ+zϵ)/(x2+xy+xz+yz) | | │ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.out │ │ │ @@ -16,18 +16,18 @@ │ │ │ │ │ │ 2 │ │ │ o6 = {1, dx, dy, dy } │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : elapsedTime A = connectionMatrices I; │ │ │ - -- .217449s elapsed │ │ │ + -- .192663s elapsed │ │ │ │ │ │ i8 : elapsedTime assert isIntegrable A │ │ │ - -- .281512s elapsed │ │ │ + -- .213428s elapsed │ │ │ │ │ │ i9 : netList A │ │ │ │ │ │ +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ o9 = || 0 1 0 0 || │ │ │ || 0 -1/x 1/x y/x || │ │ │ || -1/2xy -1/y (-x-3y+1)/2xy (-x-y+1)/2x || │ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.html │ │ │ @@ -123,21 +123,21 @@ │ │ │
    │ │ │

    Then, we compute the system in connection form and verify that it meets the integrability conditions.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -82,15 +82,15 @@ │ │ │ │ 2 2 │ │ │ │ - a*c + e - b*c + f │ │ │ │ ----------*v, x + ----------*v) │ │ │ │ d*e - a*f d*e - a*f │ │ │ │ │ │ │ │ o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v] │ │ │ │ i6 : time phi^** q │ │ │ │ - -- used 0.270847s (cpu); 0.194712s (thread); 0s (gc) │ │ │ │ + -- used 0.306466s (cpu); 0.215409s (thread); 0s (gc) │ │ │ │ │ │ │ │ e d c b a │ │ │ │ o6 = ideal (u - -*v, t - -*v, z - -*v, y - -*v, x - -*v) │ │ │ │ f f f f f │ │ │ │ │ │ │ │ o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v] │ │ │ │ i7 : oo == p │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Segre__Class.html │ │ │ @@ -139,59 +139,59 @@ │ │ │ x x - 2x x x x + x x - 2x x x x - 2x x x x + 4x x x x + x x + 4x x x x - 2x x x x - 2x x x x - 2x x x x + x x │ │ │ 3 4 2 3 4 5 2 5 1 3 4 6 1 2 5 6 0 3 5 6 1 6 1 2 4 7 0 3 4 7 0 2 5 7 0 1 6 7 0 7 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i9 : elapsedTime A = connectionMatrices I;
│ │ │ - -- 3.01135s elapsed
    │ │ │ + -- 2.51128s elapsed │ │ │
    │ │ │ 
    i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.57359s elapsed
    │ │ │ + -- 4.10474s elapsed │ │ │
    │ │ │ 
    i11 : netList A
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -172,30 +172,30 @@
│ │ │
    		│ │ │ 
    i13 : B = {1_D,dx,dy,dx*dy};
    │ │ │
    │ │ │ 
    i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ - -- .666811s elapsed
│ │ │ + -- .512682s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
    │ │ │
    │ │ │ 
    i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.61881s elapsed
    │ │ │ + -- 1.0998s elapsed │ │ │
    │ │ │ 
    i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- 1.04705s elapsed
    │ │ │ + -- .862916s elapsed │ │ │
    │ │ │ 
    i17 : netList A1
│ │ │  
│ │ │        +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -232,21 +232,21 @@
│ │ │                4      4
│ │ │  o18 : Matrix F  <-- F
    │ │ │
    │ │ │ 
    i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ - -- .546223s elapsed
    │ │ │ + -- .416064s elapsed │ │ │
    │ │ │ 
    i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- .724237s elapsed
    │ │ │ + -- .662088s elapsed │ │ │
    │ │ │ 
    i21 : netList A2
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,17 +41,17 @@
│ │ │ │  
│ │ │ │  o7 : Ideal of D
│ │ │ │  First, we check that the system has finite holonomic rank using _h_o_l_o_n_o_m_i_c_R_a_n_k.
│ │ │ │  i8 : assert(holonomicRank I == 4)
│ │ │ │  Then, we compute the system in connection form and verify that it meets the
│ │ │ │  integrability conditions.
│ │ │ │  i9 : elapsedTime A = connectionMatrices I;
│ │ │ │ - -- 3.01135s elapsed
│ │ │ │ + -- 2.51128s elapsed
│ │ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ │ - -- 5.57359s elapsed
│ │ │ │ + -- 4.10474s elapsed
│ │ │ │  i11 : netList A
│ │ │ │  
│ │ │ │        +----------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │ @@ -227,22 +227,22 @@
│ │ │ │  -----------------------------------------------------------------------------------+
│ │ │ │  Next, we use _g_a_u_g_e_ _m_a_t_r_i_x for changing base to a base given by suitable set of
│ │ │ │  standard monomials, and compute the _g_a_u_g_e_ _t_r_a_n_s_f_o_r_m with respect to this gauge
│ │ │ │  matrix.
│ │ │ │  i12 : F = baseFractionField D;
│ │ │ │  i13 : B = {1_D,dx,dy,dx*dy};
│ │ │ │  i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ │ - -- .666811s elapsed
│ │ │ │ + -- .512682s elapsed
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o14 : Matrix F  <-- F
│ │ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ │ - -- 1.61881s elapsed
│ │ │ │ + -- 1.0998s elapsed
│ │ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ │ - -- 1.04705s elapsed
│ │ │ │ + -- .862916s elapsed
│ │ │ │  i17 : netList A1
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  --------------------------------------------------------------------------+
│ │ │ │  o17 = || 0                            1                      0
│ │ │ │  0                                                      |
│ │ │ │ @@ -300,17 +300,17 @@
│ │ │ │                {0, ϵ*(x^2-z^2), 0, ϵ*(x+y)*(x+z)},
│ │ │ │                {0, 0, ϵ*(y^2-z^2), ϵ*(x+y)*(y+z)},
│ │ │ │                {0, 0, 0, -(x+y)*(x+z)*(y+z)}});
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o18 : Matrix F  <-- F
│ │ │ │  i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ │ - -- .546223s elapsed
│ │ │ │ + -- .416064s elapsed
│ │ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ │ - -- .724237s elapsed
│ │ │ │ + -- .662088s elapsed
│ │ │ │  i21 : netList A2
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------+
│ │ │ │  o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0       0                      |
│ │ │ │  |
│ │ │ │        || 0       ϵ/(x-z)     0       ϵ/(x+y)                |
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html
│ │ │ @@ -105,21 +105,21 @@
│ │ │          
    
│ │ │            
    Finally, we can compute the connection matrices.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : elapsedTime A = connectionMatrices I;
│ │ │ - -- .217449s elapsed
    
│ │ │ + -- .192663s elapsed
│ │ │              
    
│ │ │                
    i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .281512s elapsed
    
│ │ │ + -- .213428s elapsed
│ │ │              
    
│ │ │                
    i9 : netList A
│ │ │  
│ │ │       +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,17 +20,17 @@
│ │ │ │  
│ │ │ │                     2
│ │ │ │  o6 = {1, dx, dy, dy }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  Finally, we can compute the connection matrices.
│ │ │ │  i7 : elapsedTime A = connectionMatrices I;
│ │ │ │ - -- .217449s elapsed
│ │ │ │ + -- .192663s elapsed
│ │ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ │ - -- .281512s elapsed
│ │ │ │ + -- .213428s elapsed
│ │ │ │  i9 : netList A
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------+
│ │ │ │  o9 = || 0                                                       1
│ │ ├── ./usr/share/doc/Macaulay2/CorrespondenceScrolls/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon May 18 11:29:46 2026
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon May 18 11:29:47 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │  #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=52
│ │ │  cHJvZHVjdE9mUHJvamVjdGl2ZVNwYWNlcyguLi4sQ29lZmZpY2llbnRGaWVsZD0+Li4uKQ==
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out
│ │ │ @@ -13,27 +13,27 @@
│ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │  
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
│ │ │  
│ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 1.23521s (cpu); 0.911576s (thread); 0s (gc)
│ │ │ + -- used 1.30323s (cpu); 1.04349s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o3 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o3 : -----
│ │ │          5
│ │ │         H
│ │ │  
│ │ │  i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 1.21347s (cpu); 0.831445s (thread); 0s (gc)
│ │ │ + -- used 1.52607s (cpu); 1.00202s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          5
│ │ │ @@ -62,27 +62,27 @@
│ │ │          0,2 1,3    0,1 2,3
│ │ │  
│ │ │                  ZZ
│ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │  
│ │ │  i9 : time ChernClass G
│ │ │ - -- used 0.239068s (cpu); 0.15048s (thread); 0s (gc)
│ │ │ + -- used 0.142567s (cpu); 0.142567s (thread); 0s (gc)
│ │ │  
│ │ │          9      8      7      6      5      4     3
│ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o9 : -----
│ │ │         10
│ │ │        H
│ │ │  
│ │ │  i10 : time ChernClass(G,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0108133s (cpu); 0.0103892s (thread); 0s (gc)
│ │ │ + -- used 0.0263878s (cpu); 0.0141289s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out
│ │ │ @@ -1,56 +1,56 @@
│ │ │  -- -*- M2-comint -*- hash: 10433409267944421825
│ │ │  
│ │ │  i1 : ZZ/300007[t_0..t_6];
│ │ │  
│ │ │  i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00378639s (cpu); 0.00378433s (thread); 0s (gc)
│ │ │ + -- used 0.00514285s (cpu); 0.00514309s (thread); 0s (gc)
│ │ │  
│ │ │              ZZ              ZZ                3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o2 = map (------[t ..t ], ------[x ..x ], {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   300007  0   9       2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   9
│ │ │  
│ │ │  i3 : time J = kernel(phi,2)
│ │ │ - -- used 0.104398s (cpu); 0.0576775s (thread); 0s (gc)
│ │ │ + -- used 0.201927s (cpu); 0.10589s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x 
│ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │                  ZZ
│ │ │  o3 : Ideal of ------[x ..x ]
│ │ │                300007  0   9
│ │ │  
│ │ │  i4 : time degreeMap phi
│ │ │ - -- used 0.0284498s (cpu); 0.0284556s (thread); 0s (gc)
│ │ │ + -- used 0.044637s (cpu); 0.0446386s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projectiveDegrees phi
│ │ │ - -- used 0.819548s (cpu); 0.498944s (thread); 0s (gc)
│ │ │ + -- used 0.85738s (cpu); 0.582407s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.181184s (cpu); 0.104837s (thread); 0s (gc)
│ │ │ + -- used 0.178987s (cpu); 0.0962546s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.00225414s (cpu); 0.00225832s (thread); 0s (gc)
│ │ │ + -- used 0.00262157s (cpu); 0.00262558s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         ZZ
│ │ │                                                                       ------[x ..x ]
│ │ │              ZZ                                                       300007  0   9                                                  3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o7 = map (------[t ..t ], ----------------------------------------------------------------------------------------------------, {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )      2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │                              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -59,15 +59,15 @@
│ │ │                                                                             ------[x ..x ]
│ │ │                 ZZ                                                          300007  0   9
│ │ │  o7 : RingMap ------[t ..t ] <-- ----------------------------------------------------------------------------------------------------
│ │ │               300007  0   6      (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )
│ │ │                                    6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │  i8 : time psi = inverseMap phi
│ │ │ - -- used 0.473279s (cpu); 0.380675s (thread); 0s (gc)
│ │ │ + -- used 0.451317s (cpu); 0.451319s (thread); 0s (gc)
│ │ │  
│ │ │                                                         ZZ
│ │ │                                                       ------[x ..x ]
│ │ │                                                       300007  0   9                                                ZZ              3                2               2    2                        2                          2     2        2                               2                                   2               2             2                       3                                                 2                 2    2                                  2    2                 2                                                 3                         2      2    2      2                                              2
│ │ │  o8 = map (----------------------------------------------------------------------------------------------------, ------[t ..t ], {x  - 2x x x  + x x  - x x x  + x x  + x x  + x x x  - x x x  + x x  - 2x x x  - x x x  - 2x x , x x  - x x  - x x x  + x x x  + x x x  + x x  - 2x x x  - x x x  + x x x , x x  - x x x  + x x  - x x x  + x x  - x x x  - x x x , x  - x x x  + x x x  + x x x  - 2x x x  - x x x , x x  - x x x  + x x  + x x  - x x x  - x x x  - x x x , x x  - x x  - x x x  + x x  + x x x  + x x x  - 2x x x  - x x x  + x x x , x  - 2x x x  - x x x  + x x  + x x  + x x  + x x  + x x x  - 2x x x  - x x x  - x x x  - 2x x })
│ │ │            (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )  300007  0   6     2     1 2 3    0 3    1 2 5    0 5    1 6    0 2 6    0 4 6    1 7     0 2 7    0 4 7     0 9   2 3    1 3    1 2 6    0 3 6    0 5 6    1 8     0 2 8    0 4 8    0 1 9   2 3    1 3 6    0 6    0 3 8    1 9    0 2 9    0 4 9   3    1 3 8    0 6 8    1 2 9     0 3 9    0 5 9   3 6    2 3 8    0 8    2 9    1 3 9    0 6 9    0 7 9   3 6    3 8    2 6 8    1 8    2 3 9    2 5 9     1 6 9    1 7 9    0 8 9   6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
│ │ │              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -76,32 +76,32 @@
│ │ │                                                          ------[x ..x ]
│ │ │                                                          300007  0   9                                                   ZZ
│ │ │  o8 : RingMap ---------------------------------------------------------------------------------------------------- <-- ------[t ..t ]
│ │ │               (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )     300007  0   6
│ │ │                 6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │  i9 : time isInverseMap(phi,psi)
│ │ │ - -- used 0.00964078s (cpu); 0.00964299s (thread); 0s (gc)
│ │ │ + -- used 0.0173789s (cpu); 0.0173802s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time degreeMap psi
│ │ │ - -- used 0.345797s (cpu); 0.198392s (thread); 0s (gc)
│ │ │ + -- used 0.552568s (cpu); 0.2807s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
│ │ │  
│ │ │  i11 : time projectiveDegrees psi
│ │ │ - -- used 5.38156s (cpu); 4.44342s (thread); 0s (gc)
│ │ │ + -- used 6.7628s (cpu); 6.33728s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00210419s (cpu); 0.00210521s (thread); 0s (gc)
│ │ │ + -- used 0.00260705s (cpu); 0.00261086s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -147,15 +147,15 @@
│ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │                        }
│ │ │  
│ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
│ │ │  
│ │ │  i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ - -- used 0.0521222s (cpu); 0.0521315s (thread); 0s (gc)
│ │ │ + -- used 0.0651734s (cpu); 0.0651766s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                                     ZZ
│ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -217,15 +217,15 @@
│ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │                        }
│ │ │  
│ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i14 : time phi^(-1)
│ │ │ - -- used 0.495675s (cpu); 0.427528s (thread); 0s (gc)
│ │ │ + -- used 0.697532s (cpu); 0.599202s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                                     ZZ
│ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │                {
│ │ │                 x x  - x x  + x x ,
│ │ │ @@ -275,71 +275,71 @@
│ │ │                         x  - 2x x x  - x x x  + x x  + x x  + x x  + x x  + x x x  - 2x x x  - x x x  - x x x  - 2x x
│ │ │                          6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
│ │ │                        }
│ │ │  
│ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
│ │ │  
│ │ │  i15 : time degrees phi^(-1)
│ │ │ - -- used 0.470017s (cpu); 0.310654s (thread); 0s (gc)
│ │ │ + -- used 0.327871s (cpu); 0.315312s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : time degrees phi
│ │ │ - -- used 0.0179277s (cpu); 0.0174843s (thread); 0s (gc)
│ │ │ + -- used 0.0849139s (cpu); 0.0281904s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
│ │ │  
│ │ │  i17 : time describe phi
│ │ │ - -- used 0.00296318s (cpu); 0.00296427s (thread); 0s (gc)
│ │ │ + -- used 0.0041231s (cpu); 0.00412799s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = rational map defined by forms of degree 3
│ │ │        source variety: PP^6
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │        coefficient ring: ZZ/300007
│ │ │  
│ │ │  i18 : time describe phi^(-1)
│ │ │ - -- used 0.009622s (cpu); 0.00962345s (thread); 0s (gc)
│ │ │ + -- used 0.0117614s (cpu); 0.0117673s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = rational map defined by forms of degree 3
│ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        target variety: PP^6
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │        number of minimal representatives: 1
│ │ │        dimension base locus: 4
│ │ │        degree base locus: 24
│ │ │        coefficient ring: ZZ/300007
│ │ │  
│ │ │  i19 : time (f,g) = graph phi^-1; f;
│ │ │ - -- used 0.0090019s (cpu); 0.00900307s (thread); 0s (gc)
│ │ │ + -- used 0.0112566s (cpu); 0.0112618s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i21 : time degrees f
│ │ │ - -- used 1.35287s (cpu); 0.917392s (thread); 0s (gc)
│ │ │ + -- used 1.37914s (cpu); 1.03853s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
│ │ │  
│ │ │  i22 : time degree f
│ │ │ - -- used 1.6852e-05s (cpu); 1.6531e-05s (thread); 0s (gc)
│ │ │ + -- used 2.1838e-05s (cpu); 2.0995e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
│ │ │  
│ │ │  i23 : time describe f
│ │ │ - -- used 0.001505s (cpu); 0.0015062s (thread); 0s (gc)
│ │ │ + -- used 0.00196678s (cpu); 0.00197264s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20 hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2, 0},{2, 0})
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true
│ │ │        projective degrees: {904, 508, 268, 130, 56, 20, 5}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Euler__Characteristic.out
│ │ │ @@ -3,18 +3,18 @@
│ │ │  i1 : I = Grassmannian(1,4,CoefficientRing=>ZZ/190181);
│ │ │  
│ │ │                  ZZ
│ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │  
│ │ │  i2 : time EulerCharacteristic I
│ │ │ - -- used 0.427945s (cpu); 0.193961s (thread); 0s (gc)
│ │ │ + -- used 0.541737s (cpu); 0.268955s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0129232s (cpu); 0.0122924s (thread); 0s (gc)
│ │ │ + -- used 0.0567221s (cpu); 0.0191547s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 10
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp!.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │  
│ │ │  o3 = rational map defined by forms of degree 2
│ │ │       source variety: PP^5
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
│ │ │  
│ │ │  i4 : time phi! ;
│ │ │ - -- used 0.0537946s (cpu); 0.0533655s (thread); 0s (gc)
│ │ │ + -- used 0.0829391s (cpu); 0.0706775s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
│ │ │  
│ │ │  i5 : describe phi
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 2
│ │ │       source variety: PP^5
│ │ │ @@ -37,15 +37,15 @@
│ │ │  
│ │ │  o8 = rational map defined by forms of degree 2
│ │ │       source variety: PP^4
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
│ │ │  
│ │ │  i9 : time phi! ;
│ │ │ - -- used 0.202479s (cpu); 0.118999s (thread); 0s (gc)
│ │ │ + -- used 0.305442s (cpu); 0.134329s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
│ │ │  
│ │ │  i10 : describe phi
│ │ │  
│ │ │  o10 = rational map defined by forms of degree 2
│ │ │        source variety: PP^4
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp^_st_st_sp__Ideal.out
│ │ │ @@ -67,15 +67,15 @@
│ │ │       - a*c + e         - b*c + f
│ │ │       ----------*v, x + ----------*v)
│ │ │        d*e - a*f         d*e - a*f
│ │ │  
│ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │  
│ │ │  i6 : time phi^** q
│ │ │ - -- used 0.270847s (cpu); 0.194712s (thread); 0s (gc)
│ │ │ + -- used 0.306466s (cpu); 0.215409s (thread); 0s (gc)
│ │ │  
│ │ │                  e        d        c        b        a
│ │ │  o6 = ideal (u - -*v, t - -*v, z - -*v, y - -*v, x - -*v)
│ │ │                  f        f        f        f        f
│ │ │  
│ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Segre__Class.out
│ │ │ @@ -47,50 +47,50 @@
│ │ │                                                                            P7
│ │ │  o3 : Ideal of -------------------------------------------------------------------------------------------------------------------------
│ │ │                 2 2                2 2                                        2 2                                                    2 2
│ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1 6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │  
│ │ │  i4 : time SegreClass X
│ │ │ - -- used 0.997559s (cpu); 0.636987s (thread); 0s (gc)
│ │ │ + -- used 0.836973s (cpu); 0.566927s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i5 : time SegreClass lift(X,P7)
│ │ │ - -- used 0.592502s (cpu); 0.324675s (thread); 0s (gc)
│ │ │ + -- used 0.695602s (cpu); 0.401301s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o5 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i6 : time SegreClass(X,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0217502s (cpu); 0.0211891s (thread); 0s (gc)
│ │ │ + -- used 0.0369694s (cpu); 0.0251108s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0959594s (cpu); 0.0955959s (thread); 0s (gc)
│ │ │ + -- used 0.338649s (cpu); 0.170653s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -98,22 +98,22 @@
│ │ │  
│ │ │  i8 : o4 == o6 and o5 == o7
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : use ZZ/100003[x_0..x_6]
│ │ │  
│ │ │ -o9     ZZ
│ │ │ - = ------[x ..x ]
│ │ │ -   100003  0   6
│ │ │ +o9 =   ZZ
│ │ │ + ------[x ..x ]
│ │ │ + 100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
│ │ │  
│ │ │  i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ - -- used 0.226996s (cpu); 0.102455s (thread); 0s (gc)
│ │ │ + -- used 0.0700194s (cpu); 0.0700204s (thread); 0s (gc)
│ │ │  
│ │ │                                                          ZZ
│ │ │                                                        ------[y ..y ]
│ │ │                                                        100003  0   9                                                ZZ              2                              2
│ │ │  o10 = map (----------------------------------------------------------------------------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y , y  - y y  - y y , y y  + y y  - y y , y y  - y y , y y  - y y  - y y , y y  - y y  - y y })
│ │ │             (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )  100003  0   6     3    0 5    1 6   3 4    1 7   4    2 7    0 9   2 5    3 5    1 8   4 5    1 9   4 8    2 9    3 9   7 8    4 9    6 9
│ │ │               5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │ @@ -122,15 +122,15 @@
│ │ │                                                           ------[y ..y ]
│ │ │                                                           100003  0   9                                                   ZZ
│ │ │  o10 : RingMap ---------------------------------------------------------------------------------------------------- <-- ------[x ..x ]
│ │ │                (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )     100003  0   6
│ │ │                  5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │  
│ │ │  i11 : time SegreClass phi
│ │ │ - -- used 0.174105s (cpu); 0.174116s (thread); 0s (gc)
│ │ │ + -- used 0.444354s (cpu); 0.300237s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │          10
│ │ │ @@ -150,27 +150,27 @@
│ │ │                                                            100003  0   9
│ │ │  o12 : Ideal of ----------------------------------------------------------------------------------------------------
│ │ │                 (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )
│ │ │                   5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │  
│ │ │  i13 : -- Segre class of B in G(1,4)
│ │ │        time SegreClass B
│ │ │ - -- used 0.514373s (cpu); 0.297297s (thread); 0s (gc)
│ │ │ + -- used 0.508673s (cpu); 0.340466s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o13 : -----
│ │ │          10
│ │ │         H
│ │ │  
│ │ │  i14 : -- Segre class of B in P^9
│ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ - -- used 1.55464s (cpu); 0.93541s (thread); 0s (gc)
│ │ │ + -- used 1.77895s (cpu); 1.10024s (thread); 0s (gc)
│ │ │  
│ │ │             9       8       7      6     5
│ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o14 : -----
│ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_abstract__Rational__Map.out
│ │ │ @@ -17,32 +17,32 @@
│ │ │  
│ │ │  o3 = QQ[u ..u ]
│ │ │           0   5
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ - -- used 0.000440666s (cpu); 0.000435206s (thread); 0s (gc)
│ │ │ + -- used 0.000564505s (cpu); 0.000561703s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: given by a function
│ │ │  
│ │ │  o4 : AbstractRationalMap (rational map from PP^4 to PP^5)
│ │ │  
│ │ │  i5 : time projectiveDegrees(psi,3)
│ │ │ - -- used 0.114539s (cpu); 0.1145s (thread); 0s (gc)
│ │ │ + -- used 0.148982s (cpu); 0.14899s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time rationalMap psi
│ │ │ - -- used 0.563962s (cpu); 0.397087s (thread); 0s (gc)
│ │ │ + -- used 0.51318s (cpu); 0.409301s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: {
│ │ │ @@ -113,48 +113,48 @@
│ │ │                  1    0 2     1 2    0 3     2    1 3
│ │ │  
│ │ │                   ZZ
│ │ │  o13 : Ideal of -----[x ..x ]
│ │ │                 65521  0   3
│ │ │  
│ │ │  i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ - -- used 0.0371843s (cpu); 0.0371837s (thread); 0s (gc)
│ │ │ + -- used 0.197594s (cpu); 0.117672s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │        defining forms: given by a function
│ │ │  
│ │ │  o14 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │  
│ │ │  i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 2.08417s (cpu); 1.43192s (thread); 0s (gc)
│ │ │ + -- used 2.53895s (cpu); 1.92969s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
│ │ │  
│ │ │  i16 : time T2 = T * T
│ │ │ - -- used 4.777e-05s (cpu); 4.7589e-05s (thread); 0s (gc)
│ │ │ + -- used 4.7821e-05s (cpu); 4.6393e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │        defining forms: given by a function
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │  
│ │ │  i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 3.36436s (cpu); 2.35293s (thread); 0s (gc)
│ │ │ + -- used 4.09253s (cpu); 3.06776s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
│ │ │  
│ │ │  i18 : p = apply(3,i->random(ZZ/65521))|{1}
│ │ │  
│ │ │  o18 = {-6648, -23396, -12311, 1}
│ │ │  
│ │ │ @@ -169,15 +169,15 @@
│ │ │  i20 : T q
│ │ │  
│ │ │  o20 = {-6648, -23396, -12311, 1}
│ │ │  
│ │ │  o20 : List
│ │ │  
│ │ │  i21 : time f = rationalMap T
│ │ │ - -- used 3.00105s (cpu); 2.14734s (thread); 0s (gc)
│ │ │ + -- used 3.39967s (cpu); 2.60497s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_approximate__Inverse__Map.out
│ │ │ @@ -54,15 +54,15 @@
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │  -- approximateInverseMap: step 8 of 10
│ │ │  -- approximateInverseMap: step 9 of 10
│ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ - -- used 0.352424s (cpu); 0.232237s (thread); 0s (gc)
│ │ │ + -- used 0.444317s (cpu); 0.28658s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                    ZZ
│ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │                                  ZZ
│ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  i4 : assert(phi * psi == 1 and psi * phi == 1)
│ │ │  
│ │ │  i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 0.324659s (cpu); 0.181955s (thread); 0s (gc)
│ │ │ + -- used 0.419739s (cpu); 0.261883s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │  
│ │ │  i6 : assert(psi == psi')
│ │ │  
│ │ │  i7 : phi = rationalMap map(P8,ZZ/97[x_0..x_11]/ideal(x_1*x_3-8*x_2*x_3+25*x_3^2-25*x_2*x_4-22*x_3*x_4+x_0*x_5+13*x_2*x_5+41*x_3*x_5-x_0*x_6+12*x_2*x_6+25*x_1*x_7+25*x_3*x_7+23*x_5*x_7-3*x_6*x_7+2*x_0*x_8+11*x_1*x_8-37*x_3*x_8-23*x_4*x_8-33*x_6*x_8+8*x_0*x_9+10*x_1*x_9-25*x_2*x_9-9*x_3*x_9+3*x_4*x_9+24*x_5*x_9-27*x_6*x_9-5*x_0*x_10+28*x_1*x_10+37*x_2*x_10+9*x_4*x_10+27*x_6*x_10-25*x_0*x_11+9*x_2*x_11+27*x_4*x_11-27*x_5*x_11,x_2^2+17*x_2*x_3-14*x_3^2-13*x_2*x_4+34*x_3*x_4+44*x_0*x_5-30*x_2*x_5+27*x_3*x_5+31*x_2*x_6-36*x_3*x_6-x_0*x_7+13*x_1*x_7+8*x_3*x_7+9*x_5*x_7+46*x_6*x_7+41*x_0*x_8-7*x_1*x_8-34*x_3*x_8-9*x_4*x_8-46*x_6*x_8-17*x_0*x_9+32*x_1*x_9-8*x_2*x_9-35*x_3*x_9-46*x_4*x_9+26*x_5*x_9+17*x_6*x_9+15*x_0*x_10+35*x_1*x_10+34*x_2*x_10+20*x_4*x_10+14*x_0*x_11+36*x_1*x_11+35*x_2*x_11-17*x_4*x_11,x_1*x_2-40*x_2*x_3+28*x_3^2-x_0*x_4+5*x_2*x_4-16*x_3*x_4+5*x_0*x_5-36*x_2*x_5+37*x_3*x_5+48*x_2*x_6-5*x_1*x_7-5*x_3*x_7+x_5*x_7+20*x_6*x_7+10*x_0*x_8+34*x_1*x_8+41*x_3*x_8-x_4*x_8+x_6*x_8+40*x_0*x_9-32*x_1*x_9+5*x_2*x_9-11*x_3*x_9-20*x_4*x_9+45*x_5*x_9-14*x_6*x_9-25*x_0*x_10+45*x_1*x_10-41*x_2*x_10-46*x_4*x_10+8*x_6*x_10-28*x_0*x_11+11*x_2*x_11+14*x_4*x_11-8*x_5*x_11),{t_4^2+t_0*t_5+t_1*t_5+35*t_2*t_5+10*t_3*t_5+25*t_4*t_5-5*t_5^2-14*t_0*t_6-14*t_1*t_6-5*t_2*t_6-13*t_4*t_6+37*t_5*t_6+22*t_6^2-31*t_3*t_7+26*t_4*t_7+12*t_5*t_7-45*t_6*t_7-46*t_3*t_8+37*t_4*t_8+28*t_5*t_8+33*t_6*t_8,t_3*t_4+4*t_0*t_5+39*t_1*t_5-40*t_2*t_5+40*t_3*t_5+26*t_4*t_5-20*t_5^2+41*t_0*t_6+36*t_1*t_6-22*t_2*t_6+36*t_4*t_6-30*t_5*t_6-13*t_6^2-25*t_3*t_7+5*t_4*t_7-35*t_5*t_7+10*t_6*t_7+11*t_3*t_8+46*t_4*t_8+29*t_5*t_8+28*t_6*t_8,t_2*t_4-5*t_0*t_5-40*t_1*t_5+12*t_2*t_5+47*t_3*t_5+37*t_4*t_5+25*t_5^2-27*t_0*t_6-22*t_1*t_6+27*t_2*t_6-23*t_4*t_6+5*t_5*t_6-13*t_6^2-39*t_3*t_7-29*t_4*t_7+9*t_5*t_7+39*t_6*t_7+36*t_3*t_8+13*t_4*t_8+26*t_5*t_8+37*t_6*t_8,t_0*t_4-t_0*t_5-8*t_1*t_5-35*t_2*t_5-10*t_3*t_5-33*t_4*t_5+5*t_5^2+15*t_0*t_6+15*t_1*t_6+5*t_2*t_6+15*t_4*t_6-38*t_5*t_6-22*t_6^2+31*t_3*t_7-25*t_4*t_7-19*t_5*t_7+47*t_6*t_7+46*t_3*t_8-36*t_4*t_8-35*t_5*t_8-31*t_6*t_8,t_2*t_3-t_0*t_5-t_1*t_5-35*t_2*t_5-10*t_3*t_5-33*t_4*t_5+5*t_5^2+14*t_0*t_6+14*t_1*t_6+5*t_2*t_6+14*t_4*t_6-31*t_5*t_6-24*t_6^2+32*t_3*t_7-25*t_4*t_7-19*t_5*t_7+47*t_6*t_7+46*t_3*t_8-36*t_4*t_8-35*t_5*t_8-31*t_6*t_8,t_1*t_3-7*t_1*t_5+t_1*t_6+t_4*t_6-7*t_5*t_6+2*t_6^2-t_3*t_7,t_0*t_3-46*t_0*t_5-39*t_1*t_5-43*t_2*t_5-41*t_3*t_5-26*t_4*t_5-28*t_5^2-35*t_0*t_6-36*t_1*t_6+20*t_2*t_6-36*t_4*t_6+9*t_5*t_6+15*t_6^2+26*t_3*t_7-5*t_4*t_7+35*t_5*t_7-10*t_6*t_7-10*t_3*t_8-46*t_4*t_8+47*t_5*t_8-25*t_6*t_8,t_2^2-46*t_1*t_4-33*t_0*t_5-45*t_1*t_5-39*t_2*t_5-39*t_3*t_5-46*t_4*t_5-29*t_5^2-48*t_0*t_6-38*t_1*t_6-30*t_2*t_6+19*t_4*t_6-44*t_5*t_6-47*t_6^2-36*t_0*t_7-46*t_1*t_7+t_2*t_7-44*t_3*t_7+48*t_4*t_7-14*t_5*t_7+4*t_6*t_7-36*t_0*t_8-46*t_1*t_8+47*t_2*t_8-34*t_3*t_8-24*t_4*t_8-12*t_5*t_8-47*t_6*t_8+47*t_7*t_8,t_1*t_2+6*t_1*t_5+5*t_0*t_6-2*t_1*t_6-t_4*t_6-t_5*t_6+5*t_0*t_7+t_1*t_7-2*t_2*t_7-7*t_5*t_7+2*t_6*t_7-2*t_1*t_8+3*t_7*t_8,t_0*t_2+t_1*t_4+5*t_0*t_5+32*t_1*t_5-20*t_2*t_5-47*t_3*t_5-37*t_4*t_5-25*t_5^2+19*t_0*t_6+22*t_1*t_6-25*t_2*t_6+25*t_4*t_6-5*t_5*t_6+13*t_6^2+5*t_0*t_7+t_1*t_7+39*t_3*t_7+28*t_4*t_7-9*t_5*t_7-39*t_6*t_7+4*t_0*t_8+t_1*t_8-36*t_3*t_8-14*t_4*t_8-26*t_5*t_8-37*t_6*t_8,t_0*t_1-39*t_1*t_4+40*t_1*t_5-37*t_0*t_6-39*t_1*t_6+19*t_4*t_6-39*t_5*t_6-38*t_0*t_7+39*t_1*t_7+19*t_2*t_7+18*t_5*t_7-19*t_6*t_7+19*t_1*t_8+20*t_7*t_8,t_0^2+12*t_1*t_4+20*t_0*t_5+27*t_1*t_5-8*t_2*t_5+37*t_3*t_5+28*t_4*t_5+30*t_5^2-46*t_0*t_6+24*t_1*t_6-40*t_2*t_6+25*t_4*t_6+16*t_5*t_6-35*t_6^2+29*t_0*t_7+12*t_1*t_7-35*t_2*t_7-8*t_3*t_7-18*t_4*t_7+42*t_5*t_7-12*t_6*t_7-6*t_0*t_8+12*t_1*t_8-15*t_3*t_8+9*t_4*t_8+20*t_5*t_8-30*t_6*t_8+4*t_7*t_8})
│ │ │  
│ │ │ @@ -192,15 +192,15 @@
│ │ │  o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11)
│ │ │  
│ │ │  i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 2.52647s (cpu); 1.81675s (thread); 0s (gc)
│ │ │ + -- used 2.37135s (cpu); 2.03451s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │               {
│ │ │                                  2
│ │ │ @@ -258,15 +258,15 @@
│ │ │  
│ │ │  i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │ - -- used 3.65149s (cpu); 2.6771s (thread); 0s (gc)
│ │ │ + -- used 3.45379s (cpu); 2.97496s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = -- rational map --
│ │ │                                   ZZ
│ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │                {
│ │ │                                   2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_degree__Map.out
│ │ │ @@ -9,27 +9,27 @@
│ │ │                                   2                  2                             2                                       2                                                2                                                           2                                                                       2                                                                              2                                                                                            2         2                 2                             2                                       2                                              2                                                           2                                                                   2                                                                               2                                                                                          2        2                   2                            2                                      2                                                  2                                                          2                                                                      2                                                                               2                                                                                            2        2                   2                             2                                      2                                                 2                                                          2                                                                    2                                                                               2                                                                                          2         2                2                          2                                      2                                                 2                                                           2                                                                       2                                                                            2                                                                                          2        2                  2                         2                                       2                                                2                                                           2                                                                      2                                                                               2                                                                                        2       2                  2                             2                                    2                                                2                                                          2                                                                    2                                                                               2                                                                                       2       2                 2                           2                                      2                                                 2                                                            2                                                                       2                                                                                2                                                                                           2        2                   2                             2                                       2                                                  2                                                            2                                                                   2                                                                            2                                                                                          2      2                 2                            2                                       2                                                2                                                         2                                                                        2                                                                               2                                                                                          2     2                   2                             2                                      2                                                   2                                                          2                                                                     2                                                                               2                                                                                          2         2                2                            2                                       2                                                 2                                                           2                                                                      2                                                                                  2                                                                                              2      2                  2                            2                                    2                                                2                                                            2                                                                    2                                                                                2                                                                                          2       2                  2                            2                                    2                                                   2                                                        2                                                                         2                                                                               2                                                                                           2       2                  2                             2                                       2                                                 2                                                          2                                                                       2                                                                               2                                                                                       2
│ │ │  o4 = map (ringP8, ringP14, {- 95x  + 181x x  + 1028x  - 1384x x  - 1455x x  + 559x  - 502x x  + 1264x x  - 162x x  + 1209x  - 180x x  - 504x x  - 1168x x  - 676x x  + 501x  + 73x x  + 1263x x  + 1035x x  + 844x x  + 1593x x  + 785x  + 982x x  - 412x x  + 1335x x  + 1136x x  + 826x x  + 1078x x  + 1158x  + 335x x  - 982x x  - 1479x x  - 15x x  + 1363x x  + 1397x x  - 575x x  - 71x  + 1255x x  - 1138x x  - 1590x x  + 604x x  + 1182x x  - 63x x  - 1382x x  - 1255x x  - 613x , - 1444x  + 575x x  + 767x  - 1495x x  + 1631x x  - 217x  - 294x x  - 1511x x  - 504x x  - 1284x  - 1459x x  + 152x x  + 141x x  - 10x x  - 95x  + 1056x x  + 654x x  + 1397x x  - 930x x  + 578x x  - 696x  + 759x x  + 733x x  + 505x x  - 609x x  + 526x x  - 659x x  + 846x  + 1253x x  - 1519x x  + 635x x  + 576x x  + 54x x  - 1261x x  - 822x x  - 257x  - 986x x  + 356x x  - 1488x x  - 1561x x  - 850x x  - 85x x  - 1350x x  - 783x x  - 1335x , - 871x  + 1006x x  - 1399x  - 1636x x  - 699x x  - 769x  - 307x x  - 1645x x  - 502x x  - 719x  + 1405x x  + 870x x  - 1133x x  + 425x x  - 1203x  - 1601x x  + 117x x  - 382x x  + 318x x  - 117x x  - 560x  + 1135x x  + 1468x x  + 869x x  - 943x x  - 335x x  - 1218x x  + 201x  - 11x x  + 540x x  - 710x x  - 489x x  + 1605x x  + 1663x x  - 423x x  + 1246x  + 97x x  - 644x x  + 1655x x  + 1219x x  + 1476x x  + 1355x x  + 1594x x  + 893x x  + 1150x , - 143x  + 1240x x  - 1042x  + 1649x x  + 1024x x  + 794x  + 1442x x  - 1263x x  + 537x x  - 82x  - 734x x  - 1569x x  - 798x x  - 366x x  + 1289x  - 569x x  - 254x x  + 237x x  - 1234x x  - 807x x  + 264x  - 202x x  - 616x x  + 44x x  + 1465x x  + 685x x  + 1630x x  - 406x  - 123x x  - 4x x  + 1583x x  + 1235x x  + 162x x  + 1034x x  - 1035x x  + 737x  + 660x x  + 1459x x  - 359x x  - 1291x x  + 1638x x  - 325x x  - 631x x  + 73x x  - 1471x , - 1340x  + 31x x  - 994x  - 880x x  - 89x x  + 574x  + 760x x  - 1054x x  + 772x x  - 239x  - 443x x  + 1240x x  + 637x x  - 1423x x  + 320x  - 1363x x  - 1139x x  - 158x x  - 325x x  - 1578x x  + 32x  + 695x x  + 305x x  + 1012x x  + 1492x x  + 1290x x  + 1579x x  - 342x  - 83x x  - 104x x  + 998x x  - 92x x  + 1554x x  + 201x x  - 237x x  + 160x  - 228x x  - 543x x  - 1147x x  - 376x x  + 1313x x  + 603x x  + 106x x  - 1361x x  + 699x , - 228x  - 1510x x  + 277x  - 4x x  - 22x x  - 1526x  + 234x x  + 969x x  + 1253x x  - 1426x  - 1474x x  + 947x x  + 194x x  - 316x x  - 988x  - 1211x x  + 1087x x  + 536x x  - 491x x  + 870x x  - 659x  + 1490x x  - 469x x  + 1190x x  + 807x x  + 650x x  + 448x x  - 1353x  - 218x x  + 759x x  - 253x x  + 830x x  - 1080x x  - 143x x  - 1313x x  - 374x  - 180x x  + 741x x  + 742x x  - 1254x x  + 458x x  - 345x x  + 597x x  + 1567x x  - 31x , 1120x  + 709x x  - 1538x  - 1048x x  - 162x x  - 1518x  - 73x x  + 380x x  + 533x x  - 286x  + 1374x x  - 74x x  - 22x x  + 1535x x  - 1071x  - 839x x  - 560x x  + 928x x  + 335x x  - 1008x x  + 810x  - 448x x  - 357x x  - 107x x  + 40x x  + 784x x  - 1423x x  + 1276x  + 147x x  + 443x x  - 598x x  - 1077x x  - 1214x x  + 322x x  - 1408x x  + 72x  - 63x x  - 1513x x  - 791x x  + 11x x  + 77x x  + 836x x  - 1100x x  + 1637x x  - 788x , 1331x  + 318x x  - 704x  + 51x x  + 275x x  + 1149x  + 1526x x  + 768x x  + 414x x  - 782x  - 262x x  + 686x x  - 380x x  + 1377x x  + 1077x  + 1650x x  - 1129x x  - 508x x  + 846x x  + 1513x x  + 460x  - 1626x x  - 1024x x  + 862x x  + 1352x x  - 188x x  - 1382x x  - 650x  + 55x x  - 326x x  + 1037x x  + 705x x  - 667x x  + 1483x x  + 1661x x  - 1652x  - 1052x x  - 692x x  - 542x x  + 162x x  + 582x x  - 1369x x  + 934x x  + 1392x x  + 1227x , - 346x  + 1408x x  - 1225x  - 1536x x  - 1028x x  - 985x  - 210x x  - 1312x x  + 915x x  + 1633x  - 202x x  - 1636x x  - 1653x x  - 480x x  - 1260x  - 813x x  - 1623x x  - 1429x x  + 1094x x  - 747x x  + 955x  + 898x x  - 795x x  - 35x x  - 566x x  + 1631x x  - 324x x  + 926x  - 132x x  - 9x x  - 1290x x  - 543x x  + 902x x  + 735x x  - 342x x  - 400x  + 900x x  - 463x x  + 694x x  - 1262x x  - 1449x x  - 448x x  - 1402x x  - 731x x  - 996x , 301x  + 166x x  - 955x  - 739x x  - 1199x x  - 319x  + 1047x x  - 532x x  + 902x x  + 1195x  - 663x x  + 1215x x  - 534x x  - 332x x  - 973x  + 772x x  - 308x x  + 315x x  - 454x x  - 483x x  - 239x  - 1313x x  - 419x x  - 1340x x  - 1388x x  - 1340x x  - 1665x x  - 333x  - 465x x  - 1084x x  + 676x x  - 1612x x  - 288x x  + 11x x  - 1170x x  - 189x  + 498x x  - 889x x  + 693x x  + 1460x x  - 473x x  - 414x x  - 122x x  - 1659x x  - 1421x , 14x  - 1049x x  + 1506x  + 1235x x  + 642x x  - 1034x  + 460x x  + 150x x  + 760x x  - 1246x  - 1407x x  + 1570x x  + 1403x x  - 1610x x  - 431x  + 574x x  + 893x x  - 657x x  + 417x x  + 1362x x  + 224x  + 268x x  + 1097x x  + 1132x x  + 148x x  + 1331x x  - 77x x  - 756x  + 228x x  + 136x x  - 1484x x  - 1478x x  - 13x x  + 1620x x  - 701x x  - 769x  - 760x x  - 492x x  - 1077x x  - 1249x x  - 834x x  - 395x x  - 1358x x  - 988x x  + 113x , - 1634x  - 13x x  + 805x  - 21x x  - 1655x x  + 1479x  - 1510x x  - 646x x  + 225x x  - 1411x  + 1227x x  - 1108x x  + 1291x x  - 59x x  - 142x  + 586x x  - 676x x  + 655x x  - 1476x x  + 453x x  - 1076x  - 1152x x  + 1373x x  - 1191x x  - 416x x  + 699x x  + 317x x  + 825x  - 1560x x  - 488x x  - 1035x x  - 1561x x  - 644x x  - 1178x x  - 1320x x  + 158x  + 889x x  + 1444x x  - 1486x x  - 1211x x  + 1269x x  - 1228x x  + 568x x  + 1591x x  + 1207x , 105x  - 538x x  - 1222x  - 277x x  + 716x x  - 1067x  - 428x x  + 154x x  - 469x x  + 77x  + 538x x  - 179x x  + 921x x  - 223x x  + 1093x  - 262x x  + 1299x x  + 631x x  + 1486x x  - 1280x x  - 121x  - 50x x  - 978x x  - 694x x  - 531x x  + 505x x  + 1412x x  - 1061x  + 1202x x  + 448x x  - 187x x  + 1276x x  - 121x x  + 1361x x  + 697x x  + 682x  + 1592x x  + 705x x  - 227x x  - 7x x  - 1423x x  - 1446x x  - 1578x x  + 1511x x  + 917x , 1270x  - 391x x  - 1116x  - 287x x  + 653x x  + 1643x  + 1623x x  + 514x x  - 14x x  - 90x  + 1232x x  - 1434x x  + 1296x x  + 1522x x  + 136x  - 623x x  - 607x x  + 18x x  + 896x x  - 29x x  + 1059x  - 1053x x  + 1643x x  + 1652x x  - 1190x x  - 1073x x  + 1470x x  - 944x  - 93x x  - 187x x  - 994x x  - 1415x x  - 229x x  - 796x x  + 1642x x  + 1600x  - 344x x  + 905x x  + 1032x x  - 538x x  - 891x x  + 1243x x  + 1290x x  + 490x x  - 1148x , 1613x  + 175x x  - 1346x  - 1000x x  - 1217x x  - 729x  - 1296x x  + 1456x x  + 745x x  + 539x  + 525x x  - 811x x  + 753x x  + 1362x x  + 1629x  - 840x x  + 513x x  + 429x x  + 842x x  + 1414x x  - 308x  + 1415x x  - 1461x x  - 1135x x  + 701x x  + 766x x  + 785x x  + 1503x  + 147x x  + 929x x  - 1220x x  - 853x x  + 493x x  + 226x x  + 1416x x  + 280x  - 7x x  + 1632x x  + 520x x  + 1259x x  + 157x x  + 1596x x  + 655x x  - 42x x  - 586x })
│ │ │                                   0       0 1        1        0 2        1 2       2       0 3        1 3       2 3        3       0 4       1 4        2 4       3 4       4      0 5        1 5        2 5       3 5        4 5       5       0 6       1 6        2 6        3 6       4 6        5 6        6       0 7       1 7        2 7      3 7        4 7        5 7       6 7      7        0 8        1 8        2 8       3 8        4 8      5 8        6 8        7 8       8         0       0 1       1        0 2        1 2       2       0 3        1 3       2 3        3        0 4       1 4       2 4      3 4      4        0 5       1 5        2 5       3 5       4 5       5       0 6       1 6       2 6       3 6       4 6       5 6       6        0 7        1 7       2 7       3 7      4 7        5 7       6 7       7       0 8       1 8        2 8        3 8       4 8      5 8        6 8       7 8        8        0        0 1        1        0 2       1 2       2       0 3        1 3       2 3       3        0 4       1 4        2 4       3 4        4        0 5       1 5       2 5       3 5       4 5       5        0 6        1 6       2 6       3 6       4 6        5 6       6      0 7       1 7       2 7       3 7        4 7        5 7       6 7        7      0 8       1 8        2 8        3 8        4 8        5 8        6 8       7 8        8        0        0 1        1        0 2        1 2       2        0 3        1 3       2 3      3       0 4        1 4       2 4       3 4        4       0 5       1 5       2 5        3 5       4 5       5       0 6       1 6      2 6        3 6       4 6        5 6       6       0 7     1 7        2 7        3 7       4 7        5 7        6 7       7       0 8        1 8       2 8        3 8        4 8       5 8       6 8      7 8        8         0      0 1       1       0 2      1 2       2       0 3        1 3       2 3       3       0 4        1 4       2 4        3 4       4        0 5        1 5       2 5       3 5        4 5      5       0 6       1 6        2 6        3 6        4 6        5 6       6      0 7       1 7       2 7      3 7        4 7       5 7       6 7       7       0 8       1 8        2 8       3 8        4 8       5 8       6 8        7 8       8        0        0 1       1     0 2      1 2        2       0 3       1 3        2 3        3        0 4       1 4       2 4       3 4       4        0 5        1 5       2 5       3 5       4 5       5        0 6       1 6        2 6       3 6       4 6       5 6        6       0 7       1 7       2 7       3 7        4 7       5 7        6 7       7       0 8       1 8       2 8        3 8       4 8       5 8       6 8        7 8      8       0       0 1        1        0 2       1 2        2      0 3       1 3       2 3       3        0 4      1 4      2 4        3 4        4       0 5       1 5       2 5       3 5        4 5       5       0 6       1 6       2 6      3 6       4 6        5 6        6       0 7       1 7       2 7        3 7        4 7       5 7        6 7      7      0 8        1 8       2 8      3 8      4 8       5 8        6 8        7 8       8       0       0 1       1      0 2       1 2        2        0 3       1 3       2 3       3       0 4       1 4       2 4        3 4        4        0 5        1 5       2 5       3 5        4 5       5        0 6        1 6       2 6        3 6       4 6        5 6       6      0 7       1 7        2 7       3 7       4 7        5 7        6 7        7        0 8       1 8       2 8       3 8       4 8        5 8       6 8        7 8        8        0        0 1        1        0 2        1 2       2       0 3        1 3       2 3        3       0 4        1 4        2 4       3 4        4       0 5        1 5        2 5        3 5       4 5       5       0 6       1 6      2 6       3 6        4 6       5 6       6       0 7     1 7        2 7       3 7       4 7       5 7       6 7       7       0 8       1 8       2 8        3 8        4 8       5 8        6 8       7 8       8      0       0 1       1       0 2        1 2       2        0 3       1 3       2 3        3       0 4        1 4       2 4       3 4       4       0 5       1 5       2 5       3 5       4 5       5        0 6       1 6        2 6        3 6        4 6        5 6       6       0 7        1 7       2 7        3 7       4 7      5 7        6 7       7       0 8       1 8       2 8        3 8       4 8       5 8       6 8        7 8        8     0        0 1        1        0 2       1 2        2       0 3       1 3       2 3        3        0 4        1 4        2 4        3 4       4       0 5       1 5       2 5       3 5        4 5       5       0 6        1 6        2 6       3 6        4 6      5 6       6       0 7       1 7        2 7        3 7      4 7        5 7       6 7       7       0 8       1 8        2 8        3 8       4 8       5 8        6 8       7 8       8         0      0 1       1      0 2        1 2        2        0 3       1 3       2 3        3        0 4        1 4        2 4      3 4       4       0 5       1 5       2 5        3 5       4 5        5        0 6        1 6        2 6       3 6       4 6       5 6       6        0 7       1 7        2 7        3 7       4 7        5 7        6 7       7       0 8        1 8        2 8        3 8        4 8        5 8       6 8        7 8        8      0       0 1        1       0 2       1 2        2       0 3       1 3       2 3      3       0 4       1 4       2 4       3 4        4       0 5        1 5       2 5        3 5        4 5       5      0 6       1 6       2 6       3 6       4 6        5 6        6        0 7       1 7       2 7        3 7       4 7        5 7       6 7       7        0 8       1 8       2 8     3 8        4 8        5 8        6 8        7 8       8       0       0 1        1       0 2       1 2        2        0 3       1 3      2 3      3        0 4        1 4        2 4        3 4       4       0 5       1 5      2 5       3 5      4 5        5        0 6        1 6        2 6        3 6        4 6        5 6       6      0 7       1 7       2 7        3 7       4 7       5 7        6 7        7       0 8       1 8        2 8       3 8       4 8        5 8        6 8       7 8        8       0       0 1        1        0 2        1 2       2        0 3        1 3       2 3       3       0 4       1 4       2 4        3 4        4       0 5       1 5       2 5       3 5        4 5       5        0 6        1 6        2 6       3 6       4 6       5 6        6       0 7       1 7        2 7       3 7       4 7       5 7        6 7       7     0 8        1 8       2 8        3 8       4 8        5 8       6 8      7 8       8
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
│ │ │  
│ │ │  i5 : time degreeMap phi
│ │ │ - -- used 0.0464011s (cpu); 0.0464049s (thread); 0s (gc)
│ │ │ + -- used 0.0576005s (cpu); 0.057604s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
│ │ │  
│ │ │  i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 
│ │ │       -- of dimension 5 (so that the composition phi':P^8--->P^8 must have degree equal to deg(G(1,5))=14)
│ │ │       phi'=phi*map(ringP14,ringP8,for i to 8 list random(1,ringP14))
│ │ │  
│ │ │                                   2                  2                           2                                      2                                                 2                                                           2                                                                   2                                                                              2                                                                                          2        2                  2                              2                                       2                                                2                                                             2                                                                  2                                                                              2                                                                                            2        2                  2                             2                                       2                                                2                                                           2                                                                      2                                                                              2                                                                                         2         2                 2                            2                                       2                                                  2                                                             2                                                                    2                                                                                2                                                                                             2       2                   2                            2                                     2                                                2                                                          2                                                                  2                                                                                   2                                                                                            2        2                2                           2                                      2                                                  2                                                            2                                                                      2                                                                                 2                                                                                          2   2                   2                           2                                     2                                                  2                                                           2                                                                    2                                                                              2                                                                                         2      2                  2                           2                                      2                                                  2                                                             2                                                                       2                                                                              2                                                                                          2         2                  2                            2                                     2                                                 2                                                              2                                                                    2                                                                               2                                                                                        2
│ │ │  o6 = map (ringP8, ringP8, {- 780x  - 506x x  + 1537x  - 132x x  - 928x x  + 386x  - 102x x  + 422x x  + 725x x  - 1073x  - 905x x  - 830x x  + 1500x x  + 276x x  + 1533x  - 653x x  + 1558x x  + 939x x  - 1432x x  + 462x x  - 329x  - 92x x  + 661x x  - 1298x x  - 684x x  + 70x x  - 715x x  + 1093x  + 581x x  + 329x x  + 454x x  - 911x x  - 84x x  - 1452x x  - 809x x  + 1202x  + 1353x x  + 1503x x  + 482x x  + 893x x  - 643x x  + 598x x  + 110x x  + 1064x x  - 472x , - 522x  - 583x x  + 1339x  + 1535x x  - 1317x x  + 1113x  - 169x x  + 1440x x  - 1657x x  + 721x  + 40x x  - 1576x x  - 367x x  + 257x x  - 1454x  + 1612x x  + 1529x x  - 1068x x  + 560x x  - 1441x x  + 608x  - 92x x  - 1006x x  + 285x x  + 102x x  - 397x x  + 66x x  - 643x  - 38x x  + 1380x x  + 1069x x  - 426x x  + 1147x x  + 982x x  + 10x x  - 662x  + 16x x  + 1561x x  + 1597x x  + 512x x  + 1288x x  - 1253x x  + 1317x x  + 1481x x  - 354x , - 640x  - 1551x x  + 469x  + 1482x x  - 1593x x  - 986x  + 471x x  + 612x x  + 1228x x  + 1156x  - 731x x  + 1503x x  - 628x x  + 674x x  - 799x  + 1137x x  + 844x x  + 589x x  - 666x x  + 829x x  - 1024x  - 170x x  + 450x x  + 1497x x  + 1204x x  - 907x x  + 1621x x  - 417x  + 1297x x  + 1444x x  + 4x x  + 398x x  + 996x x  - 1031x x  + 239x x  + 303x  + 1215x x  - 83x x  + 1571x x  - 1543x x  - 925x x  - 694x x  + 151x x  - 520x x  + 880x , - 1210x  - 222x x  + 185x  + 245x x  + 1059x x  - 322x  + 238x x  + 962x x  + 1260x x  - 1581x  + 50x x  + 1352x x  - 1465x x  + 1555x x  + 1333x  + 1362x x  + 1365x x  + 1168x x  - 1401x x  + 149x x  - 652x  + 1378x x  - 557x x  - 112x x  + 26x x  + 315x x  + 111x x  + 1592x  - 283x x  - 1454x x  + 907x x  + 212x x  + 400x x  + 1049x x  - 882x x  - 1429x  - 183x x  + 1571x x  - 1286x x  - 1179x x  + 1319x x  + 240x x  - 1100x x  + 1500x x  - 348x , 1051x  - 1325x x  + 1354x  - 346x x  - 1532x x  - 466x  + 163x x  - 659x x  - 291x x  + 966x  + 789x x  + 393x x  + 403x x  - 1199x x  - 570x  - 93x x  - 492x x  - 418x x  + 713x x  - 1323x x  - 1384x  - 830x x  - 54x x  - 306x x  + 709x x  + 421x x  - 954x x  - 299x  + 1053x x  - 1080x x  + 686x x  + 170x x  - 1272x x  - 1661x x  + 1235x x  + 1553x  - 1454x x  - 1411x x  - 1195x x  - 962x x  + 737x x  - 390x x  + 957x x  + 1538x x  + 1234x , - 509x  + 9x x  - 1563x  - 710x x  - 642x x  + 541x  + 220x x  - 1214x x  - 16x x  + 1008x  - 1088x x  + 755x x  - 886x x  - 1433x x  + 1154x  + 1627x x  - 1547x x  - 951x x  + 866x x  + 163x x  - 1142x  - 668x x  + 1361x x  + 1324x x  - 490x x  + 282x x  - 1133x x  - 612x  + 805x x  - 126x x  + 1296x x  - 973x x  + 1271x x  - 1646x x  + 844x x  + 1073x  - 1452x x  - 1112x x  - 141x x  + 176x x  - 1579x x  - 78x x  + 848x x  - 1365x x  + 711x , x  + 1543x x  - 1076x  + 493x x  - 526x x  + 868x  - 582x x  - 996x x  + 206x x  - 419x  + 1258x x  - 391x x  + 1002x x  - 1539x x  + 931x  - 1504x x  + 810x x  + 324x x  + 1356x x  + 313x x  + 772x  + 299x x  + 1186x x  + 718x x  + 407x x  - 64x x  - 828x x  - 1393x  + 94x x  - 290x x  - 766x x  + 950x x  - 640x x  + 265x x  - 1640x x  - 1403x  - 126x x  + 891x x  - 1519x x  - 927x x  - 1335x x  - 1448x x  - x x  - 1103x x  - 1152x , 821x  + 558x x  - 1174x  - 168x x  + 986x x  + 790x  + 549x x  + 817x x  + 1396x x  + 695x  + 1211x x  + 878x x  - 1061x x  - 1244x x  - 880x  + 1409x x  - 567x x  + 1240x x  + 1126x x  - 1262x x  + 490x  + 1553x x  + 1276x x  + 805x x  + 576x x  - 1076x x  + 1617x x  - 495x  - 750x x  - 277x x  + 544x x  + 1479x x  - 784x x  - 64x x  - 1203x x  + 405x  + 1013x x  + 604x x  + 1301x x  + 1003x x  + 235x x  + 696x x  + 939x x  - 714x x  - 879x , - 1452x  + 727x x  - 1159x  + 449x x  - 1169x x  + 732x  + 575x x  - 600x x  + 924x x  - 837x  + 1298x x  - 860x x  + 1010x x  + 774x x  + 319x  + 1087x x  - 1120x x  + 1439x x  + 1175x x  - 1648x x  + 985x  - 1317x x  - 878x x  + 399x x  - 1339x x  + 70x x  - 463x x  + 470x  - 628x x  - 907x x  + 748x x  + 98x x  + 1150x x  + 1140x x  + 1308x x  + 621x  + 369x x  - 991x x  - 1186x x  + 61x x  - 907x x  - 681x x  - 1528x x  + 717x x  + 854x })
│ │ │                                   0       0 1        1       0 2       1 2       2       0 3       1 3       2 3        3       0 4       1 4        2 4       3 4        4       0 5        1 5       2 5        3 5       4 5       5      0 6       1 6        2 6       3 6      4 6       5 6        6       0 7       1 7       2 7       3 7      4 7        5 7       6 7        7        0 8        1 8       2 8       3 8       4 8       5 8       6 8        7 8       8        0       0 1        1        0 2        1 2        2       0 3        1 3        2 3       3      0 4        1 4       2 4       3 4        4        0 5        1 5        2 5       3 5        4 5       5      0 6        1 6       2 6       3 6       4 6      5 6       6      0 7        1 7        2 7       3 7        4 7       5 7      6 7       7      0 8        1 8        2 8       3 8        4 8        5 8        6 8        7 8       8        0        0 1       1        0 2        1 2       2       0 3       1 3        2 3        3       0 4        1 4       2 4       3 4       4        0 5       1 5       2 5       3 5       4 5        5       0 6       1 6        2 6        3 6       4 6        5 6       6        0 7        1 7     2 7       3 7       4 7        5 7       6 7       7        0 8      1 8        2 8        3 8       4 8       5 8       6 8       7 8       8         0       0 1       1       0 2        1 2       2       0 3       1 3        2 3        3      0 4        1 4        2 4        3 4        4        0 5        1 5        2 5        3 5       4 5       5        0 6       1 6       2 6      3 6       4 6       5 6        6       0 7        1 7       2 7       3 7       4 7        5 7       6 7        7       0 8        1 8        2 8        3 8        4 8       5 8        6 8        7 8       8       0        0 1        1       0 2        1 2       2       0 3       1 3       2 3       3       0 4       1 4       2 4        3 4       4      0 5       1 5       2 5       3 5        4 5        5       0 6      1 6       2 6       3 6       4 6       5 6       6        0 7        1 7       2 7       3 7        4 7        5 7        6 7        7        0 8        1 8        2 8       3 8       4 8       5 8       6 8        7 8        8        0     0 1        1       0 2       1 2       2       0 3        1 3      2 3        3        0 4       1 4       2 4        3 4        4        0 5        1 5       2 5       3 5       4 5        5       0 6        1 6        2 6       3 6       4 6        5 6       6       0 7       1 7        2 7       3 7        4 7        5 7       6 7        7        0 8        1 8       2 8       3 8        4 8      5 8       6 8        7 8       8   0        0 1        1       0 2       1 2       2       0 3       1 3       2 3       3        0 4       1 4        2 4        3 4       4        0 5       1 5       2 5        3 5       4 5       5       0 6        1 6       2 6       3 6      4 6       5 6        6      0 7       1 7       2 7       3 7       4 7       5 7        6 7        7       0 8       1 8        2 8       3 8        4 8        5 8    6 8        7 8        8      0       0 1        1       0 2       1 2       2       0 3       1 3        2 3       3        0 4       1 4        2 4        3 4       4        0 5       1 5        2 5        3 5        4 5       5        0 6        1 6       2 6       3 6        4 6        5 6       6       0 7       1 7       2 7        3 7       4 7      5 7        6 7       7        0 8       1 8        2 8        3 8       4 8       5 8       6 8       7 8       8         0       0 1        1       0 2        1 2       2       0 3       1 3       2 3       3        0 4       1 4        2 4       3 4       4        0 5        1 5        2 5        3 5        4 5       5        0 6       1 6       2 6        3 6      4 6       5 6       6       0 7       1 7       2 7      3 7        4 7        5 7        6 7       7       0 8       1 8        2 8      3 8       4 8       5 8        6 8       7 8       8
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
│ │ │  
│ │ │  i7 : time degreeMap phi'
│ │ │ - -- used 1.36727s (cpu); 0.743088s (thread); 0s (gc)
│ │ │ + -- used 1.66703s (cpu); 0.873296s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 14
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_force__Image.out
│ │ │ @@ -5,14 +5,14 @@
│ │ │  o2 : Ideal of P6
│ │ │  
│ │ │  i3 : Phi = rationalMap(X,Dominant=>2);
│ │ │  
│ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ - -- used 0.000639339s (cpu); 0.000632025s (thread); 0s (gc)
│ │ │ + -- used 0.000851599s (cpu); 0.000847133s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : Phi;
│ │ │  
│ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_graph.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │                        - x  + x x
│ │ │                           3    2 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i3 : time (p1,p2) = graph phi;
│ │ │ - -- used 0.0150736s (cpu); 0.0147825s (thread); 0s (gc)
│ │ │ + -- used 0.0307073s (cpu); 0.0194373s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : p1
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │                                    ZZ                                 ZZ
│ │ │       source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y , y , y , y , y ]) defined by
│ │ │                                  190181  0   1   2   3   4          190181  0   1   2   3   4   5
│ │ │ @@ -173,15 +173,15 @@
│ │ │  i8 : projectiveDegrees p2
│ │ │  
│ │ │  o8 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : time g = graph p2;
│ │ │ - -- used 0.0299231s (cpu); 0.0296181s (thread); 0s (gc)
│ │ │ + -- used 0.0477313s (cpu); 0.0360069s (thread); 0s (gc)
│ │ │  
│ │ │  i10 : g_0;
│ │ │  
│ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to PP^4)
│ │ │  
│ │ │  i11 : g_1;
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_ideal_lp__Rational__Map_rp.out
│ │ │ @@ -33,15 +33,15 @@
│ │ │                        x  - x x
│ │ │                         1    0 3
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │  
│ │ │  i3 : time ideal phi
│ │ │ - -- used 0.0031265s (cpu); 0.00312386s (thread); 0s (gc)
│ │ │ + -- used 0.00430458s (cpu); 0.00430416s (thread); 0s (gc)
│ │ │  
│ │ │               2                                     2                      
│ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3  
│ │ │       ------------------------------------------------------------------------
│ │ │              2
│ │ │       x x , x  - x x )
│ │ │ @@ -108,15 +108,15 @@
│ │ │                        y
│ │ │                         4
│ │ │                       }
│ │ │  
│ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
│ │ │  
│ │ │  i6 : time ideal phi'
│ │ │ - -- used 0.0921941s (cpu); 0.0921971s (thread); 0s (gc)
│ │ │ + -- used 0.125207s (cpu); 0.125209s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal 1
│ │ │  
│ │ │                                                                                                              QQ[x ..x , y ..y ]
│ │ │                                                                                                                  0   5   0   4
│ │ │  o6 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │                                                                                                                                                                                                       2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse__Map.out
│ │ │ @@ -72,15 +72,15 @@
│ │ │                        w w  - w w  + w w
│ │ │                         2 4    1 5    0 6
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
│ │ │  
│ │ │  i2 : time psi = inverseMap phi
│ │ │ - -- used 0.0785s (cpu); 0.0785012s (thread); 0s (gc)
│ │ │ + -- used 0.0957506s (cpu); 0.0957508s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = -- rational map --
│ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       defining forms: {
│ │ │ @@ -158,15 +158,15 @@
│ │ │  o4 = map (QQ[w ..w  ], QQ[w ..w  ], {w  w   - w  w   - w  w   - w  w   - w w  , w  w   - w  w   - w  w   - w  w   - w w  , w  w   - w  w   - w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   + w  w   - w w  , w w   - w w   + w w   + w w   + w w  , w  w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   + w  w   - w w  , w w   - w w   + w w   + w w   + w w  , w  w   - w  w   - w  w   + w  w   - w w  , w w   - w w   - w w   + w w   + w w  , w  w   - w  w   - w  w   + w  w   - w w  , w  w   - w  w   - w  w   + w  w   - w w  , w w   - w w   - w w   + w w   + w w  , w w   - w w   - w w   + w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   + w w   - w w  , w w   - w w   + w w   + w w   - w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   + w w   - w w  , w w   - w w   - w w   + w w   - w w  , w w  - w w  - w w  + w w  - w w })
│ │ │                0   26       0   26     21 22    20 23    15 24    10 25    0 26   19 22    18 23    16 24    11 25    1 26   19 20    18 21    17 24    12 25    2 26   15 19    16 21    17 23    13 25    3 26   10 19    11 21    12 23    13 24    4 26   0 19    1 21    2 23    3 24    4 25   15 18    16 20    17 22    14 25    5 26   10 18    11 20    12 22    14 24    6 26   0 18    1 20    2 22    5 24    6 25   12 16    11 17    13 18    14 19    7 26   2 16    1 17    3 18    5 19    7 25   12 15    10 17    13 20    14 21    8 26   11 15    10 16    13 22    14 23    9 26   2 15    0 17    3 20    5 21    8 25   1 15    0 16    3 22    5 23    9 25   5 13    3 14    7 15    8 16    9 17   5 12    2 14    6 17    8 18    7 20   3 12    2 13    4 17    8 19    7 21   5 11    1 14    6 16    9 18    7 22   3 11    1 13    4 16    9 19    7 23   2 11    1 12    4 18    6 19    7 24   7 10    8 11    9 12    6 13    4 14   5 10    0 14    6 15    9 20    8 22   3 10    0 13    4 15    9 21    8 23   2 10    0 12    4 20    6 21    8 24   1 10    0 11    4 22    6 23    9 24   4 5    3 6    0 7    1 8    2 9
│ │ │  
│ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
│ │ │  
│ │ │  i5 : time psi = inverseMap phi
│ │ │ - -- used 0.288262s (cpu); 0.189505s (thread); 0s (gc)
│ │ │ + -- used 0.295668s (cpu); 0.200371s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = map (QQ[w ..w  ], QQ[w ..w  ], {- w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , - w  w   + w  w   - w  w   + w  w   - w  w  , - w  w   + w  w   - w  w   + w  w   - w  w  , w w   - w w   + w w   + w  w   - w  w  , - w w   + w w   + w  w   + w w   - w w  , - w w   + w w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , w  w   - w  w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w  - w w  - w w   + w w   - w w  , - w w  + w w  + w w   - w w   + w w  , w w  - w w  - w w  + w w   - w w  })
│ │ │                0   26       0   26       5 22    8 23    14 24    13 25    0 26     5 18    8 19    14 20    10 25    1 26     5 16    8 17    13 20    10 24    2 26     5 15    14 17    13 19    10 23    3 26     5 21    20 23    19 24    17 25    4 26     8 15    14 16    13 18    10 22    6 26     8 21    20 22    18 24    16 25    7 26   17 18    16 19    15 20    10 21    9 26     13 21    17 22    16 23    15 24    11 26     14 21    19 22    18 23    15 25    12 26   0 21    4 22    7 23    12 24    11 25     4 18    7 19    12 20    1 21    9 25     4 16    7 17    11 20    2 21    9 24     4 15    12 17    11 19    3 21    9 23     7 15    12 16    11 18    6 21    9 22   12 13    11 14    0 15    3 22    6 23   10 12    9 14    1 15    3 18    6 19   10 11    9 13    2 15    3 16    6 17   8 9    7 10    1 16    2 18    6 20   5 9    4 10    1 17    2 19    3 20   8 11    7 13    0 16    2 22    6 24   5 11    4 13    0 17    2 23    3 24   8 12    7 14    0 18    1 22    6 25   5 12    4 14    0 19    1 23    3 25   5 7    4 8    0 20    1 24    2 25     5 6    3 8    0 10    1 13    2 14   4 6    3 7    0 9    1 11    2 12
│ │ │  
│ │ │  o5 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse_lp__Rational__Map_rp.out
│ │ │ @@ -28,15 +28,15 @@
│ │ │                        - -------x  + ---------x x  + ------------x x  - ----------x x  - -----x  - -----------x x  + -------------x x x  + -------------x x x  - --------x x  - ----------x x  + -------------x x x  - ----------x x  - -----------x x  + ----------x x  + ------x  + -----------x x  + ----------x x x  - -----------x x x  - -------x x  + -------------x x x  + ------------x x x x  - -----------x x x  + -----------x x x  - ------------x x x  + ----------x x  - -----------x x  - ------------x x x  - ---------x x  - ------------x x x  - -----------x x x  + -----------x x  - ----------x x  + -------x x  + --------x x  + ------x  + ---------x x  - ------------x x x  - -------------x x x  - ----------x x  + --------------x x x  + -------------x x x x  - ------------x x x  + -------------x x x  + ------------x x x  + ----------x x  + -----------x x x  - -------------x x x x  - ----------x x x  + --------------x x x x  - -------------x x x x  + -------------x x x  - ------------x x x  + ---------x x x  - ------------x x x  + ---------x x  - ---------x x  - -----------x x x  - ----------x x  + -----------x x x  + -----------x x x  + ----------x x  - -----------x x x  - -----------x x x  - ------------x x x  - ----------x x  + ---------x x  - ------x x  - --------x x  - ----------x x  - -----x
│ │ │                           290304 0    3888000  0 1    2939328000  0 1    163296000 0 1   20250 1    228614400  0 2    41150592000  0 1 2    41150592000  0 1 2    3888000 1 2     3572100  0 2    10287648000  0 1 2    342921600 1 2    114307200  0 2    63504000  1 2    25200 2     76204800  0 3    42336000  0 1 3    428652000  0 1 3    212625 1 3     5334336000  0 2 3    9601804800  0 1 2 3    489888000  1 2 3    222264000  0 2 3    12002256000 1 2 3    66679200  2 3    666792000  0 3     666792000  0 1 3    47628000 1 3    1333584000  0 2 3    444528000  1 2 3    777924000  2 3    55566000  0 3    105840 1 3    3472875 2 3    11025 3    4665600  0 4    2939328000  0 1 4     4898880000  0 1 4    29160000  1 4     41150592000  0 2 4    20575296000  0 1 2 4    4898880000  1 2 4    20575296000  0 2 4    1371686400  1 2 4    95256000  2 4     40824000  0 3 4     8573040000  0 1 3 4    11664000  1 3 4     24004512000  0 2 3 4    34292160000  1 2 3 4    12002256000  2 3 4     333396000  0 3 4    5292000  1 3 4    1333584000  2 3 4    3969000  3 4    6804000  0 4    272160000  0 1 4    58320000  1 4    190512000  0 2 4    4898880000 1 2 4    190512000 2 4    476280000  0 3 4    204120000  1 3 4    2857680000  2 3 4    23814000  3 4    30618000 0 4    46656 1 4   12757500 2 4    51030000  3 4   30375 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │  
│ │ │  i3 : time inverse phi
│ │ │ - -- used 0.0550698s (cpu); 0.0550696s (thread); 0s (gc)
│ │ │ + -- used 0.0703872s (cpu); 0.0703871s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       defining forms: {
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Birational.out
│ │ │ @@ -40,18 +40,18 @@
│ │ │                        - t  + t t
│ │ │                           3    2 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i3 : time isBirational phi
│ │ │ - -- used 0.0199568s (cpu); 0.0199548s (thread); 0s (gc)
│ │ │ + -- used 0.0253858s (cpu); 0.0253857s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0160523s (cpu); 0.0154605s (thread); 0s (gc)
│ │ │ + -- used 0.037693s (cpu); 0.0177218s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Dominant.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  i2 : phi = rationalMap ideal jacobian ideal det matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},{x_4..x_8}};
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
│ │ │  
│ │ │  i3 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 2.86189s (cpu); 2.08567s (thread); 0s (gc)
│ │ │ + -- used 2.69805s (cpu); 2.32235s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : P7 = ZZ/101[x_0..x_7];
│ │ │  
│ │ │  i5 : -- hyperelliptic curve of genus 3
│ │ │       C = ideal(x_4*x_5+23*x_5^2-23*x_0*x_6-18*x_1*x_6+6*x_2*x_6+37*x_3*x_6+23*x_4*x_6-26*x_5*x_6+2*x_6^2-25*x_0*x_7+45*x_1*x_7+30*x_2*x_7-49*x_3*x_7-49*x_4*x_7+50*x_5*x_7,x_3*x_5-24*x_5^2+21*x_0*x_6+x_1*x_6+46*x_3*x_6+27*x_4*x_6+5*x_5*x_6+35*x_6^2+20*x_0*x_7-23*x_1*x_7+8*x_2*x_7-22*x_3*x_7+20*x_4*x_7-15*x_5*x_7,x_2*x_5+47*x_5^2-40*x_0*x_6+37*x_1*x_6-25*x_2*x_6-22*x_3*x_6-8*x_4*x_6+27*x_5*x_6+15*x_6^2-23*x_0*x_7-42*x_1*x_7+27*x_2*x_7+35*x_3*x_7+39*x_4*x_7+24*x_5*x_7,x_1*x_5+15*x_5^2+49*x_0*x_6+8*x_1*x_6-31*x_2*x_6+9*x_3*x_6+38*x_4*x_6-36*x_5*x_6-30*x_6^2-33*x_0*x_7+26*x_1*x_7+32*x_2*x_7+27*x_3*x_7+6*x_4*x_7+36*x_5*x_7,x_0*x_5+30*x_5^2-11*x_0*x_6-38*x_1*x_6+13*x_2*x_6-32*x_3*x_6-30*x_4*x_6+4*x_5*x_6-28*x_6^2-30*x_0*x_7-6*x_1*x_7-45*x_2*x_7+34*x_3*x_7+20*x_4*x_7+48*x_5*x_7,x_3*x_4+46*x_5^2-37*x_0*x_6+27*x_1*x_6+33*x_2*x_6+8*x_3*x_6-32*x_4*x_6+42*x_5*x_6-34*x_6^2-37*x_0*x_7-28*x_1*x_7+10*x_2*x_7-27*x_3*x_7-42*x_4*x_7-8*x_5*x_7,x_2*x_4-25*x_5^2-4*x_0*x_6+2*x_1*x_6-31*x_2*x_6-5*x_3*x_6+16*x_4*x_6-24*x_5*x_6+31*x_6^2-30*x_0*x_7+32*x_1*x_7+12*x_2*x_7-40*x_3*x_7+3*x_4*x_7-28*x_5*x_7,x_0*x_4+15*x_5^2+48*x_0*x_6-50*x_1*x_6+46*x_2*x_6-48*x_3*x_6-23*x_4*x_6-28*x_5*x_6+39*x_6^2+38*x_1*x_7-5*x_3*x_7+5*x_4*x_7-34*x_5*x_7,x_3^2-31*x_5^2+41*x_0*x_6-30*x_1*x_6-4*x_2*x_6+43*x_3*x_6+23*x_4*x_6+7*x_5*x_6+31*x_6^2-19*x_0*x_7+25*x_1*x_7-49*x_2*x_7-16*x_3*x_7-45*x_4*x_7+25*x_5*x_7,x_2*x_3+13*x_5^2-45*x_0*x_6-22*x_1*x_6+33*x_2*x_6-26*x_3*x_6-21*x_4*x_6+34*x_5*x_6-21*x_6^2-47*x_0*x_7-10*x_1*x_7+29*x_2*x_7-46*x_3*x_7-x_4*x_7+20*x_5*x_7,x_1*x_3+22*x_5^2+4*x_0*x_6+3*x_1*x_6+45*x_2*x_6+37*x_3*x_6+17*x_4*x_6+36*x_5*x_6-2*x_6^2-31*x_0*x_7+3*x_1*x_7-12*x_2*x_7+19*x_3*x_7+28*x_4*x_7+30*x_5*x_7,x_0*x_3-47*x_5^2-43*x_0*x_6+6*x_1*x_6-40*x_2*x_6+21*x_3*x_6+26*x_4*x_6-5*x_5*x_6-5*x_6^2+4*x_0*x_7-15*x_1*x_7+18*x_2*x_7-31*x_3*x_7+50*x_4*x_7-46*x_5*x_7,x_2^2+4*x_5^2+31*x_0*x_6+41*x_1*x_6+31*x_2*x_6+28*x_3*x_6+42*x_4*x_6-28*x_5*x_6-4*x_6^2-7*x_0*x_7+15*x_1*x_7-9*x_2*x_7+31*x_3*x_7+3*x_4*x_7+7*x_5*x_7,x_1*x_2-46*x_5^2-6*x_0*x_6-50*x_1*x_6+32*x_2*x_6-10*x_3*x_6+42*x_4*x_6+33*x_5*x_6+18*x_6^2-9*x_0*x_7-20*x_1*x_7+45*x_2*x_7-9*x_3*x_7+10*x_4*x_7-8*x_5*x_7,x_0*x_2-9*x_5^2+34*x_0*x_6-45*x_1*x_6+19*x_2*x_6+24*x_3*x_6+23*x_4*x_6-37*x_5*x_6-44*x_6^2+24*x_0*x_7-33*x_2*x_7+41*x_3*x_7-40*x_4*x_7+4*x_5*x_7,x_1^2+x_1*x_4+x_4^2-28*x_5^2-33*x_0*x_6-17*x_1*x_6+11*x_3*x_6+20*x_4*x_6+25*x_5*x_6-21*x_6^2-22*x_0*x_7+24*x_1*x_7-14*x_2*x_7+5*x_3*x_7-39*x_4*x_7-18*x_5*x_7,x_0*x_1-47*x_5^2-5*x_0*x_6-9*x_1*x_6-45*x_2*x_6+48*x_3*x_6+45*x_4*x_6-29*x_5*x_6+3*x_6^2+29*x_0*x_7+40*x_1*x_7+46*x_2*x_7+27*x_3*x_7-36*x_4*x_7-39*x_5*x_7,x_0^2-31*x_5^2+36*x_0*x_6-30*x_1*x_6-10*x_2*x_6+42*x_3*x_6+9*x_4*x_6+34*x_5*x_6-6*x_6^2+48*x_0*x_7-47*x_1*x_7-19*x_2*x_7+25*x_3*x_7+28*x_4*x_7+34*x_5*x_7);
│ │ │ @@ -21,12 +21,12 @@
│ │ │  
│ │ │  i6 : phi = rationalMap(C,3,2);
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
│ │ │  
│ │ │  i7 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 3.63761s (cpu); 2.40964s (thread); 0s (gc)
│ │ │ + -- used 4.44899s (cpu); 3.08833s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = false
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_kernel_lp__Ring__Map_cm__Z__Z_rp.out
│ │ │ @@ -6,23 +6,23 @@
│ │ │  o1 = map (QQ[x ..x ], QQ[y ..y  ], {- 5x x  + x x  + x x  + 35x x  - 7x x  + x x  - x x  - 49x  - 5x x  + 2x x  - x x  + 27x x  - 4x  + x x  - 7x x  + 2x x  - 2x x  + 14x x  - 4x x , - x x  - 6x x  - 5x x  + 2x x  + x x  + x x  - 5x x  - x x  + 2x x  + 7x x  - 2x x  + 2x x  - 3x x , - 25x  + 9x x  + 10x x  - 2x x  - x  + 29x x  - x x  - 7x x  - 13x x  + 3x x  + x x  - x x  + 2x x  - x x  + 7x x  - 2x x  - 8x x  + 2x x  - 3x x , x x  + x x  + x  + 7x x  - 9x x  + 12x x  - 4x  + 2x x  + 2x x  - 14x x  + 4x x  + x x  - x x  - 14x x  + x x , - 5x x  + x x  - 7x x  + 8x x  - 5x x  + 2x x  - x x  + x x  - x x  + 7x x  - 2x x  - x x  + 7x x  - 2x x , x x  + x  - 7x x  - 8x x  + x x  + x x  + 2x x  - x x  + x x  - 7x x  + 2x x  + x x  - 7x x  + 2x x , x x  + x  - 8x x  + x x  + 6x x  - 2x  + x x  + x x  - 7x x  + 2x x  + x x  - 7x x  + 2x x , x x  - 7x x  + x x  + x x  - 7x x  + 2x  - x x , - 4x x  + x x  - x  - 7x x  + 8x x  + x x  - x x  - 6x x  + 2x  - x x  - x x  + 7x x  - 2x x  - x x  + 7x x  - 2x x , - 5x x  + 2x  + x x  - x  - x x  + 8x x  - 10x x  + 2x x  + 2x x  - 2x x  + 14x x  - 4x x  + 5x x  - 3x x  - 2x x  + 7x x  - 2x x  - 3x x , - 5x x  + x x  + x x  - 4x x  - x x  + x x  + x x , x x  - x x  + 5x x  + x x  - 14x x  - x x  - 8x x  - 8x x  + 2x x  + 4x x  + 2x x  + 4x x  + 3x x  - 7x x  + 2x x  - 3x x })
│ │ │                0   8       0   11        0 3    2 4    3 4      0 5     2 5    3 5    4 5      5     0 6     2 6    4 6      5 6     6    4 7     5 7     6 7     4 8      5 8     6 8     1 2     1 5     0 6     1 6    4 6    5 6     0 7    1 7     2 7     5 7     6 7     1 8     7 8       0     0 2      0 4     2 4    4      0 5    2 5     4 5      0 6     4 6    5 6    0 7     2 7    4 7     5 7     6 7     0 8     4 8     7 8   2 4    3 4    4     2 5     4 5      5 6     6     3 7     4 7      5 7     6 7    3 8    4 8      5 8    6 8      0 4    2 4     2 5     4 5     0 6     2 6    4 6    5 6    4 7     5 7     6 7    4 8     5 8     6 8   0 4    4     1 5     4 5    0 6    1 6     4 6    5 6    4 7     5 7     6 7    4 8     5 8     6 8   2 3    4     4 5    4 6     5 6     6    3 7    4 7     5 7     6 7    4 8     5 8     6 8   1 3     1 5    1 6    4 6     5 6     6    3 7      0 3    3 4    4     0 5     4 5    0 6    4 6     5 6     6    3 7    4 7     5 7     6 7    4 8     5 8     6 8      0 2     2    2 4    4    2 5     4 5      0 6     5 6     2 7     4 7      5 7     6 7     0 8     2 8     4 8     5 8     6 8     7 8      0 1    1 2    1 4     0 6    1 6    4 6    0 7   0 2    1 2     0 4    1 4      1 5    2 5     4 5     0 6     1 6     4 6     2 7     0 8     1 8     5 8     6 8     7 8
│ │ │  
│ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │                   0   8          0   11
│ │ │  
│ │ │  i2 : time kernel(phi,1)
│ │ │ - -- used 0.0177116s (cpu); 0.0177066s (thread); 0s (gc)
│ │ │ + -- used 0.0223929s (cpu); 0.0223912s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
│ │ │  
│ │ │  i3 : time kernel(phi,2)
│ │ │ - -- used 1.02093s (cpu); 0.491366s (thread); 0s (gc)
│ │ │ + -- used 1.13551s (cpu); 0.545581s (thread); 0s (gc)
│ │ │  
│ │ │                             2                                                
│ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6  
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                             
│ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_parametrize_lp__Ideal_rp.out
│ │ │ @@ -26,15 +26,15 @@
│ │ │                8           9
│ │ │  
│ │ │                   ZZ
│ │ │  o2 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
│ │ │  
│ │ │  i3 : time parametrize L
│ │ │ - -- used 0.00434735s (cpu); 0.00434206s (thread); 0s (gc)
│ │ │ + -- used 0.0070726s (cpu); 0.00707334s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -116,15 +116,15 @@
│ │ │               5 9           6 9           7 9           8 9           9
│ │ │  
│ │ │                   ZZ
│ │ │  o4 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
│ │ │  
│ │ │  i5 : time parametrize Q
│ │ │ - -- used 0.749084s (cpu); 0.436364s (thread); 0s (gc)
│ │ │ + -- used 0.571376s (cpu); 0.437581s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_point_lp__Quotient__Ring_rp.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 3560583829489988690
│ │ │  
│ │ │  i1 : f = inverseMap specialQuadraticTransformation(9,ZZ/33331);
│ │ │  
│ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)
│ │ │  
│ │ │  i2 : time p = point source f
│ │ │ - -- used 0.239078s (cpu); 0.151432s (thread); 0s (gc)
│ │ │ + -- used 0.299988s (cpu); 0.205009s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal (y   - 9235y  , y  + 11075y  , y  - 5847y  , y  + 7396y  , y  +
│ │ │               10        11   9         11   8        11   7        11   6  
│ │ │       ------------------------------------------------------------------------
│ │ │       13530y  , y  + 4359y  , y  - 2924y  , y  + 13040y  , y  + 6904y  , y  -
│ │ │             11   5        11   4        11   3         11   2        11   1  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -20,12 +20,12 @@
│ │ │                                                             -----[y ..y  ]
│ │ │                                                             33331  0   11
│ │ │  o2 : Ideal of -------------------------------------------------------------------------------------------------------
│ │ │                (y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y , y y  - y y  + y y )
│ │ │                  6 7    5 8    4 11   3 7    2 8    1 11   3 5    2 6    0 11   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │  i3 : time p == f^* f p
│ │ │ - -- used 0.0935425s (cpu); 0.0935509s (thread); 0s (gc)
│ │ │ + -- used 0.137661s (cpu); 0.137665s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_projective__Degrees.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │                       0   4              0   5       1    0 2     1 2    0 3     2    1 3     1 3    0 4     2 3    1 4     3    2 4
│ │ │  
│ │ │  o2 : RingMap GF 109561[t ..t ] <-- GF 109561[x ..x ]
│ │ │                          0   4                 0   5
│ │ │  
│ │ │  i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.120186s (cpu); 0.0404089s (thread); 0s (gc)
│ │ │ + -- used 0.212012s (cpu); 0.0595769s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : psi=inverseMap(toMap(phi,Dominant=>infinity))
│ │ │  
│ │ │ @@ -30,15 +30,15 @@
│ │ │                           0   5
│ │ │  o4 : RingMap ------------------ <-- GF 109561[t ..t ]
│ │ │               x x  - x x  + x x                 0   4
│ │ │                2 3    1 4    0 5
│ │ │  
│ │ │  i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.122567s (cpu); 0.0537507s (thread); 0s (gc)
│ │ │ + -- used 0.031337s (cpu); 0.0142572s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : -- Cremona transformation of P^6 defined by the quadrics through a rational octic surface
│ │ │       phi = map specialCremonaTransformation(7,ZZ/300007)
│ │ │ @@ -48,21 +48,21 @@
│ │ │            300007  0   6   300007  0   6     2 4    1 5          0 4          1 4          4         0 5          1 5         2 5          4 5         5          3 6         4 6         5 6   2 3    0 5          1 3          1 4          4         0 5          1 5         2 5          4 5         5          3 6         4 6         5 6        0 3         1 4         3 4         4          0 5         1 5         2 5          3 5          4 5         5         3 6          4 6         5 6          0 1          1         0 2          1 2         2          1 4          1 5         2 5          0 6         1 6         2 6         0          1         0 2         1 2         2         1 4          4         0 5         1 5          2 5          4 5         5         0 6         1 6          2 6          3 6         4 6         5 6
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o6 : RingMap ------[x ..x ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   6
│ │ │  
│ │ │  i7 : time projectiveDegrees phi
│ │ │ - -- used 4.6717e-05s (cpu); 4.0846e-05s (thread); 0s (gc)
│ │ │ + -- used 5.1786e-05s (cpu); 4.5854e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.155e-05s (cpu); 2.138e-05s (thread); 0s (gc)
│ │ │ + -- used 2.5654e-05s (cpu); 2.5388e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = {4, 1}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : ZZ/33331[x_0..x_6]; V = ideal(x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3,x_6);
│ │ │  
│ │ │                  ZZ
│ │ │  o2 : Ideal of -----[x ..x ]
│ │ │                33331  0   6
│ │ │  
│ │ │  i3 : time phi = rationalMap(V,3,2)
│ │ │ - -- used 0.0982366s (cpu); 0.0982396s (thread); 0s (gc)
│ │ │ + -- used 0.115831s (cpu); 0.115829s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                      ZZ
│ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │                    33331  0   1   2   3   4   5   6
│ │ │                      ZZ
│ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  , y  ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ring_cm__Tally_rp.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │                     0         1         2         3        4         5
│ │ │  
│ │ │  o4 : Ideal of X
│ │ │  
│ │ │  i5 : D = new Tally from {H => 2,C => 1};
│ │ │  
│ │ │  i6 : time phi = rationalMap D
│ │ │ - -- used 0.0303865s (cpu); 0.0303844s (thread); 0s (gc)
│ │ │ + -- used 0.035184s (cpu); 0.0351831s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │                                    ZZ
│ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │                                  65521  0   1   2   3   4   5
│ │ │               {
│ │ │                   2                  2
│ │ │ @@ -123,13 +123,13 @@
│ │ │                        x x x  + x x x  + x x x  + x x  + x x x  - 2x x x  + x x
│ │ │                         0 1 5    0 2 5    1 2 5    2 5    1 4 5     2 4 5    4 5
│ │ │                       }
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
│ │ │  
│ │ │  i7 : time ? image(phi,"F4")
│ │ │ - -- used 1.21845s (cpu); 0.768199s (thread); 0s (gc)
│ │ │ + -- used 1.55766s (cpu); 0.717511s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │       hypersurfaces of degree 2
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cremona__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1330846641081
│ │ │  
│ │ │  i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ - -- used 1.28804s (cpu); 1.00797s (thread); 0s (gc)
│ │ │ + -- used 1.23352s (cpu); 1.12677s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │        source variety: PP^3                      
│ │ │        target variety: PP^3                      
│ │ │        dominance: true                           
│ │ │        birationality: true                       
│ │ │        projective degrees: {1, 3, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cubic__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1730018912715498288
│ │ │  
│ │ │  i1 : time specialCubicTransformation 9
│ │ │ - -- used 0.0929002s (cpu); 0.0929013s (thread); 0s (gc)
│ │ │ + -- used 0.0966357s (cpu); 0.0966331s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6
│ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -62,15 +62,15 @@
│ │ │                        8x x  - 12x x  + 24x  - 11x x  + 17x x x  - 24x x  - 10x x  + 11x x  - 3x  - 6x x  + 28x x x  - 70x x  - 21x x x  + 47x x x  - 13x x  - 14x x  + 66x x  - 22x x  - 20x  + 2x x  - 2x x x  - 10x x  - 11x x x  + 8x x x  - 5x x  + 3x x x  + 23x x x  - 11x x x  - 12x x  + 3x x  - 3x x  - 2x x  + 3x x  + x  - 11x x  + 14x x x  + 34x x  - 6x x x  - 16x x x  + 3x x  - 15x x x  - 66x x x  + 12x x x  + 30x x  - 19x x x  + 2x x x  - 5x x x  - 2x x x  - 7x x  + 6x x  + 21x x  - 3x x  - 21x x  + x x  + 5x  - 8x x  + 7x x x  - 32x x  - 13x x x  + 28x x x  - 9x x  + 70x x x  - 27x x x  - 36x x  + x x x  + 4x x x  - 7x x x  - 2x x x  + 3x x  - 25x x x  - 23x x x  + 4x x x  + 27x x x  - 14x x x  - 9x x  - 2x x  + 10x x  - 6x x  - 10x x  + 3x x  - 2x x
│ │ │                          0 1      0 1      1      0 2      0 1 2      1 2      0 2      1 2     2     0 3      0 1 3      1 3      0 2 3      1 2 3      2 3      0 3      1 3      2 3      3     0 4     0 1 4      1 4      0 2 4     1 2 4     2 4     0 3 4      1 3 4      2 3 4      3 4     0 4     1 4     2 4     3 4    4      0 5      0 1 5      1 5     0 2 5      1 2 5     2 5      0 3 5      1 3 5      2 3 5      3 5      0 4 5     1 4 5     2 4 5     3 4 5     4 5     0 5      1 5     2 5      3 5    4 5     5     0 6     0 1 6      1 6      0 2 6      1 2 6     2 6      1 3 6      2 3 6      3 6    0 4 6     1 4 6     2 4 6     3 4 6     4 6      0 5 6      1 5 6     2 5 6      3 5 6      4 5 6     5 6     0 6      1 6     2 6      3 6     4 6     5 6
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i2 : time describe oo
│ │ │ - -- used 0.0192656s (cpu); 0.0192647s (thread); 0s (gc)
│ │ │ + -- used 0.0206427s (cpu); 0.020643s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 3
│ │ │       source variety: PP^6
│ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Quadratic__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1729200582376678705
│ │ │  
│ │ │  i1 : time specialQuadraticTransformation 4
│ │ │ - -- used 0.0773877s (cpu); 0.0773884s (thread); 0s (gc)
│ │ │ + -- used 0.0811578s (cpu); 0.0811568s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6   7   8
│ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -50,15 +50,15 @@
│ │ │                        x x  - x x  + x x  - x x  - x  - x x
│ │ │                         0 1    0 4    3 6    4 6    6    5 7
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
│ │ │  
│ │ │  i2 : time describe oo
│ │ │ - -- used 0.00678678s (cpu); 0.00678413s (thread); 0s (gc)
│ │ │ + -- used 0.0114262s (cpu); 0.0114275s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 2
│ │ │       source variety: PP^8
│ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_to__External__String_lp__Rational__Map_rp.out
│ │ │ @@ -7,34 +7,34 @@
│ │ │  i2 : str = toExternalString phi;
│ │ │  
│ │ │  i3 : #str
│ │ │  
│ │ │  o3 = 6927
│ │ │  
│ │ │  i4 : time phi' = value str;
│ │ │ - -- used 0.0222032s (cpu); 0.0222031s (thread); 0s (gc)
│ │ │ + -- used 0.0269988s (cpu); 0.026998s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │  
│ │ │  i5 : time describe phi'
│ │ │ - -- used 0.00537746s (cpu); 0.00537832s (thread); 0s (gc)
│ │ │ + -- used 0.00671281s (cpu); 0.00671801s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 3
│ │ │       source variety: PP^3
│ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {1, 3, 4, 2}
│ │ │       number of minimal representatives: 1
│ │ │       dimension base locus: 1
│ │ │       degree base locus: 5
│ │ │       coefficient ring: ZZ/33331
│ │ │  
│ │ │  i6 : time describe inverse phi'
│ │ │ - -- used 0.00425598s (cpu); 0.00425702s (thread); 0s (gc)
│ │ │ + -- used 0.00599937s (cpu); 0.00600649s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = rational map defined by forms of degree 2
│ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │       target variety: PP^3
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {2, 4, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Chern__Schwartz__Mac__Pherson.html
│ │ │ @@ -102,30 +102,30 @@
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
    
│ │ │              
    
│ │ │                
    i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 1.23521s (cpu); 0.911576s (thread); 0s (gc)
│ │ │ + -- used 1.30323s (cpu); 1.04349s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o3 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o3 : -----
│ │ │          5
│ │ │         H
    
│ │ │              
    
│ │ │                
    i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 1.21347s (cpu); 0.831445s (thread); 0s (gc)
│ │ │ + -- used 1.52607s (cpu); 1.00202s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          5
│ │ │ @@ -172,30 +172,30 @@
│ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
    
│ │ │              
    
│ │ │                
    i9 : time ChernClass G
│ │ │ - -- used 0.239068s (cpu); 0.15048s (thread); 0s (gc)
│ │ │ + -- used 0.142567s (cpu); 0.142567s (thread); 0s (gc)
│ │ │  
│ │ │          9      8      7      6      5      4     3
│ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o9 : -----
│ │ │         10
│ │ │        H
    
│ │ │              
    
│ │ │                
    i10 : time ChernClass(G,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0108133s (cpu); 0.0103892s (thread); 0s (gc)
│ │ │ + -- used 0.0263878s (cpu); 0.0141289s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ │          10
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,26 +39,26 @@
│ │ │ │                 2                           2
│ │ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │ │  
│ │ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │ │                          0   4
│ │ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ │ - -- used 1.23521s (cpu); 0.911576s (thread); 0s (gc)
│ │ │ │ + -- used 1.30323s (cpu); 1.04349s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         4     3     2
│ │ │ │  o3 = 3H  + 5H  + 3H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o3 : -----
│ │ │ │          5
│ │ │ │         H
│ │ │ │  i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 1.21347s (cpu); 0.831445s (thread); 0s (gc)
│ │ │ │ + -- used 1.52607s (cpu); 1.00202s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         4     3     2
│ │ │ │  o4 = 3H  + 5H  + 3H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o4 : -----
│ │ │ │          5
│ │ │ │ @@ -88,26 +88,26 @@
│ │ │ │          0,2 1,3    0,1 2,3
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p
│ │ │ │  ]
│ │ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │ │  i9 : time ChernClass G
│ │ │ │ - -- used 0.239068s (cpu); 0.15048s (thread); 0s (gc)
│ │ │ │ + -- used 0.142567s (cpu); 0.142567s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          9      8      7      6      5      4     3
│ │ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o9 : -----
│ │ │ │         10
│ │ │ │        H
│ │ │ │  i10 : time ChernClass(G,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0108133s (cpu); 0.0103892s (thread); 0s (gc)
│ │ │ │ + -- used 0.0263878s (cpu); 0.0141289s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6      5      4     3
│ │ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o10 : -----
│ │ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Euler__Characteristic.html
│ │ │ @@ -90,24 +90,24 @@
│ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
    
│ │ │              
    
│ │ │                
    i2 : time EulerCharacteristic I
│ │ │ - -- used 0.427945s (cpu); 0.193961s (thread); 0s (gc)
│ │ │ + -- used 0.541737s (cpu); 0.268955s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    
│ │ │              
    
│ │ │                
    i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0129232s (cpu); 0.0122924s (thread); 0s (gc)
│ │ │ + -- used 0.0567221s (cpu); 0.0191547s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 10
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,20 +31,20 @@
│ │ │ │  i1 : I = Grassmannian(1,4,CoefficientRing=>ZZ/190181);
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p
│ │ │ │  ]
│ │ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │ │  i2 : time EulerCharacteristic I
│ │ │ │ - -- used 0.427945s (cpu); 0.193961s (thread); 0s (gc)
│ │ │ │ + -- used 0.541737s (cpu); 0.268955s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 10
│ │ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0129232s (cpu); 0.0122924s (thread); 0s (gc)
│ │ │ │ + -- used 0.0567221s (cpu); 0.0191547s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = 10
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  No test is made to see if the projective variety is smooth.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_u_l_e_r_(_P_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_) -- topological Euler characteristic of a
│ │ │ │        (smooth) projective variety
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp!.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    │ │ │
    │ │ │ 
    i4 : time phi! ;
│ │ │ - -- used 0.0537946s (cpu); 0.0533655s (thread); 0s (gc)
│ │ │ + -- used 0.0829391s (cpu); 0.0706775s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
    │ │ │
    │ │ │ 
    i5 : describe phi
│ │ │ @@ -132,15 +132,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    │ │ │
    │ │ │ 
    i9 : time phi! ;
│ │ │ - -- used 0.202479s (cpu); 0.118999s (thread); 0s (gc)
│ │ │ + -- used 0.305442s (cpu); 0.134329s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
    │ │ │
    │ │ │ 
    i10 : describe phi
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i3 : describe phi
│ │ │ │  
│ │ │ │  o3 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^5
│ │ │ │       target variety: PP^5
│ │ │ │       coefficient ring: QQ
│ │ │ │  i4 : time phi! ;
│ │ │ │ - -- used 0.0537946s (cpu); 0.0533655s (thread); 0s (gc)
│ │ │ │ + -- used 0.0829391s (cpu); 0.0706775s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
│ │ │ │  i5 : describe phi
│ │ │ │  
│ │ │ │  o5 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^5
│ │ │ │       target variety: PP^5
│ │ │ │ @@ -47,15 +47,15 @@
│ │ │ │  i8 : describe phi
│ │ │ │  
│ │ │ │  o8 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: PP^5
│ │ │ │       coefficient ring: QQ
│ │ │ │  i9 : time phi! ;
│ │ │ │ - -- used 0.202479s (cpu); 0.118999s (thread); 0s (gc)
│ │ │ │ + -- used 0.305442s (cpu); 0.134329s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
│ │ │ │  i10 : describe phi
│ │ │ │  
│ │ │ │  o10 = rational map defined by forms of degree 2
│ │ │ │        source variety: PP^4
│ │ │ │        target variety: PP^5
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp^_st_st_sp__Ideal.html
│ │ │ @@ -158,15 +158,15 @@
│ │ │  
│ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
    │ │ │
    │ │ │ 
    i6 : time phi^** q
│ │ │ - -- used 0.270847s (cpu); 0.194712s (thread); 0s (gc)
│ │ │ + -- used 0.306466s (cpu); 0.215409s (thread); 0s (gc)
│ │ │  
│ │ │                  e        d        c        b        a
│ │ │  o6 = ideal (u - -*v, t - -*v, z - -*v, y - -*v, x - -*v)
│ │ │                  f        f        f        f        f
│ │ │  
│ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
    │ │ │
    │ │ │ 
    i4 : time SegreClass X
│ │ │ - -- used 0.997559s (cpu); 0.636987s (thread); 0s (gc)
│ │ │ + -- used 0.836973s (cpu); 0.566927s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          8
│ │ │         H
    │ │ │
    │ │ │ 
    i5 : time SegreClass lift(X,P7)
│ │ │ - -- used 0.592502s (cpu); 0.324675s (thread); 0s (gc)
│ │ │ + -- used 0.695602s (cpu); 0.401301s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o5 : -----
│ │ │          8
│ │ │         H
    │ │ │
    │ │ │ 
    i6 : time SegreClass(X,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0217502s (cpu); 0.0211891s (thread); 0s (gc)
│ │ │ + -- used 0.0369694s (cpu); 0.0251108s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          8
│ │ │         H
    │ │ │
    │ │ │ 
    i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0959594s (cpu); 0.0955959s (thread); 0s (gc)
│ │ │ + -- used 0.338649s (cpu); 0.170653s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -208,25 +208,25 @@
│ │ │
    │ │ │

    The method also accepts as input a ring map phi representing a rational map $\Phi:X\dashrightarrow Y$ between projective varieties. In this case, the method returns the push-forward to the Chow ring of the ambient projective space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it basically computes SegreClass ideal matrix phi. In the next example, we compute the Segre class of the base locus of a birational map $\mathbb{G}(1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i9 : use ZZ/100003[x_0..x_6]
│ │ │  
│ │ │ -o9     ZZ
│ │ │ - = ------[x ..x ]
│ │ │ -   100003  0   6
│ │ │ +o9 =   ZZ
│ │ │ + ------[x ..x ]
│ │ │ + 100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
    │ │ │
    │ │ │ 
    i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ - -- used 0.226996s (cpu); 0.102455s (thread); 0s (gc)
│ │ │ + -- used 0.0700194s (cpu); 0.0700204s (thread); 0s (gc)
│ │ │  
│ │ │                                                          ZZ
│ │ │                                                        ------[y ..y ]
│ │ │                                                        100003  0   9                                                ZZ              2                              2
│ │ │  o10 = map (----------------------------------------------------------------------------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y , y  - y y  - y y , y y  + y y  - y y , y y  - y y , y y  - y y  - y y , y y  - y y  - y y })
│ │ │             (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )  100003  0   6     3    0 5    1 6   3 4    1 7   4    2 7    0 9   2 5    3 5    1 8   4 5    1 9   4 8    2 9    3 9   7 8    4 9    6 9
│ │ │               5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │ @@ -238,15 +238,15 @@
│ │ │                (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )     100003  0   6
│ │ │                  5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
    │ │ │
    │ │ │ 
    i11 : time SegreClass phi
│ │ │ - -- used 0.174105s (cpu); 0.174116s (thread); 0s (gc)
│ │ │ + -- used 0.444354s (cpu); 0.300237s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │          10
│ │ │ @@ -272,30 +272,30 @@
│ │ │                   5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
    │ │ │
    │ │ │ 
    i13 : -- Segre class of B in G(1,4)
│ │ │        time SegreClass B
│ │ │ - -- used 0.514373s (cpu); 0.297297s (thread); 0s (gc)
│ │ │ + -- used 0.508673s (cpu); 0.340466s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o13 : -----
│ │ │          10
│ │ │         H
    │ │ │
    │ │ │ 
    i14 : -- Segre class of B in P^9
│ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ - -- used 1.55464s (cpu); 0.93541s (thread); 0s (gc)
│ │ │ + -- used 1.77895s (cpu); 1.10024s (thread); 0s (gc)
│ │ │  
│ │ │             9       8       7      6     5
│ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o14 : -----
│ │ │          10
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,47 +81,47 @@
│ │ │ │                 2 2                2 2                                        2
│ │ │ │  2                                                    2 2
│ │ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x
│ │ │ │  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1
│ │ │ │  6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │ │  i4 : time SegreClass X
│ │ │ │ - -- used 0.997559s (cpu); 0.636987s (thread); 0s (gc)
│ │ │ │ + -- used 0.836973s (cpu); 0.566927s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5       4      3
│ │ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o4 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i5 : time SegreClass lift(X,P7)
│ │ │ │ - -- used 0.592502s (cpu); 0.324675s (thread); 0s (gc)
│ │ │ │ + -- used 0.695602s (cpu); 0.401301s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o5 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i6 : time SegreClass(X,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0217502s (cpu); 0.0211891s (thread); 0s (gc)
│ │ │ │ + -- used 0.0369694s (cpu); 0.0251108s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5       4      3
│ │ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o6 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0959594s (cpu); 0.0955959s (thread); 0s (gc)
│ │ │ │ + -- used 0.338649s (cpu); 0.170653s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o7 : -----
│ │ │ │          8
│ │ │ │ @@ -134,22 +134,22 @@
│ │ │ │  method returns the push-forward to the Chow ring of the ambient projective
│ │ │ │  space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it
│ │ │ │  basically computes SegreClass ideal matrix phi. In the next example, we compute
│ │ │ │  the Segre class of the base locus of a birational map $\mathbb{G}
│ │ │ │  (1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.
│ │ │ │  i9 : use ZZ/100003[x_0..x_6]
│ │ │ │  
│ │ │ │ -o9     ZZ
│ │ │ │ - = ------[x ..x ]
│ │ │ │ -   100003  0   6
│ │ │ │ +o9 =   ZZ
│ │ │ │ + ------[x ..x ]
│ │ │ │ + 100003  0   6
│ │ │ │  
│ │ │ │  o9 : PolynomialRing
│ │ │ │  i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},
│ │ │ │  {x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ │ - -- used 0.226996s (cpu); 0.102455s (thread); 0s (gc)
│ │ │ │ + -- used 0.0700194s (cpu); 0.0700204s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                          ZZ
│ │ │ │                                                        ------[y ..y ]
│ │ │ │                                                        100003  0   9
│ │ │ │  ZZ              2                              2
│ │ │ │  o10 = map (--------------------------------------------------------------------
│ │ │ │  --------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y
│ │ │ │ @@ -169,15 +169,15 @@
│ │ │ │  o10 : RingMap -----------------------------------------------------------------
│ │ │ │  ----------------------------------- <-- ------[x ..x ]
│ │ │ │                (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y
│ │ │ │  - y y  + y y , y y  - y y  + y y )     100003  0   6
│ │ │ │                  5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6
│ │ │ │  1 7    0 8   2 3    1 4    0 5
│ │ │ │  i11 : time SegreClass phi
│ │ │ │ - -- used 0.174105s (cpu); 0.174116s (thread); 0s (gc)
│ │ │ │ + -- used 0.444354s (cpu); 0.300237s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6     5
│ │ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │          10
│ │ │ │ @@ -198,26 +198,26 @@
│ │ │ │  ------------------------------------
│ │ │ │                 (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y
│ │ │ │  - y y  + y y , y y  - y y  + y y )
│ │ │ │                   5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2
│ │ │ │  6    1 7    0 8   2 3    1 4    0 5
│ │ │ │  i13 : -- Segre class of B in G(1,4)
│ │ │ │        time SegreClass B
│ │ │ │ - -- used 0.514373s (cpu); 0.297297s (thread); 0s (gc)
│ │ │ │ + -- used 0.508673s (cpu); 0.340466s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6     5
│ │ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o13 : -----
│ │ │ │          10
│ │ │ │         H
│ │ │ │  i14 : -- Segre class of B in P^9
│ │ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ │ - -- used 1.55464s (cpu); 0.93541s (thread); 0s (gc)
│ │ │ │ + -- used 1.77895s (cpu); 1.10024s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             9       8       7      6     5
│ │ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o14 : -----
│ │ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_abstract__Rational__Map.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
    │ │ │
    │ │ │ 
    i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ - -- used 0.000440666s (cpu); 0.000435206s (thread); 0s (gc)
│ │ │ + -- used 0.000564505s (cpu); 0.000561703s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: given by a function
│ │ │ @@ -124,23 +124,23 @@
│ │ │
    │ │ │

    Now we compute first the degree of the forms defining the abstract map psi and then the corresponding concrete rational map.

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : time projectiveDegrees(psi,3)
│ │ │ - -- used 0.114539s (cpu); 0.1145s (thread); 0s (gc)
│ │ │ + -- used 0.148982s (cpu); 0.14899s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    │ │ │
    │ │ │ 
    i6 : time rationalMap psi
│ │ │ - -- used 0.563962s (cpu); 0.397087s (thread); 0s (gc)
│ │ │ + -- used 0.51318s (cpu); 0.409301s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: {
│ │ │ @@ -238,15 +238,15 @@
│ │ │  o13 : Ideal of -----[x ..x ]
│ │ │                 65521  0   3
    │ │ │
    │ │ │ 
    i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ - -- used 0.0371843s (cpu); 0.0371837s (thread); 0s (gc)
│ │ │ + -- used 0.197594s (cpu); 0.117672s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ @@ -258,26 +258,26 @@
│ │ │
    │ │ │

    The degree of the forms defining the abstract map T can be obtained by the following command:

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 2.08417s (cpu); 1.43192s (thread); 0s (gc)
│ │ │ + -- used 2.53895s (cpu); 1.92969s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
    │ │ │
    │ │ │

    We verify that the composition of T with itself is defined by linear forms:

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i16 : time T2 = T * T
│ │ │ - -- used 4.777e-05s (cpu); 4.7589e-05s (thread); 0s (gc)
│ │ │ + -- used 4.7821e-05s (cpu); 4.6393e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ @@ -286,15 +286,15 @@
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
    │ │ │
    │ │ │ 
    i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 3.36436s (cpu); 2.35293s (thread); 0s (gc)
│ │ │ + -- used 4.09253s (cpu); 3.06776s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
    │ │ │
    │ │ │

    We verify that the composition of T with itself leaves a random point fixed:

    │ │ │ │ │ │ @@ -327,15 +327,15 @@ │ │ │ │ │ │
    │ │ │

    We now compute the concrete rational map corresponding to T:

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i21 : time f = rationalMap T
│ │ │ - -- used 3.00105s (cpu); 2.14734s (thread); 0s (gc)
│ │ │ + -- used 3.39967s (cpu); 2.60497s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,32 +35,32 @@
│ │ │ │  i3 : P5 := QQ[u_0..u_5]
│ │ │ │  
│ │ │ │  o3 = QQ[u ..u ]
│ │ │ │           0   5
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ │ - -- used 0.000440666s (cpu); 0.000435206s (thread); 0s (gc)
│ │ │ │ + -- used 0.000564505s (cpu); 0.000561703s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = -- rational map --
│ │ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │ │                        0   1   2   3   4   5
│ │ │ │       defining forms: given by a function
│ │ │ │  
│ │ │ │  o4 : AbstractRationalMap (rational map from PP^4 to PP^5)
│ │ │ │  Now we compute first the degree of the forms defining the abstract map psi and
│ │ │ │  then the corresponding concrete rational map.
│ │ │ │  i5 : time projectiveDegrees(psi,3)
│ │ │ │ - -- used 0.114539s (cpu); 0.1145s (thread); 0s (gc)
│ │ │ │ + -- used 0.148982s (cpu); 0.14899s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time rationalMap psi
│ │ │ │ - -- used 0.563962s (cpu); 0.397087s (thread); 0s (gc)
│ │ │ │ + -- used 0.51318s (cpu); 0.409301s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = -- rational map --
│ │ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │ │                        0   1   2   3   4   5
│ │ │ │       defining forms: {
│ │ │ │ @@ -139,48 +139,48 @@
│ │ │ │  o13 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │ │                  1    0 2     1 2    0 3     2    1 3
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o13 : Ideal of -----[x ..x ]
│ │ │ │                 65521  0   3
│ │ │ │  i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ │ - -- used 0.0371843s (cpu); 0.0371837s (thread); 0s (gc)
│ │ │ │ + -- used 0.197594s (cpu); 0.117672s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │        defining forms: given by a function
│ │ │ │  
│ │ │ │  o14 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │ │  The degree of the forms defining the abstract map T can be obtained by the
│ │ │ │  following command:
│ │ │ │  i15 : time projectiveDegrees(T,2)
│ │ │ │ - -- used 2.08417s (cpu); 1.43192s (thread); 0s (gc)
│ │ │ │ + -- used 2.53895s (cpu); 1.92969s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3
│ │ │ │  We verify that the composition of T with itself is defined by linear forms:
│ │ │ │  i16 : time T2 = T * T
│ │ │ │ - -- used 4.777e-05s (cpu); 4.7589e-05s (thread); 0s (gc)
│ │ │ │ + -- used 4.7821e-05s (cpu); 4.6393e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │        defining forms: given by a function
│ │ │ │  
│ │ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │ │  i17 : time projectiveDegrees(T2,2)
│ │ │ │ - -- used 3.36436s (cpu); 2.35293s (thread); 0s (gc)
│ │ │ │ + -- used 4.09253s (cpu); 3.06776s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = 1
│ │ │ │  We verify that the composition of T with itself leaves a random point fixed:
│ │ │ │  i18 : p = apply(3,i->random(ZZ/65521))|{1}
│ │ │ │  
│ │ │ │  o18 = {-6648, -23396, -12311, 1}
│ │ │ │  
│ │ │ │ @@ -193,15 +193,15 @@
│ │ │ │  i20 : T q
│ │ │ │  
│ │ │ │  o20 = {-6648, -23396, -12311, 1}
│ │ │ │  
│ │ │ │  o20 : List
│ │ │ │  We now compute the concrete rational map corresponding to T:
│ │ │ │  i21 : time f = rationalMap T
│ │ │ │ - -- used 3.00105s (cpu); 2.14734s (thread); 0s (gc)
│ │ │ │ + -- used 3.39967s (cpu); 2.60497s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_approximate__Inverse__Map.html
│ │ │ @@ -144,15 +144,15 @@
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │  -- approximateInverseMap: step 8 of 10
│ │ │  -- approximateInverseMap: step 9 of 10
│ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ - -- used 0.352424s (cpu); 0.232237s (thread); 0s (gc)
│ │ │ + -- used 0.444317s (cpu); 0.28658s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                    ZZ
│ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │                                  ZZ
│ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -205,15 +205,15 @@
│ │ │
    │ │ │ 
    i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 0.324659s (cpu); 0.181955s (thread); 0s (gc)
│ │ │ + -- used 0.419739s (cpu); 0.261883s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
    │ │ │
    │ │ │ 
    i6 : assert(psi == psi')
    │ │ │ @@ -300,15 +300,15 @@ │ │ │
    │ │ │ 
    i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 2.52647s (cpu); 1.81675s (thread); 0s (gc)
│ │ │ + -- used 2.37135s (cpu); 2.03451s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │               {
│ │ │                                  2
│ │ │ @@ -372,15 +372,15 @@
│ │ │
    		│ │ │ 
    i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │ - -- used 3.65149s (cpu); 2.6771s (thread); 0s (gc)
│ │ │ + -- used 3.45379s (cpu); 2.97496s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = -- rational map --
│ │ │                                   ZZ
│ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │                {
│ │ │                                   2
│ │ │ ├── html2text {}
│ │ │ │ @@ -135,15 +135,15 @@
│ │ │ │  -- approximateInverseMap: step 4 of 10
│ │ │ │  -- approximateInverseMap: step 5 of 10
│ │ │ │  -- approximateInverseMap: step 6 of 10
│ │ │ │  -- approximateInverseMap: step 7 of 10
│ │ │ │  -- approximateInverseMap: step 8 of 10
│ │ │ │  -- approximateInverseMap: step 9 of 10
│ │ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ │ - -- used 0.352424s (cpu); 0.232237s (thread); 0s (gc)
│ │ │ │ + -- used 0.444317s (cpu); 0.28658s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                    ZZ
│ │ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │ │                                  ZZ
│ │ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -252,15 +252,15 @@
│ │ │ │  
│ │ │ │  o3 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i4 : assert(phi * psi == 1 and psi * phi == 1)
│ │ │ │  i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │ - -- used 0.324659s (cpu); 0.181955s (thread); 0s (gc)
│ │ │ │ + -- used 0.419739s (cpu); 0.261883s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i6 : assert(psi == psi')
│ │ │ │  A more complicated example is the following (here _i_n_v_e_r_s_e_M_a_p takes a lot of
│ │ │ │  time!).
│ │ │ │  i7 : phi = rationalMap map(P8,ZZ/97[x_0..x_11]/ideal(x_1*x_3-8*x_2*x_3+25*x_3^2-25*x_2*x_4-
│ │ │ │  22*x_3*x_4+x_0*x_5+13*x_2*x_5+41*x_3*x_5-x_0*x_6+12*x_2*x_6+25*x_1*x_7+25*x_3*x_7+23*x_5*x_7-
│ │ │ │ @@ -418,15 +418,15 @@
│ │ │ │  
│ │ │ │  o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11)
│ │ │ │  i8 : -- without the option 'CodimBsInv=>4', it takes about triple time
│ │ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │ - -- used 2.52647s (cpu); 1.81675s (thread); 0s (gc)
│ │ │ │ + -- used 2.37135s (cpu); 2.03451s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = -- rational map --
│ │ │ │                                  ZZ
│ │ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │ │               {
│ │ │ │                                  2
│ │ │ │ @@ -526,15 +526,15 @@
│ │ │ │  o9 = false
│ │ │ │  i10 : -- in this case we can remedy enabling the option Certify
│ │ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 3.65149s (cpu); 2.6771s (thread); 0s (gc)
│ │ │ │ + -- used 3.45379s (cpu); 2.97496s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = -- rational map --
│ │ │ │                                   ZZ
│ │ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │ │                {
│ │ │ │                                   2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_degree__Map.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
    │ │ │
    │ │ │ 
    i5 : time degreeMap phi
│ │ │ - -- used 0.0464011s (cpu); 0.0464049s (thread); 0s (gc)
│ │ │ + -- used 0.0576005s (cpu); 0.057604s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
    │ │ │
    │ │ │ 
    i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 
│ │ │ @@ -118,15 +118,15 @@
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
    │ │ │
    │ │ │ 
    i7 : time degreeMap phi'
│ │ │ - -- used 1.36727s (cpu); 0.743088s (thread); 0s (gc)
│ │ │ + -- used 1.66703s (cpu); 0.873296s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 14
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -266,15 +266,15 @@ │ │ │ │ 4 0 5 1 5 2 5 3 5 4 5 5 0 6 │ │ │ │ 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 │ │ │ │ 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 │ │ │ │ 8 3 8 4 8 5 8 6 8 7 8 8 │ │ │ │ │ │ │ │ o4 : RingMap ringP8 <-- ringP14 │ │ │ │ i5 : time degreeMap phi │ │ │ │ - -- used 0.0464011s (cpu); 0.0464049s (thread); 0s (gc) │ │ │ │ + -- used 0.0576005s (cpu); 0.057604s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 1 │ │ │ │ i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a │ │ │ │ general subspace of P^14 │ │ │ │ -- of dimension 5 (so that the composition phi':P^8--->P^8 must have │ │ │ │ degree equal to deg(G(1,5))=14) │ │ │ │ phi'=phi*map(ringP14,ringP8,for i to 8 list random(1,ringP14)) │ │ │ │ @@ -418,15 +418,15 @@ │ │ │ │ 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 │ │ │ │ 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 │ │ │ │ 7 4 7 5 7 6 7 7 0 8 1 8 2 8 │ │ │ │ 3 8 4 8 5 8 6 8 7 8 8 │ │ │ │ │ │ │ │ o6 : RingMap ringP8 <-- ringP8 │ │ │ │ i7 : time degreeMap phi' │ │ │ │ - -- used 1.36727s (cpu); 0.743088s (thread); 0s (gc) │ │ │ │ + -- used 1.66703s (cpu); 0.873296s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = 14 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_g_r_e_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map │ │ │ │ * _p_r_o_j_e_c_t_i_v_e_D_e_g_r_e_e_s -- projective degrees of a rational map between │ │ │ │ projective varieties │ │ │ │ ********** WWaayyss ttoo uussee ddeeggrreeeeMMaapp:: ********** │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_force__Image.html │ │ │ @@ -88,15 +88,15 @@ │ │ │ │ │ │ o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ - -- used 0.000639339s (cpu); 0.000632025s (thread); 0s (gc)
    │ │ │ + -- used 0.000851599s (cpu); 0.000847133s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : Phi;
│ │ │  
│ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional subvariety of PP^9)
    │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ │ │ │ │ o2 : Ideal of P6 │ │ │ │ i3 : Phi = rationalMap(X,Dominant=>2); │ │ │ │ │ │ │ │ o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of │ │ │ │ PP^9) │ │ │ │ i4 : time forceImage(Phi,ideal 0_(target Phi)) │ │ │ │ - -- used 0.000639339s (cpu); 0.000632025s (thread); 0s (gc) │ │ │ │ + -- used 0.000851599s (cpu); 0.000847133s (thread); 0s (gc) │ │ │ │ i5 : Phi; │ │ │ │ │ │ │ │ o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional │ │ │ │ subvariety of PP^9) │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ If the declaration is false, nonsensical answers may result. │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_graph.html │ │ │ @@ -118,15 +118,15 @@ │ │ │ │ │ │ o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time (p1,p2) = graph phi;
│ │ │ - -- used 0.0150736s (cpu); 0.0147825s (thread); 0s (gc)
    │ │ │ + -- used 0.0307073s (cpu); 0.0194373s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : p1
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │ @@ -277,15 +277,15 @@
│ │ │            
│ │ │          
│ │ │          
    When the source of the rational map is a multi-projective variety, the method returns all the projections.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -98,15 +98,15 @@
│ │ │ │  
│ │ │ │                        w w  - w w  + w w
│ │ │ │                         2 4    1 5    0 6
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
│ │ │ │  i2 : time psi = inverseMap phi
│ │ │ │ - -- used 0.0785s (cpu); 0.0785012s (thread); 0s (gc)
│ │ │ │ + -- used 0.0957506s (cpu); 0.0957508s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = -- rational map --
│ │ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w
│ │ │ │  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13
│ │ │ │  14   15   16   17   18   19   20
│ │ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w
│ │ │ │ @@ -216,15 +216,15 @@
│ │ │ │  15    9 20    8 22   3 10    0 13    4 15    9 21    8 23   2 10    0 12    4
│ │ │ │  20    6 21    8 24   1 10    0 11    4 22    6 23    9 24   4 5    3 6    0 7
│ │ │ │  1 8    2 9
│ │ │ │  
│ │ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │ │                   0   26          0   26
│ │ │ │  i5 : time psi = inverseMap phi
│ │ │ │ - -- used 0.288262s (cpu); 0.189505s (thread); 0s (gc)
│ │ │ │ + -- used 0.295668s (cpu); 0.200371s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = map (QQ[w ..w  ], QQ[w ..w  ], {- w w   + w w   + w  w   - w  w   - w w  ,
│ │ │ │  - w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   -
│ │ │ │  w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   -
│ │ │ │  w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   +
│ │ │ │  w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , - w  w   +
│ │ │ │  w  w   - w  w   + w  w   - w  w  , - w  w   + w  w   - w  w   + w  w   - w  w
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse_lp__Rational__Map_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : time g = graph p2;
│ │ │ - -- used 0.0299231s (cpu); 0.0296181s (thread); 0s (gc)
    
│ │ │ + -- used 0.0477313s (cpu); 0.0360069s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i10 : g_0;
│ │ │  
│ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to PP^4)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,15 +50,15 @@
│ │ │ │                        - x  + x x
│ │ │ │                           3    2 4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in
│ │ │ │  PP^5)
│ │ │ │  i3 : time (p1,p2) = graph phi;
│ │ │ │ - -- used 0.0150736s (cpu); 0.0147825s (thread); 0s (gc)
│ │ │ │ + -- used 0.0307073s (cpu); 0.0194373s (thread); 0s (gc)
│ │ │ │  i4 : p1
│ │ │ │  
│ │ │ │  o4 = -- rational map --
│ │ │ │                                    ZZ                                 ZZ
│ │ │ │       source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y
│ │ │ │  , y , y , y , y ]) defined by
│ │ │ │                                  190181  0   1   2   3   4          190181  0
│ │ │ │ @@ -192,15 +192,15 @@
│ │ │ │  
│ │ │ │  o8 = {51, 28, 14, 6, 2}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  When the source of the rational map is a multi-projective variety, the method
│ │ │ │  returns all the projections.
│ │ │ │  i9 : time g = graph p2;
│ │ │ │ - -- used 0.0299231s (cpu); 0.0296181s (thread); 0s (gc)
│ │ │ │ + -- used 0.0477313s (cpu); 0.0360069s (thread); 0s (gc)
│ │ │ │  i10 : g_0;
│ │ │ │  
│ │ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety
│ │ │ │  of PP^4 x PP^5 x PP^5 to PP^4)
│ │ │ │  i11 : g_1;
│ │ │ │  
│ │ │ │  o11 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_ideal_lp__Rational__Map_rp.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │              
    
│ │ │                
    i3 : time ideal phi
│ │ │ - -- used 0.0031265s (cpu); 0.00312386s (thread); 0s (gc)
│ │ │ + -- used 0.00430458s (cpu); 0.00430416s (thread); 0s (gc)
│ │ │  
│ │ │               2                                     2                      
│ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3  
│ │ │       ------------------------------------------------------------------------
│ │ │              2
│ │ │       x x , x  - x x )
│ │ │ @@ -200,15 +200,15 @@
│ │ │  
│ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
    
│ │ │              
    
│ │ │                
    i6 : time ideal phi'
│ │ │ - -- used 0.0921941s (cpu); 0.0921971s (thread); 0s (gc)
│ │ │ + -- used 0.125207s (cpu); 0.125209s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal 1
│ │ │  
│ │ │                                                                                                              QQ[x ..x , y ..y ]
│ │ │                                                                                                                  0   5   0   4
│ │ │  o6 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │                                                                                                                                                                                                       2
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │                         2
│ │ │ │                        x  - x x
│ │ │ │                         1    0 3
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │ │  i3 : time ideal phi
│ │ │ │ - -- used 0.0031265s (cpu); 0.00312386s (thread); 0s (gc)
│ │ │ │ + -- used 0.00430458s (cpu); 0.00430416s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2                                     2
│ │ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │              2
│ │ │ │       x x , x  - x x )
│ │ │ │ @@ -121,15 +121,15 @@
│ │ │ │                        y
│ │ │ │                         4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of
│ │ │ │  PP^5 x PP^4 to PP^4)
│ │ │ │  i6 : time ideal phi'
│ │ │ │ - -- used 0.0921941s (cpu); 0.0921971s (thread); 0s (gc)
│ │ │ │ + -- used 0.125207s (cpu); 0.125209s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = ideal 1
│ │ │ │  
│ │ │ │  
│ │ │ │  QQ[x ..x , y ..y ]
│ │ │ │  
│ │ │ │  0   5   0   4
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse__Map.html
│ │ │ @@ -158,15 +158,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
    
│ │ │              
    
│ │ │                
    i2 : time psi = inverseMap phi
│ │ │ - -- used 0.0785s (cpu); 0.0785012s (thread); 0s (gc)
│ │ │ + -- used 0.0957506s (cpu); 0.0957508s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = -- rational map --
│ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       defining forms: {
│ │ │ @@ -256,15 +256,15 @@
│ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
    
│ │ │              
    
│ │ │                
    i5 : time psi = inverseMap phi
│ │ │ - -- used 0.288262s (cpu); 0.189505s (thread); 0s (gc)
│ │ │ + -- used 0.295668s (cpu); 0.200371s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = map (QQ[w ..w  ], QQ[w ..w  ], {- w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , - w  w   + w  w   - w  w   + w  w   - w  w  , - w  w   + w  w   - w  w   + w  w   - w  w  , w w   - w w   + w w   + w  w   - w  w  , - w w   + w w   + w  w   + w w   - w w  , - w w   + w w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , w  w   - w  w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w  - w w  - w w   + w w   - w w  , - w w  + w w  + w w   - w w   + w w  , w w  - w w  - w w  + w w   - w w  })
│ │ │                0   26       0   26       5 22    8 23    14 24    13 25    0 26     5 18    8 19    14 20    10 25    1 26     5 16    8 17    13 20    10 24    2 26     5 15    14 17    13 19    10 23    3 26     5 21    20 23    19 24    17 25    4 26     8 15    14 16    13 18    10 22    6 26     8 21    20 22    18 24    16 25    7 26   17 18    16 19    15 20    10 21    9 26     13 21    17 22    16 23    15 24    11 26     14 21    19 22    18 23    15 25    12 26   0 21    4 22    7 23    12 24    11 25     4 18    7 19    12 20    1 21    9 25     4 16    7 17    11 20    2 21    9 24     4 15    12 17    11 19    3 21    9 23     7 15    12 16    11 18    6 21    9 22   12 13    11 14    0 15    3 22    6 23   10 12    9 14    1 15    3 18    6 19   10 11    9 13    2 15    3 16    6 17   8 9    7 10    1 16    2 18    6 20   5 9    4 10    1 17    2 19    3 20   8 11    7 13    0 16    2 22    6 24   5 11    4 13    0 17    2 23    3 24   8 12    7 14    0 18    1 22    6 25   5 12    4 14    0 19    1 23    3 25   5 7    4 8    0 20    1 24    2 25     5 6    3 8    0 10    1 13    2 14   4 6    3 7    0 9    1 11    2 12
│ │ │  
│ │ │  o5 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
    
│ │ │              
    
│ │ │                
    i3 : time inverse phi
│ │ │ - -- used 0.0550698s (cpu); 0.0550696s (thread); 0s (gc)
│ │ │ + -- used 0.0703872s (cpu); 0.0703871s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       defining forms: {
│ │ │ ├── html2text {}
│ │ │ │ @@ -290,15 +290,15 @@
│ │ │ │  58320000  1 4    190512000  0 2 4    4898880000 1 2 4    190512000 2 4
│ │ │ │  476280000  0 3 4    204120000  1 3 4    2857680000  2 3 4    23814000  3 4
│ │ │ │  30618000 0 4    46656 1 4   12757500 2 4    51030000  3 4   30375 4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │ │  i3 : time inverse phi
│ │ │ │ - -- used 0.0550698s (cpu); 0.0550696s (thread); 0s (gc)
│ │ │ │ + -- used 0.0703872s (cpu); 0.0703871s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       defining forms: {
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Birational.html
│ │ │ @@ -128,24 +128,24 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
    
│ │ │              
    
│ │ │                
    i3 : time isBirational phi
│ │ │ - -- used 0.0199568s (cpu); 0.0199548s (thread); 0s (gc)
│ │ │ + -- used 0.0253858s (cpu); 0.0253857s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
    
│ │ │                
    i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0160523s (cpu); 0.0154605s (thread); 0s (gc)
│ │ │ + -- used 0.037693s (cpu); 0.0177218s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -58,20 +58,20 @@ │ │ │ │ - t + t t │ │ │ │ 3 2 4 │ │ │ │ } │ │ │ │ │ │ │ │ o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in │ │ │ │ PP^5) │ │ │ │ i3 : time isBirational phi │ │ │ │ - -- used 0.0199568s (cpu); 0.0199548s (thread); 0s (gc) │ │ │ │ + -- used 0.0253858s (cpu); 0.0253857s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = true │ │ │ │ i4 : time isBirational(phi,Certify=>true) │ │ │ │ Certify: output certified! │ │ │ │ - -- used 0.0160523s (cpu); 0.0154605s (thread); 0s (gc) │ │ │ │ + -- used 0.037693s (cpu); 0.0177218s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _i_s_D_o_m_i_n_a_n_t -- whether a rational map is dominant │ │ │ │ ********** WWaayyss ttoo uussee iissBBiirraattiioonnaall:: ********** │ │ │ │ * isBirational(RationalMap) │ │ │ │ * isBirational(RingMap) │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Dominant.html │ │ │ @@ -91,15 +91,15 @@ │ │ │ o2 : RationalMap (rational map from PP^8 to PP^8) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 2.86189s (cpu); 2.08567s (thread); 0s (gc)
│ │ │ + -- used 2.69805s (cpu); 2.32235s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : P7 = ZZ/101[x_0..x_7];
    │ │ │ @@ -120,15 +120,15 @@ │ │ │ o6 : RationalMap (cubic rational map from PP^7 to PP^7) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 3.63761s (cpu); 2.40964s (thread); 0s (gc)
│ │ │ + -- used 4.44899s (cpu); 3.08833s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = false
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -20,15 +20,15 @@ │ │ │ │ i1 : P8 = ZZ/101[x_0..x_8]; │ │ │ │ i2 : phi = rationalMap ideal jacobian ideal det matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7}, │ │ │ │ {x_4..x_8}}; │ │ │ │ │ │ │ │ o2 : RationalMap (rational map from PP^8 to PP^8) │ │ │ │ i3 : time isDominant(phi,Certify=>true) │ │ │ │ Certify: output certified! │ │ │ │ - -- used 2.86189s (cpu); 2.08567s (thread); 0s (gc) │ │ │ │ + -- used 2.69805s (cpu); 2.32235s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = true │ │ │ │ i4 : P7 = ZZ/101[x_0..x_7]; │ │ │ │ i5 : -- hyperelliptic curve of genus 3 │ │ │ │ C = ideal(x_4*x_5+23*x_5^2-23*x_0*x_6-18*x_1*x_6+6*x_2*x_6+37*x_3*x_6+23*x_4*x_6- │ │ │ │ 26*x_5*x_6+2*x_6^2-25*x_0*x_7+45*x_1*x_7+30*x_2*x_7-49*x_3*x_7-49*x_4*x_7+50*x_5*x_7,x_3*x_5- │ │ │ │ 24*x_5^2+21*x_0*x_6+x_1*x_6+46*x_3*x_6+27*x_4*x_6+5*x_5*x_6+35*x_6^2+20*x_0*x_7- │ │ │ │ @@ -65,15 +65,15 @@ │ │ │ │ │ │ │ │ o5 : Ideal of P7 │ │ │ │ i6 : phi = rationalMap(C,3,2); │ │ │ │ │ │ │ │ o6 : RationalMap (cubic rational map from PP^7 to PP^7) │ │ │ │ i7 : time isDominant(phi,Certify=>true) │ │ │ │ Certify: output certified! │ │ │ │ - -- used 3.63761s (cpu); 2.40964s (thread); 0s (gc) │ │ │ │ + -- used 4.44899s (cpu); 3.08833s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = false │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _i_s_B_i_r_a_t_i_o_n_a_l -- whether a rational map is birational │ │ │ │ ********** WWaayyss ttoo uussee iissDDoommiinnaanntt:: ********** │ │ │ │ * isDominant(RationalMap) │ │ │ │ * isDominant(RingMap) │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_kernel_lp__Ring__Map_cm__Z__Z_rp.html │ │ │ @@ -95,26 +95,26 @@ │ │ │ o1 : RingMap QQ[x ..x ] <-- QQ[y ..y ] │ │ │ 0 8 0 11 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time kernel(phi,1)
│ │ │ - -- used 0.0177116s (cpu); 0.0177066s (thread); 0s (gc)
│ │ │ + -- used 0.0223929s (cpu); 0.0223912s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time kernel(phi,2)
│ │ │ - -- used 1.02093s (cpu); 0.491366s (thread); 0s (gc)
│ │ │ + -- used 1.13551s (cpu); 0.545581s (thread); 0s (gc)
│ │ │  
│ │ │                             2                                                
│ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6  
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                             
│ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ │ ├── html2text {}
│ │ │ │ @@ -69,22 +69,22 @@
│ │ │ │  4 8     5 8     6 8     7 8      0 1    1 2    1 4     0 6    1 6    4 6    0 7
│ │ │ │  0 2    1 2     0 4    1 4      1 5    2 5     4 5     0 6     1 6     4 6     2
│ │ │ │  7     0 8     1 8     5 8     6 8     7 8
│ │ │ │  
│ │ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │ │                   0   8          0   11
│ │ │ │  i2 : time kernel(phi,1)
│ │ │ │ - -- used 0.0177116s (cpu); 0.0177066s (thread); 0s (gc)
│ │ │ │ + -- used 0.0223929s (cpu); 0.0223912s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = ideal ()
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │ │                    0   11
│ │ │ │  i3 : time kernel(phi,2)
│ │ │ │ - -- used 1.02093s (cpu); 0.491366s (thread); 0s (gc)
│ │ │ │ + -- used 1.13551s (cpu); 0.545581s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                             2
│ │ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │  
│ │ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_parametrize_lp__Ideal_rp.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │  o2 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time parametrize L
│ │ │ - -- used 0.00434735s (cpu); 0.00434206s (thread); 0s (gc)
│ │ │ + -- used 0.0070726s (cpu); 0.00707334s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -206,15 +206,15 @@
│ │ │  o4 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time parametrize Q
│ │ │ - -- used 0.749084s (cpu); 0.436364s (thread); 0s (gc)
│ │ │ + -- used 0.571376s (cpu); 0.437581s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │       - 849671x  + 3034137x )
│ │ │ │                8           9
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o2 : Ideal of --------[x ..x ]
│ │ │ │                10000019  0   9
│ │ │ │  i3 : time parametrize L
│ │ │ │ - -- used 0.00434735s (cpu); 0.00434206s (thread); 0s (gc)
│ │ │ │ + -- used 0.0070726s (cpu); 0.00707334s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │ │                    10000019  0   1   2   3   4   5
│ │ │ │                       ZZ
│ │ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -136,15 +136,15 @@
│ │ │ │       1211601x x  - 2168594x x  - 1801762x x  + 3022242x x  + 3618789x )
│ │ │ │               5 9           6 9           7 9           8 9           9
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o4 : Ideal of --------[x ..x ]
│ │ │ │                10000019  0   9
│ │ │ │  i5 : time parametrize Q
│ │ │ │ - -- used 0.749084s (cpu); 0.436364s (thread); 0s (gc)
│ │ │ │ + -- used 0.571376s (cpu); 0.437581s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │ │                    10000019  0   1   2   3   4   5   6
│ │ │ │                       ZZ
│ │ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_point_lp__Quotient__Ring_rp.html
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time p = point source f
│ │ │ - -- used 0.239078s (cpu); 0.151432s (thread); 0s (gc)
│ │ │ + -- used 0.299988s (cpu); 0.205009s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal (y   - 9235y  , y  + 11075y  , y  - 5847y  , y  + 7396y  , y  +
│ │ │               10        11   9         11   8        11   7        11   6  
│ │ │       ------------------------------------------------------------------------
│ │ │       13530y  , y  + 4359y  , y  - 2924y  , y  + 13040y  , y  + 6904y  , y  -
│ │ │             11   5        11   4        11   3         11   2        11   1  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -105,15 +105,15 @@
│ │ │                (y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y , y y  - y y  + y y )
│ │ │                  6 7    5 8    4 11   3 7    2 8    1 11   3 5    2 6    0 11   3 4    1 6    0 8   2 4    1 5    0 7
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time p == f^* f p
│ │ │ - -- used 0.0935425s (cpu); 0.0935509s (thread); 0s (gc)
│ │ │ + -- used 0.137661s (cpu); 0.137665s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ documentation) , see _p_o_i_n_t_(_M_u_l_t_i_p_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_). │ │ │ │ Below we verify the birationality of a rational map. │ │ │ │ i1 : f = inverseMap specialQuadraticTransformation(9,ZZ/33331); │ │ │ │ │ │ │ │ o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to │ │ │ │ PP^8) │ │ │ │ i2 : time p = point source f │ │ │ │ - -- used 0.239078s (cpu); 0.151432s (thread); 0s (gc) │ │ │ │ + -- used 0.299988s (cpu); 0.205009s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = ideal (y - 9235y , y + 11075y , y - 5847y , y + 7396y , y + │ │ │ │ 10 11 9 11 8 11 7 11 6 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 13530y , y + 4359y , y - 2924y , y + 13040y , y + 6904y , y - │ │ │ │ 11 5 11 4 11 3 11 2 11 1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ @@ -40,15 +40,15 @@ │ │ │ │ o2 : Ideal of ----------------------------------------------------------------- │ │ │ │ -------------------------------------- │ │ │ │ (y y - y y + y y , y y - y y + y y , y y - y y + y y , y │ │ │ │ y - y y + y y , y y - y y + y y ) │ │ │ │ 6 7 5 8 4 11 3 7 2 8 1 11 3 5 2 6 0 11 │ │ │ │ 3 4 1 6 0 8 2 4 1 5 0 7 │ │ │ │ i3 : time p == f^* f p │ │ │ │ - -- used 0.0935425s (cpu); 0.0935509s (thread); 0s (gc) │ │ │ │ + -- used 0.137661s (cpu); 0.137665s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_n_d_o_m_K_R_a_t_i_o_n_a_l_P_o_i_n_t -- pick a random K rational point on the scheme X │ │ │ │ defined by I │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * point(PolynomialRing) │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_projective__Degrees.html │ │ │ @@ -94,15 +94,15 @@ │ │ │ 0 4 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.120186s (cpu); 0.0404089s (thread); 0s (gc)
│ │ │ + -- used 0.212012s (cpu); 0.0595769s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -122,15 +122,15 @@ │ │ │ 2 3 1 4 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.122567s (cpu); 0.0537507s (thread); 0s (gc)
│ │ │ + -- used 0.031337s (cpu); 0.0142572s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -148,25 +148,25 @@ │ │ │ o6 : RingMap ------[x ..x ] <-- ------[x ..x ] │ │ │ 300007 0 6 300007 0 6 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time projectiveDegrees phi
│ │ │ - -- used 4.6717e-05s (cpu); 4.0846e-05s (thread); 0s (gc)
│ │ │ + -- used 5.1786e-05s (cpu); 4.5854e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.155e-05s (cpu); 2.138e-05s (thread); 0s (gc)
│ │ │ + -- used 2.5654e-05s (cpu); 2.5388e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = {4, 1}
│ │ │  
│ │ │  o8 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -53,15 +53,15 @@ │ │ │ │ 0 4 0 5 1 0 2 1 2 0 3 │ │ │ │ 2 1 3 1 3 0 4 2 3 1 4 3 2 4 │ │ │ │ │ │ │ │ o2 : RingMap GF 109561[t ..t ] <-- GF 109561[x ..x ] │ │ │ │ 0 4 0 5 │ │ │ │ i3 : time projectiveDegrees(phi,Certify=>true) │ │ │ │ Certify: output certified! │ │ │ │ - -- used 0.120186s (cpu); 0.0404089s (thread); 0s (gc) │ │ │ │ + -- used 0.212012s (cpu); 0.0595769s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = {1, 2, 4, 4, 2} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : psi=inverseMap(toMap(phi,Dominant=>infinity)) │ │ │ │ │ │ │ │ GF 109561[x ..x ] │ │ │ │ @@ -76,15 +76,15 @@ │ │ │ │ GF 109561[x ..x ] │ │ │ │ 0 5 │ │ │ │ o4 : RingMap ------------------ <-- GF 109561[t ..t ] │ │ │ │ x x - x x + x x 0 4 │ │ │ │ 2 3 1 4 0 5 │ │ │ │ i5 : time projectiveDegrees(psi,Certify=>true) │ │ │ │ Certify: output certified! │ │ │ │ - -- used 0.122567s (cpu); 0.0537507s (thread); 0s (gc) │ │ │ │ + -- used 0.031337s (cpu); 0.0142572s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = {2, 4, 4, 2, 1} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : -- Cremona transformation of P^6 defined by the quadrics through a │ │ │ │ rational octic surface │ │ │ │ phi = map specialCremonaTransformation(7,ZZ/300007) │ │ │ │ @@ -119,21 +119,21 @@ │ │ │ │ 4 5 5 0 6 1 6 2 6 3 6 4 6 │ │ │ │ 5 6 │ │ │ │ │ │ │ │ ZZ ZZ │ │ │ │ o6 : RingMap ------[x ..x ] <-- ------[x ..x ] │ │ │ │ 300007 0 6 300007 0 6 │ │ │ │ i7 : time projectiveDegrees phi │ │ │ │ - -- used 4.6717e-05s (cpu); 4.0846e-05s (thread); 0s (gc) │ │ │ │ + -- used 5.1786e-05s (cpu); 4.5854e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = {1, 2, 4, 8, 8, 4, 1} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : time projectiveDegrees(phi,NumDegrees=>1) │ │ │ │ - -- used 2.155e-05s (cpu); 2.138e-05s (thread); 0s (gc) │ │ │ │ + -- used 2.5654e-05s (cpu); 2.5388e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 = {4, 1} │ │ │ │ │ │ │ │ o8 : List │ │ │ │ Another way to use this method is by passing an integer i as second argument. │ │ │ │ However, this is equivalent to first projectiveDegrees(phi,NumDegrees=>i) and │ │ │ │ generally it is not faster. │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.html │ │ │ @@ -93,15 +93,15 @@ │ │ │ o2 : Ideal of -----[x ..x ] │ │ │ 33331 0 6 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time phi = rationalMap(V,3,2)
│ │ │ - -- used 0.0982366s (cpu); 0.0982396s (thread); 0s (gc)
│ │ │ + -- used 0.115831s (cpu); 0.115829s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                      ZZ
│ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │                    33331  0   1   2   3   4   5   6
│ │ │                      ZZ
│ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  , y  ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  i1 : ZZ/33331[x_0..x_6]; V = ideal(x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-
│ │ │ │  x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3,x_6);
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o2 : Ideal of -----[x ..x ]
│ │ │ │                33331  0   6
│ │ │ │  i3 : time phi = rationalMap(V,3,2)
│ │ │ │ - -- used 0.0982366s (cpu); 0.0982396s (thread); 0s (gc)
│ │ │ │ + -- used 0.115831s (cpu); 0.115829s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                      ZZ
│ │ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │ │                    33331  0   1   2   3   4   5   6
│ │ │ │                      ZZ
│ │ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  ,
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ring_cm__Tally_rp.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │              
│ │ │                
    i5 : D = new Tally from {H => 2,C => 1};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time phi = rationalMap D
│ │ │ - -- used 0.0303865s (cpu); 0.0303844s (thread); 0s (gc)
│ │ │ + -- used 0.035184s (cpu); 0.0351831s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │                                    ZZ
│ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │                                  65521  0   1   2   3   4   5
│ │ │               {
│ │ │                   2                  2
│ │ │ @@ -224,15 +224,15 @@
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time ? image(phi,"F4")
│ │ │ - -- used 1.21845s (cpu); 0.768199s (thread); 0s (gc)
│ │ │ + -- used 1.55766s (cpu); 0.717511s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │       hypersurfaces of degree 2
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    See also the package WeilDivisors, which provides general tools for working with divisors.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  
│ │ │ │  o4 = ideal(- 32646x  - 28377x  + 26433x  - 29566x  + 3783x  + 26696x )
│ │ │ │                     0         1         2         3        4         5
│ │ │ │  
│ │ │ │  o4 : Ideal of X
│ │ │ │  i5 : D = new Tally from {H => 2,C => 1};
│ │ │ │  i6 : time phi = rationalMap D
│ │ │ │ - -- used 0.0303865s (cpu); 0.0303844s (thread); 0s (gc)
│ │ │ │ + -- used 0.035184s (cpu); 0.0351831s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = -- rational map --
│ │ │ │                                    ZZ
│ │ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │ │                                  65521  0   1   2   3   4   5
│ │ │ │               {
│ │ │ │                   2                  2
│ │ │ │ @@ -169,15 +169,15 @@
│ │ │ │                                                    2                         2
│ │ │ │                        x x x  + x x x  + x x x  + x x  + x x x  - 2x x x  + x x
│ │ │ │                         0 1 5    0 2 5    1 2 5    2 5    1 4 5     2 4 5    4 5
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
│ │ │ │  i7 : time ? image(phi,"F4")
│ │ │ │ - -- used 1.21845s (cpu); 0.768199s (thread); 0s (gc)
│ │ │ │ + -- used 1.55766s (cpu); 0.717511s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │ │       hypersurfaces of degree 2
│ │ │ │  See also the package _W_e_i_l_D_i_v_i_s_o_r_s, which provides general tools for working
│ │ │ │  with divisors.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_t_i_o_n_a_l_M_a_p -- makes a rational map
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cremona__Transformation.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    A Cremona transformation is said to be special if the base locus scheme is smooth and irreducible. To ensure this condition, the field K must be large enough but no check is made.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ - -- used 1.28804s (cpu); 1.00797s (thread); 0s (gc)
│ │ │ + -- used 1.23352s (cpu); 1.12677s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │        source variety: PP^3                      
│ │ │        target variety: PP^3                      
│ │ │        dominance: true                           
│ │ │        birationality: true                       
│ │ │        projective degrees: {1, 3, 3, 1}
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │              K, according to the classification given in Table 1 of _S_p_e_c_i_a_l
│ │ │ │              _c_u_b_i_c_ _C_r_e_m_o_n_a_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _o_f_ _P_6_ _a_n_d_ _P_7.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  A Cremona transformation is said to be special if the base locus scheme is
│ │ │ │  smooth and irreducible. To ensure this condition, the field K must be large
│ │ │ │  enough but no check is made.
│ │ │ │  i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ │ - -- used 1.28804s (cpu); 1.00797s (thread); 0s (gc)
│ │ │ │ + -- used 1.23352s (cpu); 1.12677s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │ │        source variety: PP^3
│ │ │ │        target variety: PP^3
│ │ │ │        dominance: true
│ │ │ │        birationality: true
│ │ │ │        projective degrees: {1, 3, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cubic__Transformation.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time specialCubicTransformation 9
│ │ │ - -- used 0.0929002s (cpu); 0.0929013s (thread); 0s (gc)
│ │ │ + -- used 0.0966357s (cpu); 0.0966331s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6
│ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -143,15 +143,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.0192656s (cpu); 0.0192647s (thread); 0s (gc)
│ │ │ + -- used 0.0206427s (cpu); 0.020643s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 3
│ │ │       source variety: PP^6
│ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            o a _r_a_t_i_o_n_a_l_ _m_a_p, an example of special cubic birational
│ │ │ │              transformation over K, according to the classification given in
│ │ │ │              Table 2 of _S_p_e_c_i_a_l_ _c_u_b_i_c_ _b_i_r_a_t_i_o_n_a_l_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _o_f_ _p_r_o_j_e_c_t_i_v_e
│ │ │ │              _s_p_a_c_e_s.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The field K is required to be large enough.
│ │ │ │  i1 : time specialCubicTransformation 9
│ │ │ │ - -- used 0.0929002s (cpu); 0.0929013s (thread); 0s (gc)
│ │ │ │ + -- used 0.0966357s (cpu); 0.0966331s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4   5   6
│ │ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ])
│ │ │ │  defined by
│ │ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │ │ @@ -323,15 +323,15 @@
│ │ │ │  6     4 6      0 5 6      1 5 6     2 5 6      3 5 6      4 5 6     5 6     0 6
│ │ │ │  1 6     2 6      3 6     4 6     5 6
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of
│ │ │ │  PP^9)
│ │ │ │  i2 : time describe oo
│ │ │ │ - -- used 0.0192656s (cpu); 0.0192647s (thread); 0s (gc)
│ │ │ │ + -- used 0.0206427s (cpu); 0.020643s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = rational map defined by forms of degree 3
│ │ │ │       source variety: PP^6
│ │ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │ │       dominance: true
│ │ │ │       birationality: true
│ │ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Quadratic__Transformation.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time specialQuadraticTransformation 4
│ │ │ - -- used 0.0773877s (cpu); 0.0773884s (thread); 0s (gc)
│ │ │ + -- used 0.0811578s (cpu); 0.0811568s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6   7   8
│ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.00678678s (cpu); 0.00678413s (thread); 0s (gc)
│ │ │ + -- used 0.0114262s (cpu); 0.0114275s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 2
│ │ │       source variety: PP^8
│ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            o a _r_a_t_i_o_n_a_l_ _m_a_p, an example of special quadratic birational
│ │ │ │              transformation over K, according to the classification given in
│ │ │ │              Table 1 of _E_x_a_m_p_l_e_s_ _o_f_ _s_p_e_c_i_a_l_ _q_u_a_d_r_a_t_i_c_ _b_i_r_a_t_i_o_n_a_l_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s
│ │ │ │              _i_n_t_o_ _c_o_m_p_l_e_t_e_ _i_n_t_e_r_s_e_c_t_i_o_n_s_ _o_f_ _q_u_a_d_r_i_c_s.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The field K is required to be large enough.
│ │ │ │  i1 : time specialQuadraticTransformation 4
│ │ │ │ - -- used 0.0773877s (cpu); 0.0773884s (thread); 0s (gc)
│ │ │ │ + -- used 0.0811578s (cpu); 0.0811568s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4   5   6   7   8
│ │ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ])
│ │ │ │  defined by
│ │ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │ │ @@ -78,15 +78,15 @@
│ │ │ │                                                     2
│ │ │ │                        x x  - x x  + x x  - x x  - x  - x x
│ │ │ │                         0 1    0 4    3 6    4 6    6    5 7
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i2 : time describe oo
│ │ │ │ - -- used 0.00678678s (cpu); 0.00678413s (thread); 0s (gc)
│ │ │ │ + -- used 0.0114262s (cpu); 0.0114275s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^8
│ │ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │ │       dominance: true
│ │ │ │       birationality: true
│ │ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_to__External__String_lp__Rational__Map_rp.html
│ │ │ @@ -93,23 +93,23 @@
│ │ │  
│ │ │  o3 = 6927
    
│ │ │              
    
│ │ │                
    i4 : time phi' = value str;
│ │ │ - -- used 0.0222032s (cpu); 0.0222031s (thread); 0s (gc)
│ │ │ + -- used 0.0269988s (cpu); 0.026998s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
    
│ │ │              
    
│ │ │                
    i5 : time describe phi'
│ │ │ - -- used 0.00537746s (cpu); 0.00537832s (thread); 0s (gc)
│ │ │ + -- used 0.00671281s (cpu); 0.00671801s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 3
│ │ │       source variety: PP^3
│ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {1, 3, 4, 2}
│ │ │ @@ -118,15 +118,15 @@
│ │ │       degree base locus: 5
│ │ │       coefficient ring: ZZ/33331
    
│ │ │              
    
│ │ │                
    i6 : time describe inverse phi'
│ │ │ - -- used 0.00425598s (cpu); 0.00425702s (thread); 0s (gc)
│ │ │ + -- used 0.00599937s (cpu); 0.00600649s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = rational map defined by forms of degree 2
│ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │       target variety: PP^3
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {2, 4, 3, 1}
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,32 +19,32 @@
│ │ │ │  
│ │ │ │  o1 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i2 : str = toExternalString phi;
│ │ │ │  i3 : #str
│ │ │ │  
│ │ │ │  o3 = 6927
│ │ │ │  i4 : time phi' = value str;
│ │ │ │ - -- used 0.0222032s (cpu); 0.0222031s (thread); 0s (gc)
│ │ │ │ + -- used 0.0269988s (cpu); 0.026998s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i5 : time describe phi'
│ │ │ │ - -- used 0.00537746s (cpu); 0.00537832s (thread); 0s (gc)
│ │ │ │ + -- used 0.00671281s (cpu); 0.00671801s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = rational map defined by forms of degree 3
│ │ │ │       source variety: PP^3
│ │ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │ │       dominance: true
│ │ │ │       birationality: true (the inverse map is already calculated)
│ │ │ │       projective degrees: {1, 3, 4, 2}
│ │ │ │       number of minimal representatives: 1
│ │ │ │       dimension base locus: 1
│ │ │ │       degree base locus: 5
│ │ │ │       coefficient ring: ZZ/33331
│ │ │ │  i6 : time describe inverse phi'
│ │ │ │ - -- used 0.00425598s (cpu); 0.00425702s (thread); 0s (gc)
│ │ │ │ + -- used 0.00599937s (cpu); 0.00600649s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = rational map defined by forms of degree 2
│ │ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │ │       target variety: PP^3
│ │ │ │       dominance: true
│ │ │ │       birationality: true (the inverse map is already calculated)
│ │ │ │       projective degrees: {2, 4, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/index.html
│ │ │ @@ -63,29 +63,29 @@
│ │ │              
    		
│ │ │                
    i1 : ZZ/300007[t_0..t_6];
    
│ │ │              
    
│ │ │                
    i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00378639s (cpu); 0.00378433s (thread); 0s (gc)
│ │ │ + -- used 0.00514285s (cpu); 0.00514309s (thread); 0s (gc)
│ │ │  
│ │ │              ZZ              ZZ                3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o2 = map (------[t ..t ], ------[x ..x ], {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   300007  0   9       2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   9
    
│ │ │              
    
│ │ │                
    i3 : time J = kernel(phi,2)
│ │ │ - -- used 0.104398s (cpu); 0.0576775s (thread); 0s (gc)
│ │ │ + -- used 0.201927s (cpu); 0.10589s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x 
│ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │ @@ -93,43 +93,43 @@
│ │ │  o3 : Ideal of ------[x ..x ]
│ │ │                300007  0   9
    
│ │ │              
    
│ │ │                
    i4 : time degreeMap phi
│ │ │ - -- used 0.0284498s (cpu); 0.0284556s (thread); 0s (gc)
│ │ │ + -- used 0.044637s (cpu); 0.0446386s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projectiveDegrees phi
│ │ │ - -- used 0.819548s (cpu); 0.498944s (thread); 0s (gc)
│ │ │ + -- used 0.85738s (cpu); 0.582407s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.181184s (cpu); 0.104837s (thread); 0s (gc)
│ │ │ + -- used 0.178987s (cpu); 0.0962546s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.00225414s (cpu); 0.00225832s (thread); 0s (gc)
│ │ │ + -- used 0.00262157s (cpu); 0.00262558s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         ZZ
│ │ │                                                                       ------[x ..x ]
│ │ │              ZZ                                                       300007  0   9                                                  3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o7 = map (------[t ..t ], ----------------------------------------------------------------------------------------------------, {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )      2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │                              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -141,15 +141,15 @@
│ │ │               300007  0   6      (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )
│ │ │                                    6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
    
│ │ │              
    
│ │ │                
    i8 : time psi = inverseMap phi
│ │ │ - -- used 0.473279s (cpu); 0.380675s (thread); 0s (gc)
│ │ │ + -- used 0.451317s (cpu); 0.451319s (thread); 0s (gc)
│ │ │  
│ │ │                                                         ZZ
│ │ │                                                       ------[x ..x ]
│ │ │                                                       300007  0   9                                                ZZ              3                2               2    2                        2                          2     2        2                               2                                   2               2             2                       3                                                 2                 2    2                                  2    2                 2                                                 3                         2      2    2      2                                              2
│ │ │  o8 = map (----------------------------------------------------------------------------------------------------, ------[t ..t ], {x  - 2x x x  + x x  - x x x  + x x  + x x  + x x x  - x x x  + x x  - 2x x x  - x x x  - 2x x , x x  - x x  - x x x  + x x x  + x x x  + x x  - 2x x x  - x x x  + x x x , x x  - x x x  + x x  - x x x  + x x  - x x x  - x x x , x  - x x x  + x x x  + x x x  - 2x x x  - x x x , x x  - x x x  + x x  + x x  - x x x  - x x x  - x x x , x x  - x x  - x x x  + x x  + x x x  + x x x  - 2x x x  - x x x  + x x x , x  - 2x x x  - x x x  + x x  + x x  + x x  + x x  + x x x  - 2x x x  - x x x  - x x x  - 2x x })
│ │ │            (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )  300007  0   6     2     1 2 3    0 3    1 2 5    0 5    1 6    0 2 6    0 4 6    1 7     0 2 7    0 4 7     0 9   2 3    1 3    1 2 6    0 3 6    0 5 6    1 8     0 2 8    0 4 8    0 1 9   2 3    1 3 6    0 6    0 3 8    1 9    0 2 9    0 4 9   3    1 3 8    0 6 8    1 2 9     0 3 9    0 5 9   3 6    2 3 8    0 8    2 9    1 3 9    0 6 9    0 7 9   3 6    3 8    2 6 8    1 8    2 3 9    2 5 9     1 6 9    1 7 9    0 8 9   6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
│ │ │              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -161,44 +161,44 @@
│ │ │               (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )     300007  0   6
│ │ │                 6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
    
│ │ │              
    
│ │ │                
    i9 : time isInverseMap(phi,psi)
│ │ │ - -- used 0.00964078s (cpu); 0.00964299s (thread); 0s (gc)
│ │ │ + -- used 0.0173789s (cpu); 0.0173802s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time degreeMap psi
│ │ │ - -- used 0.345797s (cpu); 0.198392s (thread); 0s (gc)
│ │ │ + -- used 0.552568s (cpu); 0.2807s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
    
│ │ │              
    
│ │ │                
    i11 : time projectiveDegrees psi
│ │ │ - -- used 5.38156s (cpu); 4.44342s (thread); 0s (gc)
│ │ │ + -- used 6.7628s (cpu); 6.33728s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
    
│ │ │          
    We repeat the example using the type RationalMap and using deterministic methods.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │        2
│ │ │ │  o6 = y T
│ │ │ │          1,2
│ │ │ │  
│ │ │ │  o6 : R[T   ..T   ]
│ │ │ │          1,1   1,3
│ │ │ │  i7 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.0147751s (cpu); 0.01359s
│ │ │ │ +Finding easy relations           :  -- used 0.186865s (cpu); 0.0401605s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = H
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │ │  i8 : homologyClass(KR,z1*z2)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Module.html
│ │ │ @@ -134,15 +134,15 @@
│ │ │  
│ │ │  o5 : Complex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00210419s (cpu); 0.00210521s (thread); 0s (gc)
│ │ │ + -- used 0.00260705s (cpu); 0.00261086s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -247,15 +247,15 @@
│ │ │  
│ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
    
│ │ │              
    
│ │ │                
    i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ - -- used 0.0521222s (cpu); 0.0521315s (thread); 0s (gc)
│ │ │ + -- used 0.0651734s (cpu); 0.0651766s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                                     ZZ
│ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -320,15 +320,15 @@
│ │ │  
│ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i14 : time phi^(-1)
│ │ │ - -- used 0.495675s (cpu); 0.427528s (thread); 0s (gc)
│ │ │ + -- used 0.697532s (cpu); 0.599202s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                                     ZZ
│ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │                {
│ │ │                 x x  - x x  + x x ,
│ │ │ @@ -381,49 +381,49 @@
│ │ │  
│ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
    
│ │ │              
    
│ │ │                
    i15 : time degrees phi^(-1)
│ │ │ - -- used 0.470017s (cpu); 0.310654s (thread); 0s (gc)
│ │ │ + -- used 0.327871s (cpu); 0.315312s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
    
│ │ │              
    
│ │ │                
    i16 : time degrees phi
│ │ │ - -- used 0.0179277s (cpu); 0.0174843s (thread); 0s (gc)
│ │ │ + -- used 0.0849139s (cpu); 0.0281904s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
    
│ │ │              
    
│ │ │                
    i17 : time describe phi
│ │ │ - -- used 0.00296318s (cpu); 0.00296427s (thread); 0s (gc)
│ │ │ + -- used 0.0041231s (cpu); 0.00412799s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = rational map defined by forms of degree 3
│ │ │        source variety: PP^6
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │        coefficient ring: ZZ/300007
    
│ │ │              
    
│ │ │                
    i18 : time describe phi^(-1)
│ │ │ - -- used 0.009622s (cpu); 0.00962345s (thread); 0s (gc)
│ │ │ + -- used 0.0117614s (cpu); 0.0117673s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = rational map defined by forms of degree 3
│ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        target variety: PP^6
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │ @@ -432,41 +432,41 @@
│ │ │        degree base locus: 24
│ │ │        coefficient ring: ZZ/300007
    
│ │ │              
    
│ │ │                
    i19 : time (f,g) = graph phi^-1; f;
│ │ │ - -- used 0.0090019s (cpu); 0.00900307s (thread); 0s (gc)
│ │ │ + -- used 0.0112566s (cpu); 0.0112618s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i21 : time degrees f
│ │ │ - -- used 1.35287s (cpu); 0.917392s (thread); 0s (gc)
│ │ │ + -- used 1.37914s (cpu); 1.03853s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
    
│ │ │              
    
│ │ │                
    i22 : time degree f
│ │ │ - -- used 1.6852e-05s (cpu); 1.6531e-05s (thread); 0s (gc)
│ │ │ + -- used 2.1838e-05s (cpu); 2.0995e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
    
│ │ │              
    
│ │ │                
    i23 : time describe f
│ │ │ - -- used 0.001505s (cpu); 0.0015062s (thread); 0s (gc)
│ │ │ + -- used 0.00196678s (cpu); 0.00197264s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20 hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2, 0},{2, 0})
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true
│ │ │        projective degrees: {904, 508, 268, 130, 56, 20, 5}
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  map) from a list of $m+1$ homogeneous elements of the same degree in $K
│ │ │ │  [x_0,...,x_n]/I$.
│ │ │ │  Below is an example using the methods provided by this package, dealing with a
│ │ │ │  birational transformation $\Phi:\mathbb{P}^6 \dashrightarrow \mathbb{G}
│ │ │ │  (2,4)\subset\mathbb{P}^9$ of bidegree $(3,3)$.
│ │ │ │  i1 : ZZ/300007[t_0..t_6];
│ │ │ │  i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ │ - -- used 0.00378639s (cpu); 0.00378433s (thread); 0s (gc)
│ │ │ │ + -- used 0.00514285s (cpu); 0.00514309s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              ZZ              ZZ                3                2    2
│ │ │ │  2        2                      2                  2    2                 2
│ │ │ │  3                2    2                2                                 2
│ │ │ │  2    2                                  2        2                      2
│ │ │ │  2                        2                         2    2                 2
│ │ │ │  3                2    2
│ │ │ │ @@ -52,43 +52,43 @@
│ │ │ │  0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4
│ │ │ │  3 4 5    2 5    3 6    2 4 6
│ │ │ │  
│ │ │ │                 ZZ                 ZZ
│ │ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │ │               300007  0   6      300007  0   9
│ │ │ │  i3 : time J = kernel(phi,2)
│ │ │ │ - -- used 0.104398s (cpu); 0.0576775s (thread); 0s (gc)
│ │ │ │ + -- used 0.201927s (cpu); 0.10589s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x
│ │ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o3 : Ideal of ------[x ..x ]
│ │ │ │                300007  0   9
│ │ │ │  i4 : time degreeMap phi
│ │ │ │ - -- used 0.0284498s (cpu); 0.0284556s (thread); 0s (gc)
│ │ │ │ + -- used 0.044637s (cpu); 0.0446386s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projectiveDegrees phi
│ │ │ │ - -- used 0.819548s (cpu); 0.498944s (thread); 0s (gc)
│ │ │ │ + -- used 0.85738s (cpu); 0.582407s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ │ - -- used 0.181184s (cpu); 0.104837s (thread); 0s (gc)
│ │ │ │ + -- used 0.178987s (cpu); 0.0962546s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = {5}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ │ - -- used 0.00225414s (cpu); 0.00225832s (thread); 0s (gc)
│ │ │ │ + -- used 0.00262157s (cpu); 0.00262558s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                                         ZZ
│ │ │ │                                                                       ------[x
│ │ │ │  ..x ]
│ │ │ │              ZZ                                                       300007  0
│ │ │ │  9                                                  3                2    2
│ │ │ │  2        2                      2                  2    2                 2
│ │ │ │ @@ -123,15 +123,15 @@
│ │ │ │  o7 : RingMap ------[t ..t ] <-- -----------------------------------------------
│ │ │ │  -----------------------------------------------------
│ │ │ │               300007  0   6      (x x  - x x  + x x , x x  - x x  + x x , x x  -
│ │ │ │  x x  + x x , x x  - x x  + x x , x x  - x x  + x x )
│ │ │ │                                    6 7    5 8    4 9   3 7    2 8    1 9   3 5
│ │ │ │  2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ │  i8 : time psi = inverseMap phi
│ │ │ │ - -- used 0.473279s (cpu); 0.380675s (thread); 0s (gc)
│ │ │ │ + -- used 0.451317s (cpu); 0.451319s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                         ZZ
│ │ │ │                                                       ------[x ..x ]
│ │ │ │                                                       300007  0   9
│ │ │ │  ZZ              3                2               2    2
│ │ │ │  2                          2     2        2                               2
│ │ │ │  2               2             2                       3
│ │ │ │ @@ -164,31 +164,31 @@
│ │ │ │  o8 : RingMap ------------------------------------------------------------------
│ │ │ │  ---------------------------------- <-- ------[t ..t ]
│ │ │ │               (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x
│ │ │ │  - x x  + x x , x x  - x x  + x x )     300007  0   6
│ │ │ │                 6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │ │  1 6    0 8   2 4    1 5    0 7
│ │ │ │  i9 : time isInverseMap(phi,psi)
│ │ │ │ - -- used 0.00964078s (cpu); 0.00964299s (thread); 0s (gc)
│ │ │ │ + -- used 0.0173789s (cpu); 0.0173802s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time degreeMap psi
│ │ │ │ - -- used 0.345797s (cpu); 0.198392s (thread); 0s (gc)
│ │ │ │ + -- used 0.552568s (cpu); 0.2807s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 1
│ │ │ │  i11 : time projectiveDegrees psi
│ │ │ │ - -- used 5.38156s (cpu); 4.44342s (thread); 0s (gc)
│ │ │ │ + -- used 6.7628s (cpu); 6.33728s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  We repeat the example using the type _R_a_t_i_o_n_a_l_M_a_p and using deterministic
│ │ │ │  methods.
│ │ │ │  i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ │ - -- used 0.00210419s (cpu); 0.00210521s (thread); 0s (gc)
│ │ │ │ + -- used 0.00260705s (cpu); 0.00261086s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │ │                     300007  0   1   2   3   4   5   6
│ │ │ │                       ZZ
│ │ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -233,15 +233,15 @@
│ │ │ │                            3                2    2
│ │ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
│ │ │ │  i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ │ - -- used 0.0521222s (cpu); 0.0521315s (thread); 0s (gc)
│ │ │ │ + -- used 0.0651734s (cpu); 0.0651766s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │ │                     300007  0   1   2   3   4   5   6
│ │ │ │                                     ZZ
│ │ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x
│ │ │ │ @@ -304,15 +304,15 @@
│ │ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of
│ │ │ │  PP^9)
│ │ │ │  i14 : time phi^(-1)
│ │ │ │ - -- used 0.495675s (cpu); 0.427528s (thread); 0s (gc)
│ │ │ │ + -- used 0.697532s (cpu); 0.599202s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = -- rational map --
│ │ │ │                                     ZZ
│ │ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x
│ │ │ │  ]) defined by
│ │ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │ │                {
│ │ │ │ @@ -373,67 +373,67 @@
│ │ │ │                          6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6
│ │ │ │  9     4 6 9    2 7 9    4 7 9     0 9
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9
│ │ │ │  to PP^6)
│ │ │ │  i15 : time degrees phi^(-1)
│ │ │ │ - -- used 0.470017s (cpu); 0.310654s (thread); 0s (gc)
│ │ │ │ + -- used 0.327871s (cpu); 0.315312s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : time degrees phi
│ │ │ │ - -- used 0.0179277s (cpu); 0.0174843s (thread); 0s (gc)
│ │ │ │ + -- used 0.0849139s (cpu); 0.0281904s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o16 : List
│ │ │ │  i17 : time describe phi
│ │ │ │ - -- used 0.00296318s (cpu); 0.00296427s (thread); 0s (gc)
│ │ │ │ + -- used 0.0041231s (cpu); 0.00412799s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = rational map defined by forms of degree 3
│ │ │ │        source variety: PP^6
│ │ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │        dominance: true
│ │ │ │        birationality: true (the inverse map is already calculated)
│ │ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │ │        coefficient ring: ZZ/300007
│ │ │ │  i18 : time describe phi^(-1)
│ │ │ │ - -- used 0.009622s (cpu); 0.00962345s (thread); 0s (gc)
│ │ │ │ + -- used 0.0117614s (cpu); 0.0117673s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o18 = rational map defined by forms of degree 3
│ │ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │        target variety: PP^6
│ │ │ │        dominance: true
│ │ │ │        birationality: true (the inverse map is already calculated)
│ │ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │ │        number of minimal representatives: 1
│ │ │ │        dimension base locus: 4
│ │ │ │        degree base locus: 24
│ │ │ │        coefficient ring: ZZ/300007
│ │ │ │  i19 : time (f,g) = graph phi^-1; f;
│ │ │ │ - -- used 0.0090019s (cpu); 0.00900307s (thread); 0s (gc)
│ │ │ │ + -- used 0.0112566s (cpu); 0.0112618s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety
│ │ │ │  of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
│ │ │ │  i21 : time degrees f
│ │ │ │ - -- used 1.35287s (cpu); 0.917392s (thread); 0s (gc)
│ │ │ │ + -- used 1.37914s (cpu); 1.03853s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │ │  
│ │ │ │  o21 : List
│ │ │ │  i22 : time degree f
│ │ │ │ - -- used 1.6852e-05s (cpu); 1.6531e-05s (thread); 0s (gc)
│ │ │ │ + -- used 2.1838e-05s (cpu); 2.0995e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = 1
│ │ │ │  i23 : time describe f
│ │ │ │ - -- used 0.001505s (cpu); 0.0015062s (thread); 0s (gc)
│ │ │ │ + -- used 0.00196678s (cpu); 0.00197264s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20
│ │ │ │  hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1,
│ │ │ │  1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2,
│ │ │ │  0},{2, 0})
│ │ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.out
│ │ │ @@ -155,15 +155,15 @@
│ │ │                                    2       2       2         2 2       2 2        2 2        2 2       2 2            2 2           2 2            2           2           2
│ │ │         Differential => {a, b, c, a T   , b T   , c T   , a*b c T   , b c T   , -a b T   , -a c T   , b c T   T   , -a c T   T   , b c T   T   , -a T   T   , c T   T   , b T   T   }
│ │ │                                      1,1     1,2     1,3         1,1       2,1        2,3        2,2       1,3 2,1        1,3 2,2       1,2 2,1      1,1 2,4     1,3 2,4     1,2 2,4
│ │ │  
│ │ │  o16 : DGAlgebra
│ │ │  
│ │ │  i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0163331s (cpu); 0.0150267s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0384638s (cpu); 0.0239545s (thread); 0s (gc)
│ │ │  
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebras.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │        Underlying algebra => R[S   ..S   ]
│ │ │                                 1,1   1,4
│ │ │        Differential => {a, b, c, d}
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.0207505s (cpu); 0.0196974s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0351638s (cpu); 0.0229684s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i6 : describe HB
│ │ │  
│ │ │ @@ -68,15 +68,15 @@
│ │ │                                      2
│ │ │        Differential => {a, b, c, d, a S   }
│ │ │                                        1,1
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │  
│ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.0194026s (cpu); 0.0178183s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0509454s (cpu); 0.026619s (thread); 0s (gc)
│ │ │  
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra.out
│ │ │ @@ -12,15 +12,15 @@
│ │ │        Underlying algebra => R[T   ..T   ]
│ │ │                                 1,1   1,3
│ │ │        Differential => {a, b, c}
│ │ │  
│ │ │  o2 : DGAlgebra
│ │ │  
│ │ │  i3 : HA = HH A
│ │ │ -Finding easy relations           :  -- used 0.0306115s (cpu); 0.0285865s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.318458s (cpu); 0.0767373s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = HA
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra__Map.out
│ │ │ @@ -28,15 +28,15 @@
│ │ │  
│ │ │  o4 = map (R[Y   ..Y   ], R[T   ..T   ], {Y   , Y   , a, b, c})
│ │ │               1,1   1,3      1,1   1,2     1,2   1,3
│ │ │  
│ │ │  o4 : DGAlgebraMap
│ │ │  
│ │ │  i5 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0142401s (cpu); 0.0134793s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0301678s (cpu); 0.0176895s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o5 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Module.out
│ │ │ @@ -34,15 +34,15 @@
│ │ │        Differentials on gens => {0, | xT_(1,1) |, | yT_(1,2) |}
│ │ │                                     |     0    |  |     0    |
│ │ │                                     |     0    |  |     0    |
│ │ │  
│ │ │  o4 : DGModule
│ │ │  
│ │ │  i5 : HM = homology M
│ │ │ -Finding easy relations           :  -- used 0.0119508s (cpu); 0.0107607s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0568968s (cpu); 0.0214653s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = cokernel {0, 0} | X_2 X_1 0   0   0   0   |
│ │ │                {3, 4} | 0   0   X_1 X_2 0   0   |
│ │ │                {3, 4} | 0   0   0   X_1 0   X_2 |
│ │ │                {3, 4} | 0   0   0   0   X_2 X_1 |
│ │ │  
│ │ │                                                  4
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Module__Map.out
│ │ │ @@ -41,15 +41,15 @@
│ │ │        Natural => {0, 0} | 1 0 0 |
│ │ │                   {2, 2} | 0 1 0 |
│ │ │                   {2, 2} | 0 0 1 |
│ │ │  
│ │ │  o4 : DGModuleMap
│ │ │  
│ │ │  i5 : h = homology idM
│ │ │ -Finding easy relations           :  -- used 0.0196973s (cpu); 0.0182781s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0270484s (cpu); 0.0133465s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {0, 0} | 1 0 0 0 |
│ │ │       {3, 4} | 0 1 0 0 |
│ │ │       {3, 4} | 0 0 1 0 |
│ │ │       {3, 4} | 0 0 0 1 |
│ │ │  
│ │ │  o5 : Matrix
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.out
│ │ │ @@ -49,15 +49,15 @@
│ │ │                                1                                                             {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |     3
│ │ │                                                                                      
│ │ │                                                                                     2
│ │ │  
│ │ │  o6 : Complex
│ │ │  
│ │ │  i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.0209292s (cpu); 0.0198305s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0339762s (cpu); 0.021111s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i8 : ideal HKR
│ │ │  
│ │ │ @@ -68,15 +68,15 @@
│ │ │  i9 : R' = ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^3,d^3,a*c,a*d,b*c,b*d,a^2*b^2-c^2*d^2}
│ │ │  
│ │ │  o9 = R'
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │  
│ │ │  i10 : HKR' = HH koszulComplexDGA R'
│ │ │ -Finding easy relations           :  -- used 0.738135s (cpu); 0.579486s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.871338s (cpu); 0.747221s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
│ │ │  
│ │ │  i11 : numgens HKR'
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_cycles.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i, A))
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : HA = homologyAlgebra A
│ │ │ -Finding easy relations           :  -- used 0.0192701s (cpu); 0.0170471s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0864399s (cpu); 0.0288691s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Deg__N__Module.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │        Underlying algebra => R[T   ..T   ]
│ │ │                                 1,1   1,3
│ │ │        Differential => {x, y, z}
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : HA = HH KR;
│ │ │ -Finding easy relations           :  -- used 0.0168358s (cpu); 0.0158693s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.207236s (cpu); 0.0426638s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : H0 = zerothHomology KR
│ │ │  
│ │ │  o6 = H0
│ │ │  
│ │ │  o6 : QuotientRing
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Algebra.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0180579s (cpu); 0.0170579s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.214155s (cpu); 0.0370411s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i5 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4,a^3*b^3*c^3*d^3}
│ │ │  
│ │ │ @@ -46,15 +46,15 @@
│ │ │  i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o7 = {1, 5, 10, 10, 4}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0920567s (cpu); 0.0889791s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.133084s (cpu); 0.111209s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
│ │ │  
│ │ │  i9 : numgens HA
│ │ │  
│ │ │ @@ -122,15 +122,15 @@
│ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o15 = {1, 7, 7, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.197523s (cpu); 0.0974276s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.351779s (cpu); 0.114847s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
│ │ │  
│ │ │  i17 : R = ZZ/101[a,b,c,d]
│ │ │  
│ │ │ @@ -159,14 +159,14 @@
│ │ │         Underlying algebra => S[T   ..T   ]
│ │ │                                  1,1   1,4
│ │ │         Differential => {a, b, c, d}
│ │ │  
│ │ │  o20 : DGAlgebra
│ │ │  
│ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.121169s (cpu); 0.0492778s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.343645s (cpu); 0.0812046s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i22 :
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class.out
│ │ │ @@ -43,15 +43,15 @@
│ │ │  o6 = y T
│ │ │          1,2
│ │ │  
│ │ │  o6 : R[T   ..T   ]
│ │ │          1,1   1,3
│ │ │  
│ │ │  i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0147751s (cpu); 0.01359s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.186865s (cpu); 0.0401605s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │  
│ │ │  i8 : homologyClass(KR,z1*z2)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Module.out
│ │ │ @@ -34,15 +34,15 @@
│ │ │  o5 = R  <-- R  <-- R  <-- R  <-- R
│ │ │                                    
│ │ │       0      1      2      3      4
│ │ │  
│ │ │  o5 : Complex
│ │ │  
│ │ │  i6 : HKR = HH(KR)
│ │ │ - -- used 0.104028s (cpu); 0.101615s (thread); 0s (gc)
│ │ │ + -- used 0.146792s (cpu); 0.124153s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o6 = HKR
│ │ │  
│ │ │  o6 : QuotientRing
│ │ │  
│ │ │  i7 : degList = first entries vars Q / degree / first
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product.out
│ │ │ @@ -68,15 +68,15 @@
│ │ │                 2
│ │ │  o9 = (true, x y T   T   T    - x x y T   T   T   )
│ │ │               2 2 1,1 1,2 1,3    1 2 2 1,2 1,3 1,4
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.609878s (cpu); 0.505607s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.66647s (cpu); 0.652583s (thread); 0s (gc)
│ │ │  
│ │ │               2
│ │ │  o10 = x x y z T   T   T   T
│ │ │         1 2 2   1,2 1,3 1,4 1,5
│ │ │  
│ │ │  o10 : R[T   ..T   ]
│ │ │           1,1   1,5
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.out
│ │ │ @@ -27,15 +27,15 @@
│ │ │                                 1,1   1,4
│ │ │        Differential => {t , t , t , t }
│ │ │                          1   2   3   4
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.161681s (cpu); 0.159275s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.204481s (cpu); 0.191339s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_tor__Algebra_lp__Ring_cm__Ring_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  i3 : S = R/ideal{a^3*b^3*c^3*d^3}
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ -Finding easy relations           :  -- used 0.507978s (cpu); 0.411331s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.746057s (cpu); 0.586322s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
│ │ │  
│ │ │  i5 : numgens HB
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.html
│ │ │ @@ -294,15 +294,15 @@
│ │ │          
    
│ │ │            
    One can also obtain the map on homology induced by a DGAlgebra map.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0163331s (cpu); 0.0150267s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0384638s (cpu); 0.0239545s (thread); 0s (gc)
│ │ │  
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -218,15 +218,15 @@
│ │ │ │                                      1,1     1,2     1,3         1,1       2,1
│ │ │ │  2,3        2,2       1,3 2,1        1,3 2,2       1,2 2,1      1,1 2,4     1,3
│ │ │ │  2,4     1,2 2,4
│ │ │ │  
│ │ │ │  o16 : DGAlgebra
│ │ │ │  One can also obtain the map on homology induced by a DGAlgebra map.
│ │ │ │  i17 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.0163331s (cpu); 0.0150267s
│ │ │ │ +Finding easy relations           :  -- used 0.0384638s (cpu); 0.0239545s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                            ZZ
│ │ │ │                           ---[a..c]
│ │ │ │              ZZ           101
│ │ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │             101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebras.html
│ │ │ @@ -118,15 +118,15 @@
│ │ │          
    
│ │ │            
    One can compute the homology algebra of a DGAlgebra using the homology (or HH) command.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -179,15 +179,15 @@
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,15 +42,15 @@
│ │ │ │                                 1,1   1,4
│ │ │ │        Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  One can compute the homology algebra of a DGAlgebra using the homology (or HH)
│ │ │ │  command.
│ │ │ │  i5 : HB = HH B
│ │ │ │ -Finding easy relations           :  -- used 0.0207505s (cpu); 0.0196974s
│ │ │ │ +Finding easy relations           :  -- used 0.0351638s (cpu); 0.0229684s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HB
│ │ │ │  
│ │ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i6 : describe HB
│ │ │ │  
│ │ │ │ @@ -87,15 +87,15 @@
│ │ │ │                                 1,1   1,4   2,1
│ │ │ │                                      2
│ │ │ │        Differential => {a, b, c, d, a S   }
│ │ │ │                                        1,1
│ │ │ │  
│ │ │ │  o9 : DGAlgebra
│ │ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ │ -Finding easy relations           :  -- used 0.0194026s (cpu); 0.0178183s
│ │ │ │ +Finding easy relations           :  -- used 0.0509454s (cpu); 0.026619s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │         ZZ
│ │ │ │  o10 = ---[X ..X ]
│ │ │ │        101  1   3
│ │ │ │  
│ │ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │  
│ │ │  o2 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  o2 = {Ring => R                          }
│ │ │ │        Underlying algebra => R[T   ..T   ]
│ │ │ │                                 1,1   1,3
│ │ │ │        Differential => {a, b, c}
│ │ │ │  
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : HA = HH A
│ │ │ │ -Finding easy relations           :  -- used 0.0306115s (cpu); 0.0285865s
│ │ │ │ +Finding easy relations           :  -- used 0.318458s (cpu); 0.0767373s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = HA
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ │ │  i4 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra__Map.html
│ │ │ @@ -120,15 +120,15 @@
│ │ │  
│ │ │  o4 : DGAlgebraMap
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -255,15 +255,15 @@
│ │ │  
│ │ │  o15 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.0207505s (cpu); 0.0196974s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0351638s (cpu); 0.0229684s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.0194026s (cpu); 0.0178183s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0509454s (cpu); 0.026619s (thread); 0s (gc)
│ │ │  
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i3 : HA = HH A
│ │ │ -Finding easy relations           :  -- used 0.0306115s (cpu); 0.0285865s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.318458s (cpu); 0.0767373s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = HA
│ │ │  
│ │ │  o3 : QuotientRing
    
│ │ │              
    
│ │ │                
    i5 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0142401s (cpu); 0.0134793s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0301678s (cpu); 0.0176895s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o5 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  i4 : g = dgAlgebraMap(K1, K2, matrix{{Y_(1,2), Y_(1,3)}})
│ │ │ │  
│ │ │ │  o4 = map (R[Y   ..Y   ], R[T   ..T   ], {Y   , Y   , a, b, c})
│ │ │ │               1,1   1,3      1,1   1,2     1,2   1,3
│ │ │ │  
│ │ │ │  o4 : DGAlgebraMap
│ │ │ │  i5 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.0142401s (cpu); 0.0134793s
│ │ │ │ +Finding easy relations           :  -- used 0.0301678s (cpu); 0.0176895s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                           ZZ
│ │ │ │                          ---[a..c]
│ │ │ │             ZZ           101
│ │ │ │  o5 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │            101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Module.html
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o4 : DGModule
    
│ │ │              
    
│ │ │                
    i5 : HM = homology M
│ │ │ -Finding easy relations           :  -- used 0.0119508s (cpu); 0.0107607s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0568968s (cpu); 0.0214653s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = cokernel {0, 0} | X_2 X_1 0   0   0   0   |
│ │ │                {3, 4} | 0   0   X_1 X_2 0   0   |
│ │ │                {3, 4} | 0   0   0   X_1 0   X_2 |
│ │ │                {3, 4} | 0   0   0   0   X_2 X_1 |
│ │ │  
│ │ │                                                  4
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,15 +57,15 @@
│ │ │ │        Generator degrees => {{0, 0}, {2, 2}, {2, 2}}
│ │ │ │        Differentials on gens => {0, | xT_(1,1) |, | yT_(1,2) |}
│ │ │ │                                     |     0    |  |     0    |
│ │ │ │                                     |     0    |  |     0    |
│ │ │ │  
│ │ │ │  o4 : DGModule
│ │ │ │  i5 : HM = homology M
│ │ │ │ -Finding easy relations           :  -- used 0.0119508s (cpu); 0.0107607s
│ │ │ │ +Finding easy relations           :  -- used 0.0568968s (cpu); 0.0214653s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = cokernel {0, 0} | X_2 X_1 0   0   0   0   |
│ │ │ │                {3, 4} | 0   0   X_1 X_2 0   0   |
│ │ │ │                {3, 4} | 0   0   0   X_1 0   X_2 |
│ │ │ │                {3, 4} | 0   0   0   0   X_2 X_1 |
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Module__Map.html
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o4 : DGModuleMap
    
│ │ │              
    
│ │ │                
    i5 : h = homology idM
│ │ │ -Finding easy relations           :  -- used 0.0196973s (cpu); 0.0182781s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0270484s (cpu); 0.0133465s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {0, 0} | 1 0 0 0 |
│ │ │       {3, 4} | 0 1 0 0 |
│ │ │       {3, 4} | 0 0 1 0 |
│ │ │       {3, 4} | 0 0 0 1 |
│ │ │  
│ │ │  o5 : Matrix
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -59,15 +59,15 @@
│ │ │ │                      1,1   1,2
│ │ │ │        Natural => {0, 0} | 1 0 0 |
│ │ │ │                   {2, 2} | 0 1 0 |
│ │ │ │                   {2, 2} | 0 0 1 |
│ │ │ │  
│ │ │ │  o4 : DGModuleMap
│ │ │ │  i5 : h = homology idM
│ │ │ │ -Finding easy relations           :  -- used 0.0196973s (cpu); 0.0182781s
│ │ │ │ +Finding easy relations           :  -- used 0.0270484s (cpu); 0.0133465s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {0, 0} | 1 0 0 0 |
│ │ │ │       {3, 4} | 0 1 0 0 |
│ │ │ │       {3, 4} | 0 0 1 0 |
│ │ │ │       {3, 4} | 0 0 0 1 |
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.html
│ │ │ @@ -143,15 +143,15 @@
│ │ │          
    
│ │ │            
    Since the Koszul complex is a DG algebra, its homology is itself an algebra.  One can obtain this algebra using the command homology, homologyAlgebra, or HH (all commands work).  This algebra structure can detect whether or not the ring is a complete intersection or Gorenstein.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -171,15 +171,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -76,15 +76,15 @@
│ │ │ │  
│ │ │ │  o6 : Complex
│ │ │ │  Since the Koszul complex is a DG algebra, its homology is itself an algebra.
│ │ │ │  One can obtain this algebra using the command homology, homologyAlgebra, or HH
│ │ │ │  (all commands work). This algebra structure can detect whether or not the ring
│ │ │ │  is a complete intersection or Gorenstein.
│ │ │ │  i7 : HKR = HH KR
│ │ │ │ -Finding easy relations           :  -- used 0.0209292s (cpu); 0.0198305s
│ │ │ │ +Finding easy relations           :  -- used 0.0339762s (cpu); 0.021111s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = HKR
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i8 : ideal HKR
│ │ │ │  
│ │ │ │ @@ -94,15 +94,15 @@
│ │ │ │  i9 : R' = ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^3,d^3,a*c,a*d,b*c,b*d,a^2*b^2-
│ │ │ │  c^2*d^2}
│ │ │ │  
│ │ │ │  o9 = R'
│ │ │ │  
│ │ │ │  o9 : QuotientRing
│ │ │ │  i10 : HKR' = HH koszulComplexDGA R'
│ │ │ │ -Finding easy relations           :  -- used 0.738135s (cpu); 0.579486s
│ │ │ │ +Finding easy relations           :  -- used 0.871338s (cpu); 0.747221s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = HKR'
│ │ │ │  
│ │ │ │  o10 : QuotientRing
│ │ │ │  i11 : numgens HKR'
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_cycles.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i, A))
│ │ │ │  
│ │ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : HA = homologyAlgebra A
│ │ │ │ -Finding easy relations           :  -- used 0.0192701s (cpu); 0.0170471s
│ │ │ │ +Finding easy relations           :  -- used 0.0864399s (cpu); 0.0288691s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Deg__N__Module.html
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.0209292s (cpu); 0.0198305s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0339762s (cpu); 0.021111s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : HKR' = HH koszulComplexDGA R'
│ │ │ -Finding easy relations           :  -- used 0.738135s (cpu); 0.579486s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.871338s (cpu); 0.747221s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra A
│ │ │ -Finding easy relations           :  -- used 0.0192701s (cpu); 0.0170471s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0864399s (cpu); 0.0288691s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i5 : HA = HH KR;
│ │ │ -Finding easy relations           :  -- used 0.0168358s (cpu); 0.0158693s (thread); 0s (gc)
    
│ │ │ +Finding easy relations           :  -- used 0.207236s (cpu); 0.0426638s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i6 : H0 = zerothHomology KR
│ │ │  
│ │ │  o6 = H0
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  o4 = {Ring => R                          }
│ │ │ │        Underlying algebra => R[T   ..T   ]
│ │ │ │                                 1,1   1,3
│ │ │ │        Differential => {x, y, z}
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  i5 : HA = HH KR;
│ │ │ │ -Finding easy relations           :  -- used 0.0168358s (cpu); 0.0158693s
│ │ │ │ +Finding easy relations           :  -- used 0.207236s (cpu); 0.0426638s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  i6 : H0 = zerothHomology KR
│ │ │ │  
│ │ │ │  o6 = H0
│ │ │ │  
│ │ │ │  o6 : QuotientRing
│ │ │ │  i7 : M1 = getDegNModule(1, H0, HA)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Algebra.html
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0180579s (cpu); 0.0170579s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.214155s (cpu); 0.0370411s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ @@ -153,15 +153,15 @@
│ │ │  
│ │ │  o7 : List
│ │ │              
    
│ │ │                
    i8 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0920567s (cpu); 0.0889791s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.133084s (cpu); 0.111209s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
    
│ │ │              
    
│ │ │                
    i16 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.197523s (cpu); 0.0974276s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.351779s (cpu); 0.114847s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
    
│ │ │ @@ -315,15 +315,15 @@
│ │ │  
│ │ │  o20 : DGAlgebra
    
│ │ │              
    
│ │ │                
    i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.121169s (cpu); 0.0492778s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.343645s (cpu); 0.0812046s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.0180579s (cpu); 0.0170579s
│ │ │ │ +Finding easy relations           :  -- used 0.214155s (cpu); 0.0370411s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  Note that HA is a graded commutative polynomial ring (i.e. an exterior algebra)
│ │ │ │  since R is a complete intersection.
│ │ │ │ @@ -60,15 +60,15 @@
│ │ │ │  o6 : DGAlgebra
│ │ │ │  i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o7 = {1, 5, 10, 10, 4}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.0920567s (cpu); 0.0889791s
│ │ │ │ +Finding easy relations           :  -- used 0.133084s (cpu); 0.111209s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HA
│ │ │ │  
│ │ │ │  o8 : QuotientRing
│ │ │ │  i9 : numgens HA
│ │ │ │  
│ │ │ │ @@ -130,15 +130,15 @@
│ │ │ │  o14 : DGAlgebra
│ │ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o15 = {1, 7, 7, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.197523s (cpu); 0.0974276s
│ │ │ │ +Finding easy relations           :  -- used 0.351779s (cpu); 0.114847s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = HA
│ │ │ │  
│ │ │ │  o16 : QuotientRing
│ │ │ │  One can check that HA has Poincare duality since R is Gorenstein.
│ │ │ │  If your DGAlgebra has generators in even degrees, then one must specify the
│ │ │ │ @@ -166,15 +166,15 @@
│ │ │ │  o20 = {Ring => S                          }
│ │ │ │         Underlying algebra => S[T   ..T   ]
│ │ │ │                                  1,1   1,4
│ │ │ │         Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o20 : DGAlgebra
│ │ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ │ -Finding easy relations           :  -- used 0.121169s (cpu); 0.0492778s
│ │ │ │ +Finding easy relations           :  -- used 0.343645s (cpu); 0.0812046s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = HB
│ │ │ │  
│ │ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  ********** WWaayyss ttoo uussee hhoommoollooggyyAAllggeebbrraa:: **********
│ │ │ │      * homologyAlgebra(DGAlgebra)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Class.html
│ │ │ @@ -140,15 +140,15 @@
│ │ │  o6 : R[T   ..T   ]
│ │ │          1,1   1,3
    
│ │ │              
    
│ │ │                
    i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0147751s (cpu); 0.01359s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.186865s (cpu); 0.0401605s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i6 : HKR = HH(KR)
│ │ │ - -- used 0.104028s (cpu); 0.101615s (thread); 0s (gc)
│ │ │ + -- used 0.146792s (cpu); 0.124153s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o6 = HKR
│ │ │  
│ │ │  o6 : QuotientRing
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │        1      4      6      4      1
│ │ │ │  o5 = R  <-- R  <-- R  <-- R  <-- R
│ │ │ │  
│ │ │ │       0      1      2      3      4
│ │ │ │  
│ │ │ │  o5 : Complex
│ │ │ │  i6 : HKR = HH(KR)
│ │ │ │ - -- used 0.104028s (cpu); 0.101615s (thread); 0s (gc)
│ │ │ │ + -- used 0.146792s (cpu); 0.124153s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o6 = HKR
│ │ │ │  
│ │ │ │  o6 : QuotientRing
│ │ │ │  The following is the graded canonical module of R:
│ │ │ │  i7 : degList = first entries vars Q / degree / first
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_massey__Triple__Product.html
│ │ │ @@ -197,15 +197,15 @@
│ │ │          
    
│ │ │            
    Given cycles z1,z2,z3 such that z1*z2 and z2*z3 are boundaries, the Massey triple product of the homology classes represented by z1,z2 and z3 is the homology class of lift12*z3 + z1*lift23.  To see this, we compute and check:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │        Underlying algebra => R[T   ..T   ]
│ │ │ │                                 1,1   1,4
│ │ │ │        Differential => {t , t , t , t }
│ │ │ │                          1   2   3   4
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  i5 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.161681s (cpu); 0.159275s
│ │ │ │ +Finding easy relations           :  -- used 0.204481s (cpu); 0.191339s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = H
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_tor__Algebra_lp__Ring_cm__Ring_rp.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  i2 : M = coker matrix {{a^3*b^3*c^3*d^3}};
│ │ │ │  i3 : S = R/ideal{a^3*b^3*c^3*d^3}
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ │ │  i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ │ -Finding easy relations           :  -- used 0.507978s (cpu); 0.411331s
│ │ │ │ +Finding easy relations           :  -- used 0.746057s (cpu); 0.586322s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HB
│ │ │ │  
│ │ │ │  o4 : QuotientRing
│ │ │ │  i5 : numgens HB
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_delete__Edges.out
│ │ │ @@ -14,15 +14,15 @@
│ │ │  
│ │ │  o3 = {{a, b}, {d, e}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : gprime = deleteEdges (g,T)
│ │ │  
│ │ │ -o4 = HyperGraph{"edges" => {{c, d}, {b, c}, {a, e}}}
│ │ │ +o4 = HyperGraph{"edges" => {{b, c}, {a, e}, {c, d}}}
│ │ │                  "ring" => S
│ │ │                  "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o4 : HyperGraph
│ │ │  
│ │ │  i5 : h = hyperGraph {a*b*c,c*d*e,a*e}
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Hyper__Graph.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o3 : HyperGraph
│ │ │  
│ │ │  i4 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              1   3   4     3   5     1   2   4   5
│ │ │ +                              3   4   5     2   4     1   2   3   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o4 : HyperGraph
│ │ │  
│ │ │  i5 : randomHyperGraph(R,{4,4,2,2}) -- impossible, returns null when time/branch limit reached
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_spanning__Tree.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : R = QQ[x_1..x_6];
│ │ │  
│ │ │  i2 : C = cycle R; -- a 6-cycle
│ │ │  
│ │ │  i3 : spanningTree C
│ │ │  
│ │ │  o3 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ -                         1   2     2   3     3   4     4   5     5   6
│ │ │ +                         1   2     3   4     4   5     1   6     5   6
│ │ │             "ring" => R
│ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │                             1   2   3   4   5   6
│ │ │  
│ │ │  o3 : Graph
│ │ │  
│ │ │  i4 : T = graph {x_1*x_2,x_2*x_3, x_1*x_4,x_1*x_5,x_5*x_6}; -- a tree (no cycles)
│ │ │ @@ -21,15 +21,15 @@
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : G = graph {x_1*x_2,x_2*x_3,x_3*x_1,x_4*x_5,x_5*x_6,x_6*x_4}; -- two three cycles
│ │ │  
│ │ │  i7 : spanningTree G
│ │ │  
│ │ │  o7 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ -                         1   2     1   3     4   5     4   6
│ │ │ +                         1   3     2   3     4   6     5   6
│ │ │             "ring" => R
│ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │                             1   2   3   4   5   6
│ │ │  
│ │ │  o7 : Graph
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_delete__Edges.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  i3 : T = {{a,b},{d,e}}
│ │ │ │  
│ │ │ │  o3 = {{a, b}, {d, e}}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : gprime = deleteEdges (g,T)
│ │ │ │  
│ │ │ │ -o4 = HyperGraph{"edges" => {{c, d}, {b, c}, {a, e}}}
│ │ │ │ +o4 = HyperGraph{"edges" => {{b, c}, {a, e}, {c, d}}}
│ │ │ │                  "ring" => S
│ │ │ │                  "vertices" => {a, b, c, d, e}
│ │ │ │  
│ │ │ │  o4 : HyperGraph
│ │ │ │  i5 : h = hyperGraph {a*b*c,c*d*e,a*e}
│ │ │ │  
│ │ │ │  o5 = HyperGraph{"edges" => {{a, b, c}, {a, e}, {c, d, e}}}
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │                  "vertices" => {x , x , x , x , x }
│ │ │ │                                  1   2   3   4   5
│ │ │ │  
│ │ │ │  o3 : HyperGraph
│ │ │ │  i4 : randomHyperGraph(R,{3,2,4})
│ │ │ │  
│ │ │ │  o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ │ -                              1   3   4     3   5     1   2   4   5
│ │ │ │ +                              3   4   5     2   4     1   2   3   5
│ │ │ │                  "ring" => R
│ │ │ │                  "vertices" => {x , x , x , x , x }
│ │ │ │                                  1   2   3   4   5
│ │ │ │  
│ │ │ │  o4 : HyperGraph
│ │ │ │  i5 : randomHyperGraph(R,{4,4,2,2}) -- impossible, returns null when time/branch
│ │ │ │  limit reached
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_spanning__Tree.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -117,15 +117,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,15 +13,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns a breadth first spanning tree of a graph.
│ │ │ │  i1 : R = QQ[x_1..x_6];
│ │ │ │  i2 : C = cycle R; -- a 6-cycle
│ │ │ │  i3 : spanningTree C
│ │ │ │  
│ │ │ │  o3 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ │ -                         1   2     2   3     3   4     4   5     5   6
│ │ │ │ +                         1   2     3   4     4   5     1   6     5   6
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │ │                             1   2   3   4   5   6
│ │ │ │  
│ │ │ │  o3 : Graph
│ │ │ │  i4 : T = graph {x_1*x_2,x_2*x_3, x_1*x_4,x_1*x_5,x_5*x_6}; -- a tree (no
│ │ │ │  cycles)
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : G = graph {x_1*x_2,x_2*x_3,x_3*x_1,x_4*x_5,x_5*x_6,x_6*x_4}; -- two three
│ │ │ │  cycles
│ │ │ │  i7 : spanningTree G
│ │ │ │  
│ │ │ │  o7 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ │ -                         1   2     1   3     4   5     4   6
│ │ │ │ +                         1   3     2   3     4   6     5   6
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │ │                             1   2   3   4   5   6
│ │ │ │  
│ │ │ │  o7 : Graph
│ │ │ │  ********** WWaayyss ttoo uussee ssppaannnniinnggTTrreeee:: **********
│ │ │ │      * spanningTree(Graph)
│ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/example-output/___Eigen__Solver.out
│ │ │ @@ -15,14 +15,14 @@
│ │ │       a*b*e*f + a*d*e*f + c*d*e*f, a*b*c*d*e + a*b*c*d*f + a*b*c*e*f +
│ │ │       ------------------------------------------------------------------------
│ │ │       a*b*d*e*f + a*c*d*e*f + b*c*d*e*f, a*b*c*d*e*f - 1)
│ │ │  
│ │ │  o2 : Ideal of QQ[a..f]
│ │ │  
│ │ │  i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .264129s elapsed
│ │ │ + -- .257076s elapsed
│ │ │  
│ │ │  i4 : #sols -- 156 solutions
│ │ │  
│ │ │  o4 = 156
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/html/index.html
│ │ │ @@ -85,15 +85,15 @@
│ │ │  
│ │ │  o2 : Ideal of QQ[a..f]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,22 +29,22 @@
│ │ │ │  i3 : g = x^2+c*x+d
│ │ │ │  
│ │ │ │        2
│ │ │ │  o3 = x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ │ - -- used 0.0024275s (cpu); 0.00242422s (thread); 0s (gc)
│ │ │ │ + -- used 0.0027891s (cpu); 0.00278802s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                        2    2             2           2
│ │ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.00149559s (cpu); 0.00149633s (thread); 0s (gc)
│ │ │ │ + -- used 0.00168825s (cpu); 0.00168983s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                          2    2             2           2
│ │ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_eliminate.html
│ │ │ @@ -102,26 +102,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,22 +30,22 @@
│ │ │ │  i3 : g = x^2+c*x+d
│ │ │ │  
│ │ │ │        2
│ │ │ │  o3 = x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ │ - -- used 0.00250084s (cpu); 0.00249755s (thread); 0s (gc)
│ │ │ │ + -- used 0.00322054s (cpu); 0.00321797s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                        2    2             2           2
│ │ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.00151767s (cpu); 0.00151838s (thread); 0s (gc)
│ │ │ │ + -- used 0.00188218s (cpu); 0.00188447s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                          2    2             2           2
│ │ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.609878s (cpu); 0.505607s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.66647s (cpu); 0.652583s (thread); 0s (gc)
│ │ │  
│ │ │               2
│ │ │  o10 = x x y z T   T   T   T
│ │ │         1 2 2   1,2 1,3 1,4 1,5
│ │ │  
│ │ │  o10 : R[T   ..T   ]
│ │ │           1,1   1,5
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -90,16 +90,16 @@
│ │ │ │  Note that the first return value of _g_e_t_B_o_u_n_d_a_r_y_P_r_e_i_m_a_g_e indicates that the
│ │ │ │  inputs are indeed boundaries, and the second value is the lift of the boundary
│ │ │ │  along the differential.
│ │ │ │  Given cycles z1,z2,z3 such that z1*z2 and z2*z3 are boundaries, the Massey
│ │ │ │  triple product of the homology classes represented by z1,z2 and z3 is the
│ │ │ │  homology class of lift12*z3 + z1*lift23. To see this, we compute and check:
│ │ │ │  i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ │ -Finding easy relations           :  -- used 0.609878s (cpu); 0.505607s
│ │ │ │ -(thread); 0s (gc)
│ │ │ │ +Finding easy relations           :  -- used 0.66647s (cpu); 0.652583s (thread);
│ │ │ │ +0s (gc)
│ │ │ │  
│ │ │ │               2
│ │ │ │  o10 = x x y z T   T   T   T
│ │ │ │         1 2 2   1,2 1,3 1,4 1,5
│ │ │ │  
│ │ │ │  o10 : R[T   ..T   ]
│ │ │ │           1,1   1,5
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.html
│ │ │ @@ -124,15 +124,15 @@
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │              
    
│ │ │                
    i5 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.161681s (cpu); 0.159275s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.204481s (cpu); 0.191339s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ -Finding easy relations           :  -- used 0.507978s (cpu); 0.411331s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.746057s (cpu); 0.586322s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : gprime = deleteEdges (g,T)
│ │ │  
│ │ │ -o4 = HyperGraph{"edges" => {{c, d}, {b, c}, {a, e}}}
│ │ │ +o4 = HyperGraph{"edges" => {{b, c}, {a, e}, {c, d}}}
│ │ │                  "ring" => S
│ │ │                  "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o4 : HyperGraph
    
│ │ │              
    
│ │ │                
    i4 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              1   3   4     3   5     1   2   4   5
│ │ │ +                              3   4   5     2   4     1   2   3   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o4 : HyperGraph
    
│ │ │              
    
│ │ │                
    i3 : spanningTree C
│ │ │  
│ │ │  o3 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ -                         1   2     2   3     3   4     4   5     5   6
│ │ │ +                         1   2     3   4     4   5     1   6     5   6
│ │ │             "ring" => R
│ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │                             1   2   3   4   5   6
│ │ │  
│ │ │  o3 : Graph
    
│ │ │              
    
│ │ │                
    i7 : spanningTree G
│ │ │  
│ │ │  o7 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ -                         1   2     1   3     4   5     4   6
│ │ │ +                         1   3     2   3     4   6     5   6
│ │ │             "ring" => R
│ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │                             1   2   3   4   5   6
│ │ │  
│ │ │  o7 : Graph
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .264129s elapsed
    
│ │ │ + -- .257076s elapsed
│ │ │              
    
│ │ │                
    i4 : #sols -- 156 solutions
│ │ │  
│ │ │  o4 = 156
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       a*b*e*f + a*d*e*f + c*d*e*f, a*b*c*d*e + a*b*c*d*f + a*b*c*e*f +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       a*b*d*e*f + a*c*d*e*f + b*c*d*e*f, a*b*c*d*e*f - 1)
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[a..f]
│ │ │ │  i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ │ - -- .264129s elapsed
│ │ │ │ + -- .257076s elapsed
│ │ │ │  i4 : #sols -- 156 solutions
│ │ │ │  
│ │ │ │  o4 = 156
│ │ │ │  The authors would like to acknowledge the June 2020 Macaulay2 workshop held
│ │ │ │  virtually at Warwick, where this package was first developed.
│ │ │ │  RReeffeerreenncceess:
│ │ │ │      * [1] Sturmfels, Bernd. Solving systems of polynomial equations. No. 97.
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.out
│ │ │ @@ -17,23 +17,23 @@
│ │ │  
│ │ │        2
│ │ │  o3 = x  + x*c + d
│ │ │  
│ │ │  o3 : R
│ │ │  
│ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.0024275s (cpu); 0.00242422s (thread); 0s (gc)
│ │ │ + -- used 0.0027891s (cpu); 0.00278802s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00149559s (cpu); 0.00149633s (thread); 0s (gc)
│ │ │ + -- used 0.00168825s (cpu); 0.00168983s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_eliminate.out
│ │ │ @@ -17,23 +17,23 @@
│ │ │  
│ │ │        2
│ │ │  o3 = x  + x*c + d
│ │ │  
│ │ │  o3 : R
│ │ │  
│ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00250084s (cpu); 0.00249755s (thread); 0s (gc)
│ │ │ + -- used 0.00322054s (cpu); 0.00321797s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00151767s (cpu); 0.00151838s (thread); 0s (gc)
│ │ │ + -- used 0.00188218s (cpu); 0.00188447s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out
│ │ │ @@ -17,15 +17,15 @@
│ │ │  
│ │ │        8    5
│ │ │  o3 = x  + x  + x*c + d
│ │ │  
│ │ │  o3 : R
│ │ │  
│ │ │  i4 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.72016s (cpu); 1.38832s (thread); 0s (gc)
│ │ │ + -- used 1.66134s (cpu); 1.38597s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -73,15 +73,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │                 3          4        4
│ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0173558s (cpu); 0.0173573s (thread); 0s (gc)
│ │ │ + -- used 0.0172552s (cpu); 0.0172567s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │  
│ │ │        8    5
│ │ │  o4 = x  + x  + x*c + d
│ │ │  
│ │ │  o4 : R
│ │ │  
│ │ │  i5 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.69246s (cpu); 1.40891s (thread); 0s (gc)
│ │ │ + -- used 1.65938s (cpu); 1.45867s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -75,15 +75,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │                 3          4        4
│ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0174401s (cpu); 0.0174433s (thread); 0s (gc)
│ │ │ + -- used 0.146868s (cpu); 0.0480885s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -108,26 +108,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
    
│ │ │                
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.0024275s (cpu); 0.00242422s (thread); 0s (gc)
│ │ │ + -- used 0.0027891s (cpu); 0.00278802s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00149559s (cpu); 0.00149633s (thread); 0s (gc)
│ │ │ + -- used 0.00168825s (cpu); 0.00168983s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00250084s (cpu); 0.00249755s (thread); 0s (gc)
│ │ │ + -- used 0.00322054s (cpu); 0.00321797s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00151767s (cpu); 0.00151838s (thread); 0s (gc)
│ │ │ + -- used 0.00188218s (cpu); 0.00188447s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.72016s (cpu); 1.38832s (thread); 0s (gc)
│ │ │ + -- used 1.66134s (cpu); 1.38597s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -169,15 +169,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0173558s (cpu); 0.0173573s (thread); 0s (gc)
│ │ │ + -- used 0.0172552s (cpu); 0.0172567s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  i3 : g = x^8+x^5+c*x+d
│ │ │ │  
│ │ │ │        8    5
│ │ │ │  o3 = x  + x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(ideal(f,g),x)
│ │ │ │ - -- used 1.72016s (cpu); 1.38832s (thread); 0s (gc)
│ │ │ │ + -- used 1.66134s (cpu); 1.38597s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -90,15 +90,15 @@
│ │ │ │       + 792a*b c*d - 1512a*b*c d + 648a*c d - 360a b*d  + 648a c*d  - 504b d
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                 3          4        4
│ │ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.0173558s (cpu); 0.0173573s (thread); 0s (gc)
│ │ │ │ + -- used 0.0172552s (cpu); 0.0172567s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o4 : R
    
│ │ │              
    
│ │ │                
    i5 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.69246s (cpu); 1.40891s (thread); 0s (gc)
│ │ │ + -- used 1.65938s (cpu); 1.45867s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -168,15 +168,15 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0174401s (cpu); 0.0174433s (thread); 0s (gc)
│ │ │ + -- used 0.146868s (cpu); 0.0480885s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  i4 : g = x^8+x^5+c*x+d
│ │ │ │  
│ │ │ │        8    5
│ │ │ │  o4 = x  + x  + x*c + d
│ │ │ │  
│ │ │ │  o4 : R
│ │ │ │  i5 : time eliminate(ideal(f,g),x)
│ │ │ │ - -- used 1.69246s (cpu); 1.40891s (thread); 0s (gc)
│ │ │ │ + -- used 1.65938s (cpu); 1.45867s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -85,15 +85,15 @@
│ │ │ │       + 792a*b c*d - 1512a*b*c d + 648a*c d - 360a b*d  + 648a c*d  - 504b d
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                 3          4        4
│ │ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.0174401s (cpu); 0.0174433s (thread); 0s (gc)
│ │ │ │ + -- used 0.146868s (cpu); 0.0480885s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_lines__Hypersurface.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1331975673177
│ │ │  
│ │ │  i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ - -- used 0.0289541s (cpu); 0.0289543s (thread); 0s (gc)
│ │ │ + -- used 0.0292445s (cpu); 0.0292452s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o1 : List
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out
│ │ │ @@ -37,83 +37,83 @@
│ │ │  i6 : rationalCurve(2) - rationalCurve(1)/8
│ │ │  
│ │ │  o6 = 609250
│ │ │  
│ │ │  o6 : QQ
│ │ │  
│ │ │  i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ - -- used 0.346568s (cpu); 0.280811s (thread); 0s (gc)
│ │ │ + -- used 0.35736s (cpu); 0.299146s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time rationalCurve(3)
│ │ │ - -- used 0.135744s (cpu); 0.135748s (thread); 0s (gc)
│ │ │ + -- used 0.139463s (cpu); 0.139241s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
│ │ │  
│ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 5.63777s (cpu); 4.66979s (thread); 0s (gc)
│ │ │ + -- used 5.19827s (cpu); 4.61875s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.135357s (cpu); 0.13536s (thread); 0s (gc)
│ │ │ + -- used 0.145262s (cpu); 0.145269s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
│ │ │  
│ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 5.46872s (cpu); 4.59863s (thread); 0s (gc)
│ │ │ + -- used 5.34458s (cpu); 4.66941s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time rationalCurve(4)
│ │ │ - -- used 1.71996s (cpu); 1.44473s (thread); 0s (gc)
│ │ │ + -- used 1.63162s (cpu); 1.44001s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
│ │ │  
│ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.88847s (cpu); 5.82264s (thread); 0s (gc)
│ │ │ + -- used 7.04771s (cpu); 5.79564s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
│ │ │  
│ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 1.95162s (cpu); 1.60322s (thread); 0s (gc)
│ │ │ + -- used 1.62179s (cpu); 1.42567s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
│ │ │  
│ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 8.47538s (cpu); 6.23384s (thread); 0s (gc)
│ │ │ + -- used 6.97979s (cpu); 5.81283s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
│ │ │  
│ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 7.7952s (cpu); 5.9147s (thread); 0s (gc)
│ │ │ + -- used 7.10132s (cpu); 5.78511s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
│ │ │  
│ │ │  i17 :
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │            
    Computes the number of lines on a general hypersurface of degree 2n - 3 in \mathbb P^n.
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the number of lines on a general hypersurface of degree
│ │ │ │              2n - 3 in \mathbb P^n
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Computes the number of lines on a general hypersurface of degree 2n - 3 in
│ │ │ │  \mathbb P^n.
│ │ │ │  i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ │ - -- used 0.0289541s (cpu); 0.0289543s (thread); 0s (gc)
│ │ │ │ + -- used 0.0292445s (cpu); 0.0292452s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       289139638632755625, 520764738758073845321}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  ********** WWaayyss ttoo uussee lliinneessHHyyppeerrssuurrffaaccee:: **********
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html
│ │ │ @@ -157,15 +157,15 @@
│ │ │            
    The numbers of conics on general complete intersection Calabi-Yau threefolds can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
    
│ │ │                
    i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ - -- used 0.0289541s (cpu); 0.0289543s (thread); 0s (gc)
│ │ │ + -- used 0.0292445s (cpu); 0.0292452s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o1 : List
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ - -- used 0.346568s (cpu); 0.280811s (thread); 0s (gc)
│ │ │ + -- used 0.35736s (cpu); 0.299146s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ @@ -173,27 +173,27 @@
│ │ │            
    For rational curves of degree 3:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -203,15 +203,15 @@
│ │ │            
    The number of rational curves of degree 3 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
    
│ │ │                
    i8 : time rationalCurve(3)
│ │ │ - -- used 0.135744s (cpu); 0.135748s (thread); 0s (gc)
│ │ │ + -- used 0.139463s (cpu); 0.139241s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
    
│ │ │              
    
│ │ │                
    i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 5.63777s (cpu); 4.66979s (thread); 0s (gc)
│ │ │ + -- used 5.19827s (cpu); 4.61875s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.135357s (cpu); 0.13536s (thread); 0s (gc)
│ │ │ + -- used 0.145262s (cpu); 0.145269s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
    
│ │ │              
    
│ │ │ @@ -219,15 +219,15 @@
│ │ │            
    The numbers of rational curves of degree 3 on general complete intersection Calabi-Yau threefolds can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 5.46872s (cpu); 4.59863s (thread); 0s (gc)
│ │ │ + -- used 5.34458s (cpu); 4.66941s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
    
│ │ │ @@ -235,27 +235,27 @@
│ │ │            
    For rational curves of degree 4:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time rationalCurve(4)
│ │ │ - -- used 1.71996s (cpu); 1.44473s (thread); 0s (gc)
│ │ │ + -- used 1.63162s (cpu); 1.44001s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
    
│ │ │              
    
│ │ │                
    i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.88847s (cpu); 5.82264s (thread); 0s (gc)
│ │ │ + -- used 7.04771s (cpu); 5.79564s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
    
│ │ │              
    
│ │ │ @@ -263,15 +263,15 @@
│ │ │            
    The number of rational curves of degree 4 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 1.95162s (cpu); 1.60322s (thread); 0s (gc)
│ │ │ + -- used 1.62179s (cpu); 1.42567s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
    
│ │ │              
    
│ │ │ @@ -279,25 +279,25 @@
│ │ │            
    The numbers of rational curves of degree 4 on general complete intersections of types (4,2) and (3,3) in \mathbb P^5 can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 8.47538s (cpu); 6.23384s (thread); 0s (gc)
│ │ │ + -- used 6.97979s (cpu); 5.81283s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
    
│ │ │              
    
│ │ │                
    i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 7.7952s (cpu); 5.9147s (thread); 0s (gc)
│ │ │ + -- used 7.10132s (cpu); 5.78511s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -59,85 +59,85 @@
│ │ │ │  
│ │ │ │  o6 = 609250
│ │ │ │  
│ │ │ │  o6 : QQ
│ │ │ │  The numbers of conics on general complete intersection Calabi-Yau threefolds
│ │ │ │  can be computed as follows:
│ │ │ │  i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ │ - -- used 0.346568s (cpu); 0.280811s (thread); 0s (gc)
│ │ │ │ + -- used 0.35736s (cpu); 0.299146s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  For rational curves of degree 3:
│ │ │ │  i8 : time rationalCurve(3)
│ │ │ │ - -- used 0.135744s (cpu); 0.135748s (thread); 0s (gc)
│ │ │ │ + -- used 0.139463s (cpu); 0.139241s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       8564575000
│ │ │ │  o8 = ----------
│ │ │ │           27
│ │ │ │  
│ │ │ │  o8 : QQ
│ │ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ │ - -- used 5.63777s (cpu); 4.66979s (thread); 0s (gc)
│ │ │ │ + -- used 5.19827s (cpu); 4.61875s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        8564575000  422690816           4834592  11239424
│ │ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │ │            27          27                 3        27
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  The number of rational curves of degree 3 on a general quintic threefold can be
│ │ │ │  computed as follows:
│ │ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ │ - -- used 0.135357s (cpu); 0.13536s (thread); 0s (gc)
│ │ │ │ + -- used 0.145262s (cpu); 0.145269s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 317206375
│ │ │ │  
│ │ │ │  o10 : QQ
│ │ │ │  The numbers of rational curves of degree 3 on general complete intersection
│ │ │ │  Calabi-Yau threefolds can be computed as follows:
│ │ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ │ - -- used 5.46872s (cpu); 4.59863s (thread); 0s (gc)
│ │ │ │ + -- used 5.34458s (cpu); 4.66941s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  For rational curves of degree 4:
│ │ │ │  i12 : time rationalCurve(4)
│ │ │ │ - -- used 1.71996s (cpu); 1.44473s (thread); 0s (gc)
│ │ │ │ + -- used 1.63162s (cpu); 1.44001s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        15517926796875
│ │ │ │  o12 = --------------
│ │ │ │              64
│ │ │ │  
│ │ │ │  o12 : QQ
│ │ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ │ - -- used 7.88847s (cpu); 5.82264s (thread); 0s (gc)
│ │ │ │ + -- used 7.04771s (cpu); 5.79564s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = 3883914084
│ │ │ │  
│ │ │ │  o13 : QQ
│ │ │ │  The number of rational curves of degree 4 on a general quintic threefold can be
│ │ │ │  computed as follows:
│ │ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ │ - -- used 1.95162s (cpu); 1.60322s (thread); 0s (gc)
│ │ │ │ + -- used 1.62179s (cpu); 1.42567s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = 242467530000
│ │ │ │  
│ │ │ │  o14 : QQ
│ │ │ │  The numbers of rational curves of degree 4 on general complete intersections of
│ │ │ │  types (4,2) and (3,3) in \mathbb P^5 can be computed as follows:
│ │ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ │ - -- used 8.47538s (cpu); 6.23384s (thread); 0s (gc)
│ │ │ │ + -- used 6.97979s (cpu); 5.81283s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3883902528
│ │ │ │  
│ │ │ │  o15 : QQ
│ │ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ │ - -- used 7.7952s (cpu); 5.9147s (thread); 0s (gc)
│ │ │ │ + -- used 7.10132s (cpu); 5.78511s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 1139448384
│ │ │ │  
│ │ │ │  o16 : QQ
│ │ │ │  ********** WWaayyss ttoo uussee rraattiioonnaallCCuurrvvee:: **********
│ │ │ │      * rationalCurve(ZZ)
│ │ │ │      * rationalCurve(ZZ,List)
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out
│ │ │ @@ -10,34 +10,34 @@
│ │ │  o3 = map (R, S, {x , x x , x x , x })
│ │ │                    1   1 0   1 0   0
│ │ │  
│ │ │  o3 : RingMap R <-- S
│ │ │  
│ │ │  i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .0024588 seconds
│ │ │ -     -- used .000580339 seconds
│ │ │ +     -- used .00234949 seconds
│ │ │ +     -- used .000684247 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00346417 seconds
│ │ │ -     -- used .00469058 seconds
│ │ │ +     -- used .00374067 seconds
│ │ │ +     -- used .00532927 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00825591 seconds
│ │ │ -     -- used .0242926 seconds
│ │ │ +     -- used .00833059 seconds
│ │ │ +     -- used .0435953 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0175996 seconds
│ │ │ -     -- used .287002 seconds
│ │ │ +     -- used .0191606 seconds
│ │ │ +     -- used .341589 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .0381251 seconds
│ │ │ -     -- used .930639 seconds
│ │ │ +     -- used .0421951 seconds
│ │ │ +     -- used .960634 seconds
│ │ │  (49, 217)
│ │ │  
│ │ │                                     2
│ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html
│ │ │ @@ -106,34 +106,34 @@
│ │ │  o3 : RingMap R <-- S
    
│ │ │              
    
│ │ │                
    i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .0024588 seconds
│ │ │ -     -- used .000580339 seconds
│ │ │ +     -- used .00234949 seconds
│ │ │ +     -- used .000684247 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00346417 seconds
│ │ │ -     -- used .00469058 seconds
│ │ │ +     -- used .00374067 seconds
│ │ │ +     -- used .00532927 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00825591 seconds
│ │ │ -     -- used .0242926 seconds
│ │ │ +     -- used .00833059 seconds
│ │ │ +     -- used .0435953 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0175996 seconds
│ │ │ -     -- used .287002 seconds
│ │ │ +     -- used .0191606 seconds
│ │ │ +     -- used .341589 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .0381251 seconds
│ │ │ -     -- used .930639 seconds
│ │ │ +     -- used .0421951 seconds
│ │ │ +     -- used .960634 seconds
│ │ │  (49, 217)
│ │ │  
│ │ │                                     2
│ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,34 +33,34 @@
│ │ │ │                    2               2
│ │ │ │  o3 = map (R, S, {x , x x , x x , x })
│ │ │ │                    1   1 0   1 0   0
│ │ │ │  
│ │ │ │  o3 : RingMap R <-- S
│ │ │ │  i4 : G = egbToric(m, OutFile=>stdio)
│ │ │ │  3
│ │ │ │ -     -- used .0024588 seconds
│ │ │ │ -     -- used .000580339 seconds
│ │ │ │ +     -- used .00234949 seconds
│ │ │ │ +     -- used .000684247 seconds
│ │ │ │  (9, 9)
│ │ │ │  new stuff found
│ │ │ │  4
│ │ │ │ -     -- used .00346417 seconds
│ │ │ │ -     -- used .00469058 seconds
│ │ │ │ +     -- used .00374067 seconds
│ │ │ │ +     -- used .00532927 seconds
│ │ │ │  (16, 26)
│ │ │ │  new stuff found
│ │ │ │  5
│ │ │ │ -     -- used .00825591 seconds
│ │ │ │ -     -- used .0242926 seconds
│ │ │ │ +     -- used .00833059 seconds
│ │ │ │ +     -- used .0435953 seconds
│ │ │ │  (25, 60)
│ │ │ │  6
│ │ │ │ -     -- used .0175996 seconds
│ │ │ │ -     -- used .287002 seconds
│ │ │ │ +     -- used .0191606 seconds
│ │ │ │ +     -- used .341589 seconds
│ │ │ │  (36, 120)
│ │ │ │  7
│ │ │ │ -     -- used .0381251 seconds
│ │ │ │ -     -- used .930639 seconds
│ │ │ │ +     -- used .0421951 seconds
│ │ │ │ +     -- used .960634 seconds
│ │ │ │  (49, 217)
│ │ │ │  
│ │ │ │                                     2
│ │ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out
│ │ │ @@ -414,50 +414,50 @@
│ │ │                 3 2 4     3 6
│ │ │  o27 = ideal(12x x x  - 4x x )
│ │ │                 3 7 9     3 9
│ │ │  
│ │ │  o27 : Ideal of S
│ │ │  
│ │ │  i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.109129s (cpu); 0.072939s (thread); 0s (gc)
│ │ │ + -- used 0.0687938s (cpu); 0.0691018s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
│ │ │  
│ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.331325s (cpu); 0.190181s (thread); 0s (gc)
│ │ │ + -- used 0.457845s (cpu); 0.228618s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
│ │ │  
│ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.536957s (cpu); 0.341657s (thread); 0s (gc)
│ │ │ + -- used 0.624905s (cpu); 0.389511s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
│ │ │  
│ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.441059s (cpu); 0.253006s (thread); 0s (gc)
│ │ │ + -- used 0.47653s (cpu); 0.272897s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
│ │ │  
│ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.432375s (cpu); 0.236922s (thread); 0s (gc)
│ │ │ + -- used 0.445784s (cpu); 0.234599s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
│ │ │  
│ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.46253s (cpu); 0.250274s (thread); 0s (gc)
│ │ │ + -- used 0.68374s (cpu); 0.320931s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
│ │ │  
│ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.421278s (cpu); 0.214888s (thread); 0s (gc)
│ │ │ + -- used 0.457986s (cpu); 0.200824s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
│ │ │  
│ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 15.826s (cpu); 9.47574s (thread); 0s (gc)
│ │ │ + -- used 19.6999s (cpu); 11.7271s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 1
│ │ │  
│ │ │  i36 : peek StrategyDefault
│ │ │  
│ │ │  o36 = OptionTable{GRevLexLargest => 0      }
│ │ │                    GRevLexSmallest => 16
│ │ │ @@ -466,15 +466,15 @@
│ │ │                    LexSmallest => 16
│ │ │                    LexSmallestTerm => 16
│ │ │                    Points => 0
│ │ │                    Random => 16
│ │ │                    RandomNonzero => 16
│ │ │  
│ │ │  i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault, Verbose=>true);
│ │ │ - -- used 0.401618s (cpu); 0.334425s (thread); 0s (gc)
│ │ │ + -- used 0.450193s (cpu); 0.375436s (thread); 0s (gc)
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ @@ -536,15 +536,15 @@
│ │ │  i41 : ptsStratGeometric = new OptionTable from (options chooseGoodMinors)#PointOptions;
│ │ │  
│ │ │  i42 : ptsStratGeometric#ExtendField --look at the default value
│ │ │  
│ │ │  o42 = true
│ │ │  
│ │ │  i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.679622s (cpu); 0.53845s (thread); 0s (gc)
│ │ │ + -- used 0.842847s (cpu); 0.684472s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
│ │ │  
│ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │  
│ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │                    DimensionFunction => dim
│ │ │ @@ -559,49 +559,49 @@
│ │ │  o44 : OptionTable
│ │ │  
│ │ │  i45 : ptsStratRational.ExtendField --look at our changed value
│ │ │  
│ │ │  o45 = false
│ │ │  
│ │ │  i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.593831s (cpu); 0.43992s (thread); 0s (gc)
│ │ │ + -- used 0.730506s (cpu); 0.546441s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
│ │ │  
│ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 2.02367s (cpu); 1.72198s (thread); 0s (gc)
│ │ │ + -- used 2.43107s (cpu); 2.17874s (thread); 0s (gc)
│ │ │  
│ │ │  o47 = true
│ │ │  
│ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 1.08932s (cpu); 0.963601s (thread); 0s (gc)
│ │ │ + -- used 1.52284s (cpu); 1.27789s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
│ │ │  
│ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 2.89938s (cpu); 2.56219s (thread); 0s (gc)
│ │ │ + -- used 3.47254s (cpu); 3.23815s (thread); 0s (gc)
│ │ │  
│ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.36877s (cpu); 1.90671s (thread); 0s (gc)
│ │ │ + -- used 2.88379s (cpu); 2.29808s (thread); 0s (gc)
│ │ │  
│ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.377509s (cpu); 0.309473s (thread); 0s (gc)
│ │ │ + -- used 0.327037s (cpu); 0.329037s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
│ │ │  
│ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.82641s (cpu); 2.23529s (thread); 0s (gc)
│ │ │ + -- used 3.46808s (cpu); 2.73617s (thread); 0s (gc)
│ │ │  
│ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.38701s (cpu); 2.73528s (thread); 0s (gc)
│ │ │ + -- used 4.05155s (cpu); 3.34484s (thread); 0s (gc)
│ │ │  
│ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 58.9137s (cpu); 47.7545s (thread); 0s (gc)
│ │ │ + -- used 64.5571s (cpu); 55.5124s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
│ │ │  
│ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 2.99397s (cpu); 2.41268s (thread); 0s (gc)
│ │ │ + -- used 3.99075s (cpu); 2.65792s (thread); 0s (gc)
│ │ │  
│ │ │  o55 = true
│ │ │  
│ │ │  i56 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out
│ │ │ @@ -7,20 +7,20 @@
│ │ │  o2 : Ideal of S
│ │ │  
│ │ │  i3 : dim (S/J)
│ │ │  
│ │ │  o3 = 4
│ │ │  
│ │ │  i4 : time regularInCodimension(1, S/J)
│ │ │ - -- used 3.07188s (cpu); 1.87079s (thread); 0s (gc)
│ │ │ + -- used 3.53956s (cpu); 2.14929s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 12.636s (cpu); 8.02663s (thread); 0s (gc)
│ │ │ + -- used 14.3616s (cpu); 9.30787s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time regularInCodimension(1, S/J, Verbose=>true)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 452.908 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -90,15 +90,15 @@
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 49, and computed = 38
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ -internal -- used 4.21009s (cpu); 2.78099s (thread); 0s (gc)
│ │ │ +internal -- used 3.86545s (cpu); 2.37618s (thread); 0s (gc)
│ │ │  ChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ @@ -165,15 +165,15 @@
│ │ │  regularInCodimension:  singularLocus dimension verified by isCodimAtLeast
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 2
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 110, and computed = 73.  singular locus dimension appears to be = 2
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : time regularInCodimension(1, S/J, MaxMinors=>10, Verbose=>true)
│ │ │ - -- used 0.282303s (cpu); 0.15322s (thread); 0s (gc)
│ │ │ + -- used 0.35817s (cpu); 0.222694s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ @@ -187,15 +187,15 @@
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 10, and computed = 10.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i8 : time regularInCodimension(1, S/J, MaxMinors=>10, Strategy=>StrategyRandom, Verbose=>true)
│ │ │ - -- used 0.216657s (cpu); 0.157232s (thread); 0s (gc)
│ │ │ + -- used 0.211768s (cpu); 0.146556s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -209,15 +209,15 @@
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 10, and computed = 10.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i9 : time regularInCodimension(1, S/J, MaxMinors=>10, MinMinorsFunction => t->3, Verbose=>true)
│ │ │ - -- used 0.469191s (cpu); 0.277576s (thread); 0s (gc)
│ │ │ + -- used 0.466091s (cpu); 0.286231s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -237,15 +237,15 @@
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 10, and computed = 10.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i10 : time regularInCodimension(1, S/J, MaxMinors=>25, CodimCheckFunction => t->t/5, MinMinorsFunction => t->2, Verbose=>true)
│ │ │ - -- used 1.28369s (cpu); 0.83928s (thread); 0s (gc)
│ │ │ + -- used 1.32984s (cpu); 0.891816s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 2, and computed = 2
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │ @@ -288,15 +288,15 @@
│ │ │  regularInCodimension:  singularLocus dimension verified by isCodimAtLeast
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 2
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 25, and computed = 24.  singular locus dimension appears to be = 2
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : time regularInCodimension(1, S/J, MaxMinors=>25, UseOnlyFastCodim => true, Verbose=>true)
│ │ │ - -- used 0.996134s (cpu); 0.615368s (thread); 0s (gc)
│ │ │ + -- used 1.10265s (cpu); 0.688868s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Strategy__Default.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 5509279875405941999
│ │ │  
│ │ │  i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
│ │ │  
│ │ │  i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ - -- 3.83891s elapsed
│ │ │ + -- 3.20246s elapsed
│ │ │  
│ │ │  o2 = true
│ │ │  
│ │ │  i3 : peek StrategyDefault
│ │ │  
│ │ │  o3 = OptionTable{GRevLexLargest => 0      }
│ │ │                   GRevLexSmallest => 16
│ │ │ @@ -16,12 +16,12 @@
│ │ │                   LexSmallest => 16
│ │ │                   LexSmallestTerm => 16
│ │ │                   Points => 0
│ │ │                   Random => 16
│ │ │                   RandomNonzero => 16
│ │ │  
│ │ │  i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ - -- 1.71544s elapsed
│ │ │ + -- 1.37712s elapsed
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out
│ │ │ @@ -16,29 +16,29 @@
│ │ │  i5 : r = rank myDiff;
│ │ │  
│ │ │  i6 : J = chooseGoodMinors(15, r, myDiff, Strategy=>StrategyDefaultNonRandom);
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : time isCodimAtLeast(3, J)
│ │ │ - -- used 0.00214509s (cpu); 0.00304547s (thread); 0s (gc)
│ │ │ + -- used 0.000171794s (cpu); 0.00300946s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : I = ideal(x_2^8*x_10^3-3*x_1*x_2^7*x_10^2*x_11+3*x_1^2*x_2^6*x_10*x_11^2-x_1^3*x_2^5*x_11^3,x_5^5*x_6^3*x_11^3-3*x_5^6*x_6^2*x_11^2*x_12+3*x_5^7*x_6*x_11*x_12^2-x_5^8*x_12^3,x_1^5*x_2^3*x_4^3-3*x_1^6*x_2^2*x_4^2*x_5+3*x_1^7*x_2*x_4*x_5^2-x_1^8*x_5^3,x_6^8*x_11^3-3*x_5*x_6^7*x_11^2*x_12+3*x_5^2*x_6^6*x_11*x_12^2-x_5^3*x_6^5*x_12^3,x_8^3*x_10^8-3*x_7*x_8^2*x_10^7*x_11+3*x_7^2*x_8*x_10^6*x_11^2-x_7^3*x_10^5*x_11^3,x_2^8*x_4^3-3*x_1*x_2^7*x_4^2*x_5+3*x_1^2*x_2^6*x_4*x_5^2-x_1^3*x_2^5*x_5^3,-x_6^3*x_11^8+3*x_5*x_6^2*x_11^7*x_12-3*x_5^2*x_6*x_11^6*x_12^2+x_5^3*x_11^5*x_12^3,-x_6^3*x_7^3*x_9^5+3*x_4*x_6^2*x_7^2*x_9^6-3*x_4^2*x_6*x_7*x_9^7+x_4^3*x_9^8,x_8^8*x_10^3-3*x_7*x_8^7*x_10^2*x_11+3*x_7^2*x_8^6*x_10*x_11^2-x_7^3*x_8^5*x_11^3,x_2^5*x_3^3*x_11^3-3*x_2^6*x_3^2*x_11^2*x_12+3*x_2^7*x_3*x_11*x_12^2-x_2^8*x_12^3);
│ │ │  
│ │ │                 ZZ
│ │ │  o8 : Ideal of ---[x  , x , x , x , x  , x , x , x  , x , x , x , x ]
│ │ │                127  11   8   1   9   12   6   5   10   2   4   3   7
│ │ │  
│ │ │  i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ - -- used 0.000913192s (cpu); 0.00275624s (thread); 0s (gc)
│ │ │ + -- used 0.00403142s (cpu); 0.00321987s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 0.000134112s (cpu); 0.00241578s (thread); 0s (gc)
│ │ │ + -- used 0.000148216s (cpu); 0.00281368s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_proj__Dim.out
│ │ │ @@ -7,17 +7,17 @@
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : pdim(module I)
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ - -- used 0.279958s (cpu); 0.147465s (thread); 0s (gc)
│ │ │ + -- used 0.310747s (cpu); 0.154684s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.0100349s (cpu); 0.0121944s (thread); 0s (gc)
│ │ │ + -- used 0.0125013s (cpu); 0.0145459s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_recursive__Minors.out
│ │ │ @@ -4,20 +4,20 @@
│ │ │  
│ │ │  i2 : M = random(R^{5,5,5,5,5,5}, R^7);
│ │ │  
│ │ │               6      7
│ │ │  o2 : Matrix R  <-- R
│ │ │  
│ │ │  i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ - -- used 0.522447s (cpu); 0.466363s (thread); 0s (gc)
│ │ │ + -- used 0.566697s (cpu); 0.509873s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.51962s (cpu); 1.28872s (thread); 0s (gc)
│ │ │ + -- used 1.49506s (cpu); 1.3608s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : I1 == I2
│ │ │  
│ │ │  o5 = true
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out
│ │ │ @@ -17,44 +17,44 @@
│ │ │  i6 : S = T/I;
│ │ │  
│ │ │  i7 : dim S
│ │ │  
│ │ │  o7 = 3
│ │ │  
│ │ │  i8 : time regularInCodimension(1, S)
│ │ │ - -- used 0.900604s (cpu); 0.589766s (thread); 0s (gc)
│ │ │ + -- used 1.01624s (cpu); 0.687858s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : time regularInCodimension(2, S)
│ │ │ - -- used 7.84848s (cpu); 5.11678s (thread); 0s (gc)
│ │ │ + -- used 9.15411s (cpu); 6.20367s (thread); 0s (gc)
│ │ │  
│ │ │  i10 : R = QQ[c, f, g, h]/ideal(g^3+h^3+1,f*g^3+f*h^3+f,c*g^3+c*h^3+c,f^2*g^3+f^2*h^3+f^2,c*f*g^3+c*f*h^3+c*f,c^2*g^3+c^2*h^3+c^2,f^3*g^3+f^3*h^3+f^3,c*f^2*g^3+c*f^2*h^3+c*f^2,c^2*f*g^3+c^2*f*h^3+c^2*f,c^3-f^2-c,c^3*h-f^2*h-c*h,c^3*g-f^2*g-c*g,c^3*h^2-f^2*h^2-c*h^2,c^3*g*h-f^2*g*h-c*g*h,c^3*g^2-f^2*g^2-c*g^2,c^3*h^3-f^2*h^3-c*h^3,c^3*g*h^2-f^2*g*h^2-c*g*h^2,c^3*g^2*h-f^2*g^2*h-c*g^2*h,c^3*g^3+f^2*h^3+c*h^3+f^2+c);
│ │ │  
│ │ │  i11 : dim(R)
│ │ │  
│ │ │  o11 = 2
│ │ │  
│ │ │  i12 : time (dim singularLocus (R))
│ │ │ - -- used 0.0199657s (cpu); 0.0199597s (thread); 0s (gc)
│ │ │ + -- used 0.020001s (cpu); 0.0208667s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
│ │ │  
│ │ │  i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.466768s (cpu); 0.283428s (thread); 0s (gc)
│ │ │ + -- used 0.494136s (cpu); 0.297448s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
│ │ │  
│ │ │  i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.411296s (cpu); 0.238336s (thread); 0s (gc)
│ │ │ + -- used 0.4619s (cpu); 0.238897s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : time regularInCodimension(2, R)
│ │ │ - -- used 0.301917s (cpu); 0.175812s (thread); 0s (gc)
│ │ │ + -- used 0.333789s (cpu); 0.189808s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = true
│ │ │  
│ │ │  i16 : time regularInCodimension(2, S, Verbose=>true)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 327.599 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ @@ -387,15 +387,15 @@
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │ -internalChooseMi -- used 8.21237s (cpu); 5.46331s (thread); 0s (gc)
│ │ │ +internalChooseMi -- used 8.95076s (cpu); 6.16084s (thread); 0s (gc)
│ │ │  nor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -430,15 +430,15 @@
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 328, and computed = 203
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 328, and computed = 203.  singular locus dimension appears to be = 1
│ │ │  
│ │ │  i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30)
│ │ │ - -- used 1.57612s (cpu); 1.06523s (thread); 0s (gc)
│ │ │ + -- used 1.82238s (cpu); 1.33957s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 30 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -490,57 +490,57 @@
│ │ │  i18 : StrategyCurrent#Random = 0;
│ │ │  
│ │ │  i19 : StrategyCurrent#LexSmallest = 100;
│ │ │  
│ │ │  i20 : StrategyCurrent#LexSmallestTerm = 0;
│ │ │  
│ │ │  i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.425641s (cpu); 0.230653s (thread); 0s (gc)
│ │ │ + -- used 0.440387s (cpu); 0.261722s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.145411s (cpu); 0.0804297s (thread); 0s (gc)
│ │ │ + -- used 0.153023s (cpu); 0.0880682s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.32308s (cpu); 0.838429s (thread); 0s (gc)
│ │ │ + -- used 1.53102s (cpu); 1.02432s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.780025s (cpu); 0.469287s (thread); 0s (gc)
│ │ │ + -- used 0.876236s (cpu); 0.546735s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : StrategyCurrent#LexSmallest = 0;
│ │ │  
│ │ │  i26 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │  
│ │ │  i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.93699s (cpu); 1.77436s (thread); 0s (gc)
│ │ │ + -- used 3.32183s (cpu); 2.13839s (thread); 0s (gc)
│ │ │  
│ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.83356s (cpu); 1.73621s (thread); 0s (gc)
│ │ │ + -- used 3.14873s (cpu); 2.04289s (thread); 0s (gc)
│ │ │  
│ │ │  i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.631252s (cpu); 0.430563s (thread); 0s (gc)
│ │ │ + -- used 0.615852s (cpu); 0.430122s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
│ │ │  
│ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.314175s (cpu); 0.188371s (thread); 0s (gc)
│ │ │ + -- used 0.376284s (cpu); 0.23688s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 0.573212s (cpu); 0.444614s (thread); 0s (gc)
│ │ │ + -- used 0.73238s (cpu); 0.596785s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.46196s (cpu); 1.04088s (thread); 0s (gc)
│ │ │ + -- used 1.58804s (cpu); 1.14474s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
│ │ │  
│ │ │  i33 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html
│ │ │ @@ -577,71 +577,71 @@
│ │ │          
    
│ │ │            
    Here the $1$ passed to the function says how many minors to compute.  For instance, let's compute 8 minors for each of these strategies and see if that was enough to verify that the ring is regular in codimension 1.  In other words, if the dimension of $J$ plus the ideal of partial minors is $\leq 1$ (since $S/J$ has dimension 3).
    
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│ │ │          
    
│ │ │                
    i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.109129s (cpu); 0.072939s (thread); 0s (gc)
│ │ │ + -- used 0.0687938s (cpu); 0.0691018s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
    
│ │ │              
    
│ │ │                
    i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.331325s (cpu); 0.190181s (thread); 0s (gc)
│ │ │ + -- used 0.457845s (cpu); 0.228618s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
    
│ │ │              
    
│ │ │                
    i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.536957s (cpu); 0.341657s (thread); 0s (gc)
│ │ │ + -- used 0.624905s (cpu); 0.389511s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
    
│ │ │              
    
│ │ │                
    i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.441059s (cpu); 0.253006s (thread); 0s (gc)
│ │ │ + -- used 0.47653s (cpu); 0.272897s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
    
│ │ │              
    
│ │ │                
    i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.432375s (cpu); 0.236922s (thread); 0s (gc)
│ │ │ + -- used 0.445784s (cpu); 0.234599s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
    
│ │ │              
    
│ │ │                
    i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.46253s (cpu); 0.250274s (thread); 0s (gc)
│ │ │ + -- used 0.68374s (cpu); 0.320931s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
    
│ │ │              
    
│ │ │                
    i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.421278s (cpu); 0.214888s (thread); 0s (gc)
│ │ │ + -- used 0.457986s (cpu); 0.200824s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
    
│ │ │              
    
│ │ │                
    i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 15.826s (cpu); 9.47574s (thread); 0s (gc)
│ │ │ + -- used 19.6999s (cpu); 11.7271s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 1
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Indeed, in this example, even computing determinants of 1,000 random submatrices is not typically enough to verify that $V(J)$ is regular in codimension 1.  On the other hand, Points is almost always quite effective at finding valuable submatrices, but can be quite slow.  In this particular example, we can see that LexSmallestTerm also performs very well (and does it quickly). Since different strategies work better or worse on different examples, the default strategy actually mixes and matches various strategies.  The default strategy, which we now elucidate,
    
│ │ │ @@ -666,15 +666,15 @@
│ │ │          
    
│ │ │            
    says that we should use GRevLexSmallest, GRevLexSmallestTerm, LexSmallest, LexSmallestTerm, Random, RandomNonzero all with equal probability (note RandomNonzero, which we have not yet discussed chooses random submatrices where no row or column is zero, which is good for working in sparse matrices).  For instance, if we run:
    
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│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault, Verbose=>true);
│ │ │ - -- used 0.401618s (cpu); 0.334425s (thread); 0s (gc)
│ │ │ + -- used 0.450193s (cpu); 0.375436s (thread); 0s (gc)
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ @@ -779,15 +779,15 @@
│ │ │  
│ │ │  o42 = true
    
│ │ │              
    
│ │ │                
    i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.679622s (cpu); 0.53845s (thread); 0s (gc)
│ │ │ + -- used 0.842847s (cpu); 0.684472s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
    
│ │ │              
    
│ │ │                
    i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │ @@ -811,15 +811,15 @@
│ │ │  
│ │ │  o45 = false
    
│ │ │              
    
│ │ │                
    i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.593831s (cpu); 0.43992s (thread); 0s (gc)
│ │ │ + -- used 0.730506s (cpu); 0.546441s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Other options may also be passed to the RandomPoints package via the PointOptions option.
    
│ │ │ @@ -827,71 +827,71 @@
│ │ │          
    
│ │ │            
    regularInCodimension:  It is reasonable to think that you should find a few minors (with one strategy or another), and see if perhaps the minors you have computed so far are enough to verify our ring is regular in codimension 1.  This is exactly what regularInCodimension does.  One can control at a fine level how frequently new minors are computed, and how frequently the dimension of what we have computed so far is checked, by the option codimCheckFunction.  For more on that, see RegularInCodimensionTutorial and regularInCodimension.  Let us finish running regularInCodimension on our example with several different strategies.
    
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│ │ │          
    
│ │ │                
    i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 2.02367s (cpu); 1.72198s (thread); 0s (gc)
│ │ │ + -- used 2.43107s (cpu); 2.17874s (thread); 0s (gc)
│ │ │  
│ │ │  o47 = true
    
│ │ │              
    
│ │ │                
    i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 1.08932s (cpu); 0.963601s (thread); 0s (gc)
│ │ │ + -- used 1.52284s (cpu); 1.27789s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
    
│ │ │              
    
│ │ │                
    i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 2.89938s (cpu); 2.56219s (thread); 0s (gc)
    
│ │ │ + -- used 3.47254s (cpu); 3.23815s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.36877s (cpu); 1.90671s (thread); 0s (gc)
    
│ │ │ + -- used 2.88379s (cpu); 2.29808s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.377509s (cpu); 0.309473s (thread); 0s (gc)
│ │ │ + -- used 0.327037s (cpu); 0.329037s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
    
│ │ │              
    
│ │ │                
    i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.82641s (cpu); 2.23529s (thread); 0s (gc)
    
│ │ │ + -- used 3.46808s (cpu); 2.73617s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.38701s (cpu); 2.73528s (thread); 0s (gc)
    
│ │ │ + -- used 4.05155s (cpu); 3.34484s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 58.9137s (cpu); 47.7545s (thread); 0s (gc)
│ │ │ + -- used 64.5571s (cpu); 55.5124s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
    
│ │ │              
    
│ │ │                
    i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 2.99397s (cpu); 2.41268s (thread); 0s (gc)
│ │ │ + -- used 3.99075s (cpu); 2.65792s (thread); 0s (gc)
│ │ │  
│ │ │  o55 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    If regularInCodimension outputs nothing, then it couldn't verify that the ring was regular in that codimension.  We set MaxMinors => 100 to keep it from running too long with an ineffective strategy.  Again, even though GRevLexSmallest and GRevLexSmallestTerm are not effective in this particular example, in others they perform better than other strategies.  Note similar considerations also apply to projDim.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -438,44 +438,44 @@
│ │ │ │  o27 : Ideal of S
│ │ │ │  Here the $1$ passed to the function says how many minors to compute. For
│ │ │ │  instance, let's compute 8 minors for each of these strategies and see if that
│ │ │ │  was enough to verify that the ring is regular in codimension 1. In other words,
│ │ │ │  if the dimension of $J$ plus the ideal of partial minors is $\leq 1$ (since $S/
│ │ │ │  J$ has dimension 3).
│ │ │ │  i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ │ - -- used 0.109129s (cpu); 0.072939s (thread); 0s (gc)
│ │ │ │ + -- used 0.0687938s (cpu); 0.0691018s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = 2
│ │ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ │ - -- used 0.331325s (cpu); 0.190181s (thread); 0s (gc)
│ │ │ │ + -- used 0.457845s (cpu); 0.228618s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = 3
│ │ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ │ - -- used 0.536957s (cpu); 0.341657s (thread); 0s (gc)
│ │ │ │ + -- used 0.624905s (cpu); 0.389511s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 1
│ │ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ │ - -- used 0.441059s (cpu); 0.253006s (thread); 0s (gc)
│ │ │ │ + -- used 0.47653s (cpu); 0.272897s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = 2
│ │ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ │ - -- used 0.432375s (cpu); 0.236922s (thread); 0s (gc)
│ │ │ │ + -- used 0.445784s (cpu); 0.234599s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o32 = 3
│ │ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J,
│ │ │ │  Strategy=>GRevLexSmallestTerm))
│ │ │ │ - -- used 0.46253s (cpu); 0.250274s (thread); 0s (gc)
│ │ │ │ + -- used 0.68374s (cpu); 0.320931s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o33 = 3
│ │ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ │ - -- used 0.421278s (cpu); 0.214888s (thread); 0s (gc)
│ │ │ │ + -- used 0.457986s (cpu); 0.200824s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o34 = 3
│ │ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ │ - -- used 15.826s (cpu); 9.47574s (thread); 0s (gc)
│ │ │ │ + -- used 19.6999s (cpu); 11.7271s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o35 = 1
│ │ │ │  Indeed, in this example, even computing determinants of 1,000 random
│ │ │ │  submatrices is not typically enough to verify that $V(J)$ is regular in
│ │ │ │  codimension 1. On the other hand, Points is almost always quite effective at
│ │ │ │  finding valuable submatrices, but can be quite slow. In this particular
│ │ │ │  example, we can see that LexSmallestTerm also performs very well (and does it
│ │ │ │ @@ -496,15 +496,15 @@
│ │ │ │  says that we should use GRevLexSmallest, GRevLexSmallestTerm, LexSmallest,
│ │ │ │  LexSmallestTerm, Random, RandomNonzero all with equal probability (note
│ │ │ │  RandomNonzero, which we have not yet discussed chooses random submatrices where
│ │ │ │  no row or column is zero, which is good for working in sparse matrices). For
│ │ │ │  instance, if we run:
│ │ │ │  i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault,
│ │ │ │  Verbose=>true);
│ │ │ │ - -- used 0.401618s (cpu); 0.334425s (thread); 0s (gc)
│ │ │ │ + -- used 0.450193s (cpu); 0.375436s (thread); 0s (gc)
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │ @@ -587,15 +587,15 @@
│ │ │ │  i41 : ptsStratGeometric = new OptionTable from (options
│ │ │ │  chooseGoodMinors)#PointOptions;
│ │ │ │  i42 : ptsStratGeometric#ExtendField --look at the default value
│ │ │ │  
│ │ │ │  o42 = true
│ │ │ │  i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points,
│ │ │ │  PointOptions=>ptsStratGeometric))
│ │ │ │ - -- used 0.679622s (cpu); 0.53845s (thread); 0s (gc)
│ │ │ │ + -- used 0.842847s (cpu); 0.684472s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o43 = 2
│ │ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that
│ │ │ │  value
│ │ │ │  
│ │ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │ │                    DimensionFunction => dim
│ │ │ │ @@ -609,60 +609,60 @@
│ │ │ │  
│ │ │ │  o44 : OptionTable
│ │ │ │  i45 : ptsStratRational.ExtendField --look at our changed value
│ │ │ │  
│ │ │ │  o45 = false
│ │ │ │  i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points,
│ │ │ │  PointOptions=>ptsStratRational))
│ │ │ │ - -- used 0.593831s (cpu); 0.43992s (thread); 0s (gc)
│ │ │ │ + -- used 0.730506s (cpu); 0.546441s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o46 = 2
│ │ │ │  Other options may also be passed to the _R_a_n_d_o_m_P_o_i_n_t_s package via the
│ │ │ │  _P_o_i_n_t_O_p_t_i_o_n_s option.
│ │ │ │  rreegguullaarrIInnCCooddiimmeennssiioonn:: It is reasonable to think that you should find a few
│ │ │ │  minors (with one strategy or another), and see if perhaps the minors you have
│ │ │ │  computed so far are enough to verify our ring is regular in codimension 1. This
│ │ │ │  is exactly what regularInCodimension does. One can control at a fine level how
│ │ │ │  frequently new minors are computed, and how frequently the dimension of what we
│ │ │ │  have computed so far is checked, by the option codimCheckFunction. For more on
│ │ │ │  that, see _R_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n_T_u_t_o_r_i_a_l and _r_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n. Let us finish
│ │ │ │  running regularInCodimension on our example with several different strategies.
│ │ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefault)
│ │ │ │ - -- used 2.02367s (cpu); 1.72198s (thread); 0s (gc)
│ │ │ │ + -- used 2.43107s (cpu); 2.17874s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o47 = true
│ │ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultNonRandom)
│ │ │ │ - -- used 1.08932s (cpu); 0.963601s (thread); 0s (gc)
│ │ │ │ + -- used 1.52284s (cpu); 1.27789s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o48 = true
│ │ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ │ - -- used 2.89938s (cpu); 2.56219s (thread); 0s (gc)
│ │ │ │ + -- used 3.47254s (cpu); 3.23815s (thread); 0s (gc)
│ │ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallest)
│ │ │ │ - -- used 2.36877s (cpu); 1.90671s (thread); 0s (gc)
│ │ │ │ + -- used 2.88379s (cpu); 2.29808s (thread); 0s (gc)
│ │ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallestTerm)
│ │ │ │ - -- used 0.377509s (cpu); 0.309473s (thread); 0s (gc)
│ │ │ │ + -- used 0.327037s (cpu); 0.329037s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o51 = true
│ │ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallest)
│ │ │ │ - -- used 2.82641s (cpu); 2.23529s (thread); 0s (gc)
│ │ │ │ + -- used 3.46808s (cpu); 2.73617s (thread); 0s (gc)
│ │ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallestTerm)
│ │ │ │ - -- used 3.38701s (cpu); 2.73528s (thread); 0s (gc)
│ │ │ │ + -- used 4.05155s (cpu); 3.34484s (thread); 0s (gc)
│ │ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ │ - -- used 58.9137s (cpu); 47.7545s (thread); 0s (gc)
│ │ │ │ + -- used 64.5571s (cpu); 55.5124s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o54 = true
│ │ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultWithPoints)
│ │ │ │ - -- used 2.99397s (cpu); 2.41268s (thread); 0s (gc)
│ │ │ │ + -- used 3.99075s (cpu); 2.65792s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o55 = true
│ │ │ │  If regularInCodimension outputs nothing, then it couldn't verify that the ring
│ │ │ │  was regular in that codimension. We set MaxMinors => 100 to keep it from
│ │ │ │  running too long with an ineffective strategy. Again, even though
│ │ │ │  GRevLexSmallest and GRevLexSmallestTerm are not effective in this particular
│ │ │ │  example, in others they perform better than other strategies. Note similar
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html
│ │ │ @@ -86,23 +86,23 @@
│ │ │          
    
│ │ │            
    It is the cone over $P^2 \times E$ where $E$ is an elliptic curve.  We have embedded it with a Segre embedding inside $P^8$.  In particular, this example is even regular in codimension 3.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : time regularInCodimension(1, S/J)
│ │ │ - -- used 3.07188s (cpu); 1.87079s (thread); 0s (gc)
│ │ │ + -- used 3.53956s (cpu); 2.14929s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │                
    i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 12.636s (cpu); 8.02663s (thread); 0s (gc)
│ │ │ + -- used 14.3616s (cpu); 9.30787s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    We try to verify that $S/J$ is regular in codimension 1 or 2 by computing the ideal made up of a small number of minors of the Jacobian matrix. In this example, instead of computing all relevant 1465128 minors to compute the singular locus, and then trying to compute the dimension of the ideal they generate, we instead compute a few of them.  regularInCodimension returns true if it verified that the ring is regular in codim 1 or 2 (respectively) and null if not.  Because of the randomness that exists in terms of selecting minors, the execution time can actually vary quite a bit.   Let's take a look at what is occurring by using the Verbose option.  We go through the output and explain what each line is telling us.
    
│ │ │ @@ -180,15 +180,15 @@
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 49, and computed = 38
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ -internal -- used 4.21009s (cpu); 2.78099s (thread); 0s (gc)
│ │ │ +internal -- used 3.86545s (cpu); 2.37618s (thread); 0s (gc)
│ │ │  ChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ @@ -263,15 +263,15 @@
│ │ │          
    
│ │ │            
    MaxMinors.  The first output says that we will compute up to 452.9 minors before giving up.  We can control that by setting the option MaxMinors.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : time regularInCodimension(1, S/J, MaxMinors=>10, Verbose=>true)
│ │ │ - -- used 0.282303s (cpu); 0.15322s (thread); 0s (gc)
│ │ │ + -- used 0.35817s (cpu); 0.222694s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ @@ -296,15 +296,15 @@
│ │ │          
    
│ │ │            
    Selecting submatrices of the Jacobian.  We also see output like: ``Choosing LexSmallest'' or ``Choosing Random''.  This is saying how we are selecting a given submatrix.  For instance, we can run:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i8 : time regularInCodimension(1, S/J, MaxMinors=>10, Strategy=>StrategyRandom, Verbose=>true)
│ │ │ - -- used 0.216657s (cpu); 0.157232s (thread); 0s (gc)
│ │ │ + -- used 0.211768s (cpu); 0.146556s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -329,15 +329,15 @@
│ │ │          
    
│ │ │            
    Computing minors vs considering the dimension of what has been computed.  Periodically we compute the codimension of the partial ideal of minors we have computed so far.  There are two options to control this.  First, we can tell the function when to first compute the dimension of the working partial ideal of minors.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : time regularInCodimension(1, S/J, MaxMinors=>10, MinMinorsFunction => t->3, Verbose=>true)
│ │ │ - -- used 0.469191s (cpu); 0.277576s (thread); 0s (gc)
│ │ │ + -- used 0.466091s (cpu); 0.286231s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -368,15 +368,15 @@
│ │ │          
    
│ │ │            
    CodimCheckFunction. The option CodimCheckFunction controls how frequently the dimension of the partial ideal of minors is computed.  For instance, setting CodimCheckFunction => t -> t/5 will say it should compute dimension after every 5 minors are examined.  In general, after the output of the CodimCheckFunction increases by an integer we compute the codimension again.  The default function has the space between computations grow exponentially.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time regularInCodimension(1, S/J, MaxMinors=>25, CodimCheckFunction => t->t/5, MinMinorsFunction => t->2, Verbose=>true)
│ │ │ - -- used 1.28369s (cpu); 0.83928s (thread); 0s (gc)
│ │ │ + -- used 1.32984s (cpu); 0.891816s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 2, and computed = 2
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │ @@ -427,15 +427,15 @@
│ │ │          
    
│ │ │            
    isCodimAtLeast and dim.  We see the lines about the ``isCodimAtLeast failed''.  This means that isCodimAtLeast was not enough on its own to verify that our ring is regular in codimension 1.  After this, ``partial singular locus dimension computed'' indicates we did a complete dimension computation of the partial ideal defining the singular locus.  How isCodimAtLeast is called can be controlled via the options SPairsFunction and PairLimit, which are simply passed to isCodimAtLeast.  You can force the function to only use isCodimAtLeast and not call dimension by setting UseOnlyFastCodim => true.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : time regularInCodimension(1, S/J, MaxMinors=>25, UseOnlyFastCodim => true, Verbose=>true)
│ │ │ - -- used 0.996134s (cpu); 0.615368s (thread); 0s (gc)
│ │ │ + -- used 1.10265s (cpu); 0.688868s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,19 +24,19 @@
│ │ │ │  i3 : dim (S/J)
│ │ │ │  
│ │ │ │  o3 = 4
│ │ │ │  It is the cone over $P^2 \times E$ where $E$ is an elliptic curve. We have
│ │ │ │  embedded it with a Segre embedding inside $P^8$. In particular, this example is
│ │ │ │  even regular in codimension 3.
│ │ │ │  i4 : time regularInCodimension(1, S/J)
│ │ │ │ - -- used 3.07188s (cpu); 1.87079s (thread); 0s (gc)
│ │ │ │ + -- used 3.53956s (cpu); 2.14929s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ │ - -- used 12.636s (cpu); 8.02663s (thread); 0s (gc)
│ │ │ │ + -- used 14.3616s (cpu); 9.30787s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  We try to verify that $S/J$ is regular in codimension 1 or 2 by computing the
│ │ │ │  ideal made up of a small number of minors of the Jacobian matrix. In this
│ │ │ │  example, instead of computing all relevant 1465128 minors to compute the
│ │ │ │  singular locus, and then trying to compute the dimension of the ideal they
│ │ │ │  generate, we instead compute a few of them. regularInCodimension returns true
│ │ │ │ @@ -123,15 +123,15 @@
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 49, and computed = 38
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │ -internal -- used 4.21009s (cpu); 2.78099s (thread); 0s (gc)
│ │ │ │ +internal -- used 3.86545s (cpu); 2.37618s (thread); 0s (gc)
│ │ │ │  ChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │ @@ -203,15 +203,15 @@
│ │ │ │  regularInCodimension:  Loop completed, submatrices considered = 110, and
│ │ │ │  computed = 73.  singular locus dimension appears to be = 2
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  MMaaxxMMiinnoorrss.. The first output says that we will compute up to 452.9 minors before
│ │ │ │  giving up. We can control that by setting the option MaxMinors.
│ │ │ │  i7 : time regularInCodimension(1, S/J, MaxMinors=>10, Verbose=>true)
│ │ │ │ - -- used 0.282303s (cpu); 0.15322s (thread); 0s (gc)
│ │ │ │ + -- used 0.35817s (cpu); 0.222694s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │ @@ -234,15 +234,15 @@
│ │ │ │  There are other finer ways to control the MaxMinors option, but they will not
│ │ │ │  be discussed in this tutorial. See _r_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n.
│ │ │ │  SSeelleeccttiinngg ssuubbmmaattrriicceess ooff tthhee JJaaccoobbiiaann.. We also see output like: ``Choosing
│ │ │ │  LexSmallest'' or ``Choosing Random''. This is saying how we are selecting a
│ │ │ │  given submatrix. For instance, we can run:
│ │ │ │  i8 : time regularInCodimension(1, S/J, MaxMinors=>10, Strategy=>StrategyRandom,
│ │ │ │  Verbose=>true)
│ │ │ │ - -- used 0.216657s (cpu); 0.157232s (thread); 0s (gc)
│ │ │ │ + -- used 0.211768s (cpu); 0.146556s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │ @@ -272,15 +272,15 @@
│ │ │ │  CCoommppuuttiinngg mmiinnoorrss vvss ccoonnssiiddeerriinngg tthhee ddiimmeennssiioonn ooff wwhhaatt hhaass bbeeeenn ccoommppuutteedd..
│ │ │ │  Periodically we compute the codimension of the partial ideal of minors we have
│ │ │ │  computed so far. There are two options to control this. First, we can tell the
│ │ │ │  function when to first compute the dimension of the working partial ideal of
│ │ │ │  minors.
│ │ │ │  i9 : time regularInCodimension(1, S/J, MaxMinors=>10, MinMinorsFunction => t-
│ │ │ │  >3, Verbose=>true)
│ │ │ │ - -- used 0.469191s (cpu); 0.277576s (thread); 0s (gc)
│ │ │ │ + -- used 0.466091s (cpu); 0.286231s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │ @@ -318,15 +318,15 @@
│ │ │ │  dimension of the partial ideal of minors is computed. For instance, setting
│ │ │ │  CodimCheckFunction => t -> t/5 will say it should compute dimension after every
│ │ │ │  5 minors are examined. In general, after the output of the CodimCheckFunction
│ │ │ │  increases by an integer we compute the codimension again. The default function
│ │ │ │  has the space between computations grow exponentially.
│ │ │ │  i10 : time regularInCodimension(1, S/J, MaxMinors=>25, CodimCheckFunction => t-
│ │ │ │  >t/5, MinMinorsFunction => t->2, Verbose=>true)
│ │ │ │ - -- used 1.28369s (cpu); 0.83928s (thread); 0s (gc)
│ │ │ │ + -- used 1.32984s (cpu); 0.891816s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 25 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 2, and computed = 2
│ │ │ │ @@ -385,15 +385,15 @@
│ │ │ │  dimension computed'' indicates we did a complete dimension computation of the
│ │ │ │  partial ideal defining the singular locus. How isCodimAtLeast is called can be
│ │ │ │  controlled via the options SPairsFunction and PairLimit, which are simply
│ │ │ │  passed to _i_s_C_o_d_i_m_A_t_L_e_a_s_t. You can force the function to only use isCodimAtLeast
│ │ │ │  and not call dimension by setting UseOnlyFastCodim => true.
│ │ │ │  i11 : time regularInCodimension(1, S/J, MaxMinors=>25, UseOnlyFastCodim =>
│ │ │ │  true, Verbose=>true)
│ │ │ │ - -- used 0.996134s (cpu); 0.615368s (thread); 0s (gc)
│ │ │ │ + -- used 1.10265s (cpu); 0.688868s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 25 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Strategy__Default.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │              
    		
│ │ │                
    i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
    
│ │ │              
    
│ │ │                
    i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ - -- 3.83891s elapsed
│ │ │ + -- 3.20246s elapsed
│ │ │  
│ │ │  o2 = true
    
│ │ │              
    
│ │ │  In this particular example, on one machine, we list average time to completion of each of the above strategies after 100 runs.        
      
│ │ │            
    • StrategyDefault: 1.65 seconds
      • 
│ │ │ @@ -127,15 +127,15 @@
│ │ │            
    • StrategyPoints: choose all submatrices via Points.
      • 
│ │ │            
    • StrategyDefaultWithPoints: like StrategyDefault but replaces the Random and RandomNonZero submatrices as with matrices chosen as in Points.
      • 
│ │ │          
    
│ │ │  Additionally, a MutableHashTable named StrategyCurrent is also exported.  It begins as the default strategy, but the user can modify it.

    Using a single heuristic  Alternatively, if the user only wants to use say LexSmallestTerm they can set, Strategy to point to that symbol, instead of a creating a custom strategy HashTable.  For example:         
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ - -- 1.71544s elapsed
│ │ │ + -- 1.37712s elapsed
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-
│ │ │ │  b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-
│ │ │ │  e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-
│ │ │ │  a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-
│ │ │ │  g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-
│ │ │ │  h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
│ │ │ │  i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ │ - -- 3.83891s elapsed
│ │ │ │ + -- 3.20246s elapsed
│ │ │ │  
│ │ │ │  o2 = true
│ │ │ │  In this particular example, on one machine, we list average time to completion
│ │ │ │  of each of the above strategies after 100 runs.
│ │ │ │      * StrategyDefault: 1.65 seconds
│ │ │ │      * StrategyRandom: 8.32 seconds
│ │ │ │      * StrategyDefaultNonRandom: 0.99 seconds
│ │ │ │ @@ -135,15 +135,15 @@
│ │ │ │  Additionally, a MutableHashTable named StrategyCurrent is also exported. It
│ │ │ │  begins as the default strategy, but the user can modify it.
│ │ │ │  
│ │ │ │  UUssiinngg aa ssiinnggllee hheeuurriissttiicc Alternatively, if the user only wants to use say
│ │ │ │  LexSmallestTerm they can set, Strategy to point to that symbol, instead of a
│ │ │ │  creating a custom strategy HashTable. For example:
│ │ │ │  i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ │ - -- 1.71544s elapsed
│ │ │ │ + -- 1.37712s elapsed
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _S_t_r_a_t_e_g_y_D_e_f_a_u_l_t is an _o_p_t_i_o_n_ _t_a_b_l_e.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.05+ds/M2/Macaulay2/packages/FastMinors.m2:1991:0.
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html
│ │ │ @@ -119,15 +119,15 @@
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i7 : time isCodimAtLeast(3, J)
│ │ │ - -- used 0.00214509s (cpu); 0.00304547s (thread); 0s (gc)
│ │ │ + -- used 0.000171794s (cpu); 0.00300946s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The function works by computing gb(I, PairLimit=>f(i)) for successive values of i.  Here f(i) is a function that takes t, some approximation of the base degree value of the polynomial ring (for example, in a standard graded polynomial ring, this is probably expected to be \{1\}).  And i is a counting variable. You can provide your own function by calling isCodimAtLeast(n, I, SPairsFunction=>( (i) -> f(i) ), the default function is SPairsFunction=>i->ceiling(1.5^i)   Perhaps more commonly however, the user may want to instead tell the function to compute for larger values of i.  This is done via the option PairLimit.  This is the max value of i to be plugged into SPairsFunction before the function gives up.  In other words, PairLimit=>5 will tell the function to check codimension 5 times.
    
│ │ │ @@ -141,24 +141,24 @@
│ │ │  o8 : Ideal of ---[x  , x , x , x , x  , x , x , x  , x , x , x , x ]
│ │ │                127  11   8   1   9   12   6   5   10   2   4   3   7
    
│ │ │              
    
│ │ │                
    i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ - -- used 0.000913192s (cpu); 0.00275624s (thread); 0s (gc)
│ │ │ + -- used 0.00403142s (cpu); 0.00321987s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 0.000134112s (cpu); 0.00241578s (thread); 0s (gc)
│ │ │ + -- used 0.000148216s (cpu); 0.00281368s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Notice in the first case the function returned null, because the depth of search was not high enough.  It only computed codim 5 times.  The second returned true, but it did so as soon as the answer was found (and before we hit the PairLimit limit).
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,15 +38,15 @@
│ │ │ │               30      12
│ │ │ │  o4 : Matrix R   <-- R
│ │ │ │  i5 : r = rank myDiff;
│ │ │ │  i6 : J = chooseGoodMinors(15, r, myDiff, Strategy=>StrategyDefaultNonRandom);
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : time isCodimAtLeast(3, J)
│ │ │ │ - -- used 0.00214509s (cpu); 0.00304547s (thread); 0s (gc)
│ │ │ │ + -- used 0.000171794s (cpu); 0.00300946s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  The function works by computing gb(I, PairLimit=>f(i)) for successive values of
│ │ │ │  i. Here f(i) is a function that takes t, some approximation of the base degree
│ │ │ │  value of the polynomial ring (for example, in a standard graded polynomial
│ │ │ │  ring, this is probably expected to be \{1\}). And i is a counting variable. You
│ │ │ │  can provide your own function by calling isCodimAtLeast(n, I, SPairsFunction=>
│ │ │ │ @@ -72,20 +72,20 @@
│ │ │ │  x_7^3*x_8^5*x_11^3,x_2^5*x_3^3*x_11^3-
│ │ │ │  3*x_2^6*x_3^2*x_11^2*x_12+3*x_2^7*x_3*x_11*x_12^2-x_2^8*x_12^3);
│ │ │ │  
│ │ │ │                 ZZ
│ │ │ │  o8 : Ideal of ---[x  , x , x , x , x  , x , x , x  , x , x , x , x ]
│ │ │ │                127  11   8   1   9   12   6   5   10   2   4   3   7
│ │ │ │  i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ │ - -- used 0.000913192s (cpu); 0.00275624s (thread); 0s (gc)
│ │ │ │ + -- used 0.00403142s (cpu); 0.00321987s (thread); 0s (gc)
│ │ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ │ - -- used 0.000134112s (cpu); 0.00241578s (thread); 0s (gc)
│ │ │ │ + -- used 0.000148216s (cpu); 0.00281368s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  Notice in the first case the function returned null, because the depth of
│ │ │ │  search was not high enough. It only computed codim 5 times. The second returned
│ │ │ │  true, but it did so as soon as the answer was found (and before we hit the
│ │ │ │  PairLimit limit).
│ │ │ │  ********** WWaayyss ttoo uussee iissCCooddiimmAAttLLeeaasstt:: **********
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html
│ │ │ @@ -104,23 +104,23 @@
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │                
    i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ - -- used 0.279958s (cpu); 0.147465s (thread); 0s (gc)
│ │ │ + -- used 0.310747s (cpu); 0.154684s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.0100349s (cpu); 0.0121944s (thread); 0s (gc)
│ │ │ + -- used 0.0125013s (cpu); 0.0145459s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The option MaxMinors can be used to control how many minors are computed at each step. If this is not specified, the number of minors is a function of the dimension $d$ of the polynomial ring and the possible minors $c$. Specifically it is 10 * d + 2 * log_1.3(c). Otherwise the user can set the option MaxMinors => ZZ to specify that a fixed integer is used for each step.  Alternatively, the user can control the number of minors computed at each step by setting the option MaxMinors => List.  In this case, the list specifies how many minors to be computed at each step, (working backwards). Finally, you can also set MaxMinors to be a custom function of the dimension $d$ of the polynomial ring and the maximum number of minors.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -44,19 +44,19 @@
│ │ │ │  i2 : I = ideal((x^3+y)^2, (x^2+y^2)^2, (x+y^3)^2, (x*y)^2);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : pdim(module I)
│ │ │ │  
│ │ │ │  o3 = 2
│ │ │ │  i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ │ - -- used 0.279958s (cpu); 0.147465s (thread); 0s (gc)
│ │ │ │ + -- used 0.310747s (cpu); 0.154684s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ │ - -- used 0.0100349s (cpu); 0.0121944s (thread); 0s (gc)
│ │ │ │ + -- used 0.0125013s (cpu); 0.0145459s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 1
│ │ │ │  The option MaxMinors can be used to control how many minors are computed at
│ │ │ │  each step. If this is not specified, the number of minors is a function of the
│ │ │ │  dimension $d$ of the polynomial ring and the possible minors $c$. Specifically
│ │ │ │  it is 10 * d + 2 * log_1.3(c). Otherwise the user can set the option MaxMinors
│ │ │ │  => ZZ to specify that a fixed integer is used for each step. Alternatively, the
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_recursive__Minors.html
│ │ │ @@ -97,23 +97,23 @@
│ │ │               6      7
│ │ │  o2 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ - -- used 0.522447s (cpu); 0.466363s (thread); 0s (gc)
│ │ │ + -- used 0.566697s (cpu); 0.509873s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.51962s (cpu); 1.28872s (thread); 0s (gc)
│ │ │ + -- used 1.49506s (cpu); 1.3608s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : I1 == I2
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,19 +27,19 @@
│ │ │ │  strategy for minors
│ │ │ │  i1 : R = QQ[x,y];
│ │ │ │  i2 : M = random(R^{5,5,5,5,5,5}, R^7);
│ │ │ │  
│ │ │ │               6      7
│ │ │ │  o2 : Matrix R  <-- R
│ │ │ │  i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ │ - -- used 0.522447s (cpu); 0.466363s (thread); 0s (gc)
│ │ │ │ + -- used 0.566697s (cpu); 0.509873s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ │ - -- used 1.51962s (cpu); 1.28872s (thread); 0s (gc)
│ │ │ │ + -- used 1.49506s (cpu); 1.3608s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : I1 == I2
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_o_r_s -- ideal generated by minors
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html
│ │ │ @@ -136,23 +136,23 @@
│ │ │  
│ │ │  o7 = 3
    
│ │ │              
    
│ │ │                
    i8 : time regularInCodimension(1, S)
│ │ │ - -- used 0.900604s (cpu); 0.589766s (thread); 0s (gc)
│ │ │ + -- used 1.01624s (cpu); 0.687858s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : time regularInCodimension(2, S)
│ │ │ - -- used 7.84848s (cpu); 5.11678s (thread); 0s (gc)
    
│ │ │ + -- used 9.15411s (cpu); 6.20367s (thread); 0s (gc)
│ │ │              
    
│ │ │          
    
│ │ │            
    There are numerous examples where regularInCodimension is several orders of magnitude faster that calls of dim singularLocus.
    
│ │ │          
    
│ │ │          
    
│ │ │ @@ -170,39 +170,39 @@
│ │ │  
│ │ │  o11 = 2
    
│ │ │              
    
│ │ │                
    i12 : time (dim singularLocus (R))
│ │ │ - -- used 0.0199657s (cpu); 0.0199597s (thread); 0s (gc)
│ │ │ + -- used 0.020001s (cpu); 0.0208667s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
    
│ │ │              
    
│ │ │                
    i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.466768s (cpu); 0.283428s (thread); 0s (gc)
│ │ │ + -- used 0.494136s (cpu); 0.297448s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
    
│ │ │              
    
│ │ │                
    i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.411296s (cpu); 0.238336s (thread); 0s (gc)
│ │ │ + -- used 0.4619s (cpu); 0.238897s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
    
│ │ │              
    
│ │ │                
    i15 : time regularInCodimension(2, R)
│ │ │ - -- used 0.301917s (cpu); 0.175812s (thread); 0s (gc)
│ │ │ + -- used 0.333789s (cpu); 0.189808s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The function works by choosing interesting looking submatrices, computing their determinants, and periodically (based on a logarithmic growth setting), computing the dimension of a subideal of the Jacobian. The option Verbose can be used to see this in action.
    
│ │ │ @@ -543,15 +543,15 @@
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │ -internalChooseMi -- used 8.21237s (cpu); 5.46331s (thread); 0s (gc)
│ │ │ +internalChooseMi -- used 8.95076s (cpu); 6.16084s (thread); 0s (gc)
│ │ │  nor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -594,15 +594,15 @@
│ │ │          
    
│ │ │            
    The maximum number of minors considered can be controlled by the option MaxMinors.  Alternatively, it can be controlled in a more precise way by passing a function to the option MaxMinors.  This function should have two inputs; the first is minimum number of minors needed to determine whether the ring is regular in codimension n, and the second is the total number of minors available in the Jacobian. The function regularInCodimension does not recompute determinants, so MaxMinors or is only an upper bound on the number of minors computed.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30)
│ │ │ - -- used 1.57612s (cpu); 1.06523s (thread); 0s (gc)
│ │ │ + -- used 1.82238s (cpu); 1.33957s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 30 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -671,39 +671,39 @@
│ │ │              
    		
│ │ │                
    i20 : StrategyCurrent#LexSmallestTerm = 0;
    
│ │ │              
    
│ │ │                
    i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.425641s (cpu); 0.230653s (thread); 0s (gc)
│ │ │ + -- used 0.440387s (cpu); 0.261722s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │                
    i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.145411s (cpu); 0.0804297s (thread); 0s (gc)
│ │ │ + -- used 0.153023s (cpu); 0.0880682s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
    
│ │ │              
    
│ │ │                
    i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.32308s (cpu); 0.838429s (thread); 0s (gc)
│ │ │ + -- used 1.53102s (cpu); 1.02432s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.780025s (cpu); 0.469287s (thread); 0s (gc)
│ │ │ + -- used 0.876236s (cpu); 0.546735s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
    
│ │ │              
    
│ │ │                
    i25 : StrategyCurrent#LexSmallest = 0;
    
│ │ │ @@ -713,51 +713,51 @@
│ │ │                  		
│ │ │                
    i26 : StrategyCurrent#LexSmallestTerm = 100;
    
│ │ │              
    
│ │ │                
    i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.93699s (cpu); 1.77436s (thread); 0s (gc)
    
│ │ │ + -- used 3.32183s (cpu); 2.13839s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.83356s (cpu); 1.73621s (thread); 0s (gc)
    
│ │ │ + -- used 3.14873s (cpu); 2.04289s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.631252s (cpu); 0.430563s (thread); 0s (gc)
│ │ │ + -- used 0.615852s (cpu); 0.430122s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
    
│ │ │              
    
│ │ │                
    i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.314175s (cpu); 0.188371s (thread); 0s (gc)
│ │ │ + -- used 0.376284s (cpu); 0.23688s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
    
│ │ │              
    
│ │ │                
    i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 0.573212s (cpu); 0.444614s (thread); 0s (gc)
│ │ │ + -- used 0.73238s (cpu); 0.596785s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
    
│ │ │              
    
│ │ │                
    i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.46196s (cpu); 1.04088s (thread); 0s (gc)
│ │ │ + -- used 1.58804s (cpu); 1.14474s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The minimum number of minors computed before checking the codimension can also be controlled by an option MinMinorsFunction.  This is should be a function of a single variable, the number of minors computed.  Finally, via the option CodimCheckFunction, you can pass the regularInCodimension a function which controls how frequently the codimension of the partial Jacobian ideal is computed.  By default this is the floor of 1.3^k. Finally, passing the option Modulus => p will do the computation after changing the coefficient ring to ZZ/p.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -72,19 +72,19 @@
│ │ │ │  
│ │ │ │  o5 : Ideal of T
│ │ │ │  i6 : S = T/I;
│ │ │ │  i7 : dim S
│ │ │ │  
│ │ │ │  o7 = 3
│ │ │ │  i8 : time regularInCodimension(1, S)
│ │ │ │ - -- used 0.900604s (cpu); 0.589766s (thread); 0s (gc)
│ │ │ │ + -- used 1.01624s (cpu); 0.687858s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : time regularInCodimension(2, S)
│ │ │ │ - -- used 7.84848s (cpu); 5.11678s (thread); 0s (gc)
│ │ │ │ + -- used 9.15411s (cpu); 6.20367s (thread); 0s (gc)
│ │ │ │  There are numerous examples where regularInCodimension is several orders of
│ │ │ │  magnitude faster that calls of dim singularLocus.
│ │ │ │  The following is a (pruned) affine chart on an Abelian surface obtained as a
│ │ │ │  product of two elliptic curves. It is nonsingular, as our function verifies. If
│ │ │ │  one does not prune it, then the dim singularLocus call takes an enormous amount
│ │ │ │  of time, otherwise the running times of dim singularLocus and our function are
│ │ │ │  frequently about the same.
│ │ │ │ @@ -92,27 +92,27 @@
│ │ │ │  (g^3+h^3+1,f*g^3+f*h^3+f,c*g^3+c*h^3+c,f^2*g^3+f^2*h^3+f^2,c*f*g^3+c*f*h^3+c*f,c^2*g^3+c^2*h^3+c^2,f^3*g^3+f^3*h^3+f^3,c*f^2*g^3+c*f^2*h^3+c*f^2,c^2*f*g^3+c^2*f*h^3+c^2*f,c^3-
│ │ │ │  f^2-c,c^3*h-f^2*h-c*h,c^3*g-f^2*g-c*g,c^3*h^2-f^2*h^2-c*h^2,c^3*g*h-f^2*g*h-c*g*h,c^3*g^2-f^2*g^2-c*g^2,c^3*h^3-f^2*h^3-c*h^3,c^3*g*h^2-f^2*g*h^2-c*g*h^2,c^3*g^2*h-f^2*g^2*h-
│ │ │ │  c*g^2*h,c^3*g^3+f^2*h^3+c*h^3+f^2+c);
│ │ │ │  i11 : dim(R)
│ │ │ │  
│ │ │ │  o11 = 2
│ │ │ │  i12 : time (dim singularLocus (R))
│ │ │ │ - -- used 0.0199657s (cpu); 0.0199597s (thread); 0s (gc)
│ │ │ │ + -- used 0.020001s (cpu); 0.0208667s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 = -1
│ │ │ │  i13 : time regularInCodimension(2, R)
│ │ │ │ - -- used 0.466768s (cpu); 0.283428s (thread); 0s (gc)
│ │ │ │ + -- used 0.494136s (cpu); 0.297448s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = true
│ │ │ │  i14 : time regularInCodimension(2, R)
│ │ │ │ - -- used 0.411296s (cpu); 0.238336s (thread); 0s (gc)
│ │ │ │ + -- used 0.4619s (cpu); 0.238897s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : time regularInCodimension(2, R)
│ │ │ │ - -- used 0.301917s (cpu); 0.175812s (thread); 0s (gc)
│ │ │ │ + -- used 0.333789s (cpu); 0.189808s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  The function works by choosing interesting looking submatrices, computing their
│ │ │ │  determinants, and periodically (based on a logarithmic growth setting),
│ │ │ │  computing the dimension of a subideal of the Jacobian. The option Verbose can
│ │ │ │  be used to see this in action.
│ │ │ │  i16 : time regularInCodimension(2, S, Verbose=>true)
│ │ │ │ @@ -462,15 +462,15 @@
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │ -internalChooseMi -- used 8.21237s (cpu); 5.46331s (thread); 0s (gc)
│ │ │ │ +internalChooseMi -- used 8.95076s (cpu); 6.16084s (thread); 0s (gc)
│ │ │ │  nor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │ @@ -515,15 +515,15 @@
│ │ │ │  a function to the option MaxMinors. This function should have two inputs; the
│ │ │ │  first is minimum number of minors needed to determine whether the ring is
│ │ │ │  regular in codimension n, and the second is the total number of minors
│ │ │ │  available in the Jacobian. The function regularInCodimension does not recompute
│ │ │ │  determinants, so MaxMinors or is only an upper bound on the number of minors
│ │ │ │  computed.
│ │ │ │  i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30)
│ │ │ │ - -- used 1.57612s (cpu); 1.06523s (thread); 0s (gc)
│ │ │ │ + -- used 1.82238s (cpu); 1.33957s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4
│ │ │ │  minors, we will compute up to 30 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │ @@ -590,49 +590,49 @@
│ │ │ │  because there are a small number of entries with nonzero constant terms, which
│ │ │ │  are selected repeatedly). However, in our first example, the LexSmallestTerm is
│ │ │ │  much faster, and Random does not perform well at all.
│ │ │ │  i18 : StrategyCurrent#Random = 0;
│ │ │ │  i19 : StrategyCurrent#LexSmallest = 100;
│ │ │ │  i20 : StrategyCurrent#LexSmallestTerm = 0;
│ │ │ │  i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.425641s (cpu); 0.230653s (thread); 0s (gc)
│ │ │ │ + -- used 0.440387s (cpu); 0.261722s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.145411s (cpu); 0.0804297s (thread); 0s (gc)
│ │ │ │ + -- used 0.153023s (cpu); 0.0880682s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = true
│ │ │ │  i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 1.32308s (cpu); 0.838429s (thread); 0s (gc)
│ │ │ │ + -- used 1.53102s (cpu); 1.02432s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.780025s (cpu); 0.469287s (thread); 0s (gc)
│ │ │ │ + -- used 0.876236s (cpu); 0.546735s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  i25 : StrategyCurrent#LexSmallest = 0;
│ │ │ │  i26 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │ │  i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 2.93699s (cpu); 1.77436s (thread); 0s (gc)
│ │ │ │ + -- used 3.32183s (cpu); 2.13839s (thread); 0s (gc)
│ │ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 2.83356s (cpu); 1.73621s (thread); 0s (gc)
│ │ │ │ + -- used 3.14873s (cpu); 2.04289s (thread); 0s (gc)
│ │ │ │  i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.631252s (cpu); 0.430563s (thread); 0s (gc)
│ │ │ │ + -- used 0.615852s (cpu); 0.430122s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = true
│ │ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.314175s (cpu); 0.188371s (thread); 0s (gc)
│ │ │ │ + -- used 0.376284s (cpu); 0.23688s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = true
│ │ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ │ - -- used 0.573212s (cpu); 0.444614s (thread); 0s (gc)
│ │ │ │ + -- used 0.73238s (cpu); 0.596785s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = true
│ │ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ │ - -- used 1.46196s (cpu); 1.04088s (thread); 0s (gc)
│ │ │ │ + -- used 1.58804s (cpu); 1.14474s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o32 = true
│ │ │ │  The minimum number of minors computed before checking the codimension can also
│ │ │ │  be controlled by an option MinMinorsFunction. This is should be a function of a
│ │ │ │  single variable, the number of minors computed. Finally, via the option
│ │ │ │  CodimCheckFunction, you can pass the regularInCodimension a function which
│ │ │ │  controls how frequently the codimension of the partial Jacobian ideal is
│ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out
│ │ │ @@ -81,23 +81,23 @@
│ │ │  
│ │ │  i14 : K3=nextDegree(gens ker Q2,2,S);
│ │ │  
│ │ │                8      8
│ │ │  o14 : Matrix R  <-- R
│ │ │  
│ │ │  i15 : time I=co1Fitting(K3)
│ │ │ - -- used 0.00244656s (cpu); 0.00244352s (thread); 0s (gc)
│ │ │ + -- used 0.00280428s (cpu); 0.00280016s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = ideal (a a   + a  - a  , a a   - a , a a   + a  - a  , a a   - a )
│ │ │                9 11    5    12   3 11    6   9 10    4    11   3 10    5
│ │ │  
│ │ │  o15 : Ideal of R
│ │ │  
│ │ │  i16 : time J=fittingIdeal(2-1,coker K3);
│ │ │ - -- used 0.0064853s (cpu); 0.00648559s (thread); 0s (gc)
│ │ │ + -- used 0.00671936s (cpu); 0.00672218s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : I==J
│ │ │  
│ │ │  o17 = true
│ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html
│ │ │ @@ -207,26 +207,26 @@
│ │ │                8      8
│ │ │  o14 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i15 : time I=co1Fitting(K3)
│ │ │ - -- used 0.00244656s (cpu); 0.00244352s (thread); 0s (gc)
│ │ │ + -- used 0.00280428s (cpu); 0.00280016s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = ideal (a a   + a  - a  , a a   - a , a a   + a  - a  , a a   - a )
│ │ │                9 11    5    12   3 11    6   9 10    4    11   3 10    5
│ │ │  
│ │ │  o15 : Ideal of R
    
│ │ │              
    
│ │ │                
    i16 : time J=fittingIdeal(2-1,coker K3);
│ │ │ - -- used 0.0064853s (cpu); 0.00648559s (thread); 0s (gc)
│ │ │ + -- used 0.00671936s (cpu); 0.00672218s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
    
│ │ │              
    
│ │ │                
    i17 : I==J
│ │ │ ├── html2text {}
│ │ │ │ @@ -95,22 +95,22 @@
│ │ │ │                2      6
│ │ │ │  o13 : Matrix R  <-- R
│ │ │ │  i14 : K3=nextDegree(gens ker Q2,2,S);
│ │ │ │  
│ │ │ │                8      8
│ │ │ │  o14 : Matrix R  <-- R
│ │ │ │  i15 : time I=co1Fitting(K3)
│ │ │ │ - -- used 0.00244656s (cpu); 0.00244352s (thread); 0s (gc)
│ │ │ │ + -- used 0.00280428s (cpu); 0.00280016s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = ideal (a a   + a  - a  , a a   - a , a a   + a  - a  , a a   - a )
│ │ │ │                9 11    5    12   3 11    6   9 10    4    11   3 10    5
│ │ │ │  
│ │ │ │  o15 : Ideal of R
│ │ │ │  i16 : time J=fittingIdeal(2-1,coker K3);
│ │ │ │ - -- used 0.0064853s (cpu); 0.00648559s (thread); 0s (gc)
│ │ │ │ + -- used 0.00671936s (cpu); 0.00672218s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 : Ideal of R
│ │ │ │  i17 : I==J
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  Note that our method is a bit faster for this small example, and for rank 2
│ │ │ │  quotients of S^3=\mathbb{Z}[x,y]^3 the time difference is massive.
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Object.out
│ │ │ @@ -4,19 +4,19 @@
│ │ │  
│ │ │  o1 = 5
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7fe4e7633270}
│ │ │ +o2 = int32{Address => 0x7fa30daac310}
│ │ │  
│ │ │  i3 : address x
│ │ │  
│ │ │ -o3 = 0x7fe4e7633270
│ │ │ +o3 = 0x7fa30daac310
│ │ │  
│ │ │  o3 : Pointer
│ │ │  
│ │ │  i4 : class x
│ │ │  
│ │ │  o4 = int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Array__Type.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  
│ │ │  o2 = {the, quick, brown, fox, jumps, over, the, lazy, dog}
│ │ │  
│ │ │  o2 : ForeignObject of type char**
│ │ │  
│ │ │  i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o3 = {0x7fe4e7675a10, 0x7fe4e7675a00, 0x7fe4e76759f0}
│ │ │ +o3 = {0x7fa30daeba40, 0x7fa30daeba30, 0x7fa30daeba20}
│ │ │  
│ │ │  o3 : ForeignObject of type void**
│ │ │  
│ │ │  i4 : x = charstarstar {"foo", "bar", "baz"}
│ │ │  
│ │ │  o4 = {foo, bar, baz}
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Array__Type_sp__Visible__List.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = {foo, bar}
│ │ │  
│ │ │  o1 : ForeignObject of type char**
│ │ │  
│ │ │  i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o2 = {0x7fe4e7675810, 0x7fe4e7675800, 0x7fe4e76757f0}
│ │ │ +o2 = {0x7fa30daeb880, 0x7fa30daeb870, 0x7fa30daeb860}
│ │ │  
│ │ │  o2 : ForeignObject of type void**
│ │ │  
│ │ │  i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │  
│ │ │  o3 = int32[2]*
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Type_sp__Pointer.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 1730835169888399450
│ │ │  
│ │ │  i1 : ptr = address int 0
│ │ │  
│ │ │ -o1 = 0x7fe4e7889f50
│ │ │ +o1 = 0x7fa30e32bf10
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7fe4e7889f50
│ │ │ +o2 = 0x7fa30e32bf10
│ │ │  
│ │ │  o2 : ForeignObject of type void*
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp__Pointer.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = 5
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7fe4e7651db0
│ │ │ +o2 = 0x7fa30dad2d40
│ │ │  
│ │ │  o2 : Pointer
│ │ │  
│ │ │  i3 : int ptr
│ │ │  
│ │ │  o3 = 5
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp_st_spvoidstar.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 1731230829183683930
│ │ │  
│ │ │  i1 : ptr = voidstar address int 5
│ │ │  
│ │ │ -o1 = 0x7fe4e7651b20
│ │ │ +o1 = 0x7fa30dad2b30
│ │ │  
│ │ │  o1 : ForeignObject of type void*
│ │ │  
│ │ │  i2 : int * ptr
│ │ │  
│ │ │  o2 = 5
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Union__Type_sp__Thing.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = myunion
│ │ │  
│ │ │  o1 : ForeignUnionType
│ │ │  
│ │ │  i2 : myunion 27
│ │ │  
│ │ │ -o2 = HashTable{"bar" => 6.94741e-310}
│ │ │ +o2 = HashTable{"bar" => 6.93362e-310}
│ │ │                 "foo" => 27
│ │ │  
│ │ │  o2 : ForeignObject of type myunion
│ │ │  
│ │ │  i3 : myunion pi
│ │ │  
│ │ │  o3 = HashTable{"bar" => 3.14159   }
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Pointer.out
│ │ │ @@ -4,28 +4,28 @@
│ │ │  
│ │ │  o1 = 20
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7fe4e76332d0}
│ │ │ +o2 = int32{Address => 0x7fa30daac330}
│ │ │  
│ │ │  i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7fe4e76332d0
│ │ │ +o3 = 0x7fa30daac330
│ │ │  
│ │ │  o3 : Pointer
│ │ │  
│ │ │  i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7fe4e76332d5
│ │ │ +o4 = 0x7fa30daac335
│ │ │  
│ │ │  o4 : Pointer
│ │ │  
│ │ │  i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7fe4e76332cd
│ │ │ +o5 = 0x7fa30daac32d
│ │ │  
│ │ │  o5 : Pointer
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Shared__Library.out
│ │ │ @@ -4,10 +4,10 @@
│ │ │  
│ │ │  o1 = mpfr
│ │ │  
│ │ │  o1 : SharedLibrary
│ │ │  
│ │ │  i2 : peek mpfr
│ │ │  
│ │ │ -o2 = SharedLibrary{0x7fe4f7389ac0, mpfr}
│ │ │ +o2 = SharedLibrary{0x7fa322db5ac0, mpfr}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/__st_spvoidstar_sp_eq_sp__Thing.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = 5
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7fe4e7651db0
│ │ │ +o2 = 0x7fa30dad2d70
│ │ │  
│ │ │  o2 : Pointer
│ │ │  
│ │ │  i3 : *ptr = int 6
│ │ │  
│ │ │  o3 = 6
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_address.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 1730181884377373595
│ │ │  
│ │ │  i1 : address int
│ │ │  
│ │ │ -o1 = 0x555fc7f2b400
│ │ │ +o1 = 0x56340a989400
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : address int 5
│ │ │  
│ │ │ -o2 = 0x7fe4e7651d00
│ │ │ +o2 = 0x7fa30dad2d60
│ │ │  
│ │ │  o2 : Pointer
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_foreign__Function.out
│ │ │ @@ -78,14 +78,14 @@
│ │ │  
│ │ │  o16 = free
│ │ │  
│ │ │  o16 : ForeignFunction
│ │ │  
│ │ │  i17 : x = malloc 8
│ │ │  
│ │ │ -o17 = 0x7f71f006a700
│ │ │ +o17 = 0x7fbbd406a700
│ │ │  
│ │ │  o17 : ForeignObject of type void*
│ │ │  
│ │ │  i18 : registerFinalizer(x, free)
│ │ │  
│ │ │  i19 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_get__Memory.out
│ │ │ @@ -1,21 +1,21 @@
│ │ │  -- -*- M2-comint -*- hash: 10647988412767280310
│ │ │  
│ │ │  i1 : ptr = getMemory 8
│ │ │  
│ │ │ -o1 = 0x7fe4f2bac9b0
│ │ │ +o1 = 0x7fa30ec0e860
│ │ │  
│ │ │  o1 : ForeignObject of type void*
│ │ │  
│ │ │  i2 : ptr = getMemory(8, Atomic => true)
│ │ │  
│ │ │ -o2 = 0x7fe4e7651be0
│ │ │ +o2 = 0x7fa30dad2ba0
│ │ │  
│ │ │  o2 : ForeignObject of type void*
│ │ │  
│ │ │  i3 : ptr = getMemory int
│ │ │  
│ │ │ -o3 = 0x7fe4e7651af0
│ │ │ +o3 = 0x7fa30dad2a90
│ │ │  
│ │ │  o3 : ForeignObject of type void*
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_register__Finalizer_lp__Foreign__Object_cm__Function_rp.out
│ │ │ @@ -17,18 +17,18 @@
│ │ │  o3 = finalizer
│ │ │  
│ │ │  o3 : FunctionClosure
│ │ │  
│ │ │  i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer))
│ │ │  
│ │ │  i5 : collectGarbage()
│ │ │ -freeing memory at 0x7fe4dc07fcf0
│ │ │ -freeing memory at 0x7fe4dc07f650
│ │ │ -freeing memory at 0x7fe4dc07fd30
│ │ │ -freeing memory at 0x7fe4dc07fd10
│ │ │ -freeing memory at 0x7fe4dc07fdb0
│ │ │ -freeing memory at 0x7fe4dc07fd90
│ │ │ -freeing memory at 0x7fe4dc07fd70
│ │ │ -freeing memory at 0x7fe4dc07fd50
│ │ │ -freeing memory at 0x7fe4dc07f630
│ │ │ +freeing memory at 0x7fa2f807fd50
│ │ │ +freeing memory at 0x7fa2f807fd70
│ │ │ +freeing memory at 0x7fa2f807fd90
│ │ │ +freeing memory at 0x7fa2f807fdb0
│ │ │ +freeing memory at 0x7fa2f807f630
│ │ │ +freeing memory at 0x7fa2f807fcf0
│ │ │ +freeing memory at 0x7fa2f807fd10
│ │ │ +freeing memory at 0x7fa2f807f650
│ │ │ +freeing memory at 0x7fa2f807fd30
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_value_lp__Foreign__Object_rp.out
│ │ │ @@ -20,21 +20,21 @@
│ │ │  
│ │ │  o4 = 5
│ │ │  
│ │ │  o4 : RR (of precision 53)
│ │ │  
│ │ │  i5 : x = voidstar address int 5
│ │ │  
│ │ │ -o5 = 0x7fe4e7651fa0
│ │ │ +o5 = 0x7fa30dad2010
│ │ │  
│ │ │  o5 : ForeignObject of type void*
│ │ │  
│ │ │  i6 : value x
│ │ │  
│ │ │ -o6 = 0x7fe4e7651fa0
│ │ │ +o6 = 0x7fa30dad2010
│ │ │  
│ │ │  o6 : Pointer
│ │ │  
│ │ │  i7 : x = charstar "Hello, world!"
│ │ │  
│ │ │  o7 = Hello, world!
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Object.html
│ │ │ @@ -69,27 +69,27 @@
│ │ │  o1 : ForeignObject of type int32
    
│ │ │              
    
│ │ │                
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7fe4e7633270}
    
│ │ │ +o2 = int32{Address => 0x7fa30daac310}
│ │ │              
    
│ │ │          
    
│ │ │            
    To get this, use address.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : address x
│ │ │  
│ │ │ -o3 = 0x7fe4e7633270
│ │ │ +o3 = 0x7fa30daac310
│ │ │  
│ │ │  o3 : Pointer
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Use class to determine the type of the object.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,19 +10,19 @@
│ │ │ │  i1 : x = int 5
│ │ │ │  
│ │ │ │  o1 = 5
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : peek x
│ │ │ │  
│ │ │ │ -o2 = int32{Address => 0x7fe4e7633270}
│ │ │ │ +o2 = int32{Address => 0x7fa30daac310}
│ │ │ │  To get this, use _a_d_d_r_e_s_s.
│ │ │ │  i3 : address x
│ │ │ │  
│ │ │ │ -o3 = 0x7fe4e7633270
│ │ │ │ +o3 = 0x7fa30daac310
│ │ │ │  
│ │ │ │  o3 : Pointer
│ │ │ │  Use _c_l_a_s_s to determine the type of the object.
│ │ │ │  i4 : class x
│ │ │ │  
│ │ │ │  o4 = int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │  o2 : ForeignObject of type char**
    
│ │ │              
    
│ │ │                
    i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o3 = {0x7fe4e7675a10, 0x7fe4e7675a00, 0x7fe4e76759f0}
│ │ │ +o3 = {0x7fa30daeba40, 0x7fa30daeba30, 0x7fa30daeba20}
│ │ │  
│ │ │  o3 : ForeignObject of type void**
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Foreign pointer arrays may be subscripted using _.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │           "lazy", "dog"}
│ │ │ │  
│ │ │ │  o2 = {the, quick, brown, fox, jumps, over, the, lazy, dog}
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type char**
│ │ │ │  i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │ │  
│ │ │ │ -o3 = {0x7fe4e7675a10, 0x7fe4e7675a00, 0x7fe4e76759f0}
│ │ │ │ +o3 = {0x7fa30daeba40, 0x7fa30daeba30, 0x7fa30daeba20}
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type void**
│ │ │ │  Foreign pointer arrays may be subscripted using __.
│ │ │ │  i4 : x = charstarstar {"foo", "bar", "baz"}
│ │ │ │  
│ │ │ │  o4 = {foo, bar, baz}
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type_sp__Visible__List.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │  o1 : ForeignObject of type char**
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o2 = {0x7fe4e7675810, 0x7fe4e7675800, 0x7fe4e76757f0}
│ │ │ +o2 = {0x7fa30daeb880, 0x7fa30daeb870, 0x7fa30daeb860}
│ │ │  
│ │ │  o2 : ForeignObject of type void**
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  i1 : charstarstar {"foo", "bar"}
│ │ │ │  
│ │ │ │  o1 = {foo, bar}
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type char**
│ │ │ │  i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │ │  
│ │ │ │ -o2 = {0x7fe4e7675810, 0x7fe4e7675800, 0x7fe4e76757f0}
│ │ │ │ +o2 = {0x7fa30daeb880, 0x7fa30daeb870, 0x7fa30daeb860}
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type void**
│ │ │ │  i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │ │  
│ │ │ │  o3 = int32[2]*
│ │ │ │  
│ │ │ │  o3 : ForeignPointerArrayType
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Type_sp__Pointer.html
│ │ │ @@ -78,24 +78,24 @@
│ │ │            
    To cast a Macaulay2 pointer to a foreign object with a pointer type, give the type followed by the pointer.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : ptr = address int 0
│ │ │  
│ │ │ -o1 = 0x7fe4e7889f50
│ │ │ +o1 = 0x7fa30e32bf10
│ │ │  
│ │ │  o1 : Pointer
    │ │ │
    │ │ │ 
    i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7fe4e7889f50
│ │ │ +o2 = 0x7fa30e32bf10
│ │ │  
│ │ │  o2 : ForeignObject of type void*
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -15,20 +15,20 @@ │ │ │ │ * Outputs: │ │ │ │ o a _f_o_r_e_i_g_n_ _o_b_j_e_c_t, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ To cast a Macaulay2 pointer to a foreign object with a pointer type, give the │ │ │ │ type followed by the pointer. │ │ │ │ i1 : ptr = address int 0 │ │ │ │ │ │ │ │ -o1 = 0x7fe4e7889f50 │ │ │ │ +o1 = 0x7fa30e32bf10 │ │ │ │ │ │ │ │ o1 : Pointer │ │ │ │ i2 : voidstar ptr │ │ │ │ │ │ │ │ -o2 = 0x7fe4e7889f50 │ │ │ │ +o2 = 0x7fa30e32bf10 │ │ │ │ │ │ │ │ o2 : ForeignObject of type void* │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _F_o_r_e_i_g_n_P_o_i_n_t_e_r_T_y_p_e_ _P_o_i_n_t_e_r -- cast a Macaulay2 pointer to a foreign │ │ │ │ pointer │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp__Pointer.html │ │ │ @@ -87,15 +87,15 @@ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7fe4e7651db0
│ │ │ +o2 = 0x7fa30dad2d40
│ │ │  
│ │ │  o2 : Pointer
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : int ptr
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  i1 : x = int 5
│ │ │ │  
│ │ │ │  o1 = 5
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : ptr = address x
│ │ │ │  
│ │ │ │ -o2 = 0x7fe4e7651db0
│ │ │ │ +o2 = 0x7fa30dad2d40
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  i3 : int ptr
│ │ │ │  
│ │ │ │  o3 = 5
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp_st_spvoidstar.html
│ │ │ @@ -78,15 +78,15 @@
│ │ │            
    This is syntactic sugar for T value ptr (see ForeignType Pointer) for dereferencing pointers.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : ptr = voidstar address int 5
│ │ │  
│ │ │ -o1 = 0x7fe4e7651b20
│ │ │ +o1 = 0x7fa30dad2b30
│ │ │  
│ │ │  o1 : ForeignObject of type void*
    │ │ │
    │ │ │ 
    i2 : int * ptr
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _f_o_r_e_i_g_n_ _o_b_j_e_c_t, of type T;
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This is syntactic sugar for T value ptr (see _F_o_r_e_i_g_n_T_y_p_e_ _P_o_i_n_t_e_r) for
│ │ │ │  dereferencing pointers.
│ │ │ │  i1 : ptr = voidstar address int 5
│ │ │ │  
│ │ │ │ -o1 = 0x7fe4e7651b20
│ │ │ │ +o1 = 0x7fa30dad2b30
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type void*
│ │ │ │  i2 : int * ptr
│ │ │ │  
│ │ │ │  o2 = 5
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Union__Type_sp__Thing.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │  o1 : ForeignUnionType
    │ │ │
    │ │ │ 
    i2 : myunion 27
│ │ │  
│ │ │ -o2 = HashTable{"bar" => 6.94741e-310}
│ │ │ +o2 = HashTable{"bar" => 6.93362e-310}
│ │ │                 "foo" => 27
│ │ │  
│ │ │  o2 : ForeignObject of type myunion
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -20,15 +20,15 @@ │ │ │ │ i1 : myunion = foreignUnionType("myunion", {"foo" => int, "bar" => double}) │ │ │ │ │ │ │ │ o1 = myunion │ │ │ │ │ │ │ │ o1 : ForeignUnionType │ │ │ │ i2 : myunion 27 │ │ │ │ │ │ │ │ -o2 = HashTable{"bar" => 6.94741e-310} │ │ │ │ +o2 = HashTable{"bar" => 6.93362e-310} │ │ │ │ "foo" => 27 │ │ │ │ │ │ │ │ o2 : ForeignObject of type myunion │ │ │ │ i3 : myunion pi │ │ │ │ │ │ │ │ o3 = HashTable{"bar" => 3.14159 } │ │ │ │ "foo" => 1413754136 │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Pointer.html │ │ │ @@ -69,50 +69,50 @@ │ │ │ o1 : ForeignObject of type int32 │ │ │
    │ │ │ 
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7fe4e76332d0}
    │ │ │ +o2 = int32{Address => 0x7fa30daac330} │ │ │
    │ │ │
    │ │ │

    These pointers can be accessed using address.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7fe4e76332d0
│ │ │ +o3 = 0x7fa30daac330
│ │ │  
│ │ │  o3 : Pointer
    │ │ │
    │ │ │
    │ │ │

    Simple arithmetic can be performed on pointers.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7fe4e76332d5
│ │ │ +o4 = 0x7fa30daac335
│ │ │  
│ │ │  o4 : Pointer
    │ │ │
    │ │ │ 
    i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7fe4e76332cd
│ │ │ +o5 = 0x7fa30daac32d
│ │ │  
│ │ │  o5 : Pointer
    │ │ │
    │ │ │ │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -10,30 +10,30 @@ │ │ │ │ i1 : x = int 20 │ │ │ │ │ │ │ │ o1 = 20 │ │ │ │ │ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ i2 : peek x │ │ │ │ │ │ │ │ -o2 = int32{Address => 0x7fe4e76332d0} │ │ │ │ +o2 = int32{Address => 0x7fa30daac330} │ │ │ │ These pointers can be accessed using _a_d_d_r_e_s_s. │ │ │ │ i3 : ptr = address x │ │ │ │ │ │ │ │ -o3 = 0x7fe4e76332d0 │ │ │ │ +o3 = 0x7fa30daac330 │ │ │ │ │ │ │ │ o3 : Pointer │ │ │ │ Simple arithmetic can be performed on pointers. │ │ │ │ i4 : ptr + 5 │ │ │ │ │ │ │ │ -o4 = 0x7fe4e76332d5 │ │ │ │ +o4 = 0x7fa30daac335 │ │ │ │ │ │ │ │ o4 : Pointer │ │ │ │ i5 : ptr - 3 │ │ │ │ │ │ │ │ -o5 = 0x7fe4e76332cd │ │ │ │ +o5 = 0x7fa30daac32d │ │ │ │ │ │ │ │ o5 : Pointer │ │ │ │ ******** MMeennuu ******** │ │ │ │ * _n_u_l_l_P_o_i_n_t_e_r -- the null pointer │ │ │ │ * _a_d_d_r_e_s_s -- pointer to type or object │ │ │ │ * _F_o_r_e_i_g_n_T_y_p_e_ _P_o_i_n_t_e_r -- dereference a pointer │ │ │ │ ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aa ppooiinntteerr:: ********** │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Shared__Library.html │ │ │ @@ -69,15 +69,15 @@ │ │ │ o1 : SharedLibrary │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : peek mpfr
│ │ │  
│ │ │ -o2 = SharedLibrary{0x7fe4f7389ac0, mpfr}
    │ │ │ +o2 = SharedLibrary{0x7fa322db5ac0, mpfr} │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Menu

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : address int
│ │ │  
│ │ │ -o1 = 0x555fc7f2b400
│ │ │ +o1 = 0x56340a989400
│ │ │  
│ │ │  o1 : Pointer
    │ │ │
    │ │ │
    │ │ │

    If x is a foreign object, then this returns the address to the object. It behaves like the & "address-of" operator in C.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i2 : address int 5
│ │ │  
│ │ │ -o2 = 0x7fe4e7651d00
│ │ │ +o2 = 0x7fa30dad2d60
│ │ │  
│ │ │  o2 : Pointer
    │ │ │
    │ │ │ │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -11,22 +11,22 @@ │ │ │ │ * Outputs: │ │ │ │ o a _p_o_i_n_t_e_r, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ If x is a foreign type, then this returns the address to the ffi_type struct │ │ │ │ used by libffi to identify the type. │ │ │ │ i1 : address int │ │ │ │ │ │ │ │ -o1 = 0x555fc7f2b400 │ │ │ │ +o1 = 0x56340a989400 │ │ │ │ │ │ │ │ o1 : Pointer │ │ │ │ If x is a foreign object, then this returns the address to the object. It │ │ │ │ behaves like the & "address-of" operator in C. │ │ │ │ i2 : address int 5 │ │ │ │ │ │ │ │ -o2 = 0x7fe4e7651d00 │ │ │ │ +o2 = 0x7fa30dad2d60 │ │ │ │ │ │ │ │ o2 : Pointer │ │ │ │ ********** WWaayyss ttoo uussee aaddddrreessss:: ********** │ │ │ │ * address(ForeignObject) │ │ │ │ * address(ForeignType) │ │ │ │ * address(Nothing) (missing documentation) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_foreign__Function.html │ │ │ @@ -237,15 +237,15 @@ │ │ │ o16 : ForeignFunction │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : x = malloc 8
│ │ │  
│ │ │ -o17 = 0x7f71f006a700
│ │ │ +o17 = 0x7fbbd406a700
│ │ │  
│ │ │  o17 : ForeignObject of type void*
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i18 : registerFinalizer(x, free)
    │ │ │ ├── html2text {} │ │ │ │ @@ -95,15 +95,15 @@ │ │ │ │ i16 : free = foreignFunction("free", void, voidstar) │ │ │ │ │ │ │ │ o16 = free │ │ │ │ │ │ │ │ o16 : ForeignFunction │ │ │ │ i17 : x = malloc 8 │ │ │ │ │ │ │ │ -o17 = 0x7f71f006a700 │ │ │ │ +o17 = 0x7fbbd406a700 │ │ │ │ │ │ │ │ o17 : ForeignObject of type void* │ │ │ │ i18 : registerFinalizer(x, free) │ │ │ │ ********** WWaayyss ttoo uussee ffoorreeiiggnnFFuunnccttiioonn:: ********** │ │ │ │ * foreignFunction(Pointer,String,ForeignType,VisibleList) │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,ForeignType) │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,VisibleList) │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_get__Memory.html │ │ │ @@ -82,43 +82,43 @@ │ │ │

    Allocate n bytes of memory using the GC garbage collector.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : ptr = getMemory 8
│ │ │  
│ │ │ -o1 = 0x7fe4f2bac9b0
│ │ │ +o1 = 0x7fa30ec0e860
│ │ │  
│ │ │  o1 : ForeignObject of type void*
    │ │ │
    │ │ │
    │ │ │

    If the memory will not contain any pointers, then set the Atomic option to true.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i2 : ptr = getMemory(8, Atomic => true)
│ │ │  
│ │ │ -o2 = 0x7fe4e7651be0
│ │ │ +o2 = 0x7fa30dad2ba0
│ │ │  
│ │ │  o2 : ForeignObject of type void*
    │ │ │
    │ │ │
    │ │ │

    Alternatively, a foreign object type T may be specified. In this case, the number of bytes and whether the Atomic option should be set will be determined automatically.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i3 : ptr = getMemory int
│ │ │  
│ │ │ -o3 = 0x7fe4e7651af0
│ │ │ +o3 = 0x7fa30dad2a90
│ │ │  
│ │ │  o3 : ForeignObject of type void*
    │ │ │
    │ │ │ │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -14,30 +14,30 @@ │ │ │ │ o Atomic => ..., default value false │ │ │ │ * Outputs: │ │ │ │ o an instance of the type _v_o_i_d_s_t_a_r, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ Allocate n bytes of memory using the _G_C_ _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r. │ │ │ │ i1 : ptr = getMemory 8 │ │ │ │ │ │ │ │ -o1 = 0x7fe4f2bac9b0 │ │ │ │ +o1 = 0x7fa30ec0e860 │ │ │ │ │ │ │ │ o1 : ForeignObject of type void* │ │ │ │ If the memory will not contain any pointers, then set the Atomic option to │ │ │ │ _t_r_u_e. │ │ │ │ i2 : ptr = getMemory(8, Atomic => true) │ │ │ │ │ │ │ │ -o2 = 0x7fe4e7651be0 │ │ │ │ +o2 = 0x7fa30dad2ba0 │ │ │ │ │ │ │ │ o2 : ForeignObject of type void* │ │ │ │ Alternatively, a foreign object type T may be specified. In this case, the │ │ │ │ number of bytes and whether the Atomic option should be set will be determined │ │ │ │ automatically. │ │ │ │ i3 : ptr = getMemory int │ │ │ │ │ │ │ │ -o3 = 0x7fe4e7651af0 │ │ │ │ +o3 = 0x7fa30dad2a90 │ │ │ │ │ │ │ │ o3 : ForeignObject of type void* │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_e_g_i_s_t_e_r_F_i_n_a_l_i_z_e_r_(_F_o_r_e_i_g_n_O_b_j_e_c_t_,_F_u_n_c_t_i_o_n_) -- register a finalizer for a │ │ │ │ foreign object │ │ │ │ ********** WWaayyss ttoo uussee ggeettMMeemmoorryy:: ********** │ │ │ │ * getMemory(ForeignType) │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_register__Finalizer_lp__Foreign__Object_cm__Function_rp.html │ │ │ @@ -105,23 +105,23 @@ │ │ │ │ │ │ 
    i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer))
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : collectGarbage()
│ │ │ -freeing memory at 0x7fe4dc07fcf0
│ │ │ -freeing memory at 0x7fe4dc07f650
│ │ │ -freeing memory at 0x7fe4dc07fd30
│ │ │ -freeing memory at 0x7fe4dc07fd10
│ │ │ -freeing memory at 0x7fe4dc07fdb0
│ │ │ -freeing memory at 0x7fe4dc07fd90
│ │ │ -freeing memory at 0x7fe4dc07fd70
│ │ │ -freeing memory at 0x7fe4dc07fd50
│ │ │ -freeing memory at 0x7fe4dc07f630
    │ │ │ +freeing memory at 0x7fa2f807fd50 │ │ │ +freeing memory at 0x7fa2f807fd70 │ │ │ +freeing memory at 0x7fa2f807fd90 │ │ │ +freeing memory at 0x7fa2f807fdb0 │ │ │ +freeing memory at 0x7fa2f807f630 │ │ │ +freeing memory at 0x7fa2f807fcf0 │ │ │ +freeing memory at 0x7fa2f807fd10 │ │ │ +freeing memory at 0x7fa2f807f650 │ │ │ +freeing memory at 0x7fa2f807fd30 │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : x = voidstar address int 5
│ │ │  
│ │ │ -o5 = 0x7fe4e7651fa0
│ │ │ +o5 = 0x7fa30dad2010
│ │ │  
│ │ │  o5 : ForeignObject of type void*
    │ │ │
    │ │ │ 
    i6 : value x
│ │ │  
│ │ │ -o6 = 0x7fe4e7651fa0
│ │ │ +o6 = 0x7fa30dad2010
│ │ │  
│ │ │  o6 : Pointer
    │ │ │
    │ │ │
    │ │ │

    Foreign string objects are converted to strings.

    │ │ │ ├── html2text {} │ │ │ │ @@ -34,20 +34,20 @@ │ │ │ │ │ │ │ │ o4 = 5 │ │ │ │ │ │ │ │ o4 : RR (of precision 53) │ │ │ │ Foreign pointer objects are converted to _P_o_i_n_t_e_r objects. │ │ │ │ i5 : x = voidstar address int 5 │ │ │ │ │ │ │ │ -o5 = 0x7fe4e7651fa0 │ │ │ │ +o5 = 0x7fa30dad2010 │ │ │ │ │ │ │ │ o5 : ForeignObject of type void* │ │ │ │ i6 : value x │ │ │ │ │ │ │ │ -o6 = 0x7fe4e7651fa0 │ │ │ │ +o6 = 0x7fa30dad2010 │ │ │ │ │ │ │ │ o6 : Pointer │ │ │ │ Foreign string objects are converted to strings. │ │ │ │ i7 : x = charstar "Hello, world!" │ │ │ │ │ │ │ │ o7 = Hello, world! │ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/example-output/_put__Matrix.out │ │ │ @@ -6,27 +6,27 @@ │ │ │ | 1 2 3 4 | │ │ │ │ │ │ 2 4 │ │ │ o1 : Matrix ZZ <-- ZZ │ │ │ │ │ │ i2 : s = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-15150-0/0 │ │ │ +o2 = /tmp/M2-18980-0/0 │ │ │ │ │ │ i3 : F = openOut(s) │ │ │ │ │ │ -o3 = /tmp/M2-15150-0/0 │ │ │ +o3 = /tmp/M2-18980-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : putMatrix(F,A) │ │ │ │ │ │ i5 : close(F) │ │ │ │ │ │ -o5 = /tmp/M2-15150-0/0 │ │ │ +o5 = /tmp/M2-18980-0/0 │ │ │ │ │ │ o5 : File │ │ │ │ │ │ i6 : getMatrix(s) │ │ │ │ │ │ o6 = | 1 1 1 1 | │ │ │ | 1 2 3 4 | │ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/html/_put__Matrix.html │ │ │ @@ -84,36 +84,36 @@ │ │ │ o1 : Matrix ZZ <-- ZZ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : s = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-15150-0/0
    │ │ │ +o2 = /tmp/M2-18980-0/0 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : F = openOut(s)
│ │ │  
│ │ │ -o3 = /tmp/M2-15150-0/0
│ │ │ +o3 = /tmp/M2-18980-0/0
│ │ │  
│ │ │  o3 : File
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : putMatrix(F,A)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : close(F)
│ │ │  
│ │ │ -o5 = /tmp/M2-15150-0/0
│ │ │ +o5 = /tmp/M2-18980-0/0
│ │ │  
│ │ │  o5 : File
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : getMatrix(s)
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,24 +16,24 @@
│ │ │ │  o1 = | 1 1 1 1 |
│ │ │ │       | 1 2 3 4 |
│ │ │ │  
│ │ │ │                2       4
│ │ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │ │  i2 : s = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-15150-0/0
│ │ │ │ +o2 = /tmp/M2-18980-0/0
│ │ │ │  i3 : F = openOut(s)
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-15150-0/0
│ │ │ │ +o3 = /tmp/M2-18980-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : putMatrix(F,A)
│ │ │ │  i5 : close(F)
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-15150-0/0
│ │ │ │ +o5 = /tmp/M2-18980-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : getMatrix(s)
│ │ │ │  
│ │ │ │  o6 = | 1 1 1 1 |
│ │ │ │       | 1 2 3 4 |
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out
│ │ │ @@ -155,31 +155,31 @@
│ │ │  i26 : numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true) -- FinalAttempt improves the estimate slightly
│ │ │  
│ │ │  o26 = {.142067, .144}
│ │ │  
│ │ │  o26 : List
│ │ │  
│ │ │  i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 3.69443s (cpu); 1.58499s (thread); 0s (gc)
│ │ │ + -- used 4.23399s (cpu); 1.83794s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 2.04495s (cpu); 0.987245s (thread); 0s (gc)
│ │ │ + -- used 2.3898s (cpu); 1.05364s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
│ │ │  
│ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 1.7695s (cpu); 0.778249s (thread); 0s (gc)
│ │ │ + -- used 2.02503s (cpu); 0.869862s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o29 = -
│ │ │        7
│ │ │  
│ │ │  o29 : QQ
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out
│ │ │ @@ -43,34 +43,34 @@
│ │ │  o12 = 220
│ │ │  
│ │ │  i13 : R = ZZ/17[x,y,z];
│ │ │  
│ │ │  i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
│ │ │  
│ │ │  i15 : time frobeniusNu(3, f)
│ │ │ - -- used 0.00420762s (cpu); 0.00420483s (thread); 0s (gc)
│ │ │ + -- used 0.00537491s (cpu); 0.00537429s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
│ │ │  
│ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.594472s (cpu); 0.311617s (thread); 0s (gc)
│ │ │ + -- used 0.658662s (cpu); 0.352716s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 3756
│ │ │  
│ │ │  i17 : R = ZZ/5[x,y,z];
│ │ │  
│ │ │  i18 : f = x^3 + y^3 + z^3 + x*y*z;
│ │ │  
│ │ │  i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default
│ │ │ - -- used 0.522479s (cpu); 0.242148s (thread); 0s (gc)
│ │ │ + -- used 0.585393s (cpu); 0.272607s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
│ │ │  
│ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.63159s (cpu); 1.17192s (thread); 0s (gc)
│ │ │ + -- used 1.55632s (cpu); 1.30753s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = 499
│ │ │  
│ │ │  i21 : R = ZZ/3[x,y];
│ │ │  
│ │ │  i22 : M = ideal(x, y);
│ │ │  
│ │ │ @@ -85,34 +85,34 @@
│ │ │  o24 = 8
│ │ │  
│ │ │  i25 : R = ZZ/5[x,y,z];
│ │ │  
│ │ │  i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
│ │ │  
│ │ │  i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 1.4647s (cpu); 0.759159s (thread); 0s (gc)
│ │ │ + -- used 1.64701s (cpu); 0.847701s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
│ │ │  
│ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 2.13257s (cpu); 1.03204s (thread); 0s (gc)
│ │ │ + -- used 2.37011s (cpu); 1.20719s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 1124
│ │ │  
│ │ │  i29 : M = ideal(x, y, z);
│ │ │  
│ │ │  o29 : Ideal of R
│ │ │  
│ │ │  i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ - -- used 1.94067s (cpu); 1.4349s (thread); 0s (gc)
│ │ │ + -- used 1.84578s (cpu); 1.50779s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
│ │ │  
│ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.640315s (cpu); 0.493893s (thread); 0s (gc)
│ │ │ + -- used 0.572141s (cpu); 0.494164s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 97
│ │ │  
│ │ │  i32 : R = ZZ/7[x,y];
│ │ │  
│ │ │  i33 : f = (x - 1)^3 - (y - 2)^2;
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html
│ │ │ @@ -368,37 +368,37 @@
│ │ │          
    
│ │ │            
    The computations performed when FinalAttempt is set to true are often slow, and often fail to improve the estimate, and for this reason, this option should be used sparingly. It is often more effective to increase the values of Attempts or DepthOfSearch, instead.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -228,29 +228,29 @@
│ │ │ │  
│ │ │ │  o26 : List
│ │ │ │  The computations performed when FinalAttempt is set to true are often slow, and
│ │ │ │  often fail to improve the estimate, and for this reason, this option should be
│ │ │ │  used sparingly. It is often more effective to increase the values of Attempts
│ │ │ │  or DepthOfSearch, instead.
│ │ │ │  i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ │ - -- used 3.69443s (cpu); 1.58499s (thread); 0s (gc)
│ │ │ │ + -- used 4.23399s (cpu); 1.83794s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = {.142067, .144}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ │ - -- used 2.04495s (cpu); 0.987245s (thread); 0s (gc)
│ │ │ │ + -- used 2.3898s (cpu); 1.05364s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o28 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o28 : QQ
│ │ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ │ - -- used 1.7695s (cpu); 0.778249s (thread); 0s (gc)
│ │ │ │ + -- used 2.02503s (cpu); 0.869862s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o29 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o29 : QQ
│ │ │ │  As seen in several examples above, when the exact answer is not found, a list
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html
│ │ │ @@ -197,23 +197,23 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 3.69443s (cpu); 1.58499s (thread); 0s (gc)
│ │ │ + -- used 4.23399s (cpu); 1.83794s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
    
│ │ │              
    
│ │ │                
    i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 2.04495s (cpu); 0.987245s (thread); 0s (gc)
│ │ │ + -- used 2.3898s (cpu); 1.05364s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
    
│ │ │              
    
│ │ │                
    i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 1.7695s (cpu); 0.778249s (thread); 0s (gc)
│ │ │ + -- used 2.02503s (cpu); 0.869862s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o29 = -
│ │ │        7
│ │ │  
│ │ │  o29 : QQ
    
│ │ │                  		
│ │ │                
    i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
    
│ │ │              
    
│ │ │                
    i15 : time frobeniusNu(3, f)
│ │ │ - -- used 0.00420762s (cpu); 0.00420483s (thread); 0s (gc)
│ │ │ + -- used 0.00537491s (cpu); 0.00537429s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
    
│ │ │              
    
│ │ │                
    i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.594472s (cpu); 0.311617s (thread); 0s (gc)
│ │ │ + -- used 0.658662s (cpu); 0.352716s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 3756
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The valid values for the option ContainmentTest are FrobeniusPower, FrobeniusRoot, and StandardPower. The default value of this option depends on what is passed to frobeniusNu. Indeed, by default, ContainmentTest is set to FrobeniusRoot if frobeniusNu is passed a ring element $f$, and is set to StandardPower if frobeniusNu is passed an ideal $I$. We describe the consequences of setting ContainmentTest to each of these values below.
    
│ │ │ @@ -230,23 +230,23 @@
│ │ │              
│ │ │                
    i18 : f = x^3 + y^3 + z^3 + x*y*z;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default
│ │ │ - -- used 0.522479s (cpu); 0.242148s (thread); 0s (gc)
│ │ │ + -- used 0.585393s (cpu); 0.272607s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.63159s (cpu); 1.17192s (thread); 0s (gc)
│ │ │ + -- used 1.55632s (cpu); 1.30753s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = 499
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Finally, when ContainmentTest is set to FrobeniusPower, then instead of producing the invariant $\nu_I^J(p^e)$ as defined above, frobeniusNu instead outputs the maximal integer $n$ such that the $n$^{th} (generalized) Frobenius power of $I$ is not contained in the $p^e$-th Frobenius power of $J$. Here, the $n$^{th} Frobenius power of $I$, when $n$ is a nonnegative integer, is as defined in the paper Frobenius Powers by Hernández, Teixeira, and Witt, which can be computed with the function frobeniusPower, from the TestIdeals package. In particular, frobeniusNu(e,I,J) and frobeniusNu(e,I,J,ContainmentTest=>FrobeniusPower) need not agree. However, they will agree when $I$ is a principal ideal.
    
│ │ │ @@ -292,46 +292,46 @@
│ │ │              
│ │ │                
    i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 1.4647s (cpu); 0.759159s (thread); 0s (gc)
│ │ │ + -- used 1.64701s (cpu); 0.847701s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 2.13257s (cpu); 1.03204s (thread); 0s (gc)
│ │ │ + -- used 2.37011s (cpu); 1.20719s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 1124
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : M = ideal(x, y, z);
│ │ │  
│ │ │  o29 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ - -- used 1.94067s (cpu); 1.4349s (thread); 0s (gc)
│ │ │ + -- used 1.84578s (cpu); 1.50779s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.640315s (cpu); 0.493893s (thread); 0s (gc)
│ │ │ + -- used 0.572141s (cpu); 0.494164s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 97
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    The option AtOrigin (default value true) can be turned off to tell frobeniusNu to effectively do the computation over all possible maximal ideals $J$ and take the minimum.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -106,19 +106,19 @@
│ │ │ │  algorithms, namely diagonal polynomials, binomials, forms in two variables, and
│ │ │ │  polynomials whose factors are in simple normal crossing. This feature can be
│ │ │ │  disabled by setting the option UseSpecialAlgorithms (default value true) to
│ │ │ │  false.
│ │ │ │  i13 : R = ZZ/17[x,y,z];
│ │ │ │  i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
│ │ │ │  i15 : time frobeniusNu(3, f)
│ │ │ │ - -- used 0.00420762s (cpu); 0.00420483s (thread); 0s (gc)
│ │ │ │ + -- used 0.00537491s (cpu); 0.00537429s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3756
│ │ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ │ - -- used 0.594472s (cpu); 0.311617s (thread); 0s (gc)
│ │ │ │ + -- used 0.658662s (cpu); 0.352716s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 3756
│ │ │ │  The valid values for the option ContainmentTest are FrobeniusPower,
│ │ │ │  FrobeniusRoot, and StandardPower. The default value of this option depends on
│ │ │ │  what is passed to frobeniusNu. Indeed, by default, ContainmentTest is set to
│ │ │ │  FrobeniusRoot if frobeniusNu is passed a ring element $f$, and is set to
│ │ │ │  StandardPower if frobeniusNu is passed an ideal $I$. We describe the
│ │ │ │ @@ -133,19 +133,19 @@
│ │ │ │  is contained in $J$. The output is unaffected, but this option often speeds up
│ │ │ │  computations, specially when a polynomial or principal ideal is passed as the
│ │ │ │  second argument.
│ │ │ │  i17 : R = ZZ/5[x,y,z];
│ │ │ │  i18 : f = x^3 + y^3 + z^3 + x*y*z;
│ │ │ │  i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by
│ │ │ │  default
│ │ │ │ - -- used 0.522479s (cpu); 0.242148s (thread); 0s (gc)
│ │ │ │ + -- used 0.585393s (cpu); 0.272607s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o19 = 499
│ │ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ │ - -- used 1.63159s (cpu); 1.17192s (thread); 0s (gc)
│ │ │ │ + -- used 1.55632s (cpu); 1.30753s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 = 499
│ │ │ │  Finally, when ContainmentTest is set to FrobeniusPower, then instead of
│ │ │ │  producing the invariant $\nu_I^J(p^e)$ as defined above, frobeniusNu instead
│ │ │ │  outputs the maximal integer $n$ such that the $n$^{th} (generalized) Frobenius
│ │ │ │  power of $I$ is not contained in the $p^e$-th Frobenius power of $J$. Here, the
│ │ │ │  $n$^{th} Frobenius power of $I$, when $n$ is a nonnegative integer, is as
│ │ │ │ @@ -167,31 +167,31 @@
│ │ │ │  The function frobeniusNu works by searching through the list of potential
│ │ │ │  integers $n$ and checking containments of $I^n$ in the specified Frobenius
│ │ │ │  power of $J$. The way this search is approached is specified by the option
│ │ │ │  Search, which can be set to Binary (the default value) or Linear.
│ │ │ │  i25 : R = ZZ/5[x,y,z];
│ │ │ │  i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
│ │ │ │  i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ │ - -- used 1.4647s (cpu); 0.759159s (thread); 0s (gc)
│ │ │ │ + -- used 1.64701s (cpu); 0.847701s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = 1124
│ │ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ │ - -- used 2.13257s (cpu); 1.03204s (thread); 0s (gc)
│ │ │ │ + -- used 2.37011s (cpu); 1.20719s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = 1124
│ │ │ │  i29 : M = ideal(x, y, z);
│ │ │ │  
│ │ │ │  o29 : Ideal of R
│ │ │ │  i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ │ - -- used 1.94067s (cpu); 1.4349s (thread); 0s (gc)
│ │ │ │ + -- used 1.84578s (cpu); 1.50779s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 97
│ │ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets
│ │ │ │  luckier
│ │ │ │ - -- used 0.640315s (cpu); 0.493893s (thread); 0s (gc)
│ │ │ │ + -- used 0.572141s (cpu); 0.494164s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = 97
│ │ │ │  The option AtOrigin (default value true) can be turned off to tell frobeniusNu
│ │ │ │  to effectively do the computation over all possible maximal ideals $J$ and take
│ │ │ │  the minimum.
│ │ │ │  i32 : R = ZZ/7[x,y];
│ │ │ │  i33 : f = (x - 1)^3 - (y - 2)^2;
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out
│ │ │ @@ -208,21 +208,21 @@
│ │ │        | 3/7 5/4 3/7  10   |
│ │ │        | 6/7 2/9 5    3/2  |
│ │ │  
│ │ │                 3       4
│ │ │  o26 : Matrix QQ  <-- QQ
│ │ │  
│ │ │  i27 : time C = orbitClosure(X,Mat)
│ │ │ - -- used 0.608397s (cpu); 0.358609s (thread); 0s (gc)
│ │ │ + -- used 2.07542s (cpu); 0.49664s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
│ │ │  
│ │ │  i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 1.89318s (cpu); 0.987542s (thread); 0s (gc)
│ │ │ + -- used 3.10898s (cpu); 1.03761s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o28 : KClass
│ │ │  
│ │ │  i29 :
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html
│ │ │ @@ -391,25 +391,25 @@
│ │ │                 3       4
│ │ │  o26 : Matrix QQ  <-- QQ
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i27 : time C = orbitClosure(X,Mat)
│ │ │ - -- used 0.608397s (cpu); 0.358609s (thread); 0s (gc)
│ │ │ + -- used 2.07542s (cpu); 0.49664s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 1.89318s (cpu); 0.987542s (thread); 0s (gc)
│ │ │ + -- used 3.10898s (cpu); 1.03761s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o28 : KClass
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -241,21 +241,21 @@ │ │ │ │ o26 = | 7 6 3/10 10/9 | │ │ │ │ | 3/7 5/4 3/7 10 | │ │ │ │ | 6/7 2/9 5 3/2 | │ │ │ │ │ │ │ │ 3 4 │ │ │ │ o26 : Matrix QQ <-- QQ │ │ │ │ i27 : time C = orbitClosure(X,Mat) │ │ │ │ - -- used 0.608397s (cpu); 0.358609s (thread); 0s (gc) │ │ │ │ + -- used 2.07542s (cpu); 0.49664s (thread); 0s (gc) │ │ │ │ │ │ │ │ o27 = an "equivariant K-class" on a GKM variety │ │ │ │ │ │ │ │ o27 : KClass │ │ │ │ i28 : time C = orbitClosure(X,Mat, RREFMethod => true) │ │ │ │ - -- used 1.89318s (cpu); 0.987542s (thread); 0s (gc) │ │ │ │ + -- used 3.10898s (cpu); 1.03761s (thread); 0s (gc) │ │ │ │ │ │ │ │ o28 = an "equivariant K-class" on a GKM variety │ │ │ │ │ │ │ │ o28 : KClass │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_n_e_r_a_l_i_z_e_d_F_l_a_g_V_a_r_i_e_t_y -- makes a generalized flag variety as a GKM │ │ │ │ variety │ │ ├── ./usr/share/doc/Macaulay2/Graphs/example-output/_new__Digraph.out │ │ │ @@ -32,12 +32,12 @@ │ │ │ 5 => {6} │ │ │ 6 => {} │ │ │ │ │ │ o2 : SortedDigraph │ │ │ │ │ │ i3 : keys H │ │ │ │ │ │ -o3 = {newDigraph, map, digraph} │ │ │ +o3 = {digraph, map, newDigraph} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Graphs/html/_new__Digraph.html │ │ │ @@ -100,15 +100,15 @@ │ │ │ o2 : SortedDigraph │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : keys H
│ │ │  
│ │ │ -o3 = {newDigraph, map, digraph}
│ │ │ +o3 = {digraph, map, newDigraph}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -36,15 +36,15 @@ │ │ │ │ 4 => {} │ │ │ │ 5 => {6} │ │ │ │ 6 => {} │ │ │ │ │ │ │ │ o2 : SortedDigraph │ │ │ │ i3 : keys H │ │ │ │ │ │ │ │ -o3 = {newDigraph, map, digraph} │ │ │ │ +o3 = {digraph, map, newDigraph} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_o_p_S_o_r_t -- topologically sort the vertices of a digraph │ │ │ │ * _S_o_r_t_e_d_D_i_g_r_a_p_h -- hashtable used in topSort │ │ │ │ * _t_o_p_o_l_o_g_i_c_a_l_S_o_r_t -- outputs a list of vertices in a topologically sorted │ │ │ │ order of a DAG. │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ @@ -243,67 +243,67 @@ │ │ │ | 2 2 2 2 2 2 2 2 2 2 2 2 | │ │ │ |t - t + t t t - t t + t t t + t t t - t t t t + 25t t + 4t t t - 2t t t t - 2t t t t - 2t t t t + t t t t + t t t t - 3t t + 3t t t t - 26t t t | │ │ │ | 6 23 16 20 22 14 22 23 22 13 23 16 19 16 22 13 19 16 19 23 20 21 20 22 13 21 16 20 19 21 23 13 19 21 22 13 19 21 16 13 19 21 20 21 20 13 19 21 13 19 21| │ │ │ +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ │ │ │ i12 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ -o12 = | -26 -24 6 39 17 28 -46 -12 11 -29 -10 -48 -36 -30 39 -29 -8 -22 -10 │ │ │ +o12 = | -14 48 41 -32 -39 30 48 36 2 -29 -30 -23 19 19 -10 -29 -8 -22 24 -13 │ │ │ ----------------------------------------------------------------------- │ │ │ - 11 19 19 24 -29 | │ │ │ + -36 -30 -29 -10 | │ │ │ │ │ │ 1 24 │ │ │ o12 : Matrix kk <-- kk │ │ │ │ │ │ i13 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ -o13 = | -30 -41 -39 30 -13 32 17 -12 29 25 -49 11 43 21 -42 -4 19 -47 39 -38 │ │ │ +o13 = | -39 -22 11 26 -28 31 38 24 -31 -16 -16 -7 -41 -24 18 -17 19 34 -38 39 │ │ │ ----------------------------------------------------------------------- │ │ │ - -16 0 -24 34 | │ │ │ + -16 0 -47 21 | │ │ │ │ │ │ 1 24 │ │ │ o13 : Matrix kk <-- kk │ │ │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ - 2 2 2 │ │ │ -o14 = ideal (a + 11b*c + 39c - 29a*d + 17b*d - 24c*d - 26d , a*b + 39b*c - │ │ │ + 2 2 2 │ │ │ +o14 = ideal (a + 2b*c - 32c - 29a*d - 39b*d + 48c*d - 14d , a*b - 10b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - 10c - 29a*d - 48b*d + 28c*d + 6d , b + 19b*c - 8c + 19a*d - 22b*d - │ │ │ + 2 2 2 2 │ │ │ + 30c - 29a*d - 23b*d + 30c*d + 41d , b - 36b*c - 8c - 30a*d - 22b*d + │ │ │ ----------------------------------------------------------------------- │ │ │ 2 2 2 │ │ │ - 36c*d - 46d , a*c + 24b*c - 10c - 29a*d + 11b*d - 30c*d - 12d ) │ │ │ + 19c*d + 48d , a*c - 29b*c + 24c - 10a*d - 13b*d + 19c*d + 36d ) │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ │ │ i15 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ 2 2 2 │ │ │ -o15 = ideal (a + 29b*c + 30c + 25a*d - 13b*d - 41c*d - 30d , a*b - 42b*c - │ │ │ +o15 = ideal (a - 31b*c + 26c - 16a*d - 28b*d - 22c*d - 39d , a*b + 18b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ 2 2 2 2 │ │ │ - 49c - 4a*d + 11b*d + 32c*d - 39d , b - 16b*c + 19c - 47b*d + 43c*d + │ │ │ + 16c - 17a*d - 7b*d + 31c*d + 11d , b - 16b*c + 19c + 34b*d - 41c*d + │ │ │ ----------------------------------------------------------------------- │ │ │ 2 2 2 │ │ │ - 17d , a*c - 24b*c + 39c + 34a*d - 38b*d + 21c*d - 12d ) │ │ │ + 38d , a*c - 47b*c - 38c + 21a*d + 39b*d - 24c*d + 24d ) │ │ │ │ │ │ o15 : Ideal of S │ │ │ │ │ │ i16 : decompose F1 │ │ │ │ │ │ 2 2 2 │ │ │ -o16 = {ideal (a + 24b - 10c - 17d, b + 19b*c - 8c + 27b*d - 48c*d - 26d ), │ │ │ +o16 = {ideal (a - 29b + 24c - 44d, b - 36b*c - 8c + 17b*d + 32c*d + 41d ), │ │ │ ----------------------------------------------------------------------- │ │ │ - ideal (c - 29d, b + 23d, a - 14d)} │ │ │ + ideal (c - 10d, b + 33d, a + 10d)} │ │ │ │ │ │ o16 : List │ │ │ │ │ │ i17 : decompose F2 │ │ │ │ │ │ -o17 = {ideal (b - 42c - 18d, a + 41c + 33d), ideal (b + 26c - 29d, a - 44c - │ │ │ +o17 = {ideal (b - 42c + 10d, a + 8c - 3d), ideal (b + 26c + 24d, a - 28c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 9d)} │ │ │ + 29d)} │ │ │ │ │ │ o17 : List │ │ │ │ │ │ i18 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out │ │ │ @@ -103,46 +103,46 @@ │ │ │ │ │ │ i13 : #compsJ │ │ │ │ │ │ o13 = 2 │ │ │ │ │ │ i14 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ -o14 = | -22 41 -11 19 -43 -7 15 4 21 -36 31 30 37 19 -9 -44 30 19 -38 1 47 24 │ │ │ +o14 = | 43 35 -43 7 38 31 47 48 46 21 8 10 6 -30 -40 10 -27 -10 -50 30 -21 │ │ │ ----------------------------------------------------------------------- │ │ │ - -29 -16 16 -29 -30 21 -10 -22 39 -24 -29 -8 -36 -38 | │ │ │ + -38 -16 -29 31 -36 39 -29 19 24 -24 -8 19 -29 21 -22 | │ │ │ │ │ │ 1 36 │ │ │ o14 : Matrix kk <-- kk │ │ │ │ │ │ i15 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ -o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47 │ │ │ +o15 = | 18 13 -48 10 27 -33 13 4 37 33 -15 46 42 -47 -35 23 45 -13 33 -43 1 7 │ │ │ ----------------------------------------------------------------------- │ │ │ - 28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 | │ │ │ + 2 -47 46 19 16 14 -18 34 38 -15 0 -39 22 -28 | │ │ │ │ │ │ 1 36 │ │ │ o15 : Matrix kk <-- kk │ │ │ │ │ │ i16 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ - 2 2 2 │ │ │ -o16 = ideal (a - 9b*c - 36c + 30a*d - 7b*d - 11c*d - 22d , a*b + 16b*c - │ │ │ + 2 2 2 │ │ │ +o16 = ideal (a - 40b*c + 21c + 10a*d + 31b*d - 43c*d + 43d , a*b + 31b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 38c + 47a*d + 37b*d + 21c*d + 41d , a*c - 24b*c - 29c + 21a*d + b*d + │ │ │ + 2 2 2 │ │ │ + 50c - 21a*d + 6b*d + 46c*d + 35d , a*c - 8b*c - 36c - 29a*d + 30b*d - │ │ │ ----------------------------------------------------------------------- │ │ │ 2 2 2 2 2 │ │ │ - 19c*d - 43d , b - 36b*c - 10c - 29a*d + 24b*d - 44c*d + 15d , b*c - │ │ │ + 30c*d + 38d , b + 21b*c + 19c + 19a*d - 38b*d + 10c*d + 47d , b*c + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 3 3 2 │ │ │ - 22b*c*d - 29c d - 30a*d + 30b*d + 31c*d + 19d , c - 38b*c*d + 39c d │ │ │ + 2 2 2 2 3 3 2 │ │ │ + 24b*c*d - 16c d + 39a*d - 27b*d + 8c*d + 7d , c - 22b*c*d - 24c d - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 3 │ │ │ - - 8a*d - 16b*d + 19c*d + 4d ) │ │ │ + 2 2 2 3 │ │ │ + 29a*d - 29b*d - 10c*d + 48d ) │ │ │ │ │ │ o16 : Ideal of S │ │ │ │ │ │ i17 : betti res F1 │ │ │ │ │ │ 0 1 2 3 │ │ │ o17 = total: 1 6 8 3 │ │ │ @@ -150,28 +150,28 @@ │ │ │ 1: . 4 4 1 │ │ │ 2: . 2 4 2 │ │ │ │ │ │ o17 : BettiTally │ │ │ │ │ │ i18 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ - 2 2 2 │ │ │ -o18 = ideal (a - 9b*c - 18c - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c + │ │ │ + 2 2 2 │ │ │ +o18 = ideal (a - 35b*c + 33c + 46a*d - 33b*d - 48c*d + 18d , a*b + 46b*c + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 6c + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c + 27a*d - 47b*d │ │ │ + 2 2 2 │ │ │ + 33c + a*d + 42b*d + 37c*d + 13d , a*c - 15b*c + 19c + 14a*d - 43b*d - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 2 │ │ │ - - 28c*d - d , b + 19b*c - 13c - 37b*d + 32c*d + 15d , b*c - 43b*c*d │ │ │ + 2 2 2 2 2 │ │ │ + 47c*d + 27d , b + 22b*c - 18c + 7b*d + 23c*d + 13d , b*c + 34b*c*d + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 3 3 2 2 │ │ │ - - 47c d + 34a*d - 22b*d - 6c*d + 17d , c + 2b*c*d + 22c d - 18a*d │ │ │ + 2 2 2 2 3 3 2 2 │ │ │ + 2c d + 16a*d + 45b*d - 15c*d + 10d , c - 28b*c*d + 38c d - 39a*d - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 3 │ │ │ - + 38b*d - 39c*d - d ) │ │ │ + 2 2 3 │ │ │ + 47b*d - 13c*d + 4d ) │ │ │ │ │ │ o18 : Ideal of S │ │ │ │ │ │ i19 : betti res F2 │ │ │ │ │ │ 0 1 2 3 │ │ │ o19 = total: 1 6 8 3 │ │ │ @@ -179,34 +179,26 @@ │ │ │ 1: . 4 4 1 │ │ │ 2: . 2 4 2 │ │ │ │ │ │ o19 : BettiTally │ │ │ │ │ │ i20 : netList decompose F1 │ │ │ │ │ │ - +---------------------------------------------------------------------------------------------------------+ │ │ │ -o20 = |ideal (c - 23d, b + 7d, a + 28d) | │ │ │ - +---------------------------------------------------------------------------------------------------------+ │ │ │ - |ideal (c + 21d, b + 36d, a - 12d) | │ │ │ - +---------------------------------------------------------------------------------------------------------+ │ │ │ - |ideal (c + 13d, b - 22d, a - 28d) | │ │ │ - +---------------------------------------------------------------------------------------------------------+ │ │ │ - | 2 2 2 2 2 | │ │ │ - |ideal (a - 24b - 29c + 22d, c + 33b*d + 43c*d + 41d , b*c - 16b*d + 4c*d - 2d , b - 9b*d - 8c*d - 19d )| │ │ │ - +---------------------------------------------------------------------------------------------------------+ │ │ │ + +----------------------------------------------------------------------------------------------------------+ │ │ │ +o20 = |ideal (c - 16d, b + 31d, a + 12d) | │ │ │ + +----------------------------------------------------------------------------------------------------------+ │ │ │ + |ideal (c - 29d, b + 29d, a - 27d) | │ │ │ + +----------------------------------------------------------------------------------------------------------+ │ │ │ + |ideal (c + 41d, b + 35d, a - 25d) | │ │ │ + +----------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a - 8b - 36c + 37d, c - 5b*d + 46c*d + 41d , b*c + 30b*d - 24c*d - 9d , b - 17b*d + 21c*d - 34d )| │ │ │ + +----------------------------------------------------------------------------------------------------------+ │ │ │ │ │ │ i21 : netList decompose F2 │ │ │ │ │ │ - +-------------------------------------------------------+ │ │ │ -o21 = |ideal (c - 32d, b - 5d, a - 29d) | │ │ │ - +-------------------------------------------------------+ │ │ │ - |ideal (c + 43d, b - 47d, a - 27d) | │ │ │ - +-------------------------------------------------------+ │ │ │ - |ideal (c + 24d, b - 49d, a) | │ │ │ - +-------------------------------------------------------+ │ │ │ - |ideal (c + 14d, b + 31d, a - 16d) | │ │ │ - +-------------------------------------------------------+ │ │ │ - | 2 2 | │ │ │ - |ideal (b + 11c + 22d, a + 11c + 42d, c - 43c*d + 31d )| │ │ │ - +-------------------------------------------------------+ │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 | │ │ │ +o21 = |ideal (a*c - 15b*c + 19c + 14a*d - 43b*d - 47c*d + 27d , b + 22b*c - 18c + 7b*d + 23c*d + 13d , a*b + 46b*c + 33c + a*d + 42b*d + 37c*d + 13d , a - 35b*c + 33c + 46a*d - 33b*d - 48c*d + 18d , c - 28b*c*d + 38c d - 39a*d - 47b*d - 13c*d + 4d , b*c + 34b*c*d + 2c d + 16a*d + 45b*d - 15c*d + 10d )| │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ │ │ │ i22 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out │ │ │ @@ -12,15 +12,15 @@ │ │ │ 2 │ │ │ c*e + b*f, - c*d + b*e, - c*e + b*f, - e + d*f) │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ i4 : pt = randomPointOnRationalVariety I │ │ │ │ │ │ -o4 = | -25 20 -30 -16 24 -36 | │ │ │ +o4 = | 1 49 24 -23 -36 -30 | │ │ │ │ │ │ 1 6 │ │ │ o4 : Matrix kk <-- kk │ │ │ │ │ │ i5 : sub(I, pt) == 0 │ │ │ │ │ │ o5 = true │ │ │ @@ -200,67 +200,67 @@ │ │ │ │ │ │ o12 = {11, 8} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ -o13 = | -22 30 -36 -17 -36 39 29 -49 -45 19 11 21 -29 -8 17 -38 -29 32 -24 │ │ │ +o13 = | 2 -18 6 4 20 -11 44 -13 0 19 11 -47 -29 -8 -19 -22 19 -20 -29 -38 -24 │ │ │ ----------------------------------------------------------------------- │ │ │ - -10 -29 -22 19 -16 | │ │ │ + -16 -10 -29 | │ │ │ │ │ │ 1 24 │ │ │ o13 : Matrix kk <-- kk │ │ │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ - 2 2 2 │ │ │ -o14 = ideal (a - 45b*c - 17c + 19a*d - 36b*d + 30c*d - 22d , a*b + 17b*c + │ │ │ + 2 2 2 2 │ │ │ +o14 = ideal (a + 4c + 19a*d + 20b*d - 18c*d + 2d , a*b - 19b*c + 11c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 11c - 38a*d + 21b*d + 39c*d - 36d , a*c - 29b*c - 29c - 22a*d + 32b*d │ │ │ + 2 2 │ │ │ + 22a*d - 47b*d - 11c*d + 6d , a*c - 24b*c + 19c - 16a*d - 20b*d - 29c*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - - 29c*d + 29d , b + 19b*c - 24c - 16a*d - 10b*d - 8c*d - 49d ) │ │ │ + 2 2 2 2 │ │ │ + + 44d , b - 10b*c - 29c - 29a*d - 38b*d - 8c*d - 13d ) │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ │ │ i15 : decompose F1 │ │ │ │ │ │ 2 2 2 │ │ │ -o15 = {ideal (a - 29b - 29c + 40d, b + 19b*c - 24c + 31b*d + 33c*d - 15d ), │ │ │ +o15 = {ideal (a - 24b + 19c - 28d, b - 10b*c - 29c - 27b*d + 38c*d - 17d ), │ │ │ ----------------------------------------------------------------------- │ │ │ - ideal (c - 22d, b + 22d, a + 2d)} │ │ │ + ideal (c - 16d, b - 33d, a - 32d)} │ │ │ │ │ │ o15 : List │ │ │ │ │ │ i16 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ -o16 = | -14 40 -5 26 -48 -26 -35 41 -8 -15 -38 31 -13 29 21 16 39 21 -18 19 │ │ │ +o16 = | -8 28 25 30 -12 -42 -20 -35 -7 -37 -5 45 19 20 -24 34 39 21 -47 -39 │ │ │ ----------------------------------------------------------------------- │ │ │ - -47 -39 34 0 | │ │ │ + -13 -18 34 0 | │ │ │ │ │ │ 1 24 │ │ │ o16 : Matrix kk <-- kk │ │ │ │ │ │ i17 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ - 2 2 2 │ │ │ -o17 = ideal (a - 8b*c + 26c - 15a*d - 48b*d + 40c*d - 14d , a*b + 21b*c - │ │ │ + 2 2 2 │ │ │ +o17 = ideal (a - 7b*c + 30c - 37a*d - 12b*d + 28c*d - 8d , a*b - 24b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 38c + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c - 39a*d + 21b*d │ │ │ + 2 2 2 │ │ │ + 5c + 34a*d + 45b*d - 42c*d + 25d , a*c - 13b*c + 39c - 18a*d + 21b*d │ │ │ ----------------------------------------------------------------------- │ │ │ 2 2 2 2 │ │ │ - - 13c*d - 35d , b + 34b*c - 18c + 19b*d + 29c*d + 41d ) │ │ │ + + 19c*d - 20d , b + 34b*c - 47c - 39b*d + 20c*d - 35d ) │ │ │ │ │ │ o17 : Ideal of S │ │ │ │ │ │ i18 : decompose F2 │ │ │ │ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c + │ │ │ +o18 = {ideal (b - 12c + 48d, a - 16c + d), ideal (b + 46c + 14d, a + 31c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 26d)} │ │ │ + 5d)} │ │ │ │ │ │ o18 : List │ │ │ │ │ │ i19 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ @@ -237,52 +237,52 @@ │ │ │ o13 = 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o14 = | -22 41 -11 19 -43 -7 15 4 21 -36 31 30 37 19 -9 -44 30 19 -38 1 47 24
│ │ │ +o14 = | 43 35 -43 7 38 31 47 48 46 21 8 10 6 -30 -40 10 -27 -10 -50 30 -21
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -29 -16 16 -29 -30 21 -10 -22 39 -24 -29 -8 -36 -38 |
│ │ │ +      -38 -16 -29 31 -36 39 -29 19 24 -24 -8 19 -29 21 -22 |
│ │ │  
│ │ │                 1       36
│ │ │  o14 : Matrix kk  <-- kk
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47
│ │ │ +o15 = | 18 13 -48 10 27 -33 13 4 37 33 -15 46 42 -47 -35 23 45 -13 33 -43 1 7
│ │ │        -----------------------------------------------------------------------
│ │ │ -      28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 |
│ │ │ +      2 -47 46 19 16 14 -18 34 38 -15 0 -39 22 -28 |
│ │ │  
│ │ │                 1       36
│ │ │  o15 : Matrix kk  <-- kk
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2             2                             2               
│ │ │ -o16 = ideal (a  - 9b*c - 36c  + 30a*d - 7b*d - 11c*d - 22d , a*b + 16b*c -
│ │ │ +              2              2                              2               
│ │ │ +o16 = ideal (a  - 40b*c + 21c  + 10a*d + 31b*d - 43c*d + 43d , a*b + 31b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2                   2                
│ │ │ -      38c  + 47a*d + 37b*d + 21c*d + 41d , a*c - 24b*c - 29c  + 21a*d + b*d +
│ │ │ +         2                             2                  2                  
│ │ │ +      50c  - 21a*d + 6b*d + 46c*d + 35d , a*c - 8b*c - 36c  - 29a*d + 30b*d -
│ │ │        -----------------------------------------------------------------------
│ │ │                   2   2              2                              2     2  
│ │ │ -      19c*d - 43d , b  - 36b*c - 10c  - 29a*d + 24b*d - 44c*d + 15d , b*c  -
│ │ │ +      30c*d + 38d , b  + 21b*c + 19c  + 19a*d - 38b*d + 10c*d + 47d , b*c  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      22b*c*d - 29c d - 30a*d  + 30b*d  + 31c*d  + 19d , c  - 38b*c*d + 39c d
│ │ │ +                   2         2        2       2     3   3                2   
│ │ │ +      24b*c*d - 16c d + 39a*d  - 27b*d  + 8c*d  + 7d , c  - 22b*c*d - 24c d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -            2        2        2     3
│ │ │ -      - 8a*d  - 16b*d  + 19c*d  + 4d )
│ │ │ +           2        2        2      3
│ │ │ +      29a*d  - 29b*d  - 10c*d  + 48d )
│ │ │  
│ │ │  o16 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : betti res F1
│ │ │ @@ -296,28 +296,28 @@
│ │ │  o17 : BettiTally
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i18 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2             2                              2               
│ │ │ -o18 = ideal (a  - 9b*c - 18c  - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c +
│ │ │ +              2              2                              2               
│ │ │ +o18 = ideal (a  - 35b*c + 33c  + 46a*d - 33b*d - 48c*d + 18d , a*b + 46b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2                   2                
│ │ │ -      6c  + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c  + 27a*d - 47b*d
│ │ │ +         2                            2                   2                  
│ │ │ +      33c  + a*d + 42b*d + 37c*d + 13d , a*c - 15b*c + 19c  + 14a*d - 43b*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                      2     2          
│ │ │ -      - 28c*d - d , b  + 19b*c - 13c  - 37b*d + 32c*d + 15d , b*c  - 43b*c*d
│ │ │ +                 2   2              2                     2     2            
│ │ │ +      47c*d + 27d , b  + 22b*c - 18c  + 7b*d + 23c*d + 13d , b*c  + 34b*c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2         2        2       2      3   3               2         2
│ │ │ -      - 47c d + 34a*d  - 22b*d  - 6c*d  + 17d , c  + 2b*c*d + 22c d - 18a*d 
│ │ │ +        2         2        2        2      3   3                2         2  
│ │ │ +      2c d + 16a*d  + 45b*d  - 15c*d  + 10d , c  - 28b*c*d + 38c d - 39a*d  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2    3
│ │ │ -      + 38b*d  - 39c*d  - d )
│ │ │ +           2        2     3
│ │ │ +      47b*d  - 13c*d  + 4d )
│ │ │  
│ │ │  o18 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i19 : betti res F2
│ │ │ @@ -336,42 +336,34 @@
│ │ │            
    What are the ideals F1 and F2?
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i20 : netList decompose F1
│ │ │  
│ │ │ -      +---------------------------------------------------------------------------------------------------------+
│ │ │ -o20 = |ideal (c - 23d, b + 7d, a + 28d)                                                                         |
│ │ │ -      +---------------------------------------------------------------------------------------------------------+
│ │ │ -      |ideal (c + 21d, b + 36d, a - 12d)                                                                        |
│ │ │ -      +---------------------------------------------------------------------------------------------------------+
│ │ │ -      |ideal (c + 13d, b - 22d, a - 28d)                                                                        |
│ │ │ -      +---------------------------------------------------------------------------------------------------------+
│ │ │ -      |                             2                      2                         2   2                    2 |
│ │ │ -      |ideal (a - 24b - 29c + 22d, c  + 33b*d + 43c*d + 41d , b*c - 16b*d + 4c*d - 2d , b  - 9b*d - 8c*d - 19d )|
│ │ │ -      +---------------------------------------------------------------------------------------------------------+
    │ │ │ + +----------------------------------------------------------------------------------------------------------+ │ │ │ +o20 = |ideal (c - 16d, b + 31d, a + 12d) | │ │ │ + +----------------------------------------------------------------------------------------------------------+ │ │ │ + |ideal (c - 29d, b + 29d, a - 27d) | │ │ │ + +----------------------------------------------------------------------------------------------------------+ │ │ │ + |ideal (c + 41d, b + 35d, a - 25d) | │ │ │ + +----------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a - 8b - 36c + 37d, c - 5b*d + 46c*d + 41d , b*c + 30b*d - 24c*d - 9d , b - 17b*d + 21c*d - 34d )| │ │ │ + +----------------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │ 
    i21 : netList decompose F2
│ │ │  
│ │ │ -      +-------------------------------------------------------+
│ │ │ -o21 = |ideal (c - 32d, b - 5d, a - 29d)                       |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 43d, b - 47d, a - 27d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 24d, b - 49d, a)                            |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 14d, b + 31d, a - 16d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |                                      2              2 |
│ │ │ -      |ideal (b + 11c + 22d, a + 11c + 42d, c  - 43c*d + 31d )|
│ │ │ -      +-------------------------------------------------------+
    │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 | │ │ │ +o21 = |ideal (a*c - 15b*c + 19c + 14a*d - 43b*d - 47c*d + 27d , b + 22b*c - 18c + 7b*d + 23c*d + 13d , a*b + 46b*c + 33c + a*d + 42b*d + 37c*d + 13d , a - 35b*c + 33c + 46a*d - 33b*d - 48c*d + 18d , c - 28b*c*d + 38c d - 39a*d - 47b*d - 13c*d + 4d , b*c + 34b*c*d + 2c d + 16a*d + 45b*d - 15c*d + 10d )| │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │
    │ │ │

    We can determine what these represent. One should be a set of 6 points, where 5 lie on a plane. The other should be 6 points with 3 points on one line, and the other 3 points on a skew line.

    │ │ │
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -170,71 +170,71 @@ │ │ │ │ 32 13 21 33 19 31 │ │ │ │ i12 : compsJ = decompose J; │ │ │ │ i13 : #compsJ │ │ │ │ │ │ │ │ o13 = 2 │ │ │ │ i14 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ │ │ -o14 = | -22 41 -11 19 -43 -7 15 4 21 -36 31 30 37 19 -9 -44 30 19 -38 1 47 24 │ │ │ │ +o14 = | 43 35 -43 7 38 31 47 48 46 21 8 10 6 -30 -40 10 -27 -10 -50 30 -21 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -29 -16 16 -29 -30 21 -10 -22 39 -24 -29 -8 -36 -38 | │ │ │ │ + -38 -16 -29 31 -36 39 -29 19 24 -24 -8 19 -29 21 -22 | │ │ │ │ │ │ │ │ 1 36 │ │ │ │ o14 : Matrix kk <-- kk │ │ │ │ i15 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ │ │ -o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47 │ │ │ │ +o15 = | 18 13 -48 10 27 -33 13 4 37 33 -15 46 42 -47 -35 23 45 -13 33 -43 1 7 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 | │ │ │ │ + 2 -47 46 19 16 14 -18 34 38 -15 0 -39 22 -28 | │ │ │ │ │ │ │ │ 1 36 │ │ │ │ o15 : Matrix kk <-- kk │ │ │ │ i16 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o16 = ideal (a - 9b*c - 36c + 30a*d - 7b*d - 11c*d - 22d , a*b + 16b*c - │ │ │ │ + 2 2 2 │ │ │ │ +o16 = ideal (a - 40b*c + 21c + 10a*d + 31b*d - 43c*d + 43d , a*b + 31b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 38c + 47a*d + 37b*d + 21c*d + 41d , a*c - 24b*c - 29c + 21a*d + b*d + │ │ │ │ + 2 2 2 │ │ │ │ + 50c - 21a*d + 6b*d + 46c*d + 35d , a*c - 8b*c - 36c - 29a*d + 30b*d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 2 2 2 2 2 │ │ │ │ - 19c*d - 43d , b - 36b*c - 10c - 29a*d + 24b*d - 44c*d + 15d , b*c - │ │ │ │ + 30c*d + 38d , b + 21b*c + 19c + 19a*d - 38b*d + 10c*d + 47d , b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 3 3 2 │ │ │ │ - 22b*c*d - 29c d - 30a*d + 30b*d + 31c*d + 19d , c - 38b*c*d + 39c d │ │ │ │ + 2 2 2 2 3 3 2 │ │ │ │ + 24b*c*d - 16c d + 39a*d - 27b*d + 8c*d + 7d , c - 22b*c*d - 24c d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 3 │ │ │ │ - - 8a*d - 16b*d + 19c*d + 4d ) │ │ │ │ + 2 2 2 3 │ │ │ │ + 29a*d - 29b*d - 10c*d + 48d ) │ │ │ │ │ │ │ │ o16 : Ideal of S │ │ │ │ i17 : betti res F1 │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ o17 = total: 1 6 8 3 │ │ │ │ 0: 1 . . . │ │ │ │ 1: . 4 4 1 │ │ │ │ 2: . 2 4 2 │ │ │ │ │ │ │ │ o17 : BettiTally │ │ │ │ i18 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o18 = ideal (a - 9b*c - 18c - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c + │ │ │ │ + 2 2 2 │ │ │ │ +o18 = ideal (a - 35b*c + 33c + 46a*d - 33b*d - 48c*d + 18d , a*b + 46b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 6c + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c + 27a*d - 47b*d │ │ │ │ + 2 2 2 │ │ │ │ + 33c + a*d + 42b*d + 37c*d + 13d , a*c - 15b*c + 19c + 14a*d - 43b*d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 2 │ │ │ │ - - 28c*d - d , b + 19b*c - 13c - 37b*d + 32c*d + 15d , b*c - 43b*c*d │ │ │ │ + 2 2 2 2 2 │ │ │ │ + 47c*d + 27d , b + 22b*c - 18c + 7b*d + 23c*d + 13d , b*c + 34b*c*d + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 3 3 2 2 │ │ │ │ - - 47c d + 34a*d - 22b*d - 6c*d + 17d , c + 2b*c*d + 22c d - 18a*d │ │ │ │ + 2 2 2 2 3 3 2 2 │ │ │ │ + 2c d + 16a*d + 45b*d - 15c*d + 10d , c - 28b*c*d + 38c d - 39a*d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 3 │ │ │ │ - + 38b*d - 39c*d - d ) │ │ │ │ + 2 2 3 │ │ │ │ + 47b*d - 13c*d + 4d ) │ │ │ │ │ │ │ │ o18 : Ideal of S │ │ │ │ i19 : betti res F2 │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ o19 = total: 1 6 8 3 │ │ │ │ 0: 1 . . . │ │ │ │ @@ -242,47 +242,54 @@ │ │ │ │ 2: . 2 4 2 │ │ │ │ │ │ │ │ o19 : BettiTally │ │ │ │ What are the ideals F1 and F2? │ │ │ │ i20 : netList decompose F1 │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ----------------------------------+ │ │ │ │ -o20 = |ideal (c - 23d, b + 7d, a + 28d) │ │ │ │ +----------------------------------+ │ │ │ │ +o20 = |ideal (c - 16d, b + 31d, a + 12d) │ │ │ │ | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ----------------------------------+ │ │ │ │ - |ideal (c + 21d, b + 36d, a - 12d) │ │ │ │ +----------------------------------+ │ │ │ │ + |ideal (c - 29d, b + 29d, a - 27d) │ │ │ │ | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ----------------------------------+ │ │ │ │ - |ideal (c + 13d, b - 22d, a - 28d) │ │ │ │ +----------------------------------+ │ │ │ │ + |ideal (c + 41d, b + 35d, a - 25d) │ │ │ │ | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ----------------------------------+ │ │ │ │ - | 2 2 │ │ │ │ -2 2 2 | │ │ │ │ - |ideal (a - 24b - 29c + 22d, c + 33b*d + 43c*d + 41d , b*c - 16b*d + │ │ │ │ -4c*d - 2d , b - 9b*d - 8c*d - 19d )| │ │ │ │ +----------------------------------+ │ │ │ │ + | 2 2 │ │ │ │ +2 2 2 | │ │ │ │ + |ideal (a - 8b - 36c + 37d, c - 5b*d + 46c*d + 41d , b*c + 30b*d - 24c*d │ │ │ │ +- 9d , b - 17b*d + 21c*d - 34d )| │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ----------------------------------+ │ │ │ │ +----------------------------------+ │ │ │ │ i21 : netList decompose F2 │ │ │ │ │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ -o21 = |ideal (c - 32d, b - 5d, a - 29d) | │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - |ideal (c + 43d, b - 47d, a - 27d) | │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - |ideal (c + 24d, b - 49d, a) | │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - |ideal (c + 14d, b + 31d, a - 16d) | │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - | 2 2 | │ │ │ │ - |ideal (b + 11c + 22d, a + 11c + 42d, c - 43c*d + 31d )| │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ ++ │ │ │ │ + | 2 2 2 │ │ │ │ +2 2 2 2 2 │ │ │ │ +2 2 3 2 2 2 │ │ │ │ +2 3 2 2 2 2 2 3 | │ │ │ │ +o21 = |ideal (a*c - 15b*c + 19c + 14a*d - 43b*d - 47c*d + 27d , b + 22b*c - │ │ │ │ +18c + 7b*d + 23c*d + 13d , a*b + 46b*c + 33c + a*d + 42b*d + 37c*d + 13d , a │ │ │ │ +- 35b*c + 33c + 46a*d - 33b*d - 48c*d + 18d , c - 28b*c*d + 38c d - 39a*d - │ │ │ │ +47b*d - 13c*d + 4d , b*c + 34b*c*d + 2c d + 16a*d + 45b*d - 15c*d + 10d │ │ │ │ +)| │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ ++ │ │ │ │ We can determine what these represent. One should be a set of 6 points, where 5 │ │ │ │ lie on a plane. The other should be 6 points with 3 points on one line, and the │ │ │ │ other 3 points on a skew line. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_n_d_o_m_P_o_i_n_t_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y -- find a random point on a variety that can │ │ │ │ be detected to be rational │ │ │ │ ********** WWaayyss ttoo uussee nnoonnmmiinniimmaallMMaappss:: ********** │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ @@ -101,15 +101,15 @@ │ │ │ o3 : Ideal of S     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : pt = randomPointOnRationalVariety I
│ │ │  
│ │ │ -o4 = | -25 20 -30 -16 24 -36 |
│ │ │ +o4 = | 1 49 24 -23 -36 -30 |
│ │ │  
│ │ │                1       6
│ │ │  o4 : Matrix kk  <-- kk
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -323,90 +323,90 @@ │ │ │

    There are 2 components. We attempt to find a point on the first component

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i13 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o13 = | -22 30 -36 -17 -36 39 29 -49 -45 19 11 21 -29 -8 17 -38 -29 32 -24
│ │ │ +o13 = | 2 -18 6 4 20 -11 44 -13 0 19 11 -47 -29 -8 -19 -22 19 -20 -29 -38 -24
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -10 -29 -22 19 -16 |
│ │ │ +      -16 -10 -29 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o14 = ideal (a  - 45b*c - 17c  + 19a*d - 36b*d + 30c*d - 22d , a*b + 17b*c +
│ │ │ +              2     2                             2                   2  
│ │ │ +o14 = ideal (a  + 4c  + 19a*d + 20b*d - 18c*d + 2d , a*b - 19b*c + 11c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2                   2                
│ │ │ -      11c  - 38a*d + 21b*d + 39c*d - 36d , a*c - 29b*c - 29c  - 22a*d + 32b*d
│ │ │ +                                2                   2                        
│ │ │ +      22a*d - 47b*d - 11c*d + 6d , a*c - 24b*c + 19c  - 16a*d - 20b*d - 29c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                             2
│ │ │ -      - 29c*d + 29d , b  + 19b*c - 24c  - 16a*d - 10b*d - 8c*d - 49d )
│ │ │ +           2   2              2                             2
│ │ │ +      + 44d , b  - 10b*c - 29c  - 29a*d - 38b*d - 8c*d - 13d )
│ │ │  
│ │ │  o14 : Ideal of S
    │ │ │
    │ │ │ 
    i15 : decompose F1
│ │ │  
│ │ │                                      2              2                      2
│ │ │ -o15 = {ideal (a - 29b - 29c + 40d, b  + 19b*c - 24c  + 31b*d + 33c*d - 15d ),
│ │ │ +o15 = {ideal (a - 24b + 19c - 28d, b  - 10b*c - 29c  - 27b*d + 38c*d - 17d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 22d, b + 22d, a + 2d)}
│ │ │ +      ideal (c - 16d, b - 33d, a - 32d)}
│ │ │  
│ │ │  o15 : List
    │ │ │
    │ │ │
    │ │ │

    We attempt to find a point on the second component in parameter space, and its corresponding ideal.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i16 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o16 = | -14 40 -5 26 -48 -26 -35 41 -8 -15 -38 31 -13 29 21 16 39 21 -18 19
│ │ │ +o16 = | -8 28 25 30 -12 -42 -20 -35 -7 -37 -5 45 19 20 -24 34 39 21 -47 -39
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -47 -39 34 0 |
│ │ │ +      -13 -18 34 0 |
│ │ │  
│ │ │                 1       24
│ │ │  o16 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i17 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2             2                              2               
│ │ │ -o17 = ideal (a  - 8b*c + 26c  - 15a*d - 48b*d + 40c*d - 14d , a*b + 21b*c -
│ │ │ +              2             2                             2               
│ │ │ +o17 = ideal (a  - 7b*c + 30c  - 37a*d - 12b*d + 28c*d - 8d , a*b - 24b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                
│ │ │ -      38c  + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c  - 39a*d + 21b*d
│ │ │ +        2                              2                   2                
│ │ │ +      5c  + 34a*d + 45b*d - 42c*d + 25d , a*c - 13b*c + 39c  - 18a*d + 21b*d
│ │ │        -----------------------------------------------------------------------
│ │ │                     2   2              2                      2
│ │ │ -      - 13c*d - 35d , b  + 34b*c - 18c  + 19b*d + 29c*d + 41d )
│ │ │ +      + 19c*d - 20d , b  + 34b*c - 47c  - 39b*d + 20c*d - 35d )
│ │ │  
│ │ │  o17 : Ideal of S
    │ │ │
    │ │ │ 
    i18 : decompose F2
│ │ │  
│ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c +
│ │ │ +o18 = {ideal (b - 12c + 48d, a - 16c + d), ideal (b + 46c + 14d, a + 31c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      26d)}
│ │ │ +      5d)}
│ │ │  
│ │ │  o18 : List
    │ │ │
    │ │ │
    │ │ │

    It turns out that this is the ideal of 2 skew lines, just not defined over this field.

    │ │ │ ├── html2text {} │ │ │ │ @@ -28,15 +28,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 │ │ │ │ c*e + b*f, - c*d + b*e, - c*e + b*f, - e + d*f) │ │ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ i4 : pt = randomPointOnRationalVariety I │ │ │ │ │ │ │ │ -o4 = | -25 20 -30 -16 24 -36 | │ │ │ │ +o4 = | 1 49 24 -23 -36 -30 | │ │ │ │ │ │ │ │ 1 6 │ │ │ │ o4 : Matrix kk <-- kk │ │ │ │ i5 : sub(I, pt) == 0 │ │ │ │ │ │ │ │ o5 = true │ │ │ │ i6 : S = kk[a..d]; │ │ │ │ @@ -212,67 +212,67 @@ │ │ │ │ │ │ │ │ o12 = {11, 8} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ There are 2 components. We attempt to find a point on the first component │ │ │ │ i13 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ │ │ -o13 = | -22 30 -36 -17 -36 39 29 -49 -45 19 11 21 -29 -8 17 -38 -29 32 -24 │ │ │ │ +o13 = | 2 -18 6 4 20 -11 44 -13 0 19 11 -47 -29 -8 -19 -22 19 -20 -29 -38 -24 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -10 -29 -22 19 -16 | │ │ │ │ + -16 -10 -29 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o13 : Matrix kk <-- kk │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o14 = ideal (a - 45b*c - 17c + 19a*d - 36b*d + 30c*d - 22d , a*b + 17b*c + │ │ │ │ + 2 2 2 2 │ │ │ │ +o14 = ideal (a + 4c + 19a*d + 20b*d - 18c*d + 2d , a*b - 19b*c + 11c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 11c - 38a*d + 21b*d + 39c*d - 36d , a*c - 29b*c - 29c - 22a*d + 32b*d │ │ │ │ + 2 2 │ │ │ │ + 22a*d - 47b*d - 11c*d + 6d , a*c - 24b*c + 19c - 16a*d - 20b*d - 29c*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - - 29c*d + 29d , b + 19b*c - 24c - 16a*d - 10b*d - 8c*d - 49d ) │ │ │ │ + 2 2 2 2 │ │ │ │ + + 44d , b - 10b*c - 29c - 29a*d - 38b*d - 8c*d - 13d ) │ │ │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ i15 : decompose F1 │ │ │ │ │ │ │ │ 2 2 2 │ │ │ │ -o15 = {ideal (a - 29b - 29c + 40d, b + 19b*c - 24c + 31b*d + 33c*d - 15d ), │ │ │ │ +o15 = {ideal (a - 24b + 19c - 28d, b - 10b*c - 29c - 27b*d + 38c*d - 17d ), │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - ideal (c - 22d, b + 22d, a + 2d)} │ │ │ │ + ideal (c - 16d, b - 33d, a - 32d)} │ │ │ │ │ │ │ │ o15 : List │ │ │ │ We attempt to find a point on the second component in parameter space, and its │ │ │ │ corresponding ideal. │ │ │ │ i16 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ │ │ -o16 = | -14 40 -5 26 -48 -26 -35 41 -8 -15 -38 31 -13 29 21 16 39 21 -18 19 │ │ │ │ +o16 = | -8 28 25 30 -12 -42 -20 -35 -7 -37 -5 45 19 20 -24 34 39 21 -47 -39 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -47 -39 34 0 | │ │ │ │ + -13 -18 34 0 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o16 : Matrix kk <-- kk │ │ │ │ i17 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o17 = ideal (a - 8b*c + 26c - 15a*d - 48b*d + 40c*d - 14d , a*b + 21b*c - │ │ │ │ + 2 2 2 │ │ │ │ +o17 = ideal (a - 7b*c + 30c - 37a*d - 12b*d + 28c*d - 8d , a*b - 24b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 38c + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c - 39a*d + 21b*d │ │ │ │ + 2 2 2 │ │ │ │ + 5c + 34a*d + 45b*d - 42c*d + 25d , a*c - 13b*c + 39c - 18a*d + 21b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 2 2 2 2 │ │ │ │ - - 13c*d - 35d , b + 34b*c - 18c + 19b*d + 29c*d + 41d ) │ │ │ │ + + 19c*d - 20d , b + 34b*c - 47c - 39b*d + 20c*d - 35d ) │ │ │ │ │ │ │ │ o17 : Ideal of S │ │ │ │ i18 : decompose F2 │ │ │ │ │ │ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c + │ │ │ │ +o18 = {ideal (b - 12c + 48d, a - 16c + d), ideal (b + 46c + 14d, a + 31c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 26d)} │ │ │ │ + 5d)} │ │ │ │ │ │ │ │ o18 : List │ │ │ │ It turns out that this is the ideal of 2 skew lines, just not defined over this │ │ │ │ field. │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ This routine expects the input to represent an irreducible variety │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html │ │ │ @@ -355,85 +355,85 @@ │ │ │

    We can find random points on each component, since these components are rational.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i12 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o12 = | -26 -24 6 39 17 28 -46 -12 11 -29 -10 -48 -36 -30 39 -29 -8 -22 -10
│ │ │ +o12 = | -14 48 41 -32 -39 30 48 36 2 -29 -30 -23 19 19 -10 -29 -8 -22 24 -13
│ │ │        -----------------------------------------------------------------------
│ │ │ -      11 19 19 24 -29 |
│ │ │ +      -36 -30 -29 -10 |
│ │ │  
│ │ │                 1       24
│ │ │  o12 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i13 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o13 = | -30 -41 -39 30 -13 32 17 -12 29 25 -49 11 43 21 -42 -4 19 -47 39 -38
│ │ │ +o13 = | -39 -22 11 26 -28 31 38 24 -31 -16 -16 -7 -41 -24 18 -17 19 34 -38 39
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -16 0 -24 34 |
│ │ │ +      -16 0 -47 21 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o14 = ideal (a  + 11b*c + 39c  - 29a*d + 17b*d - 24c*d - 26d , a*b + 39b*c -
│ │ │ +              2             2                              2               
│ │ │ +o14 = ideal (a  + 2b*c - 32c  - 29a*d - 39b*d + 48c*d - 14d , a*b - 10b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2             2                  
│ │ │ -      10c  - 29a*d - 48b*d + 28c*d + 6d , b  + 19b*c - 8c  + 19a*d - 22b*d -
│ │ │ +         2                              2   2             2                  
│ │ │ +      30c  - 29a*d - 23b*d + 30c*d + 41d , b  - 36b*c - 8c  - 30a*d - 22b*d +
│ │ │        -----------------------------------------------------------------------
│ │ │                   2                   2                              2
│ │ │ -      36c*d - 46d , a*c + 24b*c - 10c  - 29a*d + 11b*d - 30c*d - 12d )
│ │ │ +      19c*d + 48d , a*c - 29b*c + 24c  - 10a*d - 13b*d + 19c*d + 36d )
│ │ │  
│ │ │  o14 : Ideal of S
    │ │ │
    │ │ │ 
    i15 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │                2              2                              2               
│ │ │ -o15 = ideal (a  + 29b*c + 30c  + 25a*d - 13b*d - 41c*d - 30d , a*b - 42b*c -
│ │ │ +o15 = ideal (a  - 31b*c + 26c  - 16a*d - 28b*d - 22c*d - 39d , a*b + 18b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │           2                             2   2              2                  
│ │ │ -      49c  - 4a*d + 11b*d + 32c*d - 39d , b  - 16b*c + 19c  - 47b*d + 43c*d +
│ │ │ +      16c  - 17a*d - 7b*d + 31c*d + 11d , b  - 16b*c + 19c  + 34b*d - 41c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │           2                   2                              2
│ │ │ -      17d , a*c - 24b*c + 39c  + 34a*d - 38b*d + 21c*d - 12d )
│ │ │ +      38d , a*c - 47b*c - 38c  + 21a*d + 39b*d - 24c*d + 24d )
│ │ │  
│ │ │  o15 : Ideal of S
    │ │ │
    │ │ │ 
    i16 : decompose F1
│ │ │  
│ │ │                                      2             2                      2
│ │ │ -o16 = {ideal (a + 24b - 10c - 17d, b  + 19b*c - 8c  + 27b*d - 48c*d - 26d ),
│ │ │ +o16 = {ideal (a - 29b + 24c - 44d, b  - 36b*c - 8c  + 17b*d + 32c*d + 41d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 29d, b + 23d, a - 14d)}
│ │ │ +      ideal (c - 10d, b + 33d, a + 10d)}
│ │ │  
│ │ │  o16 : List
    │ │ │
    │ │ │ 
    i17 : decompose F2
│ │ │  
│ │ │ -o17 = {ideal (b - 42c - 18d, a + 41c + 33d), ideal (b + 26c - 29d, a - 44c -
│ │ │ +o17 = {ideal (b - 42c + 10d, a + 8c - 3d), ideal (b + 26c + 24d, a - 28c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      9d)}
│ │ │ +      29d)}
│ │ │  
│ │ │  o17 : List
    │ │ │
    │ │ │
    │ │ │

    Note, the general element of one component is a plane conic union a point. (The dimension of the locus of all such is: (choice of plane) + (choice of degree 2 in plane) + choice of point. This is 3 + 5 + 3 = 11.

    │ │ │ ├── html2text {} │ │ │ │ @@ -424,65 +424,65 @@ │ │ │ │ -----------------------------------------------------------+ │ │ │ │ This tells us that there are 2 components (at least over the given field). │ │ │ │ Their dimensions are 11, 8. │ │ │ │ We can find random points on each component, since these components are │ │ │ │ rational. │ │ │ │ i12 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ │ │ -o12 = | -26 -24 6 39 17 28 -46 -12 11 -29 -10 -48 -36 -30 39 -29 -8 -22 -10 │ │ │ │ +o12 = | -14 48 41 -32 -39 30 48 36 2 -29 -30 -23 19 19 -10 -29 -8 -22 24 -13 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 11 19 19 24 -29 | │ │ │ │ + -36 -30 -29 -10 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o12 : Matrix kk <-- kk │ │ │ │ i13 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ │ │ -o13 = | -30 -41 -39 30 -13 32 17 -12 29 25 -49 11 43 21 -42 -4 19 -47 39 -38 │ │ │ │ +o13 = | -39 -22 11 26 -28 31 38 24 -31 -16 -16 -7 -41 -24 18 -17 19 34 -38 39 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -16 0 -24 34 | │ │ │ │ + -16 0 -47 21 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o13 : Matrix kk <-- kk │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o14 = ideal (a + 11b*c + 39c - 29a*d + 17b*d - 24c*d - 26d , a*b + 39b*c - │ │ │ │ + 2 2 2 │ │ │ │ +o14 = ideal (a + 2b*c - 32c - 29a*d - 39b*d + 48c*d - 14d , a*b - 10b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 10c - 29a*d - 48b*d + 28c*d + 6d , b + 19b*c - 8c + 19a*d - 22b*d - │ │ │ │ + 2 2 2 2 │ │ │ │ + 30c - 29a*d - 23b*d + 30c*d + 41d , b - 36b*c - 8c - 30a*d - 22b*d + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 2 2 2 │ │ │ │ - 36c*d - 46d , a*c + 24b*c - 10c - 29a*d + 11b*d - 30c*d - 12d ) │ │ │ │ + 19c*d + 48d , a*c - 29b*c + 24c - 10a*d - 13b*d + 19c*d + 36d ) │ │ │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ i15 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ │ │ 2 2 2 │ │ │ │ -o15 = ideal (a + 29b*c + 30c + 25a*d - 13b*d - 41c*d - 30d , a*b - 42b*c - │ │ │ │ +o15 = ideal (a - 31b*c + 26c - 16a*d - 28b*d - 22c*d - 39d , a*b + 18b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 2 2 2 2 │ │ │ │ - 49c - 4a*d + 11b*d + 32c*d - 39d , b - 16b*c + 19c - 47b*d + 43c*d + │ │ │ │ + 16c - 17a*d - 7b*d + 31c*d + 11d , b - 16b*c + 19c + 34b*d - 41c*d + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 2 2 2 │ │ │ │ - 17d , a*c - 24b*c + 39c + 34a*d - 38b*d + 21c*d - 12d ) │ │ │ │ + 38d , a*c - 47b*c - 38c + 21a*d + 39b*d - 24c*d + 24d ) │ │ │ │ │ │ │ │ o15 : Ideal of S │ │ │ │ i16 : decompose F1 │ │ │ │ │ │ │ │ 2 2 2 │ │ │ │ -o16 = {ideal (a + 24b - 10c - 17d, b + 19b*c - 8c + 27b*d - 48c*d - 26d ), │ │ │ │ +o16 = {ideal (a - 29b + 24c - 44d, b - 36b*c - 8c + 17b*d + 32c*d + 41d ), │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - ideal (c - 29d, b + 23d, a - 14d)} │ │ │ │ + ideal (c - 10d, b + 33d, a + 10d)} │ │ │ │ │ │ │ │ o16 : List │ │ │ │ i17 : decompose F2 │ │ │ │ │ │ │ │ -o17 = {ideal (b - 42c - 18d, a + 41c + 33d), ideal (b + 26c - 29d, a - 44c - │ │ │ │ +o17 = {ideal (b - 42c + 10d, a + 8c - 3d), ideal (b + 26c + 24d, a - 28c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 9d)} │ │ │ │ + 29d)} │ │ │ │ │ │ │ │ o17 : List │ │ │ │ Note, the general element of one component is a plane conic union a point. (The │ │ │ │ dimension of the locus of all such is: (choice of plane) + (choice of degree 2 │ │ │ │ in plane) + choice of point. This is 3 + 5 + 3 = 11. │ │ │ │ The other component consists of two skew lines. This has dimension (choice of │ │ │ │ line) + (choice of line). This is 4 + 4 = 8. Also notice that the 2 skew lines │ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/___Groebner__Walk.out │ │ │ @@ -11,21 +11,21 @@ │ │ │ i3 : R2 = QQ[x,y,z,u,v, MonomialOrder=>Weights=>{0,0,0,1,1}]; │ │ │ │ │ │ i4 : I2 = sub(I1, R2); │ │ │ │ │ │ o4 : Ideal of R2 │ │ │ │ │ │ i5 : elapsedTime gb I2 │ │ │ - -- 3.21715s elapsed │ │ │ + -- 2.23216s elapsed │ │ │ │ │ │ o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16] │ │ │ │ │ │ o5 : GroebnerBasis │ │ │ │ │ │ i6 : elapsedTime groebnerWalk(gb I1, R2) │ │ │ - -- 2.48394s elapsed │ │ │ + -- 1.69554s elapsed │ │ │ │ │ │ o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0] │ │ │ │ │ │ o6 : GroebnerBasis │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/html/index.html │ │ │ @@ -97,30 +97,30 @@ │ │ │ │ │ │ o4 : Ideal of R2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime gb I2
│ │ │ - -- 3.21715s elapsed
│ │ │ + -- 2.23216s elapsed
│ │ │  
│ │ │  o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16]
│ │ │  
│ │ │  o5 : GroebnerBasis
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    but it is faster to compute directly in the first order and then use the Groebner walk.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : elapsedTime groebnerWalk(gb I1, R2)
│ │ │ - -- 2.48394s elapsed
│ │ │ + -- 1.69554s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]
│ │ │  
│ │ │  o6 : GroebnerBasis
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -38,23 +38,23 @@ │ │ │ │ using a different weight vector and then graded reverse lexicographic we could │ │ │ │ substitute and compute directly, │ │ │ │ i3 : R2 = QQ[x,y,z,u,v, MonomialOrder=>Weights=>{0,0,0,1,1}]; │ │ │ │ i4 : I2 = sub(I1, R2); │ │ │ │ │ │ │ │ o4 : Ideal of R2 │ │ │ │ i5 : elapsedTime gb I2 │ │ │ │ - -- 3.21715s elapsed │ │ │ │ + -- 2.23216s elapsed │ │ │ │ │ │ │ │ o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16] │ │ │ │ │ │ │ │ o5 : GroebnerBasis │ │ │ │ but it is faster to compute directly in the first order and then use the │ │ │ │ Groebner walk. │ │ │ │ i6 : elapsedTime groebnerWalk(gb I1, R2) │ │ │ │ - -- 2.48394s elapsed │ │ │ │ + -- 1.69554s elapsed │ │ │ │ │ │ │ │ o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0] │ │ │ │ │ │ │ │ o6 : GroebnerBasis │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ The target ring must be the same ring as the ring of the starting ideal, except │ │ │ │ with different monomial order. The ring must be a polynomial ring over a field. │ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out │ │ │ @@ -44,19 +44,19 @@ │ │ │ o5 = {9, 1, 99999, 9999999, 3, 999} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : netList cssLeadTerm(Hbeta, w) │ │ │ Warning: F4 Algorithm not available over current coefficient ring or inhomogeneous ideal. │ │ │ Converting to Naive algorithm. │ │ │ - -- .000003707s elapsed │ │ │ - -- .000003346s elapsed │ │ │ - -- .000007374s elapsed │ │ │ - -- .000002865s elapsed │ │ │ - -- .000001883s elapsed │ │ │ + -- .000007508s elapsed │ │ │ + -- .000007365s elapsed │ │ │ + -- .000010498s elapsed │ │ │ + -- .000005585s elapsed │ │ │ + -- .000006624s elapsed │ │ │ │ │ │ +----------------------------------------------------+ │ │ │ | 1 5 5 5 | │ │ │ | - - - - - - | │ │ │ | 2 2 2 2 | │ │ │ o6 = |x x x x | │ │ │ | 1 2 4 5 | │ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_solve__Frobenius__Ideal.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : I = ideal(t_1+t_2+t_3+t_4+t_5, t_1+t_2-t_4, t_2+t_3-t_4, t_1*t_3, t_2*t_4); │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ i3 : solveFrobeniusIdeal I │ │ │ Warning: F4 Algorithm not available over current coefficient ring or inhomogeneous ideal. │ │ │ Converting to Naive algorithm. │ │ │ - -- .000004288s elapsed │ │ │ + -- .000007134s elapsed │ │ │ │ │ │ │ │ │ o3 = {1, - 2logX + 3logX - 2logX + logX , - logX + logX - logX + logX , │ │ │ 0 1 2 3 0 1 2 4 │ │ │ ------------------------------------------------------------------------ │ │ │ 1 1 2 1 1 1 1 2 │ │ │ -logX logX - -logX + -logX logX + -logX logX + -logX logX + -logX │ │ │ @@ -26,15 +26,15 @@ │ │ │ o3 : List │ │ │ │ │ │ i4 : W = makeWeylAlgebra(QQ[x_1..x_5]); │ │ │ │ │ │ i5 : solveFrobeniusIdeal(I, W) │ │ │ Warning: F4 Algorithm not available over current coefficient ring or inhomogeneous ideal. │ │ │ Converting to Naive algorithm. │ │ │ - -- .000003657s elapsed │ │ │ + -- .000007215s elapsed │ │ │ │ │ │ │ │ │ o5 = {1, - 2logX + 3logX - 2logX + logX , - logX + logX - logX + logX , │ │ │ 0 1 2 3 0 1 2 4 │ │ │ ------------------------------------------------------------------------ │ │ │ 1 1 2 1 1 1 1 2 │ │ │ -logX logX - -logX + -logX logX + -logX logX + -logX logX + -logX │ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html │ │ │ @@ -139,19 +139,19 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : netList cssLeadTerm(Hbeta, w)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000003707s elapsed
│ │ │ - -- .000003346s elapsed
│ │ │ - -- .000007374s elapsed
│ │ │ - -- .000002865s elapsed
│ │ │ - -- .000001883s elapsed
│ │ │ + -- .000007508s elapsed
│ │ │ + -- .000007365s elapsed
│ │ │ + -- .000010498s elapsed
│ │ │ + -- .000005585s elapsed
│ │ │ + -- .000006624s elapsed
│ │ │  
│ │ │       +----------------------------------------------------+
│ │ │       |   1 5   5 5                                        |
│ │ │       | - - - - - -                                        |
│ │ │       |   2 2   2 2                                        |
│ │ │  o6 = |x   x x   x                                         |
│ │ │       | 1   2 4   5                                        |
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,19 +57,19 @@
│ │ │ │  o5 = {9, 1, 99999, 9999999, 3, 999}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : netList cssLeadTerm(Hbeta, w)
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │ - -- .000003707s elapsed
│ │ │ │ - -- .000003346s elapsed
│ │ │ │ - -- .000007374s elapsed
│ │ │ │ - -- .000002865s elapsed
│ │ │ │ - -- .000001883s elapsed
│ │ │ │ + -- .000007508s elapsed
│ │ │ │ + -- .000007365s elapsed
│ │ │ │ + -- .000010498s elapsed
│ │ │ │ + -- .000005585s elapsed
│ │ │ │ + -- .000006624s elapsed
│ │ │ │  
│ │ │ │       +----------------------------------------------------+
│ │ │ │       |   1 5   5 5                                        |
│ │ │ │       | - - - - - -                                        |
│ │ │ │       |   2 2   2 2                                        |
│ │ │ │  o6 = |x   x x   x                                         |
│ │ │ │       | 1   2 4   5                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_solve__Frobenius__Ideal.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : solveFrobeniusIdeal I
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000004288s elapsed
│ │ │ + -- .000007134s elapsed
│ │ │  
│ │ │                                                                               
│ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │       1             1    2   1             1             1             1    2
│ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX 
│ │ │ @@ -120,15 +120,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : solveFrobeniusIdeal(I, W)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000003657s elapsed
│ │ │ + -- .000007215s elapsed
│ │ │  
│ │ │                                                                               
│ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │       1             1    2   1             1             1             1    2
│ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  t_2*t_4);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : solveFrobeniusIdeal I
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │ - -- .000004288s elapsed
│ │ │ │ + -- .000007134s elapsed
│ │ │ │  
│ │ │ │  
│ │ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1             1    2   1             1             1             1    2
│ │ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
│ │ │ │  i5 : solveFrobeniusIdeal(I, W)
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │ - -- .000003657s elapsed
│ │ │ │ + -- .000007215s elapsed
│ │ │ │  
│ │ │ │  
│ │ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1             1    2   1             1             1             1    2
│ │ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_cone_lp__Arrangement_cm__Ring__Element_rp.out
│ │ │ @@ -44,15 +44,15 @@
│ │ │  
│ │ │  o13 = {x, y, x - y, 0, - x + y, x}
│ │ │  
│ │ │  o13 : Hyperplane Arrangement 
│ │ │  
│ │ │  i14 : cA'' = trim cone(A, x)
│ │ │  
│ │ │ -o14 = {x - y, y, x}
│ │ │ +o14 = {y, x, x - y}
│ │ │  
│ │ │  o14 : Hyperplane Arrangement 
│ │ │  
│ │ │  i15 : assert isCentral cA''
│ │ │  
│ │ │  i16 : assert(# hyperplanes cA'' =!= 1 + # hyperplanes A)
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_euler__Restriction_lp__Central__Arrangement_cm__List_cm__Z__Z_rp.out
│ │ │ @@ -10,35 +10,35 @@
│ │ │  
│ │ │  o2 = {x, y, z, x - y, x - z}
│ │ │  
│ │ │  o2 : Hyperplane Arrangement 
│ │ │  
│ │ │  i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │  
│ │ │ -o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │ +o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │  i4 : restriction(A,1)
│ │ │  
│ │ │  o4 = {x, z, x, x - z}
│ │ │  
│ │ │  o4 : Hyperplane Arrangement 
│ │ │  
│ │ │  i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │  
│ │ │ -o5 = {z, x, x - z}
│ │ │ +o5 = {x - z, z, x}
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
│ │ │  
│ │ │  i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
│ │ │  
│ │ │  i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │  
│ │ │ -o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │ +o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │  
│ │ │  o8 : Sequence
│ │ │  
│ │ │  i9 : prune image der(A,m)
│ │ │  
│ │ │        3
│ │ │  o9 = R
│ │ │ @@ -59,16 +59,16 @@
│ │ │  
│ │ │  o11 : QQ[y..z]-module, free, degrees {2:3}
│ │ │  
│ │ │  i12 : A = arrangement "bracelet";
│ │ │  
│ │ │  i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │  
│ │ │ -o13 = ({x  + x  + x , x  + x , x , x  + x , x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ -         2    3    4   2    4   2   3    4   3   4
│ │ │ +o13 = ({x , x  + x , x , x , x  + x  + x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ +         2   3    4   3   4   2    3    4   2    4
│ │ │  
│ │ │  o13 : Sequence
│ │ │  
│ │ │  i14 : C = restriction(A,0)
│ │ │  
│ │ │  o14 = {x , x , x , x  + x , x  + x , x  + x , x  + x , x  + x  + x }
│ │ │          2   3   4   2    4   3    4   2    4   3    4   2    3    4
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_trim_lp__Arrangement_rp.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  o2 = {x, x, 0, y, y, y, x + y, x + y, x + y, x + y, x + y}
│ │ │  
│ │ │  o2 : Hyperplane Arrangement 
│ │ │  
│ │ │  i3 : A' = trim A
│ │ │  
│ │ │ -o3 = {y, x, x + y}
│ │ │ +o3 = {x + y, y, x}
│ │ │  
│ │ │  o3 : Hyperplane Arrangement 
│ │ │  
│ │ │  i4 : assert(ring A' === R)
│ │ │  
│ │ │  i5 : assert(trim A' == A')
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_type__B_lp__Z__Z_cm__Ring_rp.out
│ │ │ @@ -33,16 +33,16 @@
│ │ │  o5 = {x , x  + x , x  + x , x }
│ │ │         1   1    2   1    2   2
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
│ │ │  
│ │ │  i6 : trim A3
│ │ │  
│ │ │ -o6 = {x  + x , x , x }
│ │ │ -       1    2   2   1
│ │ │ +o6 = {x , x , x  + x }
│ │ │ +       2   1   1    2
│ │ │  
│ │ │  o6 : Hyperplane Arrangement 
│ │ │  
│ │ │  i7 : ring A3
│ │ │  
│ │ │       ZZ
│ │ │  o7 = --[x ..x ]
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_cone_lp__Arrangement_cm__Ring__Element_rp.html
│ │ │ @@ -172,15 +172,15 @@
│ │ │  o13 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : cA'' = trim cone(A, x)
│ │ │  
│ │ │ -o14 = {x - y, y, x}
│ │ │ +o14 = {y, x, x - y}
│ │ │  
│ │ │  o14 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : assert isCentral cA''
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -60,15 +60,15 @@
│ │ │ │  i13 : cone(A, x)
│ │ │ │  
│ │ │ │  o13 = {x, y, x - y, 0, - x + y, x}
│ │ │ │  
│ │ │ │  o13 : Hyperplane Arrangement
│ │ │ │  i14 : cA'' = trim cone(A, x)
│ │ │ │  
│ │ │ │ -o14 = {x - y, y, x}
│ │ │ │ +o14 = {y, x, x - y}
│ │ │ │  
│ │ │ │  o14 : Hyperplane Arrangement
│ │ │ │  i15 : assert isCentral cA''
│ │ │ │  i16 : assert(# hyperplanes cA'' =!= 1 + # hyperplanes A)
│ │ │ │  When the second input is a _S_y_m_b_o_l, this method creates a new ring from the
│ │ │ │  underlying ring of $A$ by adjoining the symbol as a variable and constructs the
│ │ │ │  cone in this new ring.
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_euler__Restriction_lp__Central__Arrangement_cm__List_cm__Z__Z_rp.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  o2 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │  
│ │ │ -o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │ +o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │  
│ │ │  o3 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : restriction(A,1)
│ │ │ @@ -118,15 +118,15 @@
│ │ │  o4 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │  
│ │ │ -o5 = {z, x, x - z}
│ │ │ +o5 = {x - z, z, x}
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    If $({\mathcal A},m)$ is a free multiarrangement and so is $({\mathcal A},m')$, where $m'$ is obtained from $m$ by lowering a single multiplicity by one, the Euler restriction is free as well, and the modules of logarithmic derivations form a short exact sequence.  See the paper of Abe, Terao and Wakefield for details.
    
│ │ │ @@ -137,15 +137,15 @@
│ │ │                
    i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │  
│ │ │ -o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │ +o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │  
│ │ │  o8 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : prune image der(A,m)
│ │ │ @@ -186,16 +186,16 @@
│ │ │                
    i12 : A = arrangement "bracelet";
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │  
│ │ │ -o13 = ({x  + x  + x , x  + x , x , x  + x , x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ -         2    3    4   2    4   2   3    4   3   4
│ │ │ +o13 = ({x , x  + x , x , x , x  + x  + x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ +         2   3    4   3   4   2    3    4   2    4
│ │ │  
│ │ │  o13 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : C = restriction(A,0)
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,36 +33,36 @@
│ │ │ │  i2 : A = arrangement {x,y,z,x-y,x-z}
│ │ │ │  
│ │ │ │  o2 = {x, y, z, x - y, x - z}
│ │ │ │  
│ │ │ │  o2 : Hyperplane Arrangement
│ │ │ │  i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │ │  
│ │ │ │ -o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │ │ +o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : restriction(A,1)
│ │ │ │  
│ │ │ │  o4 = {x, z, x, x - z}
│ │ │ │  
│ │ │ │  o4 : Hyperplane Arrangement
│ │ │ │  i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │ │  
│ │ │ │ -o5 = {z, x, x - z}
│ │ │ │ +o5 = {x - z, z, x}
│ │ │ │  
│ │ │ │  o5 : Hyperplane Arrangement
│ │ │ │  If $({\mathcal A},m)$ is a free multiarrangement and so is $({\mathcal A},m')$,
│ │ │ │  where $m'$ is obtained from $m$ by lowering a single multiplicity by one, the
│ │ │ │  Euler restriction is free as well, and the modules of _l_o_g_a_r_i_t_h_m_i_c_ _d_e_r_i_v_a_t_i_o_n_s
│ │ │ │  form a short exact sequence. See the paper of Abe, Terao and Wakefield for
│ │ │ │  details.
│ │ │ │  i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
│ │ │ │  i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │ │  
│ │ │ │ -o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │ │ +o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ │ │  i9 : prune image der(A,m)
│ │ │ │  
│ │ │ │        3
│ │ │ │  o9 = R
│ │ │ │  
│ │ │ │ @@ -80,16 +80,16 @@
│ │ │ │  
│ │ │ │  o11 : QQ[y..z]-module, free, degrees {2:3}
│ │ │ │  It may be the case that the Euler restriction is free, while the naive
│ │ │ │  restriction is not:
│ │ │ │  i12 : A = arrangement "bracelet";
│ │ │ │  i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │ │  
│ │ │ │ -o13 = ({x  + x  + x , x  + x , x , x  + x , x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ │ -         2    3    4   2    4   2   3    4   3   4
│ │ │ │ +o13 = ({x , x  + x , x , x , x  + x  + x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ │ +         2   3    4   3   4   2    3    4   2    4
│ │ │ │  
│ │ │ │  o13 : Sequence
│ │ │ │  i14 : C = restriction(A,0)
│ │ │ │  
│ │ │ │  o14 = {x , x , x , x  + x , x  + x , x  + x , x  + x , x  + x  + x }
│ │ │ │          2   3   4   2    4   3    4   2    4   3    4   2    3    4
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_trim_lp__Arrangement_rp.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  o2 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : A' = trim A
│ │ │  
│ │ │ -o3 = {y, x, x + y}
│ │ │ +o3 = {x + y, y, x}
│ │ │  
│ │ │  o3 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : assert(ring A' === R)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  i2 : A = arrangement{x,x,0_R,y,y,y,x+y,x+y,x+y,x+y,x+y}
│ │ │ │  
│ │ │ │  o2 = {x, x, 0, y, y, y, x + y, x + y, x + y, x + y, x + y}
│ │ │ │  
│ │ │ │  o2 : Hyperplane Arrangement
│ │ │ │  i3 : A' = trim A
│ │ │ │  
│ │ │ │ -o3 = {y, x, x + y}
│ │ │ │ +o3 = {x + y, y, x}
│ │ │ │  
│ │ │ │  o3 : Hyperplane Arrangement
│ │ │ │  i4 : assert(ring A' === R)
│ │ │ │  i5 : assert(trim A' == A')
│ │ │ │  i6 : assert(trim A' == A')
│ │ │ │  Some natural operations produce non-simple hyperplane arrangements.
│ │ │ │  i7 : A'' = restriction(A, y)
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_type__B_lp__Z__Z_cm__Ring_rp.html
│ │ │ @@ -130,16 +130,16 @@
│ │ │  o5 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : trim A3
│ │ │  
│ │ │ -o6 = {x  + x , x , x }
│ │ │ -       1    2   2   1
│ │ │ +o6 = {x , x , x  + x }
│ │ │ +       2   1   1    2
│ │ │  
│ │ │  o6 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : ring A3
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,16 +51,16 @@
│ │ │ │  
│ │ │ │  o5 = {x , x  + x , x  + x , x }
│ │ │ │         1   1    2   1    2   2
│ │ │ │  
│ │ │ │  o5 : Hyperplane Arrangement
│ │ │ │  i6 : trim A3
│ │ │ │  
│ │ │ │ -o6 = {x  + x , x , x }
│ │ │ │ -       1    2   2   1
│ │ │ │ +o6 = {x , x , x  + x }
│ │ │ │ +       2   1   1    2
│ │ │ │  
│ │ │ │  o6 : Hyperplane Arrangement
│ │ │ │  i7 : ring A3
│ │ │ │  
│ │ │ │       ZZ
│ │ │ │  o7 = --[x ..x ]
│ │ │ │        2  1   2
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -16,15 +16,15 @@
│ │ │  i3 : R = S/f
│ │ │  
│ │ │  o3 = R
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : time R' = integralClosure R
│ │ │ - -- used 0.366041s (cpu); 0.288454s (thread); 0s (gc)
│ │ │ + -- used 0.469282s (cpu); 0.387216s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
│ │ │  
│ │ │  i5 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -83,15 +83,15 @@
│ │ │  i9 : R = S/f
│ │ │  
│ │ │  o9 = R
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │  
│ │ │  i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.478178s (cpu); 0.311505s (thread); 0s (gc)
│ │ │ + -- used 0.554875s (cpu); 0.371216s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
│ │ │  
│ │ │  i11 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -150,15 +150,15 @@
│ │ │  i15 : R = S/f
│ │ │  
│ │ │  o15 = R
│ │ │  
│ │ │  o15 : QuotientRing
│ │ │  
│ │ │  i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.370585s (cpu); 0.28775s (thread); 0s (gc)
│ │ │ + -- used 0.436842s (cpu); 0.358315s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
│ │ │  
│ │ │  i17 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -208,15 +208,15 @@
│ │ │  i20 : R = S/f
│ │ │  
│ │ │  o20 = R
│ │ │  
│ │ │  o20 : QuotientRing
│ │ │  
│ │ │  i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 0.535235s (cpu); 0.382352s (thread); 0s (gc)
│ │ │ + -- used 0.5723s (cpu); 0.408477s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
│ │ │  
│ │ │  i22 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -266,15 +266,15 @@
│ │ │  i25 : R = S/f
│ │ │  
│ │ │  o25 = R
│ │ │  
│ │ │  o25 : QuotientRing
│ │ │  
│ │ │  i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.804597s (cpu); 0.56172s (thread); 0s (gc)
│ │ │ + -- used 0.943304s (cpu); 0.671041s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
│ │ │  
│ │ │  i27 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -324,15 +324,15 @@
│ │ │  i30 : R = S/f
│ │ │  
│ │ │  o30 = R
│ │ │  
│ │ │  o30 : QuotientRing
│ │ │  
│ │ │  i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.383461s (cpu); 0.301121s (thread); 0s (gc)
│ │ │ + -- used 0.423338s (cpu); 0.34941s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
│ │ │  
│ │ │  i32 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -382,15 +382,15 @@
│ │ │  i35 : R = S/f
│ │ │  
│ │ │  o35 = R
│ │ │  
│ │ │  o35 : QuotientRing
│ │ │  
│ │ │  i36 : time R' = integralClosure R
│ │ │ - -- used 0.0463207s (cpu); 0.0463219s (thread); 0s (gc)
│ │ │ + -- used 0.0540181s (cpu); 0.0540162s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
│ │ │  
│ │ │  i37 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -432,15 +432,15 @@
│ │ │  i40 : R = S/I
│ │ │  
│ │ │  o40 = R
│ │ │  
│ │ │  o40 : QuotientRing
│ │ │  
│ │ │  i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.163449s (cpu); 0.0764506s (thread); 0s (gc)
│ │ │ + -- used 0.162604s (cpu); 0.088759s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
│ │ │  
│ │ │  i42 : icFractions R
│ │ │  
│ │ │ @@ -467,15 +467,15 @@
│ │ │  i45 : R = S/I
│ │ │  
│ │ │  o45 = R
│ │ │  
│ │ │  o45 : QuotientRing
│ │ │  
│ │ │  i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.0621086s (cpu); 0.0621083s (thread); 0s (gc)
│ │ │ + -- used 0.0773663s (cpu); 0.0773612s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
│ │ │  
│ │ │  i47 : icFractions R
│ │ │  
│ │ │ @@ -501,15 +501,15 @@
│ │ │  i50 : R = S/I
│ │ │  
│ │ │  o50 = R
│ │ │  
│ │ │  o50 : QuotientRing
│ │ │  
│ │ │  i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.16126s (cpu); 0.0999423s (thread); 0s (gc)
│ │ │ + -- used 0.172513s (cpu); 0.0916116s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
│ │ │  
│ │ │  i52 : icFractions R
│ │ │  
│ │ │ @@ -536,15 +536,15 @@
│ │ │  i55 : R = S/I
│ │ │  
│ │ │  o55 = R
│ │ │  
│ │ │  o55 : QuotientRing
│ │ │  
│ │ │  i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.0412101s (cpu); 0.0412113s (thread); 0s (gc)
│ │ │ + -- used 0.0530849s (cpu); 0.0528756s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
│ │ │  
│ │ │  i57 : icFractions R
│ │ │  
│ │ │ @@ -633,15 +633,15 @@
│ │ │  i66 : R = S/I
│ │ │  
│ │ │  o66 = R
│ │ │  
│ │ │  o66 : QuotientRing
│ │ │  
│ │ │  i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0626564s (cpu); 0.0626563s (thread); 0s (gc)
│ │ │ + -- used 0.0727781s (cpu); 0.0727786s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
│ │ │  
│ │ │  i68 : icFractions R
│ │ │  
│ │ │ @@ -722,15 +722,15 @@
│ │ │  i77 : R = S/I
│ │ │  
│ │ │  o77 = R
│ │ │  
│ │ │  o77 : QuotientRing
│ │ │  
│ │ │  i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.499367s (cpu); 0.320693s (thread); 0s (gc)
│ │ │ + -- used 0.543678s (cpu); 0.372757s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
│ │ │  
│ │ │  i79 : icFractions R
│ │ │  
│ │ │ @@ -750,15 +750,15 @@
│ │ │  i81 : R = S/sub(I,S)
│ │ │  
│ │ │  o81 = R
│ │ │  
│ │ │  o81 : QuotientRing
│ │ │  
│ │ │  i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.476305s (cpu); 0.308585s (thread); 0s (gc)
│ │ │ + -- used 0.519666s (cpu); 0.337567s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
│ │ │  
│ │ │  i83 : icFractions R
│ │ │  
│ │ │ @@ -778,20 +778,20 @@
│ │ │  i85 : R = S/sub(I,S)
│ │ │  
│ │ │  o85 = R
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │  
│ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000553267 sec #minors 4]
│ │ │ + [jacobian time .000643787 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .1009 sec  #fractions 6]
│ │ │ - [step 1:   time .260327 sec  #fractions 6]
│ │ │ - -- used 0.365058s (cpu); 0.26596s (thread); 0s (gc)
│ │ │ + [step 0:   time .133137 sec  #fractions 6]
│ │ │ + [step 1:   time .281863 sec  #fractions 6]
│ │ │ + -- used 0.419595s (cpu); 0.31504s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
│ │ │  
│ │ │  i87 : icFractions R
│ │ │  
│ │ │ @@ -811,20 +811,20 @@
│ │ │  i89 : R = S/sub(I,S)
│ │ │  
│ │ │  o89 = R
│ │ │  
│ │ │  o89 : QuotientRing
│ │ │  
│ │ │  i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000540333 sec #minors 4]
│ │ │ + [jacobian time .000739733 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .239665 sec  #fractions 6]
│ │ │ - [step 1:   time .253198 sec  #fractions 6]
│ │ │ - -- used 0.496594s (cpu); 0.316763s (thread); 0s (gc)
│ │ │ + [step 0:   time .256294 sec  #fractions 6]
│ │ │ + [step 1:   time .306873 sec  #fractions 6]
│ │ │ + -- used 0.567699s (cpu); 0.363369s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
│ │ │  
│ │ │  i91 : icFractions R
│ │ │  
│ │ │ @@ -844,20 +844,20 @@
│ │ │  i93 : R = S/sub(I,S)
│ │ │  
│ │ │  o93 = R
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │  
│ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000872496 sec #minors 1]
│ │ │ + [jacobian time .000850143 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .268434 sec  #fractions 6]
│ │ │ - [step 1:   time .706635 sec  #fractions 6]
│ │ │ - -- used 0.979619s (cpu); 0.587571s (thread); 0s (gc)
│ │ │ + [step 0:   time .301558 sec  #fractions 6]
│ │ │ + [step 1:   time .826583 sec  #fractions 6]
│ │ │ + -- used 1.13287s (cpu); 0.754169s (thread); 0s (gc)
│ │ │  
│ │ │  o94 = R'
│ │ │  
│ │ │  o94 : QuotientRing
│ │ │  
│ │ │  i95 : icFractions R
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.out
│ │ │ @@ -1,50 +1,50 @@
│ │ │  -- -*- M2-comint -*- hash: 13177954069434615273
│ │ │  
│ │ │  i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
│ │ │  
│ │ │  i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ - [jacobian time .000540924 sec #minors 3]
│ │ │ + [jacobian time .000748322 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00237406 seconds
│ │ │ -      idlizer1:  .00724052 seconds
│ │ │ -      idlizer2:  .00796856 seconds
│ │ │ -      minpres:   .00754598 seconds
│ │ │ -  time .0356617 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00291356 seconds
│ │ │ +      idlizer1:  .00925263 seconds
│ │ │ +      idlizer2:  .00989739 seconds
│ │ │ +      minpres:   .00904284 seconds
│ │ │ +  time .0444321 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00216054 seconds
│ │ │ -      idlizer1:  .0107394 seconds
│ │ │ -      idlizer2:  .00978004 seconds
│ │ │ -      minpres:   .0110795 seconds
│ │ │ -  time .0439125 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00273876 seconds
│ │ │ +      idlizer1:  .0134947 seconds
│ │ │ +      idlizer2:  .0123191 seconds
│ │ │ +      minpres:   .0135625 seconds
│ │ │ +  time .0546821 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00221557 seconds
│ │ │ -      idlizer1:  .13596 seconds
│ │ │ -      idlizer2:  .0103733 seconds
│ │ │ -      minpres:   .0091835 seconds
│ │ │ -  time .168397 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00288056 seconds
│ │ │ +      idlizer1:  .117649 seconds
│ │ │ +      idlizer2:  .0120434 seconds
│ │ │ +      minpres:   .0116378 seconds
│ │ │ +  time .157476 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00251998 seconds
│ │ │ -      idlizer1:  .0133362 seconds
│ │ │ -      idlizer2:  .0143524 seconds
│ │ │ -      minpres:   .016478 seconds
│ │ │ -  time .0594049 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00291486 seconds
│ │ │ +      idlizer1:  .0150719 seconds
│ │ │ +      idlizer2:  .0170579 seconds
│ │ │ +      minpres:   .0194682 seconds
│ │ │ +  time .0690217 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .00238962 seconds
│ │ │ -      idlizer1:  .00884565 seconds
│ │ │ -      idlizer2:  .0164711 seconds
│ │ │ -      minpres:   .0117843 seconds
│ │ │ -  time .0519156 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .0031077 seconds
│ │ │ +      idlizer1:  .0111216 seconds
│ │ │ +      idlizer2:  .0195059 seconds
│ │ │ +      minpres:   .0146837 seconds
│ │ │ +  time .0646093 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00253525 seconds
│ │ │ -      idlizer1:  .0091469 seconds
│ │ │ -  time .0186257 sec  #fractions 5]
│ │ │ - -- used 0.381825s (cpu); 0.30269s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .00303395 seconds
│ │ │ +      idlizer1:  .0118419 seconds
│ │ │ +  time .0249294 sec  #fractions 5]
│ │ │ + -- used 0.421092s (cpu); 0.351719s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = R'
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │  i3 : trim ideal R'
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.out
│ │ │ @@ -13,26 +13,26 @@
│ │ │  
│ │ │                  2      2    2        2   2 2     2
│ │ │  o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │  
│ │ │  i4 : time integralClosure J
│ │ │ - -- used 0.938509s (cpu); 0.685911s (thread); 0s (gc)
│ │ │ + -- used 1.32272s (cpu); 0.823029s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │  
│ │ │  o4 : Ideal of S
│ │ │  
│ │ │  i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ - -- used 0.638792s (cpu); 0.465638s (thread); 0s (gc)
│ │ │ + -- used 1.08644s (cpu); 0.570408s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o3 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time R' = integralClosure R
│ │ │ - -- used 0.366041s (cpu); 0.288454s (thread); 0s (gc)
│ │ │ + -- used 0.469282s (cpu); 0.387216s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -191,15 +191,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.478178s (cpu); 0.311505s (thread); 0s (gc)
│ │ │ + -- used 0.554875s (cpu); 0.371216s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -278,15 +278,15 @@ │ │ │ │ │ │ o15 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.370585s (cpu); 0.28775s (thread); 0s (gc)
│ │ │ + -- used 0.436842s (cpu); 0.358315s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -353,15 +353,15 @@ │ │ │ │ │ │ o20 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 0.535235s (cpu); 0.382352s (thread); 0s (gc)
│ │ │ + -- used 0.5723s (cpu); 0.408477s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -428,15 +428,15 @@ │ │ │ │ │ │ o25 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.804597s (cpu); 0.56172s (thread); 0s (gc)
│ │ │ + -- used 0.943304s (cpu); 0.671041s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -503,15 +503,15 @@ │ │ │ │ │ │ o30 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.383461s (cpu); 0.301121s (thread); 0s (gc)
│ │ │ + -- used 0.423338s (cpu); 0.34941s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -578,15 +578,15 @@ │ │ │ │ │ │ o35 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i36 : time R' = integralClosure R
│ │ │ - -- used 0.0463207s (cpu); 0.0463219s (thread); 0s (gc)
│ │ │ + -- used 0.0540181s (cpu); 0.0540162s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -648,15 +648,15 @@ │ │ │ │ │ │ o40 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.163449s (cpu); 0.0764506s (thread); 0s (gc)
│ │ │ + -- used 0.162604s (cpu); 0.088759s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -700,15 +700,15 @@ │ │ │ │ │ │ o45 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.0621086s (cpu); 0.0621083s (thread); 0s (gc)
│ │ │ + -- used 0.0773663s (cpu); 0.0773612s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -751,15 +751,15 @@ │ │ │ │ │ │ o50 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.16126s (cpu); 0.0999423s (thread); 0s (gc)
│ │ │ + -- used 0.172513s (cpu); 0.0916116s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -803,15 +803,15 @@ │ │ │ │ │ │ o55 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.0412101s (cpu); 0.0412113s (thread); 0s (gc)
│ │ │ + -- used 0.0530849s (cpu); 0.0528756s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -938,15 +938,15 @@ │ │ │ │ │ │ o66 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0626564s (cpu); 0.0626563s (thread); 0s (gc)
│ │ │ + -- used 0.0727781s (cpu); 0.0727786s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1062,15 +1062,15 @@ │ │ │ │ │ │ o77 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.499367s (cpu); 0.320693s (thread); 0s (gc)
│ │ │ + -- used 0.543678s (cpu); 0.372757s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1104,15 +1104,15 @@ │ │ │ │ │ │ o81 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.476305s (cpu); 0.308585s (thread); 0s (gc)
│ │ │ + -- used 0.519666s (cpu); 0.337567s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1146,20 +1146,20 @@ │ │ │ │ │ │ o85 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000553267 sec #minors 4]
│ │ │ + [jacobian time .000643787 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .1009 sec  #fractions 6]
│ │ │ - [step 1:   time .260327 sec  #fractions 6]
│ │ │ - -- used 0.365058s (cpu); 0.26596s (thread); 0s (gc)
│ │ │ + [step 0:   time .133137 sec  #fractions 6]
│ │ │ + [step 1:   time .281863 sec  #fractions 6]
│ │ │ + -- used 0.419595s (cpu); 0.31504s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1193,20 +1193,20 @@ │ │ │ │ │ │ o89 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000540333 sec #minors 4]
│ │ │ + [jacobian time .000739733 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .239665 sec  #fractions 6]
│ │ │ - [step 1:   time .253198 sec  #fractions 6]
│ │ │ - -- used 0.496594s (cpu); 0.316763s (thread); 0s (gc)
│ │ │ + [step 0:   time .256294 sec  #fractions 6]
│ │ │ + [step 1:   time .306873 sec  #fractions 6]
│ │ │ + -- used 0.567699s (cpu); 0.363369s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1243,20 +1243,20 @@ │ │ │ │ │ │ o93 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000872496 sec #minors 1]
│ │ │ + [jacobian time .000850143 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .268434 sec  #fractions 6]
│ │ │ - [step 1:   time .706635 sec  #fractions 6]
│ │ │ - -- used 0.979619s (cpu); 0.587571s (thread); 0s (gc)
│ │ │ + [step 0:   time .301558 sec  #fractions 6]
│ │ │ + [step 1:   time .826583 sec  #fractions 6]
│ │ │ + -- used 1.13287s (cpu); 0.754169s (thread); 0s (gc)
│ │ │  
│ │ │  o94 = R'
│ │ │  
│ │ │  o94 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -48,15 +48,15 @@ │ │ │ │ o2 : Ideal of S │ │ │ │ i3 : R = S/f │ │ │ │ │ │ │ │ o3 = R │ │ │ │ │ │ │ │ o3 : QuotientRing │ │ │ │ i4 : time R' = integralClosure R │ │ │ │ - -- used 0.366041s (cpu); 0.288454s (thread); 0s (gc) │ │ │ │ + -- used 0.469282s (cpu); 0.387216s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = R' │ │ │ │ │ │ │ │ o4 : QuotientRing │ │ │ │ i5 : netList (ideal R')_* │ │ │ │ │ │ │ │ +------------------------------------------------------------------------+ │ │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ o8 : Ideal of S │ │ │ │ i9 : R = S/f │ │ │ │ │ │ │ │ o9 = R │ │ │ │ │ │ │ │ o9 : QuotientRing │ │ │ │ i10 : time R' = integralClosure(R, Strategy => Radical) │ │ │ │ - -- used 0.478178s (cpu); 0.311505s (thread); 0s (gc) │ │ │ │ + -- used 0.554875s (cpu); 0.371216s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 = R' │ │ │ │ │ │ │ │ o10 : QuotientRing │ │ │ │ i11 : netList (ideal R')_* │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ @@ -199,15 +199,15 @@ │ │ │ │ o14 : Ideal of S │ │ │ │ i15 : R = S/f │ │ │ │ │ │ │ │ o15 = R │ │ │ │ │ │ │ │ o15 : QuotientRing │ │ │ │ i16 : time R' = integralClosure(R, Strategy => AllCodimensions) │ │ │ │ - -- used 0.370585s (cpu); 0.28775s (thread); 0s (gc) │ │ │ │ + -- used 0.436842s (cpu); 0.358315s (thread); 0s (gc) │ │ │ │ │ │ │ │ o16 = R' │ │ │ │ │ │ │ │ o16 : QuotientRing │ │ │ │ i17 : netList (ideal R')_* │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ @@ -281,15 +281,15 @@ │ │ │ │ o19 : Ideal of S │ │ │ │ i20 : R = S/f │ │ │ │ │ │ │ │ o20 = R │ │ │ │ │ │ │ │ o20 : QuotientRing │ │ │ │ i21 : time R' = integralClosure(R, Strategy => SimplifyFractions) │ │ │ │ - -- used 0.535235s (cpu); 0.382352s (thread); 0s (gc) │ │ │ │ + -- used 0.5723s (cpu); 0.408477s (thread); 0s (gc) │ │ │ │ │ │ │ │ o21 = R' │ │ │ │ │ │ │ │ o21 : QuotientRing │ │ │ │ i22 : netList (ideal R')_* │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ @@ -363,15 +363,15 @@ │ │ │ │ o24 : Ideal of S │ │ │ │ i25 : R = S/f │ │ │ │ │ │ │ │ o25 = R │ │ │ │ │ │ │ │ o25 : QuotientRing │ │ │ │ i26 : time R' = integralClosure (R, Strategy => RadicalCodim1) │ │ │ │ - -- used 0.804597s (cpu); 0.56172s (thread); 0s (gc) │ │ │ │ + -- used 0.943304s (cpu); 0.671041s (thread); 0s (gc) │ │ │ │ │ │ │ │ o26 = R' │ │ │ │ │ │ │ │ o26 : QuotientRing │ │ │ │ i27 : netList (ideal R')_* │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ @@ -445,15 +445,15 @@ │ │ │ │ o29 : Ideal of S │ │ │ │ i30 : R = S/f │ │ │ │ │ │ │ │ o30 = R │ │ │ │ │ │ │ │ o30 : QuotientRing │ │ │ │ i31 : time R' = integralClosure (R, Strategy => Vasconcelos) │ │ │ │ - -- used 0.383461s (cpu); 0.301121s (thread); 0s (gc) │ │ │ │ + -- used 0.423338s (cpu); 0.34941s (thread); 0s (gc) │ │ │ │ │ │ │ │ o31 = R' │ │ │ │ │ │ │ │ o31 : QuotientRing │ │ │ │ i32 : netList (ideal R')_* │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ @@ -527,15 +527,15 @@ │ │ │ │ o34 : Ideal of S │ │ │ │ i35 : R = S/f │ │ │ │ │ │ │ │ o35 = R │ │ │ │ │ │ │ │ o35 : QuotientRing │ │ │ │ i36 : time R' = integralClosure R │ │ │ │ - -- used 0.0463207s (cpu); 0.0463219s (thread); 0s (gc) │ │ │ │ + -- used 0.0540181s (cpu); 0.0540162s (thread); 0s (gc) │ │ │ │ │ │ │ │ o36 = R' │ │ │ │ │ │ │ │ o36 : QuotientRing │ │ │ │ i37 : netList (ideal R')_* │ │ │ │ │ │ │ │ +-----------+ │ │ │ │ @@ -573,15 +573,15 @@ │ │ │ │ o39 : Ideal of S │ │ │ │ i40 : R = S/I │ │ │ │ │ │ │ │ o40 = R │ │ │ │ │ │ │ │ o40 : QuotientRing │ │ │ │ i41 : time R' = integralClosure(R, Strategy => Radical) │ │ │ │ - -- used 0.163449s (cpu); 0.0764506s (thread); 0s (gc) │ │ │ │ + -- used 0.162604s (cpu); 0.088759s (thread); 0s (gc) │ │ │ │ │ │ │ │ o41 = R' │ │ │ │ │ │ │ │ o41 : QuotientRing │ │ │ │ i42 : icFractions R │ │ │ │ │ │ │ │ 2 │ │ │ │ @@ -603,15 +603,15 @@ │ │ │ │ o44 : Ideal of S │ │ │ │ i45 : R = S/I │ │ │ │ │ │ │ │ o45 = R │ │ │ │ │ │ │ │ o45 : QuotientRing │ │ │ │ i46 : time R' = integralClosure(R, Strategy => AllCodimensions) │ │ │ │ - -- used 0.0621086s (cpu); 0.0621083s (thread); 0s (gc) │ │ │ │ + -- used 0.0773663s (cpu); 0.0773612s (thread); 0s (gc) │ │ │ │ │ │ │ │ o46 = R' │ │ │ │ │ │ │ │ o46 : QuotientRing │ │ │ │ i47 : icFractions R │ │ │ │ │ │ │ │ b*d │ │ │ │ @@ -632,15 +632,15 @@ │ │ │ │ o49 : Ideal of S │ │ │ │ i50 : R = S/I │ │ │ │ │ │ │ │ o50 = R │ │ │ │ │ │ │ │ o50 : QuotientRing │ │ │ │ i51 : time R' = integralClosure (R, Strategy => RadicalCodim1) │ │ │ │ - -- used 0.16126s (cpu); 0.0999423s (thread); 0s (gc) │ │ │ │ + -- used 0.172513s (cpu); 0.0916116s (thread); 0s (gc) │ │ │ │ │ │ │ │ o51 = R' │ │ │ │ │ │ │ │ o51 : QuotientRing │ │ │ │ i52 : icFractions R │ │ │ │ │ │ │ │ 2 │ │ │ │ @@ -662,15 +662,15 @@ │ │ │ │ o54 : Ideal of S │ │ │ │ i55 : R = S/I │ │ │ │ │ │ │ │ o55 = R │ │ │ │ │ │ │ │ o55 : QuotientRing │ │ │ │ i56 : time R' = integralClosure (R, Strategy => Vasconcelos) │ │ │ │ - -- used 0.0412101s (cpu); 0.0412113s (thread); 0s (gc) │ │ │ │ + -- used 0.0530849s (cpu); 0.0528756s (thread); 0s (gc) │ │ │ │ │ │ │ │ o56 = R' │ │ │ │ │ │ │ │ o56 : QuotientRing │ │ │ │ i57 : icFractions R │ │ │ │ │ │ │ │ 2 │ │ │ │ @@ -755,15 +755,15 @@ │ │ │ │ o65 : BettiTally │ │ │ │ i66 : R = S/I │ │ │ │ │ │ │ │ o66 = R │ │ │ │ │ │ │ │ o66 : QuotientRing │ │ │ │ i67 : time R' = integralClosure(R, Strategy => Radical) │ │ │ │ - -- used 0.0626564s (cpu); 0.0626563s (thread); 0s (gc) │ │ │ │ + -- used 0.0727781s (cpu); 0.0727786s (thread); 0s (gc) │ │ │ │ │ │ │ │ o67 = R' │ │ │ │ │ │ │ │ o67 : QuotientRing │ │ │ │ i68 : icFractions R │ │ │ │ │ │ │ │ 2 2 │ │ │ │ @@ -839,15 +839,15 @@ │ │ │ │ o76 : BettiTally │ │ │ │ i77 : R = S/I │ │ │ │ │ │ │ │ o77 = R │ │ │ │ │ │ │ │ o77 : QuotientRing │ │ │ │ i78 : time R' = integralClosure(R, Strategy => Radical) │ │ │ │ - -- used 0.499367s (cpu); 0.320693s (thread); 0s (gc) │ │ │ │ + -- used 0.543678s (cpu); 0.372757s (thread); 0s (gc) │ │ │ │ │ │ │ │ o78 = R' │ │ │ │ │ │ │ │ o78 : QuotientRing │ │ │ │ i79 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -863,15 +863,15 @@ │ │ │ │ o80 : PolynomialRing │ │ │ │ i81 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o81 = R │ │ │ │ │ │ │ │ o81 : QuotientRing │ │ │ │ i82 : time R' = integralClosure(R, Strategy => AllCodimensions) │ │ │ │ - -- used 0.476305s (cpu); 0.308585s (thread); 0s (gc) │ │ │ │ + -- used 0.519666s (cpu); 0.337567s (thread); 0s (gc) │ │ │ │ │ │ │ │ o82 = R' │ │ │ │ │ │ │ │ o82 : QuotientRing │ │ │ │ i83 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -887,20 +887,20 @@ │ │ │ │ o84 : PolynomialRing │ │ │ │ i85 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o85 = R │ │ │ │ │ │ │ │ o85 : QuotientRing │ │ │ │ i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1) │ │ │ │ - [jacobian time .000553267 sec #minors 4] │ │ │ │ + [jacobian time .000643787 sec #minors 4] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .1009 sec #fractions 6] │ │ │ │ - [step 1: time .260327 sec #fractions 6] │ │ │ │ - -- used 0.365058s (cpu); 0.26596s (thread); 0s (gc) │ │ │ │ + [step 0: time .133137 sec #fractions 6] │ │ │ │ + [step 1: time .281863 sec #fractions 6] │ │ │ │ + -- used 0.419595s (cpu); 0.31504s (thread); 0s (gc) │ │ │ │ │ │ │ │ o86 = R' │ │ │ │ │ │ │ │ o86 : QuotientRing │ │ │ │ i87 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -916,20 +916,20 @@ │ │ │ │ o88 : PolynomialRing │ │ │ │ i89 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o89 = R │ │ │ │ │ │ │ │ o89 : QuotientRing │ │ │ │ i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1) │ │ │ │ - [jacobian time .000540333 sec #minors 4] │ │ │ │ + [jacobian time .000739733 sec #minors 4] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .239665 sec #fractions 6] │ │ │ │ - [step 1: time .253198 sec #fractions 6] │ │ │ │ - -- used 0.496594s (cpu); 0.316763s (thread); 0s (gc) │ │ │ │ + [step 0: time .256294 sec #fractions 6] │ │ │ │ + [step 1: time .306873 sec #fractions 6] │ │ │ │ + -- used 0.567699s (cpu); 0.363369s (thread); 0s (gc) │ │ │ │ │ │ │ │ o90 = R' │ │ │ │ │ │ │ │ o90 : QuotientRing │ │ │ │ i91 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -948,20 +948,20 @@ │ │ │ │ i93 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o93 = R │ │ │ │ │ │ │ │ o93 : QuotientRing │ │ │ │ i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, │ │ │ │ StartWithOneMinor}, Verbosity => 1) │ │ │ │ - [jacobian time .000872496 sec #minors 1] │ │ │ │ + [jacobian time .000850143 sec #minors 1] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .268434 sec #fractions 6] │ │ │ │ - [step 1: time .706635 sec #fractions 6] │ │ │ │ - -- used 0.979619s (cpu); 0.587571s (thread); 0s (gc) │ │ │ │ + [step 0: time .301558 sec #fractions 6] │ │ │ │ + [step 1: time .826583 sec #fractions 6] │ │ │ │ + -- used 1.13287s (cpu); 0.754169s (thread); 0s (gc) │ │ │ │ │ │ │ │ o94 = R' │ │ │ │ │ │ │ │ o94 : QuotientRing │ │ │ │ i95 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 2 3 2 │ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.html │ │ │ @@ -76,52 +76,52 @@ │ │ │ │ │ │ 
    i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ - [jacobian time .000540924 sec #minors 3]
│ │ │ + [jacobian time .000748322 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00237406 seconds
│ │ │ -      idlizer1:  .00724052 seconds
│ │ │ -      idlizer2:  .00796856 seconds
│ │ │ -      minpres:   .00754598 seconds
│ │ │ -  time .0356617 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00291356 seconds
│ │ │ +      idlizer1:  .00925263 seconds
│ │ │ +      idlizer2:  .00989739 seconds
│ │ │ +      minpres:   .00904284 seconds
│ │ │ +  time .0444321 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00216054 seconds
│ │ │ -      idlizer1:  .0107394 seconds
│ │ │ -      idlizer2:  .00978004 seconds
│ │ │ -      minpres:   .0110795 seconds
│ │ │ -  time .0439125 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00273876 seconds
│ │ │ +      idlizer1:  .0134947 seconds
│ │ │ +      idlizer2:  .0123191 seconds
│ │ │ +      minpres:   .0135625 seconds
│ │ │ +  time .0546821 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00221557 seconds
│ │ │ -      idlizer1:  .13596 seconds
│ │ │ -      idlizer2:  .0103733 seconds
│ │ │ -      minpres:   .0091835 seconds
│ │ │ -  time .168397 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00288056 seconds
│ │ │ +      idlizer1:  .117649 seconds
│ │ │ +      idlizer2:  .0120434 seconds
│ │ │ +      minpres:   .0116378 seconds
│ │ │ +  time .157476 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00251998 seconds
│ │ │ -      idlizer1:  .0133362 seconds
│ │ │ -      idlizer2:  .0143524 seconds
│ │ │ -      minpres:   .016478 seconds
│ │ │ -  time .0594049 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00291486 seconds
│ │ │ +      idlizer1:  .0150719 seconds
│ │ │ +      idlizer2:  .0170579 seconds
│ │ │ +      minpres:   .0194682 seconds
│ │ │ +  time .0690217 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .00238962 seconds
│ │ │ -      idlizer1:  .00884565 seconds
│ │ │ -      idlizer2:  .0164711 seconds
│ │ │ -      minpres:   .0117843 seconds
│ │ │ -  time .0519156 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .0031077 seconds
│ │ │ +      idlizer1:  .0111216 seconds
│ │ │ +      idlizer2:  .0195059 seconds
│ │ │ +      minpres:   .0146837 seconds
│ │ │ +  time .0646093 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00253525 seconds
│ │ │ -      idlizer1:  .0091469 seconds
│ │ │ -  time .0186257 sec  #fractions 5]
│ │ │ - -- used 0.381825s (cpu); 0.30269s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .00303395 seconds
│ │ │ +      idlizer1:  .0118419 seconds
│ │ │ +  time .0249294 sec  #fractions 5]
│ │ │ + -- used 0.421092s (cpu); 0.351719s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = R'
│ │ │  
│ │ │  o2 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -12,52 +12,52 @@ │ │ │ │ displayed. A value of 0 means: keep quiet. │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ When the computation takes a considerable time, this function can be used to │ │ │ │ decide if it will ever finish, or to get a feel for what is happening during │ │ │ │ the computation. │ │ │ │ i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3); │ │ │ │ i2 : time R' = integralClosure(R, Verbosity => 2) │ │ │ │ - [jacobian time .000540924 sec #minors 3] │ │ │ │ + [jacobian time .000748322 sec #minors 3] │ │ │ │ integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ [step 0: │ │ │ │ - radical (use minprimes) .00237406 seconds │ │ │ │ - idlizer1: .00724052 seconds │ │ │ │ - idlizer2: .00796856 seconds │ │ │ │ - minpres: .00754598 seconds │ │ │ │ - time .0356617 sec #fractions 4] │ │ │ │ + radical (use minprimes) .00291356 seconds │ │ │ │ + idlizer1: .00925263 seconds │ │ │ │ + idlizer2: .00989739 seconds │ │ │ │ + minpres: .00904284 seconds │ │ │ │ + time .0444321 sec #fractions 4] │ │ │ │ [step 1: │ │ │ │ - radical (use minprimes) .00216054 seconds │ │ │ │ - idlizer1: .0107394 seconds │ │ │ │ - idlizer2: .00978004 seconds │ │ │ │ - minpres: .0110795 seconds │ │ │ │ - time .0439125 sec #fractions 4] │ │ │ │ + radical (use minprimes) .00273876 seconds │ │ │ │ + idlizer1: .0134947 seconds │ │ │ │ + idlizer2: .0123191 seconds │ │ │ │ + minpres: .0135625 seconds │ │ │ │ + time .0546821 sec #fractions 4] │ │ │ │ [step 2: │ │ │ │ - radical (use minprimes) .00221557 seconds │ │ │ │ - idlizer1: .13596 seconds │ │ │ │ - idlizer2: .0103733 seconds │ │ │ │ - minpres: .0091835 seconds │ │ │ │ - time .168397 sec #fractions 5] │ │ │ │ + radical (use minprimes) .00288056 seconds │ │ │ │ + idlizer1: .117649 seconds │ │ │ │ + idlizer2: .0120434 seconds │ │ │ │ + minpres: .0116378 seconds │ │ │ │ + time .157476 sec #fractions 5] │ │ │ │ [step 3: │ │ │ │ - radical (use minprimes) .00251998 seconds │ │ │ │ - idlizer1: .0133362 seconds │ │ │ │ - idlizer2: .0143524 seconds │ │ │ │ - minpres: .016478 seconds │ │ │ │ - time .0594049 sec #fractions 5] │ │ │ │ + radical (use minprimes) .00291486 seconds │ │ │ │ + idlizer1: .0150719 seconds │ │ │ │ + idlizer2: .0170579 seconds │ │ │ │ + minpres: .0194682 seconds │ │ │ │ + time .0690217 sec #fractions 5] │ │ │ │ [step 4: │ │ │ │ - radical (use minprimes) .00238962 seconds │ │ │ │ - idlizer1: .00884565 seconds │ │ │ │ - idlizer2: .0164711 seconds │ │ │ │ - minpres: .0117843 seconds │ │ │ │ - time .0519156 sec #fractions 5] │ │ │ │ + radical (use minprimes) .0031077 seconds │ │ │ │ + idlizer1: .0111216 seconds │ │ │ │ + idlizer2: .0195059 seconds │ │ │ │ + minpres: .0146837 seconds │ │ │ │ + time .0646093 sec #fractions 5] │ │ │ │ [step 5: │ │ │ │ - radical (use minprimes) .00253525 seconds │ │ │ │ - idlizer1: .0091469 seconds │ │ │ │ - time .0186257 sec #fractions 5] │ │ │ │ - -- used 0.381825s (cpu); 0.30269s (thread); 0s (gc) │ │ │ │ + radical (use minprimes) .00303395 seconds │ │ │ │ + idlizer1: .0118419 seconds │ │ │ │ + time .0249294 sec #fractions 5] │ │ │ │ + -- used 0.421092s (cpu); 0.351719s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = R' │ │ │ │ │ │ │ │ o2 : QuotientRing │ │ │ │ i3 : trim ideal R' │ │ │ │ │ │ │ │ 3 2 2 2 4 4 │ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.html │ │ │ @@ -114,29 +114,29 @@ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time integralClosure J
│ │ │ - -- used 0.938509s (cpu); 0.685911s (thread); 0s (gc)
│ │ │ + -- used 1.32272s (cpu); 0.823029s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │  
│ │ │  o4 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ - -- used 0.638792s (cpu); 0.465638s (thread); 0s (gc)
│ │ │ + -- used 1.08644s (cpu); 0.570408s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,25 +46,25 @@
│ │ │ │  i3 : J = ideal jacobian ideal F
│ │ │ │  
│ │ │ │                  2      2    2        2   2 2     2
│ │ │ │  o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : time integralClosure J
│ │ │ │ - -- used 0.938509s (cpu); 0.685911s (thread); 0s (gc)
│ │ │ │ + -- used 1.32272s (cpu); 0.823029s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2 2              2 2                2          2   2
│ │ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2   3               2 2     2   5
│ │ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ │ - -- used 0.638792s (cpu); 0.465638s (thread); 0s (gc)
│ │ │ │ + -- used 1.08644s (cpu); 0.570408s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2 2              2 2                2          2   2
│ │ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2   3               2 2     2   5
│ │ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon May 18 11:29:47 2026
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon May 18 11:29:46 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │  #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  c2NocmVpZXJHcmFwaA==
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_equivariant__Hilbert.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o3 : DiagonalAction
│ │ │  
│ │ │  i4 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o4 = false
│ │ │  
│ │ │  i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00692953s elapsed
│ │ │ + -- .00348985s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (ζ ζ  + ζ   + ζ  )T + (ζ ζ  + ζ  + ζ  + ζ   + ζ  ζ   + ζ  )T  +
│ │ │             0 1    1     0        0 1    0    1    1     0  1     0      
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │       (ζ ζ  + ζ ζ  + ζ ζ  + ζ ζ   + 1 + ζ   + ζ  ζ  + ζ  ζ   + ζ  ζ   + ζ  )T 
│ │ │ @@ -51,10 +51,10 @@
│ │ │           0   1
│ │ │  
│ │ │  i6 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .00397228s elapsed
│ │ │ + -- .000744086s elapsed
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hsop_spalgorithms.out
│ │ │ @@ -23,23 +23,23 @@
│ │ │  o3 = QQ[x..z] <- {| 0 -1 0  |, | 0 -1 0 |}
│ │ │                    | 1 0  0  |  | 1 0  0 |
│ │ │                    | 0 0  -1 |  | 0 0  1 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : time P1=primaryInvariants C4xC2
│ │ │ - -- used 1.1388s (cpu); 0.557706s (thread); 0s (gc)
│ │ │ + -- used 1.09318s (cpu); 0.467521s (thread); 0s (gc)
│ │ │  
│ │ │ -       2    2   2   3       3
│ │ │ -o4 = {x  + y , z , x y - x*y }
│ │ │ +       2   2    2   3       3
│ │ │ +o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 1.29621s (cpu); 0.589812s (thread); 0s (gc)
│ │ │ + -- used 1.22395s (cpu); 0.487875s (thread); 0s (gc)
│ │ │  
│ │ │                     8                 7                   6 2  
│ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │       ------------------------------------------------------------------------
│ │ │                     5 3                  4 4                 3 5  
│ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -90,23 +90,23 @@
│ │ │       ------------------------------------------------------------------------
│ │ │          2 6    8
│ │ │       90y z  + z }
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ - -- used 0.0169186s (cpu); 0.0169215s (thread); 0s (gc)
│ │ │ + -- used 0.0208381s (cpu); 0.0208417s (thread); 0s (gc)
│ │ │  
│ │ │            4    4
│ │ │  o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 2.88612s (cpu); 1.36918s (thread); 0s (gc)
│ │ │ + -- used 2.85822s (cpu); 1.2461s (thread); 0s (gc)
│ │ │  
│ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4  
│ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │       ------------------------------------------------------------------------
│ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x 
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.out
│ │ │ @@ -14,15 +14,15 @@
│ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : elapsedTime invariants S4
│ │ │ - -- .795837s elapsed
│ │ │ + -- .449597s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -32,15 +32,15 @@
│ │ │  
│ │ │  i5 : elapsedTime invariants(S4, DegreeBound => 4)
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │ - -- .542414s elapsed
│ │ │ + -- .34748s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -14,28 +14,28 @@
│ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : elapsedTime invariants S4
│ │ │ - -- .724053s elapsed
│ │ │ + -- .454824s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │        4
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime invariants(S4, Strategy => "LinearAlgebra")
│ │ │ - -- .0774921s elapsed
│ │ │ + -- .0562124s elapsed
│ │ │  
│ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_primary__Invariants.out
│ │ │ @@ -16,16 +16,16 @@
│ │ │                    | 0 0 1 |  | 1 0 0 |
│ │ │                    | 1 0 0 |  | 0 0 1 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : primaryInvariants S3
│ │ │  
│ │ │ -                                   3    3    3
│ │ │ -o4 = {x + y + z, x*y + x*z + y*z, x  + y  + z }
│ │ │ +                  2    2    2
│ │ │ +o4 = {x + y + z, x  + y  + z , x*y*z}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : K=GF(101)
│ │ │  
│ │ │  o5 = K
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_primary__Invariants_lp..._cm__Degree__Vector_eq_gt..._rp.out
│ │ │ @@ -16,16 +16,13 @@
│ │ │                    | 0 0 1 |  | 1 0 0 |
│ │ │                    | 1 0 0 |  | 0 0 1 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : primaryInvariants(S3,DegreeVector=>{3,3,4})
│ │ │  
│ │ │ -       3    3    3   2       2    2     2       2      2   3       3    3   
│ │ │ -o4 = {x  + y  + z , x y + x*y  + x z + y z + x*z  + y*z , x y + x*y  + x z +
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -      3       3      3
│ │ │ -     y z + x*z  + y*z }
│ │ │ +              3    3    3   2 2    2 2    2 2
│ │ │ +o4 = {x*y*z, x  + y  + z , x y  + x z  + y z }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 = false
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00692953s elapsed
│ │ │ + -- .00348985s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (ζ ζ  + ζ   + ζ  )T + (ζ ζ  + ζ  + ζ  + ζ   + ζ  ζ   + ζ  )T  +
│ │ │             0 1    1     0        0 1    0    1    1     0  1     0      
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │       (ζ ζ  + ζ ζ  + ζ ζ  + ζ ζ   + 1 + ζ   + ζ  ζ  + ζ  ζ   + ζ  ζ   + ζ  )T 
│ │ │ @@ -129,15 +129,15 @@
│ │ │  
│ │ │  o6 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .00397228s elapsed
    │ │ │ + -- .000744086s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    For the programmer

    │ │ │ ├── html2text {} │ │ │ │ @@ -30,15 +30,15 @@ │ │ │ │ | 0 -1 1 | │ │ │ │ │ │ │ │ o3 : DiagonalAction │ │ │ │ i4 : T.cache.?equivariantHilbert │ │ │ │ │ │ │ │ o4 = false │ │ │ │ i5 : elapsedTime equivariantHilbertSeries(T, Order => 5) │ │ │ │ - -- .00692953s elapsed │ │ │ │ + -- .00348985s elapsed │ │ │ │ │ │ │ │ -1 -1 2 2 -2 -1 -1 -2 2 │ │ │ │ o5 = 1 + (ζ ζ + ζ + ζ )T + (ζ ζ + ζ + ζ + ζ + ζ ζ + ζ │ │ │ │ )T + │ │ │ │ 0 1 1 0 0 1 0 1 1 0 1 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 3 2 2 -1 -3 -1 -1 -2 -2 -1 -3 3 │ │ │ │ @@ -57,13 +57,13 @@ │ │ │ │ │ │ │ │ o5 : ZZ[ζ ..ζ ][T] │ │ │ │ 0 1 │ │ │ │ i6 : T.cache.?equivariantHilbert │ │ │ │ │ │ │ │ o6 = true │ │ │ │ i7 : elapsedTime equivariantHilbertSeries(T, Order => 5); │ │ │ │ - -- .00397228s elapsed │ │ │ │ + -- .000744086s elapsed │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _e_q_u_i_v_a_r_i_a_n_t_H_i_l_b_e_r_t is a _s_y_m_b_o_l. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.05+ds/M2/Macaulay2/packages/InvariantRing/AbelianGroupsDoc.m2:217:0. │ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_hsop_spalgorithms.html │ │ │ @@ -97,26 +97,26 @@ │ │ │ │ │ │ │ │ │

    The two algorithms used in primaryInvariants are timed. One sees that the Dade algorithm is faster, however the primary invariants output are all of degree 8 and have ugly coefficients.

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : time P1=primaryInvariants C4xC2
│ │ │ - -- used 1.1388s (cpu); 0.557706s (thread); 0s (gc)
│ │ │ + -- used 1.09318s (cpu); 0.467521s (thread); 0s (gc)
│ │ │  
│ │ │ -       2    2   2   3       3
│ │ │ -o4 = {x  + y , z , x y - x*y }
│ │ │ +       2   2    2   3       3
│ │ │ +o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │ 
    i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 1.29621s (cpu); 0.589812s (thread); 0s (gc)
│ │ │ + -- used 1.22395s (cpu); 0.487875s (thread); 0s (gc)
│ │ │  
│ │ │                     8                 7                   6 2  
│ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │       ------------------------------------------------------------------------
│ │ │                     5 3                  4 4                 3 5  
│ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -173,26 +173,26 @@
│ │ │
    │ │ │

    The extra work done by the default algorithm to ensure an optimal hsop is rewarded by needing to calculate a smaller collection of corresponding secondary invariants. In fact, it has proved quicker overall to calculate the invariant ring based on the optimal algorithm rather than the Dade algorithm.

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ - -- used 0.0169186s (cpu); 0.0169215s (thread); 0s (gc)
│ │ │ + -- used 0.0208381s (cpu); 0.0208417s (thread); 0s (gc)
│ │ │  
│ │ │            4    4
│ │ │  o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 2.88612s (cpu); 1.36918s (thread); 0s (gc)
│ │ │ + -- used 2.85822s (cpu); 1.2461s (thread); 0s (gc)
│ │ │  
│ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4  
│ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │       ------------------------------------------------------------------------
│ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x 
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -69,22 +69,22 @@
│ │ │ │                    | 0 0  -1 |  | 0 0  1 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  The two algorithms used in _p_r_i_m_a_r_y_I_n_v_a_r_i_a_n_t_s are timed. One sees that the Dade
│ │ │ │  algorithm is faster, however the primary invariants output are all of degree 8
│ │ │ │  and have ugly coefficients.
│ │ │ │  i4 : time P1=primaryInvariants C4xC2
│ │ │ │ - -- used 1.1388s (cpu); 0.557706s (thread); 0s (gc)
│ │ │ │ + -- used 1.09318s (cpu); 0.467521s (thread); 0s (gc)
│ │ │ │  
│ │ │ │ -       2    2   2   3       3
│ │ │ │ -o4 = {x  + y , z , x y - x*y }
│ │ │ │ +       2   2    2   3       3
│ │ │ │ +o4 = {z , x  + y , x y - x*y }
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ │ - -- used 1.29621s (cpu); 0.589812s (thread); 0s (gc)
│ │ │ │ + -- used 1.22395s (cpu); 0.487875s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                     8                 7                   6 2
│ │ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                     5 3                  4 4                 3 5
│ │ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -138,22 +138,22 @@
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  The extra work done by the default algorithm to ensure an optimal hsop is
│ │ │ │  rewarded by needing to calculate a smaller collection of corresponding
│ │ │ │  secondary invariants. In fact, it has proved quicker overall to calculate the
│ │ │ │  invariant ring based on the optimal algorithm rather than the Dade algorithm.
│ │ │ │  i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ │ - -- used 0.0169186s (cpu); 0.0169215s (thread); 0s (gc)
│ │ │ │ + -- used 0.0208381s (cpu); 0.0208417s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            4    4
│ │ │ │  o6 = {1, x  + y }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ │ - -- used 2.88612s (cpu); 1.36918s (thread); 0s (gc)
│ │ │ │ + -- used 2.85822s (cpu); 1.2461s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4
│ │ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    │ │ │
    │ │ │ 
    i4 : elapsedTime invariants S4
│ │ │ - -- .795837s elapsed
│ │ │ + -- .449597s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -122,15 +122,15 @@
│ │ │
    		│ │ │ 
    i5 : elapsedTime invariants(S4, DegreeBound => 4)
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │ - -- .542414s elapsed
│ │ │ + -- .34748s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  o3 = R <- {| 0 1 0 0 |, | 0 0 0 1 |}
│ │ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  i4 : elapsedTime invariants S4
│ │ │ │ - -- .795837s elapsed
│ │ │ │ + -- .449597s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ │ │ @@ -50,15 +50,15 @@
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime invariants(S4, DegreeBound => 4)
│ │ │ │  
│ │ │ │  Warning: stopping condition not met!
│ │ │ │  Output may not generate the entire ring of invariants.
│ │ │ │  Increase value of DegreeBound.
│ │ │ │  
│ │ │ │ - -- .542414s elapsed
│ │ │ │ + -- .34748s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    │ │ │
    │ │ │ 
    i4 : elapsedTime invariants S4
│ │ │ - -- .724053s elapsed
│ │ │ + -- .454824s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -100,15 +100,15 @@
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │ 
    i5 : elapsedTime invariants(S4, Strategy => "LinearAlgebra")
│ │ │ - -- .0774921s elapsed
│ │ │ + -- .0562124s elapsed
│ │ │  
│ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,27 +26,27 @@
│ │ │ │  o3 = R <- {| 0 1 0 0 |, | 0 0 0 1 |}
│ │ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  i4 : elapsedTime invariants S4
│ │ │ │ - -- .724053s elapsed
│ │ │ │ + -- .454824s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ │ │        4
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime invariants(S4, Strategy => "LinearAlgebra")
│ │ │ │ - -- .0774921s elapsed
│ │ │ │ + -- .0562124s elapsed
│ │ │ │  
│ │ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_primary__Invariants.html
│ │ │ @@ -106,16 +106,16 @@
│ │ │  o3 : FiniteGroupAction
    │ │ │
    │ │ │ 
    i4 : primaryInvariants S3
│ │ │  
│ │ │ -                                   3    3    3
│ │ │ -o4 = {x + y + z, x*y + x*z + y*z, x  + y  + z }
│ │ │ +                  2    2    2
│ │ │ +o4 = {x + y + z, x  + y  + z , x*y*z}
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │

    Below, the invariant ring QQ[x,y,z]S3 is calculated with K being the field with 101 elements.

    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -38,16 +38,16 @@ │ │ │ │ o3 = QQ[x..z] <- {| 0 1 0 |, | 0 1 0 |} │ │ │ │ | 0 0 1 | | 1 0 0 | │ │ │ │ | 1 0 0 | | 0 0 1 | │ │ │ │ │ │ │ │ o3 : FiniteGroupAction │ │ │ │ i4 : primaryInvariants S3 │ │ │ │ │ │ │ │ - 3 3 3 │ │ │ │ -o4 = {x + y + z, x*y + x*z + y*z, x + y + z } │ │ │ │ + 2 2 2 │ │ │ │ +o4 = {x + y + z, x + y + z , x*y*z} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ Below, the invariant ring QQ[x,y,z]S3 is calculated with K being the field with │ │ │ │ 101 elements. │ │ │ │ i5 : K=GF(101) │ │ │ │ │ │ │ │ o5 = K │ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_primary__Invariants_lp..._cm__Degree__Vector_eq_gt..._rp.html │ │ │ @@ -102,19 +102,16 @@ │ │ │ o3 : FiniteGroupAction │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : primaryInvariants(S3,DegreeVector=>{3,3,4})
│ │ │  
│ │ │ -       3    3    3   2       2    2     2       2      2   3       3    3   
│ │ │ -o4 = {x  + y  + z , x y + x*y  + x z + y z + x*z  + y*z , x y + x*y  + x z +
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -      3       3      3
│ │ │ -     y z + x*z  + y*z }
│ │ │ +              3    3    3   2 2    2 2    2 2
│ │ │ +o4 = {x*y*z, x  + y  + z , x y  + x z  + y z }
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -34,19 +34,16 @@ │ │ │ │ o3 = QQ[x..z] <- {| 0 1 0 |, | 0 1 0 |} │ │ │ │ | 0 0 1 | | 1 0 0 | │ │ │ │ | 1 0 0 | | 0 0 1 | │ │ │ │ │ │ │ │ o3 : FiniteGroupAction │ │ │ │ i4 : primaryInvariants(S3,DegreeVector=>{3,3,4}) │ │ │ │ │ │ │ │ - 3 3 3 2 2 2 2 2 2 3 3 3 │ │ │ │ -o4 = {x + y + z , x y + x*y + x z + y z + x*z + y*z , x y + x*y + x z + │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - 3 3 3 │ │ │ │ - y z + x*z + y*z } │ │ │ │ + 3 3 3 2 2 2 2 2 2 │ │ │ │ +o4 = {x*y*z, x + y + z , x y + x z + y z } │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ Currently users can only use _p_r_i_m_a_r_y_I_n_v_a_r_i_a_n_t_s to calculate a hsop for the │ │ │ │ invariant ring over a finite field by using the Dade algorithm. Users should │ │ │ │ enter the finite field as a _G_a_l_o_i_s_F_i_e_l_d or a quotient field of the form _Z_Z/ │ │ │ │ p and are advised to ensure that the ground field has cardinality greater than │ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/example-output/_is__Isomorphic.out │ │ │ @@ -156,20 +156,20 @@ │ │ │ {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_2 x_1 x_0 | {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_0^2 -x_2 x_1 | │ │ │ {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_0 -x_2 x_1 -x_3 x_2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_1^2 -x_3 x_2 | │ │ │ │ │ │ 40 │ │ │ o22 : S-module, subquotient of S │ │ │ │ │ │ i23 : elapsedTime isIsomorphic(T1, T2) │ │ │ - -- 1.5376s elapsed │ │ │ + -- 1.68122s elapsed │ │ │ │ │ │ o23 = true │ │ │ │ │ │ i24 : elapsedTime isomorphism(T1, T2) │ │ │ - -- .000020339s elapsed │ │ │ + -- .000022738s elapsed │ │ │ │ │ │ o24 = {-1} | 1 -3976 -13490 13495 -2886 2577 14757 -881 7677 │ │ │ {-1} | -2527 -13566 2778 -6934 -14806 4619 -13099 6022 -10907 │ │ │ {-1} | -15420 5642 1489 1354 4591 11881 -5253 7296 -1098 │ │ │ {-1} | 7909 -12428 -2260 -8465 12113 -6893 8411 4186 -9393 │ │ │ {-1} | -9615 2934 10440 5015 8145 -5585 1360 3295 12851 │ │ │ {-1} | -4881 -7984 12700 -10391 -10009 -14538 13207 262 -6500 │ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/html/_is__Isomorphic.html │ │ │ @@ -333,23 +333,23 @@ │ │ │ 40 │ │ │ o22 : S-module, subquotient of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ - -- 1.5376s elapsed
│ │ │ + -- 1.68122s elapsed
│ │ │  
│ │ │  o23 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i24 : elapsedTime isomorphism(T1, T2)
│ │ │ - -- .000020339s elapsed
│ │ │ + -- .000022738s elapsed
│ │ │  
│ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677  
│ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098 
│ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393 
│ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851 
│ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ │ ├── html2text {}
│ │ │ │ @@ -724,19 +724,19 @@
│ │ │ │  0      0    0   0      0    0   0      0    0   0      0    0   0      0    0
│ │ │ │  0      0    0   0      0    0   0      0    0   0      0    0   -x_1^2 -x_3 x_2
│ │ │ │  |
│ │ │ │  
│ │ │ │                                  40
│ │ │ │  o22 : S-module, subquotient of S
│ │ │ │  i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ │ - -- 1.5376s elapsed
│ │ │ │ + -- 1.68122s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isomorphism(T1, T2)
│ │ │ │ - -- .000020339s elapsed
│ │ │ │ + -- .000022738s elapsed
│ │ │ │  
│ │ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677
│ │ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098
│ │ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393
│ │ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851
│ │ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ ├── ./usr/share/doc/Macaulay2/JSON/example-output/_from__J__S__O__N.out
│ │ │ @@ -37,19 +37,19 @@
│ │ │  
│ │ │  o8 = {1, 2, 3}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-34711-0/0.json
│ │ │ +o9 = /tmp/M2-47358-0/0.json
│ │ │  
│ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-34711-0/0.json
│ │ │ +o10 = /tmp/M2-47358-0/0.json
│ │ │  
│ │ │  o10 : File
│ │ │  
│ │ │  i11 : fromJSON openIn jsonFile
│ │ │  
│ │ │  o11 = {1, 2, 3}
│ │ ├── ./usr/share/doc/Macaulay2/JSON/html/_from__J__S__O__N.html
│ │ │ @@ -170,22 +170,22 @@
│ │ │            
    The input may also be a file containing JSON data.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-34711-0/0.json
    │ │ │ +o9 = /tmp/M2-47358-0/0.json │ │ │
    │ │ │ 
    i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-34711-0/0.json
│ │ │ +o10 = /tmp/M2-47358-0/0.json
│ │ │  
│ │ │  o10 : File
    │ │ │
    │ │ │ 
    i11 : fromJSON openIn jsonFile
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,18 +49,18 @@
│ │ │ │  
│ │ │ │  o8 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  The input may also be a file containing JSON data.
│ │ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-34711-0/0.json
│ │ │ │ +o9 = /tmp/M2-47358-0/0.json
│ │ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-34711-0/0.json
│ │ │ │ +o10 = /tmp/M2-47358-0/0.json
│ │ │ │  
│ │ │ │  o10 : File
│ │ │ │  i11 : fromJSON openIn jsonFile
│ │ │ │  
│ │ │ │  o11 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o11 : List
│ │ ├── ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp1.out
│ │ │ @@ -17,24 +17,24 @@
│ │ │  o3 = ideal (y0*z0*x2 + x0*z0*y2 + x0*y0*z2 + z0*x1*y1 + y0*x1*z1 + x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1 + x0*z0*y1 + x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o3 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │  
│ │ │  i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .00212636s elapsed
│ │ │ + -- .00279941s elapsed
│ │ │  
│ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │  
│ │ │  i5 : elapsedTime radical J2I
│ │ │ - -- .35661s elapsed
│ │ │ + -- .26109s elapsed
│ │ │  
│ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │  
│ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ ├── ./usr/share/doc/Macaulay2/Jets/example-output/___Storing_sp__Computations.out
│ │ │ @@ -33,15 +33,15 @@
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : I.cache.?jet
│ │ │  
│ │ │  o7 = false
│ │ │  
│ │ │  i8 : elapsedTime jets(3,I)
│ │ │ - -- .00697468s elapsed
│ │ │ + -- .00835421s elapsed
│ │ │  
│ │ │                                                    2                 2
│ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │  
│ │ │  i9 : I.cache.?jet
│ │ │ @@ -53,23 +53,23 @@
│ │ │  o10 = CacheTable{jetsMatrix => | 2x0x3-y3+2x1x2 |}
│ │ │                                 | 2x0x2-y2+x1^2  |
│ │ │                                 | 2x0x1-y1       |
│ │ │                                 | x0^2-y0        |
│ │ │                   jetsMaxOrder => 3
│ │ │  
│ │ │  i11 : elapsedTime jets(3,I)
│ │ │ - -- .00234008s elapsed
│ │ │ + -- .00283042s elapsed
│ │ │  
│ │ │                                                     2                 2
│ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │  
│ │ │  i12 : elapsedTime jets(2,I)
│ │ │ - -- .00208205s elapsed
│ │ │ + -- .00260251s elapsed
│ │ │  
│ │ │                               2                 2
│ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
│ │ │  
│ │ │  i13 : Q = R/I
│ │ │ @@ -148,15 +148,15 @@
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : f.cache.?jet
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : elapsedTime jets(3,f)
│ │ │ - -- .0103041s elapsed
│ │ │ + -- .0132318s elapsed
│ │ │  
│ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3, y3]                                                      2                    2
│ │ │  o24 = map (QQ[t0][t1][t2][t3], ----------------------------------------------------------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                                                        2                 2
│ │ │                                 (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                                      QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ @@ -173,15 +173,15 @@
│ │ │  o26 = CacheTable{jetsMatrix => | t3 2t0t3+2t1t2 |}
│ │ │                                 | t2 2t0t2+t1^2  |
│ │ │                                 | t1 2t0t1       |
│ │ │                                 | t0 t0^2        |
│ │ │                   jetsMaxOrder => 3
│ │ │  
│ │ │  i27 : elapsedTime jets(2,f)
│ │ │ - -- .000643914s elapsed
│ │ │ + -- .000784127s elapsed
│ │ │  
│ │ │                                     QQ[x0, y0][x1, y1][x2, y2]                          2                    2
│ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                              2                 2
│ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                           QQ[x0, y0][x1, y1][x2, y2]
│ │ ├── ./usr/share/doc/Macaulay2/Jets/html/___Example_sp1.html
│ │ │ @@ -92,27 +92,27 @@
│ │ │          
    
│ │ │            
    However, by [GS06, Theorem 3.1], the radical is always a (squarefree) monomial ideal. In fact, the proof of [GS06, Theorem 3.2] shows that the radical is generated by the individual terms in the generators of the ideal of jets. This observation provides an alternative algorithm for computing radicals of jets of monomial ideals, which can be faster than the default radical computation in Macaulay2.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,23 +27,23 @@
│ │ │ │  However, by [GS06, Theorem 3.1], the radical is always a (squarefree) monomial
│ │ │ │  ideal. In fact, the proof of [GS06, Theorem 3.2] shows that the radical is
│ │ │ │  generated by the individual terms in the generators of the ideal of jets. This
│ │ │ │  observation provides an alternative algorithm for computing radicals of jets of
│ │ │ │  monomial ideals, which can be faster than the default radical computation in
│ │ │ │  Macaulay2.
│ │ │ │  i4 : elapsedTime jetsRadical(2,I)
│ │ │ │ - -- .00212636s elapsed
│ │ │ │ + -- .00279941s elapsed
│ │ │ │  
│ │ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │ │  
│ │ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │ │  i5 : elapsedTime radical J2I
│ │ │ │ - -- .35661s elapsed
│ │ │ │ + -- .26109s elapsed
│ │ │ │  
│ │ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │ │  
│ │ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │ │  For a monomial hypersurface, [GS06, Theorem 3.2] describes the minimal primes
│ │ ├── ./usr/share/doc/Macaulay2/Jets/html/___Storing_sp__Computations.html
│ │ │ @@ -122,15 +122,15 @@
│ │ │  
│ │ │  o7 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -151,26 +151,26 @@
│ │ │                                 | x0^2-y0        |
│ │ │                   jetsMaxOrder => 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -295,15 +295,15 @@
│ │ │  
│ │ │  o23 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .00212636s elapsed
│ │ │ + -- .00279941s elapsed
│ │ │  
│ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime radical J2I
│ │ │ - -- .35661s elapsed
│ │ │ + -- .26109s elapsed
│ │ │  
│ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │  
│ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime jets(3,I)
│ │ │ - -- .00697468s elapsed
│ │ │ + -- .00835421s elapsed
│ │ │  
│ │ │                                                    2                 2
│ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
    
│ │ │              
    
│ │ │                
    i11 : elapsedTime jets(3,I)
│ │ │ - -- .00234008s elapsed
│ │ │ + -- .00283042s elapsed
│ │ │  
│ │ │                                                     2                 2
│ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
    
│ │ │              
    
│ │ │                
    i12 : elapsedTime jets(2,I)
│ │ │ - -- .00208205s elapsed
│ │ │ + -- .00260251s elapsed
│ │ │  
│ │ │                               2                 2
│ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime jets(3,f)
│ │ │ - -- .0103041s elapsed
│ │ │ + -- .0132318s elapsed
│ │ │  
│ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3, y3]                                                      2                    2
│ │ │  o24 = map (QQ[t0][t1][t2][t3], ----------------------------------------------------------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                                                        2                 2
│ │ │                                 (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                                      QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ @@ -329,15 +329,15 @@
│ │ │                                 | t0 t0^2        |
│ │ │                   jetsMaxOrder => 3
    
│ │ │              
    
│ │ │                
    i27 : elapsedTime jets(2,f)
│ │ │ - -- .000643914s elapsed
│ │ │ + -- .000784127s elapsed
│ │ │  
│ │ │                                     QQ[x0, y0][x1, y1][x2, y2]                          2                    2
│ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                              2                 2
│ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                           QQ[x0, y0][x1, y1][x2, y2]
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  o6 = ideal(x  - y)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : I.cache.?jet
│ │ │ │  
│ │ │ │  o7 = false
│ │ │ │  i8 : elapsedTime jets(3,I)
│ │ │ │ - -- .00697468s elapsed
│ │ │ │ + -- .00835421s elapsed
│ │ │ │  
│ │ │ │                                                    2                 2
│ │ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ │  i9 : I.cache.?jet
│ │ │ │  
│ │ │ │ @@ -58,22 +58,22 @@
│ │ │ │  
│ │ │ │  o10 = CacheTable{jetsMatrix => | 2x0x3-y3+2x1x2 |}
│ │ │ │                                 | 2x0x2-y2+x1^2  |
│ │ │ │                                 | 2x0x1-y1       |
│ │ │ │                                 | x0^2-y0        |
│ │ │ │                   jetsMaxOrder => 3
│ │ │ │  i11 : elapsedTime jets(3,I)
│ │ │ │ - -- .00234008s elapsed
│ │ │ │ + -- .00283042s elapsed
│ │ │ │  
│ │ │ │                                                     2                 2
│ │ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ │  i12 : elapsedTime jets(2,I)
│ │ │ │ - -- .00208205s elapsed
│ │ │ │ + -- .00260251s elapsed
│ │ │ │  
│ │ │ │                               2                 2
│ │ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
│ │ │ │  For quotient rings, data is stored under *.jet. Each jets order gives rise to a
│ │ │ │  different quotient that is stored separately under *.jet.jetsRing (order zero
│ │ │ │ @@ -153,15 +153,15 @@
│ │ │ │  i22 : isWellDefined f
│ │ │ │  
│ │ │ │  o22 = true
│ │ │ │  i23 : f.cache.?jet
│ │ │ │  
│ │ │ │  o23 = false
│ │ │ │  i24 : elapsedTime jets(3,f)
│ │ │ │ - -- .0103041s elapsed
│ │ │ │ + -- .0132318s elapsed
│ │ │ │  
│ │ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3,
│ │ │ │  y3]                                                      2                    2
│ │ │ │  o24 = map (QQ[t0][t1][t2][t3], ------------------------------------------------
│ │ │ │  ----------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │ │                                                                        2
│ │ │ │  2
│ │ │ │ @@ -183,15 +183,15 @@
│ │ │ │  
│ │ │ │  o26 = CacheTable{jetsMatrix => | t3 2t0t3+2t1t2 |}
│ │ │ │                                 | t2 2t0t2+t1^2  |
│ │ │ │                                 | t1 2t0t1       |
│ │ │ │                                 | t0 t0^2        |
│ │ │ │                   jetsMaxOrder => 3
│ │ │ │  i27 : elapsedTime jets(2,f)
│ │ │ │ - -- .000643914s elapsed
│ │ │ │ + -- .000784127s elapsed
│ │ │ │  
│ │ │ │                                     QQ[x0, y0][x1, y1][x2, y2]
│ │ │ │  2                    2
│ │ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2,
│ │ │ │  2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │ │                                              2                 2
│ │ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │        32003  0   5   0   5         32003  0   5   0   5          32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5          32003  0   5   0   5
│ │ │                                                                                                                                                                                                                                                                                           
│ │ │       0                            1                             2                              3                              4                              5                              6                              7                              8                             9
│ │ │  
│ │ │  o3 : Complex
│ │ │  
│ │ │  i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .028913s elapsed
│ │ │ + -- .0298936s elapsed
│ │ │  
│ │ │  o5 = 350
│ │ │  
│ │ │  i6 : betti F_a, betti F
│ │ │  
│ │ │                 0         0  1   2   3   4   5   6   7  8 9
│ │ │  o6 = (total: 833, total: 1 36 187 491 793 833 573 250 63 7)
│ │ │ @@ -46,19 +46,19 @@
│ │ │  o7 : Expression of class Product
│ │ │  
│ │ │  i8 : L3 = select(L,c->c%3==0); #L3
│ │ │  
│ │ │  o9 = 14
│ │ │  
│ │ │  i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .00232397s elapsed
│ │ │ - -- .0148621s elapsed
│ │ │ - -- .038596s elapsed
│ │ │ - -- .0137092s elapsed
│ │ │ - -- .00357395s elapsed
│ │ │ + -- .00254306s elapsed
│ │ │ + -- .0072685s elapsed
│ │ │ + -- .038707s elapsed
│ │ │ + -- .014147s elapsed
│ │ │ + -- .00499268s elapsed
│ │ │  
│ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out
│ │ │ @@ -3,20 +3,20 @@
│ │ │  i1 : a=5,b=5
│ │ │  
│ │ │  o1 = (5, 5)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ - -- .0105404s elapsed
│ │ │ - -- .00619773s elapsed
│ │ │ - -- .023188s elapsed
│ │ │ - -- .0141825s elapsed
│ │ │ - -- .00380241s elapsed
│ │ │ - -- .424134s elapsed
│ │ │ + -- .00329851s elapsed
│ │ │ + -- .00761337s elapsed
│ │ │ + -- .0287925s elapsed
│ │ │ + -- .0198581s elapsed
│ │ │ + -- .00433678s elapsed
│ │ │ + -- .451029s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -26,15 +26,15 @@
│ │ │  i3 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │  
│ │ │                ZZ
│ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
│ │ │  
│ │ │  i4 : elapsedTime T'=minimalBetti J
│ │ │ - -- .143596s elapsed
│ │ │ + -- .219515s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -48,22 +48,22 @@
│ │ │           1: . . . . . . . . . .
│ │ │           2: . . . . . . . . . .
│ │ │           3: . . . . . . . . . .
│ │ │  
│ │ │  o5 : BettiTally
│ │ │  
│ │ │  i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00468647s elapsed
│ │ │ - -- .0638308s elapsed
│ │ │ - -- .18071s elapsed
│ │ │ - -- 1.21616s elapsed
│ │ │ - -- .436472s elapsed
│ │ │ - -- .175244s elapsed
│ │ │ - -- .00697834s elapsed
│ │ │ - -- 6.19532s elapsed
│ │ │ + -- .0048627s elapsed
│ │ │ + -- .0265366s elapsed
│ │ │ + -- .121958s elapsed
│ │ │ + -- 1.07848s elapsed
│ │ │ + -- .566785s elapsed
│ │ │ + -- .0456508s elapsed
│ │ │ + -- .00840914s elapsed
│ │ │ + -- 6.40393s elapsed
│ │ │  
│ │ │  i7 : carpetBettiTable(h,7)
│ │ │  
│ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │  o7 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
│ │ │           0: 1  .   .   .    .    .    .    .   .   .  .  .
│ │ │           1: . 55 320 891 1408 1155    .    .   .   .  .  .
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out
│ │ │ @@ -3,19 +3,19 @@
│ │ │  i1 : a=5,b=5
│ │ │  
│ │ │  o1 = (5, 5)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : h=carpetBettiTables(a,b)
│ │ │ - -- .00242979s elapsed
│ │ │ - -- .0150993s elapsed
│ │ │ - -- .0236144s elapsed
│ │ │ - -- .0144779s elapsed
│ │ │ - -- .00380124s elapsed
│ │ │ + -- .00280904s elapsed
│ │ │ + -- .0128015s elapsed
│ │ │ + -- .0479294s elapsed
│ │ │ + -- .0205234s elapsed
│ │ │ + -- .00439878s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -48,15 +48,15 @@
│ │ │  i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │  
│ │ │                ZZ
│ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
│ │ │  
│ │ │  i5 : elapsedTime T'=minimalBetti J
│ │ │ - -- .140448s elapsed
│ │ │ + -- .212728s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -70,22 +70,22 @@
│ │ │           1: . . . . . . . . . .
│ │ │           2: . . . . . . . . . .
│ │ │           3: . . . . . . . . . .
│ │ │  
│ │ │  o6 : BettiTally
│ │ │  
│ │ │  i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00456157s elapsed
│ │ │ - -- .0235407s elapsed
│ │ │ - -- .116726s elapsed
│ │ │ - -- 1.19264s elapsed
│ │ │ - -- .585079s elapsed
│ │ │ - -- .0500128s elapsed
│ │ │ - -- .00661932s elapsed
│ │ │ - -- 5.86451s elapsed
│ │ │ + -- .00511795s elapsed
│ │ │ + -- .0194595s elapsed
│ │ │ + -- .130859s elapsed
│ │ │ + -- 1.01929s elapsed
│ │ │ + -- .450226s elapsed
│ │ │ + -- .0434708s elapsed
│ │ │ + -- .00833544s elapsed
│ │ │ + -- 6.15902s elapsed
│ │ │  
│ │ │  i8 : keys h
│ │ │  
│ │ │  o8 = {0, 2, 3, 5}
│ │ │  
│ │ │  o8 : List
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out
│ │ │ @@ -3,82 +3,82 @@
│ │ │  i1 : a=4,b=4
│ │ │  
│ │ │  o1 = (4, 4)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : d=carpetDet(a,b)
│ │ │ - -- .0113392s elapsed
│ │ │ - -- .0234744s elapsed
│ │ │ + -- .00831894s elapsed
│ │ │ + -- .0140857s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000288178s elapsed
│ │ │ + -- .000297302s elapsed
│ │ │  1
│ │ │ - -- .000170288s elapsed
│ │ │ + -- .00024295s elapsed
│ │ │  1
│ │ │ - -- .000165309s elapsed
│ │ │ + -- .000197737s elapsed
│ │ │  1
│ │ │ - -- .000132348s elapsed
│ │ │ + -- .000192069s elapsed
│ │ │  1
│ │ │ - -- .000151593s elapsed
│ │ │ + -- .000204196s elapsed
│ │ │  2
│ │ │ - -- .000145241s elapsed
│ │ │ + -- .000220844s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .0467632s elapsed
│ │ │ + -- .0261779s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000172282s elapsed
│ │ │ + -- .000372196s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000133429s elapsed
│ │ │ + -- .000205455s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000129542s elapsed
│ │ │ + -- .000290631s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000128781s elapsed
│ │ │ + -- .000184796s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000125975s elapsed
│ │ │ + -- .000175119s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000136625s elapsed
│ │ │ + -- .00016957s elapsed
│ │ │  2
│ │ │ - -- .000138749s elapsed
│ │ │ + -- .000175123s elapsed
│ │ │  2
│ │ │ - -- .000142055s elapsed
│ │ │ + -- .000272867s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .00012871s elapsed
│ │ │ + -- .000275296s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000143899s elapsed
│ │ │ + -- .000194141s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000140524s elapsed
│ │ │ + -- .000202042s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000140683s elapsed
│ │ │ + -- .000297606s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .00018196s elapsed
│ │ │ + -- .000342989s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000143639s elapsed
│ │ │ + -- .00029877s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .0041553s elapsed
│ │ │ + -- .00028431s elapsed
│ │ │  2
│ │ │ - -- .000136294s elapsed
│ │ │ + -- .000269412s elapsed
│ │ │  1
│ │ │ - -- .000135064s elapsed
│ │ │ + -- .000181092s elapsed
│ │ │  1
│ │ │ - -- .000140032s elapsed
│ │ │ + -- .000178374s elapsed
│ │ │  1
│ │ │ - -- .00446562s elapsed
│ │ │ + -- .000165729s elapsed
│ │ │  1
│ │ │  
│ │ │  o2 = 3131031158784
│ │ │  
│ │ │  i3 : factor d
│ │ │  
│ │ │        32 6
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out
│ │ │ @@ -3,17 +3,17 @@
│ │ │  i1 : (a,b)=computeBound(6,4,3)
│ │ │  
│ │ │  o1 = (9, 7)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : computeBound 3
│ │ │ - -- .200616s elapsed
│ │ │ - -- .224191s elapsed
│ │ │ - -- .217225s elapsed
│ │ │ - -- .21372s elapsed
│ │ │ - -- .256605s elapsed
│ │ │ - -- .319156s elapsed
│ │ │ + -- .165673s elapsed
│ │ │ + -- .170565s elapsed
│ │ │ + -- .175863s elapsed
│ │ │ + -- .210087s elapsed
│ │ │ + -- .216758s elapsed
│ │ │ + -- .190721s elapsed
│ │ │  
│ │ │  o2 = 6
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out
│ │ │ @@ -9,19 +9,19 @@
│ │ │  i2 : e=(-1,5)
│ │ │  
│ │ │  o2 = (-1, 5)
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00895853s elapsed
│ │ │ - -- .0106852s elapsed
│ │ │ - -- .0250247s elapsed
│ │ │ - -- .00988981s elapsed
│ │ │ - -- .00380772s elapsed
│ │ │ + -- .0032664s elapsed
│ │ │ + -- .00781772s elapsed
│ │ │ + -- .0292052s elapsed
│ │ │ + -- .0138733s elapsed
│ │ │ + -- .00461877s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -49,15 +49,15 @@
│ │ │  i4 : keys h
│ │ │  
│ │ │  o4 = {0, 2, 3, 5}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ - -- .147586s elapsed
│ │ │ + -- .235423s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -77,19 +77,19 @@
│ │ │  i7 : e=(-1,5^2)
│ │ │  
│ │ │  o7 = (-1, 25)
│ │ │  
│ │ │  o7 : Sequence
│ │ │  
│ │ │  i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00258108s elapsed
│ │ │ - -- .00686158s elapsed
│ │ │ - -- .0244236s elapsed
│ │ │ - -- .00913852s elapsed
│ │ │ - -- .00344125s elapsed
│ │ │ + -- .00300207s elapsed
│ │ │ + -- .00745699s elapsed
│ │ │ + -- .0279157s elapsed
│ │ │ + -- .0104353s elapsed
│ │ │ + -- .00427333s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out
│ │ │ @@ -1,172 +1,172 @@
│ │ │  -- -*- M2-comint -*- hash: 1729182891690704738
│ │ │  
│ │ │  i1 : a=4
│ │ │  
│ │ │  o1 = 4
│ │ │  
│ │ │  i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0213231s elapsed
│ │ │ + -- .0305235s elapsed
│ │ │  (number of blocks= , 18)
│ │ │  (size of the matrices, Tally{1 => 4})
│ │ │                               2 => 6
│ │ │                               3 => 2
│ │ │                               4 => 6
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      7: 1 1
│ │ │ - -- .000044764s elapsed
│ │ │ + -- .000046948s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000082483s elapsed
│ │ │ + -- .000108106s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000068808s elapsed
│ │ │ + -- .000088203s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000076702s elapsed
│ │ │ + -- .000122223s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .00009138s elapsed
│ │ │ + -- .000115303s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000087173s elapsed
│ │ │ + -- .000102445s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000024476s elapsed
│ │ │ + -- .000037428s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .00006923s elapsed
│ │ │ + -- .000079821s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000081171s elapsed
│ │ │ + -- .000105172s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000109105s elapsed
│ │ │ + -- .000100179s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000083966s elapsed
│ │ │ + -- .000135631s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .00008596s elapsed
│ │ │ + -- .000112036s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .00007515s elapsed
│ │ │ + -- .000108502s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000068008s elapsed
│ │ │ + -- .000082539s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .000067106s elapsed
│ │ │ + -- .000095404s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000024396s elapsed
│ │ │ + -- .000034936s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000071524s elapsed
│ │ │ + -- .000090862s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000025659s elapsed
│ │ │ + -- .000034004s elapsed
│ │ │  (e )
│ │ │    1
│ │ │  
│ │ │         6      32    32
│ │ │  o2 = (3 , (e )  (e )  )
│ │ │              1     2
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html
│ │ │ @@ -107,15 +107,15 @@
│ │ │  
│ │ │  o3 : Complex
    
│ │ │              
    
│ │ │                
    i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .028913s elapsed
│ │ │ + -- .0298936s elapsed
│ │ │  
│ │ │  o5 = 350
    
│ │ │              
    
│ │ │                
    i6 : betti F_a, betti F
│ │ │ @@ -146,19 +146,19 @@
│ │ │  
│ │ │  o9 = 14
    
│ │ │              
    
│ │ │                
    i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .00232397s elapsed
│ │ │ - -- .0148621s elapsed
│ │ │ - -- .038596s elapsed
│ │ │ - -- .0137092s elapsed
│ │ │ - -- .00357395s elapsed
│ │ │ + -- .00254306s elapsed
│ │ │ + -- .0072685s elapsed
│ │ │ + -- .038707s elapsed
│ │ │ + -- .014147s elapsed
│ │ │ + -- .00499268s elapsed
│ │ │  
│ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │       0                            1                             2
│ │ │ │  3                              4                              5
│ │ │ │  6                              7                              8
│ │ │ │  9
│ │ │ │  
│ │ │ │  o3 : Complex
│ │ │ │  i4 : L = analyzeStrand(F,a); #L
│ │ │ │ - -- .028913s elapsed
│ │ │ │ + -- .0298936s elapsed
│ │ │ │  
│ │ │ │  o5 = 350
│ │ │ │  i6 : betti F_a, betti F
│ │ │ │  
│ │ │ │                 0         0  1   2   3   4   5   6   7  8 9
│ │ │ │  o6 = (total: 833, total: 1 36 187 491 793 833 573 250 63 7)
│ │ │ │            6: 350      0: 1  .   .   .   .   .   .   .  . .
│ │ │ │ @@ -72,19 +72,19 @@
│ │ │ │  o7 = 2   3
│ │ │ │  
│ │ │ │  o7 : Expression of class Product
│ │ │ │  i8 : L3 = select(L,c->c%3==0); #L3
│ │ │ │  
│ │ │ │  o9 = 14
│ │ │ │  i10 : carpetBettiTable(a,b,3)
│ │ │ │ - -- .00232397s elapsed
│ │ │ │ - -- .0148621s elapsed
│ │ │ │ - -- .038596s elapsed
│ │ │ │ - -- .0137092s elapsed
│ │ │ │ - -- .00357395s elapsed
│ │ │ │ + -- .00254306s elapsed
│ │ │ │ + -- .0072685s elapsed
│ │ │ │ + -- .038707s elapsed
│ │ │ │ + -- .014147s elapsed
│ │ │ │ + -- .00499268s elapsed
│ │ │ │  
│ │ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html
│ │ │ @@ -88,20 +88,20 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ - -- .0105404s elapsed
│ │ │ - -- .00619773s elapsed
│ │ │ - -- .023188s elapsed
│ │ │ - -- .0141825s elapsed
│ │ │ - -- .00380241s elapsed
│ │ │ - -- .424134s elapsed
│ │ │ + -- .00329851s elapsed
│ │ │ + -- .00761337s elapsed
│ │ │ + -- .0287925s elapsed
│ │ │ + -- .0198581s elapsed
│ │ │ + -- .00433678s elapsed
│ │ │ + -- .451029s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -117,15 +117,15 @@
│ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime T'=minimalBetti J
│ │ │ - -- .143596s elapsed
│ │ │ + -- .219515s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -145,22 +145,22 @@
│ │ │  
│ │ │  o5 : BettiTally
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00468647s elapsed
│ │ │ - -- .0638308s elapsed
│ │ │ - -- .18071s elapsed
│ │ │ - -- 1.21616s elapsed
│ │ │ - -- .436472s elapsed
│ │ │ - -- .175244s elapsed
│ │ │ - -- .00697834s elapsed
│ │ │ - -- 6.19532s elapsed
    
│ │ │ + -- .0048627s elapsed
│ │ │ + -- .0265366s elapsed
│ │ │ + -- .121958s elapsed
│ │ │ + -- 1.07848s elapsed
│ │ │ + -- .566785s elapsed
│ │ │ + -- .0456508s elapsed
│ │ │ + -- .00840914s elapsed
│ │ │ + -- 6.40393s elapsed
│ │ │              
    
│ │ │                
    i7 : carpetBettiTable(h,7)
│ │ │  
│ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,20 +25,20 @@
│ │ │ │  resulting data allow us to compute the Betti tables for arbitrary primes.
│ │ │ │  i1 : a=5,b=5
│ │ │ │  
│ │ │ │  o1 = (5, 5)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ │ - -- .0105404s elapsed
│ │ │ │ - -- .00619773s elapsed
│ │ │ │ - -- .023188s elapsed
│ │ │ │ - -- .0141825s elapsed
│ │ │ │ - -- .00380241s elapsed
│ │ │ │ - -- .424134s elapsed
│ │ │ │ + -- .00329851s elapsed
│ │ │ │ + -- .00761337s elapsed
│ │ │ │ + -- .0287925s elapsed
│ │ │ │ + -- .0198581s elapsed
│ │ │ │ + -- .00433678s elapsed
│ │ │ │ + -- .451029s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │  o2 : BettiTally
│ │ │ │  i3 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │ │                 3  0   5   0   5
│ │ │ │  i4 : elapsedTime T'=minimalBetti J
│ │ │ │ - -- .143596s elapsed
│ │ │ │ + -- .219515s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -66,22 +66,22 @@
│ │ │ │  o5 = total: . . . . . . . . . .
│ │ │ │           1: . . . . . . . . . .
│ │ │ │           2: . . . . . . . . . .
│ │ │ │           3: . . . . . . . . . .
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ │ │  i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ │ - -- .00468647s elapsed
│ │ │ │ - -- .0638308s elapsed
│ │ │ │ - -- .18071s elapsed
│ │ │ │ - -- 1.21616s elapsed
│ │ │ │ - -- .436472s elapsed
│ │ │ │ - -- .175244s elapsed
│ │ │ │ - -- .00697834s elapsed
│ │ │ │ - -- 6.19532s elapsed
│ │ │ │ + -- .0048627s elapsed
│ │ │ │ + -- .0265366s elapsed
│ │ │ │ + -- .121958s elapsed
│ │ │ │ + -- 1.07848s elapsed
│ │ │ │ + -- .566785s elapsed
│ │ │ │ + -- .0456508s elapsed
│ │ │ │ + -- .00840914s elapsed
│ │ │ │ + -- 6.40393s elapsed
│ │ │ │  i7 : carpetBettiTable(h,7)
│ │ │ │  
│ │ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │ │  o7 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
│ │ │ │           0: 1  .   .   .    .    .    .    .   .   .  .  .
│ │ │ │           1: . 55 320 891 1408 1155    .    .   .   .  .  .
│ │ │ │           2: .  .   .   .    .    . 1155 1408 891 320 55  .
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html
│ │ │ @@ -85,19 +85,19 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : h=carpetBettiTables(a,b)
│ │ │ - -- .00242979s elapsed
│ │ │ - -- .0150993s elapsed
│ │ │ - -- .0236144s elapsed
│ │ │ - -- .0144779s elapsed
│ │ │ - -- .00380124s elapsed
│ │ │ + -- .00280904s elapsed
│ │ │ + -- .0128015s elapsed
│ │ │ + -- .0479294s elapsed
│ │ │ + -- .0205234s elapsed
│ │ │ + -- .00439878s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -139,15 +139,15 @@
│ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime T'=minimalBetti J
│ │ │ - -- .140448s elapsed
│ │ │ + -- .212728s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -167,22 +167,22 @@
│ │ │  
│ │ │  o6 : BettiTally
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00456157s elapsed
│ │ │ - -- .0235407s elapsed
│ │ │ - -- .116726s elapsed
│ │ │ - -- 1.19264s elapsed
│ │ │ - -- .585079s elapsed
│ │ │ - -- .0500128s elapsed
│ │ │ - -- .00661932s elapsed
│ │ │ - -- 5.86451s elapsed
    
│ │ │ + -- .00511795s elapsed
│ │ │ + -- .0194595s elapsed
│ │ │ + -- .130859s elapsed
│ │ │ + -- 1.01929s elapsed
│ │ │ + -- .450226s elapsed
│ │ │ + -- .0434708s elapsed
│ │ │ + -- .00833544s elapsed
│ │ │ + -- 6.15902s elapsed
│ │ │              
    
│ │ │                
    i8 : keys h
│ │ │  
│ │ │  o8 = {0, 2, 3, 5}
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,19 +21,19 @@
│ │ │ │  resulting data allow us to compute the Betti tables for arbitrary primes.
│ │ │ │  i1 : a=5,b=5
│ │ │ │  
│ │ │ │  o1 = (5, 5)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : h=carpetBettiTables(a,b)
│ │ │ │ - -- .00242979s elapsed
│ │ │ │ - -- .0150993s elapsed
│ │ │ │ - -- .0236144s elapsed
│ │ │ │ - -- .0144779s elapsed
│ │ │ │ - -- .00380124s elapsed
│ │ │ │ + -- .00280904s elapsed
│ │ │ │ + -- .0128015s elapsed
│ │ │ │ + -- .0479294s elapsed
│ │ │ │ + -- .0205234s elapsed
│ │ │ │ + -- .00439878s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  o3 : BettiTally
│ │ │ │  i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │ │                 3  0   5   0   5
│ │ │ │  i5 : elapsedTime T'=minimalBetti J
│ │ │ │ - -- .140448s elapsed
│ │ │ │ + -- .212728s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -83,22 +83,22 @@
│ │ │ │  o6 = total: . . . . . . . . . .
│ │ │ │           1: . . . . . . . . . .
│ │ │ │           2: . . . . . . . . . .
│ │ │ │           3: . . . . . . . . . .
│ │ │ │  
│ │ │ │  o6 : BettiTally
│ │ │ │  i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ │ - -- .00456157s elapsed
│ │ │ │ - -- .0235407s elapsed
│ │ │ │ - -- .116726s elapsed
│ │ │ │ - -- 1.19264s elapsed
│ │ │ │ - -- .585079s elapsed
│ │ │ │ - -- .0500128s elapsed
│ │ │ │ - -- .00661932s elapsed
│ │ │ │ - -- 5.86451s elapsed
│ │ │ │ + -- .00511795s elapsed
│ │ │ │ + -- .0194595s elapsed
│ │ │ │ + -- .130859s elapsed
│ │ │ │ + -- 1.01929s elapsed
│ │ │ │ + -- .450226s elapsed
│ │ │ │ + -- .0434708s elapsed
│ │ │ │ + -- .00833544s elapsed
│ │ │ │ + -- 6.15902s elapsed
│ │ │ │  i8 : keys h
│ │ │ │  
│ │ │ │  o8 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  i9 : carpetBettiTable(h,7)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html
│ │ │ @@ -85,82 +85,82 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : d=carpetDet(a,b)
│ │ │ - -- .0113392s elapsed
│ │ │ - -- .0234744s elapsed
│ │ │ + -- .00831894s elapsed
│ │ │ + -- .0140857s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000288178s elapsed
│ │ │ + -- .000297302s elapsed
│ │ │  1
│ │ │ - -- .000170288s elapsed
│ │ │ + -- .00024295s elapsed
│ │ │  1
│ │ │ - -- .000165309s elapsed
│ │ │ + -- .000197737s elapsed
│ │ │  1
│ │ │ - -- .000132348s elapsed
│ │ │ + -- .000192069s elapsed
│ │ │  1
│ │ │ - -- .000151593s elapsed
│ │ │ + -- .000204196s elapsed
│ │ │  2
│ │ │ - -- .000145241s elapsed
│ │ │ + -- .000220844s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .0467632s elapsed
│ │ │ + -- .0261779s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000172282s elapsed
│ │ │ + -- .000372196s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000133429s elapsed
│ │ │ + -- .000205455s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000129542s elapsed
│ │ │ + -- .000290631s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000128781s elapsed
│ │ │ + -- .000184796s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000125975s elapsed
│ │ │ + -- .000175119s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000136625s elapsed
│ │ │ + -- .00016957s elapsed
│ │ │  2
│ │ │ - -- .000138749s elapsed
│ │ │ + -- .000175123s elapsed
│ │ │  2
│ │ │ - -- .000142055s elapsed
│ │ │ + -- .000272867s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .00012871s elapsed
│ │ │ + -- .000275296s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000143899s elapsed
│ │ │ + -- .000194141s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000140524s elapsed
│ │ │ + -- .000202042s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000140683s elapsed
│ │ │ + -- .000297606s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .00018196s elapsed
│ │ │ + -- .000342989s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000143639s elapsed
│ │ │ + -- .00029877s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .0041553s elapsed
│ │ │ + -- .00028431s elapsed
│ │ │  2
│ │ │ - -- .000136294s elapsed
│ │ │ + -- .000269412s elapsed
│ │ │  1
│ │ │ - -- .000135064s elapsed
│ │ │ + -- .000181092s elapsed
│ │ │  1
│ │ │ - -- .000140032s elapsed
│ │ │ + -- .000178374s elapsed
│ │ │  1
│ │ │ - -- .00446562s elapsed
│ │ │ + -- .000165729s elapsed
│ │ │  1
│ │ │  
│ │ │  o2 = 3131031158784
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,82 +19,82 @@
│ │ │ │  determinants and return their product.
│ │ │ │  i1 : a=4,b=4
│ │ │ │  
│ │ │ │  o1 = (4, 4)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : d=carpetDet(a,b)
│ │ │ │ - -- .0113392s elapsed
│ │ │ │ - -- .0234744s elapsed
│ │ │ │ + -- .00831894s elapsed
│ │ │ │ + -- .0140857s elapsed
│ │ │ │  (number Of blocks, 26)
│ │ │ │ - -- .000288178s elapsed
│ │ │ │ + -- .000297302s elapsed
│ │ │ │  1
│ │ │ │ - -- .000170288s elapsed
│ │ │ │ + -- .00024295s elapsed
│ │ │ │  1
│ │ │ │ - -- .000165309s elapsed
│ │ │ │ + -- .000197737s elapsed
│ │ │ │  1
│ │ │ │ - -- .000132348s elapsed
│ │ │ │ + -- .000192069s elapsed
│ │ │ │  1
│ │ │ │ - -- .000151593s elapsed
│ │ │ │ + -- .000204196s elapsed
│ │ │ │  2
│ │ │ │ - -- .000145241s elapsed
│ │ │ │ + -- .000220844s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .0467632s elapsed
│ │ │ │ + -- .0261779s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000172282s elapsed
│ │ │ │ + -- .000372196s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000133429s elapsed
│ │ │ │ + -- .000205455s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000129542s elapsed
│ │ │ │ + -- .000290631s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000128781s elapsed
│ │ │ │ + -- .000184796s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000125975s elapsed
│ │ │ │ + -- .000175119s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000136625s elapsed
│ │ │ │ + -- .00016957s elapsed
│ │ │ │  2
│ │ │ │ - -- .000138749s elapsed
│ │ │ │ + -- .000175123s elapsed
│ │ │ │  2
│ │ │ │ - -- .000142055s elapsed
│ │ │ │ + -- .000272867s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .00012871s elapsed
│ │ │ │ + -- .000275296s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000143899s elapsed
│ │ │ │ + -- .000194141s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000140524s elapsed
│ │ │ │ + -- .000202042s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000140683s elapsed
│ │ │ │ + -- .000297606s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .00018196s elapsed
│ │ │ │ + -- .000342989s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000143639s elapsed
│ │ │ │ + -- .00029877s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .0041553s elapsed
│ │ │ │ + -- .00028431s elapsed
│ │ │ │  2
│ │ │ │ - -- .000136294s elapsed
│ │ │ │ + -- .000269412s elapsed
│ │ │ │  1
│ │ │ │ - -- .000135064s elapsed
│ │ │ │ + -- .000181092s elapsed
│ │ │ │  1
│ │ │ │ - -- .000140032s elapsed
│ │ │ │ + -- .000178374s elapsed
│ │ │ │  1
│ │ │ │ - -- .00446562s elapsed
│ │ │ │ + -- .000165729s elapsed
│ │ │ │  1
│ │ │ │  
│ │ │ │  o2 = 3131031158784
│ │ │ │  i3 : factor d
│ │ │ │  
│ │ │ │        32 6
│ │ │ │  o3 = 2  3
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html
│ │ │ @@ -90,20 +90,20 @@
│ │ │  
│ │ │  o1 : Sequence
│ │ │              
    
│ │ │                
    i2 : computeBound 3
│ │ │ - -- .200616s elapsed
│ │ │ - -- .224191s elapsed
│ │ │ - -- .217225s elapsed
│ │ │ - -- .21372s elapsed
│ │ │ - -- .256605s elapsed
│ │ │ - -- .319156s elapsed
│ │ │ + -- .165673s elapsed
│ │ │ + -- .170565s elapsed
│ │ │ + -- .175863s elapsed
│ │ │ + -- .210087s elapsed
│ │ │ + -- .216758s elapsed
│ │ │ + -- .190721s elapsed
│ │ │  
│ │ │  o2 = 6
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,20 +25,20 @@
│ │ │ │  classes mod k. We conjecture that c=k^2-k.
│ │ │ │  i1 : (a,b)=computeBound(6,4,3)
│ │ │ │  
│ │ │ │  o1 = (9, 7)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : computeBound 3
│ │ │ │ - -- .200616s elapsed
│ │ │ │ - -- .224191s elapsed
│ │ │ │ - -- .217225s elapsed
│ │ │ │ - -- .21372s elapsed
│ │ │ │ - -- .256605s elapsed
│ │ │ │ - -- .319156s elapsed
│ │ │ │ + -- .165673s elapsed
│ │ │ │ + -- .170565s elapsed
│ │ │ │ + -- .175863s elapsed
│ │ │ │ + -- .210087s elapsed
│ │ │ │ + -- .216758s elapsed
│ │ │ │ + -- .190721s elapsed
│ │ │ │  
│ │ │ │  o2 = 6
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_l_a_t_i_v_e_E_q_u_a_t_i_o_n_s -- compute the relative quadrics
│ │ │ │  ********** WWaayyss ttoo uussee ccoommppuutteeBBoouunndd:: **********
│ │ │ │      * computeBound(ZZ)
│ │ │ │      * computeBound(ZZ,ZZ,ZZ)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html
│ │ │ @@ -95,19 +95,19 @@
│ │ │  
│ │ │  o2 : Sequence
    │ │ │
    │ │ │ 
    i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00895853s elapsed
│ │ │ - -- .0106852s elapsed
│ │ │ - -- .0250247s elapsed
│ │ │ - -- .00988981s elapsed
│ │ │ - -- .00380772s elapsed
│ │ │ + -- .0032664s elapsed
│ │ │ + -- .00781772s elapsed
│ │ │ + -- .0292052s elapsed
│ │ │ + -- .0138733s elapsed
│ │ │ + -- .00461877s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -141,15 +141,15 @@
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │ 
    i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ - -- .147586s elapsed
│ │ │ + -- .235423s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -183,19 +183,19 @@
│ │ │  
│ │ │  o7 : Sequence
    │ │ │
    │ │ │ 
    i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00258108s elapsed
│ │ │ - -- .00686158s elapsed
│ │ │ - -- .0244236s elapsed
│ │ │ - -- .00913852s elapsed
│ │ │ - -- .00344125s elapsed
│ │ │ + -- .00300207s elapsed
│ │ │ + -- .00745699s elapsed
│ │ │ + -- .0279157s elapsed
│ │ │ + -- .0104353s elapsed
│ │ │ + -- .00427333s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,19 +27,19 @@
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : e=(-1,5)
│ │ │ │  
│ │ │ │  o2 = (-1, 5)
│ │ │ │  
│ │ │ │  o2 : Sequence
│ │ │ │  i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ │ - -- .00895853s elapsed
│ │ │ │ - -- .0106852s elapsed
│ │ │ │ - -- .0250247s elapsed
│ │ │ │ - -- .00988981s elapsed
│ │ │ │ - -- .00380772s elapsed
│ │ │ │ + -- .0032664s elapsed
│ │ │ │ + -- .00781772s elapsed
│ │ │ │ + -- .0292052s elapsed
│ │ │ │ + -- .0138733s elapsed
│ │ │ │ + -- .00461877s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -65,15 +65,15 @@
│ │ │ │  o3 : HashTable
│ │ │ │  i4 : keys h
│ │ │ │  
│ │ │ │  o4 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ │ - -- .147586s elapsed
│ │ │ │ + -- .235423s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -94,19 +94,19 @@
│ │ │ │  these mistakes.
│ │ │ │  i7 : e=(-1,5^2)
│ │ │ │  
│ │ │ │  o7 = (-1, 25)
│ │ │ │  
│ │ │ │  o7 : Sequence
│ │ │ │  i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ │ - -- .00258108s elapsed
│ │ │ │ - -- .00686158s elapsed
│ │ │ │ - -- .0244236s elapsed
│ │ │ │ - -- .00913852s elapsed
│ │ │ │ - -- .00344125s elapsed
│ │ │ │ + -- .00300207s elapsed
│ │ │ │ + -- .00745699s elapsed
│ │ │ │ + -- .0279157s elapsed
│ │ │ │ + -- .0104353s elapsed
│ │ │ │ + -- .00427333s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html
│ │ │ @@ -83,172 +83,172 @@
│ │ │  
│ │ │  o1 = 4
    │ │ │
    │ │ │ 
    i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0213231s elapsed
│ │ │ + -- .0305235s elapsed
│ │ │  (number of blocks= , 18)
│ │ │  (size of the matrices, Tally{1 => 4})
│ │ │                               2 => 6
│ │ │                               3 => 2
│ │ │                               4 => 6
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      7: 1 1
│ │ │ - -- .000044764s elapsed
│ │ │ + -- .000046948s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000082483s elapsed
│ │ │ + -- .000108106s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000068808s elapsed
│ │ │ + -- .000088203s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000076702s elapsed
│ │ │ + -- .000122223s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .00009138s elapsed
│ │ │ + -- .000115303s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000087173s elapsed
│ │ │ + -- .000102445s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000024476s elapsed
│ │ │ + -- .000037428s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .00006923s elapsed
│ │ │ + -- .000079821s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000081171s elapsed
│ │ │ + -- .000105172s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000109105s elapsed
│ │ │ + -- .000100179s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000083966s elapsed
│ │ │ + -- .000135631s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .00008596s elapsed
│ │ │ + -- .000112036s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .00007515s elapsed
│ │ │ + -- .000108502s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000068008s elapsed
│ │ │ + -- .000082539s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .000067106s elapsed
│ │ │ + -- .000095404s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000024396s elapsed
│ │ │ + -- .000034936s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000071524s elapsed
│ │ │ + -- .000090862s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000025659s elapsed
│ │ │ + -- .000034004s elapsed
│ │ │  (e )
│ │ │    1
│ │ │  
│ │ │         6      32    32
│ │ │  o2 = (3 , (e )  (e )  )
│ │ │              1     2
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,172 +19,172 @@
│ │ │ │  grading. Viewed as a resolution over QQ(e_1,e_2), this resolution is non-
│ │ │ │  minimal and carries further gradings. We decompose the crucial map of the a-th
│ │ │ │  strand into blocks, compute their determinants, and factor the product.
│ │ │ │  i1 : a=4
│ │ │ │  
│ │ │ │  o1 = 4
│ │ │ │  i2 : (d1,d2)=resonanceDet(a)
│ │ │ │ - -- .0213231s elapsed
│ │ │ │ + -- .0305235s elapsed
│ │ │ │  (number of blocks= , 18)
│ │ │ │  (size of the matrices, Tally{1 => 4})
│ │ │ │                               2 => 6
│ │ │ │                               3 => 2
│ │ │ │                               4 => 6
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │      7: 1 1
│ │ │ │ - -- .000044764s elapsed
│ │ │ │ + -- .000046948s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . 2
│ │ │ │ - -- .000082483s elapsed
│ │ │ │ + -- .000108106s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . .
│ │ │ │      9: . 2
│ │ │ │ - -- .000068808s elapsed
│ │ │ │ + -- .000088203s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │      7: 2 .
│ │ │ │      8: 1 .
│ │ │ │      9: . 1
│ │ │ │     10: . 2
│ │ │ │ - -- .000076702s elapsed
│ │ │ │ + -- .000122223s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (-3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      7: 1 .
│ │ │ │      8: 1 .
│ │ │ │      9: 2 2
│ │ │ │     10: . 1
│ │ │ │     11: . 1
│ │ │ │ - -- .00009138s elapsed
│ │ │ │ + -- .000115303s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      8: 1 .
│ │ │ │      9: 2 1
│ │ │ │     10: 1 2
│ │ │ │     11: . 1
│ │ │ │ - -- .000087173s elapsed
│ │ │ │ + -- .000102445s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │      9: 1 1
│ │ │ │ - -- .000024476s elapsed
│ │ │ │ + -- .000037428s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      9: 1 1
│ │ │ │     10: 1 1
│ │ │ │ - -- .00006923s elapsed
│ │ │ │ + -- .000079821s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .000081171s elapsed
│ │ │ │ + -- .000105172s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 2 1
│ │ │ │     11: 1 2
│ │ │ │     12: . 1
│ │ │ │ - -- .000109105s elapsed
│ │ │ │ + -- .000100179s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 1 .
│ │ │ │     11: 2 2
│ │ │ │     12: . 1
│ │ │ │     13: . 1
│ │ │ │ - -- .000083966s elapsed
│ │ │ │ + -- .000135631s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .00008596s elapsed
│ │ │ │ + -- .000112036s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │     10: 2 .
│ │ │ │     11: 1 .
│ │ │ │     12: . 1
│ │ │ │     13: . 2
│ │ │ │ - -- .00007515s elapsed
│ │ │ │ + -- .000108502s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     10: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000068008s elapsed
│ │ │ │ + -- .000082539s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     11: 2 .
│ │ │ │     12: . .
│ │ │ │     13: . 2
│ │ │ │ - -- .000067106s elapsed
│ │ │ │ + -- .000095404s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000024396s elapsed
│ │ │ │ + -- .000034936s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     12: 2 .
│ │ │ │     13: . 2
│ │ │ │ - -- .000071524s elapsed
│ │ │ │ + -- .000090862s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     13: 1 1
│ │ │ │ - -- .000025659s elapsed
│ │ │ │ + -- .000034004s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │  
│ │ │ │         6      32    32
│ │ │ │  o2 = (3 , (e )  (e )  )
│ │ │ │              1     2
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -7,55 +7,55 @@
│ │ │  
│ │ │  i2 : m = syz m1;
│ │ │  
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i3 : time LLL m;
│ │ │ - -- used 0.0089479s (cpu); 0.00894067s (thread); 0s (gc)
│ │ │ + -- used 0.0104735s (cpu); 0.0104737s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0311152s (cpu); 0.0311183s (thread); 0s (gc)
│ │ │ + -- used 0.0329577s (cpu); 0.0329644s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.110634s (cpu); 0.11064s (thread); 0s (gc)
│ │ │ + -- used 0.133479s (cpu); 0.133301s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0110928s (cpu); 0.0110928s (thread); 0s (gc)
│ │ │ + -- used 0.0129679s (cpu); 0.0129737s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0478669s (cpu); 0.047867s (thread); 0s (gc)
│ │ │ + -- used 0.0647632s (cpu); 0.0647689s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0571961s (cpu); 0.0571964s (thread); 0s (gc)
│ │ │ + -- used 0.0682582s (cpu); 0.0682643s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.370151s (cpu); 0.370133s (thread); 0s (gc)
│ │ │ + -- used 0.425894s (cpu); 0.425901s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.163434s (cpu); 0.163438s (thread); 0s (gc)
│ │ │ + -- used 0.175747s (cpu); 0.175753s (thread); 0s (gc)
│ │ │  
│ │ │                 50       47
│ │ │  o10 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -144,78 +144,78 @@
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i3 : time LLL m;
│ │ │ - -- used 0.0089479s (cpu); 0.00894067s (thread); 0s (gc)
│ │ │ + -- used 0.0104735s (cpu); 0.0104737s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0311152s (cpu); 0.0311183s (thread); 0s (gc)
│ │ │ + -- used 0.0329577s (cpu); 0.0329644s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.110634s (cpu); 0.11064s (thread); 0s (gc)
│ │ │ + -- used 0.133479s (cpu); 0.133301s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0110928s (cpu); 0.0110928s (thread); 0s (gc)
│ │ │ + -- used 0.0129679s (cpu); 0.0129737s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0478669s (cpu); 0.047867s (thread); 0s (gc)
│ │ │ + -- used 0.0647632s (cpu); 0.0647689s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0571961s (cpu); 0.0571964s (thread); 0s (gc)
│ │ │ + -- used 0.0682582s (cpu); 0.0682643s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.370151s (cpu); 0.370133s (thread); 0s (gc)
│ │ │ + -- used 0.425894s (cpu); 0.425901s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.163434s (cpu); 0.163438s (thread); 0s (gc)
│ │ │ + -- used 0.175747s (cpu); 0.175753s (thread); 0s (gc)
│ │ │  
│ │ │                 50       47
│ │ │  o10 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -115,50 +115,50 @@ │ │ │ │ 50 50 │ │ │ │ o1 : Matrix ZZ <-- ZZ │ │ │ │ i2 : m = syz m1; │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o2 : Matrix ZZ <-- ZZ │ │ │ │ i3 : time LLL m; │ │ │ │ - -- used 0.0089479s (cpu); 0.00894067s (thread); 0s (gc) │ │ │ │ + -- used 0.0104735s (cpu); 0.0104737s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o3 : Matrix ZZ <-- ZZ │ │ │ │ i4 : time LLL(m, Strategy=>CohenEngine); │ │ │ │ - -- used 0.0311152s (cpu); 0.0311183s (thread); 0s (gc) │ │ │ │ + -- used 0.0329577s (cpu); 0.0329644s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o4 : Matrix ZZ <-- ZZ │ │ │ │ i5 : time LLL(m, Strategy=>CohenTopLevel); │ │ │ │ - -- used 0.110634s (cpu); 0.11064s (thread); 0s (gc) │ │ │ │ + -- used 0.133479s (cpu); 0.133301s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o5 : Matrix ZZ <-- ZZ │ │ │ │ i6 : time LLL(m, Strategy=>{Givens,RealFP}); │ │ │ │ - -- used 0.0110928s (cpu); 0.0110928s (thread); 0s (gc) │ │ │ │ + -- used 0.0129679s (cpu); 0.0129737s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o6 : Matrix ZZ <-- ZZ │ │ │ │ i7 : time LLL(m, Strategy=>{Givens,RealQP}); │ │ │ │ - -- used 0.0478669s (cpu); 0.047867s (thread); 0s (gc) │ │ │ │ + -- used 0.0647632s (cpu); 0.0647689s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o7 : Matrix ZZ <-- ZZ │ │ │ │ i8 : time LLL(m, Strategy=>{Givens,RealXD}); │ │ │ │ - -- used 0.0571961s (cpu); 0.0571964s (thread); 0s (gc) │ │ │ │ + -- used 0.0682582s (cpu); 0.0682643s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o8 : Matrix ZZ <-- ZZ │ │ │ │ i9 : time LLL(m, Strategy=>{Givens,RealRR}); │ │ │ │ - -- used 0.370151s (cpu); 0.370133s (thread); 0s (gc) │ │ │ │ + -- used 0.425894s (cpu); 0.425901s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o9 : Matrix ZZ <-- ZZ │ │ │ │ i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP}); │ │ │ │ - -- used 0.163434s (cpu); 0.163438s (thread); 0s (gc) │ │ │ │ + -- used 0.175747s (cpu); 0.175753s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o10 : Matrix ZZ <-- ZZ │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ For most of the options, the columns do not need to be linearly independent. │ │ │ │ The strategies CohenEngine and CohenTopLevel currently require the columns to │ │ │ │ be linearly independent. │ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ @@ -16,14 +16,14 @@ │ │ │ │ │ │ 3 8 │ │ │ o4 : Matrix ZZ <-- ZZ │ │ │ │ │ │ i5 : P = convexHull(M); │ │ │ │ │ │ i6 : time areIsomorphic(P,P); │ │ │ - -- used 0.902646s (cpu); 0.664865s (thread); 0s (gc) │ │ │ + -- used 0.930233s (cpu); 0.539662s (thread); 0s (gc) │ │ │ │ │ │ i7 : time areIsomorphic(P,P,smoothTest=>false); │ │ │ - -- used 0.337734s (cpu); 0.260958s (thread); 0s (gc) │ │ │ + -- used 0.514451s (cpu); 0.30669s (thread); 0s (gc) │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ @@ -125,21 +125,21 @@ │ │ │ │ │ │ 
    i5 : P = convexHull(M);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time areIsomorphic(P,P);
│ │ │ - -- used 0.902646s (cpu); 0.664865s (thread); 0s (gc)
    │ │ │ + -- used 0.930233s (cpu); 0.539662s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.337734s (cpu); 0.260958s (thread); 0s (gc)
    │ │ │ + -- used 0.514451s (cpu); 0.30669s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Ways to use areIsomorphic:

    │ │ │ ├── html2text {} │ │ │ │ @@ -35,17 +35,17 @@ │ │ │ │ | 0 0 1 0 1 0 1 1 | │ │ │ │ | 0 0 0 1 0 1 1 1 | │ │ │ │ │ │ │ │ 3 8 │ │ │ │ o4 : Matrix ZZ <-- ZZ │ │ │ │ i5 : P = convexHull(M); │ │ │ │ i6 : time areIsomorphic(P,P); │ │ │ │ - -- used 0.902646s (cpu); 0.664865s (thread); 0s (gc) │ │ │ │ + -- used 0.930233s (cpu); 0.539662s (thread); 0s (gc) │ │ │ │ i7 : time areIsomorphic(P,P,smoothTest=>false); │ │ │ │ - -- used 0.337734s (cpu); 0.260958s (thread); 0s (gc) │ │ │ │ + -- used 0.514451s (cpu); 0.30669s (thread); 0s (gc) │ │ │ │ ********** WWaayyss ttoo uussee aarreeIIssoommoorrpphhiicc:: ********** │ │ │ │ * areIsomorphic(Matrix,Matrix) │ │ │ │ * areIsomorphic(Polyhedron,Polyhedron) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _a_r_e_I_s_o_m_o_r_p_h_i_c is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out │ │ │ @@ -29,21 +29,21 @@ │ │ │ i5 : findRegion({{0,0},{4,4}},M,f) │ │ │ │ │ │ o5 = {{1, 2}, {3, 1}} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : elapsedTime findRegion({{0,0},{4,4}},M,f) │ │ │ - -- .0925824s elapsed │ │ │ + -- .0715313s elapsed │ │ │ │ │ │ o6 = {{1, 2}, {3, 1}} │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{{1,1}}) │ │ │ - -- .0932051s elapsed │ │ │ + -- .0310418s elapsed │ │ │ │ │ │ o7 = {{1, 2}, {3, 1}} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_linear__Truncations__Bound.out │ │ │ @@ -30,21 +30,21 @@ │ │ │ i5 : apply(L, d -> isLinearComplex res prune truncate(d,M)) │ │ │ │ │ │ o5 = {true, true} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M) │ │ │ - -- 4.38488s elapsed │ │ │ + -- 3.36406s elapsed │ │ │ │ │ │ o6 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : elapsedTime linearTruncationsBound M │ │ │ - -- .0259108s elapsed │ │ │ + -- .0310548s elapsed │ │ │ │ │ │ o7 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Region.html │ │ │ @@ -130,25 +130,25 @@ │ │ │
    │ │ │

    If some degrees d are known to satisfy f(d,M), then they can be specified using the option Inner in order to expedite the computation. Similarly, degrees not above those given in Outer will be assumed not to satisfy f(d,M). If f takes options these can also be given to findRegion.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : elapsedTime findRegion({{0,0},{4,4}},M,f)
│ │ │ - -- .0925824s elapsed
│ │ │ + -- .0715313s elapsed
│ │ │  
│ │ │  o6 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{{1,1}})
│ │ │ - -- .0932051s elapsed
│ │ │ + -- .0310418s elapsed
│ │ │  
│ │ │  o7 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o7 : List
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -48,22 +48,22 @@ │ │ │ │ │ │ │ │ o5 : List │ │ │ │ If some degrees d are known to satisfy f(d,M), then they can be specified using │ │ │ │ the option Inner in order to expedite the computation. Similarly, degrees not │ │ │ │ above those given in Outer will be assumed not to satisfy f(d,M). If f takes │ │ │ │ options these can also be given to findRegion. │ │ │ │ i6 : elapsedTime findRegion({{0,0},{4,4}},M,f) │ │ │ │ - -- .0925824s elapsed │ │ │ │ + -- .0715313s elapsed │ │ │ │ │ │ │ │ o6 = {{1, 2}, {3, 1}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{ │ │ │ │ {1,1}}) │ │ │ │ - -- .0932051s elapsed │ │ │ │ + -- .0310418s elapsed │ │ │ │ │ │ │ │ o7 = {{1, 2}, {3, 1}} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** CCoonnttrriibbuuttoorrss ********** │ │ │ │ Mahrud Sayrafi contributed to the code for this function. │ │ │ │ ********** CCaavveeaatt ********** │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html │ │ │ @@ -128,25 +128,25 @@ │ │ │
    │ │ │

    The output is a list of the minimal multidegrees $d$ such that the sum of the positive coordinates of $b-d$ is at most $i$ for all degrees $b$ appearing in the i-th step of the resolution of $M$.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M)
│ │ │ - -- 4.38488s elapsed
│ │ │ + -- 3.36406s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .0259108s elapsed
│ │ │ + -- .0310548s elapsed
│ │ │  
│ │ │  o7 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o7 : List
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -48,21 +48,21 @@ │ │ │ │ o5 = {true, true} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ The output is a list of the minimal multidegrees $d$ such that the sum of the │ │ │ │ positive coordinates of $b-d$ is at most $i$ for all degrees $b$ appearing in │ │ │ │ the i-th step of the resolution of $M$. │ │ │ │ i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M) │ │ │ │ - -- 4.38488s elapsed │ │ │ │ + -- 3.36406s elapsed │ │ │ │ │ │ │ │ o6 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : elapsedTime linearTruncationsBound M │ │ │ │ - -- .0259108s elapsed │ │ │ │ + -- .0310548s elapsed │ │ │ │ │ │ │ │ o7 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ In general linearTruncationsBound will not find the minimal degrees where $M$ │ │ │ │ has a linear resolution but will be faster than repeatedly truncating $M$. │ │ ├── ./usr/share/doc/Macaulay2/LocalRings/example-output/_hilbert__Samuel__Function.out │ │ │ @@ -15,15 +15,15 @@ │ │ │ │ │ │ o4 = cokernel | x5+y3+z3 y5+x3+z3 z5+x3+y3 | │ │ │ │ │ │ 1 │ │ │ o4 : RP-module, quotient of RP │ │ │ │ │ │ i5 : elapsedTime hilbertSamuelFunction(M, 0, 6) │ │ │ - -- .294135s elapsed │ │ │ + -- .221058s elapsed │ │ │ │ │ │ o5 = {1, 3, 6, 7, 6, 3, 1} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : oo//sum │ │ │ │ │ │ @@ -44,21 +44,21 @@ │ │ │ │ │ │ 2 3 │ │ │ o10 = ideal (x , y ) │ │ │ │ │ │ o10 : Ideal of RP │ │ │ │ │ │ i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds │ │ │ - -- .0119755s elapsed │ │ │ + -- .0147003s elapsed │ │ │ │ │ │ o11 = {1, 2, 3, 4, 5, 6} │ │ │ │ │ │ o11 : List │ │ │ │ │ │ i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds │ │ │ - -- .186217s elapsed │ │ │ + -- .233771s elapsed │ │ │ │ │ │ o12 = {6, 12, 18, 24, 30, 36} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/LocalRings/html/_hilbert__Samuel__Function.html │ │ │ @@ -116,15 +116,15 @@ │ │ │ 1 │ │ │ o4 : RP-module, quotient of RP │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime hilbertSamuelFunction(M, 0, 6)
│ │ │ - -- .294135s elapsed
│ │ │ + -- .221058s elapsed
│ │ │  
│ │ │  o5 = {1, 3, 6, 7, 6, 3, 1}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -168,25 +168,25 @@ │ │ │ │ │ │ o10 : Ideal of RP │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds
│ │ │ - -- .0119755s elapsed
│ │ │ + -- .0147003s elapsed
│ │ │  
│ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │  
│ │ │  o11 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ - -- .186217s elapsed
│ │ │ + -- .233771s elapsed
│ │ │  
│ │ │  o12 = {6, 12, 18, 24, 30, 36}
│ │ │  
│ │ │  o12 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -41,15 +41,15 @@ │ │ │ │ i4 : M = RP^1/I │ │ │ │ │ │ │ │ o4 = cokernel | x5+y3+z3 y5+x3+z3 z5+x3+y3 | │ │ │ │ │ │ │ │ 1 │ │ │ │ o4 : RP-module, quotient of RP │ │ │ │ i5 : elapsedTime hilbertSamuelFunction(M, 0, 6) │ │ │ │ - -- .294135s elapsed │ │ │ │ + -- .221058s elapsed │ │ │ │ │ │ │ │ o5 = {1, 3, 6, 7, 6, 3, 1} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : oo//sum │ │ │ │ │ │ │ │ o6 = 27 │ │ │ │ @@ -65,21 +65,21 @@ │ │ │ │ i10 : q = ideal"x2,y3" │ │ │ │ │ │ │ │ 2 3 │ │ │ │ o10 = ideal (x , y ) │ │ │ │ │ │ │ │ o10 : Ideal of RP │ │ │ │ i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds │ │ │ │ - -- .0119755s elapsed │ │ │ │ + -- .0147003s elapsed │ │ │ │ │ │ │ │ o11 = {1, 2, 3, 4, 5, 6} │ │ │ │ │ │ │ │ o11 : List │ │ │ │ i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds │ │ │ │ - -- .186217s elapsed │ │ │ │ + -- .233771s elapsed │ │ │ │ │ │ │ │ o12 = {6, 12, 18, 24, 30, 36} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ Hilbert-Samuel function with respect to a parameter ideal other than the │ │ │ │ maximal ideal can be slower. │ │ ├── ./usr/share/doc/Macaulay2/MRDI/example-output/_load__M__R__D__I.out │ │ │ @@ -57,15 +57,15 @@ │ │ │ │ │ │ i9 : I === J │ │ │ │ │ │ o9 = true │ │ │ │ │ │ i10 : fn = temporaryFileName() | ".mrdi" │ │ │ │ │ │ -o10 = /tmp/M2-39441-0/0.mrdi │ │ │ +o10 = /tmp/M2-55059-0/0.mrdi │ │ │ │ │ │ i11 : saveMRDI(I, FileName => fn) │ │ │ │ │ │ o11 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name": │ │ │ "Ideal"}, "data": [[[["0", "0", "2", "0"], ["1", "1"]], [["0", "1", │ │ │ "0", "1"], ["-1", "1"]]], [[["0", "1", "1", "0"], ["1", "1"]], [["1", │ │ │ "0", "0", "1"], ["-1", "1"]]], [[["0", "2", "0", "0"], ["1", "1"]], │ │ ├── ./usr/share/doc/Macaulay2/MRDI/example-output/_save__M__R__D__I.out │ │ │ @@ -39,15 +39,15 @@ │ │ │ "1"]]]], "_ns": {"Macaulay2": ["https://macaulay2.com", "1.26.05"]}, │ │ │ "_refs": {"6caf806b-9118-4741-bec7-217a71096848": {"_type": {"params": │ │ │ {"_type": "Ring", "data": "QQ"}, "name": "PolynomialRing"}, "data": │ │ │ {"variables": ["x", "y", "z"]}}}} │ │ │ │ │ │ i7 : fn = temporaryFileName() | ".mrdi" │ │ │ │ │ │ -o7 = /tmp/M2-39460-0/0.mrdi │ │ │ +o7 = /tmp/M2-55098-0/0.mrdi │ │ │ │ │ │ i8 : saveMRDI(f, FileName => fn) │ │ │ │ │ │ o8 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name": │ │ │ "RingElement"}, "data": [[["2", "0", "0"], ["1", "1"]], [["0", "1", │ │ │ "1"], ["1", "1"]], [["1", "0", "0"], ["-3", "1"]]], "_ns": {"Macaulay2": │ │ │ ["https://macaulay2.com", "1.26.05"]}, "_refs": │ │ │ @@ -69,22 +69,22 @@ │ │ │ "_type" => ZZ │ │ │ "data" => 5 │ │ │ │ │ │ o11 : HashTable │ │ │ │ │ │ i12 : methods {"Macaulay2", saveMRDI} │ │ │ │ │ │ -o12 = {0 => ({Macaulay2, saveMRDI}, QuotientRing) } │ │ │ - {1 => ({Macaulay2, saveMRDI}, ZZ) } │ │ │ - {2 => ({Macaulay2, saveMRDI}, RingElement) } │ │ │ - {3 => ({Macaulay2, saveMRDI}, Ring) } │ │ │ - {4 => ({Macaulay2, saveMRDI}, Matrix) } │ │ │ - {5 => ({Macaulay2, saveMRDI}, GaloisField) } │ │ │ - {6 => ({Macaulay2, saveMRDI}, Ideal) } │ │ │ - {7 => ({Macaulay2, saveMRDI}, PolynomialRing)} │ │ │ +o12 = {0 => ({Macaulay2, saveMRDI}, RingElement) } │ │ │ + {1 => ({Macaulay2, saveMRDI}, Ring) } │ │ │ + {2 => ({Macaulay2, saveMRDI}, Matrix) } │ │ │ + {3 => ({Macaulay2, saveMRDI}, GaloisField) } │ │ │ + {4 => ({Macaulay2, saveMRDI}, Ideal) } │ │ │ + {5 => ({Macaulay2, saveMRDI}, PolynomialRing)} │ │ │ + {6 => ({Macaulay2, saveMRDI}, QuotientRing) } │ │ │ + {7 => ({Macaulay2, saveMRDI}, ZZ) } │ │ │ │ │ │ o12 : NumberedVerticalList │ │ │ │ │ │ i13 : methods {"Oscar", saveMRDI} │ │ │ │ │ │ o13 = {0 => ({Oscar, saveMRDI}, RingElement) } │ │ │ {1 => ({Oscar, saveMRDI}, Ring) } │ │ ├── ./usr/share/doc/Macaulay2/MRDI/html/_load__M__R__D__I.html │ │ │ @@ -176,15 +176,15 @@ │ │ │

    Objects can be loaded from a file as well using get.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i10 : fn = temporaryFileName() | ".mrdi"
│ │ │  
│ │ │ -o10 = /tmp/M2-39441-0/0.mrdi
    │ │ │ +o10 = /tmp/M2-55059-0/0.mrdi │ │ │
    │ │ │ 
    i11 : saveMRDI(I, FileName => fn)
│ │ │  
│ │ │  o11 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ ├── html2text {}
│ │ │ │ @@ -69,15 +69,15 @@
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : I === J
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  Objects can be loaded from a file as well using _g_e_t.
│ │ │ │  i10 : fn = temporaryFileName() | ".mrdi"
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-39441-0/0.mrdi
│ │ │ │ +o10 = /tmp/M2-55059-0/0.mrdi
│ │ │ │  i11 : saveMRDI(I, FileName => fn)
│ │ │ │  
│ │ │ │  o11 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ │        "Ideal"}, "data": [[[["0", "0", "2", "0"], ["1", "1"]], [["0", "1",
│ │ │ │        "0", "1"], ["-1", "1"]]], [[["0", "1", "1", "0"], ["1", "1"]], [["1",
│ │ │ │        "0", "0", "1"], ["-1", "1"]]], [[["0", "2", "0", "0"], ["1", "1"]],
│ │ │ │        [["1", "0", "1", "0"], ["-1", "1"]]]], "_ns": {"Macaulay2":
│ │ ├── ./usr/share/doc/Macaulay2/MRDI/html/_save__M__R__D__I.html
│ │ │ @@ -160,15 +160,15 @@
│ │ │            
    The output can be written directly to a file using the FileName option.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : fn = temporaryFileName() | ".mrdi"
│ │ │  
│ │ │ -o7 = /tmp/M2-39460-0/0.mrdi
    
│ │ │ +o7 = /tmp/M2-55098-0/0.mrdi
│ │ │              
    
│ │ │                
    i8 : saveMRDI(f, FileName => fn)
│ │ │  
│ │ │  o8 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ @@ -220,22 +220,22 @@
│ │ │            
    To see which types have built-in save methods for a given namespace, call methods as follows.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : methods {"Macaulay2", saveMRDI}
│ │ │  
│ │ │ -o12 = {0 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │ -      {1 => ({Macaulay2, saveMRDI}, ZZ)            }
│ │ │ -      {2 => ({Macaulay2, saveMRDI}, RingElement)   }
│ │ │ -      {3 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ -      {4 => ({Macaulay2, saveMRDI}, Matrix)        }
│ │ │ -      {5 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ -      {6 => ({Macaulay2, saveMRDI}, Ideal)         }
│ │ │ -      {7 => ({Macaulay2, saveMRDI}, PolynomialRing)}
│ │ │ +o12 = {0 => ({Macaulay2, saveMRDI}, RingElement)   }
│ │ │ +      {1 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ +      {2 => ({Macaulay2, saveMRDI}, Matrix)        }
│ │ │ +      {3 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ +      {4 => ({Macaulay2, saveMRDI}, Ideal)         }
│ │ │ +      {5 => ({Macaulay2, saveMRDI}, PolynomialRing)}
│ │ │ +      {6 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │ +      {7 => ({Macaulay2, saveMRDI}, ZZ)            }
│ │ │  
│ │ │  o12 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i13 : methods {"Oscar", saveMRDI}
│ │ │ ├── html2text {}
│ │ │ │ @@ -66,15 +66,15 @@
│ │ │ │       "1"]]]], "_ns": {"Macaulay2": ["https://macaulay2.com", "1.26.05"]},
│ │ │ │       "_refs": {"6caf806b-9118-4741-bec7-217a71096848": {"_type": {"params":
│ │ │ │       {"_type": "Ring", "data": "QQ"}, "name": "PolynomialRing"}, "data":
│ │ │ │       {"variables": ["x", "y", "z"]}}}}
│ │ │ │  The output can be written directly to a file using the _F_i_l_e_N_a_m_e option.
│ │ │ │  i7 : fn = temporaryFileName() | ".mrdi"
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-39460-0/0.mrdi
│ │ │ │ +o7 = /tmp/M2-55098-0/0.mrdi
│ │ │ │  i8 : saveMRDI(f, FileName => fn)
│ │ │ │  
│ │ │ │  o8 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ │       "RingElement"}, "data": [[["2", "0", "0"], ["1", "1"]], [["0", "1",
│ │ │ │       "1"], ["1", "1"]], [["1", "0", "0"], ["-3", "1"]]], "_ns": {"Macaulay2":
│ │ │ │       ["https://macaulay2.com", "1.26.05"]}, "_refs":
│ │ │ │       {"6caf806b-9118-4741-bec7-217a71096848": {"_type": {"params": {"_type":
│ │ │ │ @@ -98,22 +98,22 @@
│ │ │ │                  "data" => 5
│ │ │ │  
│ │ │ │  o11 : HashTable
│ │ │ │  To see which types have built-in save methods for a given namespace, call
│ │ │ │  _m_e_t_h_o_d_s as follows.
│ │ │ │  i12 : methods {"Macaulay2", saveMRDI}
│ │ │ │  
│ │ │ │ -o12 = {0 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │ │ -      {1 => ({Macaulay2, saveMRDI}, ZZ)            }
│ │ │ │ -      {2 => ({Macaulay2, saveMRDI}, RingElement)   }
│ │ │ │ -      {3 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ │ -      {4 => ({Macaulay2, saveMRDI}, Matrix)        }
│ │ │ │ -      {5 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ │ -      {6 => ({Macaulay2, saveMRDI}, Ideal)         }
│ │ │ │ -      {7 => ({Macaulay2, saveMRDI}, PolynomialRing)}
│ │ │ │ +o12 = {0 => ({Macaulay2, saveMRDI}, RingElement)   }
│ │ │ │ +      {1 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ │ +      {2 => ({Macaulay2, saveMRDI}, Matrix)        }
│ │ │ │ +      {3 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ │ +      {4 => ({Macaulay2, saveMRDI}, Ideal)         }
│ │ │ │ +      {5 => ({Macaulay2, saveMRDI}, PolynomialRing)}
│ │ │ │ +      {6 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │ │ +      {7 => ({Macaulay2, saveMRDI}, ZZ)            }
│ │ │ │  
│ │ │ │  o12 : NumberedVerticalList
│ │ │ │  i13 : methods {"Oscar", saveMRDI}
│ │ │ │  
│ │ │ │  o13 = {0 => ({Oscar, saveMRDI}, RingElement)   }
│ │ │ │        {1 => ({Oscar, saveMRDI}, Ring)          }
│ │ │ │        {2 => ({Oscar, saveMRDI}, QQ)            }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Command.out
│ │ │ @@ -5,12 +5,12 @@
│ │ │  i2 : f
│ │ │  
│ │ │  o2 = 1073741824
│ │ │  
│ │ │  i3 : (c = Command "date";)
│ │ │  
│ │ │  i4 : c
│ │ │ -Mon May 18 12:32:59 UTC 2026
│ │ │ +Wed May 20 17:22:02 UTC 2026
│ │ │  
│ │ │  o4 = 0
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Database.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 9579076464446459296
│ │ │  
│ │ │  i1 : filename = temporaryFileName () | ".dbm"
│ │ │  
│ │ │ -o1 = /tmp/M2-12329-0/0.dbm
│ │ │ +o1 = /tmp/M2-14079-0/0.dbm
│ │ │  
│ │ │  i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-12329-0/0.dbm
│ │ │ +o2 = /tmp/M2-14079-0/0.dbm
│ │ │  
│ │ │  o2 : Database
│ │ │  
│ │ │  i3 : x#"first" = "hi there"
│ │ │  
│ │ │  o3 = hi there
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 1731899428494721487
│ │ │  
│ │ │  i1 : s = GCstats()
│ │ │  
│ │ │ -o1 = HashTable{"bytesAlloc" => 64675195162        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 64778770938        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 222048256
│ │ │ -               "numGCs" => 945
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 233422848
│ │ │ +               "numGCs" => 816
│ │ │ +               "numGCThreads" => 16
│ │ │  
│ │ │  o1 : HashTable
│ │ │  
│ │ │  i2 : s#"heapSize"
│ │ │  
│ │ │ -o2 = 222048256
│ │ │ +o2 = 233422848
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Minimal__Generators.out
│ │ │ @@ -40,20 +40,20 @@
│ │ │  o6 : PolynomialRing
│ │ │  
│ │ │  i7 : I = monomialCurveIdeal(R, {1,4,5,9});
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ - -- used 0.00855388s (cpu); 0.00854629s (thread); 0s (gc)
│ │ │ + -- used 0.00552555s (cpu); 0.00552121s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0676429s (cpu); 0.0676547s (thread); 0s (gc)
│ │ │ + -- used 0.0497031s (cpu); 0.0497124s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : numgens J
│ │ │  
│ │ │  o10 = 1067
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutable__List.out
│ │ │ @@ -37,19 +37,19 @@
│ │ │  i10 : s = new MutableList
│ │ │  
│ │ │  o10 = MutableList{}
│ │ │  
│ │ │  o10 : MutableList
│ │ │  
│ │ │  i11 : elapsedTime scan(1000, i -> s#i = i^2) -- quadratic, since we grow s at each step
│ │ │ - -- .00341642s elapsed
│ │ │ + -- .00516619s elapsed
│ │ │  
│ │ │  i12 : t = new MutableList from 1000
│ │ │  
│ │ │  o12 = MutableList{...1000...}
│ │ │  
│ │ │  o12 : MutableList
│ │ │  
│ │ │  i13 : elapsedTime scan(1000, i -> t#i = i^2) -- linear
│ │ │ - -- .000457796s elapsed
│ │ │ + -- .000386983s elapsed
│ │ │  
│ │ │  i14 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutex.out
│ │ │ @@ -20,23 +20,19 @@
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : while not all(T, isReady) do null
│ │ │  
│ │ │  i5 : stack sort lines msgs
│ │ │  
│ │ │ -o5 = hello from thread #0
│ │ │ -     hello from thread #1
│ │ │ +o5 = hello from thread #1
│ │ │       hello from thread #2
│ │ │       hello from thread #3
│ │ │       hello from thread #4
│ │ │ -     hello from thread #5
│ │ │       hello from thread #6
│ │ │ -     hello from thread #7
│ │ │ -     hello from thread #8
│ │ │       hello from thread #9
│ │ │  
│ │ │  i6 : m = new Mutex
│ │ │  
│ │ │  o6 = m
│ │ │  
│ │ │  o6 : Mutex
│ │ │ @@ -47,17 +43,19 @@
│ │ │  
│ │ │  i8 : T = apply(10, i -> schedule(() -> (lock m; sayhello i; unlock m)))
│ │ │  
│ │ │  o8 = {<>, <>, <>, <>,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     <>, <>, <>, <>, <>, <>, <>, <>}
│ │ │ +     available, task done>>, <>, <>, <>}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : while not all(T, isReady) do null
│ │ │  
│ │ │  i10 : stack sort lines msgs
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out
│ │ │ @@ -3,13 +3,13 @@
│ │ │  i1 : M = random(RR^200, RR^200);
│ │ │  
│ │ │                  200         200
│ │ │  o1 : Matrix RR      <-- RR
│ │ │                53          53
│ │ │  
│ │ │  i2 : time SVD(M);
│ │ │ - -- used 0.0225511s (cpu); 0.0225436s (thread); 0s (gc)
│ │ │ + -- used 0.0414148s (cpu); 0.0414151s (thread); 0s (gc)
│ │ │  
│ │ │  i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0220907s (cpu); 0.0220933s (thread); 0s (gc)
│ │ │ + -- used 0.0393494s (cpu); 0.0393606s (thread); 0s (gc)
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_a_spfirst_sp__Macaulay2_spsession.out
│ │ │ @@ -347,15 +347,15 @@
│ │ │                 | b e h k n q |
│ │ │                 | c f i l o r |
│ │ │  
│ │ │                               3
│ │ │  o58 : R-module, quotient of R
│ │ │  
│ │ │  i59 : time C = resolution M
│ │ │ - -- used 0.00293873s (cpu); 0.00293155s (thread); 0s (gc)
│ │ │ + -- used 0.00396274s (cpu); 0.00396264s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R
│ │ │                                       
│ │ │        0      1      2       3       4
│ │ │  
│ │ │  o59 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out
│ │ │ @@ -14,10 +14,10 @@
│ │ │  
│ │ │  i4 : peek read f
│ │ │  
│ │ │  o4 = "hi there"
│ │ │  
│ │ │  i5 : atEndOfFile f
│ │ │  
│ │ │ -o5 = false
│ │ │ +o5 = true
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_benchmark.out
│ │ │ @@ -1,9 +1,9 @@
│ │ │  -- -*- M2-comint -*- hash: 1330379359420
│ │ │  
│ │ │  i1 : benchmark "sqrt 2p100000"
│ │ │  
│ │ │ -o1 = .0002914831290148481
│ │ │ +o1 = .0003543037189214616
│ │ │  
│ │ │  o1 : RR (of precision 53)
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_caching_spcomputation_spresults.out
│ │ │ @@ -4,20 +4,20 @@
│ │ │  
│ │ │  i2 : R = QQ[x,y,z];
│ │ │  
│ │ │  i3 : M = coker vars R;
│ │ │  
│ │ │  i4 : elapsedTime pdim' M
│ │ │   -- computing pdim'
│ │ │ - -- .00641207s elapsed
│ │ │ + -- .00439629s elapsed
│ │ │  
│ │ │  o4 = 3
│ │ │  
│ │ │  i5 : elapsedTime pdim' M
│ │ │ - -- .000001934s elapsed
│ │ │ + -- .000003154s elapsed
│ │ │  
│ │ │  o5 = 3
│ │ │  
│ │ │  i6 : peek M.cache
│ │ │  
│ │ │  o6 = CacheTable{cache => MutableHashTable{}                                              }
│ │ │                  isHomogeneous => true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out
│ │ │ @@ -18,57 +18,57 @@
│ │ │  
│ │ │  o4 = <>
│ │ │  
│ │ │  o4 : Task
│ │ │  
│ │ │  i5 : n
│ │ │  
│ │ │ -o5 = 569438
│ │ │ +o5 = 904196
│ │ │  
│ │ │  i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
│ │ │  
│ │ │  i7 : t
│ │ │  
│ │ │  o7 = <>
│ │ │  
│ │ │  o7 : Task
│ │ │  
│ │ │  i8 : n
│ │ │  
│ │ │ -o8 = 1244881
│ │ │ +o8 = 1952215
│ │ │  
│ │ │  i9 : isReady t
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 : cancelTask t
│ │ │  
│ │ │  i11 : sleep 2
│ │ │ -stdio:2:39:(3):[1]: error: interrupted
│ │ │ +stdio:2:25:(3):[1]: error: interrupted
│ │ │  
│ │ │  o11 = 0
│ │ │  
│ │ │  i12 : t
│ │ │  
│ │ │  o12 = <>
│ │ │  
│ │ │  o12 : Task
│ │ │  
│ │ │  i13 : n
│ │ │  
│ │ │ -o13 = 1245096
│ │ │ +o13 = 1952563
│ │ │  
│ │ │  i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
│ │ │  
│ │ │  i15 : n
│ │ │  
│ │ │ -o15 = 1245096
│ │ │ +o15 = 1952563
│ │ │  
│ │ │  i16 : isReady t
│ │ │  
│ │ │  o16 = false
│ │ │  
│ │ │  i17 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Directory.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 8535510246140175278
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11094-0/0
│ │ │ +o1 = /tmp/M2-11564-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11094-0/0
│ │ │ +o2 = /tmp/M2-11564-0/0
│ │ │  
│ │ │  i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-11094-0/0/
│ │ │ +o3 = /tmp/M2-11564-0/0/
│ │ │  
│ │ │  i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-11094-0/0/
│ │ │ +o4 = /tmp/M2-11564-0/0/
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out
│ │ │ @@ -4,51 +4,51 @@
│ │ │  
│ │ │  o1 = FirstPackage
│ │ │  
│ │ │  o1 : Package
│ │ │  
│ │ │  i2 : check_1 FirstPackage
│ │ │   -- warning: reloading FirstPackage; recreate instances of types from this package
│ │ │ - -- capturing check(1, "FirstPackage")        -- .173698s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .219084s elapsed
│ │ │  
│ │ │  i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .180088s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .179579s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .162842s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .16334s elapsed
│ │ │  
│ │ │  i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .178872s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .16582s elapsed
│ │ │  
│ │ │  i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .178948s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .181166s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .162585s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .161662s elapsed
│ │ │  
│ │ │  i6 : tests(1, "FirstPackage")
│ │ │  
│ │ │  o6 = TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]
│ │ │  
│ │ │  o6 : TestInput
│ │ │  
│ │ │  i7 : check oo
│ │ │ - -- capturing check(1, "FirstPackage")        -- .179778s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .163704s elapsed
│ │ │  
│ │ │  i8 : tests "FirstPackage"
│ │ │  
│ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │       {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │  
│ │ │  o8 : NumberedVerticalList
│ │ │  
│ │ │  i9 : check oo
│ │ │ - -- capturing check(0, "FirstPackage")        -- .183757s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .181583s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .161488s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .164806s elapsed
│ │ │  
│ │ │  i10 : tests "FirstPackage"
│ │ │  
│ │ │  o10 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │        {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │  
│ │ │  o10 : NumberedVerticalList
│ │ │  
│ │ │  i11 : check 1
│ │ │ - -- capturing check(1, "FirstPackage")        -- .288312s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .157437s elapsed
│ │ │  
│ │ │  i12 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out
│ │ │ @@ -1,25 +1,25 @@
│ │ │  -- -*- M2-comint -*- hash: 10365735446967377456
│ │ │  
│ │ │  i1 : run "uname -a"
│ │ │ -Linux sbuild 6.12.88+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.88-1 (2026-05-15) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.88+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.88-1 (2026-05-15) x86_64 GNU/Linux
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : "!grep a" << " ba \n bc \n ad \n ef \n" << close
│ │ │   ba 
│ │ │   ad 
│ │ │  
│ │ │  o2 = !grep a
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : peek get "!uname -a"
│ │ │  
│ │ │ -o3 = "Linux sbuild 6.12.88+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ +o3 = "Linux sbuild 6.12.88+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │       6.12.88-1 (2026-05-15) x86_64 GNU/Linux\n"
│ │ │  
│ │ │  i4 : f = openInOut "!grep -E '^in'"
│ │ │  
│ │ │  o4 = !grep -E '^in'
│ │ │  
│ │ │  o4 : File
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │                  ZZ
│ │ │  o23 : Ideal of ----[x..z, w]
│ │ │                 1277
│ │ │  
│ │ │  i24 : gb I
│ │ │  
│ │ │ -   -- registering gb 5 at 0x7fad8cc0fc40
│ │ │ +   -- registering gb 5 at 0x7f50c5372700
│ │ │  
│ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │     -- number of monomials                = 8
│ │ │     -- #reduction steps = 2
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -177,15 +177,15 @@
│ │ │  
│ │ │  i32 : f = random(R^1,R^{-3,-3,-5,-6});
│ │ │  
│ │ │                1      4
│ │ │  o32 : Matrix R  <-- R
│ │ │  
│ │ │  i33 : time betti gb f
│ │ │ - -- used 0.307614s (cpu); 0.201161s (thread); 0s (gc)
│ │ │ + -- used 0.2119s (cpu); 0.21109s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o33 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ @@ -208,15 +208,15 @@
│ │ │  
│ │ │              3    5     8     9    12     14    17
│ │ │  o35 = 1 - 2T  - T  + 2T  + 2T  - T   - 2T   + T
│ │ │  
│ │ │  o35 : ZZ[T]
│ │ │  
│ │ │  i36 : time betti gb f
│ │ │ - -- used 0.00651741s (cpu); 0.00681373s (thread); 0s (gc)
│ │ │ + -- used 0.00800134s (cpu); 0.00806052s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o36 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__Directory_lp__String_cm__String_rp.out
│ │ │ @@ -1,76 +1,76 @@
│ │ │  -- -*- M2-comint -*- hash: 11422793294564310273
│ │ │  
│ │ │  i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11854-0/0/
│ │ │ +o1 = /tmp/M2-13104-0/0/
│ │ │  
│ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11854-0/1/
│ │ │ +o2 = /tmp/M2-13104-0/1/
│ │ │  
│ │ │  i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11854-0/0/a/
│ │ │ +o3 = /tmp/M2-13104-0/0/a/
│ │ │  
│ │ │  i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11854-0/0/b/
│ │ │ +o4 = /tmp/M2-13104-0/0/b/
│ │ │  
│ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11854-0/0/b/c/
│ │ │ +o5 = /tmp/M2-13104-0/0/b/c/
│ │ │  
│ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11854-0/0/a/f
│ │ │ +o6 = /tmp/M2-13104-0/0/a/f
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11854-0/0/a/g
│ │ │ +o7 = /tmp/M2-13104-0/0/a/g
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11854-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13104-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-11854-0/0/
│ │ │ -     /tmp/M2-11854-0/0/b/
│ │ │ -     /tmp/M2-11854-0/0/b/c/
│ │ │ -     /tmp/M2-11854-0/0/b/c/g
│ │ │ -     /tmp/M2-11854-0/0/a/
│ │ │ -     /tmp/M2-11854-0/0/a/g
│ │ │ -     /tmp/M2-11854-0/0/a/f
│ │ │ +o9 = /tmp/M2-13104-0/0/
│ │ │ +     /tmp/M2-13104-0/0/a/
│ │ │ +     /tmp/M2-13104-0/0/a/g
│ │ │ +     /tmp/M2-13104-0/0/a/f
│ │ │ +     /tmp/M2-13104-0/0/b/
│ │ │ +     /tmp/M2-13104-0/0/b/c/
│ │ │ +     /tmp/M2-13104-0/0/b/c/g
│ │ │  
│ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11854-0/0/b/c/g -> /tmp/M2-11854-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-11854-0/0/a/g -> /tmp/M2-11854-0/1/a/g
│ │ │ - -- copying: /tmp/M2-11854-0/0/a/f -> /tmp/M2-11854-0/1/a/f
│ │ │ + -- copying: /tmp/M2-13104-0/0/a/g -> /tmp/M2-13104-0/1/a/g
│ │ │ + -- copying: /tmp/M2-13104-0/0/a/f -> /tmp/M2-13104-0/1/a/f
│ │ │ + -- copying: /tmp/M2-13104-0/0/b/c/g -> /tmp/M2-13104-0/1/b/c/g
│ │ │  
│ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11854-0/0/b/c/g not newer than /tmp/M2-11854-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-11854-0/0/a/g not newer than /tmp/M2-11854-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-11854-0/0/a/f not newer than /tmp/M2-11854-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-13104-0/0/a/g not newer than /tmp/M2-13104-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-13104-0/0/a/f not newer than /tmp/M2-13104-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-13104-0/0/b/c/g not newer than /tmp/M2-13104-0/1/b/c/g
│ │ │  
│ │ │  i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-11854-0/1/
│ │ │ -      /tmp/M2-11854-0/1/a/
│ │ │ -      /tmp/M2-11854-0/1/a/f
│ │ │ -      /tmp/M2-11854-0/1/a/g
│ │ │ -      /tmp/M2-11854-0/1/b/
│ │ │ -      /tmp/M2-11854-0/1/b/c/
│ │ │ -      /tmp/M2-11854-0/1/b/c/g
│ │ │ +o12 = /tmp/M2-13104-0/1/
│ │ │ +      /tmp/M2-13104-0/1/a/
│ │ │ +      /tmp/M2-13104-0/1/a/g
│ │ │ +      /tmp/M2-13104-0/1/a/f
│ │ │ +      /tmp/M2-13104-0/1/b/
│ │ │ +      /tmp/M2-13104-0/1/b/c/
│ │ │ +      /tmp/M2-13104-0/1/b/c/g
│ │ │  
│ │ │  i13 : get (dst|"b/c/g")
│ │ │  
│ │ │  o13 = ho there
│ │ │  
│ │ │  i14 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__File_lp__String_cm__String_rp.out
│ │ │ @@ -1,41 +1,41 @@
│ │ │  -- -*- M2-comint -*- hash: 11539475420155775110
│ │ │  
│ │ │  i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11639-0/0
│ │ │ +o1 = /tmp/M2-12669-0/0
│ │ │  
│ │ │  i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11639-0/1
│ │ │ +o2 = /tmp/M2-12669-0/1
│ │ │  
│ │ │  i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11639-0/0
│ │ │ +o3 = /tmp/M2-12669-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11639-0/0 -> /tmp/M2-11639-0/1
│ │ │ + -- copying: /tmp/M2-12669-0/0 -> /tmp/M2-12669-0/1
│ │ │  
│ │ │  i5 : get dst
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11639-0/0 not newer than /tmp/M2-11639-0/1
│ │ │ + -- skipping: /tmp/M2-12669-0/0 not newer than /tmp/M2-12669-0/1
│ │ │  
│ │ │  i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11639-0/0
│ │ │ +o7 = /tmp/M2-12669-0/0
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11639-0/0 not newer than /tmp/M2-11639-0/1
│ │ │ + -- skipping: /tmp/M2-12669-0/0 not newer than /tmp/M2-12669-0/1
│ │ │  
│ │ │  i9 : get dst
│ │ │  
│ │ │  o9 = hi there
│ │ │  
│ │ │  i10 : removeFile src
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cpu__Time.out
│ │ │ @@ -1,23 +1,23 @@
│ │ │  -- -*- M2-comint -*- hash: 15508153783232232453
│ │ │  
│ │ │  i1 : t1 = cpuTime()
│ │ │  
│ │ │ -o1 = 489.590864066
│ │ │ +o1 = 425.123678766
│ │ │  
│ │ │  o1 : RR (of precision 53)
│ │ │  
│ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │  
│ │ │  i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 491.560329477
│ │ │ +o3 = 426.081552997
│ │ │  
│ │ │  o3 : RR (of precision 53)
│ │ │  
│ │ │  i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.969465411000044
│ │ │ +o4 = .9578742310000052
│ │ │  
│ │ │  o4 : RR (of precision 53)
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Time.out
│ │ │ @@ -1,24 +1,24 @@
│ │ │  -- -*- M2-comint -*- hash: 3660839476107967259
│ │ │  
│ │ │  i1 : currentTime()
│ │ │  
│ │ │ -o1 = 1779107685
│ │ │ +o1 = 1779297808
│ │ │  
│ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 56.37767820542365
│ │ │ +o2 = 56.38370296345478
│ │ │  
│ │ │  o2 : RR (of precision 53)
│ │ │  
│ │ │  i3 : 12 * (oo - floor oo)
│ │ │  
│ │ │ -o3 = 4.53213846508379
│ │ │ +o3 = 4.604435561457365
│ │ │  
│ │ │  o3 : RR (of precision 53)
│ │ │  
│ │ │  i4 : run "date"
│ │ │ -Mon May 18 12:34:45 UTC 2026
│ │ │ +Wed May 20 17:23:28 UTC 2026
│ │ │  
│ │ │  o4 = 0
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Time.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1330565958025
│ │ │  
│ │ │  i1 : elapsedTime sleep 1
│ │ │ - -- 1.00069s elapsed
│ │ │ + -- 1.00014s elapsed
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 1731106803207298715
│ │ │  
│ │ │  i1 : elapsedTiming sleep 1
│ │ │  
│ │ │  o1 = 0
│ │ │ -     -- 1.00016 seconds
│ │ │ +     -- 1.00012 seconds
│ │ │  
│ │ │  o1 : Time
│ │ │  
│ │ │  i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{1.00016, 0}
│ │ │ +o2 = Time{1.00012, 0}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │                 3                   3     2               3
│ │ │  o2 = ideal (- s  - s*t + x - 1, - t  - 3t  - t + y, - s*t  + z)
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time leadTerm gens gb I
│ │ │ - -- used 0.778878s (cpu); 0.330916s (thread); 0s (gc)
│ │ │ + -- used 0.15204s (cpu); 0.152039s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │       ------------------------------------------------------------------------
│ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │       ------------------------------------------------------------------------
│ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -85,15 +85,15 @@
│ │ │  
│ │ │                 3                   3     2               3
│ │ │  o7 = ideal (- s  - s*t + x - 1, - t  - 3t  + y - t, - s*t  + z)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : time G = eliminate(I,{s,t})
│ │ │ - -- used 0.474974s (cpu); 0.256002s (thread); 0s (gc)
│ │ │ + -- used 5.1295s (cpu); 0.481068s (thread); 0s (gc)
│ │ │  
│ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │       ------------------------------------------------------------------------
│ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -154,15 +154,15 @@
│ │ │  i10 : R1 = QQ[x,y,z,s,t, Degrees=>{3,3,4,1,1}];
│ │ │  
│ │ │  i11 : I1 = substitute(I,R1);
│ │ │  
│ │ │  o11 : Ideal of R1
│ │ │  
│ │ │  i12 : time G = eliminate(I1,{s,t})
│ │ │ - -- used 0.0626994s (cpu); 0.0627103s (thread); 0s (gc)
│ │ │ + -- used 0.0312705s (cpu); 0.031271s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7  
│ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2  
│ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -228,15 +228,15 @@
│ │ │  
│ │ │                     3             3     2         3
│ │ │  o16 = map (A, B, {s  + s*t + 1, t  + 3t  + t, s*t })
│ │ │  
│ │ │  o16 : RingMap A <-- B
│ │ │  
│ │ │  i17 : time G = kernel F
│ │ │ - -- used 0.84521s (cpu); 0.352873s (thread); 0s (gc)
│ │ │ + -- used 0.110857s (cpu); 0.110862s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │        -----------------------------------------------------------------------
│ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -297,23 +297,23 @@
│ │ │  i19 : use ring I
│ │ │  
│ │ │  o19 = R
│ │ │  
│ │ │  o19 : PolynomialRing
│ │ │  
│ │ │  i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ - -- used 0.00256747s (cpu); 0.00256862s (thread); 0s (gc)
│ │ │ + -- used 0.00179999s (cpu); 0.00179763s (thread); 0s (gc)
│ │ │  
│ │ │           9    9      7    3
│ │ │  o20 = x*t  - t  - z*t  - z
│ │ │  
│ │ │  o20 : R
│ │ │  
│ │ │  i21 : time f2 = resultant(I_1,f1,t)
│ │ │ - -- used 0.0727722s (cpu); 0.0727847s (thread); 0s (gc)
│ │ │ + -- used 0.0396745s (cpu); 0.0396846s (thread); 0s (gc)
│ │ │  
│ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2  
│ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │        -----------------------------------------------------------------------
│ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8   
│ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out
│ │ │ @@ -59,15 +59,15 @@
│ │ │                                      Version => 0.0
│ │ │               package prefix => /usr/
│ │ │               PackageIsLoaded => true
│ │ │               pkgname => Foo
│ │ │               private dictionary => Foo#"private dictionary"
│ │ │               processed documentation => MutableHashTable{}
│ │ │               raw documentation => MutableHashTable{}
│ │ │ -             source directory => /tmp/M2-10822-0/94-rundir/
│ │ │ +             source directory => /tmp/M2-11042-0/94-rundir/
│ │ │               source file => stdio
│ │ │               test inputs => MutableList{}
│ │ │  
│ │ │  i7 : dictionaryPath
│ │ │  
│ │ │  o7 = {Foo.Dictionary, Varieties.Dictionary, Isomorphism.Dictionary,
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Exists.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 7475038936570224899
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11189-0/0
│ │ │ +o1 = /tmp/M2-11759-0/0
│ │ │  
│ │ │  i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
│ │ │  
│ │ │  i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11189-0/0
│ │ │ +o3 = /tmp/M2-11759-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : fileExists fn
│ │ │  
│ │ │  o4 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Length.out
│ │ │ @@ -1,28 +1,28 @@
│ │ │  -- -*- M2-comint -*- hash: 1216695447195237994
│ │ │  
│ │ │  i1 : f = temporaryFileName() << "hi there"
│ │ │  
│ │ │ -o1 = /tmp/M2-12838-0/0
│ │ │ +o1 = /tmp/M2-15118-0/0
│ │ │  
│ │ │  o1 : File
│ │ │  
│ │ │  i2 : fileLength f
│ │ │  
│ │ │  o2 = 8
│ │ │  
│ │ │  i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-12838-0/0
│ │ │ +o3 = /tmp/M2-15118-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-12838-0/0
│ │ │ +o4 = /tmp/M2-15118-0/0
│ │ │  
│ │ │  i5 : fileLength filename
│ │ │  
│ │ │  o5 = 8
│ │ │  
│ │ │  i6 : get filename
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__File_rp.out
│ │ │ @@ -1,25 +1,25 @@
│ │ │  -- -*- M2-comint -*- hash: 11202140621123993633
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12044-0/0
│ │ │ +o1 = /tmp/M2-13494-0/0
│ │ │  
│ │ │  i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-12044-0/0
│ │ │ +o2 = /tmp/M2-13494-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fileMode f
│ │ │  
│ │ │  o3 = 420
│ │ │  
│ │ │  i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-12044-0/0
│ │ │ +o4 = /tmp/M2-13494-0/0
│ │ │  
│ │ │  o4 : File
│ │ │  
│ │ │  i5 : removeFile fn
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__String_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 4782570202197464532
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11658-0/0
│ │ │ +o1 = /tmp/M2-12708-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11658-0/0
│ │ │ +o2 = /tmp/M2-12708-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fileMode fn
│ │ │  
│ │ │  o3 = 420
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__File_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 17473878267845575442
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11523-0/0
│ │ │ +o1 = /tmp/M2-12433-0/0
│ │ │  
│ │ │  i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11523-0/0
│ │ │ +o2 = /tmp/M2-12433-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : m = 7 + 7*8 + 7*64
│ │ │  
│ │ │  o3 = 511
│ │ │  
│ │ │ @@ -18,15 +18,15 @@
│ │ │  
│ │ │  i5 : fileMode f
│ │ │  
│ │ │  o5 = 511
│ │ │  
│ │ │  i6 : close f
│ │ │  
│ │ │ -o6 = /tmp/M2-11523-0/0
│ │ │ +o6 = /tmp/M2-12433-0/0
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : fileMode fn
│ │ │  
│ │ │  o7 = 511
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__String_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 16772784390799334723
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12665-0/0
│ │ │ +o1 = /tmp/M2-14765-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12665-0/0
│ │ │ +o2 = /tmp/M2-14765-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : m = fileMode fn
│ │ │  
│ │ │  o3 = 420
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Time.out
│ │ │ @@ -1,7 +1,7 @@
│ │ │  -- -*- M2-comint -*- hash: 1331310711075
│ │ │  
│ │ │  i1 : currentTime() - fileTime "."
│ │ │  
│ │ │ -o1 = 83
│ │ │ +o1 = 71
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_force__G__B_lp..._cm__Syzygy__Matrix_eq_gt..._rp.out
│ │ │ @@ -29,15 +29,15 @@
│ │ │       {4} | 0     x2-3  y3-1  |
│ │ │  
│ │ │               3      3
│ │ │  o6 : Matrix R  <-- R
│ │ │  
│ │ │  i7 : syz f
│ │ │  
│ │ │ -   -- registering gb 0 at 0x7fa67066e000
│ │ │ +   -- registering gb 0 at 0x7fa3403fa540
│ │ │  
│ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb elements = 3
│ │ │     -- number of monomials                = 9
│ │ │     -- #reduction steps = 6
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_get.out
│ │ │ @@ -10,11 +10,11 @@
│ │ │  
│ │ │  o2 = hi there
│ │ │  
│ │ │  i3 : removeFile "test-file"
│ │ │  
│ │ │  i4 : get "!date"
│ │ │  
│ │ │ -o4 = Mon May 18 12:33:37 UTC 2026
│ │ │ +o4 = Wed May 20 17:22:31 UTC 2026
│ │ │  
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │                 defaultPrecision => 53
│ │ │                 engineDebugLevel => 0
│ │ │                 errorDepth => 0
│ │ │                 gbTrace => 0
│ │ │                 interpreterDepth => 1
│ │ │                 lineNumber => 2
│ │ │                 loadDepth => 3
│ │ │ -               maxAllowableThreads => 7
│ │ │ +               maxAllowableThreads => 17
│ │ │                 maxExponent => 1073741823
│ │ │                 minExponent => -1073741824
│ │ │                 numTBBThreads => 0
│ │ │                 o1 => 2432902008176640000
│ │ │                 oo => 2432902008176640000
│ │ │                 printingAccuracy => -1
│ │ │                 printingLeadLimit => 5
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Directory.out
│ │ │ @@ -2,19 +2,19 @@
│ │ │  
│ │ │  i1 : isDirectory "."
│ │ │  
│ │ │  o1 = true
│ │ │  
│ │ │  i2 : fn = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11011-0/0
│ │ │ +o2 = /tmp/M2-11401-0/0
│ │ │  
│ │ │  i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11011-0/0
│ │ │ +o3 = /tmp/M2-11401-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : isDirectory fn
│ │ │  
│ │ │  o4 = false
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out
│ │ │ @@ -75,15 +75,15 @@
│ │ │  o17 = false
│ │ │  
│ │ │  i18 : isPrime(m*m*m1*m1*m2^6)
│ │ │  
│ │ │  o18 = false
│ │ │  
│ │ │  i19 : elapsedTime facs = factor(m*m1)
│ │ │ - -- 3.70437s elapsed
│ │ │ + -- 5.14056s elapsed
│ │ │  
│ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │  
│ │ │  o19 : Expression of class Product
│ │ │  
│ │ │  i20 : facs = facs//toList/toList
│ │ │  
│ │ │ @@ -97,17 +97,17 @@
│ │ │  
│ │ │  i22 : m3 = nextPrime (m^3)
│ │ │  
│ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │        00000000000000185613
│ │ │  
│ │ │  i23 : elapsedTime isPrime m3
│ │ │ - -- .0563182s elapsed
│ │ │ + -- .0602779s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000101339s elapsed
│ │ │ + -- .000124916s elapsed
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Regular__File.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 4782205245758053629
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12876-0/0
│ │ │ +o1 = /tmp/M2-15196-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12876-0/0
│ │ │ +o2 = /tmp/M2-15196-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : isRegularFile fn
│ │ │  
│ │ │  o3 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_make__Directory_lp__String_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 5113372159204571746
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11391-0/0
│ │ │ +o1 = /tmp/M2-12161-0/0
│ │ │  
│ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-11391-0/0/a/b/c
│ │ │ +o2 = /tmp/M2-12161-0/0/a/b/c
│ │ │  
│ │ │  i3 : removeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │  i4 : removeDirectory (dir|"/a/b")
│ │ │  
│ │ │  i5 : removeDirectory (dir|"/a")
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out
│ │ │ @@ -1,7 +1,7 @@
│ │ │  -- -*- M2-comint -*- hash: 1331887830690
│ │ │  
│ │ │  i1 : maxAllowableThreads
│ │ │  
│ │ │ -o1 = 7
│ │ │ +o1 = 17
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out
│ │ │ @@ -3,31 +3,31 @@
│ │ │  i1 : fib = n -> if n <= 1 then 1 else fib(n-1) + fib(n-2)
│ │ │  
│ │ │  o1 = fib
│ │ │  
│ │ │  o1 : FunctionClosure
│ │ │  
│ │ │  i2 : time fib 28
│ │ │ - -- used 1.62905s (cpu); 0.809345s (thread); 0s (gc)
│ │ │ + -- used 1.04466s (cpu); 0.621649s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
│ │ │  
│ │ │  i3 : fib = memoize fib
│ │ │  
│ │ │  o3 = fib
│ │ │  
│ │ │  o3 : FunctionClosure
│ │ │  
│ │ │  i4 : time fib 28
│ │ │ - -- used 6.8288e-05s (cpu); 6.7878e-05s (thread); 0s (gc)
│ │ │ + -- used 7.2385e-05s (cpu); 7.0083e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
│ │ │  
│ │ │  i5 : time fib 28
│ │ │ - -- used 4.348e-06s (cpu); 3.998e-06s (thread); 0s (gc)
│ │ │ + -- used 3.748e-06s (cpu); 3.183e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 514229
│ │ │  
│ │ │  i6 : fib = memoize( n -> fib(n-1) + fib(n-2) )
│ │ │  
│ │ │  o6 = fib
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods.out
│ │ │ @@ -17,20 +17,20 @@
│ │ │       {12 => (poincare, BettiTally)                                }
│ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {14 => (degree, BettiTally)                                  }
│ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │       {17 => (regularity, BettiTally)                              }
│ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ -     {19 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {20 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {21 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {22 => (codim, BettiTally)                                   }
│ │ │ -     {23 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ -     {24 => (dual, BettiTally)                                    }
│ │ │ +     {19 => (codim, BettiTally)                                   }
│ │ │ +     {20 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {21 => (dual, BettiTally)                                    }
│ │ │ +     {22 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ +     {23 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ +     {24 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
│ │ │  
│ │ │  i2 : methods resolution
│ │ │  
│ │ │  o2 = {0 => (freeResolution, Ideal)        }
│ │ │ @@ -60,20 +60,20 @@
│ │ │  
│ │ │  o4 = {0 => (++, Module, Module)}
│ │ │  
│ │ │  o4 : NumberedVerticalList
│ │ │  
│ │ │  i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │ -o5 = {0 => (+, Matrix, Matrix)                                   }
│ │ │ -     {1 => (-, Matrix, Matrix)                                   }
│ │ │ -     {2 => (contract, Matrix, Matrix)                            }
│ │ │ -     {3 => (diff', Matrix, Matrix)                               }
│ │ │ +o5 = {0 => (diff, Matrix, Matrix)                                }
│ │ │ +     {1 => (contract, Matrix, Matrix)                            }
│ │ │ +     {2 => (+, Matrix, Matrix)                                   }
│ │ │ +     {3 => (-, Matrix, Matrix)                                   }
│ │ │       {4 => (contract', Matrix, Matrix)                           }
│ │ │ -     {5 => (diff, Matrix, Matrix)                                }
│ │ │ +     {5 => (diff', Matrix, Matrix)                               }
│ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ @@ -90,16 +90,16 @@
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │       {24 => (%, Matrix, Matrix)                                  }
│ │ │       {25 => (remainder, Matrix, Matrix)                          }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │       {28 => (tensor, Matrix, Matrix)                             }
│ │ │       {29 => (intersect, Matrix, Matrix)                          }
│ │ │ -     {30 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ -     {31 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {30 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {31 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = minimalBetti I
│ │ │ - -- 1.8029s elapsed
│ │ │ + -- 2.39876s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -26,44 +26,44 @@
│ │ │  o3 : BettiTally
│ │ │  
│ │ │  i4 : I = ideal I_*;
│ │ │  
│ │ │  o4 : Ideal of S
│ │ │  
│ │ │  i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ - -- .759513s elapsed
│ │ │ + -- 1.00799s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7
│ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │           0: 1  .   .   .   .    .   .   .
│ │ │           1: . 35 140 189  84    .   .   .
│ │ │           2: .  .   . 196 735 1080 735 196
│ │ │  
│ │ │  o5 : BettiTally
│ │ │  
│ │ │  i6 : I = ideal I_*;
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │  
│ │ │  i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ - -- .0955615s elapsed
│ │ │ + -- .0418058s elapsed
│ │ │  
│ │ │              0  1   2   3  4
│ │ │  o7 = total: 1 35 140 189 84
│ │ │           0: 1  .   .   .  .
│ │ │           1: . 35 140 189 84
│ │ │  
│ │ │  o7 : BettiTally
│ │ │  
│ │ │  i8 : I = ideal I_*;
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │  
│ │ │  i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ - -- 1.28856s elapsed
│ │ │ + -- 1.64957s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5
│ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │           0: 1  .   .   .   .    .
│ │ │           1: . 35 140 189  84    .
│ │ │           2: .  .   . 196 735 1080
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mkdir.out
│ │ │ @@ -1,22 +1,22 @@
│ │ │  -- -*- M2-comint -*- hash: 15555226809509933135
│ │ │  
│ │ │  i1 : p = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11410-0/0/
│ │ │ +o1 = /tmp/M2-12200-0/0/
│ │ │  
│ │ │  i2 : mkdir p
│ │ │  
│ │ │  i3 : isDirectory p
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11410-0/0/foo
│ │ │ +o4 = /tmp/M2-12200-0/0/foo
│ │ │  
│ │ │  o4 : File
│ │ │  
│ │ │  i5 : get fn
│ │ │  
│ │ │  o5 = hi there
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_move__File_lp__String_cm__String_rp.out
│ │ │ @@ -1,31 +1,31 @@
│ │ │  -- -*- M2-comint -*- hash: 4857944042471093218
│ │ │  
│ │ │  i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11246-0/0
│ │ │ +o1 = /tmp/M2-11876-0/0
│ │ │  
│ │ │  i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11246-0/1
│ │ │ +o2 = /tmp/M2-11876-0/1
│ │ │  
│ │ │  i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11246-0/0
│ │ │ +o3 = /tmp/M2-11876-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-11246-0/0 -> /tmp/M2-11246-0/1
│ │ │ +--moving: /tmp/M2-11876-0/0 -> /tmp/M2-11876-0/1
│ │ │  
│ │ │  i5 : get dst
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-11246-0/1.bak
│ │ │ +--backup file created: /tmp/M2-11876-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-11246-0/1.bak
│ │ │ +o6 = /tmp/M2-11876-0/1.bak
│ │ │  
│ │ │  i7 : removeFile bak
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_nanosleep.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1331114612441
│ │ │  
│ │ │  i1 : elapsedTime nanosleep 500000000
│ │ │ - -- .500144s elapsed
│ │ │ + -- .500162s elapsed
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_options_lp__Function_rp.out
│ │ │ @@ -20,34 +20,34 @@
│ │ │  
│ │ │  o3 = OptionTable{Generic => false}
│ │ │  
│ │ │  o3 : OptionTable
│ │ │  
│ │ │  i4 : methods codim
│ │ │  
│ │ │ -o4 = {0 => (codim, Module)        }
│ │ │ -     {1 => (codim, CoherentSheaf) }
│ │ │ -     {2 => (codim, Variety)       }
│ │ │ -     {3 => (codim, MonomialIdeal) }
│ │ │ -     {4 => (codim, Ideal)         }
│ │ │ -     {5 => (codim, PolynomialRing)}
│ │ │ -     {6 => (codim, BettiTally)    }
│ │ │ -     {7 => (codim, QuotientRing)  }
│ │ │ +o4 = {0 => (codim, Ideal)         }
│ │ │ +     {1 => (codim, PolynomialRing)}
│ │ │ +     {2 => (codim, BettiTally)    }
│ │ │ +     {3 => (codim, QuotientRing)  }
│ │ │ +     {4 => (codim, Module)        }
│ │ │ +     {5 => (codim, CoherentSheaf) }
│ │ │ +     {6 => (codim, Variety)       }
│ │ │ +     {7 => (codim, MonomialIdeal) }
│ │ │  
│ │ │  o4 : NumberedVerticalList
│ │ │  
│ │ │  i5 : options oo
│ │ │  
│ │ │  o5 = {0 => (OptionTable{Generic => false})}
│ │ │       {1 => (OptionTable{Generic => false})}
│ │ │ -     {2 => (OptionTable{Generic => false})}
│ │ │ +     {2 => (OptionTable{})                }
│ │ │       {3 => (OptionTable{Generic => false})}
│ │ │       {4 => (OptionTable{Generic => false})}
│ │ │       {5 => (OptionTable{Generic => false})}
│ │ │ -     {6 => (OptionTable{})                }
│ │ │ +     {6 => (OptionTable{Generic => false})}
│ │ │       {7 => (OptionTable{Generic => false})}
│ │ │  
│ │ │  o5 : NumberedVerticalList
│ │ │  
│ │ │  i6 : methods intersect
│ │ │  
│ │ │  o6 = {0 => (intersect, List)                           }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out
│ │ │ @@ -5,26 +5,26 @@
│ │ │  o1 = {1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : L = shuffle toList (1..10000);
│ │ │  
│ │ │  i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ - -- .58009s elapsed
│ │ │ + -- .692008s elapsed
│ │ │  
│ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .275306s elapsed
│ │ │ + -- .186927s elapsed
│ │ │  
│ │ │  i5 : allowableThreads
│ │ │  
│ │ │  o5 = 5
│ │ │  
│ │ │  i6 : allowableThreads = maxAllowableThreads
│ │ │  
│ │ │ -o6 = 7
│ │ │ +o6 = 17
│ │ │  
│ │ │  i7 : R = QQ[x,y,z];
│ │ │  
│ │ │  i8 : I = ideal(x^2 + 2*y^2 - y - 2*z, x^2 - 8*y^2 + 10*z - 1, x^2 - 7*y*z)
│ │ │  
│ │ │               2     2            2     2             2
│ │ │  o8 = ideal (x  + 2y  - y - 2z, x  - 8y  + 10z - 1, x  - 7y*z)
│ │ │ @@ -41,15 +41,15 @@
│ │ │  
│ │ │  o10 = <>
│ │ │  
│ │ │  o10 : Task
│ │ │  
│ │ │  i11 : t
│ │ │  
│ │ │ -o11 = <>
│ │ │ +o11 = <>
│ │ │  
│ │ │  o11 : Task
│ │ │  
│ │ │  i12 : isReady t
│ │ │  
│ │ │  o12 = false
│ │ │  
│ │ │ @@ -74,15 +74,15 @@
│ │ │  
│ │ │  o16 : Task
│ │ │  
│ │ │  i17 : schedule t';
│ │ │  
│ │ │  i18 : t'
│ │ │  
│ │ │ -o18 = <>
│ │ │ +o18 = <>
│ │ │  
│ │ │  o18 : Task
│ │ │  
│ │ │  i19 : taskResult t'
│ │ │  
│ │ │  o19 = | 980z2-18y-201z+13 35yz-4y+2z-1 10y2-y-12z+1 5x2-4y+2z-1 |
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out
│ │ │ @@ -67,15 +67,15 @@
│ │ │  i3 : S = ring I
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : elapsedTime minimalBetti I
│ │ │ - -- 1.88605s elapsed
│ │ │ + -- 2.44512s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -84,15 +84,15 @@
│ │ │  o4 : BettiTally
│ │ │  
│ │ │  i5 : I = ideal I_*;
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ - -- 1.73807s elapsed
│ │ │ + -- 2.41731s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o7 : Ideal of S
│ │ │  
│ │ │  i8 : numTBBThreads = 1
│ │ │  
│ │ │  o8 = 1
│ │ │  
│ │ │  i9 : elapsedTime minimalBetti(I)
│ │ │ - -- 1.63859s elapsed
│ │ │ + -- 2.50868s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -132,15 +132,15 @@
│ │ │  o11 = 0
│ │ │  
│ │ │  i12 : I = ideal I_*;
│ │ │  
│ │ │  o12 : Ideal of S
│ │ │  
│ │ │  i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.16228s elapsed
│ │ │ + -- 2.76842s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o13 : Complex
│ │ │ @@ -150,15 +150,15 @@
│ │ │  o14 = 1
│ │ │  
│ │ │  i15 : I = ideal I_*;
│ │ │  
│ │ │  o15 : Ideal of S
│ │ │  
│ │ │  i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.01403s elapsed
│ │ │ + -- 2.86879s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o16 : Complex
│ │ │ @@ -174,43 +174,43 @@
│ │ │  o18 : PolynomialRing
│ │ │  
│ │ │  i19 : I = ideal random(S^1, S^{4:-5});
│ │ │  
│ │ │  o19 : Ideal of S
│ │ │  
│ │ │  i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.83683s elapsed
│ │ │ + -- 4.34814s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o20 : Matrix S  <-- S
│ │ │  
│ │ │  i21 : numTBBThreads = 1
│ │ │  
│ │ │  o21 = 1
│ │ │  
│ │ │  i22 : I = ideal I_*;
│ │ │  
│ │ │  o22 : Ideal of S
│ │ │  
│ │ │  i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 7.15932s elapsed
│ │ │ + -- 8.45343s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o23 : Matrix S  <-- S
│ │ │  
│ │ │  i24 : numTBBThreads = 10
│ │ │  
│ │ │  o24 = 10
│ │ │  
│ │ │  i25 : I = ideal I_*;
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │  
│ │ │  i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 5.9065s elapsed
│ │ │ + -- 3.74261s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o26 : Matrix S  <-- S
│ │ │  
│ │ │  i27 : needsPackage "AssociativeAlgebras"
│ │ │  
│ │ │  o27 = AssociativeAlgebras
│ │ │ @@ -233,15 +233,15 @@
│ │ │  o30 = ideal (5a  + 2b*c + 3c*b, 3a*c + 5b  + 2c*a, 2a*b + 3b*a + 5c )
│ │ │  
│ │ │                  ZZ
│ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │                 101
│ │ │  
│ │ │  i31 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.0633s elapsed
│ │ │ + -- 1.06129s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
│ │ │  
│ │ │  i32 : I = ideal I_*
│ │ │  
│ │ │ @@ -253,14 +253,14 @@
│ │ │                 101
│ │ │  
│ │ │  i33 : numTBBThreads = 1
│ │ │  
│ │ │  o33 = 1
│ │ │  
│ │ │  i34 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.22532s elapsed
│ │ │ + -- 1.67904s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
│ │ │  
│ │ │  i35 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out
│ │ │ @@ -146,65 +146,65 @@
│ │ │  o26 : ZZ[T]
│ │ │  
│ │ │  i27 : gbTrace = 3
│ │ │  
│ │ │  o27 = 3
│ │ │  
│ │ │  i28 : time poincare I
│ │ │ - -- used 0.00344891s (cpu); 1.8505e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00296572s (cpu); 1.3502e-05s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
│ │ │  
│ │ │  i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 16 at 0x7f3752420380
│ │ │ +   -- registering gb 16 at 0x7fa3a54508c0
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 4186
│ │ │     -- #reduction steps = 38
│ │ │     -- #spairs done = 11
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 29
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0165272s (cpu); 0.0183577s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00903562s (cpu); 0.0120044s (thread); 0s (gc)
│ │ │  
│ │ │                1      11
│ │ │  o29 : Matrix R  <-- R
│ │ │  
│ │ │  i30 : R = QQ[a..d];
│ │ │  
│ │ │  i31 : I = ideal random(R^1, R^{3:-3});
│ │ │  
│ │ │ -   -- registering gb 17 at 0x7f37524201c0
│ │ │ +   -- registering gb 17 at 0x7fa3a5450700
│ │ │  
│ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │     -- number of monomials                = 0
│ │ │     -- #reduction steps = 0
│ │ │     -- #spairs done = 0
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │     -- nsaved = 0
│ │ │     -- 
│ │ │  o31 : Ideal of R
│ │ │  
│ │ │  i32 : time p = poincare I
│ │ │  
│ │ │ -   -- registering gb 18 at 0x7f3752420000
│ │ │ +   -- registering gb 18 at 0x7fa3a5450540
│ │ │  
│ │ │     -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 267
│ │ │     -- #reduction steps = 236
│ │ │     -- #spairs done = 30
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 20
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.00799635s (cpu); 0.00712039s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00398872s (cpu); 0.00434384s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
│ │ │  
│ │ │  i33 : S = QQ[a..d, MonomialOrder => Eliminate 2]
│ │ │ @@ -254,27 +254,27 @@
│ │ │  
│ │ │  i36 : gbTrace = 3
│ │ │  
│ │ │  o36 = 3
│ │ │  
│ │ │  i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7f3754432e00
│ │ │ +   -- registering gb 19 at 0x7fa3a5450380
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}(3,9)m
│ │ │     -- mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
│ │ │     -- {17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m{24}(1,3)m
│ │ │     -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements = 39
│ │ │     -- number of monomials                = 1051
│ │ │     -- #reduction steps = 284
│ │ │     -- #spairs done = 53
│ │ │     -- ncalls = 46
│ │ │     -- nloop = 54
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0759331s (cpu); 0.0769218s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0520013s (cpu); 0.0517201s (thread); 0s (gc)
│ │ │  
│ │ │                1      39
│ │ │  o37 : Matrix S  <-- S
│ │ │  
│ │ │  i38 : selectInSubring(1, gens gb J)
│ │ │  
│ │ │  o38 = | 188529931266160087758259645374082357642621166724936033369975727480205
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_printing_spto_spa_spfile.out
│ │ │ @@ -12,19 +12,19 @@
│ │ │  
│ │ │  o2 = stdio
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fn = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11601-0/0
│ │ │ +o3 = /tmp/M2-12591-0/0
│ │ │  
│ │ │  i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11601-0/0
│ │ │ +o4 = /tmp/M2-12591-0/0
│ │ │  
│ │ │  o4 : File
│ │ │  
│ │ │  i5 : get fn
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │ @@ -49,27 +49,27 @@
│ │ │  x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x + 1
│ │ │  o8 = stdio
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : fn << f << close
│ │ │  
│ │ │ -o9 = /tmp/M2-11601-0/0
│ │ │ +o9 = /tmp/M2-12591-0/0
│ │ │  
│ │ │  o9 : File
│ │ │  
│ │ │  i10 : get fn
│ │ │  
│ │ │  o10 =  10      9      8       7       6       5       4       3      2
│ │ │        x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x
│ │ │        + 1
│ │ │  
│ │ │  i11 : fn << toExternalString f << close
│ │ │  
│ │ │ -o11 = /tmp/M2-11601-0/0
│ │ │ +o11 = /tmp/M2-12591-0/0
│ │ │  
│ │ │  o11 : File
│ │ │  
│ │ │  i12 : get fn
│ │ │  
│ │ │  o12 = x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+
│ │ │        1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_process__I__D.out
│ │ │ @@ -1,7 +1,7 @@
│ │ │  -- -*- M2-comint -*- hash: 1330513630563
│ │ │  
│ │ │  i1 : processID()
│ │ │  
│ │ │ -o1 = 10822
│ │ │ +o1 = 11042
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_profile.out
│ │ │ @@ -9,35 +9,35 @@
│ │ │  
│ │ │                4       5
│ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i2 : profileSummary
│ │ │  
│ │ │  o2 = #run  %time   position                         
│ │ │ -     1     94.41   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ -     1     91.82   ../../m2/matrix1.m2:275:22-275:43
│ │ │ -     1     44.77   ../../m2/matrix1.m2:183:25-183:52
│ │ │ -     1     30.99   ../../m2/matrix1.m2:104:5-146:72 
│ │ │ -     1     29.81   ../../m2/matrix1.m2:130:10-145:16
│ │ │ -     1     22.38   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ -     1     21.25   ../../m2/matrix1.m2:35:10-39:22  
│ │ │ -     1     21.09   ../../m2/set.m2:129:5-129:61     
│ │ │ -     1     3.31    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ -     1     2.27    ../../m2/matrix1.m2:131:13-131:78
│ │ │ -     1     2.12    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ -     1     1.43    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ -     1     1.35    ../../m2/matrix1.m2:137:20-137:64
│ │ │ -     1     1.2     ../../m2/matrix1.m2:270:4-271:73 
│ │ │ -     1     1.14    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ -     1     1.10    ../../m2/matrix1.m2:88:10-88:46  
│ │ │ -     1     1.03    ../../m2/matrix1.m2:172:4-174:74 
│ │ │ -     1     .77     ../../m2/modules.m2:282:4-282:52 
│ │ │ -     20    .62     ../../m2/matrix1.m2:181:14-182:67
│ │ │ -     20    .53     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ -     1     .0046s  elapsed total                    
│ │ │ +     1     93.28   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ +     1     90.3    ../../m2/matrix1.m2:275:22-275:43
│ │ │ +     1     43.71   ../../m2/matrix1.m2:183:25-183:52
│ │ │ +     1     30.68   ../../m2/matrix1.m2:104:5-146:72 
│ │ │ +     1     29.53   ../../m2/matrix1.m2:130:10-145:16
│ │ │ +     1     22.96   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ +     1     21.42   ../../m2/set.m2:129:5-129:61     
│ │ │ +     1     21.16   ../../m2/matrix1.m2:35:10-39:22  
│ │ │ +     1     3.14    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ +     1     2.40    ../../m2/matrix1.m2:131:13-131:78
│ │ │ +     1     1.91    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ +     1     1.49    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ +     1     1.32    ../../m2/matrix1.m2:137:20-137:64
│ │ │ +     1     1.31    ../../m2/matrix1.m2:270:4-271:73 
│ │ │ +     1     1.08    ../../m2/matrix1.m2:172:4-174:74 
│ │ │ +     1     1.05    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ +     1     .98     ../../m2/matrix1.m2:88:10-88:46  
│ │ │ +     20    .95     ../../m2/matrix1.m2:181:14-182:67
│ │ │ +     19    .71     ../../m2/set.m2:129:36-129:41    
│ │ │ +     20    .68     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ +     1     .0039s  elapsed total                    
│ │ │  
│ │ │  i3 : coverageSummary
│ │ │  
│ │ │  o3 = covered lines:
│ │ │       ../../m2/lists.m2:146:24-146:32
│ │ │       ../../m2/lists.m2:146:34-146:58
│ │ │       ../../m2/matrix.m2:30:5-30:35
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │  i5 : codim I, degree I
│ │ │  
│ │ │  o5 = (2, 10)
│ │ │  
│ │ │  o5 : Sequence
│ │ │  
│ │ │  i6 : time randomKRationalPoint(I)
│ │ │ - -- used 0.441238s (cpu); 0.177282s (thread); 0s (gc)
│ │ │ + -- used 0.585026s (cpu); 0.154271s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal (x  - 53x , x  + 8x , x  - 4x )
│ │ │               2      3   1     3   0     3
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : R=kk[x_0..x_5];
│ │ │ @@ -33,15 +33,15 @@
│ │ │  i9 : codim I, degree I
│ │ │  
│ │ │  o9 = (3, 10)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : time randomKRationalPoint(I)
│ │ │ - -- used 0.254059s (cpu); 0.19724s (thread); 0s (gc)
│ │ │ + -- used 0.455107s (cpu); 0.273439s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : p=10007,kk=ZZ/p,R=kk[x_0..x_2]
│ │ │ @@ -58,12 +58,12 @@
│ │ │  
│ │ │  i14 : I=ideal random(n,R);
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c->degree c);
│ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ - -- used 4.21024s (cpu); 2.05106s (thread); 0s (gc)
│ │ │ + -- used 4.65365s (cpu); 2.24557s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 58
│ │ │  
│ │ │  i16 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_read__Directory.out
│ │ │ @@ -1,26 +1,26 @@
│ │ │  -- -*- M2-comint -*- hash: 20910736704070514
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12234-0/0
│ │ │ +o1 = /tmp/M2-13884-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-12234-0/0
│ │ │ +o2 = /tmp/M2-13884-0/0
│ │ │  
│ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-12234-0/0/foo
│ │ │ +o3 = /tmp/M2-13884-0/0/foo
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : readDirectory dir
│ │ │  
│ │ │ -o4 = {., .., foo}
│ │ │ +o4 = {.., ., foo}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : removeFile fn
│ │ │  
│ │ │  i6 : removeDirectory dir
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_reading_spfiles.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 13513555104200944796
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11776-0/0
│ │ │ +o1 = /tmp/M2-12946-0/0
│ │ │  
│ │ │  i2 : fn << "z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3" << endl << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11776-0/0
│ │ │ +o2 = /tmp/M2-12946-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : get fn
│ │ │  
│ │ │  o3 = z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2
│ │ │       +8*y^3
│ │ │ @@ -38,15 +38,15 @@
│ │ │  
│ │ │  o6 : Expression of class Product
│ │ │  
│ │ │  i7 : fn << "sample = 2^100
│ │ │       print sample
│ │ │       " << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11776-0/0
│ │ │ +o7 = /tmp/M2-12946-0/0
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : get fn
│ │ │  
│ │ │  o8 = sample = 2^100
│ │ │       print sample
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_readlink.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 4408639611478781130
│ │ │  
│ │ │  i1 : p = temporaryFileName ()
│ │ │  
│ │ │ -o1 = /tmp/M2-12494-0/0
│ │ │ +o1 = /tmp/M2-14404-0/0
│ │ │  
│ │ │  i2 : symlinkFile ("foo", p)
│ │ │  
│ │ │  i3 : readlink p
│ │ │  
│ │ │  o3 = foo
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_realpath.out
│ │ │ @@ -1,39 +1,39 @@
│ │ │  -- -*- M2-comint -*- hash: 324072347213224656
│ │ │  
│ │ │  i1 : realpath "."
│ │ │  
│ │ │ -o1 = /tmp/M2-10822-0/89-rundir/
│ │ │ +o1 = /tmp/M2-11042-0/89-rundir/
│ │ │  
│ │ │  i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-12513-0/0
│ │ │ +o2 = /tmp/M2-14443-0/0
│ │ │  
│ │ │  i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-12513-0/1
│ │ │ +o3 = /tmp/M2-14443-0/1
│ │ │  
│ │ │  i4 : symlinkFile(p,q)
│ │ │  
│ │ │  i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-12513-0/0
│ │ │ +o5 = /tmp/M2-14443-0/0
│ │ │  
│ │ │  o5 : File
│ │ │  
│ │ │  i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-12513-0/0
│ │ │ +o6 = /tmp/M2-14443-0/0
│ │ │  
│ │ │  i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-12513-0/0
│ │ │ +o7 = /tmp/M2-14443-0/0
│ │ │  
│ │ │  i8 : removeFile p
│ │ │  
│ │ │  i9 : removeFile q
│ │ │  
│ │ │  i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-10822-0/89-rundir/
│ │ │ +o10 = /tmp/M2-11042-0/89-rundir/
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_register__Finalizer.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 1729384374372662693
│ │ │  
│ │ │  i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --"))
│ │ │  
│ │ │  i2 : collectGarbage() 
│ │ │ ---finalization: (1)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (2)[8]: -- finalizing sequence #9 --
│ │ │ +--finalization: (1)[6]: -- finalizing sequence #7 --
│ │ │ +--finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │  --finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (4)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (5)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (6)[1]: -- finalizing sequence #2 --
│ │ │ ---finalization: (7)[7]: -- finalizing sequence #8 --
│ │ │ +--finalization: (4)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (5)[7]: -- finalizing sequence #8 --
│ │ │ +--finalization: (6)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (7)[2]: -- finalizing sequence #3 --
│ │ │  --finalization: (8)[0]: -- finalizing sequence #1 --
│ │ │ ---finalization: (9)[6]: -- finalizing sequence #7 --
│ │ │ +--finalization: (9)[8]: -- finalizing sequence #9 --
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remove__Directory.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 8532980310097060089
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11448-0/0
│ │ │ +o1 = /tmp/M2-12278-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11448-0/0
│ │ │ +o2 = /tmp/M2-12278-0/0
│ │ │  
│ │ │  i3 : readDirectory dir
│ │ │  
│ │ │ -o3 = {., ..}
│ │ │ +o3 = {.., .}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : removeDirectory dir
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_root__Path.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1731420232148149387
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10914-0/0
│ │ │ +o1 = /tmp/M2-11204-0/0
│ │ │  
│ │ │  i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-10914-0/0
│ │ │ +o2 = /tmp/M2-11204-0/0
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_root__U__R__I.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1731420231525572968
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12177-0/0
│ │ │ +o1 = /tmp/M2-13767-0/0
│ │ │  
│ │ │  i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-12177-0/0
│ │ │ +o2 = file:///tmp/M2-13767-0/0
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_saving_sppolynomials_spand_spmatrices_spin_spfiles.out
│ │ │ @@ -25,19 +25,19 @@
│ │ │  o4 = image | x2 x2-y2 xyz7 |
│ │ │  
│ │ │                               1
│ │ │  o4 : R-module, submodule of R
│ │ │  
│ │ │  i5 : f = temporaryFileName()
│ │ │  
│ │ │ -o5 = /tmp/M2-12025-0/0
│ │ │ +o5 = /tmp/M2-13455-0/0
│ │ │  
│ │ │  i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12025-0/0
│ │ │ +o6 = /tmp/M2-13455-0/0
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : get f
│ │ │  
│ │ │  o7 = (x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix {{x^2,
│ │ │       x^2-y^2, x*y*z^7}},image matrix {{x^2, x^2-y^2, x*y*z^7}})
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out
│ │ │ @@ -189,18 +189,18 @@
│ │ │  o25 = 40
│ │ │  
│ │ │  i26 : A = mutableMatrix(CC_53, N, N); fillMatrix A;
│ │ │  
│ │ │  i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
│ │ │  
│ │ │  i30 : time X = solve(A,B);
│ │ │ - -- used 0.000195457s (cpu); 0.00018674s (thread); 0s (gc)
│ │ │ + -- used 0.000252739s (cpu); 0.000243923s (thread); 0s (gc)
│ │ │  
│ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000140844s (cpu); 0.000141115s (thread); 0s (gc)
│ │ │ + -- used 0.000122025s (cpu); 0.000121211s (thread); 0s (gc)
│ │ │  
│ │ │  i32 : norm(A*X-B)
│ │ │  
│ │ │  o32 = 5.111850690840453e-15
│ │ │  
│ │ │  o32 : RR (of precision 53)
│ │ │  
│ │ │ @@ -209,18 +209,18 @@
│ │ │  o33 = 100
│ │ │  
│ │ │  i34 : A = mutableMatrix(CC_100, N, N); fillMatrix A;
│ │ │  
│ │ │  i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
│ │ │  
│ │ │  i38 : time X = solve(A,B);
│ │ │ - -- used 0.239427s (cpu); 0.239428s (thread); 0s (gc)
│ │ │ + -- used 0.148123s (cpu); 0.148126s (thread); 0s (gc)
│ │ │  
│ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.222489s (cpu); 0.222489s (thread); 0s (gc)
│ │ │ + -- used 0.146445s (cpu); 0.146458s (thread); 0s (gc)
│ │ │  
│ │ │  i40 : norm(A*X-B)
│ │ │  
│ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │  
│ │ │  o40 : RR (of precision 100)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_symlink__Directory_lp__String_cm__String_rp.out
│ │ │ @@ -1,60 +1,60 @@
│ │ │  -- -*- M2-comint -*- hash: 2989513528213557691
│ │ │  
│ │ │  i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11816-0/0/
│ │ │ +o1 = /tmp/M2-13026-0/0/
│ │ │  
│ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11816-0/1/
│ │ │ +o2 = /tmp/M2-13026-0/1/
│ │ │  
│ │ │  i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11816-0/0/a/
│ │ │ +o3 = /tmp/M2-13026-0/0/a/
│ │ │  
│ │ │  i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11816-0/0/b/
│ │ │ +o4 = /tmp/M2-13026-0/0/b/
│ │ │  
│ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11816-0/0/b/c/
│ │ │ +o5 = /tmp/M2-13026-0/0/b/c/
│ │ │  
│ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11816-0/0/a/f
│ │ │ +o6 = /tmp/M2-13026-0/0/a/f
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11816-0/0/a/g
│ │ │ +o7 = /tmp/M2-13026-0/0/a/g
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11816-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13026-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11816-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11816-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11816-0/1/a/f
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-13026-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-13026-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-13026-0/1/b/c/g
│ │ │  
│ │ │  i10 : get (dst|"b/c/g")
│ │ │  
│ │ │  o10 = ho there
│ │ │  
│ │ │  i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
│ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-11816-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11816-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11816-0/1/a/f
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-13026-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-13026-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-13026-0/1/b/c/g
│ │ │  
│ │ │  i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │  
│ │ │  o12 = rm
│ │ │  
│ │ │  o12 : FunctionClosure
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_symlink__File.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 9343844672940306595
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11873-0/0
│ │ │ +o1 = /tmp/M2-13143-0/0
│ │ │  
│ │ │  i2 : symlinkFile("qwert", fn)
│ │ │  
│ │ │  i3 : fileExists fn
│ │ │  
│ │ │  o3 = false
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_temporary__File__Name.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1731926531291302106
│ │ │  
│ │ │  i1 : temporaryFileName () | ".tex"
│ │ │  
│ │ │ -o1 = /tmp/M2-12857-0/0.tex
│ │ │ +o1 = /tmp/M2-15157-0/0.tex
│ │ │  
│ │ │  i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-12857-0/1.html
│ │ │ +o2 = /tmp/M2-15157-0/1.html
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1332435500723
│ │ │  
│ │ │  i1 : time 3^30
│ │ │ - -- used 2.0629e-05s (cpu); 1.2153e-05s (thread); 0s (gc)
│ │ │ + -- used 1.8835e-05s (cpu); 6.374e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_timing.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 1730988300469098603
│ │ │  
│ │ │  i1 : timing 3^30
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │ -     -- .000015449 seconds
│ │ │ +     -- .000019145 seconds
│ │ │  
│ │ │  o1 : Time
│ │ │  
│ │ │  i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000015449, 205891132094649}
│ │ │ +o2 = Time{.000019145, 205891132094649}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_version.out
│ │ │ @@ -36,15 +36,15 @@
│ │ │                 "memtailor version" => 1.3
│ │ │                 "mpfi version" => 1.5.4
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.11.1
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.88+deb13-amd64
│ │ │ +               "operating system release" => 6.12.88+deb13-cloud-amd64
│ │ │                 "operating system" => Linux
│ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone ChainComplexExtras Varieties Schubert2 PushForward LocalRings BoijSoederberg BGG Bruns InvolutiveBases ConwayPolynomials EdgeIdeals FourTiTwo StatePolytope Polyhedra Truncations Polymake gfanInterface PieriMaps Normaliz Posets XML OpenMath SCSCP RationalPoints MapleInterface ConvexInterface SRdeformations NumericalAlgebraicGeometry BeginningMacaulay2 FormalGroupLaws Graphics WeylGroups HodgeIntegrals Cyclotomic Binomials Kronecker Nauty ToricVectorBundles ModuleDeformations PHCpack SimplicialDecomposability BooleanGB AdjointIdeal Parametrization Serialization NAGtypes NormalToricVarieties DGAlgebras Graphs GraphicalModels BIBasis KustinMiller Units NautyGraphs VersalDeformations CharacteristicClasses RandomIdeals RandomObjects RandomPlaneCurves RandomSpaceCurves RandomGenus14Curves RandomCanonicalCurves RandomCurves TensorComplexes MonomialAlgebras QthPower EliminationMatrices EllipticIntegrals Triplets CompleteIntersectionResolutions EagonResolution MCMApproximations MultiplierIdeals InvariantRing QuillenSuslin EnumerationCurves Book3264Examples WeilDivisors EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieAlgebraRepresentations ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties TestIdeals FrobeniusThresholds NonPrincipalTestIdeals Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes OldChainComplexes GroebnerWalk RandomMonomialIdeals Matroids NumericalImplicitization CoincidentRootLoci RelativeCanonicalResolution RandomCurvesOverVerySmallFiniteFields StronglyStableIdeals SLnEquivariantMatrices CorrespondenceScrolls NCAlgebra SpaceCurves ExteriorIdeals ToricInvariants SegreClasses SemidefiniteProgramming SumsOfSquares MultiGradedRationalMap AssociativeAlgebras VirtualResolutions Quasidegrees DiffAlg DeterminantalRepresentations FGLM SpechtModule SchurComplexes SimplicialPosets SlackIdeals PositivityToricBundles SparseResultants DecomposableSparseSystems MixedMultiplicity PencilsOfQuadrics ThreadedGB AdjunctionForSurfaces VectorGraphics GKMVarieties MonomialIntegerPrograms NoetherianOperators Hadamard StatGraphs GraphicalModelsMLE EigenSolver MultiplicitySequence ResolutionsOfStanleyReisnerRings NumericalLinearAlgebra ResLengthThree MonomialOrbits MultiprojectiveVarieties SpecialFanoFourfolds RationalPoints2 SuperLinearAlgebra SubalgebraBases AInfinity LinearTruncations ThinSincereQuivers Python BettiCharacters Jets FunctionFieldDesingularization HomotopyLieAlgebra TSpreadIdeals RealRoots ExteriorModules K3Surfaces GroebnerStrata QuaternaryQuartics CotangentSchubert OnlineLookup MergeTeX Probability Isomorphism CodingTheory WhitneyStratifications JSON ForeignFunctions GeometricDecomposability PseudomonomialPrimaryDecomposition PolyominoIdeals MatchingFields CellularResolutions SagbiGbDetection A1BrouwerDegrees QuadraticIdealExamplesByRoos TerraciniLoci MatrixSchubert RInterface OIGroebnerBases PlaneCurveLinearSeries Valuations SchurVeronese VNumber TropicalToric MultigradedBGG AbstractSimplicialComplexes MultigradedImplicitization Msolve Permutations SCMAlgebras NumericalSemigroups ExteriorExtensions Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases Tableaux CpMackeyFunctors JSONRPC SimplicialModules MatrixFactorizations PathSignatures MacaulayPosets MRDI EliminationTemplates WittVectors Padic
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.13.12
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2022.3
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Command.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │              
    		
│ │ │                
    i3 : (c = Command "date";)
    
│ │ │              
    
│ │ │                
    i4 : c
│ │ │ -Mon May 18 12:32:59 UTC 2026
│ │ │ +Wed May 20 17:22:02 UTC 2026
│ │ │  
│ │ │  o4 = 0
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  in a file), then it gets executed with empty argument list.
│ │ │ │  i1 : (f = Command ( () -> 2^30 );)
│ │ │ │  i2 : f
│ │ │ │  
│ │ │ │  o2 = 1073741824
│ │ │ │  i3 : (c = Command "date";)
│ │ │ │  i4 : c
│ │ │ │ -Mon May 18 12:32:59 UTC 2026
│ │ │ │ +Wed May 20 17:22:02 UTC 2026
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_u_n -- run an external command
│ │ │ │      * _A_f_t_e_r_E_v_a_l -- top level method applied after evaluation
│ │ │ │  ********** MMeetthhooddss tthhaatt uussee aa ccoommmmaanndd:: **********
│ │ │ │      * code(Command) -- see _c_o_d_e -- display source code
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Database.html
│ │ │ @@ -57,22 +57,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  A database file is just like a hash table, except both the keys and values have to be strings.  In this example we create a database file, store a few entries, remove an entry with remove, close the file, and then remove the file.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : filename = temporaryFileName () | ".dbm"
│ │ │  
│ │ │ -o1 = /tmp/M2-12329-0/0.dbm
    
│ │ │ +o1 = /tmp/M2-14079-0/0.dbm
│ │ │              
    
│ │ │                
    i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-12329-0/0.dbm
│ │ │ +o2 = /tmp/M2-14079-0/0.dbm
│ │ │  
│ │ │  o2 : Database
    
│ │ │              
    
│ │ │                
    i3 : x#"first" = "hi there"
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,18 +7,18 @@
│ │ │ │  ************ DDaattaabbaassee ---- tthhee ccllaassss ooff aallll ddaattaabbaassee ffiilleess ************
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  A database file is just like a hash table, except both the keys and values have
│ │ │ │  to be strings. In this example we create a database file, store a few entries,
│ │ │ │  remove an entry with _r_e_m_o_v_e, close the file, and then remove the file.
│ │ │ │  i1 : filename = temporaryFileName () | ".dbm"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12329-0/0.dbm
│ │ │ │ +o1 = /tmp/M2-14079-0/0.dbm
│ │ │ │  i2 : x = openDatabaseOut filename
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12329-0/0.dbm
│ │ │ │ +o2 = /tmp/M2-14079-0/0.dbm
│ │ │ │  
│ │ │ │  o2 : Database
│ │ │ │  i3 : x#"first" = "hi there"
│ │ │ │  
│ │ │ │  o3 = hi there
│ │ │ │  i4 : x#"first"
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__Cstats.html
│ │ │ @@ -58,33 +58,33 @@
│ │ │          
    Description
    
│ │ │          
    Macaulay2 uses the Hans Boehm garbage collector to reclaim unused memory.  The function GCstats provides information about its status, such as the total number of bytes allocated, the current heap size, the number of garbage collections done, the number of threads used in each collection, the total cpu time spent in garbage collection, etc.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : s = GCstats()
│ │ │  
│ │ │ -o1 = HashTable{"bytesAlloc" => 64675195162        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 64778770938        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 222048256
│ │ │ -               "numGCs" => 945
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 233422848
│ │ │ +               "numGCs" => 816
│ │ │ +               "numGCThreads" => 16
│ │ │  
│ │ │  o1 : HashTable
    
│ │ │              
    
│ │ │          
    The value returned is a hash table, from which individual bits of information can be easily extracted, as follows.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : s#"heapSize"
│ │ │  
│ │ │ -o2 = 222048256
    
│ │ │ +o2 = 233422848
│ │ │              
    
│ │ │          
    Any entries whose keys are all upper case give the values of environment variables affecting the operation of the garbage collector that have been specified by the user.
    
│ │ │          
    For further information about the individual items in the table, we refer the user to the source code and documentation of the garbage collector.
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,28 +9,28 @@
│ │ │ │  Macaulay2 uses the Hans Boehm _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r to reclaim unused memory. The
│ │ │ │  function GCstats provides information about its status, such as the total
│ │ │ │  number of bytes allocated, the current heap size, the number of garbage
│ │ │ │  collections done, the number of threads used in each collection, the total cpu
│ │ │ │  time spent in garbage collection, etc.
│ │ │ │  i1 : s = GCstats()
│ │ │ │  
│ │ │ │ -o1 = HashTable{"bytesAlloc" => 64675195162        }
│ │ │ │ +o1 = HashTable{"bytesAlloc" => 64778770938        }
│ │ │ │                 "GC_free_space_divisor" => 3
│ │ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │ │                 "gcCpuTimeSecs" => 0
│ │ │ │ -               "heapSize" => 222048256
│ │ │ │ -               "numGCs" => 945
│ │ │ │ -               "numGCThreads" => 6
│ │ │ │ +               "heapSize" => 233422848
│ │ │ │ +               "numGCs" => 816
│ │ │ │ +               "numGCThreads" => 16
│ │ │ │  
│ │ │ │  o1 : HashTable
│ │ │ │  The value returned is a hash table, from which individual bits of information
│ │ │ │  can be easily extracted, as follows.
│ │ │ │  i2 : s#"heapSize"
│ │ │ │  
│ │ │ │ -o2 = 222048256
│ │ │ │ +o2 = 233422848
│ │ │ │  Any entries whose keys are all upper case give the values of environment
│ │ │ │  variables affecting the operation of the garbage collector that have been
│ │ │ │  specified by the user.
│ │ │ │  For further information about the individual items in the table, we refer the
│ │ │ │  user to the source code and documentation of the garbage collector.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _G_C_ _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Minimal__Generators.html
│ │ │ @@ -133,23 +133,23 @@
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │                
    i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ - -- used 0.00855388s (cpu); 0.00854629s (thread); 0s (gc)
│ │ │ + -- used 0.00552555s (cpu); 0.00552121s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
    
│ │ │              
    
│ │ │                
    i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0676429s (cpu); 0.0676547s (thread); 0s (gc)
│ │ │ + -- used 0.0497031s (cpu); 0.0497124s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
    
│ │ │              
    
│ │ │                
    i10 : numgens J
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,19 +46,19 @@
│ │ │ │  o6 = R
│ │ │ │  
│ │ │ │  o6 : PolynomialRing
│ │ │ │  i7 : I = monomialCurveIdeal(R, {1,4,5,9});
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ │ - -- used 0.00855388s (cpu); 0.00854629s (thread); 0s (gc)
│ │ │ │ + -- used 0.00552555s (cpu); 0.00552121s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ │ - -- used 0.0676429s (cpu); 0.0676547s (thread); 0s (gc)
│ │ │ │ + -- used 0.0497031s (cpu); 0.0497124s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : numgens J
│ │ │ │  
│ │ │ │  o10 = 1067
│ │ │ │  i11 : numgens K
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Mutable__List.html
│ │ │ @@ -145,30 +145,30 @@
│ │ │  
│ │ │  o10 : MutableList
    
│ │ │              
    
│ │ │                
    i11 : elapsedTime scan(1000, i -> s#i = i^2) -- quadratic, since we grow s at each step
│ │ │ - -- .00341642s elapsed
    
│ │ │ + -- .00516619s elapsed
│ │ │              
    
│ │ │                
    i12 : t = new MutableList from 1000
│ │ │  
│ │ │  o12 = MutableList{...1000...}
│ │ │  
│ │ │  o12 : MutableList
    
│ │ │              
    
│ │ │                
    i13 : elapsedTime scan(1000, i -> t#i = i^2) -- linear
│ │ │ - -- .000457796s elapsed
    
│ │ │ + -- .000386983s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,22 +43,22 @@
│ │ │ │  i10 : s = new MutableList
│ │ │ │  
│ │ │ │  o10 = MutableList{}
│ │ │ │  
│ │ │ │  o10 : MutableList
│ │ │ │  i11 : elapsedTime scan(1000, i -> s#i = i^2) -- quadratic, since we grow s at
│ │ │ │  each step
│ │ │ │ - -- .00341642s elapsed
│ │ │ │ + -- .00516619s elapsed
│ │ │ │  i12 : t = new MutableList from 1000
│ │ │ │  
│ │ │ │  o12 = MutableList{...1000...}
│ │ │ │  
│ │ │ │  o12 : MutableList
│ │ │ │  i13 : elapsedTime scan(1000, i -> t#i = i^2) -- linear
│ │ │ │ - -- .000457796s elapsed
│ │ │ │ + -- .000386983s elapsed
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _B_a_s_i_c_L_i_s_t -- the class of all basic lists
│ │ │ │  ******** MMeennuu ********
│ │ │ │      * _B_a_g -- the class of all bags
│ │ │ │  ********** TTyyppeess ooff mmuuttaabbllee lliisstt:: **********
│ │ │ │      * _B_a_g -- the class of all bags
│ │ │ │  ********** MMeetthhooddss tthhaatt uussee aa mmuuttaabbllee lliisstt:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Mutex.html
│ │ │ @@ -96,23 +96,19 @@
│ │ │                
      i4 : while not all(T, isReady) do null
      
│ │ │              
    
│ │ │                
    i5 : stack sort lines msgs
│ │ │  
│ │ │ -o5 = hello from thread #0
│ │ │ -     hello from thread #1
│ │ │ +o5 = hello from thread #1
│ │ │       hello from thread #2
│ │ │       hello from thread #3
│ │ │       hello from thread #4
│ │ │ -     hello from thread #5
│ │ │       hello from thread #6
│ │ │ -     hello from thread #7
│ │ │ -     hello from thread #8
│ │ │       hello from thread #9
    
│ │ │              
    
│ │ │          
    
│ │ │            
    We likely ended up with fewer than the expected number of 10 messages. We can get around this issue by using a mutex to lock the string so that only one thread can modify it at a time.
    
│ │ │          
    
│ │ │ @@ -137,17 +133,19 @@
│ │ │
    		│ │ │ 
    i8 : T = apply(10, i -> schedule(() -> (lock m; sayhello i; unlock m)))
│ │ │  
│ │ │  o8 = {<<task, result available, task done>>, <<task, result available, task
│ │ │       ------------------------------------------------------------------------
│ │ │       done>>, <<task, result available, task done>>, <<task, running>>,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     <<task, running>>, <<task, result available, task done>>, <<task,
│ │ │ +     <<task, running>>, <<task, result available, task done>>, <<task, result
│ │ │       ------------------------------------------------------------------------
│ │ │ -     running>>, <<task, running>>, <<task, running>>, <<task, created>>}
│ │ │ +     available, task done>>, <<task, running>>, <<task, result available,
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     task done>>, <<task, running>>}
│ │ │  
│ │ │  o8 : List
    │ │ │
    │ │ │ 
    i9 : while not all(T, isReady) do null
    │ │ │ ├── html2text {} │ │ │ │ @@ -33,23 +33,19 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ <>, <>, <>} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : while not all(T, isReady) do null │ │ │ │ i5 : stack sort lines msgs │ │ │ │ │ │ │ │ -o5 = hello from thread #0 │ │ │ │ - hello from thread #1 │ │ │ │ +o5 = hello from thread #1 │ │ │ │ hello from thread #2 │ │ │ │ hello from thread #3 │ │ │ │ hello from thread #4 │ │ │ │ - hello from thread #5 │ │ │ │ hello from thread #6 │ │ │ │ - hello from thread #7 │ │ │ │ - hello from thread #8 │ │ │ │ hello from thread #9 │ │ │ │ We likely ended up with fewer than the expected number of 10 messages. We can │ │ │ │ get around this issue by using a mutex to lock the string so that only one │ │ │ │ thread can modify it at a time. │ │ │ │ i6 : m = new Mutex │ │ │ │ │ │ │ │ o6 = m │ │ │ │ @@ -60,17 +56,19 @@ │ │ │ │ o7 = │ │ │ │ i8 : T = apply(10, i -> schedule(() -> (lock m; sayhello i; unlock m))) │ │ │ │ │ │ │ │ o8 = {<>, <>, <>, <>, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - <>, <>, <>, <>, <>, <>, <>, <>} │ │ │ │ + available, task done>>, <>, <>, <>} │ │ │ │ │ │ │ │ o8 : List │ │ │ │ i9 : while not all(T, isReady) do null │ │ │ │ i10 : stack sort lines msgs │ │ │ │ │ │ │ │ o10 = hello from thread #0 │ │ │ │ hello from thread #1 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.html │ │ │ @@ -73,21 +73,21 @@ │ │ │ o1 : Matrix RR <-- RR │ │ │ 53 53 │ │ │
    │ │ │ 
    i2 : time SVD(M);
│ │ │ - -- used 0.0225511s (cpu); 0.0225436s (thread); 0s (gc)
    │ │ │ + -- used 0.0414148s (cpu); 0.0414151s (thread); 0s (gc) │ │ │
    │ │ │ 
    i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0220907s (cpu); 0.0220933s (thread); 0s (gc)
    │ │ │ + -- used 0.0393494s (cpu); 0.0393606s (thread); 0s (gc) │ │ │
    │ │ │
    │ │ │
    │ │ │
    │ │ │

    Functions with optional argument named DivideConquer:

    │ │ │ ├── html2text {} │ │ │ │ @@ -11,17 +11,17 @@ │ │ │ │ For large matrices, this algorithm is often much faster. │ │ │ │ i1 : M = random(RR^200, RR^200); │ │ │ │ │ │ │ │ 200 200 │ │ │ │ o1 : Matrix RR <-- RR │ │ │ │ 53 53 │ │ │ │ i2 : time SVD(M); │ │ │ │ - -- used 0.0225511s (cpu); 0.0225436s (thread); 0s (gc) │ │ │ │ + -- used 0.0414148s (cpu); 0.0414151s (thread); 0s (gc) │ │ │ │ i3 : time SVD(M, DivideConquer=>true); │ │ │ │ - -- used 0.0220907s (cpu); 0.0220933s (thread); 0s (gc) │ │ │ │ + -- used 0.0393494s (cpu); 0.0393606s (thread); 0s (gc) │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd DDiivviiddeeCCoonnqquueerr:: ********** │ │ │ │ * _S_V_D_(_._._._,_D_i_v_i_d_e_C_o_n_q_u_e_r_=_>_._._._) -- whether to use the LAPACK divide and │ │ │ │ conquer SVD algorithm │ │ │ │ ********** FFuurrtthheerr iinnffoorrmmaattiioonn ********** │ │ │ │ * Default value: _t_r_u_e │ │ │ │ * Function: _S_V_D -- singular value decomposition of a matrix │ │ │ │ * Option key: _D_i_v_i_d_e_C_o_n_q_u_e_r -- an optional argument │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_a_spfirst_sp__Macaulay2_spsession.html │ │ │ @@ -827,15 +827,15 @@ │ │ │
    │ │ │

    We may use resolution to produce a projective resolution of it, and time to report the time required.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i59 : time C = resolution M
│ │ │ - -- used 0.00293873s (cpu); 0.00293155s (thread); 0s (gc)
│ │ │ + -- used 0.00396274s (cpu); 0.00396264s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R
│ │ │                                       
│ │ │        0      1      2       3       4
│ │ │  
│ │ │  o59 : Complex
    │ │ │ ├── html2text {} │ │ │ │ @@ -386,15 +386,15 @@ │ │ │ │ | c f i l o r | │ │ │ │ │ │ │ │ 3 │ │ │ │ o58 : R-module, quotient of R │ │ │ │ We may use _r_e_s_o_l_u_t_i_o_n to produce a projective resolution of it, and _t_i_m_e to │ │ │ │ report the time required. │ │ │ │ i59 : time C = resolution M │ │ │ │ - -- used 0.00293873s (cpu); 0.00293155s (thread); 0s (gc) │ │ │ │ + -- used 0.00396274s (cpu); 0.00396264s (thread); 0s (gc) │ │ │ │ │ │ │ │ 3 6 15 18 6 │ │ │ │ o59 = R <-- R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ │ │ o59 : Complex │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_at__End__Of__File_lp__File_rp.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ o4 = "hi there" │ │ │
    │ │ │ 
    i5 : atEndOfFile f
│ │ │  
│ │ │ -o5 = false
    │ │ │ +o5 = true │ │ │
    │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use this method:

    │ │ │ ├── html2text {} │ │ │ │ @@ -23,13 +23,13 @@ │ │ │ │ │ │ │ │ o3 = false │ │ │ │ i4 : peek read f │ │ │ │ │ │ │ │ o4 = "hi there" │ │ │ │ i5 : atEndOfFile f │ │ │ │ │ │ │ │ -o5 = false │ │ │ │ +o5 = true │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _a_t_E_n_d_O_f_F_i_l_e_(_F_i_l_e_) -- test for end of file │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.05+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_files.m2:374:0. │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_benchmark.html │ │ │ @@ -73,15 +73,15 @@ │ │ │
    │ │ │

    Description

    │ │ │ Produces an accurate timing for the code contained in the string s. The value returned is the number of seconds. │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : benchmark "sqrt 2p100000"
│ │ │  
│ │ │ -o1 = .0002914831290148481
│ │ │ +o1 = .0003543037189214616
│ │ │  
│ │ │  o1 : RR (of precision 53)
    │ │ │
    │ │ │ The snippet of code provided will be run enough times to register meaningfully on the clock, and the garbage collector will be called beforehand.
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -12,15 +12,15 @@ │ │ │ │ o a _r_e_a_l_ _n_u_m_b_e_r, the number of seconds it takes to evaluate the code │ │ │ │ in s │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ Produces an accurate timing for the code contained in the string s. The value │ │ │ │ returned is the number of seconds. │ │ │ │ i1 : benchmark "sqrt 2p100000" │ │ │ │ │ │ │ │ -o1 = .0002914831290148481 │ │ │ │ +o1 = .0003543037189214616 │ │ │ │ │ │ │ │ o1 : RR (of precision 53) │ │ │ │ The snippet of code provided will be run enough times to register meaningfully │ │ │ │ on the clock, and the garbage collector will be called beforehand. │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _b_e_n_c_h_m_a_r_k is a _f_u_n_c_t_i_o_n_ _c_l_o_s_u_r_e. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_caching_spcomputation_spresults.html │ │ │ @@ -74,23 +74,23 @@ │ │ │ 
    i3 : M = coker vars R;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime pdim' M
│ │ │   -- computing pdim'
│ │ │ - -- .00641207s elapsed
│ │ │ + -- .00439629s elapsed
│ │ │  
│ │ │  o4 = 3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime pdim' M
│ │ │ - -- .000001934s elapsed
│ │ │ + -- .000003154s elapsed
│ │ │  
│ │ │  o5 = 3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : peek M.cache
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,19 +8,19 @@
│ │ │ │  Here is a simple example of caching a computation in a _C_a_c_h_e_T_a_b_l_e, using the
│ │ │ │  augmented null coalescing operator _?_?_=.
│ │ │ │  i1 : pdim' = M -> M.cache.pdim' ??= ( printerr "computing pdim'"; pdim M );
│ │ │ │  i2 : R = QQ[x,y,z];
│ │ │ │  i3 : M = coker vars R;
│ │ │ │  i4 : elapsedTime pdim' M
│ │ │ │   -- computing pdim'
│ │ │ │ - -- .00641207s elapsed
│ │ │ │ + -- .00439629s elapsed
│ │ │ │  
│ │ │ │  o4 = 3
│ │ │ │  i5 : elapsedTime pdim' M
│ │ │ │ - -- .000001934s elapsed
│ │ │ │ + -- .000003154s elapsed
│ │ │ │  
│ │ │ │  o5 = 3
│ │ │ │  i6 : peek M.cache
│ │ │ │  
│ │ │ │  o6 = CacheTable{cache => MutableHashTable{}
│ │ │ │  }
│ │ │ │                  isHomogeneous => true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cancel__Task_lp__Task_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  o4 : Task
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : n
│ │ │  
│ │ │ -o5 = 569438
    │ │ │ +o5 = 904196 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
    │ │ │ @@ -132,15 +132,15 @@ │ │ │ o7 : Task │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : n
│ │ │  
│ │ │ -o8 = 1244881
    │ │ │ +o8 = 1952215 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : isReady t
│ │ │  
│ │ │  o9 = false
    │ │ │ @@ -150,15 +150,15 @@ │ │ │ │ │ │ 
    i10 : cancelTask t
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : sleep 2
│ │ │ -stdio:2:39:(3):[1]: error: interrupted
│ │ │ +stdio:2:25:(3):[1]: error: interrupted
│ │ │  
│ │ │  o11 = 0
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : t
│ │ │ @@ -168,29 +168,29 @@
│ │ │  o12 : Task
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : n
│ │ │  
│ │ │ -o13 = 1245096
    │ │ │ +o13 = 1952563 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : n
│ │ │  
│ │ │ -o15 = 1245096
    │ │ │ +o15 = 1952563 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : isReady t
│ │ │  
│ │ │  o16 = false
    │ │ │ ├── html2text {} │ │ │ │ @@ -28,48 +28,48 @@ │ │ │ │ i4 : t │ │ │ │ │ │ │ │ o4 = <> │ │ │ │ │ │ │ │ o4 : Task │ │ │ │ i5 : n │ │ │ │ │ │ │ │ -o5 = 569438 │ │ │ │ +o5 = 904196 │ │ │ │ i6 : sleep 1 │ │ │ │ │ │ │ │ o6 = 0 │ │ │ │ i7 : t │ │ │ │ │ │ │ │ o7 = <> │ │ │ │ │ │ │ │ o7 : Task │ │ │ │ i8 : n │ │ │ │ │ │ │ │ -o8 = 1244881 │ │ │ │ +o8 = 1952215 │ │ │ │ i9 : isReady t │ │ │ │ │ │ │ │ o9 = false │ │ │ │ i10 : cancelTask t │ │ │ │ i11 : sleep 2 │ │ │ │ -stdio:2:39:(3):[1]: error: interrupted │ │ │ │ +stdio:2:25:(3):[1]: error: interrupted │ │ │ │ │ │ │ │ o11 = 0 │ │ │ │ i12 : t │ │ │ │ │ │ │ │ o12 = <> │ │ │ │ │ │ │ │ o12 : Task │ │ │ │ i13 : n │ │ │ │ │ │ │ │ -o13 = 1245096 │ │ │ │ +o13 = 1952563 │ │ │ │ i14 : sleep 1 │ │ │ │ │ │ │ │ o14 = 0 │ │ │ │ i15 : n │ │ │ │ │ │ │ │ -o15 = 1245096 │ │ │ │ +o15 = 1952563 │ │ │ │ i16 : isReady t │ │ │ │ │ │ │ │ o16 = false │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _c_a_n_c_e_l_T_a_s_k_(_T_a_s_k_) -- stop a task │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_change__Directory.html │ │ │ @@ -76,36 +76,36 @@ │ │ │

    Change the current working directory to dir.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11094-0/0
    │ │ │ +o1 = /tmp/M2-11564-0/0 │ │ │
    │ │ │ 
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11094-0/0
    │ │ │ +o2 = /tmp/M2-11564-0/0 │ │ │
    │ │ │ 
    i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-11094-0/0/
    │ │ │ +o3 = /tmp/M2-11564-0/0/ │ │ │
    │ │ │ 
    i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-11094-0/0/
    │ │ │ +o4 = /tmp/M2-11564-0/0/ │ │ │
    │ │ │
    │ │ │

    If dir is omitted, then the current working directory is changed to the user's home directory.

    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -11,24 +11,24 @@ │ │ │ │ o dir, a _s_t_r_i_n_g, │ │ │ │ * Outputs: │ │ │ │ o a _s_t_r_i_n_g, the new working directory; │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ Change the current working directory to dir. │ │ │ │ i1 : dir = temporaryFileName() │ │ │ │ │ │ │ │ -o1 = /tmp/M2-11094-0/0 │ │ │ │ +o1 = /tmp/M2-11564-0/0 │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ │ │ -o2 = /tmp/M2-11094-0/0 │ │ │ │ +o2 = /tmp/M2-11564-0/0 │ │ │ │ i3 : changeDirectory dir │ │ │ │ │ │ │ │ -o3 = /tmp/M2-11094-0/0/ │ │ │ │ +o3 = /tmp/M2-11564-0/0/ │ │ │ │ i4 : currentDirectory() │ │ │ │ │ │ │ │ -o4 = /tmp/M2-11094-0/0/ │ │ │ │ +o4 = /tmp/M2-11564-0/0/ │ │ │ │ If dir is omitted, then the current working directory is changed to the user's │ │ │ │ home directory. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_u_r_r_e_n_t_D_i_r_e_c_t_o_r_y -- current working directory │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_h_a_n_g_e_D_i_r_e_c_t_o_r_y is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_check.html │ │ │ @@ -100,40 +100,40 @@ │ │ │ o1 : Package │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : check_1 FirstPackage
│ │ │   -- warning: reloading FirstPackage; recreate instances of types from this package
│ │ │ - -- capturing check(1, "FirstPackage")        -- .173698s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .219084s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .180088s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .179579s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .162842s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .16334s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    Alternatively, if the package is installed somewhere accessible, one can do the following.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .178872s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .16582s elapsed │ │ │
    │ │ │ 
    i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .178948s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .181166s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .162585s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .161662s elapsed │ │ │
    │ │ │
    │ │ │

    A TestInput object (or a list of such objects) can also be run directly.

    │ │ │
    │ │ │ │ │ │ @@ -145,15 +145,15 @@ │ │ │ │ │ │ o6 : TestInput │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i7 : check oo
│ │ │ - -- capturing check(1, "FirstPackage")        -- .179778s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .163704s elapsed │ │ │
    │ │ │ 
    i8 : tests "FirstPackage"
│ │ │  
│ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │ @@ -161,16 +161,16 @@
│ │ │  
│ │ │  o8 : NumberedVerticalList
    │ │ │
    │ │ │ 
    i9 : check oo
│ │ │ - -- capturing check(0, "FirstPackage")        -- .183757s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .181583s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .161488s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .164806s elapsed │ │ │
    │ │ │
    │ │ │

    If only an integer is passed as an argument, then the test with that index from the last call to tests is run.

    │ │ │
    │ │ │ │ │ │ @@ -183,15 +183,15 @@ │ │ │ │ │ │ o10 : NumberedVerticalList │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i11 : check 1
│ │ │ - -- capturing check(1, "FirstPackage")        -- .288312s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .157437s elapsed │ │ │
    │ │ │
    │ │ │
    │ │ │

    Caveat

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -42,52 +42,52 @@ │ │ │ │ │ │ │ │ o1 = FirstPackage │ │ │ │ │ │ │ │ o1 : Package │ │ │ │ i2 : check_1 FirstPackage │ │ │ │ -- warning: reloading FirstPackage; recreate instances of types from this │ │ │ │ package │ │ │ │ - -- capturing check(1, "FirstPackage") -- .173698s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .219084s elapsed │ │ │ │ i3 : check FirstPackage │ │ │ │ - -- capturing check(0, "FirstPackage") -- .180088s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .179579s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .162842s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .16334s elapsed │ │ │ │ Alternatively, if the package is installed somewhere accessible, one can do the │ │ │ │ following. │ │ │ │ i4 : check_1 "FirstPackage" │ │ │ │ - -- capturing check(1, "FirstPackage") -- .178872s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .16582s elapsed │ │ │ │ i5 : check "FirstPackage" │ │ │ │ - -- capturing check(0, "FirstPackage") -- .178948s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .181166s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .162585s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .161662s elapsed │ │ │ │ A _T_e_s_t_I_n_p_u_t object (or a list of such objects) can also be run directly. │ │ │ │ i6 : tests(1, "FirstPackage") │ │ │ │ │ │ │ │ o6 = TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3] │ │ │ │ │ │ │ │ o6 : TestInput │ │ │ │ i7 : check oo │ │ │ │ - -- capturing check(1, "FirstPackage") -- .179778s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .163704s elapsed │ │ │ │ i8 : tests "FirstPackage" │ │ │ │ │ │ │ │ o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]} │ │ │ │ {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]} │ │ │ │ │ │ │ │ o8 : NumberedVerticalList │ │ │ │ i9 : check oo │ │ │ │ - -- capturing check(0, "FirstPackage") -- .183757s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .181583s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .161488s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .164806s elapsed │ │ │ │ If only an integer is passed as an argument, then the test with that index from │ │ │ │ the last call to _t_e_s_t_s is run. │ │ │ │ i10 : tests "FirstPackage" │ │ │ │ │ │ │ │ o10 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]} │ │ │ │ {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]} │ │ │ │ │ │ │ │ o10 : NumberedVerticalList │ │ │ │ i11 : check 1 │ │ │ │ - -- capturing check(1, "FirstPackage") -- .288312s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .157437s elapsed │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ Currently, if the package was only partially loaded because the documentation │ │ │ │ was obtainable from a database (see _b_e_g_i_n_D_o_c_u_m_e_n_t_a_t_i_o_n), then the package will │ │ │ │ be reloaded, this time completely, to ensure that all tests are considered; │ │ │ │ this may affect user objects of types declared by the package, as they may be │ │ │ │ not usable by the new instance of the package. In a future version, either the │ │ │ │ tests and the documentation will both be cached, or neither will. │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_communicating_spwith_spprograms.html │ │ │ @@ -55,15 +55,15 @@ │ │ │
    │ │ │

    communicating with programs

    │ │ │
    │ │ │ The most naive way to interact with another program is simply to run it, let it communicate directly with the user, and wait for it to finish. This is done with the run command. │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : run "uname -a"
│ │ │ -Linux sbuild 6.12.88+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.88-1 (2026-05-15) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.88+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.88-1 (2026-05-15) x86_64 GNU/Linux
│ │ │  
│ │ │  o1 = 0
    │ │ │
    │ │ │ To run a program and provide it with input, one way is use the operator <<, with a file name whose first character is an exclamation point; the rest of the file name will be taken as the command to run, as in the following example. │ │ │ │ │ │ @@ -79,15 +79,15 @@ │ │ │ │ │ │
    │ │ │ More often, one wants to write Macaulay2 code to obtain and manipulate the output from the other program. If the program requires no input data, then we can use get with a file name whose first character is an exclamation point. In the following example, we also peek at the string to see whether it includes a newline character. │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i3 : peek get "!uname -a"
│ │ │  
│ │ │ -o3 = "Linux sbuild 6.12.88+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ +o3 = "Linux sbuild 6.12.88+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │       6.12.88-1 (2026-05-15) x86_64 GNU/Linux\n"
    │ │ │
    │ │ │ Bidirectional communication with a program is also possible. We use openInOut to create a file that serves as a bidirectional connection to a program. That file is called an input output file. In this example we open a connection to the unix utility grep and use it to locate the symbol names in Macaulay2 that begin with in. │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -33,15 +33,15 @@ │ │ │ │ take(N,-2)); │ │ │ │ │ │ │ │ o4 : ProjectiveVariety, curve in PP^8 │ │ │ │ i5 : ? X │ │ │ │ │ │ │ │ o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 │ │ │ │ i6 : time f = X ===> Y; │ │ │ │ - -- used 3.33307s (cpu); 1.85339s (thread); 0s (gc) │ │ │ │ + -- used 4.03039s (cpu); 2.27251s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 : MultirationalMap (automorphism of PP^8) │ │ │ │ i7 : f X │ │ │ │ │ │ │ │ o7 = Y │ │ │ │ │ │ │ │ o7 : ProjectiveVariety, curve in PP^8 │ │ │ │ @@ -53,15 +53,15 @@ │ │ │ │ i9 : V = random({{2},{1}},X); │ │ │ │ │ │ │ │ o9 : ProjectiveVariety, 6-dimensional subvariety of PP^8 │ │ │ │ i10 : W = random({{2},{1}},Y); │ │ │ │ │ │ │ │ o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8 │ │ │ │ i11 : time g = V ===> W; │ │ │ │ - -- used 3.28744s (cpu); 1.89691s (thread); 0s (gc) │ │ │ │ + -- used 3.77874s (cpu); 2.24785s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 : MultirationalMap (automorphism of PP^8) │ │ │ │ i12 : g||W │ │ │ │ │ │ │ │ o12 = multi-rational map consisting of one single rational map │ │ │ │ source variety: 6-dimensional subvariety of PP^8 cut out by 2 │ │ │ │ hypersurfaces of degrees 1^1 2^1 │ │ │ │ @@ -144,15 +144,15 @@ │ │ │ │ i15 : Z = projectiveVariety pfaffians(4,A); │ │ │ │ │ │ │ │ o15 : ProjectiveVariety, 6-dimensional subvariety of PP^9 │ │ │ │ i16 : ? Z │ │ │ │ │ │ │ │ o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2 │ │ │ │ i17 : time h = Z ===> GG_K(1,4) │ │ │ │ - -- used 7.61784s (cpu); 4.58879s (thread); 0s (gc) │ │ │ │ + -- used 7.26752s (cpu); 4.98001s (thread); 0s (gc) │ │ │ │ │ │ │ │ o17 = h │ │ │ │ │ │ │ │ o17 : MultirationalMap (isomorphism from PP^9 to PP^9) │ │ │ │ i18 : h || GG_K(1,4) │ │ │ │ │ │ │ │ o18 = multi-rational map consisting of one single rational map │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.html │ │ │ @@ -94,15 +94,15 @@ │ │ │ │ │ │ o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -25,24 +25,24 @@ │ │ │ │ │ │ │ │ o3 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x │ │ │ │ PP^7 to PP^7 x PP^7) │ │ │ │ i4 : Z = source Phi; │ │ │ │ │ │ │ │ o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7 │ │ │ │ i5 : time Phi Z; │ │ │ │ - -- used 0.0919907s (cpu); 0.0882659s (thread); 0s (gc) │ │ │ │ + -- used 0.19467s (cpu); 0.138129s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7 │ │ │ │ i6 : dim oo, degree oo, degrees oo │ │ │ │ │ │ │ │ o6 = (4, 80, {({0, 2}, 5), ({1, 1}, 33), ({2, 0}, 5)}) │ │ │ │ │ │ │ │ o6 : Sequence │ │ │ │ i7 : time Phi (point Z + point Z + point Z) │ │ │ │ - -- used 1.8643s (cpu); 1.27944s (thread); 0s (gc) │ │ │ │ + -- used 2.29452s (cpu); 1.4883s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of │ │ │ │ multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3 │ │ │ │ │ │ │ │ o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7 │ │ │ │ i8 : dim oo, degree oo, degrees oo │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_cm__Option_rp.html │ │ │ @@ -98,31 +98,31 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -5,16 +5,16 @@ │ │ │ │ _n_e_x_t | _p_r_e_v_i_o_u_s | _f_o_r_w_a_r_d | _b_a_c_k_w_a_r_d | _u_p | _i_n_d_e_x | _t_o_c │ │ │ │ =============================================================================== │ │ │ │ ************ ccoommmmuunniiccaattiinngg wwiitthh pprrooggrraammss ************ │ │ │ │ The most naive way to interact with another program is simply to run it, let it │ │ │ │ communicate directly with the user, and wait for it to finish. This is done │ │ │ │ with the _r_u_n command. │ │ │ │ i1 : run "uname -a" │ │ │ │ -Linux sbuild 6.12.88+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.88-1 (2026- │ │ │ │ -05-15) x86_64 GNU/Linux │ │ │ │ +Linux sbuild 6.12.88+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.88-1 │ │ │ │ +(2026-05-15) x86_64 GNU/Linux │ │ │ │ │ │ │ │ o1 = 0 │ │ │ │ To run a program and provide it with input, one way is use the operator _<_<, │ │ │ │ with a file name whose first character is an exclamation point; the rest of the │ │ │ │ file name will be taken as the command to run, as in the following example. │ │ │ │ i2 : "!grep a" << " ba \n bc \n ad \n ef \n" << close │ │ │ │ ba │ │ │ │ @@ -26,15 +26,15 @@ │ │ │ │ More often, one wants to write Macaulay2 code to obtain and manipulate the │ │ │ │ output from the other program. If the program requires no input data, then we │ │ │ │ can use _g_e_t with a file name whose first character is an exclamation point. In │ │ │ │ the following example, we also peek at the string to see whether it includes a │ │ │ │ newline character. │ │ │ │ i3 : peek get "!uname -a" │ │ │ │ │ │ │ │ -o3 = "Linux sbuild 6.12.88+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian │ │ │ │ +o3 = "Linux sbuild 6.12.88+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian │ │ │ │ 6.12.88-1 (2026-05-15) x86_64 GNU/Linux\n" │ │ │ │ Bidirectional communication with a program is also possible. We use _o_p_e_n_I_n_O_u_t │ │ │ │ to create a file that serves as a bidirectional connection to a program. That │ │ │ │ file is called an input output file. In this example we open a connection to │ │ │ │ the unix utility grep and use it to locate the symbol names in Macaulay2 that │ │ │ │ begin with in. │ │ │ │ i4 : f = openInOut "!grep -E '^in'" │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_computing_sp__Groebner_spbases.html │ │ │ @@ -274,15 +274,15 @@ │ │ │ 1277 │ │ │
    │ │ │ 
    i24 : gb I
│ │ │  
│ │ │ -   -- registering gb 5 at 0x7fad8cc0fc40
│ │ │ +   -- registering gb 5 at 0x7f50c5372700
│ │ │  
│ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │     -- number of monomials                = 8
│ │ │     -- #reduction steps = 2
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -378,15 +378,15 @@
│ │ │                1      4
│ │ │  o32 : Matrix R  <-- R
    │ │ │
    │ │ │ 
    i33 : time betti gb f
│ │ │ - -- used 0.307614s (cpu); 0.201161s (thread); 0s (gc)
│ │ │ + -- used 0.2119s (cpu); 0.21109s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o33 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ @@ -422,15 +422,15 @@
│ │ │  
│ │ │  o35 : ZZ[T]
    │ │ │
    │ │ │ 
    i36 : time betti gb f
│ │ │ - -- used 0.00651741s (cpu); 0.00681373s (thread); 0s (gc)
│ │ │ + -- used 0.00800134s (cpu); 0.00806052s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o36 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ ├── html2text {}
│ │ │ │ @@ -140,15 +140,15 @@
│ │ │ │  o23 = ideal (x*y - z , y  - w )
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o23 : Ideal of ----[x..z, w]
│ │ │ │                 1277
│ │ │ │  i24 : gb I
│ │ │ │  
│ │ │ │ -   -- registering gb 5 at 0x7fad8cc0fc40
│ │ │ │ +   -- registering gb 5 at 0x7f50c5372700
│ │ │ │  
│ │ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │ │     -- number of monomials                = 8
│ │ │ │     -- #reduction steps = 2
│ │ │ │     -- #spairs done = 6
│ │ │ │     -- ncalls = 0
│ │ │ │     -- nloop = 0
│ │ │ │ @@ -213,15 +213,15 @@
│ │ │ │  
│ │ │ │  o31 : ZZ[T]
│ │ │ │  i32 : f = random(R^1,R^{-3,-3,-5,-6});
│ │ │ │  
│ │ │ │                1      4
│ │ │ │  o32 : Matrix R  <-- R
│ │ │ │  i33 : time betti gb f
│ │ │ │ - -- used 0.307614s (cpu); 0.201161s (thread); 0s (gc)
│ │ │ │ + -- used 0.2119s (cpu); 0.21109s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o33 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ │ │ @@ -245,15 +245,15 @@
│ │ │ │  i35 : poincare cokernel f = (1-T^3)*(1-T^3)*(1-T^5)*(1-T^6) -- cache poincare
│ │ │ │  
│ │ │ │              3    5     8     9    12     14    17
│ │ │ │  o35 = 1 - 2T  - T  + 2T  + 2T  - T   - 2T   + T
│ │ │ │  
│ │ │ │  o35 : ZZ[T]
│ │ │ │  i36 : time betti gb f
│ │ │ │ - -- used 0.00651741s (cpu); 0.00681373s (thread); 0s (gc)
│ │ │ │ + -- used 0.00800134s (cpu); 0.00806052s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o36 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -85,112 +85,112 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
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│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11854-0/0/
    
│ │ │ +o1 = /tmp/M2-13104-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11854-0/1/
    
│ │ │ +o2 = /tmp/M2-13104-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11854-0/0/a/
    
│ │ │ +o3 = /tmp/M2-13104-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11854-0/0/b/
    
│ │ │ +o4 = /tmp/M2-13104-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11854-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-13104-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11854-0/0/a/f
│ │ │ +o6 = /tmp/M2-13104-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11854-0/0/a/g
│ │ │ +o7 = /tmp/M2-13104-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11854-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13104-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-11854-0/0/
│ │ │ -     /tmp/M2-11854-0/0/b/
│ │ │ -     /tmp/M2-11854-0/0/b/c/
│ │ │ -     /tmp/M2-11854-0/0/b/c/g
│ │ │ -     /tmp/M2-11854-0/0/a/
│ │ │ -     /tmp/M2-11854-0/0/a/g
│ │ │ -     /tmp/M2-11854-0/0/a/f
    
│ │ │ +o9 = /tmp/M2-13104-0/0/
│ │ │ +     /tmp/M2-13104-0/0/a/
│ │ │ +     /tmp/M2-13104-0/0/a/g
│ │ │ +     /tmp/M2-13104-0/0/a/f
│ │ │ +     /tmp/M2-13104-0/0/b/
│ │ │ +     /tmp/M2-13104-0/0/b/c/
│ │ │ +     /tmp/M2-13104-0/0/b/c/g
│ │ │              
    
│ │ │                
    i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11854-0/0/b/c/g -> /tmp/M2-11854-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-11854-0/0/a/g -> /tmp/M2-11854-0/1/a/g
│ │ │ - -- copying: /tmp/M2-11854-0/0/a/f -> /tmp/M2-11854-0/1/a/f
    
│ │ │ + -- copying: /tmp/M2-13104-0/0/a/g -> /tmp/M2-13104-0/1/a/g
│ │ │ + -- copying: /tmp/M2-13104-0/0/a/f -> /tmp/M2-13104-0/1/a/f
│ │ │ + -- copying: /tmp/M2-13104-0/0/b/c/g -> /tmp/M2-13104-0/1/b/c/g
│ │ │              
    
│ │ │                
    i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11854-0/0/b/c/g not newer than /tmp/M2-11854-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-11854-0/0/a/g not newer than /tmp/M2-11854-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-11854-0/0/a/f not newer than /tmp/M2-11854-0/1/a/f
    
│ │ │ + -- skipping: /tmp/M2-13104-0/0/a/g not newer than /tmp/M2-13104-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-13104-0/0/a/f not newer than /tmp/M2-13104-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-13104-0/0/b/c/g not newer than /tmp/M2-13104-0/1/b/c/g
│ │ │              
    
│ │ │                
    i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-11854-0/1/
│ │ │ -      /tmp/M2-11854-0/1/a/
│ │ │ -      /tmp/M2-11854-0/1/a/f
│ │ │ -      /tmp/M2-11854-0/1/a/g
│ │ │ -      /tmp/M2-11854-0/1/b/
│ │ │ -      /tmp/M2-11854-0/1/b/c/
│ │ │ -      /tmp/M2-11854-0/1/b/c/g
    
│ │ │ +o12 = /tmp/M2-13104-0/1/
│ │ │ +      /tmp/M2-13104-0/1/a/
│ │ │ +      /tmp/M2-13104-0/1/a/g
│ │ │ +      /tmp/M2-13104-0/1/a/f
│ │ │ +      /tmp/M2-13104-0/1/b/
│ │ │ +      /tmp/M2-13104-0/1/b/c/
│ │ │ +      /tmp/M2-13104-0/1/b/c/g
│ │ │              
    
│ │ │                
    i13 : get (dst|"b/c/g")
│ │ │  
│ │ │  o13 = ho there
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,68 +25,68 @@
│ │ │ │              individual file operations
│ │ │ │      * Consequences:
│ │ │ │            o a copy of the directory tree rooted at src is created, rooted at
│ │ │ │              dst
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11854-0/0/
│ │ │ │ +o1 = /tmp/M2-13104-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11854-0/1/
│ │ │ │ +o2 = /tmp/M2-13104-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11854-0/0/a/
│ │ │ │ +o3 = /tmp/M2-13104-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11854-0/0/b/
│ │ │ │ +o4 = /tmp/M2-13104-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11854-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-13104-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11854-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-13104-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11854-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-13104-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11854-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-13104-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : stack findFiles src
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-11854-0/0/
│ │ │ │ -     /tmp/M2-11854-0/0/b/
│ │ │ │ -     /tmp/M2-11854-0/0/b/c/
│ │ │ │ -     /tmp/M2-11854-0/0/b/c/g
│ │ │ │ -     /tmp/M2-11854-0/0/a/
│ │ │ │ -     /tmp/M2-11854-0/0/a/g
│ │ │ │ -     /tmp/M2-11854-0/0/a/f
│ │ │ │ +o9 = /tmp/M2-13104-0/0/
│ │ │ │ +     /tmp/M2-13104-0/0/a/
│ │ │ │ +     /tmp/M2-13104-0/0/a/g
│ │ │ │ +     /tmp/M2-13104-0/0/a/f
│ │ │ │ +     /tmp/M2-13104-0/0/b/
│ │ │ │ +     /tmp/M2-13104-0/0/b/c/
│ │ │ │ +     /tmp/M2-13104-0/0/b/c/g
│ │ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-11854-0/0/b/c/g -> /tmp/M2-11854-0/1/b/c/g
│ │ │ │ - -- copying: /tmp/M2-11854-0/0/a/g -> /tmp/M2-11854-0/1/a/g
│ │ │ │ - -- copying: /tmp/M2-11854-0/0/a/f -> /tmp/M2-11854-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-13104-0/0/a/g -> /tmp/M2-13104-0/1/a/g
│ │ │ │ + -- copying: /tmp/M2-13104-0/0/a/f -> /tmp/M2-13104-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-13104-0/0/b/c/g -> /tmp/M2-13104-0/1/b/c/g
│ │ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11854-0/0/b/c/g not newer than /tmp/M2-11854-0/1/b/c/g
│ │ │ │ - -- skipping: /tmp/M2-11854-0/0/a/g not newer than /tmp/M2-11854-0/1/a/g
│ │ │ │ - -- skipping: /tmp/M2-11854-0/0/a/f not newer than /tmp/M2-11854-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-13104-0/0/a/g not newer than /tmp/M2-13104-0/1/a/g
│ │ │ │ + -- skipping: /tmp/M2-13104-0/0/a/f not newer than /tmp/M2-13104-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-13104-0/0/b/c/g not newer than /tmp/M2-13104-0/1/b/c/g
│ │ │ │  i12 : stack findFiles dst
│ │ │ │  
│ │ │ │ -o12 = /tmp/M2-11854-0/1/
│ │ │ │ -      /tmp/M2-11854-0/1/a/
│ │ │ │ -      /tmp/M2-11854-0/1/a/f
│ │ │ │ -      /tmp/M2-11854-0/1/a/g
│ │ │ │ -      /tmp/M2-11854-0/1/b/
│ │ │ │ -      /tmp/M2-11854-0/1/b/c/
│ │ │ │ -      /tmp/M2-11854-0/1/b/c/g
│ │ │ │ +o12 = /tmp/M2-13104-0/1/
│ │ │ │ +      /tmp/M2-13104-0/1/a/
│ │ │ │ +      /tmp/M2-13104-0/1/a/g
│ │ │ │ +      /tmp/M2-13104-0/1/a/f
│ │ │ │ +      /tmp/M2-13104-0/1/b/
│ │ │ │ +      /tmp/M2-13104-0/1/b/c/
│ │ │ │ +      /tmp/M2-13104-0/1/b/c/g
│ │ │ │  i13 : get (dst|"b/c/g")
│ │ │ │  
│ │ │ │  o13 = ho there
│ │ │ │  Now we remove the files and directories we created.
│ │ │ │  i14 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ │  
│ │ │ │  o14 = rm
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__File_lp__String_cm__String_rp.html
│ │ │ @@ -83,65 +83,65 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,41 +22,41 @@
│ │ │ │            o npaths, an _i_n_t_e_g_e_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Output is verbose. For other dynamic strategies, see _M_o_n_o_d_r_o_m_y_S_o_l_v_e_r_O_p_t_i_o_n_s.
│ │ │ │  i1 : R = CC[a,b,c,d][x,y];
│ │ │ │  i2 : polys = polySystem {a*x+b*y^2,c*x*y+d};
│ │ │ │  i3 : (p0, x0) = createSeedPair polys;
│ │ │ │  i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ │ ---backup directory created: /tmp/M2-25234-0/2
│ │ │ │ - -- .00306279s elapsed
│ │ │ │ +--backup directory created: /tmp/M2-32854-0/2
│ │ │ │ + -- .00389925s elapsed
│ │ │ │    H01: 1
│ │ │ │ - -- .00278769s elapsed
│ │ │ │ + -- .00377125s elapsed
│ │ │ │    H10: 1
│ │ │ │ - -- .000547082s elapsed
│ │ │ │ + -- .000698392s elapsed
│ │ │ │  number of paths tracked: 2
│ │ │ │  found 1 points in the fiber so far
│ │ │ │ - -- .00273952s elapsed
│ │ │ │ + -- .00368601s elapsed
│ │ │ │    H01: 1
│ │ │ │ - -- .00695454s elapsed
│ │ │ │ + -- .00394449s elapsed
│ │ │ │    H10: 1
│ │ │ │ - -- .000459249s elapsed
│ │ │ │ + -- .000642978s elapsed
│ │ │ │  number of paths tracked: 4
│ │ │ │  found 1 points in the fiber so far
│ │ │ │ - -- .00680025s elapsed
│ │ │ │ + -- .00377715s elapsed
│ │ │ │    H01: 1
│ │ │ │ - -- .00775431s elapsed
│ │ │ │ + -- .00380252s elapsed
│ │ │ │    H10: 1
│ │ │ │ - -- .00043848s elapsed
│ │ │ │ + -- .000648029s elapsed
│ │ │ │  number of paths tracked: 6
│ │ │ │  found 1 points in the fiber so far
│ │ │ │ - -- .00273619s elapsed
│ │ │ │ + -- .00368081s elapsed
│ │ │ │    H01: 1
│ │ │ │ - -- .00769911s elapsed
│ │ │ │ + -- .00374297s elapsed
│ │ │ │    H10: 1
│ │ │ │ - -- .000447176s elapsed
│ │ │ │ + -- .000662382s elapsed
│ │ │ │  number of paths tracked: 8
│ │ │ │  found 1 points in the fiber so far
│ │ │ │  
│ │ │ │  o4 = ({{.892712+.673395*ii, .29398+.632944*ii}}, 8)
│ │ │ │  
│ │ │ │  o4 : Sequence
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/example-output/___Msolve.out
│ │ │ @@ -9,16 +9,16 @@
│ │ │  i2 : I = ideal(x, y, z)
│ │ │  
│ │ │  o2 = ideal (x, y, z)
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : msolveGB(I, Verbosity => 2, Threads => 6) 
│ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-37973-0/0-in.ms -o /tmp/M2-37973-0/0-out.ms
│ │ │ -Initial seed for pseudo-random number generator is 1779109822
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-53061-0/0-in.ms -o /tmp/M2-53061-0/0-out.ms
│ │ │ +Initial seed for pseudo-random number generator is 1779299622
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -29,15 +29,15 @@
│ │ │  initial hash table size     131072 (2^17)
│ │ │  max pair selection             ALL
│ │ │  reduce gb                        1
│ │ │  #threads                         6
│ │ │  info level                       2
│ │ │  generate pbm files               0
│ │ │  ------------------------------------------
│ │ │ -Initial prime = 1232765221
│ │ │ +Initial prime = 1175563589
│ │ │  
│ │ │  Legend for f4 information
│ │ │  --------------------------------------------------------
│ │ │  deg       current degree of pairs selected in this round
│ │ │  sel       number of pairs selected in this round
│ │ │  pairs     total number of pairs in pair list
│ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ @@ -47,25 +47,25 @@
│ │ │  time(rd)  time of the current f4 round in seconds given
│ │ │            for real and cpu time
│ │ │  --------------------------------------------------------
│ │ │  
│ │ │  deg     sel   pairs        mat          density            new data         time(rd) in sec (real|cpu)
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │ -reduce final basis        3 x 3          33.33%        3 new       0 zero         0.03 | 0.08         
│ │ │ +reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -overall(elapsed)        0.07 sec
│ │ │ -overall(cpu)            0.20 sec
│ │ │ +overall(elapsed)        0.00 sec
│ │ │ +overall(cpu)            0.00 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.04 sec  57.6%
│ │ │ -convert                 0.03 sec  42.3%
│ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ +symbolic prep.          0.00 sec   0.3%
│ │ │ +update                  0.00 sec  70.9%
│ │ │ +convert                 0.00 sec   1.9%
│ │ │ +linear algebra          0.00 sec   1.5%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -79,18 +79,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1096643591
│ │ │ +New prime = 1233588359
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.03 sec
│ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │  learning phase                  0.00 Gops/sec
│ │ │  application phase               0.00 Gops/sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  multi-modular steps
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  {1}{2}<100.00%> 
│ │ │ @@ -106,15 +106,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.15 sec (elapsed) /  0.44 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │  
│ │ │  o3 = | z y x |
│ │ │  
│ │ │               1      3
│ │ │  o3 : Matrix R  <-- R
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/html/index.html
│ │ │ @@ -83,16 +83,16 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,17 +31,17 @@
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I = ideal(x, y, z)
│ │ │ │  
│ │ │ │  o2 = ideal (x, y, z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : msolveGB(I, Verbosity => 2, Threads => 6)
│ │ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-37973-0/0-in.ms -o /tmp/
│ │ │ │ -M2-37973-0/0-out.ms
│ │ │ │ -Initial seed for pseudo-random number generator is 1779109822
│ │ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-53061-0/0-in.ms -o /tmp/
│ │ │ │ +M2-53061-0/0-out.ms
│ │ │ │ +Initial seed for pseudo-random number generator is 1779299622
│ │ │ │  
│ │ │ │  --------------- INPUT DATA ---------------
│ │ │ │  #variables                       3
│ │ │ │  #equations                       3
│ │ │ │  #invalid equations               0
│ │ │ │  field characteristic             0
│ │ │ │  homogeneous input?               1
│ │ │ │ @@ -52,15 +52,15 @@
│ │ │ │  initial hash table size     131072 (2^17)
│ │ │ │  max pair selection             ALL
│ │ │ │  reduce gb                        1
│ │ │ │  #threads                         6
│ │ │ │  info level                       2
│ │ │ │  generate pbm files               0
│ │ │ │  ------------------------------------------
│ │ │ │ -Initial prime = 1232765221
│ │ │ │ +Initial prime = 1175563589
│ │ │ │  
│ │ │ │  Legend for f4 information
│ │ │ │  --------------------------------------------------------
│ │ │ │  deg       current degree of pairs selected in this round
│ │ │ │  sel       number of pairs selected in this round
│ │ │ │  pairs     total number of pairs in pair list
│ │ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ │ @@ -74,26 +74,26 @@
│ │ │ │  deg     sel   pairs        mat          density            new data
│ │ │ │  time(rd) in sec (real|cpu)
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  reduce final basis        3 x 3          33.33%        3 new       0 zero
│ │ │ │ -0.03 | 0.08
│ │ │ │ +0.00 | 0.00
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │ -overall(elapsed)        0.07 sec
│ │ │ │ -overall(cpu)            0.20 sec
│ │ │ │ +overall(elapsed)        0.00 sec
│ │ │ │ +overall(cpu)            0.00 sec
│ │ │ │  select                  0.00 sec   0.0%
│ │ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ │ -update                  0.04 sec  57.6%
│ │ │ │ -convert                 0.03 sec  42.3%
│ │ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ │ +symbolic prep.          0.00 sec   0.3%
│ │ │ │ +update                  0.00 sec  70.9%
│ │ │ │ +convert                 0.00 sec   1.9%
│ │ │ │ +linear algebra          0.00 sec   1.5%
│ │ │ │  reduce gb               0.00 sec   0.0%
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │ │  size of basis                     3
│ │ │ │  #terms in basis                   3
│ │ │ │  #pairs reduced                    0
│ │ │ │ @@ -107,18 +107,18 @@
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  
│ │ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │ │  [3]
│ │ │ │  #polynomials to lift              3
│ │ │ │  -----------------------------------------
│ │ │ │ -New prime = 1096643591
│ │ │ │ +New prime = 1233588359
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │ -multi-mod overall(elapsed)      0.03 sec
│ │ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │ │  learning phase                  0.00 Gops/sec
│ │ │ │  application phase               0.00 Gops/sec
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  multi-modular steps
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │ @@ -137,15 +137,15 @@
│ │ │ │  CRT     (elapsed)               0.00 sec
│ │ │ │  ratrecon(elapsed)               0.00 sec
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │ -msolve overall time           0.15 sec (elapsed) /  0.44 sec (cpu)
│ │ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │  
│ │ │ │  o3 = | z y x |
│ │ │ │  
│ │ │ │               1      3
│ │ │ │  o3 : Matrix R  <-- R
│ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out
│ │ │ @@ -23,19 +23,19 @@
│ │ │  o4 : RingMap S <-- R
│ │ │  
│ │ │  i5 : peek componentsOfKernel(2, F)
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 6
│ │ │  number of distinct multidegrees = 6
│ │ │ - -- .00212365s elapsed
│ │ │ + -- .002321s elapsed
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ - -- .00813618s elapsed
│ │ │ + -- .0102353s elapsed
│ │ │  
│ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │                        {0, 2, 0, 2, 0} => {}
│ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html
│ │ │ @@ -122,19 +122,19 @@
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,23 +20,23 @@
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I = ideal"xy,yz,zx"
│ │ │ │  
│ │ │ │  o2 = ideal (x*y, y*z, x*z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : elapsedTime jMult I
│ │ │ │ - -- .0219807s elapsed
│ │ │ │ + -- .0312243s elapsed
│ │ │ │  
│ │ │ │  o3 = 2
│ │ │ │  i4 : elapsedTime monjMult I
│ │ │ │ - -- .134903s elapsed
│ │ │ │ + -- .0906035s elapsed
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ │ - -- .122417s elapsed
│ │ │ │ + -- .156195s elapsed
│ │ │ │  
│ │ │ │  o5 = HashTable{2 => 3}
│ │ │ │                 3 => 2
│ │ │ │  
│ │ │ │  o5 : HashTable
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_u_l_t_i_p_l_i_c_i_t_y_S_e_q_u_e_n_c_e -- the multiplicity sequence of an ideal
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11639-0/0
    
│ │ │ +o1 = /tmp/M2-12669-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11639-0/1
    
│ │ │ +o2 = /tmp/M2-12669-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11639-0/0
│ │ │ +o3 = /tmp/M2-12669-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11639-0/0 -> /tmp/M2-11639-0/1
    
│ │ │ + -- copying: /tmp/M2-12669-0/0 -> /tmp/M2-12669-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11639-0/0 not newer than /tmp/M2-11639-0/1
    
│ │ │ + -- skipping: /tmp/M2-12669-0/0 not newer than /tmp/M2-12669-0/1
│ │ │              
    
│ │ │                
    i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11639-0/0
│ │ │ +o7 = /tmp/M2-12669-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11639-0/0 not newer than /tmp/M2-11639-0/1
    
│ │ │ + -- skipping: /tmp/M2-12669-0/0 not newer than /tmp/M2-12669-0/1
│ │ │              
    
│ │ │                
    i9 : get dst
│ │ │  
│ │ │  o9 = hi there
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,37 +18,37 @@
│ │ │ │            o Verbose => a _B_o_o_l_e_a_n_ _v_a_l_u_e, default value false, whether to report
│ │ │ │              individual file operations
│ │ │ │      * Consequences:
│ │ │ │            o the file may be copied
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11639-0/0
│ │ │ │ +o1 = /tmp/M2-12669-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11639-0/1
│ │ │ │ +o2 = /tmp/M2-12669-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11639-0/0
│ │ │ │ +o3 = /tmp/M2-12669-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-11639-0/0 -> /tmp/M2-11639-0/1
│ │ │ │ + -- copying: /tmp/M2-12669-0/0 -> /tmp/M2-12669-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11639-0/0 not newer than /tmp/M2-11639-0/1
│ │ │ │ + -- skipping: /tmp/M2-12669-0/0 not newer than /tmp/M2-12669-0/1
│ │ │ │  i7 : src << "ho there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11639-0/0
│ │ │ │ +o7 = /tmp/M2-12669-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11639-0/0 not newer than /tmp/M2-11639-0/1
│ │ │ │ + -- skipping: /tmp/M2-12669-0/0 not newer than /tmp/M2-12669-0/1
│ │ │ │  i9 : get dst
│ │ │ │  
│ │ │ │  o9 = hi there
│ │ │ │  i10 : removeFile src
│ │ │ │  i11 : removeFile dst
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_p_y_D_i_r_e_c_t_o_r_y
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cpu__Time.html
│ │ │ @@ -69,38 +69,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : t1 = cpuTime()
│ │ │  
│ │ │ -o1 = 489.590864066
│ │ │ +o1 = 425.123678766
│ │ │  
│ │ │  o1 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i2 : for i from 0 to 1000000 do 223131321321*324234324324;
    
│ │ │              
    
│ │ │                
    i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 491.560329477
│ │ │ +o3 = 426.081552997
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.969465411000044
│ │ │ +o4 = .9578742310000052
│ │ │  
│ │ │  o4 : RR (of precision 53)
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,26 +9,26 @@
│ │ │ │              cpuTime()
│ │ │ │      * Outputs:
│ │ │ │            o a _r_e_a_l_ _n_u_m_b_e_r, the number of seconds of cpu time used since the
│ │ │ │              program was started
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : t1 = cpuTime()
│ │ │ │  
│ │ │ │ -o1 = 489.590864066
│ │ │ │ +o1 = 425.123678766
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │ │  i3 : t2 = cpuTime()
│ │ │ │  
│ │ │ │ -o3 = 491.560329477
│ │ │ │ +o3 = 426.081552997
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  i4 : t2-t1
│ │ │ │  
│ │ │ │ -o4 = 1.969465411000044
│ │ │ │ +o4 = .9578742310000052
│ │ │ │  
│ │ │ │  o4 : RR (of precision 53)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _c_u_r_r_e_n_t_T_i_m_e -- get the current time
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_current__Time.html
│ │ │ @@ -69,48 +69,48 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : currentTime()
│ │ │  
│ │ │ -o1 = 1779107685
    
│ │ │ +o1 = 1779297808
│ │ │              
    
│ │ │          
    We can compute, roughly, how many years ago the epoch began as follows.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 56.37767820542365
│ │ │ +o2 = 56.38370296345478
│ │ │  
│ │ │  o2 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    We can also compute how many months account for the fractional part of that number.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : 12 * (oo - floor oo)
│ │ │  
│ │ │ -o3 = 4.53213846508379
│ │ │ +o3 = 4.604435561457365
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    Compare that to the current date, available from a standard Unix command.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : run "date"
│ │ │ -Mon May 18 12:34:45 UTC 2026
│ │ │ +Wed May 20 17:23:28 UTC 2026
│ │ │  
│ │ │  o4 = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,31 +9,31 @@
│ │ │ │              currentTime()
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the current time, in seconds since 00:00:00 1970-01-01
│ │ │ │              UTC, the beginning of the epoch
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : currentTime()
│ │ │ │  
│ │ │ │ -o1 = 1779107685
│ │ │ │ +o1 = 1779297808
│ │ │ │  We can compute, roughly, how many years ago the epoch began as follows.
│ │ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │ │  
│ │ │ │ -o2 = 56.37767820542365
│ │ │ │ +o2 = 56.38370296345478
│ │ │ │  
│ │ │ │  o2 : RR (of precision 53)
│ │ │ │  We can also compute how many months account for the fractional part of that
│ │ │ │  number.
│ │ │ │  i3 : 12 * (oo - floor oo)
│ │ │ │  
│ │ │ │ -o3 = 4.53213846508379
│ │ │ │ +o3 = 4.604435561457365
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  Compare that to the current date, available from a standard Unix command.
│ │ │ │  i4 : run "date"
│ │ │ │ -Mon May 18 12:34:45 UTC 2026
│ │ │ │ +Wed May 20 17:23:28 UTC 2026
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_u_r_r_e_n_t_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.05+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:1849:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Time.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  elapsedTime e evaluates e, prints the amount of time elapsed, and returns the value of e.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime sleep 1
│ │ │ - -- 1.00069s elapsed
│ │ │ + -- 1.00014s elapsed
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,15 +7,15 @@
│ │ │ │  ************ eellaappsseeddTTiimmee ---- ttiimmee aa ccoommppuuttaattiioonn iinncclluuddiinngg ttiimmee eellaappsseedd ************
│ │ │ │      *   Usage:
│ │ │ │              elapsedTime e
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  elapsedTime e evaluates e, prints the amount of time elapsed, and returns the
│ │ │ │  value of e.
│ │ │ │  i1 : elapsedTime sleep 1
│ │ │ │ - -- 1.00069s elapsed
│ │ │ │ + -- 1.00014s elapsed
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _G_C_s_t_a_t_s -- information about the status of the garbage collector
│ │ │ │      * _p_a_r_a_l_l_e_l_ _p_r_o_g_r_a_m_m_i_n_g_ _w_i_t_h_ _t_h_r_e_a_d_s_ _a_n_d_ _t_a_s_k_s
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Timing.html
│ │ │ @@ -59,24 +59,24 @@
│ │ │  elapsedTiming e evaluates e and returns a list of type Time of the form {t,v}, where t is the number of seconds of time elapsed, and v is the value of the expression.        
    
│ │ │  The default method for printing such timing results is to display the timing separately in a comment below the computed value.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTiming sleep 1
│ │ │  
│ │ │  o1 = 0
│ │ │ -     -- 1.00016 seconds
│ │ │ +     -- 1.00012 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{1.00016, 0}
    
│ │ │ +o2 = Time{1.00012, 0}
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,20 +10,20 @@
│ │ │ │  where t is the number of seconds of time elapsed, and v is the value of the
│ │ │ │  expression.
│ │ │ │  The default method for printing such timing results is to display the timing
│ │ │ │  separately in a comment below the computed value.
│ │ │ │  i1 : elapsedTiming sleep 1
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │ -     -- 1.00016 seconds
│ │ │ │ +     -- 1.00012 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{1.00016, 0}
│ │ │ │ +o2 = Time{1.00012, 0}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _T_i_m_e -- the class of all timing results
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elimination_spof_spvariables.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
    
│ │ │                
    i3 : time leadTerm gens gb I
│ │ │ - -- used 0.778878s (cpu); 0.330916s (thread); 0s (gc)
│ │ │ + -- used 0.15204s (cpu); 0.152039s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │       ------------------------------------------------------------------------
│ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │       ------------------------------------------------------------------------
│ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -167,15 +167,15 @@
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │                
    i8 : time G = eliminate(I,{s,t})
│ │ │ - -- used 0.474974s (cpu); 0.256002s (thread); 0s (gc)
│ │ │ + -- used 5.1295s (cpu); 0.481068s (thread); 0s (gc)
│ │ │  
│ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │       ------------------------------------------------------------------------
│ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -250,15 +250,15 @@
│ │ │  
│ │ │  o11 : Ideal of R1
    
│ │ │              
    
│ │ │                
    i12 : time G = eliminate(I1,{s,t})
│ │ │ - -- used 0.0626994s (cpu); 0.0627103s (thread); 0s (gc)
│ │ │ + -- used 0.0312705s (cpu); 0.031271s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7  
│ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2  
│ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -342,15 +342,15 @@
│ │ │  
│ │ │  o16 : RingMap A <-- B
    
│ │ │              
    
│ │ │                
    i17 : time G = kernel F
│ │ │ - -- used 0.84521s (cpu); 0.352873s (thread); 0s (gc)
│ │ │ + -- used 0.110857s (cpu); 0.110862s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │        -----------------------------------------------------------------------
│ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -423,26 +423,26 @@
│ │ │  
│ │ │  o19 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ - -- used 0.00256747s (cpu); 0.00256862s (thread); 0s (gc)
│ │ │ + -- used 0.00179999s (cpu); 0.00179763s (thread); 0s (gc)
│ │ │  
│ │ │           9    9      7    3
│ │ │  o20 = x*t  - t  - z*t  - z
│ │ │  
│ │ │  o20 : R
    
│ │ │              
    
│ │ │                
    i21 : time f2 = resultant(I_1,f1,t)
│ │ │ - -- used 0.0727722s (cpu); 0.0727847s (thread); 0s (gc)
│ │ │ + -- used 0.0396745s (cpu); 0.0396846s (thread); 0s (gc)
│ │ │  
│ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2  
│ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │        -----------------------------------------------------------------------
│ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8   
│ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │        -----------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,15 +13,15 @@
│ │ │ │  i2 : I = ideal(x-s^3-s*t-1, y-t^3-3*t^2-t, z-s*t^3)
│ │ │ │  
│ │ │ │                 3                   3     2               3
│ │ │ │  o2 = ideal (- s  - s*t + x - 1, - t  - 3t  - t + y, - s*t  + z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time leadTerm gens gb I
│ │ │ │ - -- used 0.778878s (cpu); 0.330916s (thread); 0s (gc)
│ │ │ │ + -- used 0.15204s (cpu); 0.152039s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -89,15 +89,15 @@
│ │ │ │  i7 : I = ideal(x-s^3-s*t-1, y-t^3-3*t^2-t, z-s*t^3)
│ │ │ │  
│ │ │ │                 3                   3     2               3
│ │ │ │  o7 = ideal (- s  - s*t + x - 1, - t  - 3t  + y - t, - s*t  + z)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : time G = eliminate(I,{s,t})
│ │ │ │ - -- used 0.474974s (cpu); 0.256002s (thread); 0s (gc)
│ │ │ │ + -- used 5.1295s (cpu); 0.481068s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8
│ │ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7
│ │ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -156,15 +156,15 @@
│ │ │ │  Sometimes giving the variables different degrees will speed up the
│ │ │ │  computations. Here, we set the degrees of x, y, and z to be the total degrees.
│ │ │ │  i10 : R1 = QQ[x,y,z,s,t, Degrees=>{3,3,4,1,1}];
│ │ │ │  i11 : I1 = substitute(I,R1);
│ │ │ │  
│ │ │ │  o11 : Ideal of R1
│ │ │ │  i12 : time G = eliminate(I1,{s,t})
│ │ │ │ - -- used 0.0626994s (cpu); 0.0627103s (thread); 0s (gc)
│ │ │ │ + -- used 0.0312705s (cpu); 0.031271s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7
│ │ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2
│ │ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -227,15 +227,15 @@
│ │ │ │  i16 : F = map(A,B,{s^3+s*t+1, t^3+3*t^2+t, s*t^3})
│ │ │ │  
│ │ │ │                     3             3     2         3
│ │ │ │  o16 = map (A, B, {s  + s*t + 1, t  + 3t  + t, s*t })
│ │ │ │  
│ │ │ │  o16 : RingMap A <-- B
│ │ │ │  i17 : time G = kernel F
│ │ │ │ - -- used 0.84521s (cpu); 0.352873s (thread); 0s (gc)
│ │ │ │ + -- used 0.110857s (cpu); 0.110862s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8
│ │ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7
│ │ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -296,22 +296,22 @@
│ │ │ │  involve the variables s and t.
│ │ │ │  i19 : use ring I
│ │ │ │  
│ │ │ │  o19 = R
│ │ │ │  
│ │ │ │  o19 : PolynomialRing
│ │ │ │  i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ │ - -- used 0.00256747s (cpu); 0.00256862s (thread); 0s (gc)
│ │ │ │ + -- used 0.00179999s (cpu); 0.00179763s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9    9      7    3
│ │ │ │  o20 = x*t  - t  - z*t  - z
│ │ │ │  
│ │ │ │  o20 : R
│ │ │ │  i21 : time f2 = resultant(I_1,f1,t)
│ │ │ │ - -- used 0.0727722s (cpu); 0.0727847s (thread); 0s (gc)
│ │ │ │ + -- used 0.0396745s (cpu); 0.0396846s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2
│ │ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8
│ │ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_end__Package.html
│ │ │ @@ -159,15 +159,15 @@
│ │ │                                      Version => 0.0
│ │ │               package prefix => /usr/
│ │ │               PackageIsLoaded => true
│ │ │               pkgname => Foo
│ │ │               private dictionary => Foo#"private dictionary"
│ │ │               processed documentation => MutableHashTable{}
│ │ │               raw documentation => MutableHashTable{}
│ │ │ -             source directory => /tmp/M2-10822-0/94-rundir/
│ │ │ +             source directory => /tmp/M2-11042-0/94-rundir/
│ │ │               source file => stdio
│ │ │               test inputs => MutableList{}
    
│ │ │              
    
│ │ │                
    i7 : dictionaryPath
│ │ │ ├── html2text {}
│ │ │ │ @@ -77,15 +77,15 @@
│ │ │ │                                      Version => 0.0
│ │ │ │               package prefix => /usr/
│ │ │ │               PackageIsLoaded => true
│ │ │ │               pkgname => Foo
│ │ │ │               private dictionary => Foo#"private dictionary"
│ │ │ │               processed documentation => MutableHashTable{}
│ │ │ │               raw documentation => MutableHashTable{}
│ │ │ │ -             source directory => /tmp/M2-10822-0/94-rundir/
│ │ │ │ +             source directory => /tmp/M2-11042-0/94-rundir/
│ │ │ │               source file => stdio
│ │ │ │               test inputs => MutableList{}
│ │ │ │  i7 : dictionaryPath
│ │ │ │  
│ │ │ │  o7 = {Foo.Dictionary, Varieties.Dictionary, Isomorphism.Dictionary,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       Truncations.Dictionary, Polyhedra.Dictionary, Saturation.Dictionary,
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Exists.html
│ │ │ @@ -73,29 +73,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -515,27 +515,27 @@
│ │ │  o36 = 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11189-0/0
    
│ │ │ +o1 = /tmp/M2-11759-0/0
│ │ │              
    
│ │ │                
    i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
    
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11189-0/0
│ │ │ +o3 = /tmp/M2-11759-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : fileExists fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,21 +10,21 @@
│ │ │ │      * Inputs:
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Outputs:
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether a file with the filename or path fn exists
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11189-0/0
│ │ │ │ +o1 = /tmp/M2-11759-0/0
│ │ │ │  i2 : fileExists fn
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11189-0/0
│ │ │ │ +o3 = /tmp/M2-11759-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : fileExists fn
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : removeFile fn
│ │ │ │  If fn refers to a symbolic link, then whether the file exists is determined by
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Length.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │          
    Description
    
│ │ │          
    The length of an open output file is determined from the internal count of the number of bytes written so far.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : f = temporaryFileName() << "hi there"
│ │ │  
│ │ │ -o1 = /tmp/M2-12838-0/0
│ │ │ +o1 = /tmp/M2-15118-0/0
│ │ │  
│ │ │  o1 : File
    
│ │ │              
    
│ │ │                
    i2 : fileLength f
│ │ │ @@ -90,24 +90,24 @@
│ │ │  o2 = 8
    
│ │ │              
    
│ │ │                
    i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-12838-0/0
│ │ │ +o3 = /tmp/M2-15118-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-12838-0/0
    
│ │ │ +o4 = /tmp/M2-15118-0/0
│ │ │              
    
│ │ │                
    i5 : fileLength filename
│ │ │  
│ │ │  o5 = 8
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,28 +12,28 @@
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the length of the file f or the file whose name is f
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The length of an open output file is determined from the internal count of the
│ │ │ │  number of bytes written so far.
│ │ │ │  i1 : f = temporaryFileName() << "hi there"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12838-0/0
│ │ │ │ +o1 = /tmp/M2-15118-0/0
│ │ │ │  
│ │ │ │  o1 : File
│ │ │ │  i2 : fileLength f
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : close f
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12838-0/0
│ │ │ │ +o3 = /tmp/M2-15118-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : filename = toString f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12838-0/0
│ │ │ │ +o4 = /tmp/M2-15118-0/0
│ │ │ │  i5 : fileLength filename
│ │ │ │  
│ │ │ │  o5 = 8
│ │ │ │  i6 : get filename
│ │ │ │  
│ │ │ │  o6 = hi there
│ │ │ │  i7 : length oo
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__File_rp.html
│ │ │ @@ -74,22 +74,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -430,15 +430,15 @@
│ │ │  
│ │ │  o33 = 1
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12044-0/0
    
│ │ │ +o1 = /tmp/M2-13494-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-12044-0/0
│ │ │ +o2 = /tmp/M2-13494-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : fileMode f
│ │ │ @@ -97,15 +97,15 @@
│ │ │  o3 = 420
    
│ │ │              
    
│ │ │                
    i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-12044-0/0
│ │ │ +o4 = /tmp/M2-13494-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : removeFile fn
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,26 +11,26 @@
│ │ │ │      * Inputs:
│ │ │ │            o f, a _f_i_l_e
│ │ │ │      * Outputs:
│ │ │ │            o the mode of the open file f
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12044-0/0
│ │ │ │ +o1 = /tmp/M2-13494-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12044-0/0
│ │ │ │ +o2 = /tmp/M2-13494-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode f
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : close f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12044-0/0
│ │ │ │ +o4 = /tmp/M2-13494-0/0
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _f_i_l_e_M_o_d_e_(_F_i_l_e_) -- get file mode
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__String_rp.html
│ │ │ @@ -74,22 +74,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11658-0/0
    
│ │ │ +o1 = /tmp/M2-12708-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11658-0/0
│ │ │ +o2 = /tmp/M2-12708-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │      * Inputs:
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the mode of the file located at the filename or path fn
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11658-0/0
│ │ │ │ +o1 = /tmp/M2-12708-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11658-0/0
│ │ │ │ +o2 = /tmp/M2-12708-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode fn
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__Z__Z_cm__File_rp.html
│ │ │ @@ -78,22 +78,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -250,23 +250,23 @@
│ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │        00000000000000185613
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11523-0/0
    
│ │ │ +o1 = /tmp/M2-12433-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11523-0/0
│ │ │ +o2 = /tmp/M2-12433-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : m = 7 + 7*8 + 7*64
│ │ │ @@ -113,15 +113,15 @@
│ │ │  o5 = 511
    
│ │ │              
    
│ │ │                
    i6 : close f
│ │ │  
│ │ │ -o6 = /tmp/M2-11523-0/0
│ │ │ +o6 = /tmp/M2-12433-0/0
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,30 +12,30 @@
│ │ │ │            o mo, an _i_n_t_e_g_e_r
│ │ │ │            o f, a _f_i_l_e
│ │ │ │      * Consequences:
│ │ │ │            o the mode of the open file f is set to mo
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11523-0/0
│ │ │ │ +o1 = /tmp/M2-12433-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11523-0/0
│ │ │ │ +o2 = /tmp/M2-12433-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : m = 7 + 7*8 + 7*64
│ │ │ │  
│ │ │ │  o3 = 511
│ │ │ │  i4 : fileMode(m,f)
│ │ │ │  i5 : fileMode f
│ │ │ │  
│ │ │ │  o5 = 511
│ │ │ │  i6 : close f
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11523-0/0
│ │ │ │ +o6 = /tmp/M2-12433-0/0
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : fileMode fn
│ │ │ │  
│ │ │ │  o7 = 511
│ │ │ │  i8 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__Z__Z_cm__String_rp.html
│ │ │ @@ -78,22 +78,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12665-0/0
    
│ │ │ +o1 = /tmp/M2-14765-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12665-0/0
│ │ │ +o2 = /tmp/M2-14765-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : m = fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Consequences:
│ │ │ │            o the mode of the file located at the filename or path fn is set to
│ │ │ │              mo
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12665-0/0
│ │ │ │ +o1 = /tmp/M2-14765-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12665-0/0
│ │ │ │ +o2 = /tmp/M2-14765-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : m = fileMode fn
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : fileMode(m|7,fn)
│ │ │ │  i5 : fileMode fn
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Time.html
│ │ │ @@ -81,15 +81,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  The value is the number of seconds since 00:00:00 1970-01-01 UTC, the beginning of the epoch, so the number of seconds ago a file or directory was modified may be found by using the following code.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : currentTime() - fileTime "."
│ │ │  
│ │ │ -o1 = 83
    
│ │ │ +o1 = 71
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │              returns null if no error occurs
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The value is the number of seconds since 00:00:00 1970-01-01 UTC, the beginning
│ │ │ │  of the epoch, so the number of seconds ago a file or directory was modified may
│ │ │ │  be found by using the following code.
│ │ │ │  i1 : currentTime() - fileTime "."
│ │ │ │  
│ │ │ │ -o1 = 83
│ │ │ │ +o1 = 71
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_u_r_r_e_n_t_T_i_m_e -- get the current time
│ │ │ │      * _f_i_l_e_ _m_a_n_i_p_u_l_a_t_i_o_n -- Unix file manipulation functions
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _f_i_l_e_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_force__G__B_lp..._cm__Syzygy__Matrix_eq_gt..._rp.html
│ │ │ @@ -125,15 +125,15 @@
│ │ │  o6 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i7 : syz f
│ │ │  
│ │ │ -   -- registering gb 0 at 0x7fa67066e000
│ │ │ +   -- registering gb 0 at 0x7fa3403fa540
│ │ │  
│ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb elements = 3
│ │ │     -- number of monomials                = 9
│ │ │     -- #reduction steps = 6
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │       {3} | x2-3  0     -z4+2 |
│ │ │ │       {4} | 0     x2-3  y3-1  |
│ │ │ │  
│ │ │ │               3      3
│ │ │ │  o6 : Matrix R  <-- R
│ │ │ │  i7 : syz f
│ │ │ │  
│ │ │ │ -   -- registering gb 0 at 0x7fa67066e000
│ │ │ │ +   -- registering gb 0 at 0x7fa3403fa540
│ │ │ │  
│ │ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb
│ │ │ │  elements = 3
│ │ │ │     -- number of monomials                = 9
│ │ │ │     -- #reduction steps = 6
│ │ │ │     -- #spairs done = 6
│ │ │ │     -- ncalls = 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_get.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │                
    i3 : removeFile "test-file"
    
│ │ │              
    
│ │ │                
    i4 : get "!date"
│ │ │  
│ │ │ -o4 = Mon May 18 12:33:37 UTC 2026
    
│ │ │ +o4 = Wed May 20 17:22:31 UTC 2026
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  o1 : File
│ │ │ │  i2 : get "test-file"
│ │ │ │  
│ │ │ │  o2 = hi there
│ │ │ │  i3 : removeFile "test-file"
│ │ │ │  i4 : get "!date"
│ │ │ │  
│ │ │ │ -o4 = Mon May 18 12:33:37 UTC 2026
│ │ │ │ +o4 = Wed May 20 17:22:31 UTC 2026
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d -- read from a file
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ │ │      * _c_l_o_s_e -- close a file
│ │ │ │      * _F_i_l_e_ _<_<_ _T_h_i_n_g -- print to a file
│ │ │ │  ********** WWaayyss ttoo uussee ggeett:: **********
│ │ │ │      * get(File)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_instances.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │                 defaultPrecision => 53
│ │ │                 engineDebugLevel => 0
│ │ │                 errorDepth => 0
│ │ │                 gbTrace => 0
│ │ │                 interpreterDepth => 1
│ │ │                 lineNumber => 2
│ │ │                 loadDepth => 3
│ │ │ -               maxAllowableThreads => 7
│ │ │ +               maxAllowableThreads => 17
│ │ │                 maxExponent => 1073741823
│ │ │                 minExponent => -1073741824
│ │ │                 numTBBThreads => 0
│ │ │                 o1 => 2432902008176640000
│ │ │                 oo => 2432902008176640000
│ │ │                 printingAccuracy => -1
│ │ │                 printingLeadLimit => 5
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │                 defaultPrecision => 53
│ │ │ │                 engineDebugLevel => 0
│ │ │ │                 errorDepth => 0
│ │ │ │                 gbTrace => 0
│ │ │ │                 interpreterDepth => 1
│ │ │ │                 lineNumber => 2
│ │ │ │                 loadDepth => 3
│ │ │ │ -               maxAllowableThreads => 7
│ │ │ │ +               maxAllowableThreads => 17
│ │ │ │                 maxExponent => 1073741823
│ │ │ │                 minExponent => -1073741824
│ │ │ │                 numTBBThreads => 0
│ │ │ │                 o1 => 2432902008176640000
│ │ │ │                 oo => 2432902008176640000
│ │ │ │                 printingAccuracy => -1
│ │ │ │                 printingLeadLimit => 5
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Directory.html
│ │ │ @@ -80,22 +80,22 @@
│ │ │  o1 = true
    
│ │ │              
    
│ │ │                
    i2 : fn = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11011-0/0
    
│ │ │ +o2 = /tmp/M2-11401-0/0
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11011-0/0
│ │ │ +o3 = /tmp/M2-11401-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : isDirectory fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether fn is the path to a directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : isDirectory "."
│ │ │ │  
│ │ │ │  o1 = true
│ │ │ │  i2 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11011-0/0
│ │ │ │ +o2 = /tmp/M2-11401-0/0
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11011-0/0
│ │ │ │ +o3 = /tmp/M2-11401-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : isDirectory fn
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Pseudoprime_lp__Z__Z_rp.html
│ │ │ @@ -216,15 +216,15 @@
│ │ │  
│ │ │  o18 = false
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime facs = factor(m*m1)
│ │ │ - -- 3.70437s elapsed
│ │ │ + -- 5.14056s elapsed
│ │ │  
│ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │  
│ │ │  o19 : Expression of class Product
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isPrime m3
│ │ │ - -- .0563182s elapsed
│ │ │ + -- .0602779s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000101339s elapsed
│ │ │ + -- .000124916s elapsed
│ │ │  
│ │ │  o24 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -80,15 +80,15 @@
│ │ │ │  i17 : isPrime (m*m1)
│ │ │ │  
│ │ │ │  o17 = false
│ │ │ │  i18 : isPrime(m*m*m1*m1*m2^6)
│ │ │ │  
│ │ │ │  o18 = false
│ │ │ │  i19 : elapsedTime facs = factor(m*m1)
│ │ │ │ - -- 3.70437s elapsed
│ │ │ │ + -- 5.14056s elapsed
│ │ │ │  
│ │ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │ │  
│ │ │ │  o19 : Expression of class Product
│ │ │ │  i20 : facs = facs//toList/toList
│ │ │ │  
│ │ │ │  o20 = {{1000000000000000000000000000057, 1},
│ │ │ │ @@ -98,19 +98,19 @@
│ │ │ │  o20 : List
│ │ │ │  i21 : assert(set facs === set {{m,1}, {m1,1}})
│ │ │ │  i22 : m3 = nextPrime (m^3)
│ │ │ │  
│ │ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │ │        00000000000000185613
│ │ │ │  i23 : elapsedTime isPrime m3
│ │ │ │ - -- .0563182s elapsed
│ │ │ │ + -- .0602779s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ │ - -- .000101339s elapsed
│ │ │ │ + -- .000124916s elapsed
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_P_r_i_m_e_(_Z_Z_) -- whether a integer or polynomial is prime
│ │ │ │      * _f_a_c_t_o_r_(_Z_Z_) -- factor a ring element
│ │ │ │      * _n_e_x_t_P_r_i_m_e_(_N_u_m_b_e_r_) -- compute the smallest prime greater than or equal to
│ │ │ │        a given number
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Regular__File.html
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  In UNIX, a regular file is one that is not special in some way.  Special files include symbolic links and directories.  A regular file is a sequence of bytes stored permanently in a file system.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12876-0/0
    
│ │ │ +o1 = /tmp/M2-15196-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12876-0/0
│ │ │ +o2 = /tmp/M2-15196-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : isRegularFile fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether fn is the path to a regular file
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  In UNIX, a regular file is one that is not special in some way. Special files
│ │ │ │  include symbolic links and directories. A regular file is a sequence of bytes
│ │ │ │  stored permanently in a file system.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12876-0/0
│ │ │ │ +o1 = /tmp/M2-15196-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12876-0/0
│ │ │ │ +o2 = /tmp/M2-15196-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : isRegularFile fn
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_make__Directory_lp__String_rp.html
│ │ │ @@ -81,22 +81,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the name of the newly made directory
│ │ │ │      * Consequences:
│ │ │ │            o the directory is made, with as many new path components as needed
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11391-0/0
│ │ │ │ +o1 = /tmp/M2-12161-0/0
│ │ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11391-0/0/a/b/c
│ │ │ │ +o2 = /tmp/M2-12161-0/0/a/b/c
│ │ │ │  i3 : removeDirectory (dir|"/a/b/c")
│ │ │ │  i4 : removeDirectory (dir|"/a/b")
│ │ │ │  i5 : removeDirectory (dir|"/a")
│ │ │ │  A filename starting with ~/ will have the tilde replaced by the home directory.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_k_d_i_r
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_max__Allowable__Threads.html
│ │ │ @@ -69,15 +69,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11391-0/0
    
│ │ │ +o1 = /tmp/M2-12161-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-11391-0/0/a/b/c
    
│ │ │ +o2 = /tmp/M2-12161-0/0/a/b/c
│ │ │              
    
│ │ │                
    i3 : removeDirectory (dir|"/a/b/c")
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : maxAllowableThreads
│ │ │  
│ │ │ -o1 = 7
    
│ │ │ +o1 = 17
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,15 +9,15 @@
│ │ │ │      *   Usage:
│ │ │ │              maxAllowableThreads
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the maximum number to which _a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s can be set
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : maxAllowableThreads
│ │ │ │  
│ │ │ │ -o1 = 7
│ │ │ │ +o1 = 17
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_a_r_a_l_l_e_l_ _p_r_o_g_r_a_m_m_i_n_g_ _w_i_t_h_ _t_h_r_e_a_d_s_ _a_n_d_ _t_a_s_k_s
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _m_a_x_A_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s is an _i_n_t_e_g_e_r.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.05+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_threads.m2:502:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_memoize.html
│ │ │ @@ -66,15 +66,15 @@
│ │ │  
│ │ │  o1 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i2 : time fib 28
│ │ │ - -- used 1.62905s (cpu); 0.809345s (thread); 0s (gc)
│ │ │ + -- used 1.04466s (cpu); 0.621649s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
    
│ │ │              
    
│ │ │                
    i3 : fib = memoize fib
│ │ │ @@ -83,23 +83,23 @@
│ │ │  
│ │ │  o3 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i4 : time fib 28
│ │ │ - -- used 6.8288e-05s (cpu); 6.7878e-05s (thread); 0s (gc)
│ │ │ + -- used 7.2385e-05s (cpu); 7.0083e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
    
│ │ │              
    
│ │ │                
    i5 : time fib 28
│ │ │ - -- used 4.348e-06s (cpu); 3.998e-06s (thread); 0s (gc)
│ │ │ + -- used 3.748e-06s (cpu); 3.183e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 514229
    
│ │ │              
    
│ │ │          
    An optional second argument to memoize provides a list of initial values, each of the form x => v, where v is the value to be provided for the argument x.
    
│ │ │          
    Alternatively, values can be provided after defining the memoized function using the syntax f x = v. A slightly more efficient implementation of the above would be
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,28 +11,28 @@
│ │ │ │  arguments are presented.
│ │ │ │  i1 : fib = n -> if n <= 1 then 1 else fib(n-1) + fib(n-2)
│ │ │ │  
│ │ │ │  o1 = fib
│ │ │ │  
│ │ │ │  o1 : FunctionClosure
│ │ │ │  i2 : time fib 28
│ │ │ │ - -- used 1.62905s (cpu); 0.809345s (thread); 0s (gc)
│ │ │ │ + -- used 1.04466s (cpu); 0.621649s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 514229
│ │ │ │  i3 : fib = memoize fib
│ │ │ │  
│ │ │ │  o3 = fib
│ │ │ │  
│ │ │ │  o3 : FunctionClosure
│ │ │ │  i4 : time fib 28
│ │ │ │ - -- used 6.8288e-05s (cpu); 6.7878e-05s (thread); 0s (gc)
│ │ │ │ + -- used 7.2385e-05s (cpu); 7.0083e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 514229
│ │ │ │  i5 : time fib 28
│ │ │ │ - -- used 4.348e-06s (cpu); 3.998e-06s (thread); 0s (gc)
│ │ │ │ + -- used 3.748e-06s (cpu); 3.183e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 514229
│ │ │ │  An optional second argument to memoize provides a list of initial values, each
│ │ │ │  of the form x => v, where v is the value to be provided for the argument x.
│ │ │ │  Alternatively, values can be provided after defining the memoized function
│ │ │ │  using the syntax f x = v. A slightly more efficient implementation of the above
│ │ │ │  would be
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_methods.html
│ │ │ @@ -94,20 +94,20 @@
│ │ │       {12 => (poincare, BettiTally)                                }
│ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {14 => (degree, BettiTally)                                  }
│ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │       {17 => (regularity, BettiTally)                              }
│ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ -     {19 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {20 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {21 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {22 => (codim, BettiTally)                                   }
│ │ │ -     {23 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ -     {24 => (dual, BettiTally)                                    }
│ │ │ +     {19 => (codim, BettiTally)                                   }
│ │ │ +     {20 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {21 => (dual, BettiTally)                                    }
│ │ │ +     {22 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ +     {23 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ +     {24 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
    
│ │ │                
    
│ │ │ @@ -193,20 +193,20 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                  
    i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │ -o5 = {0 => (+, Matrix, Matrix)                                   }
│ │ │ -     {1 => (-, Matrix, Matrix)                                   }
│ │ │ -     {2 => (contract, Matrix, Matrix)                            }
│ │ │ -     {3 => (diff', Matrix, Matrix)                               }
│ │ │ +o5 = {0 => (diff, Matrix, Matrix)                                }
│ │ │ +     {1 => (contract, Matrix, Matrix)                            }
│ │ │ +     {2 => (+, Matrix, Matrix)                                   }
│ │ │ +     {3 => (-, Matrix, Matrix)                                   }
│ │ │       {4 => (contract', Matrix, Matrix)                           }
│ │ │ -     {5 => (diff, Matrix, Matrix)                                }
│ │ │ +     {5 => (diff', Matrix, Matrix)                               }
│ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ @@ -223,16 +223,16 @@
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │       {24 => (%, Matrix, Matrix)                                  }
│ │ │       {25 => (remainder, Matrix, Matrix)                          }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │       {28 => (tensor, Matrix, Matrix)                             }
│ │ │       {29 => (intersect, Matrix, Matrix)                          }
│ │ │ -     {30 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ -     {31 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {30 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {31 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,20 +29,20 @@
│ │ │ │       {12 => (poincare, BettiTally)                                }
│ │ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │ │       {14 => (degree, BettiTally)                                  }
│ │ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │ │       {17 => (regularity, BettiTally)                              }
│ │ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ │ -     {19 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │ -     {20 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ -     {21 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ -     {22 => (codim, BettiTally)                                   }
│ │ │ │ -     {23 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ -     {24 => (dual, BettiTally)                                    }
│ │ │ │ +     {19 => (codim, BettiTally)                                   }
│ │ │ │ +     {20 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ +     {21 => (dual, BettiTally)                                    }
│ │ │ │ +     {22 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │ +     {23 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ +     {24 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │ │  
│ │ │ │  o1 : NumberedVerticalList
│ │ │ │  i2 : methods resolution
│ │ │ │  
│ │ │ │  o2 = {0 => (freeResolution, Ideal)        }
│ │ │ │       {1 => (freeResolution, MonomialIdeal)}
│ │ │ │ @@ -85,20 +85,20 @@
│ │ │ │      * Inputs:
│ │ │ │            o X, a _t_y_p_e
│ │ │ │            o Y, a _t_y_p_e
│ │ │ │      * Outputs:
│ │ │ │            o a _v_e_r_t_i_c_a_l_ _l_i_s_t of those methods associated with
│ │ │ │  i5 : methods( Matrix, Matrix )
│ │ │ │  
│ │ │ │ -o5 = {0 => (+, Matrix, Matrix)                                   }
│ │ │ │ -     {1 => (-, Matrix, Matrix)                                   }
│ │ │ │ -     {2 => (contract, Matrix, Matrix)                            }
│ │ │ │ -     {3 => (diff', Matrix, Matrix)                               }
│ │ │ │ +o5 = {0 => (diff, Matrix, Matrix)                                }
│ │ │ │ +     {1 => (contract, Matrix, Matrix)                            }
│ │ │ │ +     {2 => (+, Matrix, Matrix)                                   }
│ │ │ │ +     {3 => (-, Matrix, Matrix)                                   }
│ │ │ │       {4 => (contract', Matrix, Matrix)                           }
│ │ │ │ -     {5 => (diff, Matrix, Matrix)                                }
│ │ │ │ +     {5 => (diff', Matrix, Matrix)                               }
│ │ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ │ @@ -115,16 +115,16 @@
│ │ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ │       {24 => (%, Matrix, Matrix)                                  }
│ │ │ │       {25 => (remainder, Matrix, Matrix)                          }
│ │ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ │       {28 => (tensor, Matrix, Matrix)                             }
│ │ │ │       {29 => (intersect, Matrix, Matrix)                          }
│ │ │ │ -     {30 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │ -     {31 => (pullback, Matrix, Matrix)                           }
│ │ │ │ +     {30 => (pullback, Matrix, Matrix)                           }
│ │ │ │ +     {31 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_minimal__Betti.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime C = minimalBetti I
│ │ │ - -- 1.8029s elapsed
│ │ │ + -- 2.39876s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -130,15 +130,15 @@
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ - -- .759513s elapsed
│ │ │ + -- 1.00799s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7
│ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │           0: 1  .   .   .   .    .   .   .
│ │ │           1: . 35 140 189  84    .   .   .
│ │ │           2: .  .   . 196 735 1080 735 196
│ │ │  
│ │ │ @@ -151,15 +151,15 @@
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ - -- .0955615s elapsed
│ │ │ + -- .0418058s elapsed
│ │ │  
│ │ │              0  1   2   3  4
│ │ │  o7 = total: 1 35 140 189 84
│ │ │           0: 1  .   .   .  .
│ │ │           1: . 35 140 189 84
│ │ │  
│ │ │  o7 : BettiTally
    
│ │ │ @@ -171,15 +171,15 @@
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │              
    
│ │ │                
    i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ - -- 1.28856s elapsed
│ │ │ + -- 1.64957s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5
│ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │           0: 1  .   .   .   .    .
│ │ │           1: . 35 140 189  84    .
│ │ │           2: .  .   . 196 735 1080
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = minimalBetti I
│ │ │ │ - -- 1.8029s elapsed
│ │ │ │ + -- 2.39876s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -60,40 +60,40 @@
│ │ │ │  o3 : BettiTally
│ │ │ │  One can compute smaller parts of the Betti table, by using _D_e_g_r_e_e_L_i_m_i_t and/or
│ │ │ │  _L_e_n_g_t_h_L_i_m_i_t.
│ │ │ │  i4 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ │ - -- .759513s elapsed
│ │ │ │ + -- 1.00799s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7
│ │ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │ │           0: 1  .   .   .   .    .   .   .
│ │ │ │           1: . 35 140 189  84    .   .   .
│ │ │ │           2: .  .   . 196 735 1080 735 196
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ │ │  i6 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ │ - -- .0955615s elapsed
│ │ │ │ + -- .0418058s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3  4
│ │ │ │  o7 = total: 1 35 140 189 84
│ │ │ │           0: 1  .   .   .  .
│ │ │ │           1: . 35 140 189 84
│ │ │ │  
│ │ │ │  o7 : BettiTally
│ │ │ │  i8 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o8 : Ideal of S
│ │ │ │  i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ │ - -- 1.28856s elapsed
│ │ │ │ + -- 1.64957s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5
│ │ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │ │           0: 1  .   .   .   .    .
│ │ │ │           1: . 35 140 189  84    .
│ │ │ │           2: .  .   . 196 735 1080
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_mkdir.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │          
    Description
    
│ │ │          
    Only one directory will be made, so the components of the path p other than the last must already exist.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -96,15 +96,15 @@
│ │ │  o3 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11410-0/0/
    
│ │ │ +o1 = /tmp/M2-12200-0/0/
│ │ │              
    
│ │ │                
    i2 : mkdir p
    
│ │ │              
    
│ │ │                
    i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11410-0/0/foo
│ │ │ +o4 = /tmp/M2-12200-0/0/foo
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : get fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,22 +12,22 @@
│ │ │ │      * Consequences:
│ │ │ │            o a directory will be created at the path p
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Only one directory will be made, so the components of the path p other than the
│ │ │ │  last must already exist.
│ │ │ │  i1 : p = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11410-0/0/
│ │ │ │ +o1 = /tmp/M2-12200-0/0/
│ │ │ │  i2 : mkdir p
│ │ │ │  i3 : isDirectory p
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11410-0/0/foo
│ │ │ │ +o4 = /tmp/M2-12200-0/0/foo
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : get fn
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : removeFile fn
│ │ │ │  i7 : removeDirectory p
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_move__File_lp__String_cm__String_rp.html
│ │ │ @@ -86,52 +86,52 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,32 +20,32 @@
│ │ │ │            o the name of the backup file if one was created, or _n_u_l_l
│ │ │ │      * Consequences:
│ │ │ │            o the file will be moved by creating a new link to the file and
│ │ │ │              removing the old one
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11246-0/0
│ │ │ │ +o1 = /tmp/M2-11876-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11246-0/1
│ │ │ │ +o2 = /tmp/M2-11876-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11246-0/0
│ │ │ │ +o3 = /tmp/M2-11876-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ │ ---moving: /tmp/M2-11246-0/0 -> /tmp/M2-11246-0/1
│ │ │ │ +--moving: /tmp/M2-11876-0/0 -> /tmp/M2-11876-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ │ ---backup file created: /tmp/M2-11246-0/1.bak
│ │ │ │ +--backup file created: /tmp/M2-11876-0/1.bak
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11246-0/1.bak
│ │ │ │ +o6 = /tmp/M2-11876-0/1.bak
│ │ │ │  i7 : removeFile bak
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_p_y_F_i_l_e
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * moveFile(String)
│ │ │ │      * _m_o_v_e_F_i_l_e_(_S_t_r_i_n_g_,_S_t_r_i_n_g_)
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_nanosleep.html
│ │ │ @@ -56,15 +56,15 @@
│ │ │        
    nanosleep -- sleep for a given number of nanoseconds
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │  nanosleep n -- sleeps for n nanoseconds.        
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11246-0/0
    
│ │ │ +o1 = /tmp/M2-11876-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11246-0/1
    
│ │ │ +o2 = /tmp/M2-11876-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11246-0/0
│ │ │ +o3 = /tmp/M2-11876-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-11246-0/0 -> /tmp/M2-11246-0/1
    
│ │ │ +--moving: /tmp/M2-11876-0/0 -> /tmp/M2-11876-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-11246-0/1.bak
│ │ │ +--backup file created: /tmp/M2-11876-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-11246-0/1.bak
    
│ │ │ +o6 = /tmp/M2-11876-0/1.bak
│ │ │              
    
│ │ │                
    i7 : removeFile bak
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime nanosleep 500000000
│ │ │ - -- .500144s elapsed
│ │ │ + -- .500162s elapsed
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -4,15 +4,15 @@
│ │ │ │  [q                   ]
│ │ │ │  _n_e_x_t | _p_r_e_v_i_o_u_s | _f_o_r_w_a_r_d | _b_a_c_k_w_a_r_d | _u_p | _i_n_d_e_x | _t_o_c
│ │ │ │  ===============================================================================
│ │ │ │  ************ nnaannoosslleeeepp ---- sslleeeepp ffoorr aa ggiivveenn nnuummbbeerr ooff nnaannoosseeccoonnddss ************
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  nanosleep n -- sleeps for n nanoseconds.
│ │ │ │  i1 : elapsedTime nanosleep 500000000
│ │ │ │ - -- .500144s elapsed
│ │ │ │ + -- .500162s elapsed
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_l_e_e_p -- sleep for a while
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _n_a_n_o_s_l_e_e_p is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_options_lp__Function_rp.html
│ │ │ @@ -108,37 +108,37 @@
│ │ │  o3 : OptionTable
    
│ │ │              
    
│ │ │                
    i4 : methods codim
│ │ │  
│ │ │ -o4 = {0 => (codim, Module)        }
│ │ │ -     {1 => (codim, CoherentSheaf) }
│ │ │ -     {2 => (codim, Variety)       }
│ │ │ -     {3 => (codim, MonomialIdeal) }
│ │ │ -     {4 => (codim, Ideal)         }
│ │ │ -     {5 => (codim, PolynomialRing)}
│ │ │ -     {6 => (codim, BettiTally)    }
│ │ │ -     {7 => (codim, QuotientRing)  }
│ │ │ +o4 = {0 => (codim, Ideal)         }
│ │ │ +     {1 => (codim, PolynomialRing)}
│ │ │ +     {2 => (codim, BettiTally)    }
│ │ │ +     {3 => (codim, QuotientRing)  }
│ │ │ +     {4 => (codim, Module)        }
│ │ │ +     {5 => (codim, CoherentSheaf) }
│ │ │ +     {6 => (codim, Variety)       }
│ │ │ +     {7 => (codim, MonomialIdeal) }
│ │ │  
│ │ │  o4 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i5 : options oo
│ │ │  
│ │ │  o5 = {0 => (OptionTable{Generic => false})}
│ │ │       {1 => (OptionTable{Generic => false})}
│ │ │ -     {2 => (OptionTable{Generic => false})}
│ │ │ +     {2 => (OptionTable{})                }
│ │ │       {3 => (OptionTable{Generic => false})}
│ │ │       {4 => (OptionTable{Generic => false})}
│ │ │       {5 => (OptionTable{Generic => false})}
│ │ │ -     {6 => (OptionTable{})                }
│ │ │ +     {6 => (OptionTable{Generic => false})}
│ │ │       {7 => (OptionTable{Generic => false})}
│ │ │  
│ │ │  o5 : NumberedVerticalList
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,33 +34,33 @@
│ │ │ │  i3 : options(codim, Ideal)
│ │ │ │  
│ │ │ │  o3 = OptionTable{Generic => false}
│ │ │ │  
│ │ │ │  o3 : OptionTable
│ │ │ │  i4 : methods codim
│ │ │ │  
│ │ │ │ -o4 = {0 => (codim, Module)        }
│ │ │ │ -     {1 => (codim, CoherentSheaf) }
│ │ │ │ -     {2 => (codim, Variety)       }
│ │ │ │ -     {3 => (codim, MonomialIdeal) }
│ │ │ │ -     {4 => (codim, Ideal)         }
│ │ │ │ -     {5 => (codim, PolynomialRing)}
│ │ │ │ -     {6 => (codim, BettiTally)    }
│ │ │ │ -     {7 => (codim, QuotientRing)  }
│ │ │ │ +o4 = {0 => (codim, Ideal)         }
│ │ │ │ +     {1 => (codim, PolynomialRing)}
│ │ │ │ +     {2 => (codim, BettiTally)    }
│ │ │ │ +     {3 => (codim, QuotientRing)  }
│ │ │ │ +     {4 => (codim, Module)        }
│ │ │ │ +     {5 => (codim, CoherentSheaf) }
│ │ │ │ +     {6 => (codim, Variety)       }
│ │ │ │ +     {7 => (codim, MonomialIdeal) }
│ │ │ │  
│ │ │ │  o4 : NumberedVerticalList
│ │ │ │  i5 : options oo
│ │ │ │  
│ │ │ │  o5 = {0 => (OptionTable{Generic => false})}
│ │ │ │       {1 => (OptionTable{Generic => false})}
│ │ │ │ -     {2 => (OptionTable{Generic => false})}
│ │ │ │ +     {2 => (OptionTable{})                }
│ │ │ │       {3 => (OptionTable{Generic => false})}
│ │ │ │       {4 => (OptionTable{Generic => false})}
│ │ │ │       {5 => (OptionTable{Generic => false})}
│ │ │ │ -     {6 => (OptionTable{})                }
│ │ │ │ +     {6 => (OptionTable{Generic => false})}
│ │ │ │       {7 => (OptionTable{Generic => false})}
│ │ │ │  
│ │ │ │  o5 : NumberedVerticalList
│ │ │ │  i6 : methods intersect
│ │ │ │  
│ │ │ │  o6 = {0 => (intersect, List)                           }
│ │ │ │       {1 => (intersect, RRi, RRi)                       }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallel_spprogramming_spwith_spthreads_spand_sptasks.html
│ │ │ @@ -77,21 +77,21 @@
│ │ │              
│ │ │                
    i2 : L = shuffle toList (1..10000);
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ - -- .58009s elapsed
    
│ │ │ + -- .692008s elapsed
│ │ │              
    
│ │ │                
    i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .275306s elapsed
    
│ │ │ + -- .186927s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    You will have to try it on your examples to see how much they speed up.
    
│ │ │            
    Warning: Threads computing in parallel can give wrong answers if their code is not "thread safe", meaning they make modifications to memory without ensuring the modifications get safely communicated to other threads. (Thread safety can slow computations some.) Currently, modifications to Macaulay2 variables and mutable hash tables are thread safe, but not changes inside mutable lists. Also, access to external libraries such as singular, etc., may not currently be thread safe.
    
│ │ │            
    The rest of this document describes how to control parallel tasks more directly.
    
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o5 = 5
    
│ │ │              
    
│ │ │                
    i6 : allowableThreads = maxAllowableThreads
│ │ │  
│ │ │ -o6 = 7
    
│ │ │ +o6 = 17
│ │ │              
    
│ │ │          
    
│ │ │            
    To run a function in another thread use schedule, as in the following example.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -155,15 +155,15 @@
│ │ │            
    Note that schedule returns a task, not the result of the computation, which will be accessible only after the task has completed the computation.
    
│ │ │          
│ │ │          
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : t
│ │ │  
│ │ │ -o11 = <<task, running>>
│ │ │ +o11 = <<task, created>>
│ │ │  
│ │ │  o11 : Task
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Use isReady to check whether the result is available yet.
    
│ │ │ @@ -229,15 +229,15 @@
│ │ │                
    i17 : schedule t';
    
│ │ │              
    
│ │ │                
    i18 : t'
│ │ │  
│ │ │ -o18 = <<task, running>>
│ │ │ +o18 = <<task, created>>
│ │ │  
│ │ │  o18 : Task
    
│ │ │              
    
│ │ │                
    i19 : taskResult t'
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,17 +17,17 @@
│ │ │ │  big computation. If the list is long, it will be split into chunks for each
│ │ │ │  core, reducing the overhead. But the speedup is still limited by the different
│ │ │ │  threads competing for memory, including cpu caches; it is like running
│ │ │ │  Macaulay2 on a computer that is running other big programs at the same time. We
│ │ │ │  can see this using _e_l_a_p_s_e_d_T_i_m_e.
│ │ │ │  i2 : L = shuffle toList (1..10000);
│ │ │ │  i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ │ - -- .58009s elapsed
│ │ │ │ + -- .692008s elapsed
│ │ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ │ - -- .275306s elapsed
│ │ │ │ + -- .186927s elapsed
│ │ │ │  You will have to try it on your examples to see how much they speed up.
│ │ │ │  Warning: Threads computing in parallel can give wrong answers if their code is
│ │ │ │  not "thread safe", meaning they make modifications to memory without ensuring
│ │ │ │  the modifications get safely communicated to other threads. (Thread safety can
│ │ │ │  slow computations some.) Currently, modifications to Macaulay2 variables and
│ │ │ │  mutable hash tables are thread safe, but not changes inside mutable lists.
│ │ │ │  Also, access to external libraries such as singular, etc., may not currently be
│ │ │ │ @@ -39,15 +39,15 @@
│ │ │ │  _a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s, and may be examined and changed as follows. (_a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s
│ │ │ │  is temporarily increased if necessary inside _p_a_r_a_l_l_e_l_A_p_p_l_y.)
│ │ │ │  i5 : allowableThreads
│ │ │ │  
│ │ │ │  o5 = 5
│ │ │ │  i6 : allowableThreads = maxAllowableThreads
│ │ │ │  
│ │ │ │ -o6 = 7
│ │ │ │ +o6 = 17
│ │ │ │  To run a function in another thread use _s_c_h_e_d_u_l_e, as in the following example.
│ │ │ │  i7 : R = QQ[x,y,z];
│ │ │ │  i8 : I = ideal(x^2 + 2*y^2 - y - 2*z, x^2 - 8*y^2 + 10*z - 1, x^2 - 7*y*z)
│ │ │ │  
│ │ │ │               2     2            2     2             2
│ │ │ │  o8 = ideal (x  + 2y  - y - 2z, x  - 8y  + 10z - 1, x  - 7y*z)
│ │ │ │  
│ │ │ │ @@ -62,15 +62,15 @@
│ │ │ │  o10 = <>
│ │ │ │  
│ │ │ │  o10 : Task
│ │ │ │  Note that _s_c_h_e_d_u_l_e returns a task, not the result of the computation, which
│ │ │ │  will be accessible only after the task has completed the computation.
│ │ │ │  i11 : t
│ │ │ │  
│ │ │ │ -o11 = <>
│ │ │ │ +o11 = <>
│ │ │ │  
│ │ │ │  o11 : Task
│ │ │ │  Use _i_s_R_e_a_d_y to check whether the result is available yet.
│ │ │ │  i12 : isReady t
│ │ │ │  
│ │ │ │  o12 = false
│ │ │ │  To wait for the result and then retrieve it, use _t_a_s_k_R_e_s_u_l_t.
│ │ │ │ @@ -92,15 +92,15 @@
│ │ │ │  o16 = <>
│ │ │ │  
│ │ │ │  o16 : Task
│ │ │ │  Start it running with _s_c_h_e_d_u_l_e.
│ │ │ │  i17 : schedule t';
│ │ │ │  i18 : t'
│ │ │ │  
│ │ │ │ -o18 = <>
│ │ │ │ +o18 = <>
│ │ │ │  
│ │ │ │  o18 : Task
│ │ │ │  i19 : taskResult t'
│ │ │ │  
│ │ │ │  o19 = | 980z2-18y-201z+13 35yz-4y+2z-1 10y2-y-12z+1 5x2-4y+2z-1 |
│ │ │ │  
│ │ │ │                1      4
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallelism_spin_spengine_spcomputations.html
│ │ │ @@ -142,15 +142,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalBetti I
│ │ │ - -- 1.88605s elapsed
│ │ │ + -- 2.44512s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -165,15 +165,15 @@
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ - -- 1.73807s elapsed
│ │ │ + -- 2.41731s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -195,15 +195,15 @@
│ │ │  
│ │ │  o8 = 1
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime minimalBetti(I)
│ │ │ - -- 1.63859s elapsed
│ │ │ + -- 2.50868s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -236,15 +236,15 @@
│ │ │  
│ │ │  o12 : Ideal of S
    
│ │ │              
    
│ │ │                
    i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.16228s elapsed
│ │ │ + -- 2.76842s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o13 : Complex
    
│ │ │ @@ -263,15 +263,15 @@
│ │ │  
│ │ │  o15 : Ideal of S
│ │ │              
    
│ │ │                
    i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.01403s elapsed
│ │ │ + -- 2.86879s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o16 : Complex
    
│ │ │ @@ -304,15 +304,15 @@
│ │ │  
│ │ │  o19 : Ideal of S
│ │ │              
    
│ │ │                
    i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.83683s elapsed
│ │ │ + -- 4.34814s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o20 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -327,15 +327,15 @@
│ │ │  
│ │ │  o22 : Ideal of S
│ │ │              
    
│ │ │                
    i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 7.15932s elapsed
│ │ │ + -- 8.45343s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o23 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -350,15 +350,15 @@
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │              
    
│ │ │                
    i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 5.9065s elapsed
│ │ │ + -- 3.74261s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o26 : Matrix S  <-- S
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -401,15 +401,15 @@
│ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │                 101
│ │ │              
    
│ │ │                
    i31 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.0633s elapsed
│ │ │ + -- 1.06129s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │                
    i34 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.22532s elapsed
│ │ │ + -- 1.67904s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -93,30 +93,30 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i3 : S = ring I
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : elapsedTime minimalBetti I
│ │ │ │ - -- 1.88605s elapsed
│ │ │ │ + -- 2.44512s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │           4: .  .   .   .   .    .   .   .   .  .  1
│ │ │ │  
│ │ │ │  o4 : BettiTally
│ │ │ │  i5 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ │ - -- 1.73807s elapsed
│ │ │ │ + -- 2.41731s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -126,15 +126,15 @@
│ │ │ │  i7 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o7 : Ideal of S
│ │ │ │  i8 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o8 = 1
│ │ │ │  i9 : elapsedTime minimalBetti(I)
│ │ │ │ - -- 1.63859s elapsed
│ │ │ │ + -- 2.50868s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -149,15 +149,15 @@
│ │ │ │  i11 : numTBBThreads = 0
│ │ │ │  
│ │ │ │  o11 = 0
│ │ │ │  i12 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o12 : Ideal of S
│ │ │ │  i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ │ - -- 2.16228s elapsed
│ │ │ │ + -- 2.76842s elapsed
│ │ │ │  
│ │ │ │         1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-
│ │ │ │  - S    <-- S    <-- S
│ │ │ │  
│ │ │ │  
│ │ │ │ @@ -168,15 +168,15 @@
│ │ │ │  i14 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o14 = 1
│ │ │ │  i15 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o15 : Ideal of S
│ │ │ │  i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ │ - -- 2.01403s elapsed
│ │ │ │ + -- 2.86879s elapsed
│ │ │ │  
│ │ │ │         1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-
│ │ │ │  - S    <-- S    <-- S
│ │ │ │  
│ │ │ │  
│ │ │ │ @@ -195,37 +195,37 @@
│ │ │ │  o18 = S
│ │ │ │  
│ │ │ │  o18 : PolynomialRing
│ │ │ │  i19 : I = ideal random(S^1, S^{4:-5});
│ │ │ │  
│ │ │ │  o19 : Ideal of S
│ │ │ │  i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 4.83683s elapsed
│ │ │ │ + -- 4.34814s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o20 : Matrix S  <-- S
│ │ │ │  i21 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o21 = 1
│ │ │ │  i22 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o22 : Ideal of S
│ │ │ │  i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 7.15932s elapsed
│ │ │ │ + -- 8.45343s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o23 : Matrix S  <-- S
│ │ │ │  i24 : numTBBThreads = 10
│ │ │ │  
│ │ │ │  o24 = 10
│ │ │ │  i25 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o25 : Ideal of S
│ │ │ │  i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 5.9065s elapsed
│ │ │ │ + -- 3.74261s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o26 : Matrix S  <-- S
│ │ │ │  For Gröbner basis computation in associative algebras, ParallelizeByDegree is
│ │ │ │  not relevant. In this case, use numTBBThreads to control the amount of
│ │ │ │  parallelism.
│ │ │ │  i27 : needsPackage "AssociativeAlgebras"
│ │ │ │ @@ -246,15 +246,15 @@
│ │ │ │                 2                         2                         2
│ │ │ │  o30 = ideal (5a  + 2b*c + 3c*b, 3a*c + 5b  + 2c*a, 2a*b + 3b*a + 5c )
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │ │                 101
│ │ │ │  i31 : elapsedTime NCGB(I, 22);
│ │ │ │ - -- 1.0633s elapsed
│ │ │ │ + -- 1.06129s elapsed
│ │ │ │  
│ │ │ │                 ZZ            1       ZZ            148
│ │ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │ │                101                   101
│ │ │ │  i32 : I = ideal I_*
│ │ │ │  
│ │ │ │                 2                         2                         2
│ │ │ │ @@ -263,15 +263,15 @@
│ │ │ │                  ZZ
│ │ │ │  o32 : Ideal of ---<|a, b, c|>
│ │ │ │                 101
│ │ │ │  i33 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o33 = 1
│ │ │ │  i34 : elapsedTime NCGB(I, 22);
│ │ │ │ - -- 1.22532s elapsed
│ │ │ │ + -- 1.67904s elapsed
│ │ │ │  
│ │ │ │                 ZZ            1       ZZ            148
│ │ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │ │                101                   101
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_i_m_a_l_B_e_t_t_i -- minimal betti numbers of (the minimal free resolution of)
│ │ │ │        a homogeneous ideal or module
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_poincare.html
│ │ │ @@ -375,36 +375,36 @@
│ │ │  
│ │ │  o27 = 3
│ │ │              
    
│ │ │                
    i28 : time poincare I
│ │ │ - -- used 0.00344891s (cpu); 1.8505e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00296572s (cpu); 1.3502e-05s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 16 at 0x7f3752420380
│ │ │ +   -- registering gb 16 at 0x7fa3a54508c0
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 4186
│ │ │     -- #reduction steps = 38
│ │ │     -- #spairs done = 11
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 29
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0165272s (cpu); 0.0183577s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00903562s (cpu); 0.0120044s (thread); 0s (gc)
│ │ │  
│ │ │                1      11
│ │ │  o29 : Matrix R  <-- R
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -416,15 +416,15 @@
│ │ │                
    i30 : R = QQ[a..d];
    
│ │ │              
    
│ │ │                
    i31 : I = ideal random(R^1, R^{3:-3});
│ │ │  
│ │ │ -   -- registering gb 17 at 0x7f37524201c0
│ │ │ +   -- registering gb 17 at 0x7fa3a5450700
│ │ │  
│ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │     -- number of monomials                = 0
│ │ │     -- #reduction steps = 0
│ │ │     -- #spairs done = 0
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -433,24 +433,24 @@
│ │ │  o31 : Ideal of R
    
│ │ │              
    
│ │ │                
    i32 : time p = poincare I
│ │ │  
│ │ │ -   -- registering gb 18 at 0x7f3752420000
│ │ │ +   -- registering gb 18 at 0x7fa3a5450540
│ │ │  
│ │ │     -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 267
│ │ │     -- #reduction steps = 236
│ │ │     -- #spairs done = 30
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 20
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.00799635s (cpu); 0.00712039s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00398872s (cpu); 0.00434384s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7f3754432e00
│ │ │ +   -- registering gb 19 at 0x7fa3a5450380
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}(3,9)m
│ │ │     -- mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
│ │ │     -- {17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m{24}(1,3)m
│ │ │     -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements = 39
│ │ │     -- number of monomials                = 1051
│ │ │     -- #reduction steps = 284
│ │ │     -- #spairs done = 53
│ │ │     -- ncalls = 46
│ │ │     -- nloop = 54
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0759331s (cpu); 0.0769218s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0520013s (cpu); 0.0517201s (thread); 0s (gc)
│ │ │  
│ │ │                1      39
│ │ │  o37 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -177,66 +177,66 @@
│ │ │ │  o26 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o26 : ZZ[T]
│ │ │ │  i27 : gbTrace = 3
│ │ │ │  
│ │ │ │  o27 = 3
│ │ │ │  i28 : time poincare I
│ │ │ │ - -- used 0.00344891s (cpu); 1.8505e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00296572s (cpu); 1.3502e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o28 : ZZ[T]
│ │ │ │  i29 : time gens gb I;
│ │ │ │  
│ │ │ │ -   -- registering gb 16 at 0x7f3752420380
│ │ │ │ +   -- registering gb 16 at 0x7fa3a54508c0
│ │ │ │  
│ │ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number of
│ │ │ │  (nonminimal) gb elements = 11
│ │ │ │     -- number of monomials                = 4186
│ │ │ │     -- #reduction steps = 38
│ │ │ │     -- #spairs done = 11
│ │ │ │     -- ncalls = 10
│ │ │ │     -- nloop = 29
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.0165272s (cpu); 0.0183577s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.00903562s (cpu); 0.0120044s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      11
│ │ │ │  o29 : Matrix R  <-- R
│ │ │ │  In this case, the savings is minimal, but often it can be dramatic. Another
│ │ │ │  important situation is to compute a Gröbner basis using a different monomial
│ │ │ │  order.
│ │ │ │  i30 : R = QQ[a..d];
│ │ │ │  i31 : I = ideal random(R^1, R^{3:-3});
│ │ │ │  
│ │ │ │ -   -- registering gb 17 at 0x7f37524201c0
│ │ │ │ +   -- registering gb 17 at 0x7fa3a5450700
│ │ │ │  
│ │ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │ │     -- number of monomials                = 0
│ │ │ │     -- #reduction steps = 0
│ │ │ │     -- #spairs done = 0
│ │ │ │     -- ncalls = 0
│ │ │ │     -- nloop = 0
│ │ │ │     -- nsaved = 0
│ │ │ │     --
│ │ │ │  o31 : Ideal of R
│ │ │ │  i32 : time p = poincare I
│ │ │ │  
│ │ │ │ -   -- registering gb 18 at 0x7f3752420000
│ │ │ │ +   -- registering gb 18 at 0x7fa3a5450540
│ │ │ │  
│ │ │ │     -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of
│ │ │ │  (nonminimal) gb elements = 11
│ │ │ │     -- number of monomials                = 267
│ │ │ │     -- #reduction steps = 236
│ │ │ │     -- #spairs done = 30
│ │ │ │     -- ncalls = 10
│ │ │ │     -- nloop = 20
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.00799635s (cpu); 0.00712039s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.00398872s (cpu); 0.00434384s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o32 : ZZ[T]
│ │ │ │  i33 : S = QQ[a..d, MonomialOrder => Eliminate 2]
│ │ │ │  
│ │ │ │ @@ -281,30 +281,30 @@
│ │ │ │  
│ │ │ │  o35 : ZZ[T]
│ │ │ │  i36 : gbTrace = 3
│ │ │ │  
│ │ │ │  o36 = 3
│ │ │ │  i37 : time gens gb J;
│ │ │ │  
│ │ │ │ -   -- registering gb 19 at 0x7f3754432e00
│ │ │ │ +   -- registering gb 19 at 0x7fa3a5450380
│ │ │ │  
│ │ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}
│ │ │ │  (3,9)m
│ │ │ │     -- mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
│ │ │ │     -- {17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m
│ │ │ │  {24}(1,3)m
│ │ │ │     -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements
│ │ │ │  = 39
│ │ │ │     -- number of monomials                = 1051
│ │ │ │     -- #reduction steps = 284
│ │ │ │     -- #spairs done = 53
│ │ │ │     -- ncalls = 46
│ │ │ │     -- nloop = 54
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.0759331s (cpu); 0.0769218s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0520013s (cpu); 0.0517201s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      39
│ │ │ │  o37 : Matrix S  <-- S
│ │ │ │  i38 : selectInSubring(1, gens gb J)
│ │ │ │  
│ │ │ │  o38 = | 188529931266160087758259645374082357642621166724936033369975727480205
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_printing_spto_spa_spfile.html
│ │ │ @@ -102,22 +102,22 @@
│ │ │  o2 : File
│ │ │              
    
│ │ │                
    i3 : fn = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11601-0/0
    
│ │ │ +o3 = /tmp/M2-12591-0/0
│ │ │              
    
│ │ │                
    i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11601-0/0
│ │ │ +o4 = /tmp/M2-12591-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : get fn
│ │ │ @@ -156,15 +156,15 @@
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : fn << f << close
│ │ │  
│ │ │ -o9 = /tmp/M2-11601-0/0
│ │ │ +o9 = /tmp/M2-12591-0/0
│ │ │  
│ │ │  o9 : File
    
│ │ │              
    
│ │ │                
    i10 : get fn
│ │ │ @@ -174,15 +174,15 @@
│ │ │        + 1
    
│ │ │              
    
│ │ │                
    i11 : fn << toExternalString f << close
│ │ │  
│ │ │ -o11 = /tmp/M2-11601-0/0
│ │ │ +o11 = /tmp/M2-12591-0/0
│ │ │  
│ │ │  o11 : File
    
│ │ │              
    
│ │ │                
    i12 : get fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,18 +36,18 @@
│ │ │ │  -- ho there --
│ │ │ │  
│ │ │ │  o2 = stdio
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11601-0/0
│ │ │ │ +o3 = /tmp/M2-12591-0/0
│ │ │ │  i4 : fn << "hi there" << endl << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11601-0/0
│ │ │ │ +o4 = /tmp/M2-12591-0/0
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : get fn
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : R = QQ[x]
│ │ │ │  
│ │ │ │ @@ -66,25 +66,25 @@
│ │ │ │   10      9      8       7       6       5       4       3      2
│ │ │ │  x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x + 1
│ │ │ │  o8 = stdio
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : fn << f << close
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-11601-0/0
│ │ │ │ +o9 = /tmp/M2-12591-0/0
│ │ │ │  
│ │ │ │  o9 : File
│ │ │ │  i10 : get fn
│ │ │ │  
│ │ │ │  o10 =  10      9      8       7       6       5       4       3      2
│ │ │ │        x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x
│ │ │ │        + 1
│ │ │ │  i11 : fn << toExternalString f << close
│ │ │ │  
│ │ │ │ -o11 = /tmp/M2-11601-0/0
│ │ │ │ +o11 = /tmp/M2-12591-0/0
│ │ │ │  
│ │ │ │  o11 : File
│ │ │ │  i12 : get fn
│ │ │ │  
│ │ │ │  o12 = x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+
│ │ │ │        1
│ │ │ │  i13 : value get fn
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_process__I__D.html
│ │ │ @@ -69,15 +69,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : processID()
│ │ │  
│ │ │ -o1 = 10822
    
│ │ │ +o1 = 11042
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,15 +8,15 @@
│ │ │ │      *   Usage:
│ │ │ │              processID()
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the process identifier of the current Macaulay2 process
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : processID()
│ │ │ │  
│ │ │ │ -o1 = 10822
│ │ │ │ +o1 = 11042
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _g_r_o_u_p_I_D -- the process group identifier
│ │ │ │      * _s_e_t_G_r_o_u_p_I_D -- set the process group identifier
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _p_r_o_c_e_s_s_I_D is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_profile.html
│ │ │ @@ -96,35 +96,35 @@
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -136,15 +136,15 @@
│ │ │  
│ │ │  o9 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -178,15 +178,15 @@
│ │ │  o14 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : profileSummary
│ │ │  
│ │ │  o2 = #run  %time   position                         
│ │ │ -     1     94.41   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ -     1     91.82   ../../m2/matrix1.m2:275:22-275:43
│ │ │ -     1     44.77   ../../m2/matrix1.m2:183:25-183:52
│ │ │ -     1     30.99   ../../m2/matrix1.m2:104:5-146:72 
│ │ │ -     1     29.81   ../../m2/matrix1.m2:130:10-145:16
│ │ │ -     1     22.38   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ -     1     21.25   ../../m2/matrix1.m2:35:10-39:22  
│ │ │ -     1     21.09   ../../m2/set.m2:129:5-129:61     
│ │ │ -     1     3.31    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ -     1     2.27    ../../m2/matrix1.m2:131:13-131:78
│ │ │ -     1     2.12    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ -     1     1.43    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ -     1     1.35    ../../m2/matrix1.m2:137:20-137:64
│ │ │ -     1     1.2     ../../m2/matrix1.m2:270:4-271:73 
│ │ │ -     1     1.14    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ -     1     1.10    ../../m2/matrix1.m2:88:10-88:46  
│ │ │ -     1     1.03    ../../m2/matrix1.m2:172:4-174:74 
│ │ │ -     1     .77     ../../m2/modules.m2:282:4-282:52 
│ │ │ -     20    .62     ../../m2/matrix1.m2:181:14-182:67
│ │ │ -     20    .53     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ -     1     .0046s  elapsed total                    
    
│ │ │ +     1     93.28   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ +     1     90.3    ../../m2/matrix1.m2:275:22-275:43
│ │ │ +     1     43.71   ../../m2/matrix1.m2:183:25-183:52
│ │ │ +     1     30.68   ../../m2/matrix1.m2:104:5-146:72 
│ │ │ +     1     29.53   ../../m2/matrix1.m2:130:10-145:16
│ │ │ +     1     22.96   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ +     1     21.42   ../../m2/set.m2:129:5-129:61     
│ │ │ +     1     21.16   ../../m2/matrix1.m2:35:10-39:22  
│ │ │ +     1     3.14    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ +     1     2.40    ../../m2/matrix1.m2:131:13-131:78
│ │ │ +     1     1.91    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ +     1     1.49    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ +     1     1.32    ../../m2/matrix1.m2:137:20-137:64
│ │ │ +     1     1.31    ../../m2/matrix1.m2:270:4-271:73 
│ │ │ +     1     1.08    ../../m2/matrix1.m2:172:4-174:74 
│ │ │ +     1     1.05    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ +     1     .98     ../../m2/matrix1.m2:88:10-88:46  
│ │ │ +     20    .95     ../../m2/matrix1.m2:181:14-182:67
│ │ │ +     19    .71     ../../m2/set.m2:129:36-129:41    
│ │ │ +     20    .68     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ +     1     .0039s  elapsed total                    
│ │ │              
    
│ │ │                
    i3 : coverageSummary
│ │ │  
│ │ │  o3 = covered lines:
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,35 +25,35 @@
│ │ │ │                4       5
│ │ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │ │  Afterwards, running profileSummary and coverageSummary produces easy to read
│ │ │ │  tables summarizing the accumulated data so far in different ways.
│ │ │ │  i2 : profileSummary
│ │ │ │  
│ │ │ │  o2 = #run  %time   position
│ │ │ │ -     1     94.41   ../../m2/matrix1.m2:273:4-276:58
│ │ │ │ -     1     91.82   ../../m2/matrix1.m2:275:22-275:43
│ │ │ │ -     1     44.77   ../../m2/matrix1.m2:183:25-183:52
│ │ │ │ -     1     30.99   ../../m2/matrix1.m2:104:5-146:72
│ │ │ │ -     1     29.81   ../../m2/matrix1.m2:130:10-145:16
│ │ │ │ -     1     22.38   ../../m2/matrix1.m2:171:4-171:42
│ │ │ │ -     1     21.25   ../../m2/matrix1.m2:35:10-39:22
│ │ │ │ -     1     21.09   ../../m2/set.m2:129:5-129:61
│ │ │ │ -     1     3.31    ../../m2/matrix1.m2:102:5-102:29
│ │ │ │ -     1     2.27    ../../m2/matrix1.m2:131:13-131:78
│ │ │ │ -     1     2.12    ../../m2/matrix1.m2:86:5-99:11
│ │ │ │ -     1     1.43    ../../m2/matrix1.m2:275:7-275:16
│ │ │ │ -     1     1.35    ../../m2/matrix1.m2:137:20-137:64
│ │ │ │ -     1     1.2     ../../m2/matrix1.m2:270:4-271:73
│ │ │ │ -     1     1.14    ../../m2/matrix1.m2:101:5-101:91
│ │ │ │ -     1     1.10    ../../m2/matrix1.m2:88:10-88:46
│ │ │ │ -     1     1.03    ../../m2/matrix1.m2:172:4-174:74
│ │ │ │ -     1     .77     ../../m2/modules.m2:282:4-282:52
│ │ │ │ -     20    .62     ../../m2/matrix1.m2:181:14-182:67
│ │ │ │ -     20    .53     ../../m2/matrix1.m2:37:43-37:71
│ │ │ │ -     1     .0046s  elapsed total
│ │ │ │ +     1     93.28   ../../m2/matrix1.m2:273:4-276:58
│ │ │ │ +     1     90.3    ../../m2/matrix1.m2:275:22-275:43
│ │ │ │ +     1     43.71   ../../m2/matrix1.m2:183:25-183:52
│ │ │ │ +     1     30.68   ../../m2/matrix1.m2:104:5-146:72
│ │ │ │ +     1     29.53   ../../m2/matrix1.m2:130:10-145:16
│ │ │ │ +     1     22.96   ../../m2/matrix1.m2:171:4-171:42
│ │ │ │ +     1     21.42   ../../m2/set.m2:129:5-129:61
│ │ │ │ +     1     21.16   ../../m2/matrix1.m2:35:10-39:22
│ │ │ │ +     1     3.14    ../../m2/matrix1.m2:102:5-102:29
│ │ │ │ +     1     2.40    ../../m2/matrix1.m2:131:13-131:78
│ │ │ │ +     1     1.91    ../../m2/matrix1.m2:86:5-99:11
│ │ │ │ +     1     1.49    ../../m2/matrix1.m2:275:7-275:16
│ │ │ │ +     1     1.32    ../../m2/matrix1.m2:137:20-137:64
│ │ │ │ +     1     1.31    ../../m2/matrix1.m2:270:4-271:73
│ │ │ │ +     1     1.08    ../../m2/matrix1.m2:172:4-174:74
│ │ │ │ +     1     1.05    ../../m2/matrix1.m2:101:5-101:91
│ │ │ │ +     1     .98     ../../m2/matrix1.m2:88:10-88:46
│ │ │ │ +     20    .95     ../../m2/matrix1.m2:181:14-182:67
│ │ │ │ +     19    .71     ../../m2/set.m2:129:36-129:41
│ │ │ │ +     20    .68     ../../m2/matrix1.m2:37:43-37:71
│ │ │ │ +     1     .0039s  elapsed total
│ │ │ │  i3 : coverageSummary
│ │ │ │  
│ │ │ │  o3 = covered lines:
│ │ │ │       ../../m2/lists.m2:146:24-146:32
│ │ │ │       ../../m2/lists.m2:146:34-146:58
│ │ │ │       ../../m2/matrix.m2:30:5-30:35
│ │ │ │       ../../m2/matrix.m2:31:5-31:46
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random__K__Rational__Point.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o5 : Sequence
    
│ │ │              
    
│ │ │                
    i6 : time randomKRationalPoint(I)
│ │ │ - -- used 0.441238s (cpu); 0.177282s (thread); 0s (gc)
│ │ │ + -- used 0.585026s (cpu); 0.154271s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal (x  - 53x , x  + 8x , x  - 4x )
│ │ │               2      3   1     3   0     3
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i10 : time randomKRationalPoint(I)
│ │ │ - -- used 0.254059s (cpu); 0.19724s (thread); 0s (gc)
│ │ │ + -- used 0.455107s (cpu); 0.273439s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │  
│ │ │  o10 : Ideal of R
    
│ │ │              
    
│ │ │                
    i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c->degree c);
│ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ - -- used 4.21024s (cpu); 2.05106s (thread); 0s (gc)
│ │ │ + -- used 4.65365s (cpu); 2.24557s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 58
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : codim I, degree I
│ │ │ │  
│ │ │ │  o5 = (2, 10)
│ │ │ │  
│ │ │ │  o5 : Sequence
│ │ │ │  i6 : time randomKRationalPoint(I)
│ │ │ │ - -- used 0.441238s (cpu); 0.177282s (thread); 0s (gc)
│ │ │ │ + -- used 0.585026s (cpu); 0.154271s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = ideal (x  - 53x , x  + 8x , x  - 4x )
│ │ │ │               2      3   1     3   0     3
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : R=kk[x_0..x_5];
│ │ │ │  i8 : I=minors(3,random(R^5,R^{3:-1}));
│ │ │ │ @@ -45,15 +45,15 @@
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : codim I, degree I
│ │ │ │  
│ │ │ │  o9 = (3, 10)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : time randomKRationalPoint(I)
│ │ │ │ - -- used 0.254059s (cpu); 0.19724s (thread); 0s (gc)
│ │ │ │ + -- used 0.455107s (cpu); 0.273439s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  The claim that $63 \%$ of the intersections contain a K-rational point can be
│ │ │ │  experimentally tested:
│ │ │ │ @@ -69,15 +69,15 @@
│ │ │ │  o13 : RR (of precision 53)
│ │ │ │  i14 : I=ideal random(n,R);
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c-
│ │ │ │  >degree c);
│ │ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ │ - -- used 4.21024s (cpu); 2.05106s (thread); 0s (gc)
│ │ │ │ + -- used 4.65365s (cpu); 2.24557s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 58
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommKKRRaattiioonnaallPPooiinntt:: **********
│ │ │ │      * randomKRationalPoint(Ideal)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_a_n_d_o_m_K_R_a_t_i_o_n_a_l_P_o_i_n_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_read__Directory.html
│ │ │ @@ -73,38 +73,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12234-0/0
    
│ │ │ +o1 = /tmp/M2-13884-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-12234-0/0
    
│ │ │ +o2 = /tmp/M2-13884-0/0
│ │ │              
    
│ │ │                
    i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-12234-0/0/foo
│ │ │ +o3 = /tmp/M2-13884-0/0/foo
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : readDirectory dir
│ │ │  
│ │ │ -o4 = {., .., foo}
│ │ │ +o4 = {.., ., foo}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : removeFile fn
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,26 +10,26 @@
│ │ │ │      * Inputs:
│ │ │ │            o dir, a _s_t_r_i_n_g, a filename or path to a directory
│ │ │ │      * Outputs:
│ │ │ │            o a _l_i_s_t, the list of filenames stored in the directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12234-0/0
│ │ │ │ +o1 = /tmp/M2-13884-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12234-0/0
│ │ │ │ +o2 = /tmp/M2-13884-0/0
│ │ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12234-0/0/foo
│ │ │ │ +o3 = /tmp/M2-13884-0/0/foo
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : readDirectory dir
│ │ │ │  
│ │ │ │ -o4 = {., .., foo}
│ │ │ │ +o4 = {.., ., foo}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : removeFile fn
│ │ │ │  i6 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_m_o_v_e_D_i_r_e_c_t_o_r_y -- remove a directory
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_reading_spfiles.html
│ │ │ @@ -57,22 +57,22 @@
│ │ │        
    
│ │ │  Sometimes a file will contain a single expression whose value you wish to have access to.  For example, it might be a polynomial produced by another program.  The function get can be used to obtain the entire contents of a file as a single string.  We illustrate this here with a file whose name is expression.        
    
│ │ │  First we create the file by writing the desired text to it.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11776-0/0
    
│ │ │ +o1 = /tmp/M2-12946-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3" << endl << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11776-0/0
│ │ │ +o2 = /tmp/M2-12946-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │  Now we get the contents of the file, as a single string.        
│ │ │            
│ │ │ @@ -121,15 +121,15 @@
│ │ │  Often a file will contain code written in the Macaulay2 language. Let's create such a file.        
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : fn << "sample = 2^100
│ │ │       print sample
│ │ │       " << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11776-0/0
│ │ │ +o7 = /tmp/M2-12946-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │  Now verify that it contains the desired text with get.        
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,20 +8,20 @@
│ │ │ │  Sometimes a file will contain a single expression whose value you wish to have
│ │ │ │  access to. For example, it might be a polynomial produced by another program.
│ │ │ │  The function _g_e_t can be used to obtain the entire contents of a file as a
│ │ │ │  single string. We illustrate this here with a file whose name is expression.
│ │ │ │  First we create the file by writing the desired text to it.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11776-0/0
│ │ │ │ +o1 = /tmp/M2-12946-0/0
│ │ │ │  i2 : fn <<
│ │ │ │  "z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3"
│ │ │ │  << endl << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11776-0/0
│ │ │ │ +o2 = /tmp/M2-12946-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  Now we get the contents of the file, as a single string.
│ │ │ │  i3 : get fn
│ │ │ │  
│ │ │ │  o3 = z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2
│ │ │ │       +8*y^3
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  o6 : Expression of class Product
│ │ │ │  Often a file will contain code written in the Macaulay2 language. Let's create
│ │ │ │  such a file.
│ │ │ │  i7 : fn << "sample = 2^100
│ │ │ │       print sample
│ │ │ │       " << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11776-0/0
│ │ │ │ +o7 = /tmp/M2-12946-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  Now verify that it contains the desired text with _g_e_t.
│ │ │ │  i8 : get fn
│ │ │ │  
│ │ │ │  o8 = sample = 2^100
│ │ │ │       print sample
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_readlink.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │            o fn, a _s_t_r_i_n_g, a filename or path to a file
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the resolved path to a symbolic link, or null if the file
│ │ │ │              was not a symbolic link.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : p = temporaryFileName ()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12494-0/0
│ │ │ │ +o1 = /tmp/M2-14404-0/0
│ │ │ │  i2 : symlinkFile ("foo", p)
│ │ │ │  i3 : readlink p
│ │ │ │  
│ │ │ │  o3 = foo
│ │ │ │  i4 : removeFile p
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_l_p_a_t_h -- convert a filename to one passing through no symbolic links
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_realpath.html
│ │ │ @@ -73,57 +73,57 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName ()
│ │ │  
│ │ │ -o1 = /tmp/M2-12494-0/0
    
│ │ │ +o1 = /tmp/M2-14404-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile ("foo", p)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -135,15 +135,15 @@
│ │ │          
    
│ │ │                
    i1 : realpath "."
│ │ │  
│ │ │ -o1 = /tmp/M2-10822-0/89-rundir/
    
│ │ │ +o1 = /tmp/M2-11042-0/89-rundir/
│ │ │              
    
│ │ │                
    i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-12513-0/0
    
│ │ │ +o2 = /tmp/M2-14443-0/0
│ │ │              
    
│ │ │                
    i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-12513-0/1
    
│ │ │ +o3 = /tmp/M2-14443-0/1
│ │ │              
    
│ │ │                
    i4 : symlinkFile(p,q)
    
│ │ │              
    
│ │ │                
    i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-12513-0/0
│ │ │ +o5 = /tmp/M2-14443-0/0
│ │ │  
│ │ │  o5 : File
    
│ │ │              
    
│ │ │                
    i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-12513-0/0
    
│ │ │ +o6 = /tmp/M2-14443-0/0
│ │ │              
    
│ │ │                
    i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-12513-0/0
    
│ │ │ +o7 = /tmp/M2-14443-0/0
│ │ │              
    
│ │ │                
    i8 : removeFile p
    
│ │ │              
    
│ │ │          
    The empty string is interpreted as a reference to the current directory.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-10822-0/89-rundir/
    
│ │ │ +o10 = /tmp/M2-11042-0/89-rundir/
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │  Every component of the path must exist in the file system and be accessible to the user. Terminal slashes will be dropped.  Warning: under most operating systems, the value returned is an absolute path (one starting at the root of the file system), but under Solaris, this system call may, in certain circumstances, return a relative path when given a relative path.      
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,39 +12,39 @@
│ │ │ │            o fn, a _s_t_r_i_n_g, a filename, or path to a file
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, a pathname to fn passing through no symbolic links, and
│ │ │ │              ending with a slash if fn refers to a directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : realpath "."
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10822-0/89-rundir/
│ │ │ │ +o1 = /tmp/M2-11042-0/89-rundir/
│ │ │ │  i2 : p = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12513-0/0
│ │ │ │ +o2 = /tmp/M2-14443-0/0
│ │ │ │  i3 : q = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12513-0/1
│ │ │ │ +o3 = /tmp/M2-14443-0/1
│ │ │ │  i4 : symlinkFile(p,q)
│ │ │ │  i5 : p << close
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-12513-0/0
│ │ │ │ +o5 = /tmp/M2-14443-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : readlink q
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12513-0/0
│ │ │ │ +o6 = /tmp/M2-14443-0/0
│ │ │ │  i7 : realpath q
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-12513-0/0
│ │ │ │ +o7 = /tmp/M2-14443-0/0
│ │ │ │  i8 : removeFile p
│ │ │ │  i9 : removeFile q
│ │ │ │  The empty string is interpreted as a reference to the current directory.
│ │ │ │  i10 : realpath ""
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-10822-0/89-rundir/
│ │ │ │ +o10 = /tmp/M2-11042-0/89-rundir/
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  Every component of the path must exist in the file system and be accessible to
│ │ │ │  the user. Terminal slashes will be dropped. Warning: under most operating
│ │ │ │  systems, the value returned is an absolute path (one starting at the root of
│ │ │ │  the file system), but under Solaris, this system call may, in certain
│ │ │ │  circumstances, return a relative path when given a relative path.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_register__Finalizer.html
│ │ │ @@ -81,23 +81,23 @@
│ │ │                  		
│ │ │                
    i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --"))
    
│ │ │              
    
│ │ │                
    i2 : collectGarbage() 
│ │ │ ---finalization: (1)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (2)[8]: -- finalizing sequence #9 --
│ │ │ +--finalization: (1)[6]: -- finalizing sequence #7 --
│ │ │ +--finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │  --finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (4)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (5)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (6)[1]: -- finalizing sequence #2 --
│ │ │ ---finalization: (7)[7]: -- finalizing sequence #8 --
│ │ │ +--finalization: (4)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (5)[7]: -- finalizing sequence #8 --
│ │ │ +--finalization: (6)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (7)[2]: -- finalizing sequence #3 --
│ │ │  --finalization: (8)[0]: -- finalizing sequence #1 --
│ │ │ ---finalization: (9)[6]: -- finalizing sequence #7 --
    
│ │ │ +--finalization: (9)[8]: -- finalizing sequence #9 --
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │  This function should mainly be used for debugging.  Having a large number of finalizers might degrade the performance of the program.  Moreover, registering two or more objects that are members of a circular chain of pointers for finalization will result in a memory leak, with none of the objects in the chain being freed, even if nothing else points to any of them.      
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,23 +14,23 @@
│ │ │ │      * Consequences:
│ │ │ │            o A finalizer is registered with the garbage collector to print a
│ │ │ │              string when that object is collected as garbage
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-
│ │ │ │  - finalizing sequence #"|i|" --"))
│ │ │ │  i2 : collectGarbage()
│ │ │ │ ---finalization: (1)[2]: -- finalizing sequence #3 --
│ │ │ │ ---finalization: (2)[8]: -- finalizing sequence #9 --
│ │ │ │ +--finalization: (1)[6]: -- finalizing sequence #7 --
│ │ │ │ +--finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │ │  --finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ │ ---finalization: (4)[5]: -- finalizing sequence #6 --
│ │ │ │ ---finalization: (5)[4]: -- finalizing sequence #5 --
│ │ │ │ ---finalization: (6)[1]: -- finalizing sequence #2 --
│ │ │ │ ---finalization: (7)[7]: -- finalizing sequence #8 --
│ │ │ │ +--finalization: (4)[1]: -- finalizing sequence #2 --
│ │ │ │ +--finalization: (5)[7]: -- finalizing sequence #8 --
│ │ │ │ +--finalization: (6)[4]: -- finalizing sequence #5 --
│ │ │ │ +--finalization: (7)[2]: -- finalizing sequence #3 --
│ │ │ │  --finalization: (8)[0]: -- finalizing sequence #1 --
│ │ │ │ ---finalization: (9)[6]: -- finalizing sequence #7 --
│ │ │ │ +--finalization: (9)[8]: -- finalizing sequence #9 --
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  This function should mainly be used for debugging. Having a large number of
│ │ │ │  finalizers might degrade the performance of the program. Moreover, registering
│ │ │ │  two or more objects that are members of a circular chain of pointers for
│ │ │ │  finalization will result in a memory leak, with none of the objects in the
│ │ │ │  chain being freed, even if nothing else points to any of them.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_remove__Directory.html
│ │ │ @@ -76,29 +76,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11448-0/0
    
│ │ │ +o1 = /tmp/M2-12278-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11448-0/0
    
│ │ │ +o2 = /tmp/M2-12278-0/0
│ │ │              
    
│ │ │                
    i3 : readDirectory dir
│ │ │  
│ │ │ -o3 = {., ..}
│ │ │ +o3 = {.., .}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : removeDirectory dir
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,21 +10,21 @@
│ │ │ │      * Inputs:
│ │ │ │            o dir, a _s_t_r_i_n_g, a filename or path to a directory
│ │ │ │      * Consequences:
│ │ │ │            o the directory is removed
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11448-0/0
│ │ │ │ +o1 = /tmp/M2-12278-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11448-0/0
│ │ │ │ +o2 = /tmp/M2-12278-0/0
│ │ │ │  i3 : readDirectory dir
│ │ │ │  
│ │ │ │ -o3 = {., ..}
│ │ │ │ +o3 = {.., .}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d_D_i_r_e_c_t_o_r_y -- read the contents of a directory
│ │ │ │      * _m_a_k_e_D_i_r_e_c_t_o_r_y -- make a directory
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__Path.html
│ │ │ @@ -70,22 +70,22 @@
│ │ │          
    Description
    
│ │ │          
    This string may be concatenated with an absolute path to get one understandable by external programs.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10914-0/0
    
│ │ │ +o1 = /tmp/M2-11204-0/0
│ │ │              
    
│ │ │                
    i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-10914-0/0
    
│ │ │ +o2 = /tmp/M2-11204-0/0
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            o a _s_t_r_i_n_g, the path, as seen by external programs, to the root of
│ │ │ │              the file system seen by Macaulay2
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This string may be concatenated with an absolute path to get one understandable
│ │ │ │  by external programs.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10914-0/0
│ │ │ │ +o1 = /tmp/M2-11204-0/0
│ │ │ │  i2 : rootPath | fn
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10914-0/0
│ │ │ │ +o2 = /tmp/M2-11204-0/0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_o_o_t_U_R_I
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_o_o_t_P_a_t_h is a _s_t_r_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.05+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2025:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__U__R__I.html
│ │ │ @@ -70,22 +70,22 @@
│ │ │          
      Description
      
│ │ │          
      This string may be concatenated with an absolute path to get one understandable by an external browser.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12177-0/0
      
│ │ │ +o1 = /tmp/M2-13767-0/0
│ │ │              
      
│ │ │                
      i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-12177-0/0
      
│ │ │ +o2 = file:///tmp/M2-13767-0/0
│ │ │              
      
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            o a _s_t_r_i_n_g, the path, as seen by an external browser, to the root of
│ │ │ │              the file system seen by Macaulay2
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This string may be concatenated with an absolute path to get one understandable
│ │ │ │  by an external browser.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12177-0/0
│ │ │ │ +o1 = /tmp/M2-13767-0/0
│ │ │ │  i2 : rootURI | fn
│ │ │ │  
│ │ │ │ -o2 = file:///tmp/M2-12177-0/0
│ │ │ │ +o2 = file:///tmp/M2-13767-0/0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_o_o_t_P_a_t_h
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_o_o_t_U_R_I is a _s_t_r_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.05+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2041:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_saving_sppolynomials_spand_spmatrices_spin_spfiles.html
│ │ │ @@ -95,22 +95,22 @@
│ │ │  o4 : R-module, submodule of R
│ │ │              
    
│ │ │                
    i5 : f = temporaryFileName()
│ │ │  
│ │ │ -o5 = /tmp/M2-12025-0/0
    
│ │ │ +o5 = /tmp/M2-13455-0/0
│ │ │              
    
│ │ │                
    i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12025-0/0
│ │ │ +o6 = /tmp/M2-13455-0/0
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : get f
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,18 +28,18 @@
│ │ │ │  
│ │ │ │  o4 = image | x2 x2-y2 xyz7 |
│ │ │ │  
│ │ │ │                               1
│ │ │ │  o4 : R-module, submodule of R
│ │ │ │  i5 : f = temporaryFileName()
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-12025-0/0
│ │ │ │ +o5 = /tmp/M2-13455-0/0
│ │ │ │  i6 : f << toString (p,m,M) << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12025-0/0
│ │ │ │ +o6 = /tmp/M2-13455-0/0
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : get f
│ │ │ │  
│ │ │ │  o7 = (x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix {{x^2,
│ │ │ │       x^2-y^2, x*y*z^7}},image matrix {{x^2, x^2-y^2, x*y*z^7}})
│ │ │ │  i8 : (p',m',M') = value get f
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_solve.html
│ │ │ @@ -371,21 +371,21 @@
│ │ │              
    		
│ │ │                
    i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i30 : time X = solve(A,B);
│ │ │ - -- used 0.000195457s (cpu); 0.00018674s (thread); 0s (gc)
    
│ │ │ + -- used 0.000252739s (cpu); 0.000243923s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000140844s (cpu); 0.000141115s (thread); 0s (gc)
    
│ │ │ + -- used 0.000122025s (cpu); 0.000121211s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i32 : norm(A*X-B)
│ │ │  
│ │ │  o32 = 5.111850690840453e-15
│ │ │ @@ -416,21 +416,21 @@
│ │ │              
    		
│ │ │                
    i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i38 : time X = solve(A,B);
│ │ │ - -- used 0.239427s (cpu); 0.239428s (thread); 0s (gc)
    
│ │ │ + -- used 0.148123s (cpu); 0.148126s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.222489s (cpu); 0.222489s (thread); 0s (gc)
    
│ │ │ + -- used 0.146445s (cpu); 0.146458s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i40 : norm(A*X-B)
│ │ │  
│ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │ ├── html2text {}
│ │ │ │ @@ -192,33 +192,33 @@
│ │ │ │  i24 : printingPrecision = 4;
│ │ │ │  i25 : N = 40
│ │ │ │  
│ │ │ │  o25 = 40
│ │ │ │  i26 : A = mutableMatrix(CC_53, N, N); fillMatrix A;
│ │ │ │  i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
│ │ │ │  i30 : time X = solve(A,B);
│ │ │ │ - -- used 0.000195457s (cpu); 0.00018674s (thread); 0s (gc)
│ │ │ │ + -- used 0.000252739s (cpu); 0.000243923s (thread); 0s (gc)
│ │ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.000140844s (cpu); 0.000141115s (thread); 0s (gc)
│ │ │ │ + -- used 0.000122025s (cpu); 0.000121211s (thread); 0s (gc)
│ │ │ │  i32 : norm(A*X-B)
│ │ │ │  
│ │ │ │  o32 = 5.111850690840453e-15
│ │ │ │  
│ │ │ │  o32 : RR (of precision 53)
│ │ │ │  Over higher precision RR or CC, these routines will be much slower than the
│ │ │ │  lower precision LAPACK routines.
│ │ │ │  i33 : N = 100
│ │ │ │  
│ │ │ │  o33 = 100
│ │ │ │  i34 : A = mutableMatrix(CC_100, N, N); fillMatrix A;
│ │ │ │  i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
│ │ │ │  i38 : time X = solve(A,B);
│ │ │ │ - -- used 0.239427s (cpu); 0.239428s (thread); 0s (gc)
│ │ │ │ + -- used 0.148123s (cpu); 0.148126s (thread); 0s (gc)
│ │ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.222489s (cpu); 0.222489s (thread); 0s (gc)
│ │ │ │ + -- used 0.146445s (cpu); 0.146458s (thread); 0s (gc)
│ │ │ │  i40 : norm(A*X-B)
│ │ │ │  
│ │ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │ │  
│ │ │ │  o40 : RR (of precision 100)
│ │ │ │  Giving the option ClosestFit=>true, in the case when the field is RR or CC,
│ │ │ │  uses a least squares algorithm to find a best fit solution.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -85,93 +85,93 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11816-0/0/
    
│ │ │ +o1 = /tmp/M2-13026-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11816-0/1/
    
│ │ │ +o2 = /tmp/M2-13026-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11816-0/0/a/
    
│ │ │ +o3 = /tmp/M2-13026-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11816-0/0/b/
    
│ │ │ +o4 = /tmp/M2-13026-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11816-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-13026-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11816-0/0/a/f
│ │ │ +o6 = /tmp/M2-13026-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11816-0/0/a/g
│ │ │ +o7 = /tmp/M2-13026-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11816-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13026-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11816-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11816-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11816-0/1/a/f
    
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-13026-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-13026-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-13026-0/1/b/c/g
│ │ │              
    
│ │ │                
    i10 : get (dst|"b/c/g")
│ │ │  
│ │ │  o10 = ho there
    
│ │ │              
    
│ │ │                
    i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
│ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-11816-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11816-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11816-0/1/a/f
    
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-13026-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-13026-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-13026-0/1/b/c/g
│ │ │              
    
│ │ │  Now we remove the files and directories we created.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,53 +30,53 @@
│ │ │ │            o The directory tree rooted at src is duplicated by a directory tree
│ │ │ │              rooted at dst. The files in the source tree are represented by
│ │ │ │              relative symbolic links in the destination tree to the original
│ │ │ │              files in the source tree.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11816-0/0/
│ │ │ │ +o1 = /tmp/M2-13026-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11816-0/1/
│ │ │ │ +o2 = /tmp/M2-13026-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11816-0/0/a/
│ │ │ │ +o3 = /tmp/M2-13026-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11816-0/0/b/
│ │ │ │ +o4 = /tmp/M2-13026-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11816-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-13026-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11816-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-13026-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11816-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-13026-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11816-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-13026-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11816-0/1/b/c/g
│ │ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11816-0/1/a/g
│ │ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11816-0/1/a/f
│ │ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-13026-0/1/a/g
│ │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-13026-0/1/a/f
│ │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-13026-0/1/b/c/g
│ │ │ │  i10 : get (dst|"b/c/g")
│ │ │ │  
│ │ │ │  o10 = ho there
│ │ │ │  i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
│ │ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-11816-0/1/b/c/g
│ │ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11816-0/1/a/g
│ │ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11816-0/1/a/f
│ │ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-13026-0/1/a/g
│ │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-13026-0/1/a/f
│ │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-13026-0/1/b/c/g
│ │ │ │  Now we remove the files and directories we created.
│ │ │ │  i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ │  
│ │ │ │  o12 = rm
│ │ │ │  
│ │ │ │  o12 : FunctionClosure
│ │ │ │  i13 : scan(reverse findFiles src, rm)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__File.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │            o dst, a _s_t_r_i_n_g
│ │ │ │      * Consequences:
│ │ │ │            o a symbolic link at the location in the directory tree specified by
│ │ │ │              dst is created, pointing to src
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11873-0/0
│ │ │ │ +o1 = /tmp/M2-13143-0/0
│ │ │ │  i2 : symlinkFile("qwert", fn)
│ │ │ │  i3 : fileExists fn
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : readlink fn
│ │ │ │  
│ │ │ │  o4 = qwert
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_temporary__File__Name.html
│ │ │ @@ -69,22 +69,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  The file name is so unique that even with various suffixes appended, no collision with existing files will occur.  The files will be removed when the program terminates, unless it terminates as the result of an error.        
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11873-0/0
    
│ │ │ +o1 = /tmp/M2-13143-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile("qwert", fn)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : temporaryFileName () | ".tex"
│ │ │  
│ │ │ -o1 = /tmp/M2-12857-0/0.tex
    
│ │ │ +o1 = /tmp/M2-15157-0/0.tex
│ │ │              
    
│ │ │                
    i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-12857-0/1.html
    
│ │ │ +o2 = /tmp/M2-15157-0/1.html
│ │ │              
    
│ │ │          
    This function will work under Unix, and also under Windows if you have a directory on the same drive called /tmp.
    
│ │ │          
    If the name of the temporary file will be given to an external program, it may be necessary to concatenate it with rootPath or rootURI to enable the external program to find the file.
    
│ │ │          
    The temporary file name is derived from the value of the environment variable TMPDIR, if it has one.
    
│ │ │          
    If fork is used, then the parent and child Macaulay2 processes will each remove their own temporary files upon termination, with the parent removing any files created before fork was called.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            o a unique temporary file name.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The file name is so unique that even with various suffixes appended, no
│ │ │ │  collision with existing files will occur. The files will be removed when the
│ │ │ │  program terminates, unless it terminates as the result of an error.
│ │ │ │  i1 : temporaryFileName () | ".tex"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12857-0/0.tex
│ │ │ │ +o1 = /tmp/M2-15157-0/0.tex
│ │ │ │  i2 : temporaryFileName () | ".html"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12857-0/1.html
│ │ │ │ +o2 = /tmp/M2-15157-0/1.html
│ │ │ │  This function will work under Unix, and also under Windows if you have a
│ │ │ │  directory on the same drive called /tmp.
│ │ │ │  If the name of the temporary file will be given to an external program, it may
│ │ │ │  be necessary to concatenate it with _r_o_o_t_P_a_t_h or _r_o_o_t_U_R_I to enable the external
│ │ │ │  program to find the file.
│ │ │ │  The temporary file name is derived from the value of the environment variable
│ │ │ │  TMPDIR, if it has one.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_time.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  time e evaluates e, prints the amount of cpu time used, and returns the value of e.  The time used by the the current thread and garbage collection during the evaluation of e is also shown.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time 3^30
│ │ │ - -- used 2.0629e-05s (cpu); 1.2153e-05s (thread); 0s (gc)
│ │ │ + -- used 1.8835e-05s (cpu); 6.374e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = 205891132094649
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,15 +7,15 @@
│ │ │ │      *   Usage:
│ │ │ │              time e
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  time e evaluates e, prints the amount of cpu time used, and returns the value
│ │ │ │  of e. The time used by the the current thread and garbage collection during the
│ │ │ │  evaluation of e is also shown.
│ │ │ │  i1 : time 3^30
│ │ │ │ - -- used 2.0629e-05s (cpu); 1.2153e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.8835e-05s (cpu); 6.374e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = 205891132094649
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_timing.html
│ │ │ @@ -59,24 +59,24 @@
│ │ │  timing e evaluates e and returns a list of type Time of the form {t,v}, where t is the number of seconds of cpu timing used, and v is the value of the expression.        
    
│ │ │  The default method for printing such timing results is to display the timing separately in a comment below the computed value.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : timing 3^30
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │ -     -- .000015449 seconds
│ │ │ +     -- .000019145 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000015449, 205891132094649}
    
│ │ │ +o2 = Time{.000019145, 205891132094649}
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,20 +10,20 @@
│ │ │ │  is the number of seconds of cpu timing used, and v is the value of the
│ │ │ │  expression.
│ │ │ │  The default method for printing such timing results is to display the timing
│ │ │ │  separately in a comment below the computed value.
│ │ │ │  i1 : timing 3^30
│ │ │ │  
│ │ │ │  o1 = 205891132094649
│ │ │ │ -     -- .000015449 seconds
│ │ │ │ +     -- .000019145 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{.000015449, 205891132094649}
│ │ │ │ +o2 = Time{.000019145, 205891132094649}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _T_i_m_e -- the class of all timing results
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_version.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │                 "memtailor version" => 1.3
│ │ │                 "mpfi version" => 1.5.4
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.11.1
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.88+deb13-amd64
│ │ │ +               "operating system release" => 6.12.88+deb13-cloud-amd64
│ │ │                 "operating system" => Linux
│ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone ChainComplexExtras Varieties Schubert2 PushForward LocalRings BoijSoederberg BGG Bruns InvolutiveBases ConwayPolynomials EdgeIdeals FourTiTwo StatePolytope Polyhedra Truncations Polymake gfanInterface PieriMaps Normaliz Posets XML OpenMath SCSCP RationalPoints MapleInterface ConvexInterface SRdeformations NumericalAlgebraicGeometry BeginningMacaulay2 FormalGroupLaws Graphics WeylGroups HodgeIntegrals Cyclotomic Binomials Kronecker Nauty ToricVectorBundles ModuleDeformations PHCpack SimplicialDecomposability BooleanGB AdjointIdeal Parametrization Serialization NAGtypes NormalToricVarieties DGAlgebras Graphs GraphicalModels BIBasis KustinMiller Units NautyGraphs VersalDeformations CharacteristicClasses RandomIdeals RandomObjects RandomPlaneCurves RandomSpaceCurves RandomGenus14Curves RandomCanonicalCurves RandomCurves TensorComplexes MonomialAlgebras QthPower EliminationMatrices EllipticIntegrals Triplets CompleteIntersectionResolutions EagonResolution MCMApproximations MultiplierIdeals InvariantRing QuillenSuslin EnumerationCurves Book3264Examples WeilDivisors EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieAlgebraRepresentations ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties TestIdeals FrobeniusThresholds NonPrincipalTestIdeals Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes OldChainComplexes GroebnerWalk RandomMonomialIdeals Matroids NumericalImplicitization CoincidentRootLoci RelativeCanonicalResolution RandomCurvesOverVerySmallFiniteFields StronglyStableIdeals SLnEquivariantMatrices CorrespondenceScrolls NCAlgebra SpaceCurves ExteriorIdeals ToricInvariants SegreClasses SemidefiniteProgramming SumsOfSquares MultiGradedRationalMap AssociativeAlgebras VirtualResolutions Quasidegrees DiffAlg DeterminantalRepresentations FGLM SpechtModule SchurComplexes SimplicialPosets SlackIdeals PositivityToricBundles SparseResultants DecomposableSparseSystems MixedMultiplicity PencilsOfQuadrics ThreadedGB AdjunctionForSurfaces VectorGraphics GKMVarieties MonomialIntegerPrograms NoetherianOperators Hadamard StatGraphs GraphicalModelsMLE EigenSolver MultiplicitySequence ResolutionsOfStanleyReisnerRings NumericalLinearAlgebra ResLengthThree MonomialOrbits MultiprojectiveVarieties SpecialFanoFourfolds RationalPoints2 SuperLinearAlgebra SubalgebraBases AInfinity LinearTruncations ThinSincereQuivers Python BettiCharacters Jets FunctionFieldDesingularization HomotopyLieAlgebra TSpreadIdeals RealRoots ExteriorModules K3Surfaces GroebnerStrata QuaternaryQuartics CotangentSchubert OnlineLookup MergeTeX Probability Isomorphism CodingTheory WhitneyStratifications JSON ForeignFunctions GeometricDecomposability PseudomonomialPrimaryDecomposition PolyominoIdeals MatchingFields CellularResolutions SagbiGbDetection A1BrouwerDegrees QuadraticIdealExamplesByRoos TerraciniLoci MatrixSchubert RInterface OIGroebnerBases PlaneCurveLinearSeries Valuations SchurVeronese VNumber TropicalToric MultigradedBGG AbstractSimplicialComplexes MultigradedImplicitization Msolve Permutations SCMAlgebras NumericalSemigroups ExteriorExtensions Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases Tableaux CpMackeyFunctors JSONRPC SimplicialModules MatrixFactorizations PathSignatures MacaulayPosets MRDI EliminationTemplates WittVectors Padic
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.13.12
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2022.3
│ │ │ ├── html2text {}
│ │ │ │ @@ -66,15 +66,15 @@
│ │ │ │                 "memtailor version" => 1.3
│ │ │ │                 "mpfi version" => 1.5.4
│ │ │ │                 "mpfr version" => 4.2.2
│ │ │ │                 "mpsolve version" => 3.2.2
│ │ │ │                 "mysql version" => not present
│ │ │ │                 "normaliz version" => 3.11.1
│ │ │ │                 "ntl version" => 11.5.1
│ │ │ │ -               "operating system release" => 6.12.88+deb13-amd64
│ │ │ │ +               "operating system release" => 6.12.88+deb13-cloud-amd64
│ │ │ │                 "operating system" => Linux
│ │ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic
│ │ │ │  Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition
│ │ │ │  FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato
│ │ │ │  ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure
│ │ │ │  HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra
│ │ │ │  Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes
│ │ ├── ./usr/share/doc/Macaulay2/Markov/example-output/___Markov.out
│ │ │ @@ -70,15 +70,15 @@
│ │ │       |   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2|   1,2,2,2 2,2,2,1    1,2,2,1 2,2,2,2|
│ │ │       +-------------------------------------+-------------------------------------+
│ │ │       |- p       p        + p       p       |- p       p        + p       p       |
│ │ │       |   1,1,2,1 1,2,1,1    1,1,1,1 1,2,2,1|   1,1,2,2 1,2,1,2    1,1,1,2 1,2,2,2|
│ │ │       +-------------------------------------+-------------------------------------+
│ │ │  
│ │ │  i8 : time netList primaryDecomposition J
│ │ │ - -- used 1.73002s (cpu); 1.20682s (thread); 0s (gc)
│ │ │ + -- used 1.98805s (cpu); 1.38141s (thread); 0s (gc)
│ │ │  
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o8 = |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,2,1,2   1,2,1,1   1,1,2,2 2,1,2,1    1,1,2,1 2,1,2,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │       |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,1,2,2   1,1,2,1   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ ├── ./usr/share/doc/Macaulay2/Markov/html/index.html
│ │ │ @@ -166,15 +166,15 @@
│ │ │          
      
│ │ │            
      This ideal has 5 primary components.  The first is the one that has statistical significance. The significance of the other components is still poorly understood.
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i8 : time netList primaryDecomposition J
│ │ │ - -- used 1.73002s (cpu); 1.20682s (thread); 0s (gc)
│ │ │ + -- used 1.98805s (cpu); 1.38141s (thread); 0s (gc)
│ │ │  
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o8 = |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,2,1,2   1,2,1,1   1,1,2,2 2,1,2,1    1,1,2,1 2,1,2,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │       |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,1,2,2   1,1,2,1   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │ ├── html2text {}
│ │ │ │ @@ -102,15 +102,15 @@
│ │ │ │  1,2,2,2|
│ │ │ │       +-------------------------------------+-----------------------------------
│ │ │ │  --+
│ │ │ │  This ideal has 5 primary components. The first is the one that has statistical
│ │ │ │  significance. The significance of the other components is still poorly
│ │ │ │  understood.
│ │ │ │  i8 : time netList primaryDecomposition J
│ │ │ │ - -- used 1.73002s (cpu); 1.20682s (thread); 0s (gc)
│ │ │ │ + -- used 1.98805s (cpu); 1.38141s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_sp__A__S__M_spvarieties.out
│ │ │ @@ -212,17 +212,17 @@
│ │ │        | 1 -1 1 |
│ │ │        | 0 1  0 |
│ │ │  
│ │ │                 3       3
│ │ │  o22 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i23 : time schubertRegularity B
│ │ │ - -- used 0.10575s (cpu); 0.0415231s (thread); 0s (gc)
│ │ │ + -- used 0.104869s (cpu); 0.0427732s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
│ │ │  
│ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0152096s (cpu); 0.0152123s (thread); 0s (gc)
│ │ │ + -- used 0.027162s (cpu); 0.0271678s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = 1
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_spmatrix_sp__Schubert_spvarieties.out
│ │ │ @@ -178,17 +178,17 @@
│ │ │        z   z   z   , z   z   z    - z   z   , z   z   z    - z   z   )
│ │ │         1,2 1,3 2,4   1,2 1,4 2,2    1,2 2,4   1,2 1,3 2,2    1,2 2,3
│ │ │  
│ │ │  o15 : Ideal of QQ[z   ..z   ]
│ │ │                     1,1   5,5
│ │ │  
│ │ │  i16 : time schubertRegularity p
│ │ │ - -- used 0.00028812s (cpu); 0.000285085s (thread); 0s (gc)
│ │ │ + -- used 0.000404882s (cpu); 0.000399915s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
│ │ │  
│ │ │  i17 : time regularity comodule I
│ │ │ - -- used 0.0196653s (cpu); 0.0196691s (thread); 0s (gc)
│ │ │ + -- used 0.0230738s (cpu); 0.0230805s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 5
│ │ │  
│ │ │  i18 :
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_grothendieck__Polynomial.out
│ │ │ @@ -3,25 +3,25 @@
│ │ │  i1 : w = {2,1,4,3}
│ │ │  
│ │ │  o1 = {2, 1, 4, 3}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : time grothendieckPolynomial w
│ │ │ - -- used 0.00343384s (cpu); 0.00343448s (thread); 0s (gc)
│ │ │ + -- used 0.00577835s (cpu); 0.00577789s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o2 : QQ[x ..x ]
│ │ │           1   4
│ │ │  
│ │ │  i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ - -- used 0.0019653s (cpu); 0.00196918s (thread); 0s (gc)
│ │ │ + -- used 0.00269888s (cpu); 0.00269941s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o3 : QQ[x ..x ]
│ │ │           1   4
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_sp__A__S__M_spvarieties.html
│ │ │ @@ -388,23 +388,23 @@
│ │ │          
      
│ │ │            
      Additionally, this package facilitates investigating homological invariants of ASM ideals such as regularity (schubertRegularity) and codimension (schubertCodim). efficiently by computing the associated invariants for their antidiagonal initial ideals, which are known to be squarefree by [Wei17]. Therefore the extremal Betti numbers (which encode regularity, depth, and projective dimension) of ASM ideals coincide with those of their antidiagonal initial ideals by [CV20].
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i23 : time schubertRegularity B
│ │ │ - -- used 0.10575s (cpu); 0.0415231s (thread); 0s (gc)
│ │ │ + -- used 0.104869s (cpu); 0.0427732s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
      
│ │ │              
      
│ │ │                
      i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0152096s (cpu); 0.0152123s (thread); 0s (gc)
│ │ │ + -- used 0.027162s (cpu); 0.0271678s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = 1
      
│ │ │              
      
│ │ │          
      
│ │ │            
      Functions for investigating ASM varieties
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -244,19 +244,19 @@
│ │ │ │  ASM ideals such as regularity (_s_c_h_u_b_e_r_t_R_e_g_u_l_a_r_i_t_y) and codimension
│ │ │ │  (_s_c_h_u_b_e_r_t_C_o_d_i_m). efficiently by computing the associated invariants for their
│ │ │ │  antidiagonal initial ideals, which are known to be squarefree by [Wei17].
│ │ │ │  Therefore the extremal Betti numbers (which encode regularity, depth, and
│ │ │ │  projective dimension) of ASM ideals coincide with those of their antidiagonal
│ │ │ │  initial ideals by [CV20].
│ │ │ │  i23 : time schubertRegularity B
│ │ │ │ - -- used 0.10575s (cpu); 0.0415231s (thread); 0s (gc)
│ │ │ │ + -- used 0.104869s (cpu); 0.0427732s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = 1
│ │ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ │ - -- used 0.0152096s (cpu); 0.0152123s (thread); 0s (gc)
│ │ │ │ + -- used 0.027162s (cpu); 0.0271678s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = 1
│ │ │ │  ********** FFuunnccttiioonnss ffoorr iinnvveessttiiggaattiinngg AASSMM vvaarriieettiieess **********
│ │ │ │      * _i_s_P_a_r_t_i_a_l_A_S_M_(_M_a_t_r_i_x_) -- whether a matrix is a partial alternating sign
│ │ │ │        matrix
│ │ │ │      * _p_a_r_t_i_a_l_A_S_M_T_o_A_S_M_(_M_a_t_r_i_x_) -- extend a partial alternating sign matrix to an
│ │ │ │        alternating sign matrix
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_spmatrix_sp__Schubert_spvarieties.html
│ │ │ @@ -320,23 +320,23 @@
│ │ │          
      
│ │ │            
      Finally, this package contains functions for investigating homological invariants of matrix Schubert varieties efficiently through combinatorial algorithms produced in [PSW24] via schubertRegularityschubertCodim.
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i16 : time schubertRegularity p
│ │ │ - -- used 0.00028812s (cpu); 0.000285085s (thread); 0s (gc)
│ │ │ + -- used 0.000404882s (cpu); 0.000399915s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
      
│ │ │              
      
│ │ │                
      i17 : time regularity comodule I
│ │ │ - -- used 0.0196653s (cpu); 0.0196691s (thread); 0s (gc)
│ │ │ + -- used 0.0230738s (cpu); 0.0230805s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 5
      
│ │ │              
      
│ │ │          
      
│ │ │            
      Functions for investigating matrix Schubert varieties
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -545,19 +545,19 @@
│ │ │ │  
│ │ │ │  o15 : Ideal of QQ[z   ..z   ]
│ │ │ │                     1,1   5,5
│ │ │ │  Finally, this package contains functions for investigating homological
│ │ │ │  invariants of matrix Schubert varieties efficiently through combinatorial
│ │ │ │  algorithms produced in [PSW24] via _s_c_h_u_b_e_r_t_R_e_g_u_l_a_r_i_t_y_s_c_h_u_b_e_r_t_C_o_d_i_m.
│ │ │ │  i16 : time schubertRegularity p
│ │ │ │ - -- used 0.00028812s (cpu); 0.000285085s (thread); 0s (gc)
│ │ │ │ + -- used 0.000404882s (cpu); 0.000399915s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 5
│ │ │ │  i17 : time regularity comodule I
│ │ │ │ - -- used 0.0196653s (cpu); 0.0196691s (thread); 0s (gc)
│ │ │ │ + -- used 0.0230738s (cpu); 0.0230805s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = 5
│ │ │ │  ********** FFuunnccttiioonnss ffoorr iinnvveessttiiggaattiinngg mmaattrriixx SScchhuubbeerrtt vvaarriieettiieess **********
│ │ │ │      * _a_n_t_i_D_i_a_g_I_n_i_t_(_L_i_s_t_) -- compute the (unique) antidiagonal initial ideal of
│ │ │ │        an ASM ideal
│ │ │ │      * _r_a_n_k_T_a_b_l_e_(_L_i_s_t_) -- compute a table of rank conditions that determines the
│ │ │ │        corresponding ASM or matrix Schubert variety
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/_grothendieck__Polynomial.html
│ │ │ @@ -85,28 +85,28 @@
│ │ │  
│ │ │  o1 : List
      
│ │ │              
      
│ │ │                
      i2 : time grothendieckPolynomial w
│ │ │ - -- used 0.00343384s (cpu); 0.00343448s (thread); 0s (gc)
│ │ │ + -- used 0.00577835s (cpu); 0.00577789s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o2 : QQ[x ..x ]
│ │ │           1   4
      
│ │ │              
      
│ │ │                
      i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ - -- used 0.0019653s (cpu); 0.00196918s (thread); 0s (gc)
│ │ │ + -- used 0.00269888s (cpu); 0.00269941s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o3 : QQ[x ..x ]
│ │ │           1   4
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,24 +19,24 @@
│ │ │ │  PipeDream.
│ │ │ │  i1 : w = {2,1,4,3}
│ │ │ │  
│ │ │ │  o1 = {2, 1, 4, 3}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : time grothendieckPolynomial w
│ │ │ │ - -- used 0.00343384s (cpu); 0.00343448s (thread); 0s (gc)
│ │ │ │ + -- used 0.00577835s (cpu); 0.00577789s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2        2      2               2
│ │ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │ │  
│ │ │ │  o2 : QQ[x ..x ]
│ │ │ │           1   4
│ │ │ │  i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ │ - -- used 0.0019653s (cpu); 0.00196918s (thread); 0s (gc)
│ │ │ │ + -- used 0.00269888s (cpu); 0.00269941s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2        2      2               2
│ │ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │ │  
│ │ │ │  o3 : QQ[x ..x ]
│ │ │ │           1   4
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/___Matroid.out
│ │ │ @@ -51,20 +51,20 @@
│ │ │  i9 : keys R10.cache
│ │ │  
│ │ │  o9 = {groundSet, rankFunction, storedRepresentation}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time isWellDefined R10
│ │ │ - -- used 0.0544881s (cpu); 0.0544809s (thread); 0s (gc)
│ │ │ + -- used 0.0677211s (cpu); 0.0674732s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : time fVector R10
│ │ │ - -- used 0.0420488s (cpu); 0.0420536s (thread); 0s (gc)
│ │ │ + -- used 0.0507737s (cpu); 0.0507897s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ @@ -76,15 +76,15 @@
│ │ │  o12 = {hyperplanes, flatsRelationsTable, rankFunction, ideal, ranks, flats,
│ │ │        -----------------------------------------------------------------------
│ │ │        groundSet, dual, storedRepresentation}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : time fVector R10
│ │ │ - -- used 0.000193744s (cpu); 0.000193313s (thread); 0s (gc)
│ │ │ + -- used 0.000333805s (cpu); 0.000330462s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_all__Minors.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : U25 = uniformMatroid(2,5)
│ │ │  
│ │ │  o2 = a "matroid" of rank 2 on 5 elements
│ │ │  
│ │ │  o2 : Matroid
│ │ │  
│ │ │  i3 : elapsedTime L = allMinors(V, U25);
│ │ │ - -- .0424447s elapsed
│ │ │ + -- .0543837s elapsed
│ │ │  
│ │ │  i4 : #L
│ │ │  
│ │ │  o4 = 64
│ │ │  
│ │ │  i5 : netList L_{0..4}
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_get__Isos.out
│ │ │ @@ -33,14 +33,14 @@
│ │ │  i6 : F7 = specificMatroid "fano"
│ │ │  
│ │ │  o6 = a "matroid" of rank 3 on 7 elements
│ │ │  
│ │ │  o6 : Matroid
│ │ │  
│ │ │  i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.133364s (cpu); 0.0611622s (thread); 0s (gc)
│ │ │ + -- used 0.138409s (cpu); 0.0645937s (thread); 0s (gc)
│ │ │  
│ │ │  i8 : #autF7
│ │ │  
│ │ │  o8 = 168
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_has__Minor.out
│ │ │ @@ -9,12 +9,12 @@
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │  
│ │ │  o2 = false
│ │ │  
│ │ │  i3 : time hasMinor(M6, M5)
│ │ │ - -- used 2.01107s (cpu); 1.37745s (thread); 0s (gc)
│ │ │ + -- used 2.1323s (cpu); 1.45273s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_isomorphism_lp__Matroid_cm__Matroid_rp.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │  i4 : minorM6 = minor(M6, set{8}, set{4,5,6,7})
│ │ │  
│ │ │  o4 = a "matroid" of rank 4 on 10 elements
│ │ │  
│ │ │  o4 : Matroid
│ │ │  
│ │ │  i5 : time isomorphism(M5, minorM6)
│ │ │ - -- used 0.0159467s (cpu); 0.0159446s (thread); 0s (gc)
│ │ │ + -- used 0.0250298s (cpu); 0.0249367s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -56,15 +56,15 @@
│ │ │  i7 : N = relabel M6
│ │ │  
│ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │  
│ │ │  o7 : Matroid
│ │ │  
│ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 3.16616s (cpu); 2.54905s (thread); 0s (gc)
│ │ │ + -- used 3.25247s (cpu); 2.84514s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HashTable{0 => 11 }
│ │ │                 1 => 0
│ │ │                 2 => 1
│ │ │                 3 => 6
│ │ │                 4 => 9
│ │ │                 5 => 8
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_quick__Isomorphism__Test.out
│ │ │ @@ -37,15 +37,15 @@
│ │ │  o7 : Matroid
│ │ │  
│ │ │  i8 : R = ZZ[x,y]; tuttePolynomial(M0, R) == tuttePolynomial(M1, R)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time quickIsomorphismTest(M0, M1)
│ │ │ - -- used 0.000647424s (cpu); 0.000643838s (thread); 0s (gc)
│ │ │ + -- used 0.00064572s (cpu); 0.000642318s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = false
│ │ │  
│ │ │  i11 : value oo === false
│ │ │  
│ │ │  o11 = true
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_set__Representation.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │  i5 : keys M.cache
│ │ │  
│ │ │  o5 = {groundSet, rankFunction, storedRepresentation}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime fVector M
│ │ │ - -- .0188526s elapsed
│ │ │ + -- .0125707s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/___Matroid.html
│ │ │ @@ -153,23 +153,23 @@
│ │ │  
│ │ │  o9 : List
│ │ │              
      
│ │ │                
      i10 : time isWellDefined R10
│ │ │ - -- used 0.0544881s (cpu); 0.0544809s (thread); 0s (gc)
│ │ │ + -- used 0.0677211s (cpu); 0.0674732s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
      
│ │ │              
      
│ │ │                
      i11 : time fVector R10
│ │ │ - -- used 0.0420488s (cpu); 0.0420536s (thread); 0s (gc)
│ │ │ + -- used 0.0507737s (cpu); 0.0507897s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ @@ -187,15 +187,15 @@
│ │ │  
│ │ │  o12 : List
      
│ │ │              
      
│ │ │                
      i13 : time fVector R10
│ │ │ - -- used 0.000193744s (cpu); 0.000193313s (thread); 0s (gc)
│ │ │ + -- used 0.000333805s (cpu); 0.000330462s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -71,19 +71,19 @@
│ │ │ │  o8 : Matroid
│ │ │ │  i9 : keys R10.cache
│ │ │ │  
│ │ │ │  o9 = {groundSet, rankFunction, storedRepresentation}
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  i10 : time isWellDefined R10
│ │ │ │ - -- used 0.0544881s (cpu); 0.0544809s (thread); 0s (gc)
│ │ │ │ + -- used 0.0677211s (cpu); 0.0674732s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : time fVector R10
│ │ │ │ - -- used 0.0420488s (cpu); 0.0420536s (thread); 0s (gc)
│ │ │ │ + -- used 0.0507737s (cpu); 0.0507897s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = HashTable{0 => 1 }
│ │ │ │                  1 => 10
│ │ │ │                  2 => 45
│ │ │ │                  3 => 75
│ │ │ │                  4 => 30
│ │ │ │                  5 => 1
│ │ │ │ @@ -93,15 +93,15 @@
│ │ │ │  
│ │ │ │  o12 = {hyperplanes, flatsRelationsTable, rankFunction, ideal, ranks, flats,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        groundSet, dual, storedRepresentation}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : time fVector R10
│ │ │ │ - -- used 0.000193744s (cpu); 0.000193313s (thread); 0s (gc)
│ │ │ │ + -- used 0.000333805s (cpu); 0.000330462s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = HashTable{0 => 1 }
│ │ │ │                  1 => 10
│ │ │ │                  2 => 45
│ │ │ │                  3 => 75
│ │ │ │                  4 => 30
│ │ │ │                  5 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_all__Minors.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o2 : Matroid
      
│ │ │              
      
│ │ │                
      i3 : elapsedTime L = allMinors(V, U25);
│ │ │ - -- .0424447s elapsed
      
│ │ │ + -- .0543837s elapsed
│ │ │              
      
│ │ │                
      i4 : #L
│ │ │  
│ │ │  o4 = 64
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  o1 : Matroid
│ │ │ │  i2 : U25 = uniformMatroid(2,5)
│ │ │ │  
│ │ │ │  o2 = a "matroid" of rank 2 on 5 elements
│ │ │ │  
│ │ │ │  o2 : Matroid
│ │ │ │  i3 : elapsedTime L = allMinors(V, U25);
│ │ │ │ - -- .0424447s elapsed
│ │ │ │ + -- .0543837s elapsed
│ │ │ │  i4 : #L
│ │ │ │  
│ │ │ │  o4 = 64
│ │ │ │  i5 : netList L_{0..4}
│ │ │ │  
│ │ │ │       +----------+-------+
│ │ │ │  o5 = |set {5, 3}|set {2}|
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_get__Isos.html
│ │ │ @@ -140,15 +140,15 @@
│ │ │  
│ │ │  o6 : Matroid
│ │ │              
      
│ │ │                
      i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.133364s (cpu); 0.0611622s (thread); 0s (gc)
      
│ │ │ + -- used 0.138409s (cpu); 0.0645937s (thread); 0s (gc)
│ │ │              
      
│ │ │                
      i8 : #autF7
│ │ │  
│ │ │  o8 = 168
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  symmetric group S_7:
│ │ │ │  i6 : F7 = specificMatroid "fano"
│ │ │ │  
│ │ │ │  o6 = a "matroid" of rank 3 on 7 elements
│ │ │ │  
│ │ │ │  o6 : Matroid
│ │ │ │  i7 : time autF7 = getIsos(F7, F7);
│ │ │ │ - -- used 0.133364s (cpu); 0.0611622s (thread); 0s (gc)
│ │ │ │ + -- used 0.138409s (cpu); 0.0645937s (thread); 0s (gc)
│ │ │ │  i8 : #autF7
│ │ │ │  
│ │ │ │  o8 = 168
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_o_m_o_r_p_h_i_s_m_(_M_a_t_r_o_i_d_,_M_a_t_r_o_i_d_) -- computes an isomorphism between
│ │ │ │        isomorphic matroids
│ │ │ │      * _q_u_i_c_k_I_s_o_m_o_r_p_h_i_s_m_T_e_s_t -- quick checks for isomorphism between matroids
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_has__Minor.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o2 = false
│ │ │              
      
│ │ │                
      i3 : time hasMinor(M6, M5)
│ │ │ - -- used 2.01107s (cpu); 1.37745s (thread); 0s (gc)
│ │ │ + -- used 2.1323s (cpu); 1.45273s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
      
│ │ │              
      
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │       elements, a "matroid" of rank 5 on 15 elements)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : time hasMinor(M6, M5)
│ │ │ │ - -- used 2.01107s (cpu); 1.37745s (thread); 0s (gc)
│ │ │ │ + -- used 2.1323s (cpu); 1.45273s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_o_r -- minor of matroid
│ │ │ │      * _i_s_B_i_n_a_r_y -- whether a matroid is representable over F_2
│ │ │ │  ********** WWaayyss ttoo uussee hhaassMMiinnoorr:: **********
│ │ │ │      * hasMinor(Matroid,Matroid)
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_isomorphism_lp__Matroid_cm__Matroid_rp.html
│ │ │ @@ -123,15 +123,15 @@
│ │ │  
│ │ │  o4 : Matroid
    
│ │ │              
    
│ │ │                
    i5 : time isomorphism(M5, minorM6)
│ │ │ - -- used 0.0159467s (cpu); 0.0159446s (thread); 0s (gc)
│ │ │ + -- used 0.0250298s (cpu); 0.0249367s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -169,15 +169,15 @@
│ │ │  
│ │ │  o7 : Matroid
    
│ │ │              
    
│ │ │                
    i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 3.16616s (cpu); 2.54905s (thread); 0s (gc)
│ │ │ + -- used 3.25247s (cpu); 2.84514s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HashTable{0 => 11 }
│ │ │                 1 => 0
│ │ │                 2 => 1
│ │ │                 3 => 6
│ │ │                 4 => 9
│ │ │                 5 => 8
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : minorM6 = minor(M6, set{8}, set{4,5,6,7})
│ │ │ │  
│ │ │ │  o4 = a "matroid" of rank 4 on 10 elements
│ │ │ │  
│ │ │ │  o4 : Matroid
│ │ │ │  i5 : time isomorphism(M5, minorM6)
│ │ │ │ - -- used 0.0159467s (cpu); 0.0159446s (thread); 0s (gc)
│ │ │ │ + -- used 0.0250298s (cpu); 0.0249367s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{0 => 1}
│ │ │ │                 1 => 0
│ │ │ │                 2 => 3
│ │ │ │                 3 => 2
│ │ │ │                 4 => 6
│ │ │ │                 5 => 5
│ │ │ │ @@ -74,15 +74,15 @@
│ │ │ │  o6 : HashTable
│ │ │ │  i7 : N = relabel M6
│ │ │ │  
│ │ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │ │  
│ │ │ │  o7 : Matroid
│ │ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ │ - -- used 3.16616s (cpu); 2.54905s (thread); 0s (gc)
│ │ │ │ + -- used 3.25247s (cpu); 2.84514s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HashTable{0 => 11 }
│ │ │ │                 1 => 0
│ │ │ │                 2 => 1
│ │ │ │                 3 => 6
│ │ │ │                 4 => 9
│ │ │ │                 5 => 8
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_quick__Isomorphism__Test.html
│ │ │ @@ -142,15 +142,15 @@
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time quickIsomorphismTest(M0, M1)
│ │ │ - -- used 0.000647424s (cpu); 0.000643838s (thread); 0s (gc)
│ │ │ + -- used 0.00064572s (cpu); 0.000642318s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = false
    
│ │ │              
    
│ │ │                
    i11 : value oo === false
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  o7 = a "matroid" of rank 7 on 11 elements
│ │ │ │  
│ │ │ │  o7 : Matroid
│ │ │ │  i8 : R = ZZ[x,y]; tuttePolynomial(M0, R) == tuttePolynomial(M1, R)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time quickIsomorphismTest(M0, M1)
│ │ │ │ - -- used 0.000647424s (cpu); 0.000643838s (thread); 0s (gc)
│ │ │ │ + -- used 0.00064572s (cpu); 0.000642318s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = false
│ │ │ │  i11 : value oo === false
│ │ │ │  
│ │ │ │  o11 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_o_m_o_r_p_h_i_s_m_(_M_a_t_r_o_i_d_,_M_a_t_r_o_i_d_) -- computes an isomorphism between
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_set__Representation.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime fVector M
│ │ │ - -- .0188526s elapsed
│ │ │ + -- .0125707s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,15 +48,15 @@
│ │ │ │  o4 : Matrix QQ  <-- QQ
│ │ │ │  i5 : keys M.cache
│ │ │ │  
│ │ │ │  o5 = {groundSet, rankFunction, storedRepresentation}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : elapsedTime fVector M
│ │ │ │ - -- .0188526s elapsed
│ │ │ │ + -- .0125707s elapsed
│ │ │ │  
│ │ │ │  o6 = HashTable{0 => 1 }
│ │ │ │                 1 => 6
│ │ │ │                 2 => 15
│ │ │ │                 3 => 20
│ │ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/___Hybrid.out
│ │ │ @@ -5,16 +5,16 @@
│ │ │  i2 : R = ZZ/101[w..z];
│ │ │  
│ │ │  i3 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2);
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time .00106033)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0151769)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000348634)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000024296)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .0010769)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0342429)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000452517)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000019634)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000627457
│ │ │ - -- .0365148s elapsed
│ │ │ + --  Time taken : .00108048
│ │ │ + -- .0293128s elapsed
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical.out
│ │ │ @@ -30,21 +30,21 @@
│ │ │  
│ │ │               2        2   3     2
│ │ │  o5 = ideal (c , a*c, a , b , a*b )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000435395s elapsed
│ │ │ + -- .00060441s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0116827s elapsed
│ │ │ + -- .0157807s elapsed
│ │ │  
│ │ │  o7 = ideal (c, b, a)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical__Containment.out
│ │ │ @@ -29,22 +29,22 @@
│ │ │  o5 = 840
│ │ │  
│ │ │  i6 : x_0^(D-1) % I != 0 and x_0^D % I == 0
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ - -- .0877554s elapsed
│ │ │ + -- .105019s elapsed
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00176877s elapsed
│ │ │ + -- .00348436s elapsed
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .001213s elapsed
│ │ │ + -- .00274631s elapsed
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/___Hybrid.html
│ │ │ @@ -77,21 +77,21 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time .00106033)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0151769)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000348634)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000024296)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .0010769)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0342429)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000452517)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000019634)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000627457
│ │ │ - -- .0365148s elapsed
    
│ │ │ + --  Time taken : .00108048
│ │ │ + -- .0293128s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,23 +11,23 @@
│ │ │ │  i1 : debug MinimalPrimes
│ │ │ │  i2 : R = ZZ/101[w..z];
│ │ │ │  i3 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2);
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid
│ │ │ │  {Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ │ -  Strategy: Linear            (time .00106033)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Birational        (time .0151769)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Factorization     (time .000348634)  #primes = 0 #prunedViaCodim =
│ │ │ │ +  Strategy: Linear            (time .0010769)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Birational        (time .0342429)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Factorization     (time .000452517)  #primes = 0 #prunedViaCodim =
│ │ │ │  0
│ │ │ │ -  Strategy: DecomposeMonomials(time .000024296)  #primes = 1 #prunedViaCodim =
│ │ │ │ +  Strategy: DecomposeMonomials(time .000019634)  #primes = 1 #prunedViaCodim =
│ │ │ │  0
│ │ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ │ - --  Time taken : .000627457
│ │ │ │ - -- .0365148s elapsed
│ │ │ │ + --  Time taken : .00108048
│ │ │ │ + -- .0293128s elapsed
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n_(_._._._,_S_t_r_a_t_e_g_y_=_>_._._._)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _H_y_b_r_i_d is a _s_e_l_f_ _i_n_i_t_i_a_l_i_z_i_n_g_ _t_y_p_e, with ancestor classes _L_i_s_t <
│ │ │ │  _V_i_s_i_b_l_e_L_i_s_t < _B_a_s_i_c_L_i_s_t < _T_h_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical.html
│ │ │ @@ -136,25 +136,25 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000435395s elapsed
│ │ │ + -- .00060441s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0116827s elapsed
│ │ │ + -- .0157807s elapsed
│ │ │  
│ │ │  o7 = ideal (c, b, a)
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -62,21 +62,21 @@
│ │ │ │  i5 : I = intersect(ideal(a^2,b^2,c), ideal(a,b^3,c^2))
│ │ │ │  
│ │ │ │               2        2   3     2
│ │ │ │  o5 = ideal (c , a*c, a , b , a*b )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ │ - -- .000435395s elapsed
│ │ │ │ + -- .00060441s elapsed
│ │ │ │  
│ │ │ │  o6 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ │ - -- .0116827s elapsed
│ │ │ │ + -- .0157807s elapsed
│ │ │ │  
│ │ │ │  o7 = ideal (c, b, a)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  For another example, see _P_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n.
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ │ │  Eisenbud, Huneke, Vasconcelos, Invent. Math. 110 207-235 (1992).
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical__Containment.html
│ │ │ @@ -130,31 +130,31 @@
│ │ │  
│ │ │  o6 = true
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ - -- .0877554s elapsed
│ │ │ + -- .105019s elapsed
│ │ │  
│ │ │  o7 = true
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00176877s elapsed
│ │ │ + -- .00348436s elapsed
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .001213s elapsed
│ │ │ + -- .00274631s elapsed
│ │ │  
│ │ │  o9 = false
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,23 +50,23 @@
│ │ │ │  i5 : D = product(I_*/degree/sum)
│ │ │ │  
│ │ │ │  o5 = 840
│ │ │ │  i6 : x_0^(D-1) % I != 0 and x_0^D % I == 0
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ │ - -- .0877554s elapsed
│ │ │ │ + -- .105019s elapsed
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ │ - -- .00176877s elapsed
│ │ │ │ + -- .00348436s elapsed
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ │ - -- .001213s elapsed
│ │ │ │ + -- .00274631s elapsed
│ │ │ │  
│ │ │ │  o9 = false
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_d_i_c_a_l -- the radical of an ideal
│ │ │ │  ********** WWaayyss ttoo uussee rraaddiiccaallCCoonnttaaiinnmmeenntt:: **********
│ │ │ │      * radicalContainment(Ideal,Ideal)
│ │ │ │      * radicalContainment(RingElement,Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal.out
│ │ │ @@ -57,29 +57,29 @@
│ │ │  i9 : J = ideal vars U
│ │ │  
│ │ │  o9 = ideal (a, b, c)
│ │ │  
│ │ │  o9 : Ideal of U
│ │ │  
│ │ │  i10 : time multiReesIdeal J
│ │ │ - -- used 0.144587s (cpu); 0.0923636s (thread); 0s (gc)
│ │ │ + -- used 0.245885s (cpu); 0.101386s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │           0      2   1    0 2   0 1    2   0    1 2
│ │ │  
│ │ │  o10 : Ideal of U[X ..X ]
│ │ │                    0   2
│ │ │  
│ │ │  i11 : time multiReesIdeal (J, a)
│ │ │ - -- used 0.0105924s (cpu); 0.0103375s (thread); 0s (gc)
│ │ │ + -- used 0.0819176s (cpu); 0.0226811s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal.html
│ │ │ @@ -178,15 +178,15 @@
│ │ │  
│ │ │  o9 : Ideal of U
    
│ │ │              
    
│ │ │                
    i10 : time multiReesIdeal J
│ │ │ - -- used 0.144587s (cpu); 0.0923636s (thread); 0s (gc)
│ │ │ + -- used 0.245885s (cpu); 0.101386s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ @@ -195,15 +195,15 @@
│ │ │  o10 : Ideal of U[X ..X ]
│ │ │                    0   2
    
│ │ │              
    
│ │ │                
    i11 : time multiReesIdeal (J, a)
│ │ │ - -- used 0.0105924s (cpu); 0.0103375s (thread); 0s (gc)
│ │ │ + -- used 0.0819176s (cpu); 0.0226811s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ ├── html2text {}
│ │ │ │ @@ -79,28 +79,28 @@
│ │ │ │  i8 : U = T/minors(2,m);
│ │ │ │  i9 : J = ideal vars U
│ │ │ │  
│ │ │ │  o9 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o9 : Ideal of U
│ │ │ │  i10 : time multiReesIdeal J
│ │ │ │ - -- used 0.144587s (cpu); 0.0923636s (thread); 0s (gc)
│ │ │ │ + -- used 0.245885s (cpu); 0.101386s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │ │                  1      2     1      2     1      2     0      2     0      2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                      2                 2   2
│ │ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ │           0      2   1    0 2   0 1    2   0    1 2
│ │ │ │  
│ │ │ │  o10 : Ideal of U[X ..X ]
│ │ │ │                    0   2
│ │ │ │  i11 : time multiReesIdeal (J, a)
│ │ │ │ - -- used 0.0105924s (cpu); 0.0103375s (thread); 0s (gc)
│ │ │ │ + -- used 0.0819176s (cpu); 0.0226811s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │ │                  1      2     1      2     1      2     0      2     0      2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                      2                 2   2
│ │ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/_deform__M__C__M__Module_lp__Module_rp.out
│ │ │ @@ -40,15 +40,15 @@
│ │ │  
│ │ │  o7 = image | x2 y2 |
│ │ │  
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
│ │ │  
│ │ │  i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.59959s (cpu); 0.34716s (thread); 0s (gc)
│ │ │ + -- used 0.680142s (cpu); 0.402802s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │                    {8} | xxi_4-y+xi_3                                       
│ │ │       ------------------------------------------------------------------------
│ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │       -x2-xxi_2-xi_1                                           |
│ │ │  
│ │ │ @@ -70,15 +70,15 @@
│ │ │  o10 = cokernel | x2 y2  |
│ │ │                 | -y -x2 |
│ │ │  
│ │ │                               2
│ │ │  o10 : R-module, quotient of R
│ │ │  
│ │ │  i11 : (S',N') = time deformMCMModule N0'
│ │ │ - -- used 0.763073s (cpu); 0.494551s (thread); 0s (gc)
│ │ │ + -- used 0.859532s (cpu); 0.574426s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │                      | xxi_4-y+xi_3                                       
│ │ │        -----------------------------------------------------------------------
│ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │        -x2-xxi_2-xi_1                             |
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_rp.html
│ │ │ @@ -150,15 +150,15 @@
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
    
│ │ │              
    
│ │ │                
    i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.59959s (cpu); 0.34716s (thread); 0s (gc)
│ │ │ + -- used 0.680142s (cpu); 0.402802s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │                    {8} | xxi_4-y+xi_3                                       
│ │ │       ------------------------------------------------------------------------
│ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │       -x2-xxi_2-xi_1                                           |
│ │ │  
│ │ │ @@ -191,15 +191,15 @@
│ │ │                               2
│ │ │  o10 : R-module, quotient of R
    
│ │ │              
    
│ │ │                
    i11 : (S',N') = time deformMCMModule N0'
│ │ │ - -- used 0.763073s (cpu); 0.494551s (thread); 0s (gc)
│ │ │ + -- used 0.859532s (cpu); 0.574426s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │                      | xxi_4-y+xi_3                                       
│ │ │        -----------------------------------------------------------------------
│ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │        -x2-xxi_2-xi_1                             |
│ │ │ ├── html2text {}
│ │ │ │ @@ -70,15 +70,15 @@
│ │ │ │  i7 : N0 = module ideal (x^2,y^2)
│ │ │ │  
│ │ │ │  o7 = image | x2 y2 |
│ │ │ │  
│ │ │ │                               1
│ │ │ │  o7 : R-module, submodule of R
│ │ │ │  i8 : (S,N) = time deformMCMModule N0
│ │ │ │ - -- used 0.59959s (cpu); 0.34716s (thread); 0s (gc)
│ │ │ │ + -- used 0.680142s (cpu); 0.402802s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │ │                    {8} | xxi_4-y+xi_3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │ │       -x2-xxi_2-xi_1                                           |
│ │ │ │  
│ │ │ │ @@ -103,15 +103,15 @@
│ │ │ │  
│ │ │ │  o10 = cokernel | x2 y2  |
│ │ │ │                 | -y -x2 |
│ │ │ │  
│ │ │ │                               2
│ │ │ │  o10 : R-module, quotient of R
│ │ │ │  i11 : (S',N') = time deformMCMModule N0'
│ │ │ │ - -- used 0.763073s (cpu); 0.494551s (thread); 0s (gc)
│ │ │ │ + -- used 0.859532s (cpu); 0.574426s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │ │                      | xxi_4-y+xi_3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │ │        -x2-xxi_2-xi_1                             |
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out
│ │ │ @@ -3,41 +3,41 @@
│ │ │  i1 : R = CC[a,b,c,d][x,y];
│ │ │  
│ │ │  i2 : polys = polySystem {a*x+b*y^2,c*x*y+d};
│ │ │  
│ │ │  i3 : (p0, x0) = createSeedPair polys;
│ │ │  
│ │ │  i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ ---backup directory created: /tmp/M2-25234-0/2
│ │ │ - -- .00306279s elapsed
│ │ │ +--backup directory created: /tmp/M2-32854-0/2
│ │ │ + -- .00389925s elapsed
│ │ │    H01: 1
│ │ │ - -- .00278769s elapsed
│ │ │ + -- .00377125s elapsed
│ │ │    H10: 1
│ │ │ - -- .000547082s elapsed
│ │ │ + -- .000698392s elapsed
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00273952s elapsed
│ │ │ + -- .00368601s elapsed
│ │ │    H01: 1
│ │ │ - -- .00695454s elapsed
│ │ │ + -- .00394449s elapsed
│ │ │    H10: 1
│ │ │ - -- .000459249s elapsed
│ │ │ + -- .000642978s elapsed
│ │ │  number of paths tracked: 4
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00680025s elapsed
│ │ │ + -- .00377715s elapsed
│ │ │    H01: 1
│ │ │ - -- .00775431s elapsed
│ │ │ + -- .00380252s elapsed
│ │ │    H10: 1
│ │ │ - -- .00043848s elapsed
│ │ │ + -- .000648029s elapsed
│ │ │  number of paths tracked: 6
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00273619s elapsed
│ │ │ + -- .00368081s elapsed
│ │ │    H01: 1
│ │ │ - -- .00769911s elapsed
│ │ │ + -- .00374297s elapsed
│ │ │    H10: 1
│ │ │ - -- .000447176s elapsed
│ │ │ + -- .000662382s elapsed
│ │ │  number of paths tracked: 8
│ │ │  found 1 points in the fiber so far
│ │ │  
│ │ │  o4 = ({{.892712+.673395*ii, .29398+.632944*ii}}, 8)
│ │ │  
│ │ │  o4 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html
│ │ │ @@ -101,41 +101,41 @@
│ │ │              
    		
│ │ │                
    i3 : (p0, x0) = createSeedPair polys;
    
│ │ │              
    
│ │ │                
    i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ ---backup directory created: /tmp/M2-25234-0/2
│ │ │ - -- .00306279s elapsed
│ │ │ +--backup directory created: /tmp/M2-32854-0/2
│ │ │ + -- .00389925s elapsed
│ │ │    H01: 1
│ │ │ - -- .00278769s elapsed
│ │ │ + -- .00377125s elapsed
│ │ │    H10: 1
│ │ │ - -- .000547082s elapsed
│ │ │ + -- .000698392s elapsed
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00273952s elapsed
│ │ │ + -- .00368601s elapsed
│ │ │    H01: 1
│ │ │ - -- .00695454s elapsed
│ │ │ + -- .00394449s elapsed
│ │ │    H10: 1
│ │ │ - -- .000459249s elapsed
│ │ │ + -- .000642978s elapsed
│ │ │  number of paths tracked: 4
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00680025s elapsed
│ │ │ + -- .00377715s elapsed
│ │ │    H01: 1
│ │ │ - -- .00775431s elapsed
│ │ │ + -- .00380252s elapsed
│ │ │    H10: 1
│ │ │ - -- .00043848s elapsed
│ │ │ + -- .000648029s elapsed
│ │ │  number of paths tracked: 6
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00273619s elapsed
│ │ │ + -- .00368081s elapsed
│ │ │    H01: 1
│ │ │ - -- .00769911s elapsed
│ │ │ + -- .00374297s elapsed
│ │ │    H10: 1
│ │ │ - -- .000447176s elapsed
│ │ │ + -- .000662382s elapsed
│ │ │  number of paths tracked: 8
│ │ │  found 1 points in the fiber so far
│ │ │  
│ │ │  o4 = ({{.892712+.673395*ii, .29398+.632944*ii}}, 8)
│ │ │  
│ │ │  o4 : Sequence
    
│ │ │              
    
│ │ │                
    i3 : msolveGB(I, Verbosity => 2, Threads => 6) 
│ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-37973-0/0-in.ms -o /tmp/M2-37973-0/0-out.ms
│ │ │ -Initial seed for pseudo-random number generator is 1779109822
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-53061-0/0-in.ms -o /tmp/M2-53061-0/0-out.ms
│ │ │ +Initial seed for pseudo-random number generator is 1779299622
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -103,15 +103,15 @@
│ │ │  initial hash table size     131072 (2^17)
│ │ │  max pair selection             ALL
│ │ │  reduce gb                        1
│ │ │  #threads                         6
│ │ │  info level                       2
│ │ │  generate pbm files               0
│ │ │  ------------------------------------------
│ │ │ -Initial prime = 1232765221
│ │ │ +Initial prime = 1175563589
│ │ │  
│ │ │  Legend for f4 information
│ │ │  --------------------------------------------------------
│ │ │  deg       current degree of pairs selected in this round
│ │ │  sel       number of pairs selected in this round
│ │ │  pairs     total number of pairs in pair list
│ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ @@ -121,25 +121,25 @@
│ │ │  time(rd)  time of the current f4 round in seconds given
│ │ │            for real and cpu time
│ │ │  --------------------------------------------------------
│ │ │  
│ │ │  deg     sel   pairs        mat          density            new data         time(rd) in sec (real|cpu)
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │ -reduce final basis        3 x 3          33.33%        3 new       0 zero         0.03 | 0.08         
│ │ │ +reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -overall(elapsed)        0.07 sec
│ │ │ -overall(cpu)            0.20 sec
│ │ │ +overall(elapsed)        0.00 sec
│ │ │ +overall(cpu)            0.00 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.04 sec  57.6%
│ │ │ -convert                 0.03 sec  42.3%
│ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ +symbolic prep.          0.00 sec   0.3%
│ │ │ +update                  0.00 sec  70.9%
│ │ │ +convert                 0.00 sec   1.9%
│ │ │ +linear algebra          0.00 sec   1.5%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -153,18 +153,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1096643591
│ │ │ +New prime = 1233588359
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.03 sec
│ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │  learning phase                  0.00 Gops/sec
│ │ │  application phase               0.00 Gops/sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  multi-modular steps
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  {1}{2}<100.00%> 
│ │ │ @@ -180,15 +180,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.15 sec (elapsed) /  0.44 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │  
│ │ │  o3 = | z y x |
│ │ │  
│ │ │               1      3
│ │ │  o3 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i5 : peek componentsOfKernel(2, F)
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 6
│ │ │  number of distinct multidegrees = 6
│ │ │ - -- .00212365s elapsed
│ │ │ + -- .002321s elapsed
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ - -- .00813618s elapsed
│ │ │ + -- .0102353s elapsed
│ │ │  
│ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │                        {0, 2, 0, 2, 0} => {}
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,19 +51,19 @@
│ │ │ │  o4 : RingMap S <-- R
│ │ │ │  i5 : peek componentsOfKernel(2, F)
│ │ │ │  warning: computation begun over finite field. resulting polynomials may not lie
│ │ │ │  in the ideal
│ │ │ │  computing total degree: 1
│ │ │ │  number of monomials = 6
│ │ │ │  number of distinct multidegrees = 6
│ │ │ │ - -- .00212365s elapsed
│ │ │ │ + -- .002321s elapsed
│ │ │ │  computing total degree: 2
│ │ │ │  number of monomials = 21
│ │ │ │  number of distinct multidegrees = 18
│ │ │ │ - -- .00813618s elapsed
│ │ │ │ + -- .0102353s elapsed
│ │ │ │  
│ │ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │ │                        {0, 2, 0, 2, 0} => {}
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out
│ │ │ @@ -9,25 +9,25 @@
│ │ │  i2 : I = ideal"xy,yz,zx"
│ │ │  
│ │ │  o2 = ideal (x*y, y*z, x*z)
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime jMult I
│ │ │ - -- .0219807s elapsed
│ │ │ + -- .0312243s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : elapsedTime monjMult I
│ │ │ - -- .134903s elapsed
│ │ │ + -- .0906035s elapsed
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .122417s elapsed
│ │ │ + -- .156195s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out
│ │ │ @@ -10,12 +10,12 @@
│ │ │  
│ │ │               2        3
│ │ │  o2 = ideal (x , x*y, y )
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .0569657s elapsed
│ │ │ + -- .0722788s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_monj__Mult.out
│ │ │ @@ -13,17 +13,17 @@
│ │ │       ------------------------------------------------------------------------
│ │ │        10 11   8 12   9 11   10 10   11 9   12 8
│ │ │       x  y  , x y  , x y  , x  y  , x  y , x  y )
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime monjMult I
│ │ │ - -- .161352s elapsed
│ │ │ + -- .115325s elapsed
│ │ │  
│ │ │  o3 = 192
│ │ │  
│ │ │  i4 : elapsedTime jMult I
│ │ │ - -- 1.53634s elapsed
│ │ │ + -- 1.53822s elapsed
│ │ │  
│ │ │  o4 = 192
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html
│ │ │ @@ -93,31 +93,31 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime jMult I
│ │ │ - -- .0219807s elapsed
│ │ │ + -- .0312243s elapsed
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime monjMult I
│ │ │ - -- .134903s elapsed
│ │ │ + -- .0906035s elapsed
│ │ │  
│ │ │  o4 = 2
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .122417s elapsed
│ │ │ + -- .156195s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .0569657s elapsed
│ │ │ + -- .0722788s elapsed
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  i2 : I = ideal"x2,xy,y3"
│ │ │ │  
│ │ │ │               2        3
│ │ │ │  o2 = ideal (x , x*y, y )
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : elapsedTime monAnalyticSpread I
│ │ │ │ - -- .0569657s elapsed
│ │ │ │ + -- .0722788s elapsed
│ │ │ │  
│ │ │ │  o3 = 2
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _N_P -- the Newton polyhedron of a monomial ideal
│ │ │ │  ********** WWaayyss ttoo uussee mmoonnAAnnaallyyttiiccSSpprreeaadd:: **********
│ │ │ │      * monAnalyticSpread(Ideal)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html
│ │ │ @@ -97,23 +97,23 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
    
│ │ │                
    i3 : elapsedTime monjMult I
│ │ │ - -- .161352s elapsed
│ │ │ + -- .115325s elapsed
│ │ │  
│ │ │  o3 = 192
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime jMult I
│ │ │ - -- 1.53634s elapsed
│ │ │ + -- 1.53822s elapsed
│ │ │  
│ │ │  o4 = 192
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,19 +24,19 @@
│ │ │ │  o2 = ideal (x y  , x y  , x y  , x y  , x y  , x y  , x y  , x y  , x y  ,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        10 11   8 12   9 11   10 10   11 9   12 8
│ │ │ │       x  y  , x y  , x y  , x  y  , x  y , x  y )
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : elapsedTime monjMult I
│ │ │ │ - -- .161352s elapsed
│ │ │ │ + -- .115325s elapsed
│ │ │ │  
│ │ │ │  o3 = 192
│ │ │ │  i4 : elapsedTime jMult I
│ │ │ │ - -- 1.53634s elapsed
│ │ │ │ + -- 1.53822s elapsed
│ │ │ │  
│ │ │ │  o4 = 192
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_u_l_t_i_p_l_i_c_i_t_y_S_e_q_u_e_n_c_e -- the multiplicity sequence of an ideal
│ │ │ │      * _j_M_u_l_t -- the j-multiplicity of an ideal
│ │ │ │      * _m_o_n_R_e_d_u_c_t_i_o_n -- the minimal monomial reduction of a monomial ideal
│ │ │ │      * _N_P -- the Newton polyhedron of a monomial ideal
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │  o4 : ProjectiveVariety, curve in PP^8
│ │ │  
│ │ │  i5 : ? X
│ │ │  
│ │ │  o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 
│ │ │  
│ │ │  i6 : time f = X ===> Y;
│ │ │ - -- used 3.33307s (cpu); 1.85339s (thread); 0s (gc)
│ │ │ + -- used 4.03039s (cpu); 2.27251s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (automorphism of PP^8)
│ │ │  
│ │ │  i7 : f X
│ │ │  
│ │ │  o7 = Y
│ │ │  
│ │ │ @@ -38,15 +38,15 @@
│ │ │  o9 : ProjectiveVariety, 6-dimensional subvariety of PP^8
│ │ │  
│ │ │  i10 : W = random({{2},{1}},Y);
│ │ │  
│ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
│ │ │  
│ │ │  i11 : time g = V ===> W;
│ │ │ - -- used 3.28744s (cpu); 1.89691s (thread); 0s (gc)
│ │ │ + -- used 3.77874s (cpu); 2.24785s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : MultirationalMap (automorphism of PP^8)
│ │ │  
│ │ │  i12 : g||W
│ │ │  
│ │ │  o12 = multi-rational map consisting of one single rational map
│ │ │        source variety: 6-dimensional subvariety of PP^8 cut out by 2 hypersurfaces of degrees 1^1 2^1 
│ │ │ @@ -129,15 +129,15 @@
│ │ │  o15 : ProjectiveVariety, 6-dimensional subvariety of PP^9
│ │ │  
│ │ │  i16 : ? Z
│ │ │  
│ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │  
│ │ │  i17 : time h = Z ===> GG_K(1,4)
│ │ │ - -- used 7.61784s (cpu); 4.58879s (thread); 0s (gc)
│ │ │ + -- used 7.26752s (cpu); 4.98001s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = h
│ │ │  
│ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
│ │ │  
│ │ │  i18 : h || GG_K(1,4)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.out
│ │ │ @@ -7,15 +7,15 @@
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │  
│ │ │  i3 : Y = projectiveVariety ideal(random({1,1},ring target Phi), random({1,1},ring target Phi));
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4
│ │ │  
│ │ │  i4 : time X = Phi^* Y;
│ │ │ - -- used 5.34676s (cpu); 3.6979s (thread); 0s (gc)
│ │ │ + -- used 4.65457s (cpu); 3.82823s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension 2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 )
│ │ │  
│ │ │  i5 : dim X, degree X, degrees X
│ │ │  
│ │ │  o5 = (1, 31, {({0, 0, 2}, 1), ({0, 0, 3}, 4), ({0, 1, 1}, 4), ({0, 4, 1}, 1),
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out
│ │ │ @@ -11,26 +11,26 @@
│ │ │  o3 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^7 x PP^7)
│ │ │  
│ │ │  i4 : Z = source Phi;
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
│ │ │  
│ │ │  i5 : time Phi Z;
│ │ │ - -- used 0.0919907s (cpu); 0.0882659s (thread); 0s (gc)
│ │ │ + -- used 0.19467s (cpu); 0.138129s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
│ │ │  
│ │ │  i6 : dim oo, degree oo, degrees oo
│ │ │  
│ │ │  o6 = (4, 80, {({0, 2}, 5), ({1, 1}, 33), ({2, 0}, 5)})
│ │ │  
│ │ │  o6 : Sequence
│ │ │  
│ │ │  i7 : time Phi (point Z + point Z + point Z)
│ │ │ - -- used 1.8643s (cpu); 1.27944s (thread); 0s (gc)
│ │ │ + -- used 2.29452s (cpu); 1.4883s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3 
│ │ │  
│ │ │  o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7
│ │ │  
│ │ │  i8 : dim oo, degree oo, degrees oo
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_cm__Option_rp.out
│ │ │ @@ -11,22 +11,22 @@
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: threefold in PP^4 x PP^4 cut out by 13 hypersurfaces of
│ │ │       target variety: hypersurface in PP^4 defined by a form of degree 2
│ │ │       ------------------------------------------------------------------------
│ │ │       multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1
│ │ │  
│ │ │  i4 : time degree(Phi,Strategy=>"random point")
│ │ │ - -- used 3.64956s (cpu); 2.49249s (thread); 0s (gc)
│ │ │ + -- used 4.28042s (cpu); 2.69353s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.33988s (cpu); 0.260128s (thread); 0s (gc)
│ │ │ + -- used 0.360237s (cpu); 0.293668s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time degree Phi
│ │ │ - -- used 0.412254s (cpu); 0.297669s (thread); 0s (gc)
│ │ │ + -- used 0.360351s (cpu); 0.291068s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = 2
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_rp.out
│ │ │ @@ -3,12 +3,12 @@
│ │ │  i1 : ZZ/300007[x_0..x_3], f = rationalMap {x_2^2-x_1*x_3, x_1*x_2-x_0*x_3, x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3, x_3^2};
│ │ │  
│ │ │  i2 : Phi = last graph rationalMap {f,g};
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │  
│ │ │  i3 : time degree Phi
│ │ │ - -- used 0.553594s (cpu); 0.394728s (thread); 0s (gc)
│ │ │ + -- used 0.576617s (cpu); 0.397189s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 1
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out
│ │ │ @@ -1,52 +1,52 @@
│ │ │  -- -*- M2-comint -*- hash: 11533721324852072161
│ │ │  
│ │ │  i1 : Phi = multirationalMap graph rationalMap PP_(ZZ/65521)^(1,4);
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4 x PP^5)
│ │ │  
│ │ │  i2 : time ? Phi
│ │ │ - -- used 0.00342768s (cpu); 0.000183434s (thread); 0s (gc)
│ │ │ + -- used 0.00202157s (cpu); 0.000187616s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │       target variety: PP^4 x PP^5
│ │ │       ------------------------------------------------------------------------
│ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │  
│ │ │  i3 : image Phi;
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5
│ │ │  
│ │ │  i4 : time ? Phi
│ │ │ - -- used 0.00288947s (cpu); 0.000375835s (thread); 0s (gc)
│ │ │ + -- used 0.00130634s (cpu); 0.00028587s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │  
│ │ │  i5 : time describe Phi
│ │ │ - -- used 1.79181s (cpu); 1.38453s (thread); 0s (gc)
│ │ │ + -- used 1.27427s (cpu); 1.02924s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       multidegree: {51, 51, 51, 51, 51}
│ │ │       degree: 1
│ │ │       degree sequence (map 1/2): [(1,0), (0,2)]
│ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │       coefficient ring: ZZ/65521
│ │ │  
│ │ │  i6 : time ? Phi
│ │ │ - -- used 0.000152105s (cpu); 0.000376657s (thread); 0s (gc)
│ │ │ + -- used 0.000140518s (cpu); 0.000463679s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_graph_lp__Multirational__Map_rp.out
│ │ │ @@ -3,45 +3,45 @@
│ │ │  i1 : Phi = rationalMap(PP_(ZZ/333331)^(1,4),Dominant=>true)
│ │ │  
│ │ │  o1 = Phi
│ │ │  
│ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i2 : time (Phi1,Phi2) = graph Phi
│ │ │ - -- used 0.0242798s (cpu); 0.020939s (thread); 0s (gc)
│ │ │ + -- used 0.091949s (cpu); 0.0354259s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (Phi1, Phi2)
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 : Phi1;
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4)
│ │ │  
│ │ │  i4 : Phi2;
│ │ │  
│ │ │  o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
│ │ │  
│ │ │  i5 : time (Phi21,Phi22) = graph Phi2
│ │ │ - -- used 0.146281s (cpu); 0.0709226s (thread); 0s (gc)
│ │ │ + -- used 0.19681s (cpu); 0.0712912s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = (Phi21, Phi22)
│ │ │  
│ │ │  o5 : Sequence
│ │ │  
│ │ │  i6 : Phi21;
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │  
│ │ │  i7 : Phi22;
│ │ │  
│ │ │  o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to hypersurface in PP^5)
│ │ │  
│ │ │  i8 : time (Phi211,Phi212) = graph Phi21
│ │ │ - -- used 0.113295s (cpu); 0.112444s (thread); 0s (gc)
│ │ │ + -- used 0.193167s (cpu); 0.13781s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (Phi211, Phi212)
│ │ │  
│ │ │  o8 : Sequence
│ │ │  
│ │ │  i9 : Phi211;
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_image_lp__Multirational__Map_rp.out
│ │ │ @@ -11,25 +11,25 @@
│ │ │  o3 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │  
│ │ │  i4 : Phi = rationalMap {f,g};
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4)
│ │ │  
│ │ │  i5 : time Z = image Phi;
│ │ │ - -- used 0.105129s (cpu); 0.102279s (thread); 0s (gc)
│ │ │ + -- used 0.145995s (cpu); 0.13377s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │  
│ │ │  i6 : dim Z, degree Z, degrees Z
│ │ │  
│ │ │  o6 = (4, 151, {({1, 1}, 4), ({1, 2}, 3), ({2, 0}, 5), ({2, 1}, 13)})
│ │ │  
│ │ │  o6 : Sequence
│ │ │  
│ │ │  i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ - -- used 4.71458s (cpu); 2.51596s (thread); 0s (gc)
│ │ │ + -- used 10.1193s (cpu); 2.80453s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │  
│ │ │  i8 : assert(Z == Z')
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse2.out
│ │ │ @@ -4,25 +4,25 @@
│ │ │  
│ │ │  i2 : -- map defined by the cubics through the secant variety to the rational normal curve of degree 6
│ │ │       Phi = multirationalMap rationalMap(ring PP_K^6,ring GG_K(2,4),gens ideal PP_K([6],2));
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │  
│ │ │  i3 : time Psi = inverse2 Phi;
│ │ │ - -- used 0.378754s (cpu); 0.296528s (thread); 0s (gc)
│ │ │ + -- used 0.380273s (cpu); 0.308995s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
│ │ │  
│ │ │  i4 : assert(Phi * Psi == 1)
│ │ │  
│ │ │  i5 : Phi' = Phi || Phi;
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │  
│ │ │  i6 : time Psi' = inverse2 Phi';
│ │ │ - -- used 1.599s (cpu); 1.09s (thread); 0s (gc)
│ │ │ + -- used 1.30234s (cpu); 1.13542s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
│ │ │  
│ │ │  i7 : assert(Phi' * Psi' == 1)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse_lp__Multirational__Map_rp.out
│ │ │ @@ -7,33 +7,33 @@
│ │ │  
│ │ │  i2 : -- we see Phi as a dominant map
│ │ │       Phi = rationalMap(Phi,image Phi);
│ │ │  
│ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i3 : time inverse Phi;
│ │ │ - -- used 0.153397s (cpu); 0.0708753s (thread); 0s (gc)
│ │ │ + -- used 0.162984s (cpu); 0.0803958s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4)
│ │ │  
│ │ │  i4 : Psi = last graph Phi;
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
│ │ │  
│ │ │  i5 : time inverse Psi;
│ │ │ - -- used 0.1864s (cpu); 0.103535s (thread); 0s (gc)
│ │ │ + -- used 0.213217s (cpu); 0.111382s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │  
│ │ │  i6 : Eta = first graph Psi;
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │  
│ │ │  i7 : time inverse Eta;
│ │ │ - -- used 0.352284s (cpu); 0.27461s (thread); 0s (gc)
│ │ │ + -- used 0.461885s (cpu); 0.307729s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5)
│ │ │  
│ │ │  i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
│ │ │  
│ │ │  i9 : assert(Psi * Psi^-1 == 1 and Psi^-1 * Psi == 1)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Isomorphism_lp__Multirational__Map_rp.out
│ │ │ @@ -6,32 +6,32 @@
│ │ │  o2 : RationalMap (quadratic rational map from PP^3 to PP^2)
│ │ │  
│ │ │  i3 : Phi = rationalMap {f,f};
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │  
│ │ │  i4 : time isIsomorphism Phi
│ │ │ - -- used 0.00123848s (cpu); 9.188e-06s (thread); 0s (gc)
│ │ │ + -- used 0.00341899s (cpu); 9.82e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
│ │ │  
│ │ │  i5 : Psi = first graph Phi;
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to PP^3)
│ │ │  
│ │ │  i6 : time isIsomorphism Psi
│ │ │ - -- used 0.452327s (cpu); 0.228819s (thread); 0s (gc)
│ │ │ + -- used 0.624242s (cpu); 0.241401s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = false
│ │ │  
│ │ │  i7 : Eta = first graph Psi;
│ │ │  
│ │ │  o7 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 x PP^3 to threefold in PP^3 x PP^2 x PP^2)
│ │ │  
│ │ │  i8 : time isIsomorphism Eta
│ │ │ - -- used 1.64153s (cpu); 0.898134s (thread); 0s (gc)
│ │ │ + -- used 2.05633s (cpu); 1.0353s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : assert(o8 and (not o6) and (not o4))
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Morphism_lp__Multirational__Map_rp.out
│ │ │ @@ -3,24 +3,24 @@
│ │ │  i1 : ZZ/300007[a..e], f = first graph rationalMap ideal(c^2-b*d,b*c-a*d,b^2-a*c,e), g = rationalMap submatrix(matrix f,{0..2});
│ │ │  
│ │ │  i2 : Phi = rationalMap {f,g};
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^4 x PP^2)
│ │ │  
│ │ │  i3 : time isMorphism Phi
│ │ │ - -- used 0.372762s (cpu); 0.223282s (thread); 0s (gc)
│ │ │ + -- used 0.445085s (cpu); 0.265907s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.171617s (cpu); 0.10156s (thread); 0s (gc)
│ │ │ + -- used 0.250477s (cpu); 0.114506s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^7 x PP^4 x PP^2 to 4-dimensional subvariety of PP^4 x PP^7)
│ │ │  
│ │ │  i5 : time isMorphism Psi
│ │ │ - -- used 4.159s (cpu); 3.00876s (thread); 0s (gc)
│ │ │ + -- used 4.10963s (cpu); 3.41703s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : assert((not o3) and o5)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_linearly__Normal__Embedding.out
│ │ │ @@ -3,24 +3,24 @@
│ │ │  i1 : K = ZZ/333331;
│ │ │  
│ │ │  i2 : X = PP_K^(1,7); -- rational normal curve of degree 7
│ │ │  
│ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │  
│ │ │  i3 : time f = linearlyNormalEmbedding X;
│ │ │ - -- used 0.0079494s (cpu); 0.00865425s (thread); 0s (gc)
│ │ │ + -- used 0.0121033s (cpu); 0.0110541s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (automorphism of X)
│ │ │  
│ │ │  i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an isomorphic projection of X in PP^3
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3
│ │ │  
│ │ │  i5 : time g = linearlyNormalEmbedding Y;
│ │ │ - -- used 0.575291s (cpu); 0.434308s (thread); 0s (gc)
│ │ │ + -- used 0.519796s (cpu); 0.443447s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
│ │ │  
│ │ │  i6 : assert(isIsomorphism g)
│ │ │  
│ │ │  i7 : describe g
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Multirational__Map_rp.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : ZZ/300007[x_0..x_3], f = rationalMap {x_2^2-x_1*x_3, x_1*x_2-x_0*x_3, x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3, x_3^2};
│ │ │  
│ │ │  i2 : Phi = last graph rationalMap {f,g};
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │  
│ │ │  i3 : time multidegree Phi
│ │ │ - -- used 0.53565s (cpu); 0.391645s (thread); 0s (gc)
│ │ │ + -- used 0.580115s (cpu); 0.401078s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {66, 46, 31, 20}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : (degree source Phi,degree image Phi)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.out
│ │ │ @@ -1,21 +1,21 @@
│ │ │  -- -*- M2-comint -*- hash: 16199733219210081214
│ │ │  
│ │ │  i1 : Phi = last graph rationalMap PP_(ZZ/300007)^(1,4);
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5)
│ │ │  
│ │ │  i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi)
│ │ │ - -- used 0.00200117s (cpu); 0.00145578s (thread); 0s (gc)
│ │ │ - -- used 0.202005s (cpu); 0.128646s (thread); 0s (gc)
│ │ │ - -- used 0.233863s (cpu); 0.151089s (thread); 0s (gc)
│ │ │ - -- used 0.322585s (cpu); 0.170041s (thread); 0s (gc)
│ │ │ - -- used 0.148551s (cpu); 0.0965599s (thread); 0s (gc)
│ │ │ + -- used 0.00402704s (cpu); 0.00150338s (thread); 0s (gc)
│ │ │ + -- used 0.234586s (cpu); 0.154972s (thread); 0s (gc)
│ │ │ + -- used 0.251794s (cpu); 0.176388s (thread); 0s (gc)
│ │ │ + -- used 0.350078s (cpu); 0.201292s (thread); 0s (gc)
│ │ │ + -- used 0.217942s (cpu); 0.14613s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.0498404s (cpu); 0.0522257s (thread); 0s (gc)
│ │ │ + -- used 0.133375s (cpu); 0.0749226s (thread); 0s (gc)
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_point_lp__Multiprojective__Variety_rp.out
│ │ │ @@ -3,26 +3,26 @@
│ │ │  i1 : K = ZZ/1000003;
│ │ │  
│ │ │  i2 : X = PP_K^({1,1,2},{3,2,3});
│ │ │  
│ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9
│ │ │  
│ │ │  i3 : time p := point X
│ │ │ - -- used 0.124959s (cpu); 0.0446009s (thread); 0s (gc)
│ │ │ + -- used 0.205181s (cpu); 0.0611894s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],[3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1])
│ │ │  
│ │ │  o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
│ │ │  
│ │ │  i4 : Y = random({2,1,2},X);
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9
│ │ │  
│ │ │  i5 : time q = point Y
│ │ │ - -- used 1.82924s (cpu); 1.00825s (thread); 0s (gc)
│ │ │ + -- used 1.69161s (cpu); 1.10907s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = q
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
│ │ │  
│ │ │  i6 : assert(isSubset(p,X) and isSubset(q,Y))
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multirational__Map_rp.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  o4 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │  
│ │ │  i5 : Phi = rationalMap {f,g,h};
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x PP^4)
│ │ │  
│ │ │  i6 : time segre Phi;
│ │ │ - -- used 0.885868s (cpu); 0.592017s (thread); 0s (gc)
│ │ │ + -- used 1.4408s (cpu); 0.75308s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
│ │ │  
│ │ │  i7 : describe segre Phi
│ │ │  
│ │ │  o7 = rational map defined by forms of degree 6
│ │ │       source variety: PP^4
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_show_lp__Multirational__Map_rp.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : Phi = inverse first graph last graph rationalMap PP_(ZZ/33331)^(1,3)
│ │ │  
│ │ │  o1 = Phi
│ │ │  
│ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to threefold in PP^3 x PP^2 x PP^2)
│ │ │  
│ │ │  i2 : time describe Phi
│ │ │ - -- used 0.227275s (cpu); 0.148372s (thread); 0s (gc)
│ │ │ + -- used 0.250751s (cpu); 0.171452s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of multi-degree (1,1)
│ │ │       target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2 
│ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ │       dominance: true
│ │ │       multidegree: {10, 14, 19, 25}
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.html
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │  o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15
    │ │ │
    │ │ │ 
    i6 : time f = X ===> Y;
│ │ │ - -- used 3.33307s (cpu); 1.85339s (thread); 0s (gc)
│ │ │ + -- used 4.03039s (cpu); 2.27251s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (automorphism of PP^8)
    │ │ │
    │ │ │ 
    i7 : f X
│ │ │ @@ -148,15 +148,15 @@
│ │ │  
│ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
    │ │ │
    │ │ │ 
    i11 : time g = V ===> W;
│ │ │ - -- used 3.28744s (cpu); 1.89691s (thread); 0s (gc)
│ │ │ + -- used 3.77874s (cpu); 2.24785s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : MultirationalMap (automorphism of PP^8)
    │ │ │
    │ │ │ 
    i12 : g||W
│ │ │ @@ -257,15 +257,15 @@
│ │ │  
│ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
    │ │ │
    │ │ │ 
    i17 : time h = Z ===> GG_K(1,4)
│ │ │ - -- used 7.61784s (cpu); 4.58879s (thread); 0s (gc)
│ │ │ + -- used 7.26752s (cpu); 4.98001s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = h
│ │ │  
│ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
    │ │ │
    │ │ │ 
    i4 : time X = Phi^* Y;
│ │ │ - -- used 5.34676s (cpu); 3.6979s (thread); 0s (gc)
│ │ │ + -- used 4.65457s (cpu); 3.82823s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension 2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 )
    │ │ │
    │ │ │ 
    i5 : dim X, degree X, degrees X
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to
│ │ │ │  PP^2 x PP^4)
│ │ │ │  i3 : Y = projectiveVariety ideal(random({1,1},ring target Phi), random(
│ │ │ │  {1,1},ring target Phi));
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4
│ │ │ │  i4 : time X = Phi^* Y;
│ │ │ │ - -- used 5.34676s (cpu); 3.6979s (thread); 0s (gc)
│ │ │ │ + -- used 4.65457s (cpu); 3.82823s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension
│ │ │ │  2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-
│ │ │ │  degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 )
│ │ │ │  i5 : dim X, degree X, degrees X
│ │ │ │  
│ │ │ │  o5 = (1, 31, {({0, 0, 2}, 1), ({0, 0, 3}, 4), ({0, 1, 1}, 4), ({0, 4, 1}, 1),
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp__Multiprojective__Variety.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
    │ │ │
    │ │ │ 
    i5 : time Phi Z;
│ │ │ - -- used 0.0919907s (cpu); 0.0882659s (thread); 0s (gc)
│ │ │ + -- used 0.19467s (cpu); 0.138129s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
    │ │ │
    │ │ │ 
    i6 : dim oo, degree oo, degrees oo
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o6 : Sequence
    │ │ │
    │ │ │ 
    i7 : time Phi (point Z + point Z + point Z)
│ │ │ - -- used 1.8643s (cpu); 1.27944s (thread); 0s (gc)
│ │ │ + -- used 2.29452s (cpu); 1.4883s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3 
│ │ │  
│ │ │  o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7
    │ │ │
    │ │ │ 
    i4 : time degree(Phi,Strategy=>"random point")
│ │ │ - -- used 3.64956s (cpu); 2.49249s (thread); 0s (gc)
│ │ │ + -- used 4.28042s (cpu); 2.69353s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    │ │ │
    │ │ │ 
    i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.33988s (cpu); 0.260128s (thread); 0s (gc)
│ │ │ + -- used 0.360237s (cpu); 0.293668s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    │ │ │
    │ │ │ 
    i6 : time degree Phi
│ │ │ - -- used 0.412254s (cpu); 0.297669s (thread); 0s (gc)
│ │ │ + -- used 0.360351s (cpu); 0.291068s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = 2
    │ │ │
    │ │ │

    Note, as in the example above, that calculation times may vary depending on the strategy used.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -27,23 +27,23 @@ │ │ │ │ │ │ │ │ o3 = multi-rational map consisting of one single rational map │ │ │ │ source variety: threefold in PP^4 x PP^4 cut out by 13 hypersurfaces of │ │ │ │ target variety: hypersurface in PP^4 defined by a form of degree 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1 │ │ │ │ i4 : time degree(Phi,Strategy=>"random point") │ │ │ │ - -- used 3.64956s (cpu); 2.49249s (thread); 0s (gc) │ │ │ │ + -- used 4.28042s (cpu); 2.69353s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ i5 : time degree(Phi,Strategy=>"0-th projective degree") │ │ │ │ - -- used 0.33988s (cpu); 0.260128s (thread); 0s (gc) │ │ │ │ + -- used 0.360237s (cpu); 0.293668s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 2 │ │ │ │ i6 : time degree Phi │ │ │ │ - -- used 0.412254s (cpu); 0.297669s (thread); 0s (gc) │ │ │ │ + -- used 0.360351s (cpu); 0.291068s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 = 2 │ │ │ │ Note, as in the example above, that calculation times may vary depending on the │ │ │ │ strategy used. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- degree of a multi-rational map │ │ │ │ * _d_e_g_r_e_e_M_a_p_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_rp.html │ │ │ @@ -86,15 +86,15 @@ │ │ │ │ │ │ o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time degree Phi
│ │ │ - -- used 0.553594s (cpu); 0.394728s (thread); 0s (gc)
│ │ │ + -- used 0.576617s (cpu); 0.397189s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 1
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -18,15 +18,15 @@ │ │ │ │ x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3, │ │ │ │ x_3^2}; │ │ │ │ i2 : Phi = last graph rationalMap {f,g}; │ │ │ │ │ │ │ │ o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to │ │ │ │ PP^2 x PP^4) │ │ │ │ i3 : time degree Phi │ │ │ │ - -- used 0.553594s (cpu); 0.394728s (thread); 0s (gc) │ │ │ │ + -- used 0.576617s (cpu); 0.397189s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = 1 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_,_O_p_t_i_o_n_) -- degree of a multi-rational map using a │ │ │ │ probabilistic approach │ │ │ │ * _d_e_g_r_e_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map │ │ │ │ * _m_u_l_t_i_d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a multi-rational │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_describe_lp__Multirational__Map_rp.html │ │ │ @@ -82,15 +82,15 @@ │ │ │ │ │ │ o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4 x PP^5) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time ? Phi
│ │ │ - -- used 0.00342768s (cpu); 0.000183434s (thread); 0s (gc)
│ │ │ + -- used 0.00202157s (cpu); 0.000187616s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │       target variety: PP^4 x PP^5
│ │ │       ------------------------------------------------------------------------
│ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
    │ │ │ │ │ │ @@ -101,27 +101,27 @@ │ │ │ │ │ │ o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time ? Phi
│ │ │ - -- used 0.00288947s (cpu); 0.000375835s (thread); 0s (gc)
│ │ │ + -- used 0.00130634s (cpu); 0.00028587s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time describe Phi
│ │ │ - -- used 1.79181s (cpu); 1.38453s (thread); 0s (gc)
│ │ │ + -- used 1.27427s (cpu); 1.02924s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │ @@ -131,15 +131,15 @@
│ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │       coefficient ring: ZZ/65521
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time ? Phi
│ │ │ - -- used 0.000152105s (cpu); 0.000376657s (thread); 0s (gc)
│ │ │ + -- used 0.000140518s (cpu); 0.000463679s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,36 +16,36 @@
│ │ │ │  ? Phi is a lite version of describe Phi. The latter has a different behavior
│ │ │ │  than _d_e_s_c_r_i_b_e_(_R_a_t_i_o_n_a_l_M_a_p_), since it performs computations.
│ │ │ │  i1 : Phi = multirationalMap graph rationalMap PP_(ZZ/65521)^(1,4);
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to PP^4 x PP^5)
│ │ │ │  i2 : time ? Phi
│ │ │ │ - -- used 0.00342768s (cpu); 0.000183434s (thread); 0s (gc)
│ │ │ │ + -- used 0.00202157s (cpu); 0.000187616s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │  i3 : image Phi;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5
│ │ │ │  i4 : time ? Phi
│ │ │ │ - -- used 0.00288947s (cpu); 0.000375835s (thread); 0s (gc)
│ │ │ │ + -- used 0.00130634s (cpu); 0.00028587s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │  hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       dominance: false
│ │ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces
│ │ │ │  of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │  i5 : time describe Phi
│ │ │ │ - -- used 1.79181s (cpu); 1.38453s (thread); 0s (gc)
│ │ │ │ + -- used 1.27427s (cpu); 1.02924s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │  hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │ │       dominance: false
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │  of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       multidegree: {51, 51, 51, 51, 51}
│ │ │ │       degree: 1
│ │ │ │       degree sequence (map 1/2): [(1,0), (0,2)]
│ │ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │ │       coefficient ring: ZZ/65521
│ │ │ │  i6 : time ? Phi
│ │ │ │ - -- used 0.000152105s (cpu); 0.000376657s (thread); 0s (gc)
│ │ │ │ + -- used 0.000140518s (cpu); 0.000463679s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │  hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │ │       dominance: false
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_graph_lp__Multirational__Map_rp.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time (Phi1,Phi2) = graph Phi
│ │ │ - -- used 0.0242798s (cpu); 0.020939s (thread); 0s (gc)
│ │ │ + -- used 0.091949s (cpu); 0.0354259s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (Phi1, Phi2)
│ │ │  
│ │ │  o2 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -112,15 +112,15 @@ │ │ │ │ │ │ o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time (Phi21,Phi22) = graph Phi2
│ │ │ - -- used 0.146281s (cpu); 0.0709226s (thread); 0s (gc)
│ │ │ + -- used 0.19681s (cpu); 0.0712912s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = (Phi21, Phi22)
│ │ │  
│ │ │  o5 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -136,15 +136,15 @@ │ │ │ │ │ │ o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to hypersurface in PP^5) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time (Phi211,Phi212) = graph Phi21
│ │ │ - -- used 0.113295s (cpu); 0.112444s (thread); 0s (gc)
│ │ │ + -- used 0.193167s (cpu); 0.13781s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (Phi211, Phi212)
│ │ │  
│ │ │  o8 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -19,43 +19,43 @@ │ │ │ │ Phi)^-1 * (last graph Phi) == Phi are always satisfied. │ │ │ │ i1 : Phi = rationalMap(PP_(ZZ/333331)^(1,4),Dominant=>true) │ │ │ │ │ │ │ │ o1 = Phi │ │ │ │ │ │ │ │ o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5) │ │ │ │ i2 : time (Phi1,Phi2) = graph Phi │ │ │ │ - -- used 0.0242798s (cpu); 0.020939s (thread); 0s (gc) │ │ │ │ + -- used 0.091949s (cpu); 0.0354259s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = (Phi1, Phi2) │ │ │ │ │ │ │ │ o2 : Sequence │ │ │ │ i3 : Phi1; │ │ │ │ │ │ │ │ o3 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x │ │ │ │ PP^5 to PP^4) │ │ │ │ i4 : Phi2; │ │ │ │ │ │ │ │ o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of │ │ │ │ PP^4 x PP^5 to hypersurface in PP^5) │ │ │ │ i5 : time (Phi21,Phi22) = graph Phi2 │ │ │ │ - -- used 0.146281s (cpu); 0.0709226s (thread); 0s (gc) │ │ │ │ + -- used 0.19681s (cpu); 0.0712912s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = (Phi21, Phi22) │ │ │ │ │ │ │ │ o5 : Sequence │ │ │ │ i6 : Phi21; │ │ │ │ │ │ │ │ o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x │ │ │ │ PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5) │ │ │ │ i7 : Phi22; │ │ │ │ │ │ │ │ o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of │ │ │ │ PP^4 x PP^5 x PP^5 to hypersurface in PP^5) │ │ │ │ i8 : time (Phi211,Phi212) = graph Phi21 │ │ │ │ - -- used 0.113295s (cpu); 0.112444s (thread); 0s (gc) │ │ │ │ + -- used 0.193167s (cpu); 0.13781s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 = (Phi211, Phi212) │ │ │ │ │ │ │ │ o8 : Sequence │ │ │ │ i9 : Phi211; │ │ │ │ │ │ │ │ o9 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_image_lp__Multirational__Map_rp.html │ │ │ @@ -100,15 +100,15 @@ │ │ │ │ │ │ o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time Z = image Phi;
│ │ │ - -- used 0.105129s (cpu); 0.102279s (thread); 0s (gc)
│ │ │ + -- used 0.145995s (cpu); 0.13377s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : dim Z, degree Z, degrees Z
│ │ │ @@ -120,15 +120,15 @@
│ │ │            
│ │ │          
│ │ │          
    Alternatively, the calculation can be performed using the Segre embedding as follows:
    
│ │ │          
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│ │ │            
│ │ │              
│ │ │            
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│ │ │              
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│ │ │            
│ │ │              
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│ │ │              
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│ │ │            
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│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3,
│ │ │ │  x_3^2};
│ │ │ │  i2 : Phi = last graph rationalMap {f,g};
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to
│ │ │ │  PP^2 x PP^4)
│ │ │ │  i3 : time multidegree Phi
│ │ │ │ - -- used 0.53565s (cpu); 0.391645s (thread); 0s (gc)
│ │ │ │ + -- used 0.580115s (cpu); 0.401078s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = {66, 46, 31, 20}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : (degree source Phi,degree image Phi)
│ │ │ │  
│ │ │ │  o4 = (66, 20)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.html
│ │ │ @@ -82,29 +82,29 @@
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ - -- used 4.71458s (cpu); 2.51596s (thread); 0s (gc)
│ │ │ + -- used 10.1193s (cpu); 2.80453s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    
│ │ │              
    
│ │ │                
    i8 : assert(Z == Z')
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,26 +23,26 @@
│ │ │ │  3*x_2^2+2*x_1*x_3+x_0*x_4, 2*x_1*x_2-2*x_0*x_3, -x_1^2+x_0*x_2};
│ │ │ │  
│ │ │ │  o3 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │ │  i4 : Phi = rationalMap {f,g};
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4)
│ │ │ │  i5 : time Z = image Phi;
│ │ │ │ - -- used 0.105129s (cpu); 0.102279s (thread); 0s (gc)
│ │ │ │ + -- used 0.145995s (cpu); 0.13377s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │ │  i6 : dim Z, degree Z, degrees Z
│ │ │ │  
│ │ │ │  o6 = (4, 151, {({1, 1}, 4), ({1, 2}, 3), ({2, 0}, 5), ({2, 1}, 13)})
│ │ │ │  
│ │ │ │  o6 : Sequence
│ │ │ │  Alternatively, the calculation can be performed using the Segre embedding as
│ │ │ │  follows:
│ │ │ │  i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ │ - -- used 4.71458s (cpu); 2.51596s (thread); 0s (gc)
│ │ │ │ + -- used 10.1193s (cpu); 2.80453s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │ │  i8 : assert(Z == Z')
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_ _M_u_l_t_i_p_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y -- direct image via a multi-
│ │ │ │        rational map
│ │ │ │      * _i_m_a_g_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- closure of the image of a rational map
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse2.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │              
    
│ │ │                
    i3 : time Psi = inverse2 Phi;
│ │ │ - -- used 0.378754s (cpu); 0.296528s (thread); 0s (gc)
│ │ │ + -- used 0.380273s (cpu); 0.308995s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
    
│ │ │              
    
│ │ │                
    i4 : assert(Phi * Psi == 1)
    
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │              
    
│ │ │                
    i6 : time Psi' = inverse2 Phi';
│ │ │ - -- used 1.599s (cpu); 1.09s (thread); 0s (gc)
│ │ │ + -- used 1.30234s (cpu); 1.13542s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
    
│ │ │              
    
│ │ │                
    i7 : assert(Phi' * Psi' == 1)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,23 +24,23 @@
│ │ │ │  i2 : -- map defined by the cubics through the secant variety to the rational
│ │ │ │  normal curve of degree 6
│ │ │ │       Phi = multirationalMap rationalMap(ring PP_K^6,ring GG_K(2,4),gens ideal
│ │ │ │  PP_K([6],2));
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │ │  i3 : time Psi = inverse2 Phi;
│ │ │ │ - -- used 0.378754s (cpu); 0.296528s (thread); 0s (gc)
│ │ │ │ + -- used 0.380273s (cpu); 0.308995s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
│ │ │ │  i4 : assert(Phi * Psi == 1)
│ │ │ │  i5 : Phi' = Phi || Phi;
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │ │  i6 : time Psi' = inverse2 Phi';
│ │ │ │ - -- used 1.599s (cpu); 1.09s (thread); 0s (gc)
│ │ │ │ + -- used 1.30234s (cpu); 1.13542s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
│ │ │ │  i7 : assert(Phi' * Psi' == 1)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_n_v_e_r_s_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- inverse of a birational map
│ │ │ │      * _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_ _<_=_=_>_ _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p -- equality of multi-rational maps
│ │ │ │        with checks on internal data
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse_lp__Multirational__Map_rp.html
│ │ │ @@ -93,45 +93,45 @@
│ │ │  
│ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │              
    
│ │ │                
    i3 : time inverse Phi;
│ │ │ - -- used 0.153397s (cpu); 0.0708753s (thread); 0s (gc)
│ │ │ + -- used 0.162984s (cpu); 0.0803958s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4)
    
│ │ │              
    
│ │ │                
    i4 : Psi = last graph Phi;
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
    
│ │ │              
    
│ │ │                
    i5 : time inverse Psi;
│ │ │ - -- used 0.1864s (cpu); 0.103535s (thread); 0s (gc)
│ │ │ + -- used 0.213217s (cpu); 0.111382s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
    
│ │ │              
    
│ │ │                
    i6 : Eta = first graph Psi;
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
    
│ │ │              
    
│ │ │                
    i7 : time inverse Eta;
│ │ │ - -- used 0.352284s (cpu); 0.27461s (thread); 0s (gc)
│ │ │ + -- used 0.461885s (cpu); 0.307729s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5)
    
│ │ │              
    
│ │ │                
    i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,32 +24,32 @@
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (rational map from PP^4 to PP^5)
│ │ │ │  i2 : -- we see Phi as a dominant map
│ │ │ │       Phi = rationalMap(Phi,image Phi);
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │ │  i3 : time inverse Phi;
│ │ │ │ - -- used 0.153397s (cpu); 0.0708753s (thread); 0s (gc)
│ │ │ │ + -- used 0.162984s (cpu); 0.0803958s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4)
│ │ │ │  i4 : Psi = last graph Phi;
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to hypersurface in PP^5)
│ │ │ │  i5 : time inverse Psi;
│ │ │ │ - -- used 0.1864s (cpu); 0.103535s (thread); 0s (gc)
│ │ │ │ + -- used 0.213217s (cpu); 0.111382s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-
│ │ │ │  dimensional subvariety of PP^4 x PP^5)
│ │ │ │  i6 : Eta = first graph Psi;
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │ │  i7 : time inverse Eta;
│ │ │ │ - -- used 0.352284s (cpu); 0.27461s (thread); 0s (gc)
│ │ │ │ + -- used 0.461885s (cpu); 0.307729s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5)
│ │ │ │  i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
│ │ │ │  i9 : assert(Psi * Psi^-1 == 1 and Psi^-1 * Psi == 1)
│ │ │ │  i10 : assert(Eta * Eta^-1 == 1 and Eta^-1 * Eta == 1)
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Isomorphism_lp__Multirational__Map_rp.html
│ │ │ @@ -88,45 +88,45 @@
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │              
    
│ │ │                
    i4 : time isIsomorphism Phi
│ │ │ - -- used 0.00123848s (cpu); 9.188e-06s (thread); 0s (gc)
│ │ │ + -- used 0.00341899s (cpu); 9.82e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
    
│ │ │              
    
│ │ │                
    i5 : Psi = first graph Phi;
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to PP^3)
    
│ │ │              
    
│ │ │                
    i6 : time isIsomorphism Psi
│ │ │ - -- used 0.452327s (cpu); 0.228819s (thread); 0s (gc)
│ │ │ + -- used 0.624242s (cpu); 0.241401s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = false
    
│ │ │              
    
│ │ │                
    i7 : Eta = first graph Psi;
│ │ │  
│ │ │  o7 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 x PP^3 to threefold in PP^3 x PP^2 x PP^2)
    
│ │ │              
    
│ │ │                
    i8 : time isIsomorphism Eta
│ │ │ - -- used 1.64153s (cpu); 0.898134s (thread); 0s (gc)
│ │ │ + -- used 2.05633s (cpu); 1.0353s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : assert(o8 and (not o6) and (not o4))
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,31 +17,31 @@
│ │ │ │       ZZ/33331[a..d]; f = rationalMap {c^2-b*d,b*c-a*d,b^2-a*c};
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic rational map from PP^3 to PP^2)
│ │ │ │  i3 : Phi = rationalMap {f,f};
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │ │  i4 : time isIsomorphism Phi
│ │ │ │ - -- used 0.00123848s (cpu); 9.188e-06s (thread); 0s (gc)
│ │ │ │ + -- used 0.00341899s (cpu); 9.82e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : Psi = first graph Phi;
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to
│ │ │ │  PP^3)
│ │ │ │  i6 : time isIsomorphism Psi
│ │ │ │ - -- used 0.452327s (cpu); 0.228819s (thread); 0s (gc)
│ │ │ │ + -- used 0.624242s (cpu); 0.241401s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = false
│ │ │ │  i7 : Eta = first graph Psi;
│ │ │ │  
│ │ │ │  o7 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 x
│ │ │ │  PP^3 to threefold in PP^3 x PP^2 x PP^2)
│ │ │ │  i8 : time isIsomorphism Eta
│ │ │ │ - -- used 1.64153s (cpu); 0.898134s (thread); 0s (gc)
│ │ │ │ + -- used 2.05633s (cpu); 1.0353s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : assert(o8 and (not o6) and (not o4))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_n_v_e_r_s_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- inverse of a birational map
│ │ │ │      * _i_s_M_o_r_p_h_i_s_m_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- whether a multi-rational map is a
│ │ │ │        morphism
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Morphism_lp__Multirational__Map_rp.html
│ │ │ @@ -85,31 +85,31 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^4 x PP^2)
│ │ │              
    
│ │ │                
    i3 : time isMorphism Phi
│ │ │ - -- used 0.372762s (cpu); 0.223282s (thread); 0s (gc)
│ │ │ + -- used 0.445085s (cpu); 0.265907s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
    
│ │ │              
    
│ │ │                
    i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.171617s (cpu); 0.10156s (thread); 0s (gc)
│ │ │ + -- used 0.250477s (cpu); 0.114506s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^7 x PP^4 x PP^2 to 4-dimensional subvariety of PP^4 x PP^7)
    
│ │ │              
    
│ │ │                
    i5 : time isMorphism Psi
│ │ │ - -- used 4.159s (cpu); 3.00876s (thread); 0s (gc)
│ │ │ + -- used 4.10963s (cpu); 3.41703s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
    
│ │ │                
    i6 : assert((not o3) and o5)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,24 +17,24 @@
│ │ │ │  i1 : ZZ/300007[a..e], f = first graph rationalMap ideal(c^2-b*d,b*c-a*d,b^2-
│ │ │ │  a*c,e), g = rationalMap submatrix(matrix f,{0..2});
│ │ │ │  i2 : Phi = rationalMap {f,g};
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^7 to PP^4 x PP^2)
│ │ │ │  i3 : time isMorphism Phi
│ │ │ │ - -- used 0.372762s (cpu); 0.223282s (thread); 0s (gc)
│ │ │ │ + -- used 0.445085s (cpu); 0.265907s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : time Psi = first graph Phi;
│ │ │ │ - -- used 0.171617s (cpu); 0.10156s (thread); 0s (gc)
│ │ │ │ + -- used 0.250477s (cpu); 0.114506s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^7 x PP^4 x PP^2 to 4-dimensional subvariety of PP^4 x PP^7)
│ │ │ │  i5 : time isMorphism Psi
│ │ │ │ - -- used 4.159s (cpu); 3.00876s (thread); 0s (gc)
│ │ │ │ + -- used 4.10963s (cpu); 3.41703s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : assert((not o3) and o5)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_I_s_o_m_o_r_p_h_i_s_m_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- whether a birational map is an
│ │ │ │        isomorphism
│ │ │ │      * _i_s_M_o_r_p_h_i_s_m_(_R_a_t_i_o_n_a_l_M_a_p_) -- whether a rational map is a morphism
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_linearly__Normal__Embedding.html
│ │ │ @@ -84,30 +84,30 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │              
    
│ │ │                
    i3 : time f = linearlyNormalEmbedding X;
│ │ │ - -- used 0.0079494s (cpu); 0.00865425s (thread); 0s (gc)
│ │ │ + -- used 0.0121033s (cpu); 0.0110541s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (automorphism of X)
    
│ │ │              
    
│ │ │                
    i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an isomorphic projection of X in PP^3
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3
    
│ │ │              
    
│ │ │                
    i5 : time g = linearlyNormalEmbedding Y;
│ │ │ - -- used 0.575291s (cpu); 0.434308s (thread); 0s (gc)
│ │ │ + -- used 0.519796s (cpu); 0.443447s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
    
│ │ │              
    
│ │ │                
    i6 : assert(isIsomorphism g)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,23 +13,23 @@
│ │ │ │              is a linear projection
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : K = ZZ/333331;
│ │ │ │  i2 : X = PP_K^(1,7); -- rational normal curve of degree 7
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │ │  i3 : time f = linearlyNormalEmbedding X;
│ │ │ │ - -- used 0.0079494s (cpu); 0.00865425s (thread); 0s (gc)
│ │ │ │ + -- used 0.0121033s (cpu); 0.0110541s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (automorphism of X)
│ │ │ │  i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an
│ │ │ │  isomorphic projection of X in PP^3
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, curve in PP^3
│ │ │ │  i5 : time g = linearlyNormalEmbedding Y;
│ │ │ │ - -- used 0.575291s (cpu); 0.434308s (thread); 0s (gc)
│ │ │ │ + -- used 0.519796s (cpu); 0.443447s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
│ │ │ │  i6 : assert(isIsomorphism g)
│ │ │ │  i7 : describe g
│ │ │ │  
│ │ │ │  o7 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: curve in PP^3 cut out by 6 hypersurfaces of degree 4
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Multirational__Map_rp.html
│ │ │ @@ -86,15 +86,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │              
    
│ │ │                
    i3 : time multidegree Phi
│ │ │ - -- used 0.53565s (cpu); 0.391645s (thread); 0s (gc)
│ │ │ + -- used 0.580115s (cpu); 0.401078s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {66, 46, 31, 20}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi)
│ │ │ - -- used 0.00200117s (cpu); 0.00145578s (thread); 0s (gc)
│ │ │ - -- used 0.202005s (cpu); 0.128646s (thread); 0s (gc)
│ │ │ - -- used 0.233863s (cpu); 0.151089s (thread); 0s (gc)
│ │ │ - -- used 0.322585s (cpu); 0.170041s (thread); 0s (gc)
│ │ │ - -- used 0.148551s (cpu); 0.0965599s (thread); 0s (gc)
│ │ │ + -- used 0.00402704s (cpu); 0.00150338s (thread); 0s (gc)
│ │ │ + -- used 0.234586s (cpu); 0.154972s (thread); 0s (gc)
│ │ │ + -- used 0.251794s (cpu); 0.176388s (thread); 0s (gc)
│ │ │ + -- used 0.350078s (cpu); 0.201292s (thread); 0s (gc)
│ │ │ + -- used 0.217942s (cpu); 0.14613s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.0498404s (cpu); 0.0522257s (thread); 0s (gc)
    
│ │ │ + -- used 0.133375s (cpu); 0.0749226s (thread); 0s (gc)
│ │ │              
    
│ │ │
    │ │ │
    │ │ │

    References

    │ │ │ ArXiv preprint: Computations with rational maps between multi-projective varieties.
    │ │ │ ├── html2text {} │ │ │ │ @@ -17,25 +17,25 @@ │ │ │ │ This is calculated by means of the inverse image of an appropriate random │ │ │ │ subvariety of the target. │ │ │ │ i1 : Phi = last graph rationalMap PP_(ZZ/300007)^(1,4); │ │ │ │ │ │ │ │ o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x │ │ │ │ PP^5 to PP^5) │ │ │ │ i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi) │ │ │ │ - -- used 0.00200117s (cpu); 0.00145578s (thread); 0s (gc) │ │ │ │ - -- used 0.202005s (cpu); 0.128646s (thread); 0s (gc) │ │ │ │ - -- used 0.233863s (cpu); 0.151089s (thread); 0s (gc) │ │ │ │ - -- used 0.322585s (cpu); 0.170041s (thread); 0s (gc) │ │ │ │ - -- used 0.148551s (cpu); 0.0965599s (thread); 0s (gc) │ │ │ │ + -- used 0.00402704s (cpu); 0.00150338s (thread); 0s (gc) │ │ │ │ + -- used 0.234586s (cpu); 0.154972s (thread); 0s (gc) │ │ │ │ + -- used 0.251794s (cpu); 0.176388s (thread); 0s (gc) │ │ │ │ + -- used 0.350078s (cpu); 0.201292s (thread); 0s (gc) │ │ │ │ + -- used 0.217942s (cpu); 0.14613s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = {51, 28, 14, 6, 2} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : time assert(oo == multidegree Phi) │ │ │ │ - -- used 0.0498404s (cpu); 0.0522257s (thread); 0s (gc) │ │ │ │ + -- used 0.133375s (cpu); 0.0749226s (thread); 0s (gc) │ │ │ │ ********** RReeffeerreenncceess ********** │ │ │ │ ArXiv preprint: _C_o_m_p_u_t_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _m_a_p_s_ _b_e_t_w_e_e_n_ _m_u_l_t_i_-_p_r_o_j_e_c_t_i_v_e │ │ │ │ _v_a_r_i_e_t_i_e_s. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_u_l_t_i_d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a multi-rational │ │ │ │ map │ │ │ │ * _p_r_o_j_e_c_t_i_v_e_D_e_g_r_e_e_s_(_R_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a rational map │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_point_lp__Multiprojective__Variety_rp.html │ │ │ @@ -85,15 +85,15 @@ │ │ │ │ │ │ o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time p := point X
│ │ │ - -- used 0.124959s (cpu); 0.0446009s (thread); 0s (gc)
│ │ │ + -- used 0.205181s (cpu); 0.0611894s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],[3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1])
│ │ │  
│ │ │  o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -102,15 +102,15 @@ │ │ │ │ │ │ o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time q = point Y
│ │ │ - -- used 1.82924s (cpu); 1.00825s (thread); 0s (gc)
│ │ │ + -- used 1.69161s (cpu); 1.10907s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = q
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -14,25 +14,25 @@ │ │ │ │ o a _m_u_l_t_i_-_p_r_o_j_e_c_t_i_v_e_ _v_a_r_i_e_t_y, a random rational point on $X$ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : K = ZZ/1000003; │ │ │ │ i2 : X = PP_K^({1,1,2},{3,2,3}); │ │ │ │ │ │ │ │ o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9 │ │ │ │ i3 : time p := point X │ │ │ │ - -- used 0.124959s (cpu); 0.0446009s (thread); 0s (gc) │ │ │ │ + -- used 0.205181s (cpu); 0.0611894s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1], │ │ │ │ [3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1]) │ │ │ │ │ │ │ │ o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9 │ │ │ │ i4 : Y = random({2,1,2},X); │ │ │ │ │ │ │ │ o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9 │ │ │ │ i5 : time q = point Y │ │ │ │ - -- used 1.82924s (cpu); 1.00825s (thread); 0s (gc) │ │ │ │ + -- used 1.69161s (cpu); 1.10907s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = q │ │ │ │ │ │ │ │ o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9 │ │ │ │ i6 : assert(isSubset(p,X) and isSubset(q,Y)) │ │ │ │ The list of homogeneous coordinates can be obtained with the operator |-. │ │ │ │ i7 : |- p │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre_lp__Multirational__Map_rp.html │ │ │ @@ -106,15 +106,15 @@ │ │ │ │ │ │ o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x PP^4) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time segre Phi;
│ │ │ - -- used 0.885868s (cpu); 0.592017s (thread); 0s (gc)
│ │ │ + -- used 1.4408s (cpu); 0.75308s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : describe segre Phi
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  
│ │ │ │  o4 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │ │  i5 : Phi = rationalMap {f,g,h};
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x
│ │ │ │  PP^4)
│ │ │ │  i6 : time segre Phi;
│ │ │ │ - -- used 0.885868s (cpu); 0.592017s (thread); 0s (gc)
│ │ │ │ + -- used 1.4408s (cpu); 0.75308s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
│ │ │ │  i7 : describe segre Phi
│ │ │ │  
│ │ │ │  o7 = rational map defined by forms of degree 6
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: PP^149
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_show_lp__Multirational__Map_rp.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to threefold in PP^3 x PP^2 x PP^2)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time describe Phi
│ │ │ - -- used 0.227275s (cpu); 0.148372s (thread); 0s (gc)
│ │ │ + -- used 0.250751s (cpu); 0.171452s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of multi-degree (1,1)
│ │ │       target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2 
│ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ │       dominance: true
│ │ │       multidegree: {10, 14, 19, 25}
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  i1 : Phi = inverse first graph last graph rationalMap PP_(ZZ/33331)^(1,3)
│ │ │ │  
│ │ │ │  o1 = Phi
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to
│ │ │ │  threefold in PP^3 x PP^2 x PP^2)
│ │ │ │  i2 : time describe Phi
│ │ │ │ - -- used 0.227275s (cpu); 0.148372s (thread); 0s (gc)
│ │ │ │ + -- used 0.250751s (cpu); 0.171452s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of
│ │ │ │  multi-degree (1,1)
│ │ │ │       target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces
│ │ │ │  of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2
│ │ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out
│ │ │ @@ -26,22 +26,22 @@
│ │ │  
│ │ │  i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => true};
│ │ │  
│ │ │  i8 : prob = n -> log(n)/n;
│ │ │  
│ │ │  i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (70, 88, 88, 93, 94, 94, 94, 95, 98, 98, 94, 94, 95, 95, 94, 96, 97,
│ │ │ +o9 = (59, 82, 80, 92, 94, 94, 95, 94, 95, 95, 98, 95, 98, 98, 96, 99, 97, 98,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     100, 95, 98, 98, 97, 98, 98, 95, 99, 98, 98, 96)
│ │ │ +     97, 98, 100, 95, 98, 97, 97, 97, 99, 98, 98)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (19, 5, 5, 5, 2, 2, 4, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 1, 0, 0,
│ │ │ +o10 = (25, 11, 4, 6, 4, 4, 2, 0, 2, 0, 2, 1, 1, 1, 2, 1, 0, 1, 1, 0, 0, 3, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o10 : Sequence
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Graphs.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DyK, Dvc, DJw, DKG, DKo}
│ │ │ +o2 = {DyS, DGO, DNg, Dv{, DJk}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │  
│ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Regular__Graphs.out
│ │ │ @@ -1,21 +1,21 @@
│ │ │  -- -*- M2-comint -*- hash: 1729831171060067675
│ │ │  
│ │ │  i1 : R = QQ[a..e];
│ │ │  
│ │ │  i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │ -o2 = {Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}},
│ │ │ +o2 = {Graph{"edges" => {{a, c}, {b, d}, {c, d}, {a, e}, {b, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │       Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}}}
│ │ │ +     Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_graph__Complement.out
│ │ │ @@ -13,13 +13,13 @@
│ │ │  i3 : graphComplement "Dhc"
│ │ │  
│ │ │  o3 = DUW
│ │ │  
│ │ │  i4 : G = generateBipartiteGraphs 7;
│ │ │  
│ │ │  i5 : time graphComplement G;
│ │ │ - -- used 0.000481823s (cpu); 0.000476523s (thread); 0s (gc)
│ │ │ + -- used 0.000964222s (cpu); 0.000952981s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : time (graphComplement \ G);
│ │ │ - -- used 0.142292s (cpu); 0.0701726s (thread); 0s (gc)
│ │ │ + -- used 0.150527s (cpu); 0.0797011s (thread); 0s (gc)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html
│ │ │ @@ -122,26 +122,26 @@
│ │ │                
    i8 : prob = n -> log(n)/n;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (70, 88, 88, 93, 94, 94, 94, 95, 98, 98, 94, 94, 95, 95, 94, 96, 97,
│ │ │ +o9 = (59, 82, 80, 92, 94, 94, 95, 94, 95, 95, 98, 95, 98, 98, 96, 99, 97, 98,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     100, 95, 98, 98, 97, 98, 98, 95, 99, 98, 98, 96)
│ │ │ +     97, 98, 100, 95, 98, 97, 97, 97, 99, 98, 98)
│ │ │  
│ │ │  o9 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (19, 5, 5, 5, 2, 2, 4, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 1, 0, 0,
│ │ │ +o10 = (25, 11, 4, 6, 4, 4, 2, 0, 2, 0, 2, 1, 1, 1, 2, 1, 0, 1, 1, 0, 0, 3, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o10 : Sequence
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,23 +38,23 @@
│ │ │ │  connected, at least as $n$ tends to infinity.
│ │ │ │  i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" =>
│ │ │ │  true};
│ │ │ │  i8 : prob = n -> log(n)/n;
│ │ │ │  i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)),
│ │ │ │  connected))
│ │ │ │  
│ │ │ │ -o9 = (70, 88, 88, 93, 94, 94, 94, 95, 98, 98, 94, 94, 95, 95, 94, 96, 97,
│ │ │ │ +o9 = (59, 82, 80, 92, 94, 94, 95, 94, 95, 95, 98, 95, 98, 98, 96, 99, 97, 98,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     100, 95, 98, 98, 97, 98, 98, 95, 99, 98, 98, 96)
│ │ │ │ +     97, 98, 100, 95, 98, 97, 97, 97, 99, 98, 98)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2),
│ │ │ │  connected))
│ │ │ │  
│ │ │ │ -o10 = (19, 5, 5, 5, 2, 2, 4, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 1, 0, 0,
│ │ │ │ +o10 = (25, 11, 4, 6, 4, 4, 2, 0, 2, 0, 2, 1, 1, 1, 2, 1, 0, 1, 1, 0, 0, 3, 0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        0, 0, 0, 0, 0, 0)
│ │ │ │  
│ │ │ │  o10 : Sequence
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _b_u_i_l_d_G_r_a_p_h_F_i_l_t_e_r -- creates the appropriate filter string for use with
│ │ │ │        filterGraphs and countGraphs
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Graphs.html
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o1 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DyK, Dvc, DJw, DKG, DKo}
│ │ │ +o2 = {DyS, DGO, DNg, Dv{, DJk}
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │  i1 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │ │  
│ │ │ │ -o2 = {DyK, Dvc, DJw, DKG, DKo}
│ │ │ │ +o2 = {DyS, DGO, DNg, Dv{, DJk}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o3 : List
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Regular__Graphs.html
│ │ │ @@ -92,23 +92,23 @@
│ │ │                
    i1 : R = QQ[a..e];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │ -o2 = {Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}},
│ │ │ +o2 = {Graph{"edges" => {{a, c}, {b, d}, {c, d}, {a, e}, {b, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │       Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}}}
│ │ │ +     Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,23 +24,23 @@
│ │ │ │  vertices with a given regularity. Note that some graphs may be isomorphic.
│ │ │ │  If a _P_o_l_y_n_o_m_i_a_l_R_i_n_g $R$ is supplied instead, then the number of vertices is the
│ │ │ │  number of generators. Moreover, the nauty-based strings are automatically
│ │ │ │  converted to instances of the class _G_r_a_p_h in $R$.
│ │ │ │  i1 : R = QQ[a..e];
│ │ │ │  i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │ │  
│ │ │ │ -o2 = {Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}},
│ │ │ │ +o2 = {Graph{"edges" => {{a, c}, {b, d}, {c, d}, {a, e}, {b, e}}},
│ │ │ │              "ring" => R
│ │ │ │              "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}},
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}}}
│ │ │ │ +     Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}}}
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The number of vertices $n$ must be positive as nauty cannot handle graphs with
│ │ │ │  zero vertices.
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_graph__Complement.html
│ │ │ @@ -121,21 +121,21 @@
│ │ │              
│ │ │                
    i4 : G = generateBipartiteGraphs 7;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time graphComplement G;
│ │ │ - -- used 0.000481823s (cpu); 0.000476523s (thread); 0s (gc)
    
│ │ │ + -- used 0.000964222s (cpu); 0.000952981s (thread); 0s (gc)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time (graphComplement \ G);
│ │ │ - -- used 0.142292s (cpu); 0.0701726s (thread); 0s (gc)
    │ │ │ + -- used 0.150527s (cpu); 0.0797011s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -41,17 +41,17 @@ │ │ │ │ │ │ │ │ o3 = DUW │ │ │ │ Batch calls can be performed considerably faster when using the List input │ │ │ │ format. However, care should be taken as the returned list is entirely in │ │ │ │ Graph6 or Sparse6 format. │ │ │ │ i4 : G = generateBipartiteGraphs 7; │ │ │ │ i5 : time graphComplement G; │ │ │ │ - -- used 0.000481823s (cpu); 0.000476523s (thread); 0s (gc) │ │ │ │ + -- used 0.000964222s (cpu); 0.000952981s (thread); 0s (gc) │ │ │ │ i6 : time (graphComplement \ G); │ │ │ │ - -- used 0.142292s (cpu); 0.0701726s (thread); 0s (gc) │ │ │ │ + -- used 0.150527s (cpu); 0.0797011s (thread); 0s (gc) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_o_m_p_l_e_m_e_n_t_G_r_a_p_h -- returns the complement of a graph or hypergraph │ │ │ │ ********** WWaayyss ttoo uussee ggrraapphhCCoommpplleemmeenntt:: ********** │ │ │ │ * graphComplement(Graph) │ │ │ │ * graphComplement(List) │ │ │ │ * graphComplement(String) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ @@ -26,22 +26,22 @@ │ │ │ │ │ │ i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => true}; │ │ │ │ │ │ i8 : prob = n -> log(n)/n; │ │ │ │ │ │ i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected)) │ │ │ │ │ │ -o9 = (64, 72, 87, 92, 96, 94, 93, 95, 99, 96, 98, 98, 99, 95, 98, 99, 96, 96, │ │ │ +o9 = (77, 85, 88, 94, 94, 95, 98, 95, 99, 96, 99, 95, 96, 99, 100, 99, 97, │ │ │ ------------------------------------------------------------------------ │ │ │ - 96, 99, 97, 97, 97, 98, 98, 100, 100, 97, 99) │ │ │ + 99, 100, 96, 98, 99, 99, 98, 97, 97, 98, 98, 99) │ │ │ │ │ │ o9 : Sequence │ │ │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected)) │ │ │ │ │ │ -o10 = (22, 10, 5, 5, 3, 3, 3, 2, 2, 1, 2, 2, 1, 1, 0, 2, 0, 0, 0, 0, 0, 0, 1, │ │ │ +o10 = (15, 10, 6, 6, 2, 2, 1, 3, 0, 1, 4, 0, 1, 0, 3, 1, 1, 1, 2, 0, 0, 0, 0, │ │ │ ----------------------------------------------------------------------- │ │ │ - 1, 0, 0, 1, 0, 0) │ │ │ + 0, 0, 0, 0, 0, 0) │ │ │ │ │ │ o10 : Sequence │ │ │ │ │ │ i11 : │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Graphs.out │ │ │ @@ -4,15 +4,15 @@ │ │ │ │ │ │ o1 = {DDO, Dx_, Dlw, Dx{, D_K} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : generateRandomGraphs(5, 5) │ │ │ │ │ │ -o2 = {Dq?, D{[, Dvw, Ds?, Da[} │ │ │ +o2 = {DI{, DRc, DzS, DO{, Dsc} │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : generateRandomGraphs(5, 5, RandomSeed => 314159) │ │ │ │ │ │ o3 = {DDO, Dx_, Dlw, Dx{, D_K} │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Regular__Graphs.out │ │ │ @@ -1,9 +1,9 @@ │ │ │ -- -*- M2-comint -*- hash: 1331287392268 │ │ │ │ │ │ i1 : generateRandomRegularGraphs(5, 3, 2) │ │ │ │ │ │ -o1 = {DqK, DUW, DYc} │ │ │ +o1 = {DRo, DpS, DUW} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ @@ -13,13 +13,13 @@ │ │ │ 4 => {2, 1} │ │ │ │ │ │ o2 : Graph │ │ │ │ │ │ i3 : G = generateBipartiteGraphs 7; │ │ │ │ │ │ i4 : time graphComplement G; │ │ │ - -- used 0.000605455s (cpu); 0.000607439s (thread); 0s (gc) │ │ │ + -- used 0.000641154s (cpu); 0.000620436s (thread); 0s (gc) │ │ │ │ │ │ i5 : time (graphComplement \ G); │ │ │ - -- used 0.0498549s (cpu); 0.0479399s (thread); 0s (gc) │ │ │ + -- used 0.164099s (cpu); 0.0797744s (thread); 0s (gc) │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ @@ -122,28 +122,28 @@ │ │ │ 
      i8 : prob = n -> log(n)/n;
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (64, 72, 87, 92, 96, 94, 93, 95, 99, 96, 98, 98, 99, 95, 98, 99, 96, 96,
│ │ │ +o9 = (77, 85, 88, 94, 94, 95, 98, 95, 99, 96, 99, 95, 96, 99, 100, 99, 97,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     96, 99, 97, 97, 97, 98, 98, 100, 100, 97, 99)
│ │ │ +     99, 100, 96, 98, 99, 99, 98, 97, 97, 98, 98, 99)
│ │ │  
│ │ │  o9 : Sequence
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (22, 10, 5, 5, 3, 3, 3, 2, 2, 1, 2, 2, 1, 1, 0, 2, 0, 0, 0, 0, 0, 0, 1,
│ │ │ +o10 = (15, 10, 6, 6, 2, 2, 1, 3, 0, 1, 4, 0, 1, 0, 3, 1, 1, 1, 2, 0, 0, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 0, 0, 1, 0, 0)
│ │ │ +      0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o10 : Sequence
      │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -38,25 +38,25 @@ │ │ │ │ connected, at least as $n$ tends to infinity. │ │ │ │ i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => │ │ │ │ true}; │ │ │ │ i8 : prob = n -> log(n)/n; │ │ │ │ i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o9 = (64, 72, 87, 92, 96, 94, 93, 95, 99, 96, 98, 98, 99, 95, 98, 99, 96, 96, │ │ │ │ +o9 = (77, 85, 88, 94, 94, 95, 98, 95, 99, 96, 99, 95, 96, 99, 100, 99, 97, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 96, 99, 97, 97, 97, 98, 98, 100, 100, 97, 99) │ │ │ │ + 99, 100, 96, 98, 99, 99, 98, 97, 97, 98, 98, 99) │ │ │ │ │ │ │ │ o9 : Sequence │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o10 = (22, 10, 5, 5, 3, 3, 3, 2, 2, 1, 2, 2, 1, 1, 0, 2, 0, 0, 0, 0, 0, 0, 1, │ │ │ │ +o10 = (15, 10, 6, 6, 2, 2, 1, 3, 0, 1, 4, 0, 1, 0, 3, 1, 1, 1, 2, 0, 0, 0, 0, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 1, 0, 0, 1, 0, 0) │ │ │ │ + 0, 0, 0, 0, 0, 0) │ │ │ │ │ │ │ │ o10 : Sequence │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _b_u_i_l_d_G_r_a_p_h_F_i_l_t_e_r -- creates the appropriate filter string for use with │ │ │ │ filterGraphs and countGraphs │ │ │ │ * _f_i_l_t_e_r_G_r_a_p_h_s -- filters (i.e., selects) graphs in a list for given │ │ │ │ properties │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Graphs.html │ │ │ @@ -98,15 +98,15 @@ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {Dq?, D{[, Dvw, Ds?, Da[}
│ │ │ +o2 = {DI{, DRc, DzS, DO{, Dsc}
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  i1 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │ │  
│ │ │ │ -o2 = {Dq?, D{[, Dvw, Ds?, Da[}
│ │ │ │ +o2 = {DI{, DRc, DzS, DO{, Dsc}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o3 : List
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Regular__Graphs.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │            
    This method generates a specified number of random graphs on a given number of vertices with a given regularity. Note that some graphs may be isomorphic.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : generateRandomRegularGraphs(5, 3, 2)
│ │ │  
│ │ │ -o1 = {DqK, DUW, DYc}
│ │ │ +o1 = {DRo, DpS, DUW}
│ │ │  
│ │ │  o1 : List
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -18,15 +18,15 @@ │ │ │ │ * Outputs: │ │ │ │ o G, a _l_i_s_t, the randomly generated regular graphs │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ This method generates a specified number of random graphs on a given number of │ │ │ │ vertices with a given regularity. Note that some graphs may be isomorphic. │ │ │ │ i1 : generateRandomRegularGraphs(5, 3, 2) │ │ │ │ │ │ │ │ -o1 = {DqK, DUW, DYc} │ │ │ │ +o1 = {DRo, DpS, DUW} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ The number of vertices $n$ must be positive as nauty cannot handle graphs with │ │ │ │ zero vertices. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_n_e_r_a_t_e_R_a_n_d_o_m_G_r_a_p_h_s -- generates random graphs on a given number of │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ @@ -115,21 +115,21 @@ │ │ │ │ │ │ 
    i3 : G = generateBipartiteGraphs 7;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time graphComplement G;
│ │ │ - -- used 0.000605455s (cpu); 0.000607439s (thread); 0s (gc)
    │ │ │ + -- used 0.000641154s (cpu); 0.000620436s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time (graphComplement \ G);
│ │ │ - -- used 0.0498549s (cpu); 0.0479399s (thread); 0s (gc)
    │ │ │ + -- used 0.164099s (cpu); 0.0797744s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use graphComplement:

    │ │ │ ├── html2text {} │ │ │ │ @@ -38,17 +38,17 @@ │ │ │ │ │ │ │ │ o2 : Graph │ │ │ │ Batch calls can be performed considerably faster when using the List input │ │ │ │ format. However, care should be taken as the returned list is entirely in │ │ │ │ Graph6 or Sparse6 format. │ │ │ │ i3 : G = generateBipartiteGraphs 7; │ │ │ │ i4 : time graphComplement G; │ │ │ │ - -- used 0.000605455s (cpu); 0.000607439s (thread); 0s (gc) │ │ │ │ + -- used 0.000641154s (cpu); 0.000620436s (thread); 0s (gc) │ │ │ │ i5 : time (graphComplement \ G); │ │ │ │ - -- used 0.0498549s (cpu); 0.0479399s (thread); 0s (gc) │ │ │ │ + -- used 0.164099s (cpu); 0.0797744s (thread); 0s (gc) │ │ │ │ ********** WWaayyss ttoo uussee ggrraapphhCCoommpplleemmeenntt:: ********** │ │ │ │ * graphComplement(Graph) │ │ │ │ * graphComplement(List) │ │ │ │ * graphComplement(String) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _g_r_a_p_h_C_o_m_p_l_e_m_e_n_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out │ │ │ @@ -47,15 +47,15 @@ │ │ │ o4 : Ideal of R │ │ │ │ │ │ i5 : isPrimary Q │ │ │ │ │ │ o5 = true │ │ │ │ │ │ i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot") │ │ │ - -- .181597s elapsed │ │ │ + -- .0904388s elapsed │ │ │ │ │ │ o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, | │ │ │ ------------------------------------------------------------------------ │ │ │ 2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |} │ │ │ │ │ │ o6 : List │ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ @@ -125,15 +125,15 @@ │ │ │ │ │ │ o5 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot")
│ │ │ - -- .181597s elapsed
│ │ │ + -- .0904388s elapsed
│ │ │  
│ │ │  o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, |
│ │ │       ------------------------------------------------------------------------
│ │ │       2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |}
│ │ │  
│ │ │  o6 : List
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -51,15 +51,15 @@ │ │ │ │ 1 2 3 2 3 │ │ │ │ │ │ │ │ o4 : Ideal of R │ │ │ │ i5 : isPrimary Q │ │ │ │ │ │ │ │ o5 = true │ │ │ │ i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot") │ │ │ │ - -- .181597s elapsed │ │ │ │ + -- .0904388s elapsed │ │ │ │ │ │ │ │ o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ @@ -78,15 +78,15 @@ │ │ │ i13 : for i to dim X list hilbertFunction (i, A1) │ │ │ │ │ │ o13 = {1, 2, 3, 3, 2, 1} │ │ │ │ │ │ o13 : List │ │ │ │ │ │ i14 : Y = time smoothFanoToricVariety(5,100); │ │ │ - -- used 0.369739s (cpu); 0.273607s (thread); 0s (gc) │ │ │ + -- used 0.27879s (cpu); 0.27879s (thread); 0s (gc) │ │ │ │ │ │ i15 : A2 = intersectionRing Y; │ │ │ │ │ │ i16 : assert (# rays Y === numgens A2) │ │ │ │ │ │ i17 : ideal A2 │ │ │ │ │ │ @@ -110,19 +110,19 @@ │ │ │ 2 2 2 2 2 2 2 2 3 2 │ │ │ (t + t t , t t + t , t + t t , t t , t t + t , t - t t - 3t t + t t + 2t , - t t + t + 2t t , t t , - t t + t , t t ) │ │ │ 3 3 5 3 5 5 5 5 6 3 6 5 6 6 8 8 9 8 10 9 10 10 8 9 9 9 10 8 9 8 10 10 8 10 │ │ │ │ │ │ o18 : QuotientRing │ │ │ │ │ │ i19 : for i to dim Y list time hilbertFunction (i, A2) │ │ │ - -- used 0.00147436s (cpu); 0.00146862s (thread); 0s (gc) │ │ │ - -- used 9.9947e-05s (cpu); 0.000100518s (thread); 0s (gc) │ │ │ - -- used 7.2105e-05s (cpu); 7.2466e-05s (thread); 0s (gc) │ │ │ - -- used 7.3277e-05s (cpu); 7.3578e-05s (thread); 0s (gc) │ │ │ - -- used 7.1664e-05s (cpu); 7.1945e-05s (thread); 0s (gc) │ │ │ - -- used 7.2246e-05s (cpu); 7.2686e-05s (thread); 0s (gc) │ │ │ + -- used 0.00175085s (cpu); 0.00174972s (thread); 0s (gc) │ │ │ + -- used 0.000120317s (cpu); 0.000120492s (thread); 0s (gc) │ │ │ + -- used 0.000101363s (cpu); 0.000101917s (thread); 0s (gc) │ │ │ + -- used 8.5461e-05s (cpu); 8.5639e-05s (thread); 0s (gc) │ │ │ + -- used 8.732e-05s (cpu); 8.7381e-05s (thread); 0s (gc) │ │ │ + -- used 8.0771e-05s (cpu); 8.0973e-05s (thread); 0s (gc) │ │ │ │ │ │ o19 = {1, 6, 13, 13, 6, 1} │ │ │ │ │ │ o19 : List │ │ │ │ │ │ i20 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_is__Well__Defined_lp__Normal__Toric__Variety_rp.out │ │ │ @@ -1,29 +1,29 @@ │ │ │ -- -*- M2-comint -*- hash: 16408385764843695632 │ │ │ │ │ │ i1 : assert all (5, d -> isWellDefined toricProjectiveSpace (d+1)) │ │ │ │ │ │ i2 : setRandomSeed (currentTime ()); │ │ │ - -- setting random seed to 1779108377 │ │ │ + -- setting random seed to 1779298406 │ │ │ │ │ │ i3 : a = sort apply (3, i -> random (7)) │ │ │ │ │ │ -o3 = {0, 1, 4} │ │ │ +o3 = {3, 5, 6} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : assert isWellDefined kleinschmidt (4,a) │ │ │ │ │ │ i5 : q = sort apply (5, j -> random (1,9)); │ │ │ │ │ │ i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9)); │ │ │ │ │ │ i7 : q │ │ │ │ │ │ -o7 = {1, 2, 4, 5, 7} │ │ │ +o7 = {3, 3, 6, 8, 8} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : assert isWellDefined weightedProjectiveSpace q │ │ │ │ │ │ i9 : X = new MutableHashTable; │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ @@ -6,61 +6,61 @@ │ │ │ │ │ │ o2 = 5*PP2 │ │ │ 0 │ │ │ │ │ │ o2 : ToricDivisor on PP2 │ │ │ │ │ │ i3 : M1 = elapsedTime monomials D1 │ │ │ - -- .026837s elapsed │ │ │ + -- .0358733s elapsed │ │ │ │ │ │ 5 4 4 2 3 3 2 3 3 2 2 2 2 2 3 2 4 │ │ │ o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x , │ │ │ 2 1 2 0 2 1 2 0 1 2 0 2 1 2 0 1 2 0 1 2 0 2 1 2 │ │ │ ------------------------------------------------------------------------ │ │ │ 3 2 2 3 4 5 4 2 3 3 2 4 5 │ │ │ x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x } │ │ │ 0 1 2 0 1 2 0 1 2 0 2 1 0 1 0 1 0 1 0 1 0 │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1)) │ │ │ - -- .00101458s elapsed │ │ │ + -- .00155238s elapsed │ │ │ │ │ │ i5 : FF2 = hirzebruchSurface 2; │ │ │ │ │ │ i6 : D2 = 2*FF2_0 + 3 * FF2_1 │ │ │ │ │ │ o6 = 2*FF2 + 3*FF2 │ │ │ 0 1 │ │ │ │ │ │ o6 : ToricDivisor on FF2 │ │ │ │ │ │ i7 : M2 = elapsedTime monomials D2 │ │ │ - -- .0268873s elapsed │ │ │ + -- .0521441s elapsed │ │ │ │ │ │ 2 3 2 3 2 3 │ │ │ o7 = {x x , x x , x x x , x x } │ │ │ 1 3 1 2 0 1 2 0 1 │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring variety D2)) │ │ │ - -- .000884292s elapsed │ │ │ + -- .0408103s elapsed │ │ │ │ │ │ i9 : X = kleinschmidt (5, {1,2,3}); │ │ │ │ │ │ i10 : D3 = 3*X_0 + 5*X_1 │ │ │ │ │ │ o10 = 3*X + 5*X │ │ │ 0 1 │ │ │ │ │ │ o10 : ToricDivisor on X │ │ │ │ │ │ i11 : m3 = elapsedTime # monomials D3 │ │ │ - -- 44.5515s elapsed │ │ │ + -- 30.4191s elapsed │ │ │ │ │ │ o11 = 7909 │ │ │ │ │ │ i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3)) │ │ │ - -- .0231023s elapsed │ │ │ + -- .0314611s elapsed │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ @@ -24,19 +24,19 @@ │ │ │ o3 : List │ │ │ │ │ │ i4 : X = normalToricVariety F; │ │ │ │ │ │ i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F) │ │ │ │ │ │ i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}}) │ │ │ - -- used 3.2351e-05s (cpu); 2.5398e-05s (thread); 0s (gc) │ │ │ + -- used 3.1095e-05s (cpu); 2.4747e-05s (thread); 0s (gc) │ │ │ │ │ │ o6 = X1 │ │ │ │ │ │ o6 : NormalToricVariety │ │ │ │ │ │ i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}}; │ │ │ - -- used 0.0432183s (cpu); 0.0432268s (thread); 0s (gc) │ │ │ + -- used 0.0632056s (cpu); 0.0632127s (thread); 0s (gc) │ │ │ │ │ │ i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2) │ │ │ │ │ │ i9 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ @@ -88,15 +88,15 @@ │ │ │ o18 = | 0 1 0 | │ │ │ | 0 0 1 | │ │ │ │ │ │ 2 3 │ │ │ o18 : Matrix ZZ <-- ZZ │ │ │ │ │ │ i19 : X1 = time normalToricVariety convexHull (vertMatrix); │ │ │ - -- used 0.0226572s (cpu); 0.0226567s (thread); 0s (gc) │ │ │ + -- used 0.031523s (cpu); 0.0315238s (thread); 0s (gc) │ │ │ │ │ │ i20 : X2 = time normalToricVariety vertMatrix; │ │ │ - -- used 0.0023202s (cpu); 0.00232116s (thread); 0s (gc) │ │ │ + -- used 0.00314759s (cpu); 0.00315328s (thread); 0s (gc) │ │ │ │ │ │ i21 : assert (set rays X2 === set rays X1 and max X1 === max X2) │ │ │ │ │ │ i22 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ @@ -212,15 +212,15 @@ │ │ │
    │ │ │

    We end with a slightly larger example.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -259,20 +259,20 @@ │ │ │ │ │ │ o18 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i14 : Y = time smoothFanoToricVariety(5,100);
│ │ │ - -- used 0.369739s (cpu); 0.273607s (thread); 0s (gc)
    │ │ │ + -- used 0.27879s (cpu); 0.27879s (thread); 0s (gc) │ │ │
    │ │ │ 
    i15 : A2 = intersectionRing Y;
    │ │ │
    │ │ │ 
    i19 : for i to dim Y list time hilbertFunction (i, A2)
│ │ │ - -- used 0.00147436s (cpu); 0.00146862s (thread); 0s (gc)
│ │ │ - -- used 9.9947e-05s (cpu); 0.000100518s (thread); 0s (gc)
│ │ │ - -- used 7.2105e-05s (cpu); 7.2466e-05s (thread); 0s (gc)
│ │ │ - -- used 7.3277e-05s (cpu); 7.3578e-05s (thread); 0s (gc)
│ │ │ - -- used 7.1664e-05s (cpu); 7.1945e-05s (thread); 0s (gc)
│ │ │ - -- used 7.2246e-05s (cpu); 7.2686e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00175085s (cpu); 0.00174972s (thread); 0s (gc)
│ │ │ + -- used 0.000120317s (cpu); 0.000120492s (thread); 0s (gc)
│ │ │ + -- used 0.000101363s (cpu); 0.000101917s (thread); 0s (gc)
│ │ │ + -- used 8.5461e-05s (cpu); 8.5639e-05s (thread); 0s (gc)
│ │ │ + -- used 8.732e-05s (cpu); 8.7381e-05s (thread); 0s (gc)
│ │ │ + -- used 8.0771e-05s (cpu); 8.0973e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = {1, 6, 13, 13, 6, 1}
│ │ │  
│ │ │  o19 : List
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -96,15 +96,15 @@ │ │ │ │ i13 : for i to dim X list hilbertFunction (i, A1) │ │ │ │ │ │ │ │ o13 = {1, 2, 3, 3, 2, 1} │ │ │ │ │ │ │ │ o13 : List │ │ │ │ We end with a slightly larger example. │ │ │ │ i14 : Y = time smoothFanoToricVariety(5,100); │ │ │ │ - -- used 0.369739s (cpu); 0.273607s (thread); 0s (gc) │ │ │ │ + -- used 0.27879s (cpu); 0.27879s (thread); 0s (gc) │ │ │ │ i15 : A2 = intersectionRing Y; │ │ │ │ i16 : assert (# rays Y === numgens A2) │ │ │ │ i17 : ideal A2 │ │ │ │ │ │ │ │ o17 = ideal (t t , t t , t t , t t , t t , t t , t t , t t , t t t , │ │ │ │ 2 3 2 5 4 5 3 6 4 6 1 7 7 9 8 9 0 1 10 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ @@ -129,20 +129,20 @@ │ │ │ │ (t + t t , t t + t , t + t t , t t , t t + t , t - t t - 3t t + t │ │ │ │ t + 2t , - t t + t + 2t t , t t , - t t + t , t t ) │ │ │ │ 3 3 5 3 5 5 5 5 6 3 6 5 6 6 8 8 9 8 10 │ │ │ │ 9 10 10 8 9 9 9 10 8 9 8 10 10 8 10 │ │ │ │ │ │ │ │ o18 : QuotientRing │ │ │ │ i19 : for i to dim Y list time hilbertFunction (i, A2) │ │ │ │ - -- used 0.00147436s (cpu); 0.00146862s (thread); 0s (gc) │ │ │ │ - -- used 9.9947e-05s (cpu); 0.000100518s (thread); 0s (gc) │ │ │ │ - -- used 7.2105e-05s (cpu); 7.2466e-05s (thread); 0s (gc) │ │ │ │ - -- used 7.3277e-05s (cpu); 7.3578e-05s (thread); 0s (gc) │ │ │ │ - -- used 7.1664e-05s (cpu); 7.1945e-05s (thread); 0s (gc) │ │ │ │ - -- used 7.2246e-05s (cpu); 7.2686e-05s (thread); 0s (gc) │ │ │ │ + -- used 0.00175085s (cpu); 0.00174972s (thread); 0s (gc) │ │ │ │ + -- used 0.000120317s (cpu); 0.000120492s (thread); 0s (gc) │ │ │ │ + -- used 0.000101363s (cpu); 0.000101917s (thread); 0s (gc) │ │ │ │ + -- used 8.5461e-05s (cpu); 8.5639e-05s (thread); 0s (gc) │ │ │ │ + -- used 8.732e-05s (cpu); 8.7381e-05s (thread); 0s (gc) │ │ │ │ + -- used 8.0771e-05s (cpu); 8.0973e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o19 = {1, 6, 13, 13, 6, 1} │ │ │ │ │ │ │ │ o19 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _w_o_r_k_i_n_g_ _w_i_t_h_ _s_h_e_a_v_e_s -- information about coherent sheaves and total │ │ │ │ coordinate rings (a.k.a. Cox rings) │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_is__Well__Defined_lp__Normal__Toric__Variety_rp.html │ │ │ @@ -98,22 +98,22 @@ │ │ │
    │ │ │

    The second examples show that a randomly selected Kleinschmidt toric variety and a weighted projective space are also well-defined.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i2 : setRandomSeed (currentTime ());
│ │ │ - -- setting random seed to 1779108377
    │ │ │ + -- setting random seed to 1779298406 │ │ │
    │ │ │ 
    i3 : a = sort apply (3, i -> random (7))
│ │ │  
│ │ │ -o3 = {0, 1, 4}
│ │ │ +o3 = {3, 5, 6}
│ │ │  
│ │ │  o3 : List
    │ │ │
    │ │ │ 
    i4 : assert isWellDefined kleinschmidt (4,a)
    │ │ │ @@ -131,15 +131,15 @@ │ │ │ 
    i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9));
    │ │ │
    │ │ │ 
    i7 : q
│ │ │  
│ │ │ -o7 = {1, 2, 4, 5, 7}
│ │ │ +o7 = {3, 3, 6, 8, 8}
│ │ │  
│ │ │  o7 : List
    │ │ │
    │ │ │ 
    i8 : assert isWellDefined weightedProjectiveSpace q
    │ │ │ ├── html2text {} │ │ │ │ @@ -28,27 +28,27 @@ │ │ │ │ * the intersection of the cones associated to two elements of coneList is a │ │ │ │ face of each cone. │ │ │ │ The first examples illustrate that small projective spaces are well-defined. │ │ │ │ i1 : assert all (5, d -> isWellDefined toricProjectiveSpace (d+1)) │ │ │ │ The second examples show that a randomly selected Kleinschmidt toric variety │ │ │ │ and a weighted projective space are also well-defined. │ │ │ │ i2 : setRandomSeed (currentTime ()); │ │ │ │ - -- setting random seed to 1779108377 │ │ │ │ + -- setting random seed to 1779298406 │ │ │ │ i3 : a = sort apply (3, i -> random (7)) │ │ │ │ │ │ │ │ -o3 = {0, 1, 4} │ │ │ │ +o3 = {3, 5, 6} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : assert isWellDefined kleinschmidt (4,a) │ │ │ │ i5 : q = sort apply (5, j -> random (1,9)); │ │ │ │ i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j │ │ │ │ -> random (1,9)); │ │ │ │ i7 : q │ │ │ │ │ │ │ │ -o7 = {1, 2, 4, 5, 7} │ │ │ │ +o7 = {3, 3, 6, 8, 8} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : assert isWellDefined weightedProjectiveSpace q │ │ │ │ The next ten examples illustrate various ways that two lists can fail to define │ │ │ │ a normal toric variety. By making the current debugging level greater than one, │ │ │ │ one gets some addition information about the nature of the failure. │ │ │ │ i9 : X = new MutableHashTable; │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ @@ -101,15 +101,15 @@ │ │ │ │ │ │ o2 : ToricDivisor on PP2 │ │ │
    │ │ │ 
    i3 : M1 = elapsedTime monomials D1
│ │ │ - -- .026837s elapsed
│ │ │ + -- .0358733s elapsed
│ │ │  
│ │ │         5     4     4   2 3       3   2 3   3 2     2 2   2   2   3 2   4   
│ │ │  o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x ,
│ │ │         2   1 2   0 2   1 2   0 1 2   0 2   1 2   0 1 2   0 1 2   0 2   1 2 
│ │ │       ------------------------------------------------------------------------
│ │ │          3     2 2     3       4     5     4   2 3   3 2   4     5
│ │ │       x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x }
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o3 : List
    │ │ │
    │ │ │ 
    i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1))
│ │ │ - -- .00101458s elapsed
    │ │ │ + -- .00155238s elapsed │ │ │
    │ │ │
    │ │ │

    Toric varieties of Picard-rank 2 are slightly more interesting.

    │ │ │
    │ │ │ │ │ │ @@ -143,27 +143,27 @@ │ │ │ │ │ │ o6 : ToricDivisor on FF2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -176,23 +176,23 @@ │ │ │ │ │ │ o10 : ToricDivisor on X │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i7 : M2 = elapsedTime monomials D2
│ │ │ - -- .0268873s elapsed
│ │ │ + -- .0521441s elapsed
│ │ │  
│ │ │         2     3 2     3     2 3
│ │ │  o7 = {x x , x x , x x x , x x }
│ │ │         1 3   1 2   0 1 2   0 1
│ │ │  
│ │ │  o7 : List
    │ │ │
    │ │ │ 
    i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring variety D2))
│ │ │ - -- .000884292s elapsed
    │ │ │ + -- .0408103s elapsed │ │ │
    │ │ │ 
    i9 : X = kleinschmidt (5, {1,2,3});
    │ │ │
    │ │ │ 
    i11 : m3 = elapsedTime # monomials D3
│ │ │ - -- 44.5515s elapsed
│ │ │ + -- 30.4191s elapsed
│ │ │  
│ │ │  o11 = 7909
    │ │ │
    │ │ │ 
    i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .0231023s elapsed
    │ │ │ + -- .0314611s elapsed │ │ │
    │ │ │
    │ │ │

    By exploiting latticePoints, this method function avoids using the basis function.

    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -27,61 +27,61 @@ │ │ │ │ i2 : D1 = 5*PP2_0 │ │ │ │ │ │ │ │ o2 = 5*PP2 │ │ │ │ 0 │ │ │ │ │ │ │ │ o2 : ToricDivisor on PP2 │ │ │ │ i3 : M1 = elapsedTime monomials D1 │ │ │ │ - -- .026837s elapsed │ │ │ │ + -- .0358733s elapsed │ │ │ │ │ │ │ │ 5 4 4 2 3 3 2 3 3 2 2 2 2 2 3 2 4 │ │ │ │ o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x , │ │ │ │ 2 1 2 0 2 1 2 0 1 2 0 2 1 2 0 1 2 0 1 2 0 2 1 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 2 2 3 4 5 4 2 3 3 2 4 5 │ │ │ │ x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x } │ │ │ │ 0 1 2 0 1 2 0 1 2 0 2 1 0 1 0 1 0 1 0 1 0 │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring │ │ │ │ variety D1)) │ │ │ │ - -- .00101458s elapsed │ │ │ │ + -- .00155238s elapsed │ │ │ │ Toric varieties of Picard-rank 2 are slightly more interesting. │ │ │ │ i5 : FF2 = hirzebruchSurface 2; │ │ │ │ i6 : D2 = 2*FF2_0 + 3 * FF2_1 │ │ │ │ │ │ │ │ o6 = 2*FF2 + 3*FF2 │ │ │ │ 0 1 │ │ │ │ │ │ │ │ o6 : ToricDivisor on FF2 │ │ │ │ i7 : M2 = elapsedTime monomials D2 │ │ │ │ - -- .0268873s elapsed │ │ │ │ + -- .0521441s elapsed │ │ │ │ │ │ │ │ 2 3 2 3 2 3 │ │ │ │ o7 = {x x , x x , x x x , x x } │ │ │ │ 1 3 1 2 0 1 2 0 1 │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring │ │ │ │ variety D2)) │ │ │ │ - -- .000884292s elapsed │ │ │ │ + -- .0408103s elapsed │ │ │ │ i9 : X = kleinschmidt (5, {1,2,3}); │ │ │ │ i10 : D3 = 3*X_0 + 5*X_1 │ │ │ │ │ │ │ │ o10 = 3*X + 5*X │ │ │ │ 0 1 │ │ │ │ │ │ │ │ o10 : ToricDivisor on X │ │ │ │ i11 : m3 = elapsedTime # monomials D3 │ │ │ │ - -- 44.5515s elapsed │ │ │ │ + -- 30.4191s elapsed │ │ │ │ │ │ │ │ o11 = 7909 │ │ │ │ i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety │ │ │ │ D3)) │ │ │ │ - -- .0231023s elapsed │ │ │ │ + -- .0314611s elapsed │ │ │ │ By exploiting _l_a_t_t_i_c_e_P_o_i_n_t_s, this method function avoids using the _b_a_s_i_s │ │ │ │ function. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _w_o_r_k_i_n_g_ _w_i_t_h_ _d_i_v_i_s_o_r_s -- information about toric divisors and their │ │ │ │ related groups │ │ │ │ * _r_i_n_g_(_N_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_) -- make the total coordinate ring (a.k.a. Cox │ │ │ │ ring) │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ @@ -130,25 +130,25 @@ │ │ │
    │ │ │

    The recommended method for creating a NormalToricVariety from a fan is normalToricVariety(List,List). In fact, this package avoids using objects from the Polyhedra package whenever possible. Here is a trivial example, namely projective 2-space, illustrating the substantial increase in time resulting from the use of a Polyhedra fan.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -48,22 +48,22 @@ │ │ │ │ i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F) │ │ │ │ The recommended method for creating a _N_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y from a fan is │ │ │ │ _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_L_i_s_t_,_L_i_s_t_). In fact, this package avoids using objects from │ │ │ │ the _P_o_l_y_h_e_d_r_a package whenever possible. Here is a trivial example, namely │ │ │ │ projective 2-space, illustrating the substantial increase in time resulting │ │ │ │ from the use of a _P_o_l_y_h_e_d_r_a fan. │ │ │ │ i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}}) │ │ │ │ - -- used 3.2351e-05s (cpu); 2.5398e-05s (thread); 0s (gc) │ │ │ │ + -- used 3.1095e-05s (cpu); 2.4747e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 = X1 │ │ │ │ │ │ │ │ o6 : NormalToricVariety │ │ │ │ i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull │ │ │ │ matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}}; │ │ │ │ - -- used 0.0432183s (cpu); 0.0432268s (thread); 0s (gc) │ │ │ │ + -- used 0.0632056s (cpu); 0.0632127s (thread); 0s (gc) │ │ │ │ i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_i_n_g_ _n_o_r_m_a_l_ _t_o_r_i_c_ _v_a_r_i_e_t_i_e_s -- information about the basic constructors │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y -- make a normal toric variety │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_F_a_n_) -- make a normal toric variety from a 'Polyhedra' │ │ │ │ fan │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ @@ -238,21 +238,21 @@ │ │ │ 2 3 │ │ │ o18 : Matrix ZZ <-- ZZ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -102,17 +102,17 @@ │ │ │ │ │ │ │ │ o18 = | 0 1 0 | │ │ │ │ | 0 0 1 | │ │ │ │ │ │ │ │ 2 3 │ │ │ │ o18 : Matrix ZZ <-- ZZ │ │ │ │ i19 : X1 = time normalToricVariety convexHull (vertMatrix); │ │ │ │ - -- used 0.0226572s (cpu); 0.0226567s (thread); 0s (gc) │ │ │ │ + -- used 0.031523s (cpu); 0.0315238s (thread); 0s (gc) │ │ │ │ i20 : X2 = time normalToricVariety vertMatrix; │ │ │ │ - -- used 0.0023202s (cpu); 0.00232116s (thread); 0s (gc) │ │ │ │ + -- used 0.00314759s (cpu); 0.00315328s (thread); 0s (gc) │ │ │ │ i21 : assert (set rays X2 === set rays X1 and max X1 === max X2) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_i_n_g_ _n_o_r_m_a_l_ _t_o_r_i_c_ _v_a_r_i_e_t_i_e_s -- information about the basic constructors │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_M_a_t_r_i_x_) -- make a normal toric variety from a polytope │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_P_o_l_y_h_e_d_r_o_n_) -- make a normal toric variety from a │ │ │ │ 'Polyhedra' polyhedron │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out │ │ │ @@ -21,21 +21,21 @@ │ │ │ │ │ │ i4 : (numericalHilbertFunction(F, I, 3, Verbose => false)).hilbertFunctionValue == 0 │ │ │ │ │ │ o4 = true │ │ │ │ │ │ i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true) │ │ │ Sampling image points ... │ │ │ - -- used .00672189 seconds │ │ │ + -- used .00812445 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .0058048 seconds │ │ │ + -- used .00714516 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00268928 seconds │ │ │ + -- used .00336244 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .0954805 seconds │ │ │ + -- used .0866842 seconds │ │ │ │ │ │ o5 = a "numerical interpolation table", indicating │ │ │ the space of degree 3 forms in the ideal of the image has dimension 3 │ │ │ │ │ │ o5 : NumericalInterpolationTable │ │ │ │ │ │ i6 : extractImageEquations(T, AttemptZZ => true) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out │ │ │ @@ -11,21 +11,21 @@ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ 1 4 │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true) │ │ │ Sampling image points ... │ │ │ - -- used .00321894 seconds │ │ │ + -- used .0041176 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .00233726 seconds │ │ │ + -- used .00320456 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .000982764 seconds │ │ │ + -- used .00134111 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .000271359 seconds │ │ │ + -- used .000298052 seconds │ │ │ │ │ │ o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 | │ │ │ │ │ │ 1 3 │ │ │ o3 : Matrix (CC [y ..y ]) <-- (CC [y ..y ]) │ │ │ 53 0 3 53 0 3 │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out │ │ │ @@ -11,40 +11,40 @@ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ 1 4 │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ i3 : numericalHilbertFunction(F, ideal 0_R, 4) │ │ │ Sampling image points ... │ │ │ - -- used .0107411 seconds │ │ │ + -- used .0133731 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .0109895 seconds │ │ │ + -- used .0128837 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00699058 seconds │ │ │ + -- used .00789063 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .00086927 seconds │ │ │ + -- used .000988368 seconds │ │ │ │ │ │ o3 = a "numerical interpolation table", indicating │ │ │ the space of degree 4 forms in the ideal of the image has dimension 22 │ │ │ │ │ │ o3 : NumericalInterpolationTable │ │ │ │ │ │ i4 : R = CC[x_(1,1)..x_(2,4)]; │ │ │ │ │ │ i5 : F = (minors(2, genericMatrix(R, 2, 4)))_*; │ │ │ │ │ │ i6 : S = numericalImageSample(F, ideal 0_R, 60); │ │ │ │ │ │ i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true) │ │ │ Creating interpolation matrix ... │ │ │ - -- used .10749 seconds │ │ │ + -- used .100172 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .0080657 seconds │ │ │ + -- used .00985427 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .000918353 seconds │ │ │ + -- used .00109451 seconds │ │ │ │ │ │ o7 = a "numerical interpolation table", indicating │ │ │ the space of degree 2 forms in the ideal of the image has dimension 1 │ │ │ │ │ │ o7 : NumericalInterpolationTable │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Dim.out │ │ │ @@ -20,12 +20,12 @@ │ │ │ │ │ │ i8 : F = sum(1..14, i -> basis(4, R, Variables=>toList(a_(i,1)..a_(i,5)))); │ │ │ │ │ │ 1 70 │ │ │ o8 : Matrix R <-- R │ │ │ │ │ │ i9 : time numericalImageDim(F, ideal 0_R) │ │ │ - -- used 0.0692326s (cpu); 0.0691933s (thread); 0s (gc) │ │ │ + -- used 0.0935776s (cpu); 0.0935704s (thread); 0s (gc) │ │ │ │ │ │ o9 = 69 │ │ │ │ │ │ i10 : │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_real__Point.out │ │ │ @@ -31,15 +31,15 @@ │ │ │ o5 : Ideal of R │ │ │ │ │ │ i6 : I = I1 + I2; │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ │ │ i7 : elapsedTime p = realPoint(I, Iterations => 100) │ │ │ - -- .82045s elapsed │ │ │ + -- .607407s elapsed │ │ │ │ │ │ o7 = p │ │ │ │ │ │ o7 : Point │ │ │ │ │ │ i8 : matrix pack(5, p#Coordinates) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html │ │ │ @@ -105,21 +105,21 @@ │ │ │ o4 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -33,21 +33,21 @@ │ │ │ │ o3 : Ideal of R │ │ │ │ i4 : (numericalHilbertFunction(F, I, 3, Verbose => false)).hilbertFunctionValue │ │ │ │ == 0 │ │ │ │ │ │ │ │ o4 = true │ │ │ │ i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .00672189 seconds │ │ │ │ + -- used .00812445 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .0058048 seconds │ │ │ │ + -- used .00714516 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00268928 seconds │ │ │ │ + -- used .00336244 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .0954805 seconds │ │ │ │ + -- used .0866842 seconds │ │ │ │ │ │ │ │ o5 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 3 forms in the ideal of the image has dimension 3 │ │ │ │ │ │ │ │ o5 : NumericalInterpolationTable │ │ │ │ i6 : extractImageEquations(T, AttemptZZ => true) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_extract__Image__Equations.html │ │ │ @@ -107,21 +107,21 @@ │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -38,21 +38,21 @@ │ │ │ │ │ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ │ │ 1 4 │ │ │ │ o2 : Matrix R <-- R │ │ │ │ i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .00321894 seconds │ │ │ │ + -- used .0041176 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .00233726 seconds │ │ │ │ + -- used .00320456 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .000982764 seconds │ │ │ │ + -- used .00134111 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .000271359 seconds │ │ │ │ + -- used .000298052 seconds │ │ │ │ │ │ │ │ o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 | │ │ │ │ │ │ │ │ 1 3 │ │ │ │ o3 : Matrix (CC [y ..y ]) <-- (CC [y ..y ]) │ │ │ │ 53 0 3 53 0 3 │ │ │ │ Here is how to do the same computation symbolically. │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Hilbert__Function.html │ │ │ @@ -112,21 +112,21 @@ │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -152,19 +152,19 @@ │ │ │ 
    i6 : S = numericalImageSample(F, ideal 0_R, 60);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -57,39 +57,39 @@ │ │ │ │ │ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ │ │ 1 4 │ │ │ │ o2 : Matrix R <-- R │ │ │ │ i3 : numericalHilbertFunction(F, ideal 0_R, 4) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .0107411 seconds │ │ │ │ + -- used .0133731 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .0109895 seconds │ │ │ │ + -- used .0128837 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00699058 seconds │ │ │ │ + -- used .00789063 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .00086927 seconds │ │ │ │ + -- used .000988368 seconds │ │ │ │ │ │ │ │ o3 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 4 forms in the ideal of the image has dimension 22 │ │ │ │ │ │ │ │ o3 : NumericalInterpolationTable │ │ │ │ The following example computes the dimension of Plücker quadrics in the │ │ │ │ defining ideal of the Grassmannian $Gr(2,4)$ of $P^1$'s in $P^3$, in the │ │ │ │ ambient space $P^5$. │ │ │ │ i4 : R = CC[x_(1,1)..x_(2,4)]; │ │ │ │ i5 : F = (minors(2, genericMatrix(R, 2, 4)))_*; │ │ │ │ i6 : S = numericalImageSample(F, ideal 0_R, 60); │ │ │ │ i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true) │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .10749 seconds │ │ │ │ + -- used .100172 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .0080657 seconds │ │ │ │ + -- used .00985427 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .000918353 seconds │ │ │ │ + -- used .00109451 seconds │ │ │ │ │ │ │ │ o7 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 2 forms in the ideal of the image has dimension 1 │ │ │ │ │ │ │ │ o7 : NumericalInterpolationTable │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _N_u_m_e_r_i_c_a_l_I_n_t_e_r_p_o_l_a_t_i_o_n_T_a_b_l_e -- the class of all │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html │ │ │ @@ -145,15 +145,15 @@ │ │ │ 1 70 │ │ │ o8 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
│ │ │ - -- used 3.2351e-05s (cpu); 2.5398e-05s (thread); 0s (gc)
│ │ │ + -- used 3.1095e-05s (cpu); 2.4747e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = X1
│ │ │  
│ │ │  o6 : NormalToricVariety
    │ │ │
    │ │ │ 
    i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}};
│ │ │ - -- used 0.0432183s (cpu); 0.0432268s (thread); 0s (gc)
    │ │ │ + -- used 0.0632056s (cpu); 0.0632127s (thread); 0s (gc) │ │ │
    │ │ │ 
    i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)
    │ │ │
    │ │ │ 
    i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ - -- used 0.0226572s (cpu); 0.0226567s (thread); 0s (gc)
    │ │ │ + -- used 0.031523s (cpu); 0.0315238s (thread); 0s (gc) │ │ │
    │ │ │ 
    i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 0.0023202s (cpu); 0.00232116s (thread); 0s (gc)
    │ │ │ + -- used 0.00314759s (cpu); 0.00315328s (thread); 0s (gc) │ │ │
    │ │ │ 
    i21 : assert (set rays X2 === set rays X1 and max X1 === max X2)
    │ │ │
    │ │ │ 
    i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .00672189 seconds
│ │ │ +     -- used .00812445 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0058048 seconds
│ │ │ +     -- used .00714516 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00268928 seconds
│ │ │ +     -- used .00336244 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .0954805 seconds
│ │ │ +     -- used .0866842 seconds
│ │ │  
│ │ │  o5 = a "numerical interpolation table", indicating
│ │ │       the space of degree 3 forms in the ideal of the image has dimension 3
│ │ │  
│ │ │  o5 : NumericalInterpolationTable
    │ │ │
    │ │ │ 
    i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .00321894 seconds
│ │ │ +     -- used .0041176 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00233726 seconds
│ │ │ +     -- used .00320456 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .000982764 seconds
│ │ │ +     -- used .00134111 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000271359 seconds
│ │ │ +     -- used .000298052 seconds
│ │ │  
│ │ │  o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 |
│ │ │  
│ │ │                            1                   3
│ │ │  o3 : Matrix (CC  [y ..y ])  <-- (CC  [y ..y ])
│ │ │                 53  0   3           53  0   3
    │ │ │
    │ │ │ 
    i3 : numericalHilbertFunction(F, ideal 0_R, 4)
│ │ │  Sampling image points ...
│ │ │ -     -- used .0107411 seconds
│ │ │ +     -- used .0133731 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0109895 seconds
│ │ │ +     -- used .0128837 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00699058 seconds
│ │ │ +     -- used .00789063 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .00086927 seconds
│ │ │ +     -- used .000988368 seconds
│ │ │  
│ │ │  o3 = a "numerical interpolation table", indicating
│ │ │       the space of degree 4 forms in the ideal of the image has dimension 22
│ │ │  
│ │ │  o3 : NumericalInterpolationTable
    │ │ │
    │ │ │ 
    i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true)
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .10749 seconds
│ │ │ +     -- used .100172 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .0080657 seconds
│ │ │ +     -- used .00985427 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000918353 seconds
│ │ │ +     -- used .00109451 seconds
│ │ │  
│ │ │  o7 = a "numerical interpolation table", indicating
│ │ │       the space of degree 2 forms in the ideal of the image has dimension 1
│ │ │  
│ │ │  o7 : NumericalInterpolationTable
    │ │ │
    │ │ │ 
    i9 : time numericalImageDim(F, ideal 0_R)
│ │ │ - -- used 0.0692326s (cpu); 0.0691933s (thread); 0s (gc)
│ │ │ + -- used 0.0935776s (cpu); 0.0935704s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = 69
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -43,15 +43,15 @@ │ │ │ │ 201-222. We numerically verify this below. │ │ │ │ i7 : R = CC[a_(1,1)..a_(14,5)]; │ │ │ │ i8 : F = sum(1..14, i -> basis(4, R, Variables=>toList(a_(i,1)..a_(i,5)))); │ │ │ │ │ │ │ │ 1 70 │ │ │ │ o8 : Matrix R <-- R │ │ │ │ i9 : time numericalImageDim(F, ideal 0_R) │ │ │ │ - -- used 0.0692326s (cpu); 0.0691933s (thread); 0s (gc) │ │ │ │ + -- used 0.0935776s (cpu); 0.0935704s (thread); 0s (gc) │ │ │ │ │ │ │ │ o9 = 69 │ │ │ │ ********** WWaayyss ttoo uussee nnuummeerriiccaallIImmaaggeeDDiimm:: ********** │ │ │ │ * numericalImageDim(List,Ideal) │ │ │ │ * numericalImageDim(List,Ideal,Point) │ │ │ │ * numericalImageDim(Matrix,Ideal) │ │ │ │ * numericalImageDim(Matrix,Ideal,Point) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_real__Point.html │ │ │ @@ -137,15 +137,15 @@ │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime p = realPoint(I, Iterations => 100)
│ │ │ - -- .82045s elapsed
│ │ │ + -- .607407s elapsed
│ │ │  
│ │ │  o7 = p
│ │ │  
│ │ │  o7 : Point
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -49,15 +49,15 @@ │ │ │ │ i5 : I2 = ideal apply(entries transpose A, row -> sum(row, v -> v^2) - 1); │ │ │ │ │ │ │ │ o5 : Ideal of R │ │ │ │ i6 : I = I1 + I2; │ │ │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ i7 : elapsedTime p = realPoint(I, Iterations => 100) │ │ │ │ - -- .82045s elapsed │ │ │ │ + -- .607407s elapsed │ │ │ │ │ │ │ │ o7 = p │ │ │ │ │ │ │ │ o7 : Point │ │ │ │ i8 : matrix pack(5, p#Coordinates) │ │ │ │ │ │ │ │ o8 = | .722359 .289465 -.295808 .591752 -.454678 | │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/___Lab__Book__Protocol.out │ │ │ @@ -14,35 +14,35 @@ │ │ │ │ │ │ i4 : LL7a=select(LL7,L->not knownExample L);#LL7a │ │ │ │ │ │ o5 = 2 │ │ │ │ │ │ i6 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0,Verbose=>true)) │ │ │ unfolding │ │ │ - -- .146506s elapsed │ │ │ + -- .124577s elapsed │ │ │ flatteningRelations │ │ │ - -- .164102s elapsed │ │ │ + -- .120068s elapsed │ │ │ next gb │ │ │ - -- .000820344s elapsed │ │ │ + -- .0012414s elapsed │ │ │ true │ │ │ unfolding │ │ │ - -- .062437s elapsed │ │ │ + -- .0922854s elapsed │ │ │ flatteningRelations │ │ │ - -- .106231s elapsed │ │ │ + -- .110275s elapsed │ │ │ next gb │ │ │ - -- .000505014s elapsed │ │ │ + -- .000587901s elapsed │ │ │ true │ │ │ - -- 1.575s elapsed │ │ │ + -- 1.31766s elapsed │ │ │ │ │ │ o6 = {} │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0)) │ │ │ - -- 1.65638s elapsed │ │ │ + -- 1.22788s elapsed │ │ │ │ │ │ o7 = {} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : LL7b=={} │ │ │ │ │ │ @@ -75,23 +75,23 @@ │ │ │ │ │ │ o10 : Sequence │ │ │ │ │ │ i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true) │ │ │ (13, 1) │ │ │ {5, 8, 11, 12} │ │ │ unfolding │ │ │ - -- .286719s elapsed │ │ │ + -- .193601s elapsed │ │ │ flatteningRelations │ │ │ - -- .12441s elapsed │ │ │ + -- .124742s elapsed │ │ │ next gb │ │ │ - -- .000912956s elapsed │ │ │ + -- .0009495s elapsed │ │ │ true │ │ │ - -- .888207s elapsed │ │ │ + -- .704308s elapsed │ │ │ (5, 8, all semigroups are smoothable) │ │ │ - -- .978467s elapsed │ │ │ + -- .76393s elapsed │ │ │ │ │ │ o11 = {} │ │ │ │ │ │ o11 : List │ │ │ │ │ │ i12 : L={6,8,9,11} │ │ │ │ │ │ @@ -100,22 +100,22 @@ │ │ │ o12 : List │ │ │ │ │ │ i13 : genus L │ │ │ │ │ │ o13 = 8 │ │ │ │ │ │ i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true) │ │ │ - -- .0682537s elapsed │ │ │ + -- .0592915s elapsed │ │ │ 6 │ │ │ false │ │ │ 5 │ │ │ false │ │ │ 4 │ │ │ decompose │ │ │ - -- .330823s elapsed │ │ │ + -- .335222s elapsed │ │ │ number of components: 2 │ │ │ support c, codim c: {(1, 1), (16, 3)} │ │ │ {0, -1} │ │ │ │ │ │ o14 = true │ │ │ │ │ │ i15 : │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_heuristic__Smoothness.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ -- setting random seed to 1644814534404491274313411285186041988099567563905780374824086062516559438 │ │ │ │ │ │ i4 : elapsedTime tally apply(10,i-> ( │ │ │ c=minors(2,random(S^2,S^{3:-2})); │ │ │ c=sub(c,x_0=>1); │ │ │ R=kk[support c];c=sub(c,R); │ │ │ heuristicSmoothness c)) │ │ │ - -- 3.14556s elapsed │ │ │ + -- 2.47518s elapsed │ │ │ │ │ │ o4 = Tally{false => 9} │ │ │ true => 1 │ │ │ │ │ │ o4 : Tally │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_is__Smoothable__Semigroup.out │ │ │ @@ -7,17 +7,17 @@ │ │ │ o1 : List │ │ │ │ │ │ i2 : genus L │ │ │ │ │ │ o2 = 8 │ │ │ │ │ │ i3 : elapsedTime isSmoothableSemigroup(L,0.30,0) │ │ │ - -- .920652s elapsed │ │ │ + -- .840829s elapsed │ │ │ │ │ │ o3 = false │ │ │ │ │ │ i4 : elapsedTime isSmoothableSemigroup(L,0.14,0) │ │ │ - -- 4.75674s elapsed │ │ │ + -- 3.53509s elapsed │ │ │ │ │ │ o4 = true │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_is__Weierstrass__Semigroup.out │ │ │ @@ -7,12 +7,12 @@ │ │ │ o1 : List │ │ │ │ │ │ i2 : genus L │ │ │ │ │ │ o2 = 8 │ │ │ │ │ │ i3 : elapsedTime isWeierstrassSemigroup(L,0.15) │ │ │ - -- 3.70301s elapsed │ │ │ + -- 3.36799s elapsed │ │ │ │ │ │ o3 = true │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_non__Weierstrass__Semigroups.out │ │ │ @@ -1,12 +1,12 @@ │ │ │ -- -*- M2-comint -*- hash: 6860996532851631556 │ │ │ │ │ │ i1 : elapsedTime nonWeierstrassSemigroups(6,7) │ │ │ (6, 7, all semigroups are smoothable) │ │ │ - -- 1.32446s elapsed │ │ │ + -- 1.24049s elapsed │ │ │ │ │ │ o1 = {} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : LLdifficult={{6, 8, 9, 11}} │ │ │ │ │ │ @@ -14,61 +14,61 @@ │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true) │ │ │ (17, 5) │ │ │ {6, 7, 8, 17} │ │ │ unfolding │ │ │ - -- .402156s elapsed │ │ │ + -- .366613s elapsed │ │ │ flatteningRelations │ │ │ - -- .138689s elapsed │ │ │ + -- .217766s elapsed │ │ │ next gb │ │ │ - -- .00193938s elapsed │ │ │ + -- .00246003s elapsed │ │ │ true │ │ │ - -- .956436s elapsed │ │ │ + -- .961915s elapsed │ │ │ {6, 7, 9, 17} │ │ │ unfolding │ │ │ - -- .44802s elapsed │ │ │ + -- .339742s elapsed │ │ │ flatteningRelations │ │ │ - -- .188992s elapsed │ │ │ + -- .183867s elapsed │ │ │ next gb │ │ │ - -- .00398281s elapsed │ │ │ + -- .00281018s elapsed │ │ │ decompose │ │ │ - -- .241914s elapsed │ │ │ + -- .129346s elapsed │ │ │ number of components: 2 │ │ │ support c, codim c: {(2, 2), (5, 2)} │ │ │ {0, -1} │ │ │ - -- 3.43475s elapsed │ │ │ + -- 2.61066s elapsed │ │ │ {6, 8, 9, 10} │ │ │ unfolding │ │ │ - -- .150863s elapsed │ │ │ + -- .116705s elapsed │ │ │ flatteningRelations │ │ │ - -- .114868s elapsed │ │ │ + -- .107202s elapsed │ │ │ next gb │ │ │ - -- .000478735s elapsed │ │ │ + -- .000467096s elapsed │ │ │ true │ │ │ - -- 1.26211s elapsed │ │ │ + -- .978871s elapsed │ │ │ {6, 8, 10, 11, 13} │ │ │ unfolding │ │ │ - -- .644758s elapsed │ │ │ + -- .50882s elapsed │ │ │ flatteningRelations │ │ │ - -- .256825s elapsed │ │ │ + -- .249886s elapsed │ │ │ next gb │ │ │ - -- .00377869s elapsed │ │ │ + -- .00659451s elapsed │ │ │ decompose │ │ │ - -- .880657s elapsed │ │ │ + -- .933102s elapsed │ │ │ number of components: 1 │ │ │ support c, codim c: {(5, 1)} │ │ │ {-1} │ │ │ - -- 2.97713s elapsed │ │ │ - -- 8.63053s elapsed │ │ │ + -- 2.64091s elapsed │ │ │ + -- 7.19249s elapsed │ │ │ 0 │ │ │ │ │ │ {} │ │ │ - -- .000002905s elapsed │ │ │ - -- 8.72002s elapsed │ │ │ + -- .000004504s elapsed │ │ │ + -- 7.22948s elapsed │ │ │ │ │ │ o3 = {{6, 8, 9, 11}} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/___Lab__Book__Protocol.html │ │ │ @@ -101,38 +101,38 @@ │ │ │ o5 = 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0,Verbose=>true))
│ │ │  unfolding
│ │ │ - -- .146506s elapsed
│ │ │ + -- .124577s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .164102s elapsed
│ │ │ + -- .120068s elapsed
│ │ │  next gb
│ │ │ - -- .000820344s elapsed
│ │ │ + -- .0012414s elapsed
│ │ │  true
│ │ │  unfolding
│ │ │ - -- .062437s elapsed
│ │ │ + -- .0922854s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .106231s elapsed
│ │ │ + -- .110275s elapsed
│ │ │  next gb
│ │ │ - -- .000505014s elapsed
│ │ │ + -- .000587901s elapsed
│ │ │  true
│ │ │ - -- 1.575s elapsed
│ │ │ + -- 1.31766s elapsed
│ │ │  
│ │ │  o6 = {}
│ │ │  
│ │ │  o6 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ - -- 1.65638s elapsed
│ │ │ + -- 1.22788s elapsed
│ │ │  
│ │ │  o7 = {}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -189,23 +189,23 @@ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true)
│ │ │  (13, 1)
│ │ │  {5, 8, 11, 12}
│ │ │  unfolding
│ │ │ - -- .286719s elapsed
│ │ │ + -- .193601s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .12441s elapsed
│ │ │ + -- .124742s elapsed
│ │ │  next gb
│ │ │ - -- .000912956s elapsed
│ │ │ + -- .0009495s elapsed
│ │ │  true
│ │ │ - -- .888207s elapsed
│ │ │ + -- .704308s elapsed
│ │ │  (5, 8,  all semigroups are smoothable)
│ │ │ - -- .978467s elapsed
│ │ │ + -- .76393s elapsed
│ │ │  
│ │ │  o11 = {}
│ │ │  
│ │ │  o11 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -228,22 +228,22 @@ │ │ │ │ │ │ o13 = 8 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true)
│ │ │ - -- .0682537s elapsed
│ │ │ + -- .0592915s elapsed
│ │ │  6
│ │ │  false
│ │ │  5
│ │ │  false
│ │ │  4
│ │ │  decompose
│ │ │ - -- .330823s elapsed
│ │ │ + -- .335222s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(1, 1), (16, 3)}
│ │ │  {0, -1}
│ │ │  
│ │ │  o14 = true
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -26,34 +26,34 @@ │ │ │ │ o3 = 39 │ │ │ │ i4 : LL7a=select(LL7,L->not knownExample L);#LL7a │ │ │ │ │ │ │ │ o5 = 2 │ │ │ │ i6 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup │ │ │ │ (L,0.25,0,Verbose=>true)) │ │ │ │ unfolding │ │ │ │ - -- .146506s elapsed │ │ │ │ + -- .124577s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .164102s elapsed │ │ │ │ + -- .120068s elapsed │ │ │ │ next gb │ │ │ │ - -- .000820344s elapsed │ │ │ │ + -- .0012414s elapsed │ │ │ │ true │ │ │ │ unfolding │ │ │ │ - -- .062437s elapsed │ │ │ │ + -- .0922854s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .106231s elapsed │ │ │ │ + -- .110275s elapsed │ │ │ │ next gb │ │ │ │ - -- .000505014s elapsed │ │ │ │ + -- .000587901s elapsed │ │ │ │ true │ │ │ │ - -- 1.575s elapsed │ │ │ │ + -- 1.31766s elapsed │ │ │ │ │ │ │ │ o6 = {} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0)) │ │ │ │ - -- 1.65638s elapsed │ │ │ │ + -- 1.22788s elapsed │ │ │ │ │ │ │ │ o7 = {} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : LL7b=={} │ │ │ │ │ │ │ │ o8 = true │ │ │ │ @@ -92,23 +92,23 @@ │ │ │ │ o10 = (5, 8) │ │ │ │ │ │ │ │ o10 : Sequence │ │ │ │ i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true) │ │ │ │ (13, 1) │ │ │ │ {5, 8, 11, 12} │ │ │ │ unfolding │ │ │ │ - -- .286719s elapsed │ │ │ │ + -- .193601s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .12441s elapsed │ │ │ │ + -- .124742s elapsed │ │ │ │ next gb │ │ │ │ - -- .000912956s elapsed │ │ │ │ + -- .0009495s elapsed │ │ │ │ true │ │ │ │ - -- .888207s elapsed │ │ │ │ + -- .704308s elapsed │ │ │ │ (5, 8, all semigroups are smoothable) │ │ │ │ - -- .978467s elapsed │ │ │ │ + -- .76393s elapsed │ │ │ │ │ │ │ │ o11 = {} │ │ │ │ │ │ │ │ o11 : List │ │ │ │ In the verbose mode we get timings of various computation steps and further │ │ │ │ information. The first line, (13,1), indicates that there 13 semigroups of │ │ │ │ multiplicity 5 and genus 8 of which only 1 is not flagged as smoothable by the │ │ │ │ @@ -120,22 +120,22 @@ │ │ │ │ o12 = {6, 8, 9, 11} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ i13 : genus L │ │ │ │ │ │ │ │ o13 = 8 │ │ │ │ i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true) │ │ │ │ - -- .0682537s elapsed │ │ │ │ + -- .0592915s elapsed │ │ │ │ 6 │ │ │ │ false │ │ │ │ 5 │ │ │ │ false │ │ │ │ 4 │ │ │ │ decompose │ │ │ │ - -- .330823s elapsed │ │ │ │ + -- .335222s elapsed │ │ │ │ number of components: 2 │ │ │ │ support c, codim c: {(1, 1), (16, 3)} │ │ │ │ {0, -1} │ │ │ │ │ │ │ │ o14 = true │ │ │ │ The first integer, 6, tells that in this attempt deformation parameters of │ │ │ │ degree >= 6 were used and no smooth fiber was found. Finally with all │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_heuristic__Smoothness.html │ │ │ @@ -99,15 +99,15 @@ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime tally apply(10,i-> (
│ │ │               c=minors(2,random(S^2,S^{3:-2}));
│ │ │               c=sub(c,x_0=>1);
│ │ │               R=kk[support c];c=sub(c,R);
│ │ │               heuristicSmoothness c))
│ │ │ - -- 3.14556s elapsed
│ │ │ + -- 2.47518s elapsed
│ │ │  
│ │ │  o4 = Tally{false => 9}
│ │ │             true => 1
│ │ │  
│ │ │  o4 : Tally
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -27,15 +27,15 @@ │ │ │ │ -- setting random seed to │ │ │ │ 1644814534404491274313411285186041988099567563905780374824086062516559438 │ │ │ │ i4 : elapsedTime tally apply(10,i-> ( │ │ │ │ c=minors(2,random(S^2,S^{3:-2})); │ │ │ │ c=sub(c,x_0=>1); │ │ │ │ R=kk[support c];c=sub(c,R); │ │ │ │ heuristicSmoothness c)) │ │ │ │ - -- 3.14556s elapsed │ │ │ │ + -- 2.47518s elapsed │ │ │ │ │ │ │ │ o4 = Tally{false => 9} │ │ │ │ true => 1 │ │ │ │ │ │ │ │ o4 : Tally │ │ │ │ ********** WWaayyss ttoo uussee hheeuurriissttiiccSSmmooootthhnneessss:: ********** │ │ │ │ * heuristicSmoothness(Ideal) │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_is__Smoothable__Semigroup.html │ │ │ @@ -100,23 +100,23 @@ │ │ │ │ │ │ o2 = 8 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime isSmoothableSemigroup(L,0.30,0)
│ │ │ - -- .920652s elapsed
│ │ │ + -- .840829s elapsed
│ │ │  
│ │ │  o3 = false
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ - -- 4.75674s elapsed
│ │ │ + -- 3.53509s elapsed
│ │ │  
│ │ │  o4 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,19 +29,19 @@ │ │ │ │ o1 = {6, 8, 9, 11} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : genus L │ │ │ │ │ │ │ │ o2 = 8 │ │ │ │ i3 : elapsedTime isSmoothableSemigroup(L,0.30,0) │ │ │ │ - -- .920652s elapsed │ │ │ │ + -- .840829s elapsed │ │ │ │ │ │ │ │ o3 = false │ │ │ │ i4 : elapsedTime isSmoothableSemigroup(L,0.14,0) │ │ │ │ - -- 4.75674s elapsed │ │ │ │ + -- 3.53509s elapsed │ │ │ │ │ │ │ │ o4 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_e_U_n_f_o_l_d_i_n_g -- Makes the universal homogeneous unfolding of an ideal │ │ │ │ with positive degree parameters │ │ │ │ * _f_l_a_t_t_e_n_i_n_g_R_e_l_a_t_i_o_n_s -- Compute the flattening relations of an unfolding │ │ │ │ * _g_e_t_F_l_a_t_F_a_m_i_l_y -- Compute the flat family depending on a subset of │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_is__Weierstrass__Semigroup.html │ │ │ @@ -99,15 +99,15 @@ │ │ │ │ │ │ o2 = 8 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime isWeierstrassSemigroup(L,0.15)
│ │ │ - -- 3.70301s elapsed
│ │ │ + -- 3.36799s elapsed
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ o1 = {6, 8, 9, 11} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : genus L │ │ │ │ │ │ │ │ o2 = 8 │ │ │ │ i3 : elapsedTime isWeierstrassSemigroup(L,0.15) │ │ │ │ - -- 3.70301s elapsed │ │ │ │ + -- 3.36799s elapsed │ │ │ │ │ │ │ │ o3 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_e_U_n_f_o_l_d_i_n_g -- Makes the universal homogeneous unfolding of an ideal │ │ │ │ with positive degree parameters │ │ │ │ * _f_l_a_t_t_e_n_i_n_g_R_e_l_a_t_i_o_n_s -- Compute the flattening relations of an unfolding │ │ │ │ * _g_e_t_F_l_a_t_F_a_m_i_l_y -- Compute the flat family depending on a subset of │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_non__Weierstrass__Semigroups.html │ │ │ @@ -84,15 +84,15 @@ │ │ │

    We test which semigroups of multiplicity m and genus g are smoothable. If no smoothing was found then L is a candidate for a non Weierstrass semigroup. In this search certain semigroups L in LLdifficult, where the computation is particular heavy are excluded.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -106,62 +106,62 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : elapsedTime nonWeierstrassSemigroups(6,7)
│ │ │  (6, 7,  all semigroups are smoothable)
│ │ │ - -- 1.32446s elapsed
│ │ │ + -- 1.24049s elapsed
│ │ │  
│ │ │  o1 = {}
│ │ │  
│ │ │  o1 : List
    │ │ │
    │ │ │ 
    i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
│ │ │  (17, 5)
│ │ │  {6, 7, 8, 17}
│ │ │  unfolding
│ │ │ - -- .402156s elapsed
│ │ │ + -- .366613s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .138689s elapsed
│ │ │ + -- .217766s elapsed
│ │ │  next gb
│ │ │ - -- .00193938s elapsed
│ │ │ + -- .00246003s elapsed
│ │ │  true
│ │ │ - -- .956436s elapsed
│ │ │ + -- .961915s elapsed
│ │ │  {6, 7, 9, 17}
│ │ │  unfolding
│ │ │ - -- .44802s elapsed
│ │ │ + -- .339742s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .188992s elapsed
│ │ │ + -- .183867s elapsed
│ │ │  next gb
│ │ │ - -- .00398281s elapsed
│ │ │ + -- .00281018s elapsed
│ │ │  decompose
│ │ │ - -- .241914s elapsed
│ │ │ + -- .129346s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │  {0, -1}
│ │ │ - -- 3.43475s elapsed
│ │ │ + -- 2.61066s elapsed
│ │ │  {6, 8, 9, 10}
│ │ │  unfolding
│ │ │ - -- .150863s elapsed
│ │ │ + -- .116705s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .114868s elapsed
│ │ │ + -- .107202s elapsed
│ │ │  next gb
│ │ │ - -- .000478735s elapsed
│ │ │ + -- .000467096s elapsed
│ │ │  true
│ │ │ - -- 1.26211s elapsed
│ │ │ + -- .978871s elapsed
│ │ │  {6, 8, 10, 11, 13}
│ │ │  unfolding
│ │ │ - -- .644758s elapsed
│ │ │ + -- .50882s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .256825s elapsed
│ │ │ + -- .249886s elapsed
│ │ │  next gb
│ │ │ - -- .00377869s elapsed
│ │ │ + -- .00659451s elapsed
│ │ │  decompose
│ │ │ - -- .880657s elapsed
│ │ │ + -- .933102s elapsed
│ │ │  number of components: 1
│ │ │  support c, codim c: {(5, 1)}
│ │ │  {-1}
│ │ │ - -- 2.97713s elapsed
│ │ │ - -- 8.63053s elapsed
│ │ │ + -- 2.64091s elapsed
│ │ │ + -- 7.19249s elapsed
│ │ │  0
│ │ │  
│ │ │  {}
│ │ │ - -- .000002905s elapsed
│ │ │ - -- 8.72002s elapsed
│ │ │ + -- .000004504s elapsed
│ │ │ + -- 7.22948s elapsed
│ │ │  
│ │ │  o3 = {{6, 8, 9, 11}}
│ │ │  
│ │ │  o3 : List
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -22,76 +22,76 @@ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ We test which semigroups of multiplicity m and genus g are smoothable. If no │ │ │ │ smoothing was found then L is a candidate for a non Weierstrass semigroup. In │ │ │ │ this search certain semigroups L in LLdifficult, where the computation is │ │ │ │ particular heavy are excluded. │ │ │ │ i1 : elapsedTime nonWeierstrassSemigroups(6,7) │ │ │ │ (6, 7, all semigroups are smoothable) │ │ │ │ - -- 1.32446s elapsed │ │ │ │ + -- 1.24049s elapsed │ │ │ │ │ │ │ │ o1 = {} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : LLdifficult={{6, 8, 9, 11}} │ │ │ │ │ │ │ │ o2 = {{6, 8, 9, 11}} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true) │ │ │ │ (17, 5) │ │ │ │ {6, 7, 8, 17} │ │ │ │ unfolding │ │ │ │ - -- .402156s elapsed │ │ │ │ + -- .366613s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .138689s elapsed │ │ │ │ + -- .217766s elapsed │ │ │ │ next gb │ │ │ │ - -- .00193938s elapsed │ │ │ │ + -- .00246003s elapsed │ │ │ │ true │ │ │ │ - -- .956436s elapsed │ │ │ │ + -- .961915s elapsed │ │ │ │ {6, 7, 9, 17} │ │ │ │ unfolding │ │ │ │ - -- .44802s elapsed │ │ │ │ + -- .339742s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .188992s elapsed │ │ │ │ + -- .183867s elapsed │ │ │ │ next gb │ │ │ │ - -- .00398281s elapsed │ │ │ │ + -- .00281018s elapsed │ │ │ │ decompose │ │ │ │ - -- .241914s elapsed │ │ │ │ + -- .129346s elapsed │ │ │ │ number of components: 2 │ │ │ │ support c, codim c: {(2, 2), (5, 2)} │ │ │ │ {0, -1} │ │ │ │ - -- 3.43475s elapsed │ │ │ │ + -- 2.61066s elapsed │ │ │ │ {6, 8, 9, 10} │ │ │ │ unfolding │ │ │ │ - -- .150863s elapsed │ │ │ │ + -- .116705s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .114868s elapsed │ │ │ │ + -- .107202s elapsed │ │ │ │ next gb │ │ │ │ - -- .000478735s elapsed │ │ │ │ + -- .000467096s elapsed │ │ │ │ true │ │ │ │ - -- 1.26211s elapsed │ │ │ │ + -- .978871s elapsed │ │ │ │ {6, 8, 10, 11, 13} │ │ │ │ unfolding │ │ │ │ - -- .644758s elapsed │ │ │ │ + -- .50882s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .256825s elapsed │ │ │ │ + -- .249886s elapsed │ │ │ │ next gb │ │ │ │ - -- .00377869s elapsed │ │ │ │ + -- .00659451s elapsed │ │ │ │ decompose │ │ │ │ - -- .880657s elapsed │ │ │ │ + -- .933102s elapsed │ │ │ │ number of components: 1 │ │ │ │ support c, codim c: {(5, 1)} │ │ │ │ {-1} │ │ │ │ - -- 2.97713s elapsed │ │ │ │ - -- 8.63053s elapsed │ │ │ │ + -- 2.64091s elapsed │ │ │ │ + -- 7.19249s elapsed │ │ │ │ 0 │ │ │ │ │ │ │ │ {} │ │ │ │ - -- .000002905s elapsed │ │ │ │ - -- 8.72002s elapsed │ │ │ │ + -- .000004504s elapsed │ │ │ │ + -- 7.22948s elapsed │ │ │ │ │ │ │ │ o3 = {{6, 8, 9, 11}} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ In the verbose mode we get timings of various computation steps and further │ │ │ │ information. The first line, (17,5), indicates that there 17 semigroups of │ │ │ │ multiplicity 6 and genus 8 of which only 5 is not flagged as smoothable by the │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___O__I__Resolution.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1) │ │ │ - -- used 0.080543s (cpu); 0.0805438s (thread); 0s (gc) │ │ │ + -- used 0.0981741s (cpu); 0.0981738s (thread); 0s (gc) │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4, 4}, {-4, -4}) │ │ │ │ │ │ o5 : OIResolution │ │ │ │ │ │ i6 : C.dd_0 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___O__I__Resolution_sp_us_sp__Z__Z.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ - -- used 0.216842s (cpu); 0.118275s (thread); 0s (gc) │ │ │ + -- used 0.226771s (cpu); 0.12285s (thread); 0s (gc) │ │ │ │ │ │ i6 : C_0 │ │ │ │ │ │ o6 = Basis symbol: e0 │ │ │ Basis element widths: {2} │ │ │ Degree shifts: {-2} │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Top__Nonminimal.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time oiRes({b}, 2, TopNonminimal => true) │ │ │ - -- used 0.473764s (cpu); 0.306975s (thread); 0s (gc) │ │ │ + -- used 0.482409s (cpu); 0.304752s (thread); 0s (gc) │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4}, {-4}) │ │ │ 2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5}) │ │ │ │ │ │ o5 : OIResolution │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe__Full.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ - -- used 0.079822s (cpu); 0.0798193s (thread); 0s (gc) │ │ │ + -- used 0.0971198s (cpu); 0.0971192s (thread); 0s (gc) │ │ │ │ │ │ i6 : describeFull C │ │ │ │ │ │ o6 = 0: Module: Basis symbol: e0 │ │ │ Basis element widths: {2} │ │ │ Degree shifts: {-2} │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe_lp__O__I__Resolution_rp.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ - -- used 0.0803059s (cpu); 0.0803098s (thread); 0s (gc) │ │ │ + -- used 0.0970331s (cpu); 0.0970338s (thread); 0s (gc) │ │ │ │ │ │ i6 : describe C │ │ │ │ │ │ o6 = 0: Module: Basis symbol: e0 │ │ │ Basis element widths: {2} │ │ │ Degree shifts: {-2} │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__Complex.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 2, TopNonminimal => true) │ │ │ - -- used 0.31327s (cpu); 0.248089s (thread); 0s (gc) │ │ │ + -- used 0.357147s (cpu); 0.27584s (thread); 0s (gc) │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4}, {-4}) │ │ │ 2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5}) │ │ │ │ │ │ o5 : OIResolution │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__O__I__G__B.out │ │ │ @@ -15,15 +15,15 @@ │ │ │ i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3); │ │ │ │ │ │ i10 : isOIGB {b1, b2} │ │ │ │ │ │ o10 = false │ │ │ │ │ │ i11 : time B = oiGB {b1, b2} │ │ │ - -- used 0.164268s (cpu); 0.0745431s (thread); 0s (gc) │ │ │ + -- used 0.164789s (cpu); 0.0608928s (thread); 0s (gc) │ │ │ │ │ │ o11 = {x e + x e , x x e + x x e , │ │ │ 1,1 1,{1},1 2,1 1,{1},2 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 │ │ │ ----------------------------------------------------------------------- │ │ │ x x x e - x x x e } │ │ │ 2,3 2,2 1,1 3,{2, 3},3 2,3 2,1 1,2 3,{1, 3},3 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_minimize__O__I__G__B.out │ │ │ @@ -11,15 +11,15 @@ │ │ │ i5 : installGeneratorsInWidth(F, 3); │ │ │ │ │ │ i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2); │ │ │ │ │ │ i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3); │ │ │ │ │ │ i10 : time B = oiGB {b1, b2} │ │ │ - -- used 0.0278249s (cpu); 0.0278252s (thread); 0s (gc) │ │ │ + -- used 0.0370094s (cpu); 0.0370075s (thread); 0s (gc) │ │ │ │ │ │ o10 = {x e + x e , x x e + x x e , │ │ │ 1,1 1,{1},1 2,1 1,{1},2 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 │ │ │ ----------------------------------------------------------------------- │ │ │ x x x e - x x x e } │ │ │ 2,3 2,2 1,1 3,{2, 3},3 2,3 2,1 1,2 3,{1, 3},3 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__O__I__Resolution_rp.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ - -- used 0.212876s (cpu); 0.111707s (thread); 0s (gc) │ │ │ + -- used 0.229258s (cpu); 0.12685s (thread); 0s (gc) │ │ │ │ │ │ i6 : net C │ │ │ │ │ │ o6 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4, 4}, {-4, -4}) │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__G__B.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ i4 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i5 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2); │ │ │ │ │ │ i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3); │ │ │ │ │ │ i9 : time oiGB {b1, b2} │ │ │ - -- used 0.026697s (cpu); 0.0266981s (thread); 0s (gc) │ │ │ + -- used 0.0325008s (cpu); 0.0325005s (thread); 0s (gc) │ │ │ │ │ │ o9 = {x e + x e , x x e + x x e , │ │ │ 1,1 1,{1},1 2,1 1,{1},2 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 │ │ │ ------------------------------------------------------------------------ │ │ │ x x x e - x x x e } │ │ │ 2,3 2,2 1,1 3,{2, 3},3 2,3 2,1 1,2 3,{1, 3},3 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__Res.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time oiRes({b}, 2, TopNonminimal => true) │ │ │ - -- used 0.636277s (cpu); 0.33181s (thread); 0s (gc) │ │ │ + -- used 0.6644s (cpu); 0.354789s (thread); 0s (gc) │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4}, {-4}) │ │ │ 2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5}) │ │ │ │ │ │ o5 : OIResolution │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_reduce__O__I__G__B.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ i4 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i5 : use F_1; b1 = x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2); │ │ │ │ │ │ i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2); │ │ │ │ │ │ i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal) │ │ │ - -- used 0.140081s (cpu); 0.140081s (thread); 0s (gc) │ │ │ + -- used 0.138067s (cpu); 0.138067s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ o9 = {x e + x e , x x e + x x e , │ │ │ 2,1 1,{1},2 1,1 1,{1},2 1,2 1,1 2,{2},1 2,2 1,2 2,{2},2 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 │ │ │ x x e - x x e } │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution.html │ │ │ @@ -79,15 +79,15 @@ │ │ │ │ │ │ 
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time C = oiRes({b}, 1)
│ │ │ - -- used 0.080543s (cpu); 0.0805438s (thread); 0s (gc)
│ │ │ + -- used 0.0981741s (cpu); 0.0981738s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │  
│ │ │  o5 : OIResolution
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -11,15 +11,15 @@ │ │ │ │ complex, use _i_s_C_o_m_p_l_e_x. To get the $n$th differential in an OI-resolution C, │ │ │ │ use C.dd_n. │ │ │ │ i1 : P = makePolynomialOIAlgebra(2, x, QQ); │ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ i5 : time C = oiRes({b}, 1) │ │ │ │ - -- used 0.080543s (cpu); 0.0805438s (thread); 0s (gc) │ │ │ │ + -- used 0.0981741s (cpu); 0.0981738s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ │ 1: (e1, {4, 4}, {-4, -4}) │ │ │ │ │ │ │ │ o5 : OIResolution │ │ │ │ i6 : C.dd_0 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution_sp_us_sp__Z__Z.html │ │ │ @@ -97,15 +97,15 @@ │ │ │ │ │ │ 
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.216842s (cpu); 0.118275s (thread); 0s (gc)
    │ │ │ + -- used 0.226771s (cpu); 0.12285s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : C_0
│ │ │  
│ │ │  o6 = Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Returns the free OI-module of $C$ in homological degree $n$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.216842s (cpu); 0.118275s (thread); 0s (gc)
│ │ │ │ + -- used 0.226771s (cpu); 0.12285s (thread); 0s (gc)
│ │ │ │  i6 : C_0
│ │ │ │  
│ │ │ │  o6 = Basis symbol: e0
│ │ │ │       Basis element widths: {2}
│ │ │ │       Degree shifts: {-2}
│ │ │ │       Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │       Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Top__Nonminimal.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.473764s (cpu); 0.306975s (thread); 0s (gc)
│ │ │ + -- used 0.482409s (cpu); 0.304752s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │  homological degree $n-1$ to be minimized. Therefore, use TopNonminimal => true
│ │ │ │  for no minimization of the basis in degree $n-1$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.473764s (cpu); 0.306975s (thread); 0s (gc)
│ │ │ │ + -- used 0.482409s (cpu); 0.304752s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd TTooppNNoonnmmiinniimmaall:: **********
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe__Full.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.079822s (cpu); 0.0798193s (thread); 0s (gc)
    
│ │ │ + -- used 0.0971198s (cpu); 0.0971192s (thread); 0s (gc)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : describeFull C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │  Displays the free OI-modules and describes the differentials of an OI-
│ │ │ │  resolution.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.079822s (cpu); 0.0798193s (thread); 0s (gc)
│ │ │ │ + -- used 0.0971198s (cpu); 0.0971192s (thread); 0s (gc)
│ │ │ │  i6 : describeFull C
│ │ │ │  
│ │ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ │                  Basis element widths: {2}
│ │ │ │                  Degree shifts: {-2}
│ │ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │                  Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__O__I__Resolution_rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.0803059s (cpu); 0.0803098s (thread); 0s (gc)
    
│ │ │ + -- used 0.0970331s (cpu); 0.0970338s (thread); 0s (gc)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : describe C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Displays the free OI-modules and differentials of an OI-resolution.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.0803059s (cpu); 0.0803098s (thread); 0s (gc)
│ │ │ │ + -- used 0.0970331s (cpu); 0.0970338s (thread); 0s (gc)
│ │ │ │  i6 : describe C
│ │ │ │  
│ │ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ │                  Basis element widths: {2}
│ │ │ │                  Degree shifts: {-2}
│ │ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │                  Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Complex.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.31327s (cpu); 0.248089s (thread); 0s (gc)
│ │ │ + -- used 0.357147s (cpu); 0.27584s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  option must be either true or false, depending on whether one wants debug
│ │ │ │  information printed.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.31327s (cpu); 0.248089s (thread); 0s (gc)
│ │ │ │ + -- used 0.357147s (cpu); 0.27584s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : isComplex C
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__O__I__G__B.html
│ │ │ @@ -121,15 +121,15 @@
│ │ │  
│ │ │  o10 = false
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.164268s (cpu); 0.0745431s (thread); 0s (gc)
│ │ │ + -- used 0.164789s (cpu); 0.0608928s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,15 +24,15 @@
│ │ │ │  i5 : installGeneratorsInWidth(F, 3);
│ │ │ │  i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i10 : isOIGB {b1, b2}
│ │ │ │  
│ │ │ │  o10 = false
│ │ │ │  i11 : time B = oiGB {b1, b2}
│ │ │ │ - -- used 0.164268s (cpu); 0.0745431s (thread); 0s (gc)
│ │ │ │ + -- used 0.164789s (cpu); 0.0608928s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x   x   x   e           - x   x   x   e          }
│ │ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_minimize__O__I__G__B.html
│ │ │ @@ -114,15 +114,15 @@
│ │ │              
│ │ │                
    i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0278249s (cpu); 0.0278252s (thread); 0s (gc)
│ │ │ + -- used 0.0370094s (cpu); 0.0370075s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : installGeneratorsInWidth(F, 3);
│ │ │ │  i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i10 : time B = oiGB {b1, b2}
│ │ │ │ - -- used 0.0278249s (cpu); 0.0278252s (thread); 0s (gc)
│ │ │ │ + -- used 0.0370094s (cpu); 0.0370075s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x   x   x   e           - x   x   x   e          }
│ │ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.212876s (cpu); 0.111707s (thread); 0s (gc)
    
│ │ │ + -- used 0.229258s (cpu); 0.12685s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : net C
│ │ │  
│ │ │  o6 = 0: (e0, {2}, {-2})
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  Displays the basis element widths and degree shifts of the free OI-modules in
│ │ │ │  an OI-resolution.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.212876s (cpu); 0.111707s (thread); 0s (gc)
│ │ │ │ + -- used 0.229258s (cpu); 0.12685s (thread); 0s (gc)
│ │ │ │  i6 : net C
│ │ │ │  
│ │ │ │  o6 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _n_e_t_(_O_I_R_e_s_o_l_u_t_i_o_n_) -- display an OI-resolution
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__G__B.html
│ │ │ @@ -117,15 +117,15 @@
│ │ │              
│ │ │                
    i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time oiGB {b1, b2}
│ │ │ - -- used 0.026697s (cpu); 0.0266981s (thread); 0s (gc)
│ │ │ + -- used 0.0325008s (cpu); 0.0325005s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │         1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   x   e           - x   x   x   e          }
│ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i9 : time oiGB {b1, b2}
│ │ │ │ - -- used 0.026697s (cpu); 0.0266981s (thread); 0s (gc)
│ │ │ │ + -- used 0.0325008s (cpu); 0.0325005s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │         1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x   x   x   e           - x   x   x   e          }
│ │ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Res.html
│ │ │ @@ -112,15 +112,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.636277s (cpu); 0.33181s (thread); 0s (gc)
│ │ │ + -- used 0.6644s (cpu); 0.354789s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  Therefore, use TopNonminimal => true for no minimization of the basis in degree
│ │ │ │  $n-1$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.636277s (cpu); 0.33181s (thread); 0s (gc)
│ │ │ │ + -- used 0.6644s (cpu); 0.354789s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  ********** WWaayyss ttoo uussee ooiiRReess:: **********
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_reduce__O__I__G__B.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │              
│ │ │                
    i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal)
│ │ │ - -- used 0.140081s (cpu); 0.140081s (thread); 0s (gc)
│ │ │ + -- used 0.138067s (cpu); 0.138067s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e       ,
│ │ │         2,1 1,{1},2    1,1 1,{1},2   1,2 1,1 2,{2},1    2,2 1,2 2,{2},2 
│ │ │       ------------------------------------------------------------------------
│ │ │        2                  2
│ │ │       x   x   e        - x   x   e       }
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : use F_1; b1 = x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2);
│ │ │ │  i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);
│ │ │ │  i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal)
│ │ │ │ - -- used 0.140081s (cpu); 0.140081s (thread); 0s (gc)
│ │ │ │ + -- used 0.138067s (cpu); 0.138067s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e       ,
│ │ │ │         2,1 1,{1},2    1,1 1,{1},2   1,2 1,1 2,{2},1    2,2 1,2 2,{2},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2                  2
│ │ │ │       x   x   e        - x   x   e       }
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Fast__Nonminimal.out
│ │ │ @@ -9,25 +9,25 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 2.11976s elapsed
│ │ │ + -- 2.61645s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
│ │ │  
│ │ │  i4 : elapsedTime C1 = res ideal(I_*)
│ │ │ - -- 1.53192s elapsed
│ │ │ + -- 1.47376s elapsed
│ │ │  
│ │ │        1      35      140      385      819      1080      819      385      140      35      1
│ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S    <-- S   <-- S  <-- 0
│ │ │                                                                                                      
│ │ │       0      1       2        3        4        5         6        7        8        9       10     11
│ │ │  
│ │ │  o4 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp..._cm__Minimize_eq_gt..._rp.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 2.19953s elapsed
│ │ │ + -- 2.64628s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_computing_spresolutions.out
│ │ │ @@ -36,16 +36,16 @@
│ │ │            << res M << endl << endl;
│ │ │            break;
│ │ │            ) else (
│ │ │            << "-- computation interrupted" << endl;
│ │ │            status M.cache.resolution;
│ │ │            << "-- continuing the computation" << endl;
│ │ │            ))
│ │ │ - -- used 0.941571s (cpu); 0.695161s (thread); 0s (gc)
│ │ │ - -- used 0.639919s (cpu); 0.46002s (thread); 0s (gc)
│ │ │ + -- used 1.07221s (cpu); 0.992525s (thread); 0s (gc)
│ │ │ + -- used 1.06381s (cpu); 0.887844s (thread); 0s (gc)
│ │ │  -- computation started: 
│ │ │  -- computation interrupted
│ │ │  -- continuing the computation
│ │ │  -- computation complete
│ │ │   4      11      89      122      40
│ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Fast__Nonminimal.html
│ │ │ @@ -94,28 +94,28 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 2.11976s elapsed
│ │ │ + -- 2.61645s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime C1 = res ideal(I_*)
│ │ │ - -- 1.53192s elapsed
│ │ │ + -- 1.47376s elapsed
│ │ │  
│ │ │        1      35      140      385      819      1080      819      385      140      35      1
│ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S    <-- S   <-- S  <-- 0
│ │ │                                                                                                      
│ │ │       0      1       2        3        4        5         6        7        8        9       10     11
│ │ │  
│ │ │  o4 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,28 +29,28 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ │ - -- 2.11976s elapsed
│ │ │ │ + -- 2.61645s elapsed
│ │ │ │  
│ │ │ │        1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S
│ │ │ │  <-- S    <-- S   <-- 0
│ │ │ │  
│ │ │ │  
│ │ │ │       0      1       2        3        4         5         6         7         8
│ │ │ │  9        10      11
│ │ │ │  
│ │ │ │  o3 : ChainComplex
│ │ │ │  i4 : elapsedTime C1 = res ideal(I_*)
│ │ │ │ - -- 1.53192s elapsed
│ │ │ │ + -- 1.47376s elapsed
│ │ │ │  
│ │ │ │        1      35      140      385      819      1080      819      385      140
│ │ │ │  35      1
│ │ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S
│ │ │ │  <-- S   <-- S  <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/_betti_lp..._cm__Minimize_eq_gt..._rp.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 2.19953s elapsed
│ │ │ + -- 2.64628s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ │ - -- 2.19953s elapsed
│ │ │ │ + -- 2.64628s elapsed
│ │ │ │  
│ │ │ │        1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S
│ │ │ │  <-- S    <-- S   <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html
│ │ │ @@ -117,16 +117,16 @@
│ │ │            << res M << endl << endl;
│ │ │            break;
│ │ │            ) else (
│ │ │            << "-- computation interrupted" << endl;
│ │ │            status M.cache.resolution;
│ │ │            << "-- continuing the computation" << endl;
│ │ │            ))
│ │ │ - -- used 0.941571s (cpu); 0.695161s (thread); 0s (gc)
│ │ │ - -- used 0.639919s (cpu); 0.46002s (thread); 0s (gc)
│ │ │ + -- used 1.07221s (cpu); 0.992525s (thread); 0s (gc)
│ │ │ + -- used 1.06381s (cpu); 0.887844s (thread); 0s (gc)
│ │ │  -- computation started: 
│ │ │  -- computation interrupted
│ │ │  -- continuing the computation
│ │ │  -- computation complete
│ │ │   4      11      89      122      40
│ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,16 +50,16 @@
│ │ │ │            << res M << endl << endl;
│ │ │ │            break;
│ │ │ │            ) else (
│ │ │ │            << "-- computation interrupted" << endl;
│ │ │ │            status M.cache.resolution;
│ │ │ │            << "-- continuing the computation" << endl;
│ │ │ │            ))
│ │ │ │ - -- used 0.941571s (cpu); 0.695161s (thread); 0s (gc)
│ │ │ │ - -- used 0.639919s (cpu); 0.46002s (thread); 0s (gc)
│ │ │ │ + -- used 1.07221s (cpu); 0.992525s (thread); 0s (gc)
│ │ │ │ + -- used 1.06381s (cpu); 0.887844s (thread); 0s (gc)
│ │ │ │  -- computation started:
│ │ │ │  -- computation interrupted
│ │ │ │  -- continuing the computation
│ │ │ │  -- computation complete
│ │ │ │   4      11      89      122      40
│ │ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out
│ │ │ @@ -182,25 +182,25 @@
│ │ │  o15 = 4
│ │ │  
│ │ │  i16 : for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .308205s elapsed
│ │ │ - -- .316063s elapsed
│ │ │ - -- .473982s elapsed
│ │ │ - -- .240853s elapsed
│ │ │ - -- .259009s elapsed
│ │ │ - -- .297748s elapsed
│ │ │ - -- .73903s elapsed
│ │ │ - -- .456206s elapsed
│ │ │ - -- .518789s elapsed
│ │ │ - -- .40949s elapsed
│ │ │ - -- .215279s elapsed
│ │ │ + -- .285133s elapsed
│ │ │ + -- .337462s elapsed
│ │ │ + -- .502137s elapsed
│ │ │ + -- .244018s elapsed
│ │ │ + -- .282999s elapsed
│ │ │ + -- .317288s elapsed
│ │ │ + -- .586806s elapsed
│ │ │ + -- .429265s elapsed
│ │ │ + -- .486214s elapsed
│ │ │ + -- .406767s elapsed
│ │ │ + -- .239585s elapsed
│ │ │  
│ │ │  i17 : netList oo
│ │ │  
│ │ │        +---------------+---------------+
│ │ │  o17 = |{3, 4, 4}      |{2, 3, 5}      |
│ │ │        +---------------+---------------+
│ │ │        |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ @@ -242,75 +242,75 @@
│ │ │  o22 = 15
│ │ │  
│ │ │  i23 : allcomps = for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .512744s elapsed
│ │ │ - -- .53346s elapsed
│ │ │ - -- .947268s elapsed
│ │ │ - -- 1.36888s elapsed
│ │ │ - -- .713392s elapsed
│ │ │ - -- 1.08517s elapsed
│ │ │ - -- .998329s elapsed
│ │ │ - -- 1.14996s elapsed
│ │ │ - -- .704672s elapsed
│ │ │ - -- .852667s elapsed
│ │ │ - -- .387001s elapsed
│ │ │ - -- .504836s elapsed
│ │ │ - -- .528683s elapsed
│ │ │ - -- .767751s elapsed
│ │ │ - -- 1.03422s elapsed
│ │ │ - -- 1.45198s elapsed
│ │ │ - -- 1.0926s elapsed
│ │ │ - -- .934718s elapsed
│ │ │ - -- 1.28281s elapsed
│ │ │ - -- .978316s elapsed
│ │ │ - -- .807522s elapsed
│ │ │ - -- 1.05501s elapsed
│ │ │ - -- 1.46125s elapsed
│ │ │ - -- 1.41515s elapsed
│ │ │ - -- .549585s elapsed
│ │ │ - -- .65606s elapsed
│ │ │ - -- 1.40895s elapsed
│ │ │ - -- .706024s elapsed
│ │ │ - -- .714696s elapsed
│ │ │ - -- .90422s elapsed
│ │ │ - -- 1.21962s elapsed
│ │ │ - -- 1.02135s elapsed
│ │ │ - -- .71753s elapsed
│ │ │ - -- 1.11064s elapsed
│ │ │ - -- .875974s elapsed
│ │ │ - -- 1.12442s elapsed
│ │ │ - -- 1.06557s elapsed
│ │ │ - -- 1.12262s elapsed
│ │ │ - -- 1.24296s elapsed
│ │ │ - -- .687213s elapsed
│ │ │ - -- .608936s elapsed
│ │ │ - -- 1.274s elapsed
│ │ │ - -- 1.77821s elapsed
│ │ │ - -- 2.15089s elapsed
│ │ │ - -- 1.30465s elapsed
│ │ │ - -- 1.20759s elapsed
│ │ │ - -- 1.55759s elapsed
│ │ │ - -- 1.18831s elapsed
│ │ │ - -- 1.06044s elapsed
│ │ │ - -- 1.17392s elapsed
│ │ │ - -- 1.11023s elapsed
│ │ │ - -- .920933s elapsed
│ │ │ - -- .926059s elapsed
│ │ │ - -- .975083s elapsed
│ │ │ - -- .655386s elapsed
│ │ │ - -- 1.0732s elapsed
│ │ │ - -- 1.28686s elapsed
│ │ │ - -- 1.4154s elapsed
│ │ │ - -- .78408s elapsed
│ │ │ - -- .487365s elapsed
│ │ │ - -- .322309s elapsed
│ │ │ + -- .43429s elapsed
│ │ │ + -- .488807s elapsed
│ │ │ + -- .986765s elapsed
│ │ │ + -- 1.13438s elapsed
│ │ │ + -- .66353s elapsed
│ │ │ + -- .849367s elapsed
│ │ │ + -- .96252s elapsed
│ │ │ + -- 1.00203s elapsed
│ │ │ + -- .709805s elapsed
│ │ │ + -- .755778s elapsed
│ │ │ + -- .386826s elapsed
│ │ │ + -- .506371s elapsed
│ │ │ + -- .500769s elapsed
│ │ │ + -- .746669s elapsed
│ │ │ + -- .981253s elapsed
│ │ │ + -- 1.34631s elapsed
│ │ │ + -- .993942s elapsed
│ │ │ + -- .909858s elapsed
│ │ │ + -- 1.34434s elapsed
│ │ │ + -- .979517s elapsed
│ │ │ + -- .806581s elapsed
│ │ │ + -- 1.01257s elapsed
│ │ │ + -- 1.3251s elapsed
│ │ │ + -- 1.27674s elapsed
│ │ │ + -- .479649s elapsed
│ │ │ + -- .636482s elapsed
│ │ │ + -- 1.34076s elapsed
│ │ │ + -- .671167s elapsed
│ │ │ + -- .629873s elapsed
│ │ │ + -- .806261s elapsed
│ │ │ + -- 1.03571s elapsed
│ │ │ + -- .833223s elapsed
│ │ │ + -- .595599s elapsed
│ │ │ + -- 1.09641s elapsed
│ │ │ + -- .798496s elapsed
│ │ │ + -- 1.05704s elapsed
│ │ │ + -- .979693s elapsed
│ │ │ + -- 1.20378s elapsed
│ │ │ + -- 1.27075s elapsed
│ │ │ + -- .739912s elapsed
│ │ │ + -- .667106s elapsed
│ │ │ + -- 1.11128s elapsed
│ │ │ + -- 1.49206s elapsed
│ │ │ + -- 1.8468s elapsed
│ │ │ + -- 1.12295s elapsed
│ │ │ + -- 1.02267s elapsed
│ │ │ + -- 1.34934s elapsed
│ │ │ + -- 1.19619s elapsed
│ │ │ + -- .965103s elapsed
│ │ │ + -- 1.02915s elapsed
│ │ │ + -- 1.11806s elapsed
│ │ │ + -- .74944s elapsed
│ │ │ + -- .759433s elapsed
│ │ │ + -- .980329s elapsed
│ │ │ + -- .608849s elapsed
│ │ │ + -- 1.17856s elapsed
│ │ │ + -- 1.29766s elapsed
│ │ │ + -- 1.39978s elapsed
│ │ │ + -- .710064s elapsed
│ │ │ + -- .447572s elapsed
│ │ │ + -- .343799s elapsed
│ │ │  
│ │ │  i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │  
│ │ │        +------------------------+------------------------+
│ │ │  o24 = |codimensions            |degrees                 |
│ │ │        +------------------------+------------------------+
│ │ │        |{3, 5, 5}               |{2, 4, 6}               |
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.out
│ │ │ @@ -39,15 +39,15 @@
│ │ │       .98, .98, .101, -.98, -.298, .393, .201, .201, .201, -.995, -.201,
│ │ │       ------------------------------------------------------------------------
│ │ │       .954}}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ - -- .736s elapsed
│ │ │ + -- .957s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ @@ -60,15 +60,15 @@
│ │ │  +---+---+---+---+
│ │ │  |72 |144|216|288|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │ - -- .899s elapsed
│ │ │ + -- 1.04s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {1, 1, 1, 1, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       0, 0, 0}, {.309, -.809, -.809, .309, -.951, -.588, .588, .951}}
│ │ │  
│ │ │  o6 : List
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out
│ │ │ @@ -44,19 +44,19 @@
│ │ │  
│ │ │  i5 : printingPrecision = 3
│ │ │  
│ │ │  o5 = 3
│ │ │  
│ │ │  i6 : for G in Gs list showExoticSolutions G;
│ │ │  warning: some solutions are not regular: {36, 41, 42, 43, 47, 48, 50, 51, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 74, 75, 76, 77, 79, 80, 82, 85, 86, 87, 88, 89, 90}
│ │ │ - -- .544s elapsed
│ │ │ + -- .716s elapsed
│ │ │  warning: some solutions are not regular: {49, 50, 53, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 75, 77, 78, 80, 82, 83, 84, 85, 86, 88, 91, 94, 95, 97}
│ │ │ - -- .572s elapsed
│ │ │ - -- .703s elapsed
│ │ │ - -- .853s elapsed
│ │ │ + -- .643s elapsed
│ │ │ + -- .82s elapsed
│ │ │ + -- 1.05s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │                                                  1 => {3, 4}
│ │ │                                                  2 => {0, 4}
│ │ │                                                  3 => {0, 1}
│ │ │                                                  4 => {2, 1}
│ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │  |1    |1   |1   |1    |0    |0    |0    |0    |
│ │ │ @@ -69,20 +69,20 @@
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │ - -- .924s elapsed
│ │ │   -- 1.2s elapsed
│ │ │ + -- 1.38s elapsed
│ │ │  warning: some solutions are not regular: {28, 30, 35, 37, 38, 40, 43, 44, 46, 47, 48, 53, 59, 60, 61}
│ │ │ - -- 1.27s elapsed
│ │ │ + -- 1.69s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34}
│ │ │ - -- 1.08s elapsed
│ │ │ - -- 1.05s elapsed
│ │ │ -warning: some solutions are not regular: {26, 27, 30, 31, 33}
│ │ │   -- 1.33s elapsed
│ │ │ + -- 1.33s elapsed
│ │ │ +warning: some solutions are not regular: {26, 27, 30, 31, 33}
│ │ │ + -- 1.47s elapsed
│ │ │  warning: some solutions are not regular: {38, 44, 46, 49, 52, 53, 63, 70, 74, 75, 76, 77}
│ │ │ - -- 1.06s elapsed
│ │ │ + -- 1.35s elapsed
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Linearly__Stable__Solutions.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 1729328129346969841
│ │ │  
│ │ │  i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
│ │ │  
│ │ │  i2 : getLinearlyStableSolutions(G)
│ │ │  warning: some solutions are not regular: {4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21}
│ │ │ - -- .13668s elapsed
│ │ │ + -- .233607s elapsed
│ │ │  
│ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_show__Exotic__Solutions.out
│ │ │ @@ -7,15 +7,15 @@
│ │ │             2 => {1, 3}
│ │ │             3 => {2, 4}
│ │ │             4 => {0, 3}
│ │ │  
│ │ │  o1 : Graph
│ │ │  
│ │ │  i2 : showExoticSolutions G
│ │ │ - -- .775257s elapsed
│ │ │ + -- .940037s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ @@ -48,14 +48,14 @@
│ │ │             2 => {1, 3, 4}
│ │ │             3 => {2, 4}
│ │ │             4 => {0, 2, 3}
│ │ │  
│ │ │  o3 : Graph
│ │ │  
│ │ │  i4 : showExoticSolutions G
│ │ │ - -- 1.07829s elapsed
│ │ │ + -- 1.30425s elapsed
│ │ │  
│ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html
│ │ │ @@ -300,25 +300,25 @@
│ │ │            
│ │ │              
│ │ │                
    i16 : for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .308205s elapsed
│ │ │ - -- .316063s elapsed
│ │ │ - -- .473982s elapsed
│ │ │ - -- .240853s elapsed
│ │ │ - -- .259009s elapsed
│ │ │ - -- .297748s elapsed
│ │ │ - -- .73903s elapsed
│ │ │ - -- .456206s elapsed
│ │ │ - -- .518789s elapsed
│ │ │ - -- .40949s elapsed
│ │ │ - -- .215279s elapsed
    
│ │ │ + -- .285133s elapsed
│ │ │ + -- .337462s elapsed
│ │ │ + -- .502137s elapsed
│ │ │ + -- .244018s elapsed
│ │ │ + -- .282999s elapsed
│ │ │ + -- .317288s elapsed
│ │ │ + -- .586806s elapsed
│ │ │ + -- .429265s elapsed
│ │ │ + -- .486214s elapsed
│ │ │ + -- .406767s elapsed
│ │ │ + -- .239585s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : netList oo
│ │ │  
│ │ │        +---------------+---------------+
│ │ │ @@ -385,75 +385,75 @@
│ │ │            
│ │ │              
│ │ │                
    i23 : allcomps = for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .512744s elapsed
│ │ │ - -- .53346s elapsed
│ │ │ - -- .947268s elapsed
│ │ │ - -- 1.36888s elapsed
│ │ │ - -- .713392s elapsed
│ │ │ - -- 1.08517s elapsed
│ │ │ - -- .998329s elapsed
│ │ │ - -- 1.14996s elapsed
│ │ │ - -- .704672s elapsed
│ │ │ - -- .852667s elapsed
│ │ │ - -- .387001s elapsed
│ │ │ - -- .504836s elapsed
│ │ │ - -- .528683s elapsed
│ │ │ - -- .767751s elapsed
│ │ │ - -- 1.03422s elapsed
│ │ │ - -- 1.45198s elapsed
│ │ │ - -- 1.0926s elapsed
│ │ │ - -- .934718s elapsed
│ │ │ - -- 1.28281s elapsed
│ │ │ - -- .978316s elapsed
│ │ │ - -- .807522s elapsed
│ │ │ - -- 1.05501s elapsed
│ │ │ - -- 1.46125s elapsed
│ │ │ - -- 1.41515s elapsed
│ │ │ - -- .549585s elapsed
│ │ │ - -- .65606s elapsed
│ │ │ - -- 1.40895s elapsed
│ │ │ - -- .706024s elapsed
│ │ │ - -- .714696s elapsed
│ │ │ - -- .90422s elapsed
│ │ │ - -- 1.21962s elapsed
│ │ │ - -- 1.02135s elapsed
│ │ │ - -- .71753s elapsed
│ │ │ - -- 1.11064s elapsed
│ │ │ - -- .875974s elapsed
│ │ │ - -- 1.12442s elapsed
│ │ │ - -- 1.06557s elapsed
│ │ │ - -- 1.12262s elapsed
│ │ │ - -- 1.24296s elapsed
│ │ │ - -- .687213s elapsed
│ │ │ - -- .608936s elapsed
│ │ │ - -- 1.274s elapsed
│ │ │ - -- 1.77821s elapsed
│ │ │ - -- 2.15089s elapsed
│ │ │ - -- 1.30465s elapsed
│ │ │ - -- 1.20759s elapsed
│ │ │ - -- 1.55759s elapsed
│ │ │ - -- 1.18831s elapsed
│ │ │ - -- 1.06044s elapsed
│ │ │ - -- 1.17392s elapsed
│ │ │ - -- 1.11023s elapsed
│ │ │ - -- .920933s elapsed
│ │ │ - -- .926059s elapsed
│ │ │ - -- .975083s elapsed
│ │ │ - -- .655386s elapsed
│ │ │ - -- 1.0732s elapsed
│ │ │ - -- 1.28686s elapsed
│ │ │ - -- 1.4154s elapsed
│ │ │ - -- .78408s elapsed
│ │ │ - -- .487365s elapsed
│ │ │ - -- .322309s elapsed
    
│ │ │ + -- .43429s elapsed
│ │ │ + -- .488807s elapsed
│ │ │ + -- .986765s elapsed
│ │ │ + -- 1.13438s elapsed
│ │ │ + -- .66353s elapsed
│ │ │ + -- .849367s elapsed
│ │ │ + -- .96252s elapsed
│ │ │ + -- 1.00203s elapsed
│ │ │ + -- .709805s elapsed
│ │ │ + -- .755778s elapsed
│ │ │ + -- .386826s elapsed
│ │ │ + -- .506371s elapsed
│ │ │ + -- .500769s elapsed
│ │ │ + -- .746669s elapsed
│ │ │ + -- .981253s elapsed
│ │ │ + -- 1.34631s elapsed
│ │ │ + -- .993942s elapsed
│ │ │ + -- .909858s elapsed
│ │ │ + -- 1.34434s elapsed
│ │ │ + -- .979517s elapsed
│ │ │ + -- .806581s elapsed
│ │ │ + -- 1.01257s elapsed
│ │ │ + -- 1.3251s elapsed
│ │ │ + -- 1.27674s elapsed
│ │ │ + -- .479649s elapsed
│ │ │ + -- .636482s elapsed
│ │ │ + -- 1.34076s elapsed
│ │ │ + -- .671167s elapsed
│ │ │ + -- .629873s elapsed
│ │ │ + -- .806261s elapsed
│ │ │ + -- 1.03571s elapsed
│ │ │ + -- .833223s elapsed
│ │ │ + -- .595599s elapsed
│ │ │ + -- 1.09641s elapsed
│ │ │ + -- .798496s elapsed
│ │ │ + -- 1.05704s elapsed
│ │ │ + -- .979693s elapsed
│ │ │ + -- 1.20378s elapsed
│ │ │ + -- 1.27075s elapsed
│ │ │ + -- .739912s elapsed
│ │ │ + -- .667106s elapsed
│ │ │ + -- 1.11128s elapsed
│ │ │ + -- 1.49206s elapsed
│ │ │ + -- 1.8468s elapsed
│ │ │ + -- 1.12295s elapsed
│ │ │ + -- 1.02267s elapsed
│ │ │ + -- 1.34934s elapsed
│ │ │ + -- 1.19619s elapsed
│ │ │ + -- .965103s elapsed
│ │ │ + -- 1.02915s elapsed
│ │ │ + -- 1.11806s elapsed
│ │ │ + -- .74944s elapsed
│ │ │ + -- .759433s elapsed
│ │ │ + -- .980329s elapsed
│ │ │ + -- .608849s elapsed
│ │ │ + -- 1.17856s elapsed
│ │ │ + -- 1.29766s elapsed
│ │ │ + -- 1.39978s elapsed
│ │ │ + -- .710064s elapsed
│ │ │ + -- .447572s elapsed
│ │ │ + -- .343799s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │  
│ │ │        +------------------------+------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -180,25 +180,25 @@
│ │ │ │  
│ │ │ │  o15 = 4
│ │ │ │  i16 : for G in Gs list (
│ │ │ │            IG = oscQuadrics(G, R);
│ │ │ │            elapsedTime comps := decompose IG;
│ │ │ │            {comps/codim, comps/degree}
│ │ │ │            );
│ │ │ │ - -- .308205s elapsed
│ │ │ │ - -- .316063s elapsed
│ │ │ │ - -- .473982s elapsed
│ │ │ │ - -- .240853s elapsed
│ │ │ │ - -- .259009s elapsed
│ │ │ │ - -- .297748s elapsed
│ │ │ │ - -- .73903s elapsed
│ │ │ │ - -- .456206s elapsed
│ │ │ │ - -- .518789s elapsed
│ │ │ │ - -- .40949s elapsed
│ │ │ │ - -- .215279s elapsed
│ │ │ │ + -- .285133s elapsed
│ │ │ │ + -- .337462s elapsed
│ │ │ │ + -- .502137s elapsed
│ │ │ │ + -- .244018s elapsed
│ │ │ │ + -- .282999s elapsed
│ │ │ │ + -- .317288s elapsed
│ │ │ │ + -- .586806s elapsed
│ │ │ │ + -- .429265s elapsed
│ │ │ │ + -- .486214s elapsed
│ │ │ │ + -- .406767s elapsed
│ │ │ │ + -- .239585s elapsed
│ │ │ │  i17 : netList oo
│ │ │ │  
│ │ │ │        +---------------+---------------+
│ │ │ │  o17 = |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ │        +---------------+---------------+
│ │ │ │        |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ │        +---------------+---------------+
│ │ │ │ @@ -233,75 +233,75 @@
│ │ │ │  
│ │ │ │  o22 = 15
│ │ │ │  i23 : allcomps = for G in Gs list (
│ │ │ │            IG = oscQuadrics(G, R);
│ │ │ │            elapsedTime comps := decompose IG;
│ │ │ │            {comps/codim, comps/degree}
│ │ │ │            );
│ │ │ │ - -- .512744s elapsed
│ │ │ │ - -- .53346s elapsed
│ │ │ │ - -- .947268s elapsed
│ │ │ │ - -- 1.36888s elapsed
│ │ │ │ - -- .713392s elapsed
│ │ │ │ - -- 1.08517s elapsed
│ │ │ │ - -- .998329s elapsed
│ │ │ │ - -- 1.14996s elapsed
│ │ │ │ - -- .704672s elapsed
│ │ │ │ - -- .852667s elapsed
│ │ │ │ - -- .387001s elapsed
│ │ │ │ - -- .504836s elapsed
│ │ │ │ - -- .528683s elapsed
│ │ │ │ - -- .767751s elapsed
│ │ │ │ - -- 1.03422s elapsed
│ │ │ │ - -- 1.45198s elapsed
│ │ │ │ - -- 1.0926s elapsed
│ │ │ │ - -- .934718s elapsed
│ │ │ │ - -- 1.28281s elapsed
│ │ │ │ - -- .978316s elapsed
│ │ │ │ - -- .807522s elapsed
│ │ │ │ - -- 1.05501s elapsed
│ │ │ │ - -- 1.46125s elapsed
│ │ │ │ - -- 1.41515s elapsed
│ │ │ │ - -- .549585s elapsed
│ │ │ │ - -- .65606s elapsed
│ │ │ │ - -- 1.40895s elapsed
│ │ │ │ - -- .706024s elapsed
│ │ │ │ - -- .714696s elapsed
│ │ │ │ - -- .90422s elapsed
│ │ │ │ - -- 1.21962s elapsed
│ │ │ │ - -- 1.02135s elapsed
│ │ │ │ - -- .71753s elapsed
│ │ │ │ - -- 1.11064s elapsed
│ │ │ │ - -- .875974s elapsed
│ │ │ │ - -- 1.12442s elapsed
│ │ │ │ - -- 1.06557s elapsed
│ │ │ │ - -- 1.12262s elapsed
│ │ │ │ - -- 1.24296s elapsed
│ │ │ │ - -- .687213s elapsed
│ │ │ │ - -- .608936s elapsed
│ │ │ │ - -- 1.274s elapsed
│ │ │ │ - -- 1.77821s elapsed
│ │ │ │ - -- 2.15089s elapsed
│ │ │ │ - -- 1.30465s elapsed
│ │ │ │ - -- 1.20759s elapsed
│ │ │ │ - -- 1.55759s elapsed
│ │ │ │ - -- 1.18831s elapsed
│ │ │ │ - -- 1.06044s elapsed
│ │ │ │ - -- 1.17392s elapsed
│ │ │ │ - -- 1.11023s elapsed
│ │ │ │ - -- .920933s elapsed
│ │ │ │ - -- .926059s elapsed
│ │ │ │ - -- .975083s elapsed
│ │ │ │ - -- .655386s elapsed
│ │ │ │ - -- 1.0732s elapsed
│ │ │ │ - -- 1.28686s elapsed
│ │ │ │ - -- 1.4154s elapsed
│ │ │ │ - -- .78408s elapsed
│ │ │ │ - -- .487365s elapsed
│ │ │ │ - -- .322309s elapsed
│ │ │ │ + -- .43429s elapsed
│ │ │ │ + -- .488807s elapsed
│ │ │ │ + -- .986765s elapsed
│ │ │ │ + -- 1.13438s elapsed
│ │ │ │ + -- .66353s elapsed
│ │ │ │ + -- .849367s elapsed
│ │ │ │ + -- .96252s elapsed
│ │ │ │ + -- 1.00203s elapsed
│ │ │ │ + -- .709805s elapsed
│ │ │ │ + -- .755778s elapsed
│ │ │ │ + -- .386826s elapsed
│ │ │ │ + -- .506371s elapsed
│ │ │ │ + -- .500769s elapsed
│ │ │ │ + -- .746669s elapsed
│ │ │ │ + -- .981253s elapsed
│ │ │ │ + -- 1.34631s elapsed
│ │ │ │ + -- .993942s elapsed
│ │ │ │ + -- .909858s elapsed
│ │ │ │ + -- 1.34434s elapsed
│ │ │ │ + -- .979517s elapsed
│ │ │ │ + -- .806581s elapsed
│ │ │ │ + -- 1.01257s elapsed
│ │ │ │ + -- 1.3251s elapsed
│ │ │ │ + -- 1.27674s elapsed
│ │ │ │ + -- .479649s elapsed
│ │ │ │ + -- .636482s elapsed
│ │ │ │ + -- 1.34076s elapsed
│ │ │ │ + -- .671167s elapsed
│ │ │ │ + -- .629873s elapsed
│ │ │ │ + -- .806261s elapsed
│ │ │ │ + -- 1.03571s elapsed
│ │ │ │ + -- .833223s elapsed
│ │ │ │ + -- .595599s elapsed
│ │ │ │ + -- 1.09641s elapsed
│ │ │ │ + -- .798496s elapsed
│ │ │ │ + -- 1.05704s elapsed
│ │ │ │ + -- .979693s elapsed
│ │ │ │ + -- 1.20378s elapsed
│ │ │ │ + -- 1.27075s elapsed
│ │ │ │ + -- .739912s elapsed
│ │ │ │ + -- .667106s elapsed
│ │ │ │ + -- 1.11128s elapsed
│ │ │ │ + -- 1.49206s elapsed
│ │ │ │ + -- 1.8468s elapsed
│ │ │ │ + -- 1.12295s elapsed
│ │ │ │ + -- 1.02267s elapsed
│ │ │ │ + -- 1.34934s elapsed
│ │ │ │ + -- 1.19619s elapsed
│ │ │ │ + -- .965103s elapsed
│ │ │ │ + -- 1.02915s elapsed
│ │ │ │ + -- 1.11806s elapsed
│ │ │ │ + -- .74944s elapsed
│ │ │ │ + -- .759433s elapsed
│ │ │ │ + -- .980329s elapsed
│ │ │ │ + -- .608849s elapsed
│ │ │ │ + -- 1.17856s elapsed
│ │ │ │ + -- 1.29766s elapsed
│ │ │ │ + -- 1.39978s elapsed
│ │ │ │ + -- .710064s elapsed
│ │ │ │ + -- .447572s elapsed
│ │ │ │ + -- .343799s elapsed
│ │ │ │  i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │ │  
│ │ │ │        +------------------------+------------------------+
│ │ │ │  o24 = |codimensions            |degrees                 |
│ │ │ │        +------------------------+------------------------+
│ │ │ │        |{3, 5, 5}               |{2, 4, 6}               |
│ │ │ │        +------------------------+------------------------+
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.html
│ │ │ @@ -115,15 +115,15 @@
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ - -- .736s elapsed
│ │ │ + -- .957s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ @@ -136,15 +136,15 @@
│ │ │  +---+---+---+---+
│ │ │  |72 |144|216|288|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │ - -- .899s elapsed
│ │ │ + -- 1.04s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {1, 1, 1, 1, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       0, 0, 0}, {.309, -.809, -.809, .309, -.951, -.588, .588, .951}}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       .98, .98, .101, -.98, -.298, .393, .201, .201, .201, -.995, -.201,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       .954}}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ │ - -- .736s elapsed
│ │ │ │ + -- .957s elapsed
│ │ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │ │                                                  1 => {0, 2}
│ │ │ │                                                  2 => {1, 3}
│ │ │ │                                                  3 => {2, 4}
│ │ │ │                                                  4 => {0, 3}
│ │ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ │ @@ -64,15 +64,15 @@
│ │ │ │  +---+---+---+---+
│ │ │ │  |72 |144|216|288|
│ │ │ │  +---+---+---+---+
│ │ │ │  |0  |0  |0  |0  |
│ │ │ │  +---+---+---+---+
│ │ │ │  |288|216|144|72 |
│ │ │ │  +---+---+---+---+
│ │ │ │ - -- .899s elapsed
│ │ │ │ + -- 1.04s elapsed
│ │ │ │  
│ │ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {1, 1, 1, 1, 0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       0, 0, 0}, {.309, -.809, -.809, .309, -.951, -.588, .588, .951}}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  Computing the (linearly) stable solutions for K5C5 takes a minute or two:
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html
│ │ │ @@ -120,19 +120,19 @@
│ │ │  o5 = 3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : for G in Gs list showExoticSolutions G;
│ │ │  warning: some solutions are not regular: {36, 41, 42, 43, 47, 48, 50, 51, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 74, 75, 76, 77, 79, 80, 82, 85, 86, 87, 88, 89, 90}
│ │ │ - -- .544s elapsed
│ │ │ + -- .716s elapsed
│ │ │  warning: some solutions are not regular: {49, 50, 53, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 75, 77, 78, 80, 82, 83, 84, 85, 86, 88, 91, 94, 95, 97}
│ │ │ - -- .572s elapsed
│ │ │ - -- .703s elapsed
│ │ │ - -- .853s elapsed
│ │ │ + -- .643s elapsed
│ │ │ + -- .82s elapsed
│ │ │ + -- 1.05s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │                                                  1 => {3, 4}
│ │ │                                                  2 => {0, 4}
│ │ │                                                  3 => {0, 1}
│ │ │                                                  4 => {2, 1}
│ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │  |1    |1   |1   |1    |0    |0    |0    |0    |
│ │ │ @@ -145,25 +145,25 @@
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │ - -- .924s elapsed
│ │ │   -- 1.2s elapsed
│ │ │ + -- 1.38s elapsed
│ │ │  warning: some solutions are not regular: {28, 30, 35, 37, 38, 40, 43, 44, 46, 47, 48, 53, 59, 60, 61}
│ │ │ - -- 1.27s elapsed
│ │ │ + -- 1.69s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34}
│ │ │ - -- 1.08s elapsed
│ │ │ - -- 1.05s elapsed
│ │ │ -warning: some solutions are not regular: {26, 27, 30, 31, 33}
│ │ │   -- 1.33s elapsed
│ │ │ + -- 1.33s elapsed
│ │ │ +warning: some solutions are not regular: {26, 27, 30, 31, 33}
│ │ │ + -- 1.47s elapsed
│ │ │  warning: some solutions are not regular: {38, 44, 46, 49, 52, 53, 63, 70, 74, 75, 76, 77}
│ │ │ - -- 1.06s elapsed
    │ │ │ + -- 1.35s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -48,21 +48,21 @@ │ │ │ │ i5 : printingPrecision = 3 │ │ │ │ │ │ │ │ o5 = 3 │ │ │ │ i6 : for G in Gs list showExoticSolutions G; │ │ │ │ warning: some solutions are not regular: {36, 41, 42, 43, 47, 48, 50, 51, 53, │ │ │ │ 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 74, 75, 76, 77, 79, │ │ │ │ 80, 82, 85, 86, 87, 88, 89, 90} │ │ │ │ - -- .544s elapsed │ │ │ │ + -- .716s elapsed │ │ │ │ warning: some solutions are not regular: {49, 50, 53, 56, 57, 58, 59, 60, 61, │ │ │ │ 62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 75, 77, 78, 80, 82, 83, 84, 85, 86, 88, │ │ │ │ 91, 94, 95, 97} │ │ │ │ - -- .572s elapsed │ │ │ │ - -- .703s elapsed │ │ │ │ - -- .853s elapsed │ │ │ │ + -- .643s elapsed │ │ │ │ + -- .82s elapsed │ │ │ │ + -- 1.05s elapsed │ │ │ │ -- found extra exotic solutions for graph Graph{0 => {2, 3}} -- │ │ │ │ 1 => {3, 4} │ │ │ │ 2 => {0, 4} │ │ │ │ 3 => {0, 1} │ │ │ │ 4 => {2, 1} │ │ │ │ +-----+----+----+-----+-----+-----+-----+-----+ │ │ │ │ |1 |1 |1 |1 |0 |0 |0 |0 | │ │ │ │ @@ -75,24 +75,24 @@ │ │ │ │ +---+---+---+---+ │ │ │ │ |0 |0 |0 |0 | │ │ │ │ +---+---+---+---+ │ │ │ │ |216|72 |288|144| │ │ │ │ +---+---+---+---+ │ │ │ │ |144|288|72 |216| │ │ │ │ +---+---+---+---+ │ │ │ │ - -- .924s elapsed │ │ │ │ -- 1.2s elapsed │ │ │ │ + -- 1.38s elapsed │ │ │ │ warning: some solutions are not regular: {28, 30, 35, 37, 38, 40, 43, 44, 46, │ │ │ │ 47, 48, 53, 59, 60, 61} │ │ │ │ - -- 1.27s elapsed │ │ │ │ + -- 1.69s elapsed │ │ │ │ warning: some solutions are not regular: {16, 17, 20, 21, 22, 23, 24, 26, 27, │ │ │ │ 28, 29, 30, 31, 32, 33, 34} │ │ │ │ - -- 1.08s elapsed │ │ │ │ - -- 1.05s elapsed │ │ │ │ -warning: some solutions are not regular: {26, 27, 30, 31, 33} │ │ │ │ -- 1.33s elapsed │ │ │ │ + -- 1.33s elapsed │ │ │ │ +warning: some solutions are not regular: {26, 27, 30, 31, 33} │ │ │ │ + -- 1.47s elapsed │ │ │ │ warning: some solutions are not regular: {38, 44, 46, 49, 52, 53, 63, 70, 74, │ │ │ │ 75, 76, 77} │ │ │ │ - -- 1.06s elapsed │ │ │ │ + -- 1.35s elapsed │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.05+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:812:0. │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html │ │ │ @@ -81,15 +81,15 @@ │ │ │ 
    i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : getLinearlyStableSolutions(G)
│ │ │  warning: some solutions are not regular: {4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21}
│ │ │ - -- .13668s elapsed
│ │ │ + -- .233607s elapsed
│ │ │  
│ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ of each oscillator is given by the Kuramoto model. The linear stability of a │ │ │ │ solution is determined by the eigenvalues of the Jacobian matrix of the system │ │ │ │ evaluated at the solution. │ │ │ │ i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}}); │ │ │ │ i2 : getLinearlyStableSolutions(G) │ │ │ │ warning: some solutions are not regular: {4, 5, 7, 8, 9, 10, 12, 13, 14, 15, │ │ │ │ 16, 17, 18, 19, 21} │ │ │ │ - -- .13668s elapsed │ │ │ │ + -- .233607s elapsed │ │ │ │ │ │ │ │ o2 = {{1, 1, 1, 0, 0, 0}} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _f_i_n_d_R_e_a_l_S_o_l_u_t_i_o_n_s -- find real solutions, at least one per component for │ │ │ │ well-conditioned systems │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_show__Exotic__Solutions.html │ │ │ @@ -98,15 +98,15 @@ │ │ │ │ │ │ o1 : Graph │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : showExoticSolutions G
│ │ │ - -- .775257s elapsed
│ │ │ + -- .940037s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ @@ -150,15 +150,15 @@
│ │ │  
│ │ │  o3 : Graph
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : showExoticSolutions G
│ │ │ - -- 1.07829s elapsed
│ │ │ + -- 1.30425s elapsed
│ │ │  
│ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o4 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -34,15 +34,15 @@ │ │ │ │ 1 => {0, 2} │ │ │ │ 2 => {1, 3} │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 3} │ │ │ │ │ │ │ │ o1 : Graph │ │ │ │ i2 : showExoticSolutions G │ │ │ │ - -- .775257s elapsed │ │ │ │ + -- .940037s elapsed │ │ │ │ -- found extra exotic solutions for graph Graph{0 => {1, 4}} -- │ │ │ │ 1 => {0, 2} │ │ │ │ 2 => {1, 3} │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 3} │ │ │ │ +-------+--------+--------+-------+--------+--------+--------+--------+ │ │ │ │ |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057| │ │ │ │ @@ -76,15 +76,15 @@ │ │ │ │ 1 => {0, 2} │ │ │ │ 2 => {1, 3, 4} │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 2, 3} │ │ │ │ │ │ │ │ o3 : Graph │ │ │ │ i4 : showExoticSolutions G │ │ │ │ - -- 1.07829s elapsed │ │ │ │ + -- 1.30425s elapsed │ │ │ │ │ │ │ │ o4 = {{1, 1, 1, 1, 0, 0, 0, 0}} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_t_L_i_n_e_a_r_l_y_S_t_a_b_l_e_S_o_l_u_t_i_o_n_s -- Compute linearly stable solutions for the │ │ │ │ Kuramoto oscillator system associated to a graph │ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out │ │ │ @@ -80,20 +80,20 @@ │ │ │ i19 : needsPackage "MultigradedImplicitization"; │ │ │ │ │ │ i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix {toList(9:1)}), S); │ │ │ warning: computation begun over finite field. resulting polynomials may not lie in the ideal │ │ │ computing total degree: 1 │ │ │ number of monomials = 9 │ │ │ number of distinct multidegrees = 1 │ │ │ - -- .00849315s elapsed │ │ │ + -- .00939644s elapsed │ │ │ WARNING: There are linear relations. You may want to reduce the number of variables to speed up the computation. │ │ │ computing total degree: 2 │ │ │ number of monomials = 45 │ │ │ number of distinct multidegrees = 1 │ │ │ - -- .579308s elapsed │ │ │ + -- .573461s elapsed │ │ │ │ │ │ o20 : Ideal of S │ │ │ │ │ │ i21 : dim I │ │ │ │ │ │ o21 = 5 │ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html │ │ │ @@ -213,20 +213,20 @@ │ │ │ │ │ │ │ │ │ 
    i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix {toList(9:1)}), S);
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 9
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .00849315s elapsed
│ │ │ + -- .00939644s elapsed
│ │ │  WARNING: There are linear relations. You may want to reduce the number of variables to speed up the computation.
│ │ │  computing total degree: 2
│ │ │  number of monomials = 45
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .579308s elapsed
│ │ │ + -- .573461s elapsed
│ │ │  
│ │ │  o20 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i21 : dim I
│ │ │ ├── html2text {}
│ │ │ │ @@ -77,21 +77,21 @@
│ │ │ │  i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix
│ │ │ │  {toList(9:1)}), S);
│ │ │ │  warning: computation begun over finite field. resulting polynomials may not lie
│ │ │ │  in the ideal
│ │ │ │  computing total degree: 1
│ │ │ │  number of monomials = 9
│ │ │ │  number of distinct multidegrees = 1
│ │ │ │ - -- .00849315s elapsed
│ │ │ │ + -- .00939644s elapsed
│ │ │ │  WARNING: There are linear relations. You may want to reduce the number of
│ │ │ │  variables to speed up the computation.
│ │ │ │  computing total degree: 2
│ │ │ │  number of monomials = 45
│ │ │ │  number of distinct multidegrees = 1
│ │ │ │ - -- .579308s elapsed
│ │ │ │ + -- .573461s elapsed
│ │ │ │  
│ │ │ │  o20 : Ideal of S
│ │ │ │  i21 : dim I
│ │ │ │  
│ │ │ │  o21 = 5
│ │ │ │  i22 : isPrime I
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/___Lab__Book__Protocol.out
│ │ │ @@ -41,15 +41,15 @@
│ │ │  i3 : g=3
│ │ │  
│ │ │  o3 = 3
│ │ │  
│ │ │  i4 : kk= ZZ/101;
│ │ │  
│ │ │  i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- .116945s elapsed
│ │ │ + -- .158421s elapsed
│ │ │  
│ │ │  i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o6 = CliffordModule{...6...}
│ │ │  
│ │ │  o6 : CliffordModule
│ │ │  
│ │ │ @@ -67,30 +67,30 @@
│ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │            V = vectorBundleOnE m12;
│ │ │            Ul=tensorProduct(Mor,V);
│ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │            d0=unique degrees target Ul.yAction;
│ │ │            d1=unique degrees target Ul1.yAction;
│ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ - -- .426656s elapsed
│ │ │ + -- .381112s elapsed
│ │ │  
│ │ │  i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │  
│ │ │                 0  1          0  1
│ │ │  o12 = (total: 32 32, total: 32 32)
│ │ │            -4: 16  .     -2: 32  .
│ │ │            -3: 16  .     -1:  .  .
│ │ │            -2:  .  .      0:  .  .
│ │ │            -1:  . 16      1:  . 32
│ │ │             0:  . 16
│ │ │  
│ │ │  o12 : Sequence
│ │ │  
│ │ │  i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the actions of generators
│ │ │ - -- 20.3359s elapsed
│ │ │ + -- 14.2447s elapsed
│ │ │  
│ │ │  i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │  
│ │ │                32      32
│ │ │  o14 : Matrix S   <-- S
│ │ │  
│ │ │  i15 : r=2
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_search__Ulrich.out
│ │ │ @@ -46,30 +46,30 @@
│ │ │  i11 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o11 = CliffordModule{...6...}
│ │ │  
│ │ │  o11 : CliffordModule
│ │ │  
│ │ │  i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- .738181s elapsed
│ │ │ + -- .653624s elapsed
│ │ │  
│ │ │  i13 : betti freeResolution Ulr
│ │ │  
│ │ │               0  1 2
│ │ │  o13 = total: 8 16 8
│ │ │            0: 8 16 8
│ │ │  
│ │ │  o13 : BettiTally
│ │ │  
│ │ │  i14 : ann Ulr == ideal qs
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.6622s elapsed
│ │ │ + -- 2.26655s elapsed
│ │ │  
│ │ │  i16 : betti freeResolution Ulr3
│ │ │  
│ │ │                0  1  2
│ │ │  o16 = total: 12 24 12
│ │ │            0: 12 24 12
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Lab__Book__Protocol.html
│ │ │ @@ -133,15 +133,15 @@
│ │ │              
│ │ │                
    i4 : kk= ZZ/101;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- .116945s elapsed
    
│ │ │ + -- .158421s elapsed
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o6 = CliffordModule{...6...}
│ │ │ @@ -177,15 +177,15 @@
│ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │            V = vectorBundleOnE m12;
│ │ │            Ul=tensorProduct(Mor,V);
│ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │            d0=unique degrees target Ul.yAction;
│ │ │            d1=unique degrees target Ul1.yAction;
│ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ - -- .426656s elapsed
    │ │ │ + -- .381112s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │  
│ │ │                 0  1          0  1
│ │ │ @@ -198,15 +198,15 @@
│ │ │  
│ │ │  o12 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the actions of generators
│ │ │ - -- 20.3359s elapsed
    │ │ │ + -- 14.2447s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │  
│ │ │                32      32
│ │ │ ├── html2text {}
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │              -- will give an Ulrich bundle, with betti table
│ │ │ │              -- 16 32 16
│ │ │ │  i3 : g=3
│ │ │ │  
│ │ │ │  o3 = 3
│ │ │ │  i4 : kk= ZZ/101;
│ │ │ │  i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ │ - -- .116945s elapsed
│ │ │ │ + -- .158421s elapsed
│ │ │ │  i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │ │  
│ │ │ │  o6 = CliffordModule{...6...}
│ │ │ │  
│ │ │ │  o6 : CliffordModule
│ │ │ │  i7 : Mor = vectorBundleOnE M.evenCenter;
│ │ │ │  i8 : Mor1= vectorBundleOnE M.oddCenter;
│ │ │ │ @@ -75,29 +75,29 @@
│ │ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │ │            V = vectorBundleOnE m12;
│ │ │ │            Ul=tensorProduct(Mor,V);
│ │ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │ │            d0=unique degrees target Ul.yAction;
│ │ │ │            d1=unique degrees target Ul1.yAction;
│ │ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ │ - -- .426656s elapsed
│ │ │ │ + -- .381112s elapsed
│ │ │ │  i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │ │  
│ │ │ │                 0  1          0  1
│ │ │ │  o12 = (total: 32 32, total: 32 32)
│ │ │ │            -4: 16  .     -2: 32  .
│ │ │ │            -3: 16  .     -1:  .  .
│ │ │ │            -2:  .  .      0:  .  .
│ │ │ │            -1:  . 16      1:  . 32
│ │ │ │             0:  . 16
│ │ │ │  
│ │ │ │  o12 : Sequence
│ │ │ │  i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the
│ │ │ │  actions of generators
│ │ │ │ - -- 20.3359s elapsed
│ │ │ │ + -- 14.2447s elapsed
│ │ │ │  i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │ │  
│ │ │ │                32      32
│ │ │ │  o14 : Matrix S   <-- S
│ │ │ │  i15 : r=2
│ │ │ │  
│ │ │ │  o15 = 2
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_search__Ulrich.html
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o11 : CliffordModule
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- .738181s elapsed
    │ │ │ + -- .653624s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : betti freeResolution Ulr
│ │ │  
│ │ │               0  1 2
│ │ │ @@ -190,15 +190,15 @@
│ │ │  
│ │ │  o14 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.6622s elapsed
    │ │ │ + -- 2.26655s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : betti freeResolution Ulr3
│ │ │  
│ │ │                0  1  2
│ │ │ ├── html2text {}
│ │ │ │ @@ -64,27 +64,27 @@
│ │ │ │  o10 : Matrix S  <-- S
│ │ │ │  i11 : M=cliffordModule(Mu1,Mu2,R)
│ │ │ │  
│ │ │ │  o11 = CliffordModule{...6...}
│ │ │ │  
│ │ │ │  o11 : CliffordModule
│ │ │ │  i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ │ - -- .738181s elapsed
│ │ │ │ + -- .653624s elapsed
│ │ │ │  i13 : betti freeResolution Ulr
│ │ │ │  
│ │ │ │               0  1 2
│ │ │ │  o13 = total: 8 16 8
│ │ │ │            0: 8 16 8
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : ann Ulr == ideal qs
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ │ - -- 2.6622s elapsed
│ │ │ │ + -- 2.26655s elapsed
│ │ │ │  i16 : betti freeResolution Ulr3
│ │ │ │  
│ │ │ │                0  1  2
│ │ │ │  o16 = total: 12 24 12
│ │ │ │            0: 12 24 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out
│ │ │ @@ -66,17 +66,17 @@
│ │ │  i9 : mults = {1,2,3,1,2,3,1,2,3,1,2,3}
│ │ │  
│ │ │  o9 = {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 1.84622s elapsed
│ │ │ + -- 1.52842s elapsed
│ │ │  
│ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 5.34163s elapsed
│ │ │ + -- 3.94783s elapsed
│ │ │  
│ │ │  i12 : G==H
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html
│ │ │ @@ -182,21 +182,21 @@
│ │ │  
│ │ │  o9 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 1.84622s elapsed
    │ │ │ + -- 1.52842s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 5.34163s elapsed
    │ │ │ + -- 3.94783s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : G==H
│ │ │  
│ │ │  o12 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -81,17 +81,17 @@ │ │ │ │ o8 : Matrix K <-- K │ │ │ │ i9 : mults = {1,2,3,1,2,3,1,2,3,1,2,3} │ │ │ │ │ │ │ │ o9 = {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3} │ │ │ │ │ │ │ │ o9 : List │ │ │ │ i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R); │ │ │ │ - -- 1.84622s elapsed │ │ │ │ + -- 1.52842s elapsed │ │ │ │ i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R); │ │ │ │ - -- 5.34163s elapsed │ │ │ │ + -- 3.94783s elapsed │ │ │ │ i12 : G==H │ │ │ │ │ │ │ │ o12 = true │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ For reduced points, this function may be a bit slower than _a_f_f_i_n_e_P_o_i_n_t_s. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _a_f_f_i_n_e_F_a_t_P_o_i_n_t_s_B_y_I_n_t_e_r_s_e_c_t_i_o_n_(_M_a_t_r_i_x_,_L_i_s_t_,_R_i_n_g_) -- computes ideal of fat │ │ ├── ./usr/share/doc/Macaulay2/Posets/dump/rawdocumentation.dump │ │ │ @@ -1,8 +1,8 @@ │ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon May 18 11:29:46 2026 │ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon May 18 11:29:47 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=17 │ │ │ bWF4aW1hbEFudGljaGFpbnM= │ │ ├── ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out │ │ │ @@ -31,27 +31,27 @@ │ │ │ o5 = CacheTable{name => P} │ │ │ │ │ │ i6 : C == P │ │ │ │ │ │ o6 = true │ │ │ │ │ │ i7 : time isDistributive C │ │ │ - -- used 1.1742e-05s (cpu); 8.175e-06s (thread); 0s (gc) │ │ │ + -- used 1.536e-05s (cpu); 7.546e-06s (thread); 0s (gc) │ │ │ │ │ │ o7 = true │ │ │ │ │ │ i8 : time isDistributive P │ │ │ - -- used 5.75152s (cpu); 3.6505s (thread); 0s (gc) │ │ │ + -- used 6.42552s (cpu); 4.15609s (thread); 0s (gc) │ │ │ │ │ │ o8 = true │ │ │ │ │ │ i9 : C' = dual C; │ │ │ │ │ │ i10 : time isDistributive C' │ │ │ - -- used 7.735e-06s (cpu); 6.882e-06s (thread); 0s (gc) │ │ │ + -- used 8.926e-06s (cpu); 7.008e-06s (thread); 0s (gc) │ │ │ │ │ │ o10 = true │ │ │ │ │ │ i11 : peek C'.cache │ │ │ │ │ │ o11 = CacheTable{connectedComponents => {{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}} } │ │ │ coveringRelations => {{1, 0}, {2, 1}, {3, 2}, {4, 3}, {5, 4}, {6, 5}, {7, 6}, {8, 7}, {9, 8}} │ │ ├── ./usr/share/doc/Macaulay2/Posets/example-output/_greene__Kleitman__Partition.out │ │ │ @@ -7,22 +7,22 @@ │ │ │ o2 = Partition{4, 2} │ │ │ │ │ │ o2 : Partition │ │ │ │ │ │ i3 : D = dominanceLattice 6; │ │ │ │ │ │ i4 : time greeneKleitmanPartition(D, Strategy => "antichains") │ │ │ - -- used 0.418314s (cpu); 0.256713s (thread); 0s (gc) │ │ │ + -- used 0.431946s (cpu); 0.260349s (thread); 0s (gc) │ │ │ │ │ │ o4 = Partition{9, 2} │ │ │ │ │ │ o4 : Partition │ │ │ │ │ │ i5 : time greeneKleitmanPartition(D, Strategy => "chains") │ │ │ - -- used 1.4517e-05s (cpu); 1.4006e-05s (thread); 0s (gc) │ │ │ + -- used 1.4081e-05s (cpu); 1.3228e-05s (thread); 0s (gc) │ │ │ │ │ │ o5 = Partition{9, 2} │ │ │ │ │ │ o5 : Partition │ │ │ │ │ │ i6 : greeneKleitmanPartition chain 10 │ │ ├── ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ @@ -112,23 +112,23 @@ │ │ │ │ │ │ o6 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time isDistributive C
│ │ │ - -- used 1.1742e-05s (cpu); 8.175e-06s (thread); 0s (gc)
│ │ │ + -- used 1.536e-05s (cpu); 7.546e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time isDistributive P
│ │ │ - -- used 5.75152s (cpu); 3.6505s (thread); 0s (gc)
│ │ │ + -- used 6.42552s (cpu); 4.15609s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    We also know that the dual of a distributive lattice is again a distributive lattice. Other information is copied when possible.

    │ │ │ @@ -138,15 +138,15 @@ │ │ │ │ │ │ 
    i9 : C' = dual C;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : time isDistributive C'
│ │ │ - -- used 7.735e-06s (cpu); 6.882e-06s (thread); 0s (gc)
│ │ │ + -- used 8.926e-06s (cpu); 7.008e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : peek C'.cache
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,26 +41,26 @@
│ │ │ │  i5 : peek P.cache
│ │ │ │  
│ │ │ │  o5 = CacheTable{name => P}
│ │ │ │  i6 : C == P
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : time isDistributive C
│ │ │ │ - -- used 1.1742e-05s (cpu); 8.175e-06s (thread); 0s (gc)
│ │ │ │ + -- used 1.536e-05s (cpu); 7.546e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : time isDistributive P
│ │ │ │ - -- used 5.75152s (cpu); 3.6505s (thread); 0s (gc)
│ │ │ │ + -- used 6.42552s (cpu); 4.15609s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  We also know that the dual of a distributive lattice is again a distributive
│ │ │ │  lattice. Other information is copied when possible.
│ │ │ │  i9 : C' = dual C;
│ │ │ │  i10 : time isDistributive C'
│ │ │ │ - -- used 7.735e-06s (cpu); 6.882e-06s (thread); 0s (gc)
│ │ │ │ + -- used 8.926e-06s (cpu); 7.008e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : peek C'.cache
│ │ │ │  
│ │ │ │  o11 = CacheTable{connectedComponents => {{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}}
│ │ │ │  }
│ │ │ │                   coveringRelations => {{1, 0}, {2, 1}, {3, 2}, {4, 3}, {5, 4},
│ │ ├── ./usr/share/doc/Macaulay2/Posets/html/_greene__Kleitman__Partition.html
│ │ │ @@ -107,25 +107,25 @@
│ │ │              
│ │ │                
    i3 : D = dominanceLattice 6;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ - -- used 0.418314s (cpu); 0.256713s (thread); 0s (gc)
│ │ │ + -- used 0.431946s (cpu); 0.260349s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Partition{9, 2}
│ │ │  
│ │ │  o4 : Partition
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ - -- used 1.4517e-05s (cpu); 1.4006e-05s (thread); 0s (gc)
│ │ │ + -- used 1.4081e-05s (cpu); 1.3228e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = Partition{9, 2}
│ │ │  
│ │ │  o5 : Partition
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,21 +30,21 @@
│ │ │ │  
│ │ │ │  o2 : Partition
│ │ │ │  The conjugate of $l$ has the same property, but with chains replaced by
│ │ │ │  _a_n_t_i_c_h_a_i_n_s. Because of this, it is often better to count via antichains instead
│ │ │ │  of chains. This can be done by passing "antichains" as the Strategy.
│ │ │ │  i3 : D = dominanceLattice 6;
│ │ │ │  i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ │ - -- used 0.418314s (cpu); 0.256713s (thread); 0s (gc)
│ │ │ │ + -- used 0.431946s (cpu); 0.260349s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o4 : Partition
│ │ │ │  i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ │ - -- used 1.4517e-05s (cpu); 1.4006e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.4081e-05s (cpu); 1.3228e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o5 : Partition
│ │ │ │  The Greene-Kleitman partition of the $n$ _c_h_a_i_n is the partition of $n$ with $1$
│ │ │ │  part.
│ │ │ │  i6 : greeneKleitmanPartition chain 10
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out
│ │ │ @@ -24,35 +24,35 @@
│ │ │                | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                              3
│ │ │  o3 : R-module, quotient of R
│ │ │  
│ │ │  i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .158641s elapsed
│ │ │ + -- .135779s elapsed
│ │ │  
│ │ │  o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 1 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o4 : R-module, subquotient of R
│ │ │  
│ │ │  i5 : elapsedTime kernelOfLocalization(M, I2)
│ │ │ - -- .0173439s elapsed
│ │ │ + -- .0217822s elapsed
│ │ │  
│ │ │  o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o5 : R-module, subquotient of R
│ │ │  
│ │ │  i6 : elapsedTime kernelOfLocalization(M, I3)
│ │ │ - -- .0169717s elapsed
│ │ │ + -- .0219609s elapsed
│ │ │  
│ │ │  o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 1 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 0 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o6 : R-module, subquotient of R
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       x x )
│ │ │        0 4
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime regSeqInIdeal I
│ │ │ - -- .031224s elapsed
│ │ │ + -- .0415922s elapsed
│ │ │  
│ │ │  o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
│ │ │               2 7   3 6    0 7   2 5    0 7   1 4    0 7
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : R = QQ[h,l,s,x,y,z]
│ │ │ @@ -41,15 +41,15 @@
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : isSubset(I, ideal(s,l,h)) -- implies codim I == 3
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ - -- .00758927s elapsed
│ │ │ + -- .00896314s elapsed
│ │ │  
│ │ │                     2                3    2     2    8    3    2     2
│ │ │  o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html
│ │ │ @@ -112,41 +112,41 @@
│ │ │                              3
│ │ │  o3 : R-module, quotient of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .158641s elapsed
│ │ │ + -- .135779s elapsed
│ │ │  
│ │ │  o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 1 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o4 : R-module, subquotient of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime kernelOfLocalization(M, I2)
│ │ │ - -- .0173439s elapsed
│ │ │ + -- .0217822s elapsed
│ │ │  
│ │ │  o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o5 : R-module, subquotient of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : elapsedTime kernelOfLocalization(M, I3)
│ │ │ - -- .0169717s elapsed
│ │ │ + -- .0219609s elapsed
│ │ │  
│ │ │  o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 1 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 0 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o6 : R-module, subquotient of R
    │ │ │ ├── html2text {} │ │ │ │ @@ -41,39 +41,39 @@ │ │ │ │ | │ │ │ │ | 0 0 0 0 x_1^5- │ │ │ │ x_0x_2^4 | │ │ │ │ │ │ │ │ 3 │ │ │ │ o3 : R-module, quotient of R │ │ │ │ i4 : elapsedTime kernelOfLocalization(M, I1) │ │ │ │ - -- .158641s elapsed │ │ │ │ + -- .135779s elapsed │ │ │ │ │ │ │ │ o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0 │ │ │ │ 0 |) │ │ │ │ | 1 0 | | 0 0 0 x_1^3- │ │ │ │ x_0x_2^2 0 | │ │ │ │ | 0 1 | | 0 0 0 0 │ │ │ │ x_1^5-x_0x_2^4 | │ │ │ │ │ │ │ │ 3 │ │ │ │ o4 : R-module, subquotient of R │ │ │ │ i5 : elapsedTime kernelOfLocalization(M, I2) │ │ │ │ - -- .0173439s elapsed │ │ │ │ + -- .0217822s elapsed │ │ │ │ │ │ │ │ o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0 │ │ │ │ 0 |) │ │ │ │ | 0 0 | | 0 0 0 x_1^3- │ │ │ │ x_0x_2^2 0 | │ │ │ │ | 0 1 | | 0 0 0 0 │ │ │ │ x_1^5-x_0x_2^4 | │ │ │ │ │ │ │ │ 3 │ │ │ │ o5 : R-module, subquotient of R │ │ │ │ i6 : elapsedTime kernelOfLocalization(M, I3) │ │ │ │ - -- .0169717s elapsed │ │ │ │ + -- .0219609s elapsed │ │ │ │ │ │ │ │ o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0 │ │ │ │ 0 |) │ │ │ │ | 0 1 | | 0 0 0 x_1^3- │ │ │ │ x_0x_2^2 0 | │ │ │ │ | 0 0 | | 0 0 0 0 │ │ │ │ x_1^5-x_0x_2^4 | │ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ @@ -107,15 +107,15 @@ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime regSeqInIdeal I
│ │ │ - -- .031224s elapsed
│ │ │ + -- .0415922s elapsed
│ │ │  
│ │ │  o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
│ │ │               2 7   3 6    0 7   2 5    0 7   1 4    0 7
│ │ │  
│ │ │  o3 : Ideal of R
    │ │ │ │ │ │ │ │ │ @@ -153,15 +153,15 @@ │ │ │ │ │ │ o6 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ - -- .00758927s elapsed
│ │ │ + -- .00896314s elapsed
│ │ │  
│ │ │                     2                3    2     2    8    3    2     2
│ │ │  o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)
│ │ │  
│ │ │  o7 : Ideal of R
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -40,15 +40,15 @@ │ │ │ │ 2 7 0 7 3 6 2 6 1 6 0 6 2 5 0 5 3 4 2 4 1 4 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x ) │ │ │ │ 0 4 │ │ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ i3 : elapsedTime regSeqInIdeal I │ │ │ │ - -- .031224s elapsed │ │ │ │ + -- .0415922s elapsed │ │ │ │ │ │ │ │ o3 = ideal (x x , x x + x x , x x + x x , x x + x x ) │ │ │ │ 2 7 3 6 0 7 2 5 0 7 1 4 0 7 │ │ │ │ │ │ │ │ o3 : Ideal of R │ │ │ │ If I is the unit ideal, then an ideal of variables of the ring is returned. │ │ │ │ If the codimension of I is already known, then one can specify this, along with │ │ │ │ @@ -70,15 +70,15 @@ │ │ │ │ l , s ) │ │ │ │ │ │ │ │ o5 : Ideal of R │ │ │ │ i6 : isSubset(I, ideal(s,l,h)) -- implies codim I == 3 │ │ │ │ │ │ │ │ o6 = true │ │ │ │ i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1) │ │ │ │ - -- .00758927s elapsed │ │ │ │ + -- .00896314s elapsed │ │ │ │ │ │ │ │ 2 3 2 2 8 3 2 2 │ │ │ │ o7 = ideal (h*l - l - 4l*s + h*y, h + l s - h x, s + h + l s - h x) │ │ │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_d_i_c_a_l -- the radical of an ideal │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_iterator_lp__Python__Object_rp.out │ │ │ @@ -10,12 +10,12 @@ │ │ │ │ │ │ o2 = range(0, 3) │ │ │ │ │ │ o2 : PythonObject of class range │ │ │ │ │ │ i3 : i = iterator x │ │ │ │ │ │ -o3 = │ │ │ +o3 = │ │ │ │ │ │ o3 : PythonObject of class range_iterator │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_next_lp__Python__Object_rp.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ │ │ │ o2 = range(0, 3) │ │ │ │ │ │ o2 : PythonObject of class range │ │ │ │ │ │ i3 : i = iterator x │ │ │ │ │ │ -o3 = │ │ │ +o3 = │ │ │ │ │ │ o3 : PythonObject of class range_iterator │ │ │ │ │ │ i4 : next i │ │ │ │ │ │ o4 = 0 │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_python__Run__Script.out │ │ │ @@ -1,22 +1,22 @@ │ │ │ -- -*- M2-comint -*- hash: 447449196062331972 │ │ │ │ │ │ i1 : pyfile = temporaryFileName() | ".py" │ │ │ │ │ │ -o1 = /tmp/M2-32439-0/0.py │ │ │ +o1 = /tmp/M2-43611-0/0.py │ │ │ │ │ │ i2 : pyfile << "import math" << endl │ │ │ │ │ │ -o2 = /tmp/M2-32439-0/0.py │ │ │ +o2 = /tmp/M2-43611-0/0.py │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : pyfile << "x = math.sin(3.4)" << endl << close │ │ │ │ │ │ -o3 = /tmp/M2-32439-0/0.py │ │ │ +o3 = /tmp/M2-43611-0/0.py │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : get pyfile │ │ │ │ │ │ o4 = import math │ │ │ x = math.sin(3.4) │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_to__Python.out │ │ │ @@ -72,15 +72,15 @@ │ │ │ │ │ │ o12 = m2sqrt │ │ │ │ │ │ o12 : FunctionClosure │ │ │ │ │ │ i13 : pysqrt = toPython m2sqrt │ │ │ │ │ │ -o13 = │ │ │ +o13 = │ │ │ │ │ │ o13 : PythonObject of class builtin_function_or_method │ │ │ │ │ │ i14 : pysqrt 2 │ │ │ calling Macaulay2 code from Python! │ │ │ │ │ │ o14 = 1.4142135623730951 │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_use_lp__Python__Context_rp.out │ │ │ @@ -30,15 +30,15 @@ │ │ │ │ │ │ o7 : Symbol │ │ │ │ │ │ i8 : use ctx │ │ │ │ │ │ i9 : f │ │ │ │ │ │ -o9 = at 0x7f0c0cc84d60> │ │ │ +o9 = at 0x7f7909724d60> │ │ │ │ │ │ o9 : PythonObject of class function │ │ │ │ │ │ i10 : x │ │ │ │ │ │ o10 = 5 │ │ ├── ./usr/share/doc/Macaulay2/Python/html/_iterator_lp__Python__Object_rp.html │ │ │ @@ -95,15 +95,15 @@ │ │ │ o2 : PythonObject of class range │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7f0c0cccd3e0>
│ │ │ +o3 = <range_iterator object at 0x7f790976da70>
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -22,15 +22,15 @@ │ │ │ │ i2 : x = builtins@@range 3 │ │ │ │ │ │ │ │ o2 = range(0, 3) │ │ │ │ │ │ │ │ o2 : PythonObject of class range │ │ │ │ i3 : i = iterator x │ │ │ │ │ │ │ │ -o3 = │ │ │ │ +o3 = │ │ │ │ │ │ │ │ o3 : PythonObject of class range_iterator │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _n_e_x_t_(_P_y_t_h_o_n_O_b_j_e_c_t_) -- retrieve the next item from a python iterator │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _i_t_e_r_a_t_o_r_(_P_y_t_h_o_n_O_b_j_e_c_t_) -- get iterator of iterable python object │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/Python/html/_next_lp__Python__Object_rp.html │ │ │ @@ -91,15 +91,15 @@ │ │ │ o2 : PythonObject of class range │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7f0c0ccc1950>
│ │ │ +o3 = <range_iterator object at 0x7f7909762040>
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : next i
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i2 : x = builtins@@range 3
│ │ │ │  
│ │ │ │  o2 = range(0, 3)
│ │ │ │  
│ │ │ │  o2 : PythonObject of class range
│ │ │ │  i3 : i = iterator x
│ │ │ │  
│ │ │ │ -o3 = 
│ │ │ │ +o3 = 
│ │ │ │  
│ │ │ │  o3 : PythonObject of class range_iterator
│ │ │ │  i4 : next i
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  
│ │ │ │  o4 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_python__Run__Script.html
│ │ │ @@ -81,31 +81,31 @@
│ │ │            
    The return value is a Python dictionary containing all the variables defined in the global scope.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : pyfile = temporaryFileName() | ".py"
│ │ │  
│ │ │ -o1 = /tmp/M2-32439-0/0.py
    │ │ │ +o1 = /tmp/M2-43611-0/0.py │ │ │
    │ │ │ 
    i2 : pyfile << "import math" << endl
│ │ │  
│ │ │ -o2 = /tmp/M2-32439-0/0.py
│ │ │ +o2 = /tmp/M2-43611-0/0.py
│ │ │  
│ │ │  o2 : File
    │ │ │
    │ │ │ 
    i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │  
│ │ │ -o3 = /tmp/M2-32439-0/0.py
│ │ │ +o3 = /tmp/M2-43611-0/0.py
│ │ │  
│ │ │  o3 : File
    │ │ │
    │ │ │ 
    i4 : get pyfile
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,23 +16,23 @@
│ │ │ │  Execute a sequence of statements as if they were read from a Python file. This
│ │ │ │  is for multi-line code that might contain definitions, control structures,
│ │ │ │  imports, etc. It is great for running Python code from a file.
│ │ │ │  The return value is a Python dictionary containing all the variables defined in
│ │ │ │  the global scope.
│ │ │ │  i1 : pyfile = temporaryFileName() | ".py"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-32439-0/0.py
│ │ │ │ +o1 = /tmp/M2-43611-0/0.py
│ │ │ │  i2 : pyfile << "import math" << endl
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-32439-0/0.py
│ │ │ │ +o2 = /tmp/M2-43611-0/0.py
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-32439-0/0.py
│ │ │ │ +o3 = /tmp/M2-43611-0/0.py
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : get pyfile
│ │ │ │  
│ │ │ │  o4 = import math
│ │ │ │       x = math.sin(3.4)
│ │ │ │  i5 : pythonRunScript oo
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_to__Python.html
│ │ │ @@ -186,15 +186,15 @@
│ │ │  o12 : FunctionClosure
    │ │ │
    │ │ │ 
    i13 : pysqrt = toPython m2sqrt
│ │ │  
│ │ │ -o13 = <built-in method m2sqrt of PyCapsule object at 0x7f0c0cca2ca0>
│ │ │ +o13 = <built-in method m2sqrt of PyCapsule object at 0x7f7909742ca0>
│ │ │  
│ │ │  o13 : PythonObject of class builtin_function_or_method
    │ │ │
    │ │ │ 
    i14 : pysqrt 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -72,15 +72,15 @@
│ │ │ │            sqrt x)
│ │ │ │  
│ │ │ │  o12 = m2sqrt
│ │ │ │  
│ │ │ │  o12 : FunctionClosure
│ │ │ │  i13 : pysqrt = toPython m2sqrt
│ │ │ │  
│ │ │ │ -o13 = 
│ │ │ │ +o13 = 
│ │ │ │  
│ │ │ │  o13 : PythonObject of class builtin_function_or_method
│ │ │ │  i14 : pysqrt 2
│ │ │ │  calling Macaulay2 code from Python!
│ │ │ │  
│ │ │ │  o14 = 1.4142135623730951
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_use_lp__Python__Context_rp.html
│ │ │ @@ -129,15 +129,15 @@
│ │ │                
    i8 : use ctx
    
│ │ │
    │ │ │ 
    i9 : f
│ │ │  
│ │ │ -o9 = <function <lambda> at 0x7f0c0cc84d60>
│ │ │ +o9 = <function <lambda> at 0x7f7909724d60>
│ │ │  
│ │ │  o9 : PythonObject of class function
    │ │ │
    │ │ │ 
    i10 : x
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  
│ │ │ │  o7 = y
│ │ │ │  
│ │ │ │  o7 : Symbol
│ │ │ │  i8 : use ctx
│ │ │ │  i9 : f
│ │ │ │  
│ │ │ │ -o9 =  at 0x7f0c0cc84d60>
│ │ │ │ +o9 =  at 0x7f7909724d60>
│ │ │ │  
│ │ │ │  o9 : PythonObject of class function
│ │ │ │  i10 : x
│ │ │ │  
│ │ │ │  o10 = 5
│ │ │ │  
│ │ │ │  o10 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out
│ │ │ @@ -180,15 +180,15 @@
│ │ │  i21 : L = trim groebnerStratum F;
│ │ │  
│ │ │  o21 : Ideal of T
│ │ │  
│ │ │  i22 : assert(dim L == 18)
│ │ │  
│ │ │  i23 : elapsedTime isPrime L
│ │ │ - -- 3.08156s elapsed
│ │ │ + -- 2.54797s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : I = pointsIdeal randomPoints(S, 6)
│ │ │  
│ │ │                               2                              2   2          
│ │ │  o24 = ideal (a*c - 7b*c - 49c  + 40a*d - 42b*d + 12c*d + 28d , b  - 36b*c -
│ │ │ @@ -302,15 +302,15 @@
│ │ │  o38 = true
│ │ │  
│ │ │  i39 : L441 = trim(L + ideal M1);
│ │ │  
│ │ │  o39 : Ideal of T
│ │ │  
│ │ │  i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 1.69883s elapsed
│ │ │ + -- 1.44897s elapsed
│ │ │  
│ │ │  i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
│ │ │  
│ │ │  i42 : compsL441/dim -- two components, of dimensions 14 and 16.
│ │ │  
│ │ │ @@ -320,37 +320,37 @@
│ │ │  
│ │ │  i43 : compsL441/dim == {16, 14}
│ │ │  
│ │ │  o43 = true
│ │ │  
│ │ │  i44 : pta = randomPointOnRationalVariety compsL441_0
│ │ │  
│ │ │ -o44 = | 26 9 43 -2 -49 13 -30 -49 -47 50 -31 -25 -16 -50 -23 -19 20 19 -13 38
│ │ │ +o44 = | -40 -4 -40 44 -22 -50 -30 13 -1 -16 45 -23 29 19 14 23 -21 19 14 29 6
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -20 -32 19 26 -8 49 45 10 37 18 34 -29 -10 -28 5 15 |
│ │ │ +      10 -32 18 -22 37 34 15 5 -10 -29 26 49 -50 45 -28 |
│ │ │  
│ │ │                 1       36
│ │ │  o44 : Matrix kk  <-- kk
│ │ │  
│ │ │  i45 : Fa = sub(F, (vars S) | pta)
│ │ │  
│ │ │ -              2              2                              2              
│ │ │ -o45 = ideal (a  - 23b*c + 50c  - 25a*d + 13b*d + 43c*d + 26d , a*b - 8b*c -
│ │ │ +              2              2                              2               
│ │ │ +o45 = ideal (a  + 14b*c - 16c  - 23a*d - 50b*d - 40c*d - 40d , a*b - 22b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2             2                  
│ │ │ -      13c  - 20a*d - 16b*d - 47c*d + 9d , b  + 5b*c + 10c  - 10a*d - 32b*d -
│ │ │ +         2                          2   2              2                  
│ │ │ +      14c  + 6a*d + 29b*d - c*d - 4d , b  + 45b*c + 15c  + 49a*d + 10b*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                   2                              2     2  
│ │ │ -      19c*d - 30d , a*c - 29b*c + 49c  + 37a*d + 38b*d - 50c*d - 49d , b*c  +
│ │ │ +                 2                   2                             2     2  
│ │ │ +      23c*d - 30d , a*c + 26b*c + 37c  + 5a*d + 29b*d + 19c*d - 22d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2     3   3                2 
│ │ │ -      18b*c*d + 19c d + 45a*d  + 20b*d  - 31c*d  - 2d , c  + 15b*c*d + 34c d
│ │ │ +                   2         2        2        2      3   3                2 
│ │ │ +      10b*c*d - 32c d + 34a*d  - 21b*d  + 45c*d  + 44d , c  - 28b*c*d - 29c d
│ │ │        -----------------------------------------------------------------------
│ │ │               2        2        2      3
│ │ │ -      - 28a*d  + 26b*d  + 19c*d  - 49d )
│ │ │ +      - 50a*d  + 18b*d  + 19c*d  + 13d )
│ │ │  
│ │ │  o45 : Ideal of S
│ │ │  
│ │ │  i46 : betti res Fa
│ │ │  
│ │ │               0 1 2 3
│ │ │  o46 = total: 1 6 8 3
│ │ │ @@ -358,81 +358,84 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o46 : BettiTally
│ │ │  
│ │ │  i47 : netList decompose Fa -- this one is 5 points on a plane, and another point
│ │ │  
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -o47 = |ideal (c + 37d, b - 33d, a - 11d)                                                                                                                                 |
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                             2             2                      2   3                2         2        2     3     2                2         2        2     3 |
│ │ │ -      |ideal (a - 29b + 49c - 45d, b  + 5b*c + 10c  - 19b*d - 34c*d + 25d , c  + 15b*c*d + 34c d + 22b*d  - 23c*d  + 4d , b*c  + 18b*c*d + 19c d + 12b*d  - 14c*d  + 3d )|
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      +------------------------------------------------------------------------------------------------------------+
│ │ │ +o47 = |ideal (c + 5d, b - 33d, a - 21d)                                                                            |
│ │ │ +      +------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                      2              2                                                      |
│ │ │ +      |ideal (b + 45c + 49d, a - 22c - 26d, c  + 49c*d + 42d )                                                     |
│ │ │ +      +------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                             2                      2                           2   2                     2 |
│ │ │ +      |ideal (a + 26b + 37c + 36d, c  - 21b*d + 43c*d + 27d , b*c - 30b*d + 16c*d + 26d , b  - 3b*d - 24c*d - 36d )|
│ │ │ +      +------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i48 : CFa = minimalPrimes Fa
│ │ │  
│ │ │ -                                                                       2  
│ │ │ -o48 = {ideal (c + 37d, b - 33d, a - 11d), ideal (a - 29b + 49c - 45d, b  +
│ │ │ +                                                                             
│ │ │ +o48 = {ideal (c + 5d, b - 33d, a - 21d), ideal (b + 45c + 49d, a - 22c - 26d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -                2                      2   3                2         2  
│ │ │ -      5b*c + 10c  - 19b*d - 34c*d + 25d , c  + 15b*c*d + 34c d + 22b*d  -
│ │ │ +       2              2                                2                  
│ │ │ +      c  + 49c*d + 42d ), ideal (a + 26b + 37c + 36d, c  - 21b*d + 43c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2     3     2                2         2        2     3
│ │ │ -      23c*d  + 4d , b*c  + 18b*c*d + 19c d + 12b*d  - 14c*d  + 3d )}
│ │ │ +         2                           2   2                     2
│ │ │ +      27d , b*c - 30b*d + 16c*d + 26d , b  - 3b*d - 24c*d - 36d )}
│ │ │  
│ │ │  o48 : List
│ │ │  
│ │ │  i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │  
│ │ │ -o49 = a - 29b + 49c - 45d
│ │ │ +o49 = b + 45c + 49d
│ │ │  
│ │ │  o49 : S
│ │ │  
│ │ │  i50 : CFa/degree
│ │ │  
│ │ │ -o50 = {1, 5}
│ │ │ +o50 = {1, 2, 3}
│ │ │  
│ │ │  o50 : List
│ │ │  
│ │ │  i51 : CFa/(I -> lin % I == 0) -- so 5 points on the plane.
│ │ │  
│ │ │ -o51 = {false, true}
│ │ │ +o51 = {false, true, false}
│ │ │  
│ │ │  o51 : List
│ │ │  
│ │ │  i52 : degree(Fa : (Fa : lin))  -- somewhat simpler(?) way to see the ideal of the 5 points
│ │ │  
│ │ │ -o52 = 5
│ │ │ +o52 = 2
│ │ │  
│ │ │  i53 : ptb = randomPointOnRationalVariety compsL441_1
│ │ │  
│ │ │ -o53 = | 41 19 21 -24 23 -10 -39 29 42 -26 -46 29 -34 21 23 12 -42 21 -9 17
│ │ │ +o53 = | 31 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -30 -49 2 9 -28 -13 -9 38 -28 -23 -37 -29 0 -47 -4 47 |
│ │ │ +      -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 |
│ │ │  
│ │ │                 1       36
│ │ │  o53 : Matrix kk  <-- kk
│ │ │  
│ │ │  i54 : Fb = sub(F, (vars S) | ptb)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o54 = ideal (a  + 23b*c - 26c  + 29a*d - 10b*d + 21c*d + 41d , a*b - 28b*c -
│ │ │ +              2              2                             2               
│ │ │ +o54 = ideal (a  + 50b*c - 42c  - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2   2             2                  
│ │ │ -      9c  - 30a*d - 34b*d + 42c*d + 19d , b  - 4b*c + 38c  - 49b*d + 12c*d -
│ │ │ +         2                             2   2             2                 
│ │ │ +      42c  + 13a*d + 14b*d - 3c*d + 42d , b  - 9b*c - 29c  + 31b*d + 5c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                   2                              2     2          
│ │ │ -      39d , a*c - 29b*c - 13c  - 28a*d + 17b*d + 21c*d + 23d , b*c  - 23b*c*d
│ │ │ +         2                 2                             2     2            
│ │ │ +      43d , a*c + 9b*c - 4c  - 23a*d + 47b*d + 2c*d + 19d , b*c  + 21b*c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -          2        2        2        2      3   3                2         2
│ │ │ -      + 2c d - 9a*d  - 42b*d  - 46c*d  - 24d , c  + 47b*c*d - 37c d - 47a*d 
│ │ │ +         2         2        2        2      3   3                2         2
│ │ │ +      37c d + 38a*d  + 36b*d  - 29c*d  + 25d , c  - 47b*c*d + 17c d + 21a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -            2        2      3
│ │ │ -      + 9b*d  + 21c*d  + 29d )
│ │ │ +             2        2     3
│ │ │ +      - 23b*d  - 13c*d  - 7d )
│ │ │  
│ │ │  o54 : Ideal of S
│ │ │  
│ │ │  i55 : betti res Fb
│ │ │  
│ │ │               0 1 2 3
│ │ │  o55 = total: 1 6 8 3
│ │ │ @@ -440,80 +443,114 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o55 : BettiTally
│ │ │  
│ │ │  i56 : netList decompose Fb --
│ │ │  
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     2             2                                                                                                                                                        |
│ │ │ -o56 = |ideal (b - 28c - 28d, a + 35c + 6d, c  + 32c*d - 7d )                                                                                                                                                       |
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                     2                                 2                                   2   2                           2                                   2   2                             2 |
│ │ │ -      |ideal (c  - 8a*d - 49b*d - 33d , b*c - 24a*d + b*d - 49c*d - 43d , a*c - 20a*d + 15b*d + 14c*d - 37d , b  + 6a*d - b*d + 18c*d + 33d , a*b + 34a*d - 43b*d - 17c*d + 33d , a  - 31a*d + 6b*d + 37c*d - 30d )|
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      +-------------------------------------------------------+
│ │ │ +o56 = |ideal (c - 45d, b + 16d, a + 38d)                      |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 43d, b + 10d, a + 8d)                       |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 34d, b + 15d, a + 28d)                      |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 11d, b + 39d, a + 23d)                      |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |                                      2              2 |
│ │ │ +      |ideal (b - 32c + 42d, a - 19c - 16d, c  - 28c*d - 40d )|
│ │ │ +      +-------------------------------------------------------+
│ │ │  
│ │ │  i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │  
│ │ │ -o57 = ++
│ │ │ -      ++
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                          2                      2                         2   2                      2                                                                                                             |
│ │ │ +o57 = |ideal (a - 7b + 32c + d, c  + 42b*d + 33c*d - 10d , b*c - b*d + 13c*d + 18d , b  + 28b*d - 32c*d + 16d )                                                                                                            |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                          2                      2                           2   2                      2                                                                                                           |
│ │ │ +      |ideal (a - 7b + 32c + d, c  + 40b*d - 36c*d + 33d , b*c + 45b*d - 16c*d + 39d , b  - 20b*d + 29c*d + 38d )                                                                                                          |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                          2                      2                           2   2                     2                                                                                                            |
│ │ │ +      |ideal (a - 7b + 32c + d, c  - 10b*d + 17c*d - 21d , b*c - 17b*d - 23c*d - 32d , b  - 8b*d - 12c*d - 46d )                                                                                                           |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                     3      2         2      3                                                                                                                                                      |
│ │ │ +      |ideal (b + 23c - 11d, a - 9c + 25d, c  - 13c d - 14c*d  + 23d )                                                                                                                                                     |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                     2                      2   2      2                      2   3      2         2        2      3                                                                                |
│ │ │ +      |ideal (a + 48b - 40c - 20d, b*c - 32c  + 43b*d - 21c*d - 12d , b  - 14c  + 14b*d + 18c*d + 36d , c  + 28c d - 20b*d  + 42c*d  - 50d )                                                                               |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                   2                      2   2      2                      2   3      2         2        2     3                                                                                   |
│ │ │ +      |ideal (a + b + 50c + 26d, b*c - 32c  + 34b*d - 36c*d + 14d , b  - 14c  + 34b*d - 16c*d - 33d , c  + 28c d + 39b*d  - 28c*d  + 4d )                                                                                  |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |        2                            2                                 2                                 2   2                              2                                  2   2                              2 |
│ │ │ +      |ideal (c  - 7a*d - 19b*d + 6c*d - 19d , b*c - 5a*d + 49b*d - 4c*d + 50d , a*c - 6a*d + 35b*d - 39c*d - 2d , b  - 46a*d + 22b*d + 42c*d + 43d , a*b + 3a*d - 12b*d - 49c*d + 40d , a  + 28a*d - 13b*d - 25c*d - 35d )|
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                     2                     2   2      2                      2   3      2         2        2     3                                                                                  |
│ │ │ +      |ideal (a - 46b + 39c - 29d, b*c - 32c  + 11b*d - 7c*d - 43d , b  - 14c  + 29b*d + 43c*d - 41d , c  + 28c d + 46b*d  - 50c*d  - 5d )                                                                                 |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |        2                              2                                2                                   2   2                              2                                 2   2                           2  |
│ │ │ +      |ideal (c  + 15a*d + 27b*d + 35c*d + 46d , b*c - 6a*d + b*d + 36c*d - 31d , a*c - 10a*d + 45b*d + 20c*d - 23d , b  - 23a*d + 15b*d + 31c*d - 13d , a*b - 6a*d - 40b*d + 8c*d + 18d , a  - 8a*d - 24b*d + c*d - 22d ) |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |        2                              2                                2                                  2   2                      2                                   2   2                              2      |
│ │ │ +      |ideal (c  + 37a*d + 25b*d - 16c*d + 14d , b*c - 7a*d + 47b*d - 3c*d - 2d , a*c - 14a*d + 27b*d - 35c*d - 8d , b  - 33b*d + 19c*d + 27d , a*b - 15a*d - 30b*d - 40c*d - 24d , a  - 44a*d + 16b*d + 11c*d + 12d )     |
│ │ │ +      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │  
│ │ │ -o58 = | 17 -43 -38 -37 -19 -24 15 -23 -11 48 -17 7 45 -18 31 47 28 8 -44 -30
│ │ │ +o58 = | 13 -24 9 5 -49 49 36 -36 -50 8 31 -22 49 8 35 49 6 -42 32 15 -8 -24
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -3 -33 -42 -13 33 39 -21 -24 -2 -22 -23 15 -29 46 -40 18 |
│ │ │ +      -33 -22 28 -2 -23 18 -40 -29 15 -13 39 -18 -21 46 |
│ │ │  
│ │ │                 1       36
│ │ │  o58 : Matrix kk  <-- kk
│ │ │  
│ │ │  i59 : pt1 = randomPointOnRationalVariety(compsL441_1)
│ │ │  
│ │ │ -o59 = | -39 20 -45 -4 -13 35 -45 30 -48 -16 -7 -2 -6 3 3 4 47 3 -8 -18 47 8
│ │ │ +o59 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -35 32 -16 46 -15 21 35 37 -50 10 0 33 -14 -49 |
│ │ │ +      -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 |
│ │ │  
│ │ │                 1       36
│ │ │  o59 : Matrix kk  <-- kk
│ │ │  
│ │ │  i60 : I0 = sub(sub(F, (vars ring F) | sub(pt0, ring F)), S)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o60 = ideal (a  + 31b*c + 48c  + 7a*d - 24b*d - 38c*d + 17d , a*b + 33b*c -
│ │ │ +              2             2                             2               
│ │ │ +o60 = ideal (a  + 35b*c + 8c  - 22a*d + 49b*d + 9c*d + 13d , a*b + 28b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │           2                             2   2              2                  
│ │ │ -      44c  - 3a*d + 45b*d - 11c*d - 43d , b  - 40b*c - 24c  - 29a*d - 33b*d +
│ │ │ +      32c  - 8a*d + 49b*d - 50c*d - 24d , b  - 21b*c + 18c  + 39a*d - 24b*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                   2                             2     2  
│ │ │ -      47c*d + 15d , a*c + 15b*c + 39c  - 2a*d - 30b*d - 18c*d - 19d , b*c  -
│ │ │ +                 2                  2                             2     2  
│ │ │ +      49c*d + 36d , a*c - 13b*c - 2c  - 40a*d + 15b*d + 8c*d - 49d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      22b*c*d - 42c d - 21a*d  + 28b*d  - 17c*d  - 37d , c  + 18b*c*d - 23c d
│ │ │ +                   2         2       2        2     3   3                2   
│ │ │ +      29b*c*d - 33c d - 23a*d  + 6b*d  + 31c*d  + 5d , c  + 46b*c*d + 15c d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2       2      3
│ │ │ -      + 46a*d  - 13b*d  + 8c*d  - 23d )
│ │ │ +           2        2        2      3
│ │ │ +      18a*d  - 22b*d  - 42c*d  - 36d )
│ │ │  
│ │ │  o60 : Ideal of S
│ │ │  
│ │ │  i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S)
│ │ │  
│ │ │ -              2             2                             2               
│ │ │ -o61 = ideal (a  + 3b*c - 16c  - 2a*d + 35b*d - 45c*d - 39d , a*b - 16b*c -
│ │ │ +              2              2                             2               
│ │ │ +o61 = ideal (a  - 21b*c + 45c  - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                             2   2              2                
│ │ │ -      8c  + 47a*d - 6b*d - 48c*d + 20d , b  - 14b*c + 21c  + 8b*d + 4c*d -
│ │ │ +         2                             2   2              2                  
│ │ │ +      17c  - 23a*d + 37b*d - 5c*d + 18d , b  - 15b*c + 10c  + 15b*d + 28c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                   2                             2     2          
│ │ │ -      45d , a*c + 10b*c + 46c  + 35a*d - 18b*d + 3c*d - 13d , b*c  + 37b*c*d
│ │ │ +         2                   2                              2     2         
│ │ │ +      50d , a*c + 32b*c - 14c  - 31a*d - 49b*d - 35c*d + 21d , b*c  + 3b*c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2         2        2       2     3   3                2         2
│ │ │ -      - 35c d - 15a*d  + 47b*d  - 7c*d  - 4d , c  - 49b*c*d - 50c d + 33a*d 
│ │ │ +           2         2        2        2      3   3                2        2
│ │ │ +      - 50c d + 21a*d  + 27b*d  - 47c*d  + 38d , c  + 33b*c*d - 18c d + 3a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2       2      3
│ │ │ -      + 32b*d  + 3c*d  + 30d )
│ │ │ +             2        2     3
│ │ │ +      + 37b*d  + 46c*d  - 8d )
│ │ │  
│ │ │  o61 : Ideal of S
│ │ │  
│ │ │  i62 : betti res I0
│ │ │  
│ │ │               0 1 2 3
│ │ │  o62 = total: 1 6 8 3
│ │ │ @@ -531,32 +568,38 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o63 : BettiTally
│ │ │  
│ │ │  i64 : netList decompose I0
│ │ │  
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -o64 = |ideal (c - 2d, b - 34d, a + 26d)                                                             |
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -      |ideal (c - 40d, b - 45d, a - 28d)                                                            |
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -      |                                     2                     2   2      2                    2 |
│ │ │ -      |ideal (a + 15b + 39c - 41d, b*c - 15c  - 50b*d + 6c*d - 44d , b  - 18c  + 18b*d + 4c*d - 5d )|
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ +      +-------------------------------------------------------------------------------------------------------+
│ │ │ +o64 = |ideal (c - 40d, b - 10d, a + 32d)                                                                      |
│ │ │ +      +-------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                      2              2                                                 |
│ │ │ +      |ideal (b + 10c + 25d, a + 27c - 50d, c  - 34c*d - 17d )                                                |
│ │ │ +      +-------------------------------------------------------------------------------------------------------+
│ │ │ +      |                            2                    2                          2   2                    2 |
│ │ │ +      |ideal (a - 13b - 2c + 29d, c  - 5b*d - 20c*d + 8d , b*c + 40b*d + 16c*d + 8d , b  - b*d + 15c*d + 40d )|
│ │ │ +      +-------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i65 : netList decompose I1
│ │ │  
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                    2              2                                                                                                                                                                  |
│ │ │ -o65 = |ideal (b + 24c - 6d, a - 39c - 6d, c  + 44c*d - 11d )                                                                                                                                                                 |
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                              2                                  2                                   2   2                             2                                  2   2                            2 |
│ │ │ -      |ideal (c  + 17a*d - 13b*d + 32c*d - 46d , b*c + 41a*d - 8b*d + 19c*d + 12d , a*c - 46a*d - 47b*d - 43c*d - 37d , b  + 15a*d - 33b*d + 2c*d - 22d , a*b + 31a*d - 36b*d + 7c*d + 46d , a  + 46a*d - 48b*d + 6c*d - 3d )|
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      +------------------------------------------------------+
│ │ │ +o65 = |ideal (c + 32d, b + 18d, a - 33d)                     |
│ │ │ +      +------------------------------------------------------+
│ │ │ +      |ideal (c + 29d, b - 8d, a + 50d)                      |
│ │ │ +      +------------------------------------------------------+
│ │ │ +      |ideal (c + 16d, b + 39d, a - 32d)                     |
│ │ │ +      +------------------------------------------------------+
│ │ │ +      |ideal (c + 5d, b - 14d, a + 7d)                       |
│ │ │ +      +------------------------------------------------------+
│ │ │ +      |                                     2              2 |
│ │ │ +      |ideal (b - 40c + 5d, a - 47c + 24d, c  - 27c*d + 15d )|
│ │ │ +      +------------------------------------------------------+
│ │ │  
│ │ │  i66 : L430 = (trim minors(2, M1)) + groebnerStratum F;
│ │ │  
│ │ │  o66 : Ideal of T
│ │ │  
│ │ │  i67 : C = res(I, Strategy => Nonminimal)
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html
│ │ │ @@ -349,15 +349,15 @@
│ │ │
    		│ │ │ 
    i22 : assert(dim L == 18)
    │ │ │
    │ │ │ 
    i23 : elapsedTime isPrime L
│ │ │ - -- 3.08156s elapsed
│ │ │ + -- 2.54797s elapsed
│ │ │  
│ │ │  o23 = true
    │ │ │
    │ │ │
    │ │ │

    The Schreyer resolution and minimal Betti numbers

    │ │ │ @@ -561,15 +561,15 @@ │ │ │ │ │ │ o39 : Ideal of T     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 1.69883s elapsed
    │ │ │ + -- 1.44897s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
    │ │ │ @@ -596,40 +596,40 @@ │ │ │

    Both components are rational, and here are random points, one on each component:

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i44 : pta = randomPointOnRationalVariety compsL441_0
│ │ │  
│ │ │ -o44 = | 26 9 43 -2 -49 13 -30 -49 -47 50 -31 -25 -16 -50 -23 -19 20 19 -13 38
│ │ │ +o44 = | -40 -4 -40 44 -22 -50 -30 13 -1 -16 45 -23 29 19 14 23 -21 19 14 29 6
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -20 -32 19 26 -8 49 45 10 37 18 34 -29 -10 -28 5 15 |
│ │ │ +      10 -32 18 -22 37 34 15 5 -10 -29 26 49 -50 45 -28 |
│ │ │  
│ │ │                 1       36
│ │ │  o44 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i45 : Fa = sub(F, (vars S) | pta)
│ │ │  
│ │ │ -              2              2                              2              
│ │ │ -o45 = ideal (a  - 23b*c + 50c  - 25a*d + 13b*d + 43c*d + 26d , a*b - 8b*c -
│ │ │ +              2              2                              2               
│ │ │ +o45 = ideal (a  + 14b*c - 16c  - 23a*d - 50b*d - 40c*d - 40d , a*b - 22b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2             2                  
│ │ │ -      13c  - 20a*d - 16b*d - 47c*d + 9d , b  + 5b*c + 10c  - 10a*d - 32b*d -
│ │ │ +         2                          2   2              2                  
│ │ │ +      14c  + 6a*d + 29b*d - c*d - 4d , b  + 45b*c + 15c  + 49a*d + 10b*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                   2                              2     2  
│ │ │ -      19c*d - 30d , a*c - 29b*c + 49c  + 37a*d + 38b*d - 50c*d - 49d , b*c  +
│ │ │ +                 2                   2                             2     2  
│ │ │ +      23c*d - 30d , a*c + 26b*c + 37c  + 5a*d + 29b*d + 19c*d - 22d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2     3   3                2 
│ │ │ -      18b*c*d + 19c d + 45a*d  + 20b*d  - 31c*d  - 2d , c  + 15b*c*d + 34c d
│ │ │ +                   2         2        2        2      3   3                2 
│ │ │ +      10b*c*d - 32c d + 34a*d  - 21b*d  + 45c*d  + 44d , c  - 28b*c*d - 29c d
│ │ │        -----------------------------------------------------------------------
│ │ │               2        2        2      3
│ │ │ -      - 28a*d  + 26b*d  + 19c*d  - 49d )
│ │ │ +      - 50a*d  + 18b*d  + 19c*d  + 13d )
│ │ │  
│ │ │  o45 : Ideal of S
    │ │ │
    │ │ │ 
    i46 : betti res Fa
│ │ │ @@ -643,104 +643,107 @@
│ │ │  o46 : BettiTally
    │ │ │
    │ │ │ 
    i47 : netList decompose Fa -- this one is 5 points on a plane, and another point
│ │ │  
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -o47 = |ideal (c + 37d, b - 33d, a - 11d)                                                                                                                                 |
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                             2             2                      2   3                2         2        2     3     2                2         2        2     3 |
│ │ │ -      |ideal (a - 29b + 49c - 45d, b  + 5b*c + 10c  - 19b*d - 34c*d + 25d , c  + 15b*c*d + 34c d + 22b*d  - 23c*d  + 4d , b*c  + 18b*c*d + 19c d + 12b*d  - 14c*d  + 3d )|
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    │ │ │ + +------------------------------------------------------------------------------------------------------------+ │ │ │ +o47 = |ideal (c + 5d, b - 33d, a - 21d) | │ │ │ + +------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 | │ │ │ + |ideal (b + 45c + 49d, a - 22c - 26d, c + 49c*d + 42d ) | │ │ │ + +------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a + 26b + 37c + 36d, c - 21b*d + 43c*d + 27d , b*c - 30b*d + 16c*d + 26d , b - 3b*d - 24c*d - 36d )| │ │ │ + +------------------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │ 
    i48 : CFa = minimalPrimes Fa
│ │ │  
│ │ │ -                                                                       2  
│ │ │ -o48 = {ideal (c + 37d, b - 33d, a - 11d), ideal (a - 29b + 49c - 45d, b  +
│ │ │ +                                                                             
│ │ │ +o48 = {ideal (c + 5d, b - 33d, a - 21d), ideal (b + 45c + 49d, a - 22c - 26d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -                2                      2   3                2         2  
│ │ │ -      5b*c + 10c  - 19b*d - 34c*d + 25d , c  + 15b*c*d + 34c d + 22b*d  -
│ │ │ +       2              2                                2                  
│ │ │ +      c  + 49c*d + 42d ), ideal (a + 26b + 37c + 36d, c  - 21b*d + 43c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2     3     2                2         2        2     3
│ │ │ -      23c*d  + 4d , b*c  + 18b*c*d + 19c d + 12b*d  - 14c*d  + 3d )}
│ │ │ +         2                           2   2                     2
│ │ │ +      27d , b*c - 30b*d + 16c*d + 26d , b  - 3b*d - 24c*d - 36d )}
│ │ │  
│ │ │  o48 : List
    │ │ │
    │ │ │ 
    i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │  
│ │ │ -o49 = a - 29b + 49c - 45d
│ │ │ +o49 = b + 45c + 49d
│ │ │  
│ │ │  o49 : S
    │ │ │
    │ │ │ 
    i50 : CFa/degree
│ │ │  
│ │ │ -o50 = {1, 5}
│ │ │ +o50 = {1, 2, 3}
│ │ │  
│ │ │  o50 : List
    │ │ │
    │ │ │ 
    i51 : CFa/(I -> lin % I == 0) -- so 5 points on the plane.
│ │ │  
│ │ │ -o51 = {false, true}
│ │ │ +o51 = {false, true, false}
│ │ │  
│ │ │  o51 : List
    │ │ │
    │ │ │ 
    i52 : degree(Fa : (Fa : lin))  -- somewhat simpler(?) way to see the ideal of the 5 points
│ │ │  
│ │ │ -o52 = 5
    │ │ │ +o52 = 2 │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i53 : ptb = randomPointOnRationalVariety compsL441_1
│ │ │  
│ │ │ -o53 = | 41 19 21 -24 23 -10 -39 29 42 -26 -46 29 -34 21 23 12 -42 21 -9 17
│ │ │ +o53 = | 31 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -30 -49 2 9 -28 -13 -9 38 -28 -23 -37 -29 0 -47 -4 47 |
│ │ │ +      -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 |
│ │ │  
│ │ │                 1       36
│ │ │  o53 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i54 : Fb = sub(F, (vars S) | ptb)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o54 = ideal (a  + 23b*c - 26c  + 29a*d - 10b*d + 21c*d + 41d , a*b - 28b*c -
│ │ │ +              2              2                             2               
│ │ │ +o54 = ideal (a  + 50b*c - 42c  - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2   2             2                  
│ │ │ -      9c  - 30a*d - 34b*d + 42c*d + 19d , b  - 4b*c + 38c  - 49b*d + 12c*d -
│ │ │ +         2                             2   2             2                 
│ │ │ +      42c  + 13a*d + 14b*d - 3c*d + 42d , b  - 9b*c - 29c  + 31b*d + 5c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                   2                              2     2          
│ │ │ -      39d , a*c - 29b*c - 13c  - 28a*d + 17b*d + 21c*d + 23d , b*c  - 23b*c*d
│ │ │ +         2                 2                             2     2            
│ │ │ +      43d , a*c + 9b*c - 4c  - 23a*d + 47b*d + 2c*d + 19d , b*c  + 21b*c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -          2        2        2        2      3   3                2         2
│ │ │ -      + 2c d - 9a*d  - 42b*d  - 46c*d  - 24d , c  + 47b*c*d - 37c d - 47a*d 
│ │ │ +         2         2        2        2      3   3                2         2
│ │ │ +      37c d + 38a*d  + 36b*d  - 29c*d  + 25d , c  - 47b*c*d + 17c d + 21a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -            2        2      3
│ │ │ -      + 9b*d  + 21c*d  + 29d )
│ │ │ +             2        2     3
│ │ │ +      - 23b*d  - 13c*d  - 7d )
│ │ │  
│ │ │  o54 : Ideal of S
    │ │ │
    │ │ │ 
    i55 : betti res Fb
│ │ │ @@ -754,102 +757,136 @@
│ │ │  o55 : BettiTally
    │ │ │
    │ │ │ 
    i56 : netList decompose Fb --
│ │ │  
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     2             2                                                                                                                                                        |
│ │ │ -o56 = |ideal (b - 28c - 28d, a + 35c + 6d, c  + 32c*d - 7d )                                                                                                                                                       |
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                     2                                 2                                   2   2                           2                                   2   2                             2 |
│ │ │ -      |ideal (c  - 8a*d - 49b*d - 33d , b*c - 24a*d + b*d - 49c*d - 43d , a*c - 20a*d + 15b*d + 14c*d - 37d , b  + 6a*d - b*d + 18c*d + 33d , a*b + 34a*d - 43b*d - 17c*d + 33d , a  - 31a*d + 6b*d + 37c*d - 30d )|
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    │ │ │ + +-------------------------------------------------------+ │ │ │ +o56 = |ideal (c - 45d, b + 16d, a + 38d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 43d, b + 10d, a + 8d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 34d, b + 15d, a + 28d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 11d, b + 39d, a + 23d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + | 2 2 | │ │ │ + |ideal (b - 32c + 42d, a - 19c - 16d, c - 28c*d - 40d )| │ │ │ + +-------------------------------------------------------+ │ │ │
    │ │ │ 
    i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │  
│ │ │ -o57 = ++
│ │ │ -      ++
    │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ +o57 = |ideal (a - 7b + 32c + d, c + 42b*d + 33c*d - 10d , b*c - b*d + 13c*d + 18d , b + 28b*d - 32c*d + 16d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a - 7b + 32c + d, c + 40b*d - 36c*d + 33d , b*c + 45b*d - 16c*d + 39d , b - 20b*d + 29c*d + 38d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a - 7b + 32c + d, c - 10b*d + 17c*d - 21d , b*c - 17b*d - 23c*d - 32d , b - 8b*d - 12c*d - 46d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 3 2 2 3 | │ │ │ + |ideal (b + 23c - 11d, a - 9c + 25d, c - 13c d - 14c*d + 23d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 3 2 2 2 3 | │ │ │ + |ideal (a + 48b - 40c - 20d, b*c - 32c + 43b*d - 21c*d - 12d , b - 14c + 14b*d + 18c*d + 36d , c + 28c d - 20b*d + 42c*d - 50d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 3 2 2 2 3 | │ │ │ + |ideal (a + b + 50c + 26d, b*c - 32c + 34b*d - 36c*d + 14d , b - 14c + 34b*d - 16c*d - 33d , c + 28c d + 39b*d - 28c*d + 4d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 2 2 2 2 | │ │ │ + |ideal (c - 7a*d - 19b*d + 6c*d - 19d , b*c - 5a*d + 49b*d - 4c*d + 50d , a*c - 6a*d + 35b*d - 39c*d - 2d , b - 46a*d + 22b*d + 42c*d + 43d , a*b + 3a*d - 12b*d - 49c*d + 40d , a + 28a*d - 13b*d - 25c*d - 35d )| │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 3 2 2 2 3 | │ │ │ + |ideal (a - 46b + 39c - 29d, b*c - 32c + 11b*d - 7c*d - 43d , b - 14c + 29b*d + 43c*d - 41d , c + 28c d + 46b*d - 50c*d - 5d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 2 2 2 2 | │ │ │ + |ideal (c + 15a*d + 27b*d + 35c*d + 46d , b*c - 6a*d + b*d + 36c*d - 31d , a*c - 10a*d + 45b*d + 20c*d - 23d , b - 23a*d + 15b*d + 31c*d - 13d , a*b - 6a*d - 40b*d + 8c*d + 18d , a - 8a*d - 24b*d + c*d - 22d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 2 2 2 2 | │ │ │ + |ideal (c + 37a*d + 25b*d - 16c*d + 14d , b*c - 7a*d + 47b*d - 3c*d - 2d , a*c - 14a*d + 27b*d - 35c*d - 8d , b - 33b*d + 19c*d + 27d , a*b - 15a*d - 30b*d - 40c*d - 24d , a - 44a*d + 16b*d + 11c*d + 12d ) | │ │ │ + +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │  
│ │ │ -o58 = | 17 -43 -38 -37 -19 -24 15 -23 -11 48 -17 7 45 -18 31 47 28 8 -44 -30
│ │ │ +o58 = | 13 -24 9 5 -49 49 36 -36 -50 8 31 -22 49 8 35 49 6 -42 32 15 -8 -24
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -3 -33 -42 -13 33 39 -21 -24 -2 -22 -23 15 -29 46 -40 18 |
│ │ │ +      -33 -22 28 -2 -23 18 -40 -29 15 -13 39 -18 -21 46 |
│ │ │  
│ │ │                 1       36
│ │ │  o58 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i59 : pt1 = randomPointOnRationalVariety(compsL441_1)
│ │ │  
│ │ │ -o59 = | -39 20 -45 -4 -13 35 -45 30 -48 -16 -7 -2 -6 3 3 4 47 3 -8 -18 47 8
│ │ │ +o59 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -35 32 -16 46 -15 21 35 37 -50 10 0 33 -14 -49 |
│ │ │ +      -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 |
│ │ │  
│ │ │                 1       36
│ │ │  o59 : Matrix kk  <-- kk
    │ │ │
    │ │ │
    │ │ │

    We compute the ideal of the corresponding zero dimensional scheme with length 6, corresponding to the points pt0, pt1 in Hilb.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i60 : I0 = sub(sub(F, (vars ring F) | sub(pt0, ring F)), S)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o60 = ideal (a  + 31b*c + 48c  + 7a*d - 24b*d - 38c*d + 17d , a*b + 33b*c -
│ │ │ +              2             2                             2               
│ │ │ +o60 = ideal (a  + 35b*c + 8c  - 22a*d + 49b*d + 9c*d + 13d , a*b + 28b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │           2                             2   2              2                  
│ │ │ -      44c  - 3a*d + 45b*d - 11c*d - 43d , b  - 40b*c - 24c  - 29a*d - 33b*d +
│ │ │ +      32c  - 8a*d + 49b*d - 50c*d - 24d , b  - 21b*c + 18c  + 39a*d - 24b*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                   2                             2     2  
│ │ │ -      47c*d + 15d , a*c + 15b*c + 39c  - 2a*d - 30b*d - 18c*d - 19d , b*c  -
│ │ │ +                 2                  2                             2     2  
│ │ │ +      49c*d + 36d , a*c - 13b*c - 2c  - 40a*d + 15b*d + 8c*d - 49d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      22b*c*d - 42c d - 21a*d  + 28b*d  - 17c*d  - 37d , c  + 18b*c*d - 23c d
│ │ │ +                   2         2       2        2     3   3                2   
│ │ │ +      29b*c*d - 33c d - 23a*d  + 6b*d  + 31c*d  + 5d , c  + 46b*c*d + 15c d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2       2      3
│ │ │ -      + 46a*d  - 13b*d  + 8c*d  - 23d )
│ │ │ +           2        2        2      3
│ │ │ +      18a*d  - 22b*d  - 42c*d  - 36d )
│ │ │  
│ │ │  o60 : Ideal of S
    │ │ │
    │ │ │ 
    i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S)
│ │ │  
│ │ │ -              2             2                             2               
│ │ │ -o61 = ideal (a  + 3b*c - 16c  - 2a*d + 35b*d - 45c*d - 39d , a*b - 16b*c -
│ │ │ +              2              2                             2               
│ │ │ +o61 = ideal (a  - 21b*c + 45c  - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                             2   2              2                
│ │ │ -      8c  + 47a*d - 6b*d - 48c*d + 20d , b  - 14b*c + 21c  + 8b*d + 4c*d -
│ │ │ +         2                             2   2              2                  
│ │ │ +      17c  - 23a*d + 37b*d - 5c*d + 18d , b  - 15b*c + 10c  + 15b*d + 28c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                   2                             2     2          
│ │ │ -      45d , a*c + 10b*c + 46c  + 35a*d - 18b*d + 3c*d - 13d , b*c  + 37b*c*d
│ │ │ +         2                   2                              2     2         
│ │ │ +      50d , a*c + 32b*c - 14c  - 31a*d - 49b*d - 35c*d + 21d , b*c  + 3b*c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2         2        2       2     3   3                2         2
│ │ │ -      - 35c d - 15a*d  + 47b*d  - 7c*d  - 4d , c  - 49b*c*d - 50c d + 33a*d 
│ │ │ +           2         2        2        2      3   3                2        2
│ │ │ +      - 50c d + 21a*d  + 27b*d  - 47c*d  + 38d , c  + 33b*c*d - 18c d + 3a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2       2      3
│ │ │ -      + 32b*d  + 3c*d  + 30d )
│ │ │ +             2        2     3
│ │ │ +      + 37b*d  + 46c*d  - 8d )
│ │ │  
│ │ │  o61 : Ideal of S
    │ │ │
    │ │ │ 
    i62 : betti res I0
│ │ │ @@ -878,35 +915,41 @@
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i64 : netList decompose I0
│ │ │  
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -o64 = |ideal (c - 2d, b - 34d, a + 26d)                                                             |
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -      |ideal (c - 40d, b - 45d, a - 28d)                                                            |
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -      |                                     2                     2   2      2                    2 |
│ │ │ -      |ideal (a + 15b + 39c - 41d, b*c - 15c  - 50b*d + 6c*d - 44d , b  - 18c  + 18b*d + 4c*d - 5d )|
│ │ │ -      +---------------------------------------------------------------------------------------------+
    │ │ │ + +-------------------------------------------------------------------------------------------------------+ │ │ │ +o64 = |ideal (c - 40d, b - 10d, a + 32d) | │ │ │ + +-------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 | │ │ │ + |ideal (b + 10c + 25d, a + 27c - 50d, c - 34c*d - 17d ) | │ │ │ + +-------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a - 13b - 2c + 29d, c - 5b*d - 20c*d + 8d , b*c + 40b*d + 16c*d + 8d , b - b*d + 15c*d + 40d )| │ │ │ + +-------------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │ 
    i65 : netList decompose I1
│ │ │  
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                    2              2                                                                                                                                                                  |
│ │ │ -o65 = |ideal (b + 24c - 6d, a - 39c - 6d, c  + 44c*d - 11d )                                                                                                                                                                 |
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                              2                                  2                                   2   2                             2                                  2   2                            2 |
│ │ │ -      |ideal (c  + 17a*d - 13b*d + 32c*d - 46d , b*c + 41a*d - 8b*d + 19c*d + 12d , a*c - 46a*d - 47b*d - 43c*d - 37d , b  + 15a*d - 33b*d + 2c*d - 22d , a*b + 31a*d - 36b*d + 7c*d + 46d , a  + 46a*d - 48b*d + 6c*d - 3d )|
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    │ │ │ + +------------------------------------------------------+ │ │ │ +o65 = |ideal (c + 32d, b + 18d, a - 33d) | │ │ │ + +------------------------------------------------------+ │ │ │ + |ideal (c + 29d, b - 8d, a + 50d) | │ │ │ + +------------------------------------------------------+ │ │ │ + |ideal (c + 16d, b + 39d, a - 32d) | │ │ │ + +------------------------------------------------------+ │ │ │ + |ideal (c + 5d, b - 14d, a + 7d) | │ │ │ + +------------------------------------------------------+ │ │ │ + | 2 2 | │ │ │ + |ideal (b - 40c + 5d, a - 47c + 24d, c - 27c*d + 15d )| │ │ │ + +------------------------------------------------------+ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i66 : L430 = (trim minors(2, M1)) + groebnerStratum F;
│ │ │ ├── html2text {}
│ │ │ │ @@ -251,15 +251,15 @@
│ │ │ │        |      31         33        32       34        35        36    |
│ │ │ │        +--------------------------------------------------------------+
│ │ │ │  i21 : L = trim groebnerStratum F;
│ │ │ │  
│ │ │ │  o21 : Ideal of T
│ │ │ │  i22 : assert(dim L == 18)
│ │ │ │  i23 : elapsedTime isPrime L
│ │ │ │ - -- 3.08156s elapsed
│ │ │ │ + -- 2.54797s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  ********** TThhee SScchhrreeyyeerr rreessoolluuttiioonn aanndd mmiinniimmaall BBeettttii nnuummbbeerrss **********
│ │ │ │  Schreyer's construction of a nonminimal free resolution starts with a Groebner
│ │ │ │  basis. First, one constructs the SScchhrreeyyeerr ffrraammee (see La Scala, Stillman). This
│ │ │ │  is determined solely from the initial ideal $J$ and its minimal generators (but
│ │ │ │  depends on some choices of ordering, but otherwise is combinatorial). This
│ │ │ │ @@ -415,15 +415,15 @@
│ │ │ │  We now compute the locus in $V(L)$ where the Betti diagram has no cancellation.
│ │ │ │  This is a closed subscheme of $V(L)$, which is a closed subscheme of the
│ │ │ │  Hilbert scheme. Notice that there are two components.
│ │ │ │  i39 : L441 = trim(L + ideal M1);
│ │ │ │  
│ │ │ │  o39 : Ideal of T
│ │ │ │  i40 : elapsedTime compsL441 = decompose L441;
│ │ │ │ - -- 1.69883s elapsed
│ │ │ │ + -- 1.44897s elapsed
│ │ │ │  i41 : #compsL441
│ │ │ │  
│ │ │ │  o41 = 2
│ │ │ │  i42 : compsL441/dim -- two components, of dimensions 14 and 16.
│ │ │ │  
│ │ │ │  o42 = {16, 14}
│ │ │ │  
│ │ │ │ @@ -431,36 +431,36 @@
│ │ │ │  i43 : compsL441/dim == {16, 14}
│ │ │ │  
│ │ │ │  o43 = true
│ │ │ │  Both components are rational, and here are random points, one on each
│ │ │ │  component:
│ │ │ │  i44 : pta = randomPointOnRationalVariety compsL441_0
│ │ │ │  
│ │ │ │ -o44 = | 26 9 43 -2 -49 13 -30 -49 -47 50 -31 -25 -16 -50 -23 -19 20 19 -13 38
│ │ │ │ +o44 = | -40 -4 -40 44 -22 -50 -30 13 -1 -16 45 -23 29 19 14 23 -21 19 14 29 6
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -20 -32 19 26 -8 49 45 10 37 18 34 -29 -10 -28 5 15 |
│ │ │ │ +      10 -32 18 -22 37 34 15 5 -10 -29 26 49 -50 45 -28 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o44 : Matrix kk  <-- kk
│ │ │ │  i45 : Fa = sub(F, (vars S) | pta)
│ │ │ │  
│ │ │ │                2              2                              2
│ │ │ │ -o45 = ideal (a  - 23b*c + 50c  - 25a*d + 13b*d + 43c*d + 26d , a*b - 8b*c -
│ │ │ │ +o45 = ideal (a  + 14b*c - 16c  - 23a*d - 50b*d - 40c*d - 40d , a*b - 22b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                             2   2             2
│ │ │ │ -      13c  - 20a*d - 16b*d - 47c*d + 9d , b  + 5b*c + 10c  - 10a*d - 32b*d -
│ │ │ │ +         2                          2   2              2
│ │ │ │ +      14c  + 6a*d + 29b*d - c*d - 4d , b  + 45b*c + 15c  + 49a*d + 10b*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 2                   2                              2     2
│ │ │ │ -      19c*d - 30d , a*c - 29b*c + 49c  + 37a*d + 38b*d - 50c*d - 49d , b*c  +
│ │ │ │ +                 2                   2                             2     2
│ │ │ │ +      23c*d - 30d , a*c + 26b*c + 37c  + 5a*d + 29b*d + 19c*d - 22d , b*c  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2         2        2        2     3   3                2
│ │ │ │ -      18b*c*d + 19c d + 45a*d  + 20b*d  - 31c*d  - 2d , c  + 15b*c*d + 34c d
│ │ │ │ +                   2         2        2        2      3   3                2
│ │ │ │ +      10b*c*d - 32c d + 34a*d  - 21b*d  + 45c*d  + 44d , c  - 28b*c*d - 29c d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │               2        2        2      3
│ │ │ │ -      - 28a*d  + 26b*d  + 19c*d  - 49d )
│ │ │ │ +      - 50a*d  + 18b*d  + 19c*d  + 13d )
│ │ │ │  
│ │ │ │  o45 : Ideal of S
│ │ │ │  i46 : betti res Fa
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o46 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │ @@ -468,172 +468,256 @@
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o46 : BettiTally
│ │ │ │  i47 : netList decompose Fa -- this one is 5 points on a plane, and another
│ │ │ │  point
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ ------------+
│ │ │ │ -o47 = |ideal (c + 37d, b - 33d, a - 11d)
│ │ │ │ +------------------------------------+
│ │ │ │ +o47 = |ideal (c + 5d, b - 33d, a - 21d)
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ ------------+
│ │ │ │ -      |                             2             2                      2   3
│ │ │ │ -2         2        2     3     2                2         2        2     3 |
│ │ │ │ -      |ideal (a - 29b + 49c - 45d, b  + 5b*c + 10c  - 19b*d - 34c*d + 25d , c
│ │ │ │ -+ 15b*c*d + 34c d + 22b*d  - 23c*d  + 4d , b*c  + 18b*c*d + 19c d + 12b*d  -
│ │ │ │ -14c*d  + 3d )|
│ │ │ │ +------------------------------------+
│ │ │ │ +      |                                      2              2
│ │ │ │ +|
│ │ │ │ +      |ideal (b + 45c + 49d, a - 22c - 26d, c  + 49c*d + 42d )
│ │ │ │ +|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ ------------+
│ │ │ │ +------------------------------------+
│ │ │ │ +      |                             2                      2
│ │ │ │ +2   2                     2 |
│ │ │ │ +      |ideal (a + 26b + 37c + 36d, c  - 21b*d + 43c*d + 27d , b*c - 30b*d +
│ │ │ │ +16c*d + 26d , b  - 3b*d - 24c*d - 36d )|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +------------------------------------+
│ │ │ │  i48 : CFa = minimalPrimes Fa
│ │ │ │  
│ │ │ │ -                                                                       2
│ │ │ │ -o48 = {ideal (c + 37d, b - 33d, a - 11d), ideal (a - 29b + 49c - 45d, b  +
│ │ │ │ +
│ │ │ │ +o48 = {ideal (c + 5d, b - 33d, a - 21d), ideal (b + 45c + 49d, a - 22c - 26d,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                2                      2   3                2         2
│ │ │ │ -      5b*c + 10c  - 19b*d - 34c*d + 25d , c  + 15b*c*d + 34c d + 22b*d  -
│ │ │ │ +       2              2                                2
│ │ │ │ +      c  + 49c*d + 42d ), ideal (a + 26b + 37c + 36d, c  - 21b*d + 43c*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -           2     3     2                2         2        2     3
│ │ │ │ -      23c*d  + 4d , b*c  + 18b*c*d + 19c d + 12b*d  - 14c*d  + 3d )}
│ │ │ │ +         2                           2   2                     2
│ │ │ │ +      27d , b*c - 30b*d + 16c*d + 26d , b  - 3b*d - 24c*d - 36d )}
│ │ │ │  
│ │ │ │  o48 : List
│ │ │ │  i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │ │  
│ │ │ │ -o49 = a - 29b + 49c - 45d
│ │ │ │ +o49 = b + 45c + 49d
│ │ │ │  
│ │ │ │  o49 : S
│ │ │ │  i50 : CFa/degree
│ │ │ │  
│ │ │ │ -o50 = {1, 5}
│ │ │ │ +o50 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o50 : List
│ │ │ │  i51 : CFa/(I -> lin % I == 0) -- so 5 points on the plane.
│ │ │ │  
│ │ │ │ -o51 = {false, true}
│ │ │ │ +o51 = {false, true, false}
│ │ │ │  
│ │ │ │  o51 : List
│ │ │ │  i52 : degree(Fa : (Fa : lin))  -- somewhat simpler(?) way to see the ideal of
│ │ │ │  the 5 points
│ │ │ │  
│ │ │ │ -o52 = 5
│ │ │ │ +o52 = 2
│ │ │ │  i53 : ptb = randomPointOnRationalVariety compsL441_1
│ │ │ │  
│ │ │ │ -o53 = | 41 19 21 -24 23 -10 -39 29 42 -26 -46 29 -34 21 23 12 -42 21 -9 17
│ │ │ │ +o53 = | 31 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -30 -49 2 9 -28 -13 -9 38 -28 -23 -37 -29 0 -47 -4 47 |
│ │ │ │ +      -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o53 : Matrix kk  <-- kk
│ │ │ │  i54 : Fb = sub(F, (vars S) | ptb)
│ │ │ │  
│ │ │ │ -              2              2                              2
│ │ │ │ -o54 = ideal (a  + 23b*c - 26c  + 29a*d - 10b*d + 21c*d + 41d , a*b - 28b*c -
│ │ │ │ +              2              2                             2
│ │ │ │ +o54 = ideal (a  + 50b*c - 42c  - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -        2                              2   2             2
│ │ │ │ -      9c  - 30a*d - 34b*d + 42c*d + 19d , b  - 4b*c + 38c  - 49b*d + 12c*d -
│ │ │ │ +         2                             2   2             2
│ │ │ │ +      42c  + 13a*d + 14b*d - 3c*d + 42d , b  - 9b*c - 29c  + 31b*d + 5c*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                   2                              2     2
│ │ │ │ -      39d , a*c - 29b*c - 13c  - 28a*d + 17b*d + 21c*d + 23d , b*c  - 23b*c*d
│ │ │ │ +         2                 2                             2     2
│ │ │ │ +      43d , a*c + 9b*c - 4c  - 23a*d + 47b*d + 2c*d + 19d , b*c  + 21b*c*d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -          2        2        2        2      3   3                2         2
│ │ │ │ -      + 2c d - 9a*d  - 42b*d  - 46c*d  - 24d , c  + 47b*c*d - 37c d - 47a*d
│ │ │ │ +         2         2        2        2      3   3                2         2
│ │ │ │ +      37c d + 38a*d  + 36b*d  - 29c*d  + 25d , c  - 47b*c*d + 17c d + 21a*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -            2        2      3
│ │ │ │ -      + 9b*d  + 21c*d  + 29d )
│ │ │ │ +             2        2     3
│ │ │ │ +      - 23b*d  - 13c*d  - 7d )
│ │ │ │  
│ │ │ │  o54 : Ideal of S
│ │ │ │  i55 : betti res Fb
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o55 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │            1: . 4 4 1
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o55 : BettiTally
│ │ │ │  i56 : netList decompose Fb --
│ │ │ │  
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +o56 = |ideal (c - 45d, b + 16d, a + 38d)                      |
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +      |ideal (c + 43d, b + 10d, a + 8d)                       |
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +      |ideal (c + 34d, b + 15d, a + 28d)                      |
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +      |ideal (c + 11d, b + 39d, a + 23d)                      |
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +      |                                      2              2 |
│ │ │ │ +      |ideal (b - 32c + 42d, a - 19c - 16d, c  - 28c*d - 40d )|
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │ │ +
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ------------------------------------------------------+
│ │ │ │ -      |                                     2             2
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |                          2                      2
│ │ │ │ +2   2                      2
│ │ │ │  |
│ │ │ │ -o56 = |ideal (b - 28c - 28d, a + 35c + 6d, c  + 32c*d - 7d )
│ │ │ │ +o57 = |ideal (a - 7b + 32c + d, c  + 42b*d + 33c*d - 10d , b*c - b*d + 13c*d +
│ │ │ │ +18d , b  + 28b*d - 32c*d + 16d )
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ------------------------------------------------------+
│ │ │ │ -      |        2                     2                                 2
│ │ │ │ -2   2                           2                                   2   2
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |                          2                      2
│ │ │ │ +2   2                      2
│ │ │ │ +|
│ │ │ │ +      |ideal (a - 7b + 32c + d, c  + 40b*d - 36c*d + 33d , b*c + 45b*d - 16c*d
│ │ │ │ ++ 39d , b  - 20b*d + 29c*d + 38d )
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |                          2                      2
│ │ │ │ +2   2                     2
│ │ │ │ +|
│ │ │ │ +      |ideal (a - 7b + 32c + d, c  - 10b*d + 17c*d - 21d , b*c - 17b*d - 23c*d
│ │ │ │ +- 32d , b  - 8b*d - 12c*d - 46d )
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |                                     3      2         2      3
│ │ │ │ +|
│ │ │ │ +      |ideal (b + 23c - 11d, a - 9c + 25d, c  - 13c d - 14c*d  + 23d )
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |                                     2                      2   2      2
│ │ │ │ +2   3      2         2        2      3
│ │ │ │ +|
│ │ │ │ +      |ideal (a + 48b - 40c - 20d, b*c - 32c  + 43b*d - 21c*d - 12d , b  - 14c
│ │ │ │ ++ 14b*d + 18c*d + 36d , c  + 28c d - 20b*d  + 42c*d  - 50d )
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |                                   2                      2   2      2
│ │ │ │ +2   3      2         2        2     3
│ │ │ │ +|
│ │ │ │ +      |ideal (a + b + 50c + 26d, b*c - 32c  + 34b*d - 36c*d + 14d , b  - 14c  +
│ │ │ │ +34b*d - 16c*d - 33d , c  + 28c d + 39b*d  - 28c*d  + 4d )
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |        2                            2                                 2
│ │ │ │ +2   2                              2                                  2   2
│ │ │ │  2 |
│ │ │ │ -      |ideal (c  - 8a*d - 49b*d - 33d , b*c - 24a*d + b*d - 49c*d - 43d , a*c -
│ │ │ │ -20a*d + 15b*d + 14c*d - 37d , b  + 6a*d - b*d + 18c*d + 33d , a*b + 34a*d -
│ │ │ │ -43b*d - 17c*d + 33d , a  - 31a*d + 6b*d + 37c*d - 30d )|
│ │ │ │ +      |ideal (c  - 7a*d - 19b*d + 6c*d - 19d , b*c - 5a*d + 49b*d - 4c*d + 50d
│ │ │ │ +, a*c - 6a*d + 35b*d - 39c*d - 2d , b  - 46a*d + 22b*d + 42c*d + 43d , a*b +
│ │ │ │ +3a*d - 12b*d - 49c*d + 40d , a  + 28a*d - 13b*d - 25c*d - 35d )|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ------------------------------------------------------+
│ │ │ │ -i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │ │ -
│ │ │ │ -o57 = ++
│ │ │ │ -      ++
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |                                     2                     2   2      2
│ │ │ │ +2   3      2         2        2     3
│ │ │ │ +|
│ │ │ │ +      |ideal (a - 46b + 39c - 29d, b*c - 32c  + 11b*d - 7c*d - 43d , b  - 14c
│ │ │ │ ++ 29b*d + 43c*d - 41d , c  + 28c d + 46b*d  - 50c*d  - 5d )
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |        2                              2
│ │ │ │ +2                                   2   2                              2
│ │ │ │ +2   2                           2  |
│ │ │ │ +      |ideal (c  + 15a*d + 27b*d + 35c*d + 46d , b*c - 6a*d + b*d + 36c*d - 31d
│ │ │ │ +, a*c - 10a*d + 45b*d + 20c*d - 23d , b  - 23a*d + 15b*d + 31c*d - 13d , a*b -
│ │ │ │ +6a*d - 40b*d + 8c*d + 18d , a  - 8a*d - 24b*d + c*d - 22d ) |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │ +      |        2                              2
│ │ │ │ +2                                  2   2                      2
│ │ │ │ +2   2                              2      |
│ │ │ │ +      |ideal (c  + 37a*d + 25b*d - 16c*d + 14d , b*c - 7a*d + 47b*d - 3c*d - 2d
│ │ │ │ +, a*c - 14a*d + 27b*d - 35c*d - 8d , b  - 33b*d + 19c*d + 27d , a*b - 15a*d -
│ │ │ │ +30b*d - 40c*d - 24d , a  - 44a*d + 16b*d + 11c*d + 12d )     |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------+
│ │ │ │  i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │ │  
│ │ │ │ -o58 = | 17 -43 -38 -37 -19 -24 15 -23 -11 48 -17 7 45 -18 31 47 28 8 -44 -30
│ │ │ │ +o58 = | 13 -24 9 5 -49 49 36 -36 -50 8 31 -22 49 8 35 49 6 -42 32 15 -8 -24
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -3 -33 -42 -13 33 39 -21 -24 -2 -22 -23 15 -29 46 -40 18 |
│ │ │ │ +      -33 -22 28 -2 -23 18 -40 -29 15 -13 39 -18 -21 46 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o58 : Matrix kk  <-- kk
│ │ │ │  i59 : pt1 = randomPointOnRationalVariety(compsL441_1)
│ │ │ │  
│ │ │ │ -o59 = | -39 20 -45 -4 -13 35 -45 30 -48 -16 -7 -2 -6 3 3 4 47 3 -8 -18 47 8
│ │ │ │ +o59 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -35 32 -16 46 -15 21 35 37 -50 10 0 33 -14 -49 |
│ │ │ │ +      -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o59 : Matrix kk  <-- kk
│ │ │ │  We compute the ideal of the corresponding zero dimensional scheme with length
│ │ │ │  6, corresponding to the points pt0, pt1 in Hilb.
│ │ │ │  i60 : I0 = sub(sub(F, (vars ring F) | sub(pt0, ring F)), S)
│ │ │ │  
│ │ │ │ -              2              2                             2
│ │ │ │ -o60 = ideal (a  + 31b*c + 48c  + 7a*d - 24b*d - 38c*d + 17d , a*b + 33b*c -
│ │ │ │ +              2             2                             2
│ │ │ │ +o60 = ideal (a  + 35b*c + 8c  - 22a*d + 49b*d + 9c*d + 13d , a*b + 28b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │           2                             2   2              2
│ │ │ │ -      44c  - 3a*d + 45b*d - 11c*d - 43d , b  - 40b*c - 24c  - 29a*d - 33b*d +
│ │ │ │ +      32c  - 8a*d + 49b*d - 50c*d - 24d , b  - 21b*c + 18c  + 39a*d - 24b*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 2                   2                             2     2
│ │ │ │ -      47c*d + 15d , a*c + 15b*c + 39c  - 2a*d - 30b*d - 18c*d - 19d , b*c  -
│ │ │ │ +                 2                  2                             2     2
│ │ │ │ +      49c*d + 36d , a*c - 13b*c - 2c  - 40a*d + 15b*d + 8c*d - 49d , b*c  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2         2        2        2      3   3                2
│ │ │ │ -      22b*c*d - 42c d - 21a*d  + 28b*d  - 17c*d  - 37d , c  + 18b*c*d - 23c d
│ │ │ │ +                   2         2       2        2     3   3                2
│ │ │ │ +      29b*c*d - 33c d - 23a*d  + 6b*d  + 31c*d  + 5d , c  + 46b*c*d + 15c d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2        2       2      3
│ │ │ │ -      + 46a*d  - 13b*d  + 8c*d  - 23d )
│ │ │ │ +           2        2        2      3
│ │ │ │ +      18a*d  - 22b*d  - 42c*d  - 36d )
│ │ │ │  
│ │ │ │  o60 : Ideal of S
│ │ │ │  i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S)
│ │ │ │  
│ │ │ │ -              2             2                             2
│ │ │ │ -o61 = ideal (a  + 3b*c - 16c  - 2a*d + 35b*d - 45c*d - 39d , a*b - 16b*c -
│ │ │ │ +              2              2                             2
│ │ │ │ +o61 = ideal (a  - 21b*c + 45c  - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -        2                             2   2              2
│ │ │ │ -      8c  + 47a*d - 6b*d - 48c*d + 20d , b  - 14b*c + 21c  + 8b*d + 4c*d -
│ │ │ │ +         2                             2   2              2
│ │ │ │ +      17c  - 23a*d + 37b*d - 5c*d + 18d , b  - 15b*c + 10c  + 15b*d + 28c*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                   2                             2     2
│ │ │ │ -      45d , a*c + 10b*c + 46c  + 35a*d - 18b*d + 3c*d - 13d , b*c  + 37b*c*d
│ │ │ │ +         2                   2                              2     2
│ │ │ │ +      50d , a*c + 32b*c - 14c  - 31a*d - 49b*d - 35c*d + 21d , b*c  + 3b*c*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -           2         2        2       2     3   3                2         2
│ │ │ │ -      - 35c d - 15a*d  + 47b*d  - 7c*d  - 4d , c  - 49b*c*d - 50c d + 33a*d
│ │ │ │ +           2         2        2        2      3   3                2        2
│ │ │ │ +      - 50c d + 21a*d  + 27b*d  - 47c*d  + 38d , c  + 33b*c*d - 18c d + 3a*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2       2      3
│ │ │ │ -      + 32b*d  + 3c*d  + 30d )
│ │ │ │ +             2        2     3
│ │ │ │ +      + 37b*d  + 46c*d  - 8d )
│ │ │ │  
│ │ │ │  o61 : Ideal of S
│ │ │ │  i62 : betti res I0
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o62 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │ @@ -649,50 +733,45 @@
│ │ │ │            1: . 4 4 1
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o63 : BettiTally
│ │ │ │  i64 : netList decompose I0
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ ----------------------+
│ │ │ │ -o64 = |ideal (c - 2d, b - 34d, a + 26d)
│ │ │ │ +-------------------------------+
│ │ │ │ +o64 = |ideal (c - 40d, b - 10d, a + 32d)
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ ----------------------+
│ │ │ │ -      |ideal (c - 40d, b - 45d, a - 28d)
│ │ │ │ +-------------------------------+
│ │ │ │ +      |                                      2              2
│ │ │ │ +|
│ │ │ │ +      |ideal (b + 10c + 25d, a + 27c - 50d, c  - 34c*d - 17d )
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ ----------------------+
│ │ │ │ -      |                                     2                     2   2      2
│ │ │ │ -2 |
│ │ │ │ -      |ideal (a + 15b + 39c - 41d, b*c - 15c  - 50b*d + 6c*d - 44d , b  - 18c
│ │ │ │ -+ 18b*d + 4c*d - 5d )|
│ │ │ │ +-------------------------------+
│ │ │ │ +      |                            2                    2
│ │ │ │ +2   2                    2 |
│ │ │ │ +      |ideal (a - 13b - 2c + 29d, c  - 5b*d - 20c*d + 8d , b*c + 40b*d + 16c*d
│ │ │ │ ++ 8d , b  - b*d + 15c*d + 40d )|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ ----------------------+
│ │ │ │ +-------------------------------+
│ │ │ │  i65 : netList decompose I1
│ │ │ │  
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ ----------------------------------------------------------------+
│ │ │ │ -      |                                    2              2
│ │ │ │ -|
│ │ │ │ -o65 = |ideal (b + 24c - 6d, a - 39c - 6d, c  + 44c*d - 11d )
│ │ │ │ -|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ ----------------------------------------------------------------+
│ │ │ │ -      |        2                              2
│ │ │ │ -2                                   2   2                             2
│ │ │ │ -2   2                            2 |
│ │ │ │ -      |ideal (c  + 17a*d - 13b*d + 32c*d - 46d , b*c + 41a*d - 8b*d + 19c*d +
│ │ │ │ -12d , a*c - 46a*d - 47b*d - 43c*d - 37d , b  + 15a*d - 33b*d + 2c*d - 22d , a*b
│ │ │ │ -+ 31a*d - 36b*d + 7c*d + 46d , a  + 46a*d - 48b*d + 6c*d - 3d )|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ ----------------------------------------------------------------+
│ │ │ │ +      +------------------------------------------------------+
│ │ │ │ +o65 = |ideal (c + 32d, b + 18d, a - 33d)                     |
│ │ │ │ +      +------------------------------------------------------+
│ │ │ │ +      |ideal (c + 29d, b - 8d, a + 50d)                      |
│ │ │ │ +      +------------------------------------------------------+
│ │ │ │ +      |ideal (c + 16d, b + 39d, a - 32d)                     |
│ │ │ │ +      +------------------------------------------------------+
│ │ │ │ +      |ideal (c + 5d, b - 14d, a + 7d)                       |
│ │ │ │ +      +------------------------------------------------------+
│ │ │ │ +      |                                     2              2 |
│ │ │ │ +      |ideal (b - 40c + 5d, a - 47c + 24d, c  - 27c*d + 15d )|
│ │ │ │ +      +------------------------------------------------------+
│ │ │ │  i66 : L430 = (trim minors(2, M1)) + groebnerStratum F;
│ │ │ │  
│ │ │ │  o66 : Ideal of T
│ │ │ │  i67 : C = res(I, Strategy => Nonminimal)
│ │ │ │  
│ │ │ │         1      6      8      3
│ │ │ │  o67 = S  <-- S  <-- S  <-- S
│ │ ├── ./usr/share/doc/Macaulay2/RInterface/example-output/___R__Value.out
│ │ │ @@ -14,15 +14,15 @@
│ │ │  
│ │ │  o3 = [1] 120
│ │ │  
│ │ │  o3 : RObject of type double
│ │ │  
│ │ │  i4 : env = RObject hashTable {"n" => 10_ZZ, "k" => 3_ZZ}
│ │ │  
│ │ │ -o4 = 
│ │ │ +o4 = 
│ │ │  
│ │ │  o4 : RObject of type environment
│ │ │  
│ │ │  i5 : RValue("choose(n, k)", Environment => env)
│ │ │  
│ │ │  o5 = [1] 120
│ │ ├── ./usr/share/doc/Macaulay2/RInterface/example-output/_new_sp__R__Object_spfrom_sp__Hash__Table.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 5466188237138105394
│ │ │  
│ │ │  i1 : env = RObject hashTable {"x" => 5_ZZ, "y" => 2_ZZ}
│ │ │  
│ │ │ -o1 = 
│ │ │ +o1 = 
│ │ │  
│ │ │  o1 : RObject of type environment
│ │ │  
│ │ │  i2 : RValue("x^y", Environment => env)
│ │ │  
│ │ │  o2 = [1] 25
│ │ ├── ./usr/share/doc/Macaulay2/RInterface/html/___R__Value.html
│ │ │ @@ -115,15 +115,15 @@
│ │ │            
    The Environment option specifies the R environment in which to evaluate the code.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -66,24 +66,24 @@
│ │ │ │  first in rings with more variables.
│ │ │ │  i6 : S=ZZ/103[y_0..y_30];
│ │ │ │  i7 : I=minors(2,random(S^3,S^{3:-1}));
│ │ │ │  
│ │ │ │  o7 : Ideal of S
│ │ │ │  i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection,
│ │ │ │  DecompositionStrategy=>MultiplicationTable)
│ │ │ │ - -- 4.01109s elapsed
│ │ │ │ + -- 3.12569s elapsed
│ │ │ │  
│ │ │ │  o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection,
│ │ │ │  DecompositionStrategy=>Decompose)
│ │ │ │ - -- 2.98436s elapsed
│ │ │ │ + -- 2.38132s elapsed
│ │ │ │  
│ │ │ │  o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}}
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommPPooiinnttss:: **********
│ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/example-output/_inverse__Of__Map.out
│ │ │ @@ -49,15 +49,15 @@
│ │ │  i12 : Q=QQ[x,y,z,t,u];
│ │ │  
│ │ │  i13 : phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}});
│ │ │  
│ │ │  o13 : RingMap Q <-- Q
│ │ │  
│ │ │  i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
│ │ │ - -- used 0.44766s (cpu); 0.365702s (thread); 0s (gc)
│ │ │ + -- used 0.602562s (cpu); 0.405314s (thread); 0s (gc)
│ │ │  
│ │ │                                  125   124      120 5    124    100 25     104 20       108 15 2      112 10 3     116 5 4    120 5    124      125      4 120        8 115 2        12 110 3         16 105 4         20 100 5          24 95 6          28 90 7           32 85 8           36 80 9           40 75 10           44 70 11           48 65 12           52 60 13           56 55 14           60 50 15           64 45 16           68 40 17          72 35 18          76 30 19         80 25 20         84 20 21        88 15 22       92 10 23      96 5 24    100 25     24 100        28 95          32 90 2         36 85 3          40 80 4          44 75 5           48 70 6           52 65 7           56 60 8           60 55 9           64 50 10           68 45 11           72 40 12           76 35 13           80 30 14          84 25 15          88 20 16         92 15 17        96 10 18        100 5 19      104 20       48 75 2       52 70   2        56 65 2 2        60 60 3 2         64 55 4 2         68 50 5 2         72 45 6 2         76 40 7 2         80 35 8 2         84 30 9 2         88 25 10 2         92 20 11 2        96 15 12 2        100 10 13 2       104 5 14 2      108 15 2      72 50 3       76 45   3       80 40 2 3        84 35 3 3        88 30 4 3        92 25 5 3        96 20 6 3        100 15 7 3       104 10 8 3       108 5 9 3      112 10 3     96 25 4      100 20   4      104 15 2 4      108 10 3 4      112 5 4 4     116 5 4    120 5    124
│ │ │  o14 = Proj Q - - - > Proj Q   {x   , x   y, - x   y  + x   z, x   y   - 5x   y  z + 10x   y  z  - 10x   y  z  + 5x   y z  - x   z  + x   t, - y    + 25x y   z - 300x y   z  + 2300x  y   z  - 12650x  y   z  + 53130x  y   z  - 177100x  y  z  + 480700x  y  z  - 1081575x  y  z  + 2042975x  y  z  - 3268760x  y  z   + 4457400x  y  z   - 5200300x  y  z   + 5200300x  y  z   - 4457400x  y  z   + 3268760x  y  z   - 2042975x  y  z   + 1081575x  y  z   - 480700x  y  z   + 177100x  y  z   - 53130x  y  z   + 12650x  y  z   - 2300x  y  z   + 300x  y  z   - 25x  y z   + x   z   - 5x  y   t + 100x  y  z*t - 950x  y  z t + 5700x  y  z t - 24225x  y  z t + 77520x  y  z t - 193800x  y  z t + 387600x  y  z t - 629850x  y  z t + 839800x  y  z t - 923780x  y  z  t + 839800x  y  z  t - 629850x  y  z  t + 387600x  y  z  t - 193800x  y  z  t + 77520x  y  z  t - 24225x  y  z  t + 5700x  y  z  t - 950x  y  z  t + 100x   y z  t - 5x   z  t - 10x  y  t  + 150x  y  z*t  - 1050x  y  z t  + 4550x  y  z t  - 13650x  y  z t  + 30030x  y  z t  - 50050x  y  z t  + 64350x  y  z t  - 64350x  y  z t  + 50050x  y  z t  - 30030x  y  z  t  + 13650x  y  z  t  - 4550x  y  z  t  + 1050x   y  z  t  - 150x   y z  t  + 10x   z  t  - 10x  y  t  + 100x  y  z*t  - 450x  y  z t  + 1200x  y  z t  - 2100x  y  z t  + 2520x  y  z t  - 2100x  y  z t  + 1200x   y  z t  - 450x   y  z t  + 100x   y z t  - 10x   z  t  - 5x  y  t  + 25x   y  z*t  - 50x   y  z t  + 50x   y  z t  - 25x   y z t  + 5x   z t  - x   t  + x   u}
│ │ │  
│ │ │  o14 : RationalMapping
│ │ │  
│ │ │  i15 : R=QQ[x,y,z,t]/(z-2*t);
│ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/html/_inverse__Of__Map.html
│ │ │ @@ -194,15 +194,15 @@
│ │ │  
│ │ │  o13 : RingMap Q <-- Q
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -94,15 +94,15 @@
│ │ │ │  o11 : Ideal of blowUpSubvar
│ │ │ │  The next example is a birational map on $\mathbb{P}^4$.
│ │ │ │  i12 : Q=QQ[x,y,z,t,u];
│ │ │ │  i13 : phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}});
│ │ │ │  
│ │ │ │  o13 : RingMap Q <-- Q
│ │ │ │  i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
│ │ │ │ - -- used 0.44766s (cpu); 0.365702s (thread); 0s (gc)
│ │ │ │ + -- used 0.602562s (cpu); 0.405314s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                  125   124      120 5    124    100 25     104
│ │ │ │  20       108 15 2      112 10 3     116 5 4    120 5    124      125      4 120
│ │ │ │  8 115 2        12 110 3         16 105 4         20 100 5          24 95 6
│ │ │ │  28 90 7           32 85 8           36 80 9           40 75 10           44 70
│ │ │ │  11           48 65 12           52 60 13           56 55 14           60 50 15
│ │ │ │  64 45 16           68 40 17          72 35 18          76 30 19         80 25
│ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_rational__Points.out
│ │ │ @@ -48,15 +48,15 @@
│ │ │               0    1    2    3    4    5    6    7    8    9    10
│ │ │  
│ │ │                  ZZ
│ │ │  o13 : Ideal of ---[u ..u  ]
│ │ │                 101  0   10
│ │ │  
│ │ │  i14 : time rationalPoints(I, Amount => true)
│ │ │ - -- used 0.00313113s (cpu); 0.0031276s (thread); 0s (gc)
│ │ │ + -- used 0.0038744s (cpu); 0.00387301s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 110462212541120451001
│ │ │  
│ │ │  i15 : QQ[x,y,z]; I = homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)), z);
│ │ │  
│ │ │  o16 : Ideal of QQ[x..z]
│ │ │  
│ │ │ @@ -142,23 +142,23 @@
│ │ │  
│ │ │  i31 : nodes = I + ideal jacobian I;
│ │ │  
│ │ │  o31 : Ideal of R
│ │ │  
│ │ │  i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
│ │ │ - -- used 0.983967s (cpu); 0.775123s (thread); 0s (gc)
│ │ │ + -- used 1.12634s (cpu); 0.899099s (thread); 0s (gc)
│ │ │  
│ │ │  i33 : #oo
│ │ │  
│ │ │  o33 = 31
│ │ │  
│ │ │  i34 : nodes' = baseChange_32003 nodes;
│ │ │  
│ │ │  o34 : Ideal of GF 1048969271299456081[x..z, w]
│ │ │  
│ │ │  i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ - -- used 0.28051s (cpu); 0.202638s (thread); 0s (gc)
│ │ │ + -- used 0.347821s (cpu); 0.251987s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 31
│ │ │  
│ │ │  i36 :
│ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/html/_rational__Points.html
│ │ │ @@ -183,15 +183,15 @@
│ │ │  o13 : Ideal of ---[u ..u  ]
│ │ │                 101  0   10
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : env = RObject hashTable {"n" => 10_ZZ, "k" => 3_ZZ}
│ │ │  
│ │ │ -o4 = <environment: 0x7faf8315e6b0>
│ │ │ +o4 = <environment: 0x7fc7bf15ff90>
│ │ │  
│ │ │  o4 : RObject of type environment
    
│ │ │              
    
│ │ │                
    i5 : RValue("choose(n, k)", Environment => env)
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,15 +32,15 @@
│ │ │ │  o3 = [1] 120
│ │ │ │  
│ │ │ │  o3 : RObject of type double
│ │ │ │  The Environment option specifies the R environment in which to evaluate the
│ │ │ │  code.
│ │ │ │  i4 : env = RObject hashTable {"n" => 10_ZZ, "k" => 3_ZZ}
│ │ │ │  
│ │ │ │ -o4 = 
│ │ │ │ +o4 = 
│ │ │ │  
│ │ │ │  o4 : RObject of type environment
│ │ │ │  i5 : RValue("choose(n, k)", Environment => env)
│ │ │ │  
│ │ │ │  o5 = [1] 120
│ │ │ │  
│ │ │ │  o5 : RObject of type double
│ │ ├── ./usr/share/doc/Macaulay2/RInterface/html/_new_sp__R__Object_spfrom_sp__Hash__Table.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │            
    Converts a Macaulay2 HashTable with String keys to an R environment, with each key-value pair becoming a variable binding.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : env = RObject hashTable {"x" => 5_ZZ, "y" => 2_ZZ}
│ │ │  
│ │ │ -o1 = <environment: 0x7faf82a90c00>
│ │ │ +o1 = <environment: 0x7fc7bea924e0>
│ │ │  
│ │ │  o1 : RObject of type environment
    
│ │ │              
    
│ │ │                
    i2 : RValue("x^y", Environment => env)
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _R_ _o_b_j_e_c_t, an R environment;
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Converts a Macaulay2 _H_a_s_h_T_a_b_l_e with _S_t_r_i_n_g keys to an R eennvviirroonnmmeenntt, with each
│ │ │ │  key-value pair becoming a variable binding.
│ │ │ │  i1 : env = RObject hashTable {"x" => 5_ZZ, "y" => 2_ZZ}
│ │ │ │  
│ │ │ │ -o1 = 
│ │ │ │ +o1 = 
│ │ │ │  
│ │ │ │  o1 : RObject of type environment
│ │ │ │  i2 : RValue("x^y", Environment => env)
│ │ │ │  
│ │ │ │  o2 = [1] 25
│ │ │ │  
│ │ │ │  o2 : RObject of type double
│ │ ├── ./usr/share/doc/Macaulay2/RandomCanonicalCurves/example-output/_canonical__Curve.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  i2 : g=14;
│ │ │  
│ │ │  i3 : FF=ZZ/10007;
│ │ │  
│ │ │  i4 : R=FF[x_0..x_(g-1)];
│ │ │  
│ │ │  i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 8.43057s (cpu); 5.80702s (thread); 0s (gc)
│ │ │ + -- used 7.63019s (cpu); 6.01276s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/_canonical__Curve.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │              
    		
│ │ │                
    i4 : R=FF[x_0..x_(g-1)];
    
│ │ │              
    
│ │ │                
    i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 8.43057s (cpu); 5.80702s (thread); 0s (gc)
│ │ │ + -- used 7.63019s (cpu); 6.01276s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  unirationality of $M_g$ by Severi, Sernesi, Chang-Ran and Verra.
│ │ │ │  i1 : setRandomSeed "alpha";
│ │ │ │   -- setting random seed to 10206284518
│ │ │ │  i2 : g=14;
│ │ │ │  i3 : FF=ZZ/10007;
│ │ │ │  i4 : R=FF[x_0..x_(g-1)];
│ │ │ │  i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ │ - -- used 8.43057s (cpu); 5.80702s (thread); 0s (gc)
│ │ │ │ + -- used 7.63019s (cpu); 6.01276s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              0  1
│ │ │ │  o5 = total: 1 66
│ │ │ │           0: 1  .
│ │ │ │           1: . 66
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out
│ │ │ @@ -7,42 +7,42 @@
│ │ │  
│ │ │  o2 = (10, 20)
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │  
│ │ │ -o3 = ({5, 2.91596e52, 9}, .00191125, .000840977)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00183205, .000824684)
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │  i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .0052666, .0352226)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .00461786, .058447)
│ │ │  
│ │ │  o4 : Sequence
│ │ │  
│ │ │  i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {(.00647943, .0123131), (.00529175, .0041851), (.00644371, .00670157),
│ │ │ +o5 = {(.00569843, .0143546), (.00848308, .00822127), (.00888698, .0077662),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00536156, .00990137), (.0052383, .0132064), (.00592282, .0124077),
│ │ │ +     (.00643537, .0111378), (.00781646, .0153473), (.00714585, .0151402),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00501436, .00818598), (.0053675, .00745461), (.00461355, .00533369),
│ │ │ +     (.0064064, .00921267), (.00949573, .0126932), (.00507532, .00613987),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00571401, .00832175)}
│ │ │ +     (.00636421, .00912886)}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .005544699000000009
│ │ │ +o6 = .00718078290000026
│ │ │  
│ │ │  o6 : RR (of precision 53)
│ │ │  
│ │ │  i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .008801128499999899
│ │ │ +o7 = .01091419569999994
│ │ │  
│ │ │  o7 : RR (of precision 53)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html
│ │ │ @@ -98,57 +98,57 @@
│ │ │  o2 : Sequence
│ │ │              
    
│ │ │                
    i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │  
│ │ │ -o3 = ({5, 2.91596e52, 9}, .00191125, .000840977)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00183205, .000824684)
│ │ │  
│ │ │  o3 : Sequence
    
│ │ │              
    
│ │ │                
    i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .0052666, .0352226)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .00461786, .058447)
│ │ │  
│ │ │  o4 : Sequence
    
│ │ │              
    
│ │ │                
    i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {(.00647943, .0123131), (.00529175, .0041851), (.00644371, .00670157),
│ │ │ +o5 = {(.00569843, .0143546), (.00848308, .00822127), (.00888698, .0077662),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00536156, .00990137), (.0052383, .0132064), (.00592282, .0124077),
│ │ │ +     (.00643537, .0111378), (.00781646, .0153473), (.00714585, .0151402),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00501436, .00818598), (.0053675, .00745461), (.00461355, .00533369),
│ │ │ +     (.0064064, .00921267), (.00949573, .0126932), (.00507532, .00613987),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00571401, .00832175)}
│ │ │ +     (.00636421, .00912886)}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .005544699000000009
│ │ │ +o6 = .00718078290000026
│ │ │  
│ │ │  o6 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .008801128499999899
│ │ │ +o7 = .01091419569999994
│ │ │  
│ │ │  o7 : RR (of precision 53)
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,41 +25,41 @@
│ │ │ │  i2 : r=10,n=20
│ │ │ │  
│ │ │ │  o2 = (10, 20)
│ │ │ │  
│ │ │ │  o2 : Sequence
│ │ │ │  i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │ │  
│ │ │ │ -o3 = ({5, 2.91596e52, 9}, .00191125, .000840977)
│ │ │ │ +o3 = ({5, 2.91596e52, 9}, .00183205, .000824684)
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │ │  
│ │ │ │ -o4 = ({50, 2.30853e454, 98}, .0052666, .0352226)
│ │ │ │ +o4 = ({50, 2.30853e454, 98}, .00461786, .058447)
│ │ │ │  
│ │ │ │  o4 : Sequence
│ │ │ │  i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │ │  
│ │ │ │ -o5 = {(.00647943, .0123131), (.00529175, .0041851), (.00644371, .00670157),
│ │ │ │ +o5 = {(.00569843, .0143546), (.00848308, .00822127), (.00888698, .0077662),
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     (.00536156, .00990137), (.0052383, .0132064), (.00592282, .0124077),
│ │ │ │ +     (.00643537, .0111378), (.00781646, .0153473), (.00714585, .0151402),
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     (.00501436, .00818598), (.0053675, .00745461), (.00461355, .00533369),
│ │ │ │ +     (.0064064, .00921267), (.00949573, .0126932), (.00507532, .00613987),
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     (.00571401, .00832175)}
│ │ │ │ +     (.00636421, .00912886)}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : 1/10*sum(L,t->t_0)
│ │ │ │  
│ │ │ │ -o6 = .005544699000000009
│ │ │ │ +o6 = .00718078290000026
│ │ │ │  
│ │ │ │  o6 : RR (of precision 53)
│ │ │ │  i7 : 1/10*sum(L,t->t_1)
│ │ │ │  
│ │ │ │ -o7 = .008801128499999899
│ │ │ │ +o7 = .01091419569999994
│ │ │ │  
│ │ │ │  o7 : RR (of precision 53)
│ │ │ │  ********** WWaayyss ttoo uussee tteessttTTiimmeeFFoorrLLLLLLoonnSSyyzzyyggiieess:: **********
│ │ │ │      * testTimeForLLLonSyzygies(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _t_e_s_t_T_i_m_e_F_o_r_L_L_L_o_n_S_y_z_y_g_i_e_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/_smooth__Canonical__Curve.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 11549527689790345152
│ │ │  
│ │ │  i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ - -- used 1.49274s (cpu); 1.22731s (thread); 0s (gc)
│ │ │ + -- used 1.41087s (cpu); 1.21411s (thread); 0s (gc)
│ │ │  
│ │ │                ZZ
│ │ │  o1 : Ideal of --[t ..t  ]
│ │ │                 5  0   10
│ │ │  
│ │ │  i2 : (dim ICan, genus ICan, degree ICan)
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │            
    If the option Printing is set to true then printings about the current step in the construction are displayed.
    
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  For g<=10 the curves are constructed via plane models.
│ │ │ │  For g<=13 the curves are constructed via space models.
│ │ │ │  For g=14 the curves are constructed by Verra's method.
│ │ │ │  For g=15 the curves are constructed via matrix factorizations.
│ │ │ │  If the option Printing is set to true then printings about the current step in
│ │ │ │  the construction are displayed.
│ │ │ │  i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ │ - -- used 1.49274s (cpu); 1.22731s (thread); 0s (gc)
│ │ │ │ + -- used 1.41087s (cpu); 1.21411s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o1 : Ideal of --[t ..t  ]
│ │ │ │                 5  0   10
│ │ │ │  i2 : (dim ICan, genus ICan, degree ICan)
│ │ │ │  
│ │ │ │  o2 = (2, 11, 20)
│ │ ├── ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Curve__Genus14__Degree18in__P6.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │   -- setting random seed to 10206284518
│ │ │  
│ │ │  i2 : FF=ZZ/10007;
│ │ │  
│ │ │  i3 : S=FF[x_0..x_6];
│ │ │  
│ │ │  i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ - -- used 1.75282s (cpu); 1.39315s (thread); 0s (gc)
│ │ │ + -- used 1.64553s (cpu); 1.42273s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S
│ │ │  
│ │ │  i5 : betti res I
│ │ │  
│ │ │              0  1  2  3  4 5
│ │ │  o5 = total: 1 13 45 56 25 2
│ │ ├── ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Curve__Genus14__Degree18in__P6.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ - -- used 1.49274s (cpu); 1.22731s (thread); 0s (gc)
│ │ │ + -- used 1.41087s (cpu); 1.21411s (thread); 0s (gc)
│ │ │  
│ │ │                ZZ
│ │ │  o1 : Ideal of --[t ..t  ]
│ │ │                 5  0   10
    
│ │ │              
    
│ │ │                
    i3 : S=FF[x_0..x_6];
    
│ │ │              
    
│ │ │                
    i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ - -- used 1.75282s (cpu); 1.39315s (thread); 0s (gc)
│ │ │ + -- used 1.64553s (cpu); 1.42273s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │              
    
│ │ │                
    i5 : betti res I
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  fields of the chosen finite characteristic 10007, for fields of characteristic
│ │ │ │  0 by semi-continuity, and, hence, for all but finitely many primes $p$.
│ │ │ │  i1 : setRandomSeed("alpha");
│ │ │ │   -- setting random seed to 10206284518
│ │ │ │  i2 : FF=ZZ/10007;
│ │ │ │  i3 : S=FF[x_0..x_6];
│ │ │ │  i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ │ - -- used 1.75282s (cpu); 1.39315s (thread); 0s (gc)
│ │ │ │ + -- used 1.64553s (cpu); 1.42273s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : betti res I
│ │ │ │  
│ │ │ │              0  1  2  3  4 5
│ │ │ │  o5 = total: 1 13 45 56 25 2
│ │ │ │           0: 1  .  .  .  . .
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out
│ │ │ @@ -1,24 +1,23 @@
│ │ │  -- -*- M2-comint -*- hash: 9542801742429495161
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1779108470
│ │ │ + -- setting random seed to 1779298484
│ │ │  
│ │ │ -o1 = 1779108470
│ │ │ +o1 = 1779298484
│ │ │  
│ │ │  i2 : kk=ZZ/101;
│ │ │  
│ │ │  i3 : S=kk[vars(0..5)];
│ │ │  
│ │ │  i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ - -- used 3.34044s (cpu); 1.83367s (thread); 0s (gc)
│ │ │ + -- used 3.50494s (cpu); 1.98883s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 39 }
│ │ │ -           5 => 200
│ │ │ -           6 => 184
│ │ │ -           7 => 63
│ │ │ -           8 => 12
│ │ │ -           9 => 2
│ │ │ +o4 = Tally{4 => 46 }
│ │ │ +           5 => 190
│ │ │ +           6 => 179
│ │ │ +           7 => 72
│ │ │ +           8 => 13
│ │ │  
│ │ │  o4 : Tally
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 5959465567197821046
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1779108477
│ │ │ + -- setting random seed to 1779298488
│ │ │  
│ │ │ -o1 = 1779108477
│ │ │ +o1 = 1779298488
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -15,13 +15,13 @@
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -        2
│ │ │ -o4 = a*b
│ │ │ +      3
│ │ │ +o4 = a
│ │ │  
│ │ │  o4 : S
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 8876340562021865447
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1779108488
│ │ │ + -- setting random seed to 1779298496
│ │ │  
│ │ │ -o1 = 1779108488
│ │ │ +o1 = 1779298496
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -22,18 +22,18 @@
│ │ │  o4 = {3, 5, 7}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │  low degree gens generated everything
│ │ │  
│ │ │ -o5 = ideal(a*b*e)
│ │ │ +o5 = ideal(a*c*d)
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (a*c, d*e, b*c, a*e, b*d)
│ │ │ +o6 = ideal (c*e, c*d, b*e, a*c, a*e)
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 10504911213508281315
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1779108483
│ │ │ + -- setting random seed to 1779298492
│ │ │  
│ │ │ -o1 = 1779108483
│ │ │ +o1 = 1779298492
│ │ │  
│ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │ @@ -15,17 +15,15 @@
│ │ │  
│ │ │  o3 = monomialIdeal (a*b, a*d, b*c*d)
│ │ │  
│ │ │  o3 : MonomialIdeal of S
│ │ │  
│ │ │  i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b*c, a*d, b*c*d), {a*b*c, a*d, b*c*d}, {c*d, b*d, b*c,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     a*c, a*b}}
│ │ │ +o4 = {monomialIdeal (a*b, a*d, b*d), {a*b, a*d, b*d}, {c*d, b*c, a*c}}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : setRandomSeed(1)
│ │ │   -- setting random seed to 1
│ │ │  
│ │ │  o5 = 1
│ │ │ @@ -39,15 +37,15 @@
│ │ │  i7 : J = monomialIdeal 0_S
│ │ │  
│ │ │  o7 = monomialIdeal ()
│ │ │  
│ │ │  o7 : MonomialIdeal of S
│ │ │  
│ │ │  i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 4.27518s (cpu); 2.8639s (thread); 0s (gc)
│ │ │ + -- used 4.57332s (cpu); 3.23803s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : #T
│ │ │  
│ │ │  o9 = 168
│ │ │  
│ │ │  i10 : T
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html
│ │ │ @@ -76,17 +76,17 @@
│ │ │          
    
│ │ │            
    Chooses a random monomial.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1779108477
│ │ │ + -- setting random seed to 1779298488
│ │ │  
│ │ │ -o1 = 1779108477
    
│ │ │ +o1 = 1779298488
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -103,16 +103,16 @@
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -        2
│ │ │ -o4 = a*b
│ │ │ +      3
│ │ │ +o4 = a
│ │ │  
│ │ │  o4 : S
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,31 +11,31 @@
│ │ │ │            o d, an _i_n_t_e_g_e_r, non-negative
│ │ │ │            o S, a _r_i_n_g, polynomial ring
│ │ │ │      * Outputs:
│ │ │ │            o m, a _r_i_n_g_ _e_l_e_m_e_n_t, monomial of S
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Chooses a random monomial.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1779108477
│ │ │ │ + -- setting random seed to 1779298488
│ │ │ │  
│ │ │ │ -o1 = 1779108477
│ │ │ │ +o1 = 1779298488
│ │ │ │  i2 : kk=ZZ/101
│ │ │ │  
│ │ │ │  o2 = kk
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : S=kk[a,b,c]
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : randomMonomial(3,S)
│ │ │ │  
│ │ │ │ -        2
│ │ │ │ -o4 = a*b
│ │ │ │ +      3
│ │ │ │ +o4 = a
│ │ │ │  
│ │ │ │  o4 : S
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_n_d_o_m_M_o_n_o_m_i_a_l_I_d_e_a_l -- random monomial ideal with given degree generators
│ │ │ │      * _r_a_n_d_o_m_S_q_u_a_r_e_F_r_e_e_M_o_n_o_m_i_a_l_I_d_e_a_l -- random square-free monomial ideal with
│ │ │ │        given degree generators
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommMMoonnoommiiaall:: **********
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Monomial__Ideal.html
│ │ │ @@ -76,17 +76,17 @@
│ │ │          
    
│ │ │            
    Choose a random square-free monomial ideal whose generators have the degrees specified by the list or sequence L.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1779108488
│ │ │ + -- setting random seed to 1779298496
│ │ │  
│ │ │ -o1 = 1779108488
    
│ │ │ +o1 = 1779298496
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -113,24 +113,24 @@
│ │ │              
    
│ │ │                
    i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │  low degree gens generated everything
│ │ │  
│ │ │ -o5 = ideal(a*b*e)
│ │ │ +o5 = ideal(a*c*d)
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (a*c, d*e, b*c, a*e, b*d)
│ │ │ +o6 = ideal (c*e, c*d, b*e, a*c, a*e)
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,17 +13,17 @@
│ │ │ │      * Outputs:
│ │ │ │            o I, an _i_d_e_a_l, square-free monomial ideal with generators of
│ │ │ │              specified degrees
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Choose a random square-free monomial ideal whose generators have the degrees
│ │ │ │  specified by the list or sequence L.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1779108488
│ │ │ │ + -- setting random seed to 1779298496
│ │ │ │  
│ │ │ │ -o1 = 1779108488
│ │ │ │ +o1 = 1779298496
│ │ │ │  i2 : kk=ZZ/101
│ │ │ │  
│ │ │ │  o2 = kk
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : S=kk[a..e]
│ │ │ │  
│ │ │ │ @@ -34,20 +34,20 @@
│ │ │ │  
│ │ │ │  o4 = {3, 5, 7}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │ │  low degree gens generated everything
│ │ │ │  
│ │ │ │ -o5 = ideal(a*b*e)
│ │ │ │ +o5 = ideal(a*c*d)
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │ │  
│ │ │ │ -o6 = ideal (a*c, d*e, b*c, a*e, b*d)
│ │ │ │ +o6 = ideal (c*e, c*d, b*e, a*c, a*e)
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The ideal is constructed degree by degree, starting from the lowest degree
│ │ │ │  specified. If there are not enough monomials of the next specified degree that
│ │ │ │  are not already in the ideal, the function prints a warning and returns an
│ │ │ │  ideal containing all the generators of that degree.
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Step.html
│ │ │ @@ -84,17 +84,17 @@
│ │ │            
    With probability p the routine takes the Alexander dual of I; the default value of p is .05, and it can be set with the option AlexanderProbility.
    
│ │ │            
    Otherwise uses the Metropolis algorithm to produce a random walk on the space of square-free ideals. Note that there are a LOT of square-free ideals; these are the Dedekind numbers, and the sequence (with 1,2,3,4,5,6,7,8 variables) begins 3,6,20,168,7581, 7828354, 2414682040998, 56130437228687557907788. (see the Online Encyclopedia of Integer Sequences for more information). Given I in a polynomial ring S, we make a list ISocgens of the square-free minimal monomial generators of the socle of S/(squares+I) and a list of minimal generators Igens of I. A candidate "next" ideal J is formed as follows: We choose randomly from the union of these lists; if a socle element is chosen, it's added to I; if a minimal generator is chosen, it's replaced by the square-free part of the maximal ideal times it. the chance of making the given move is then 1/(#ISocgens+#Igens), and the chance of making the move back would be the similar quantity for J, so we make the move or not depending on whether random RR < (nJ+nSocJ)/(nI+nSocI) or not; here random RR is a random number in [0,1].
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1779108483
│ │ │ + -- setting random seed to 1779298492
│ │ │  
│ │ │ -o1 = 1779108483
    
│ │ │ +o1 = 1779298492
│ │ │              
    
│ │ │                
    i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │ @@ -111,17 +111,15 @@
│ │ │  o3 : MonomialIdeal of S
    
│ │ │              
    
│ │ │                
    i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b*c, a*d, b*c*d), {a*b*c, a*d, b*c*d}, {c*d, b*d, b*c,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     a*c, a*b}}
│ │ │ +o4 = {monomialIdeal (a*b, a*d, b*d), {a*b, a*d, b*d}, {c*d, b*c, a*c}}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │          
    
│ │ │            
    With 4 variables and 168 possible monomial ideals, a run of 5000 takes less than 6 seconds on a reasonably fast machine. With 10 variables a run of 1000 takes about 2 seconds.
    
│ │ │ @@ -152,15 +150,15 @@
│ │ │  
│ │ │  o7 : MonomialIdeal of S
    
│ │ │              
    
│ │ │                
    i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 4.27518s (cpu); 2.8639s (thread); 0s (gc)
    
│ │ │ + -- used 4.57332s (cpu); 3.23803s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i9 : #T
│ │ │  
│ │ │  o9 = 168
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,32 +35,30 @@
│ │ │ │  choose randomly from the union of these lists; if a socle element is chosen,
│ │ │ │  it's added to I; if a minimal generator is chosen, it's replaced by the square-
│ │ │ │  free part of the maximal ideal times it. the chance of making the given move is
│ │ │ │  then 1/(#ISocgens+#Igens), and the chance of making the move back would be the
│ │ │ │  similar quantity for J, so we make the move or not depending on whether random
│ │ │ │  RR < (nJ+nSocJ)/(nI+nSocI) or not; here random RR is a random number in [0,1].
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1779108483
│ │ │ │ + -- setting random seed to 1779298492
│ │ │ │  
│ │ │ │ -o1 = 1779108483
│ │ │ │ +o1 = 1779298492
│ │ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : J = monomialIdeal"ab,ad, bcd"
│ │ │ │  
│ │ │ │  o3 = monomialIdeal (a*b, a*d, b*c*d)
│ │ │ │  
│ │ │ │  o3 : MonomialIdeal of S
│ │ │ │  i4 : randomSquareFreeStep J
│ │ │ │  
│ │ │ │ -o4 = {monomialIdeal (a*b*c, a*d, b*c*d), {a*b*c, a*d, b*c*d}, {c*d, b*d, b*c,
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     a*c, a*b}}
│ │ │ │ +o4 = {monomialIdeal (a*b, a*d, b*d), {a*b, a*d, b*d}, {c*d, b*c, a*c}}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  With 4 variables and 168 possible monomial ideals, a run of 5000 takes less
│ │ │ │  than 6 seconds on a reasonably fast machine. With 10 variables a run of 1000
│ │ │ │  takes about 2 seconds.
│ │ │ │  i5 : setRandomSeed(1)
│ │ │ │   -- setting random seed to 1
│ │ │ │ @@ -74,15 +72,15 @@
│ │ │ │  i7 : J = monomialIdeal 0_S
│ │ │ │  
│ │ │ │  o7 = monomialIdeal ()
│ │ │ │  
│ │ │ │  o7 : MonomialIdeal of S
│ │ │ │  i8 : time T=tally for t from 1 to 5000 list first (J=rsfs
│ │ │ │  (J,AlexanderProbability => .01));
│ │ │ │ - -- used 4.27518s (cpu); 2.8639s (thread); 0s (gc)
│ │ │ │ + -- used 4.57332s (cpu); 3.23803s (thread); 0s (gc)
│ │ │ │  i9 : #T
│ │ │ │  
│ │ │ │  o9 = 168
│ │ │ │  i10 : T
│ │ │ │  
│ │ │ │  o10 = Tally{monomialIdeal () => 45                            }
│ │ │ │              monomialIdeal (a*b*c, a*b*d) => 33
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html
│ │ │ @@ -59,17 +59,17 @@
│ │ │          
    
│ │ │            
    This package can be used to make experiments, trying many ideals, perhaps over small fields. For example...what would you expect the regularities of "typical" monomial ideals with 10 generators of degree 3 in 6 variables to be? Try a bunch of examples -- it's fast. Here we do only 500 -- this takes about a second on a fast machine -- but with a little patience, thousands can be done conveniently.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -77,22 +77,21 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1779108470
│ │ │ + -- setting random seed to 1779298484
│ │ │  
│ │ │ -o1 = 1779108470
    
│ │ │ +o1 = 1779298484
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101;
    
│ │ │              
    
│ │ │                
    i3 : S=kk[vars(0..5)];
    
│ │ │              
    
│ │ │                
    i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ - -- used 3.34044s (cpu); 1.83367s (thread); 0s (gc)
│ │ │ + -- used 3.50494s (cpu); 1.98883s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 39 }
│ │ │ -           5 => 200
│ │ │ -           6 => 184
│ │ │ -           7 => 63
│ │ │ -           8 => 12
│ │ │ -           9 => 2
│ │ │ +o4 = Tally{4 => 46 }
│ │ │ +           5 => 190
│ │ │ +           6 => 179
│ │ │ +           7 => 72
│ │ │ +           8 => 13
│ │ │  
│ │ │  o4 : Tally
    
│ │ │              
    
│ │ │          
    
│ │ │            
    How does this compare with the case of binomial ideals? or pure binomial ideals? We invite the reader to experiment, replacing "randomMonomialIdeal" above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger numbers of examples. Click the link "Finding Extreme Examples" below to see some other, more elaborate ways to search.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,28 +9,27 @@
│ │ │ │  This package can be used to make experiments, trying many ideals, perhaps over
│ │ │ │  small fields. For example...what would you expect the regularities of "typical"
│ │ │ │  monomial ideals with 10 generators of degree 3 in 6 variables to be? Try a
│ │ │ │  bunch of examples -- it's fast. Here we do only 500 -- this takes about a
│ │ │ │  second on a fast machine -- but with a little patience, thousands can be done
│ │ │ │  conveniently.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1779108470
│ │ │ │ + -- setting random seed to 1779298484
│ │ │ │  
│ │ │ │ -o1 = 1779108470
│ │ │ │ +o1 = 1779298484
│ │ │ │  i2 : kk=ZZ/101;
│ │ │ │  i3 : S=kk[vars(0..5)];
│ │ │ │  i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ │ - -- used 3.34044s (cpu); 1.83367s (thread); 0s (gc)
│ │ │ │ + -- used 3.50494s (cpu); 1.98883s (thread); 0s (gc)
│ │ │ │  
│ │ │ │ -o4 = Tally{4 => 39 }
│ │ │ │ -           5 => 200
│ │ │ │ -           6 => 184
│ │ │ │ -           7 => 63
│ │ │ │ -           8 => 12
│ │ │ │ -           9 => 2
│ │ │ │ +o4 = Tally{4 => 46 }
│ │ │ │ +           5 => 190
│ │ │ │ +           6 => 179
│ │ │ │ +           7 => 72
│ │ │ │ +           8 => 13
│ │ │ │  
│ │ │ │  o4 : Tally
│ │ │ │  How does this compare with the case of binomial ideals? or pure binomial
│ │ │ │  ideals? We invite the reader to experiment, replacing "randomMonomialIdeal"
│ │ │ │  above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger
│ │ │ │  numbers of examples. Click the link "Finding Extreme Examples" below to see
│ │ │ │  some other, more elaborate ways to search.
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_dim__Via__Bezout.out
│ │ │ @@ -5,17 +5,17 @@
│ │ │  i2 : S=kk[y_0..y_8];
│ │ │  
│ │ │  i3 : I=ideal random(S^1,S^{-2,-2,-2,-2})+(ideal random(2,S))^2;
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │  
│ │ │  i4 : elapsedTime dimViaBezout(I)
│ │ │ - -- 1.66927s elapsed
│ │ │ + -- 1.52183s elapsed
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : elapsedTime dim I
│ │ │ - -- 3.60862s elapsed
│ │ │ + -- 3.4739s elapsed
│ │ │  
│ │ │  o5 = 4
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_extend__Ideal__By__Non__Zero__Minor.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │  i8 : i = 0;
│ │ │  
│ │ │  i9 : J = I;
│ │ │  
│ │ │  o9 : Ideal of T
│ │ │  
│ │ │  i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ - -- 2.81641s elapsed
│ │ │ + -- 1.86421s elapsed
│ │ │  
│ │ │  i11 : dim J
│ │ │  
│ │ │  o11 = 1
│ │ │  
│ │ │  i12 : i
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Points.out
│ │ │ @@ -27,24 +27,24 @@
│ │ │  i6 : S=ZZ/103[y_0..y_30];
│ │ │  
│ │ │  i7 : I=minors(2,random(S^3,S^{3:-1}));
│ │ │  
│ │ │  o7 : Ideal of S
│ │ │  
│ │ │  i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable)
│ │ │ - -- 4.01109s elapsed
│ │ │ + -- 3.12569s elapsed
│ │ │  
│ │ │  o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>Decompose)
│ │ │ - -- 2.98436s elapsed
│ │ │ + -- 2.38132s elapsed
│ │ │  
│ │ │  o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │       11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}}
│ │ │  
│ │ │  o9 : List
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_dim__Via__Bezout.html
│ │ │ @@ -100,23 +100,23 @@
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │              
    
│ │ │                
    i4 : elapsedTime dimViaBezout(I)
│ │ │ - -- 1.66927s elapsed
│ │ │ + -- 1.52183s elapsed
│ │ │  
│ │ │  o4 = 4
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime dim I
│ │ │ - -- 3.60862s elapsed
│ │ │ + -- 3.4739s elapsed
│ │ │  
│ │ │  o5 = 4
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The user may set the MinimumFieldSize to ensure that the field being worked over is big enough.  For instance, there are relatively few linear spaces over a field of characteristic 2, and this can cause incorrect results to be provided.  If no size is provided, the function tries to guess a good size based on ambient ring.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,19 +32,19 @@
│ │ │ │  examples, the built in dim function is much faster.
│ │ │ │  i1 : kk=ZZ/101;
│ │ │ │  i2 : S=kk[y_0..y_8];
│ │ │ │  i3 : I=ideal random(S^1,S^{-2,-2,-2,-2})+(ideal random(2,S))^2;
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : elapsedTime dimViaBezout(I)
│ │ │ │ - -- 1.66927s elapsed
│ │ │ │ + -- 1.52183s elapsed
│ │ │ │  
│ │ │ │  o4 = 4
│ │ │ │  i5 : elapsedTime dim I
│ │ │ │ - -- 3.60862s elapsed
│ │ │ │ + -- 3.4739s elapsed
│ │ │ │  
│ │ │ │  o5 = 4
│ │ │ │  The user may set the MinimumFieldSize to ensure that the field being worked
│ │ │ │  over is big enough. For instance, there are relatively few linear spaces over a
│ │ │ │  field of characteristic 2, and this can cause incorrect results to be provided.
│ │ │ │  If no size is provided, the function tries to guess a good size based on
│ │ │ │  ambient ring.
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_extend__Ideal__By__Non__Zero__Minor.html
│ │ │ @@ -160,15 +160,15 @@
│ │ │  
│ │ │  o9 : Ideal of T
    
│ │ │              
    
│ │ │                
    i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ - -- 2.81641s elapsed
    
│ │ │ + -- 1.86421s elapsed
│ │ │              
    
│ │ │                
    i11 : dim J
│ │ │  
│ │ │  o11 = 1
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -79,15 +79,15 @@
│ │ │ │  o7 : Matrix T  <-- T
│ │ │ │  i8 : i = 0;
│ │ │ │  i9 : J = I;
│ │ │ │  
│ │ │ │  o9 : Ideal of T
│ │ │ │  i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J =
│ │ │ │  extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ │ - -- 2.81641s elapsed
│ │ │ │ + -- 1.86421s elapsed
│ │ │ │  i11 : dim J
│ │ │ │  
│ │ │ │  o11 = 1
│ │ │ │  i12 : i
│ │ │ │  
│ │ │ │  o12 = 4
│ │ │ │  In this particular example, there tend to be about 5 associated primes when
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Points.html
│ │ │ @@ -149,27 +149,27 @@
│ │ │  
│ │ │  o7 : Ideal of S
│ │ │              
    
│ │ │                
    i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable)
│ │ │ - -- 4.01109s elapsed
│ │ │ + -- 3.12569s elapsed
│ │ │  
│ │ │  o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}}
│ │ │  
│ │ │  o8 : List
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>Decompose)
│ │ │ - -- 2.98436s elapsed
│ │ │ + -- 2.38132s elapsed
│ │ │  
│ │ │  o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │       11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}}
│ │ │  
│ │ │  o9 : List
    
│ │ │              
    
│ │ │                
    i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
│ │ │ - -- used 0.44766s (cpu); 0.365702s (thread); 0s (gc)
│ │ │ + -- used 0.602562s (cpu); 0.405314s (thread); 0s (gc)
│ │ │  
│ │ │                                  125   124      120 5    124    100 25     104 20       108 15 2      112 10 3     116 5 4    120 5    124      125      4 120        8 115 2        12 110 3         16 105 4         20 100 5          24 95 6          28 90 7           32 85 8           36 80 9           40 75 10           44 70 11           48 65 12           52 60 13           56 55 14           60 50 15           64 45 16           68 40 17          72 35 18          76 30 19         80 25 20         84 20 21        88 15 22       92 10 23      96 5 24    100 25     24 100        28 95          32 90 2         36 85 3          40 80 4          44 75 5           48 70 6           52 65 7           56 60 8           60 55 9           64 50 10           68 45 11           72 40 12           76 35 13           80 30 14          84 25 15          88 20 16         92 15 17        96 10 18        100 5 19      104 20       48 75 2       52 70   2        56 65 2 2        60 60 3 2         64 55 4 2         68 50 5 2         72 45 6 2         76 40 7 2         80 35 8 2         84 30 9 2         88 25 10 2         92 20 11 2        96 15 12 2        100 10 13 2       104 5 14 2      108 15 2      72 50 3       76 45   3       80 40 2 3        84 35 3 3        88 30 4 3        92 25 5 3        96 20 6 3        100 15 7 3       104 10 8 3       108 5 9 3      112 10 3     96 25 4      100 20   4      104 15 2 4      108 10 3 4      112 5 4 4     116 5 4    120 5    124
│ │ │  o14 = Proj Q - - - > Proj Q   {x   , x   y, - x   y  + x   z, x   y   - 5x   y  z + 10x   y  z  - 10x   y  z  + 5x   y z  - x   z  + x   t, - y    + 25x y   z - 300x y   z  + 2300x  y   z  - 12650x  y   z  + 53130x  y   z  - 177100x  y  z  + 480700x  y  z  - 1081575x  y  z  + 2042975x  y  z  - 3268760x  y  z   + 4457400x  y  z   - 5200300x  y  z   + 5200300x  y  z   - 4457400x  y  z   + 3268760x  y  z   - 2042975x  y  z   + 1081575x  y  z   - 480700x  y  z   + 177100x  y  z   - 53130x  y  z   + 12650x  y  z   - 2300x  y  z   + 300x  y  z   - 25x  y z   + x   z   - 5x  y   t + 100x  y  z*t - 950x  y  z t + 5700x  y  z t - 24225x  y  z t + 77520x  y  z t - 193800x  y  z t + 387600x  y  z t - 629850x  y  z t + 839800x  y  z t - 923780x  y  z  t + 839800x  y  z  t - 629850x  y  z  t + 387600x  y  z  t - 193800x  y  z  t + 77520x  y  z  t - 24225x  y  z  t + 5700x  y  z  t - 950x  y  z  t + 100x   y z  t - 5x   z  t - 10x  y  t  + 150x  y  z*t  - 1050x  y  z t  + 4550x  y  z t  - 13650x  y  z t  + 30030x  y  z t  - 50050x  y  z t  + 64350x  y  z t  - 64350x  y  z t  + 50050x  y  z t  - 30030x  y  z  t  + 13650x  y  z  t  - 4550x  y  z  t  + 1050x   y  z  t  - 150x   y z  t  + 10x   z  t  - 10x  y  t  + 100x  y  z*t  - 450x  y  z t  + 1200x  y  z t  - 2100x  y  z t  + 2520x  y  z t  - 2100x  y  z t  + 1200x   y  z t  - 450x   y  z t  + 100x   y z t  - 10x   z  t  - 5x  y  t  + 25x   y  z*t  - 50x   y  z t  + 50x   y  z t  - 25x   y z t  + 5x   z t  - x   t  + x   u}
│ │ │  
│ │ │  o14 : RationalMapping
    
│ │ │              
    
│ │ │                
    i14 : time rationalPoints(I, Amount => true)
│ │ │ - -- used 0.00313113s (cpu); 0.0031276s (thread); 0s (gc)
│ │ │ + -- used 0.0038744s (cpu); 0.00387301s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 110462212541120451001
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Over number fields
    
│ │ │ @@ -353,15 +353,15 @@
│ │ │  o31 : Ideal of R
    │ │ │
    │ │ │ 
    i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
│ │ │ - -- used 0.983967s (cpu); 0.775123s (thread); 0s (gc)
    │ │ │ + -- used 1.12634s (cpu); 0.899099s (thread); 0s (gc) │ │ │
    │ │ │ 
    i33 : #oo
│ │ │  
│ │ │  o33 = 31
    │ │ │ @@ -378,15 +378,15 @@ │ │ │ │ │ │ o34 : Ideal of GF 1048969271299456081[x..z, w] │ │ │
    │ │ │ 
    i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ - -- used 0.28051s (cpu); 0.202638s (thread); 0s (gc)
│ │ │ + -- used 0.347821s (cpu); 0.251987s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 31
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -89,15 +89,15 @@ │ │ │ │ o13 = ideal(u + u + u + u + u + u + u + u + u + u + u ) │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ │ │ │ │ │ ZZ │ │ │ │ o13 : Ideal of ---[u ..u ] │ │ │ │ 101 0 10 │ │ │ │ i14 : time rationalPoints(I, Amount => true) │ │ │ │ - -- used 0.00313113s (cpu); 0.0031276s (thread); 0s (gc) │ │ │ │ + -- used 0.0038744s (cpu); 0.00387301s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 = 110462212541120451001 │ │ │ │ ****** OOvveerr nnuummbbeerr ffiieellddss ****** │ │ │ │ Over a number field one can use the option Bound to specify a maximal │ │ │ │ multiplicative height given by $(x_0:\dots:x_n)\mapsto \prod_{v}\max_i|x_i|_v ^ │ │ │ │ {d_v/d}$ (this is also available as a method _g_l_o_b_a_l_H_e_i_g_h_t). │ │ │ │ i15 : QQ[x,y,z]; I = homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)), z); │ │ │ │ @@ -197,24 +197,24 @@ │ │ │ │ │ │ │ │ o30 : Ideal of R │ │ │ │ i31 : nodes = I + ideal jacobian I; │ │ │ │ │ │ │ │ o31 : Ideal of R │ │ │ │ i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true); │ │ │ │ -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25) │ │ │ │ - -- used 0.983967s (cpu); 0.775123s (thread); 0s (gc) │ │ │ │ + -- used 1.12634s (cpu); 0.899099s (thread); 0s (gc) │ │ │ │ i33 : #oo │ │ │ │ │ │ │ │ o33 = 31 │ │ │ │ Still it runs a lot faster when reduced to a positive characteristic. │ │ │ │ i34 : nodes' = baseChange_32003 nodes; │ │ │ │ │ │ │ │ o34 : Ideal of GF 1048969271299456081[x..z, w] │ │ │ │ i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true) │ │ │ │ - -- used 0.28051s (cpu); 0.202638s (thread); 0s (gc) │ │ │ │ + -- used 0.347821s (cpu); 0.251987s (thread); 0s (gc) │ │ │ │ │ │ │ │ o35 = 31 │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ For a number field other than QQ, the enumeration of elements with bounded │ │ │ │ height depends on an algorithm by Doyle–Krumm, which is currently only │ │ │ │ implemented in Sage. │ │ │ │ ******** MMeennuu ******** │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/___Plane__Curve__Singularities.out │ │ │ @@ -331,15 +331,15 @@ │ │ │ 2 2 2 2 2 2 2 │ │ │ - p w , p y - p , p w y - p p , p w - p ) │ │ │ 2 1 0 1 0 0 1 2 0 0 2 │ │ │ │ │ │ o47 : Ideal of B2 │ │ │ │ │ │ i48 : time sing2 = ideal singularLocus strictTransform2; │ │ │ - -- used 0.920794s (cpu); 0.731202s (thread); 0s (gc) │ │ │ + -- used 1.03998s (cpu); 0.874856s (thread); 0s (gc) │ │ │ │ │ │ ZZ │ │ │ o48 : Ideal of -----[p ..p , w ..w , x..y] │ │ │ 32003 0 2 0 1 │ │ │ │ │ │ i49 : saturate(sing2, sub(irrelTot, ring sing2)) │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_expected__Rees__Ideal.out │ │ │ @@ -58,15 +58,15 @@ │ │ │ o5 : Matrix S <-- S │ │ │ │ │ │ i6 : I = minors(n-1, M); │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec. │ │ │ - -- used 1.29738s (cpu); 0.894035s (thread); 0s (gc) │ │ │ + -- used 1.08264s (cpu); 0.81702s (thread); 0s (gc) │ │ │ │ │ │ o7 : Ideal of S[w ..w ] │ │ │ 0 4 │ │ │ │ │ │ i8 : kk = ZZ/101; │ │ │ │ │ │ i9 : S = kk[x,y,z]; │ │ │ @@ -77,19 +77,19 @@ │ │ │ o10 : Matrix S <-- S │ │ │ │ │ │ i11 : I = minors(3,m); │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ │ │ i12 : time reesIdeal (I, I_0); │ │ │ - -- used 1.83269s (cpu); 1.46198s (thread); 0s (gc) │ │ │ + -- used 1.94446s (cpu); 1.5931s (thread); 0s (gc) │ │ │ │ │ │ o12 : Ideal of S[w ..w ] │ │ │ 0 3 │ │ │ │ │ │ i13 : time reesIdeal (I, I_0, Jacobian =>false); │ │ │ - -- used 1.65871s (cpu); 1.25818s (thread); 0s (gc) │ │ │ + -- used 1.72688s (cpu); 1.4654s (thread); 0s (gc) │ │ │ │ │ │ o13 : Ideal of S[w ..w ] │ │ │ 0 3 │ │ │ │ │ │ i14 : │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_rees__Ideal.out │ │ │ @@ -13,21 +13,21 @@ │ │ │ 3 2 │ │ │ - x x x , x - x x ) │ │ │ 0 2 4 1 0 4 │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ i4 : time V1 = reesIdeal i; │ │ │ - -- used 0.0272195s (cpu); 0.0251912s (thread); 0s (gc) │ │ │ + -- used 0.224166s (cpu); 0.053349s (thread); 0s (gc) │ │ │ │ │ │ o4 : Ideal of S[w ..w ] │ │ │ 0 6 │ │ │ │ │ │ i5 : time V2 = reesIdeal(i,i_0); │ │ │ - -- used 0.114248s (cpu); 0.11353s (thread); 0s (gc) │ │ │ + -- used 0.220129s (cpu); 0.156863s (thread); 0s (gc) │ │ │ │ │ │ o5 : Ideal of S[w ..w ] │ │ │ 0 6 │ │ │ │ │ │ i6 : S=kk[a,b,c] │ │ │ │ │ │ o6 = S │ │ │ @@ -47,21 +47,21 @@ │ │ │ │ │ │ 2 2 │ │ │ o8 = ideal (a , a*b, b ) │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ │ │ i9 : time I1 = reesIdeal i; │ │ │ - -- used 0.0199757s (cpu); 0.0185652s (thread); 0s (gc) │ │ │ + -- used 0.270631s (cpu); 0.0524086s (thread); 0s (gc) │ │ │ │ │ │ o9 : Ideal of S[w ..w ] │ │ │ 0 2 │ │ │ │ │ │ i10 : time I2 = reesIdeal(i,i_0); │ │ │ - -- used 0.00781173s (cpu); 0.00745956s (thread); 0s (gc) │ │ │ + -- used 0.0230843s (cpu); 0.0109181s (thread); 0s (gc) │ │ │ │ │ │ o10 : Ideal of S[w ..w ] │ │ │ 0 2 │ │ │ │ │ │ i11 : transpose gens I1 │ │ │ │ │ │ o11 = {-1, -3} | aw_1-bw_2 | │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/___Plane__Curve__Singularities.html │ │ │ @@ -592,15 +592,15 @@ │ │ │
    │ │ │

    We compute the singular locus once again:

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -325,15 +325,15 @@ │ │ │ │ 2 2 2 2 2 2 2 │ │ │ │ - p w , p y - p , p w y - p p , p w - p ) │ │ │ │ 2 1 0 1 0 0 1 2 0 0 2 │ │ │ │ │ │ │ │ o47 : Ideal of B2 │ │ │ │ We compute the singular locus once again: │ │ │ │ i48 : time sing2 = ideal singularLocus strictTransform2; │ │ │ │ - -- used 0.920794s (cpu); 0.731202s (thread); 0s (gc) │ │ │ │ + -- used 1.03998s (cpu); 0.874856s (thread); 0s (gc) │ │ │ │ │ │ │ │ ZZ │ │ │ │ o48 : Ideal of -----[p ..p , w ..w , x..y] │ │ │ │ 32003 0 2 0 1 │ │ │ │ i49 : saturate(sing2, sub(irrelTot, ring sing2)) │ │ │ │ │ │ │ │ o49 = ideal 1 │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/_expected__Rees__Ideal.html │ │ │ @@ -156,15 +156,15 @@ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i48 : time sing2 = ideal singularLocus strictTransform2;
│ │ │ - -- used 0.920794s (cpu); 0.731202s (thread); 0s (gc)
│ │ │ + -- used 1.03998s (cpu); 0.874856s (thread); 0s (gc)
│ │ │  
│ │ │                   ZZ
│ │ │  o48 : Ideal of -----[p ..p , w ..w , x..y]
│ │ │                 32003  0   2   0   1
    │ │ │
    │ │ │ 
    i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec.
│ │ │ - -- used 1.29738s (cpu); 0.894035s (thread); 0s (gc)
│ │ │ + -- used 1.08264s (cpu); 0.81702s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S[w ..w ]
│ │ │                   0   4
    │ │ │
    │ │ │ @@ -190,24 +190,24 @@ │ │ │ │ │ │ o11 : Ideal of S │ │ │
    │ │ │ 
    i12 : time reesIdeal (I, I_0);
│ │ │ - -- used 1.83269s (cpu); 1.46198s (thread); 0s (gc)
│ │ │ + -- used 1.94446s (cpu); 1.5931s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S[w ..w ]
│ │ │                    0   3
    │ │ │
    │ │ │ 
    i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ - -- used 1.65871s (cpu); 1.25818s (thread); 0s (gc)
│ │ │ + -- used 1.72688s (cpu); 1.4654s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of S[w ..w ]
│ │ │                    0   3
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -86,34 +86,34 @@ │ │ │ │ │ │ │ │ 5 4 │ │ │ │ o5 : Matrix S <-- S │ │ │ │ i6 : I = minors(n-1, M); │ │ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec. │ │ │ │ - -- used 1.29738s (cpu); 0.894035s (thread); 0s (gc) │ │ │ │ + -- used 1.08264s (cpu); 0.81702s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 : Ideal of S[w ..w ] │ │ │ │ 0 4 │ │ │ │ i8 : kk = ZZ/101; │ │ │ │ i9 : S = kk[x,y,z]; │ │ │ │ i10 : m = random(S^3, S^{4:-2}); │ │ │ │ │ │ │ │ 3 4 │ │ │ │ o10 : Matrix S <-- S │ │ │ │ i11 : I = minors(3,m); │ │ │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ i12 : time reesIdeal (I, I_0); │ │ │ │ - -- used 1.83269s (cpu); 1.46198s (thread); 0s (gc) │ │ │ │ + -- used 1.94446s (cpu); 1.5931s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of S[w ..w ] │ │ │ │ 0 3 │ │ │ │ i13 : time reesIdeal (I, I_0, Jacobian =>false); │ │ │ │ - -- used 1.65871s (cpu); 1.25818s (thread); 0s (gc) │ │ │ │ + -- used 1.72688s (cpu); 1.4654s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 : Ideal of S[w ..w ] │ │ │ │ 0 3 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_y_m_m_e_t_r_i_c_A_l_g_e_b_r_a_I_d_e_a_l -- Ideal of the symmetric algebra of an ideal or │ │ │ │ module │ │ │ │ * _j_a_c_o_b_i_a_n_D_u_a_l -- Computes the 'jacobian dual', part of a method of finding │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal.html │ │ │ @@ -115,24 +115,24 @@ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time V1 = reesIdeal i;
│ │ │ - -- used 0.0272195s (cpu); 0.0251912s (thread); 0s (gc)
│ │ │ + -- used 0.224166s (cpu); 0.053349s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S[w ..w ]
│ │ │                   0   6
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time V2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.114248s (cpu); 0.11353s (thread); 0s (gc)
│ │ │ + -- used 0.220129s (cpu); 0.156863s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : Ideal of S[w ..w ]
│ │ │                   0   6
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ @@ -169,24 +169,24 @@ │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : time I1 = reesIdeal i;
│ │ │ - -- used 0.0199757s (cpu); 0.0185652s (thread); 0s (gc)
│ │ │ + -- used 0.270631s (cpu); 0.0524086s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of S[w ..w ]
│ │ │                   0   2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : time I2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.00781173s (cpu); 0.00745956s (thread); 0s (gc)
│ │ │ + -- used 0.0230843s (cpu); 0.0109181s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of S[w ..w ]
│ │ │                    0   2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -51,20 +51,20 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 2 │ │ │ │ - x x x , x - x x ) │ │ │ │ 0 2 4 1 0 4 │ │ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ i4 : time V1 = reesIdeal i; │ │ │ │ - -- used 0.0272195s (cpu); 0.0251912s (thread); 0s (gc) │ │ │ │ + -- used 0.224166s (cpu); 0.053349s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 : Ideal of S[w ..w ] │ │ │ │ 0 6 │ │ │ │ i5 : time V2 = reesIdeal(i,i_0); │ │ │ │ - -- used 0.114248s (cpu); 0.11353s (thread); 0s (gc) │ │ │ │ + -- used 0.220129s (cpu); 0.156863s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 : Ideal of S[w ..w ] │ │ │ │ 0 6 │ │ │ │ The following example shows how we handle degrees │ │ │ │ i6 : S=kk[a,b,c] │ │ │ │ │ │ │ │ o6 = S │ │ │ │ @@ -81,20 +81,20 @@ │ │ │ │ i8 : i=minors(2,m) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o8 = ideal (a , a*b, b ) │ │ │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ i9 : time I1 = reesIdeal i; │ │ │ │ - -- used 0.0199757s (cpu); 0.0185652s (thread); 0s (gc) │ │ │ │ + -- used 0.270631s (cpu); 0.0524086s (thread); 0s (gc) │ │ │ │ │ │ │ │ o9 : Ideal of S[w ..w ] │ │ │ │ 0 2 │ │ │ │ i10 : time I2 = reesIdeal(i,i_0); │ │ │ │ - -- used 0.00781173s (cpu); 0.00745956s (thread); 0s (gc) │ │ │ │ + -- used 0.0230843s (cpu); 0.0109181s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 : Ideal of S[w ..w ] │ │ │ │ 0 2 │ │ │ │ i11 : transpose gens I1 │ │ │ │ │ │ │ │ o11 = {-1, -3} | aw_1-bw_2 | │ │ │ │ {-1, -3} | aw_0-bw_1 | │ │ ├── ./usr/share/doc/Macaulay2/Regularity/example-output/_m__Regularity.out │ │ │ @@ -71,15 +71,15 @@ │ │ │ x x x , x + x x - x x - x x x , x + x - x x ) │ │ │ 0 1 3 0 0 1 1 2 0 2 5 0 2 0 5 │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ │ │ i8 : benchmark "mRegularity I1" │ │ │ │ │ │ -o8 = .2807511285000001 │ │ │ +o8 = .2578586074545454 │ │ │ │ │ │ o8 : RR (of precision 53) │ │ │ │ │ │ i9 : R = QQ[x_0..x_5] │ │ │ │ │ │ o9 = R │ │ │ │ │ │ @@ -87,17 +87,17 @@ │ │ │ │ │ │ i10 : I2 = ideal ( x_0^2+x_5^2, x_0^2+x_0*x_3+x_4^2, x_0^2+x_0*x_5+x_2*x_5, x_0^2-x_0*x_3-x_3*x_5, x_0^2-x_3*x_4, x_0*x_3); │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ │ │ i11 : benchmark " mRegularity I2" │ │ │ │ │ │ -o11 = .04334666027272725 │ │ │ +o11 = .04907154652083327 │ │ │ │ │ │ o11 : RR (of precision 53) │ │ │ │ │ │ i12 : time regularity I2 │ │ │ - -- used 0.00350377s (cpu); 0.00350965s (thread); 0s (gc) │ │ │ + -- used 0.00370759s (cpu); 0.00371211s (thread); 0s (gc) │ │ │ │ │ │ o12 = 4 │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/Regularity/html/_m__Regularity.html │ │ │ @@ -181,15 +181,15 @@ │ │ │ o7 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : benchmark "mRegularity I1"
│ │ │  
│ │ │ -o8 = .2807511285000001
│ │ │ +o8 = .2578586074545454
│ │ │  
│ │ │  o8 : RR (of precision 53)
    │ │ │ │ │ │ │ │ │ │ │ │

    This is an example where regularity is faster than mRegularity.

    │ │ │ │ │ │ @@ -209,23 +209,23 @@ │ │ │ o10 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i11 : benchmark " mRegularity I2"
│ │ │  
│ │ │ -o11 = .04334666027272725
│ │ │ +o11 = .04907154652083327
│ │ │  
│ │ │  o11 : RR (of precision 53)
    │ │ │
    │ │ │ 
    i12 : time regularity I2  
│ │ │ - -- used 0.00350377s (cpu); 0.00350965s (thread); 0s (gc)
│ │ │ + -- used 0.00370759s (cpu); 0.00371211s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = 4
    │ │ │
    │ │ │

    This symbol is provided by the package Regularity.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -94,34 +94,34 @@ │ │ │ │ 3 2 2 3 3 2 │ │ │ │ x x x , x + x x - x x - x x x , x + x - x x ) │ │ │ │ 0 1 3 0 0 1 1 2 0 2 5 0 2 0 5 │ │ │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ i8 : benchmark "mRegularity I1" │ │ │ │ │ │ │ │ -o8 = .2807511285000001 │ │ │ │ +o8 = .2578586074545454 │ │ │ │ │ │ │ │ o8 : RR (of precision 53) │ │ │ │ This is an example where regularity is faster than mRegularity. │ │ │ │ i9 : R = QQ[x_0..x_5] │ │ │ │ │ │ │ │ o9 = R │ │ │ │ │ │ │ │ o9 : PolynomialRing │ │ │ │ i10 : I2 = ideal ( x_0^2+x_5^2, x_0^2+x_0*x_3+x_4^2, x_0^2+x_0*x_5+x_2*x_5, │ │ │ │ x_0^2-x_0*x_3-x_3*x_5, x_0^2-x_3*x_4, x_0*x_3); │ │ │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ i11 : benchmark " mRegularity I2" │ │ │ │ │ │ │ │ -o11 = .04334666027272725 │ │ │ │ +o11 = .04907154652083327 │ │ │ │ │ │ │ │ o11 : RR (of precision 53) │ │ │ │ i12 : time regularity I2 │ │ │ │ - -- used 0.00350377s (cpu); 0.00350965s (thread); 0s (gc) │ │ │ │ + -- used 0.00370759s (cpu); 0.00371211s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 = 4 │ │ │ │ This symbol is provided by the package Regularity. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_e_g_u_l_a_r_i_t_y -- compute the Castelnuovo-Mumford regularity │ │ │ │ ********** WWaayyss ttoo uussee mmRReegguullaarriittyy:: ********** │ │ │ │ * mRegularity(Ideal) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_cayley__Trick.out │ │ │ @@ -5,18 +5,18 @@ │ │ │ o2 = ideal(x x - x x ) │ │ │ 0 1 2 3 │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2); │ │ │ - -- used 0.146312s (cpu); 0.0851178s (thread); 0s (gc) │ │ │ + -- used 0.143875s (cpu); 0.0759174s (thread); 0s (gc) │ │ │ │ │ │ i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1) │ │ │ - -- used 0.145095s (cpu); 0.074383s (thread); 0s (gc) │ │ │ + -- used 0.163961s (cpu); 0.0935498s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ o4 = (ideal (x x - x x , x x - x x , x x - x x , │ │ │ 0,3 1,2 0,2 1,3 1,0 1,1 1,2 1,3 0,3 1,1 0,1 1,3 │ │ │ ------------------------------------------------------------------------ │ │ │ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ @@ -37,17 +37,17 @@ │ │ │ 2 2 │ │ │ 4x x x x - 2x x x x + x x )) │ │ │ 0,0 0,1 1,2 1,3 0,2 0,3 1,2 1,3 0,2 1,3 │ │ │ │ │ │ o4 : Sequence │ │ │ │ │ │ i5 : time cayleyTrick(P1xP1,1,Duality=>true); │ │ │ - -- used 0.257486s (cpu); 0.11514s (thread); 0s (gc) │ │ │ + -- used 0.28628s (cpu); 0.12188s (thread); 0s (gc) │ │ │ │ │ │ i6 : assert(oo == (P1xP1xP1,P1xP1xP1')) │ │ │ │ │ │ i7 : time cayleyTrick(P1xP1,2,Duality=>true); │ │ │ - -- used 0.168639s (cpu); 0.10303s (thread); 0s (gc) │ │ │ + -- used 0.185993s (cpu); 0.114514s (thread); 0s (gc) │ │ │ │ │ │ i8 : assert(oo == (P1xP1xP2,P1xP1xP2')) │ │ │ │ │ │ i9 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Equations.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ o2 = ideal (x + x + x + x , x x + x x + x x ) │ │ │ 0 1 2 3 0 1 1 2 2 3 │ │ │ │ │ │ o2 : Ideal of P3 │ │ │ │ │ │ i3 : -- Chow equations of C │ │ │ time eqsC = chowEquations chowForm C │ │ │ - -- used 0.135904s (cpu); 0.0637037s (thread); 0s (gc) │ │ │ + -- used 0.135231s (cpu); 0.0709316s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 2 2 4 2 2 2 2 │ │ │ o3 = ideal (x x + x x + x x + x , x x x x + x x x + x x , x x x + │ │ │ 0 3 1 3 2 3 3 0 1 2 3 1 2 3 2 3 0 2 3 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 3 2 2 3 3 2 2 2 2 │ │ │ x x x + x x - 2x x x - 2x x x - x x , x x + 2x x x - x x x + x x │ │ │ @@ -72,15 +72,15 @@ │ │ │ o5 = ideal (x - x x , x - x x x , x x - x x ) │ │ │ 1 0 2 2 0 1 3 1 2 0 3 │ │ │ │ │ │ o5 : Ideal of P3 │ │ │ │ │ │ i6 : -- Chow equations of D │ │ │ time eqsD = chowEquations chowForm D │ │ │ - -- used 0.0620687s (cpu); 0.0620762s (thread); 0s (gc) │ │ │ + -- used 0.0466663s (cpu); 0.0466682s (thread); 0s (gc) │ │ │ │ │ │ 4 3 2 3 2 2 3 2 2 2 2 2 2 │ │ │ o6 = ideal (x x - x x , x x x - x x x , x x x - x x x , x x x - x x x , │ │ │ 2 3 1 3 1 2 3 0 1 3 0 2 3 0 1 3 1 2 3 0 1 3 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 3 2 3 3 2 4 2 2 2 3 │ │ │ x x x x - x x , x x x - x x , x x - 4x x x x + 3x x x , x x x - │ │ │ @@ -117,24 +117,24 @@ │ │ │ o9 = ideal(x x + x x ) │ │ │ 0 1 2 3 │ │ │ │ │ │ o9 : Ideal of P3 │ │ │ │ │ │ i10 : -- tangential Chow forms of Q │ │ │ time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2)) │ │ │ - -- used 0.269994s (cpu); 0.136493s (thread); 0s (gc) │ │ │ + -- used 0.290456s (cpu); 0.153597s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o10 = (x x + x x , x - 4x x + 2x x + x , x x + │ │ │ 0 1 2 3 0,1 0,2 1,3 0,1 2,3 2,3 0,1,2 0,1,3 │ │ │ ----------------------------------------------------------------------- │ │ │ x x ) │ │ │ 0,2,3 1,2,3 │ │ │ │ │ │ o10 : Sequence │ │ │ │ │ │ i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2)) │ │ │ - -- used 0.189432s (cpu); 0.117846s (thread); 0s (gc) │ │ │ + -- used 0.17532s (cpu); 0.106165s (thread); 0s (gc) │ │ │ │ │ │ o11 = true │ │ │ │ │ │ i12 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Form.out │ │ │ @@ -16,15 +16,15 @@ │ │ │ │ │ │ ZZ │ │ │ o2 : Ideal of ----[x ..x ] │ │ │ 3331 0 5 │ │ │ │ │ │ i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an affine chart of the Grassmannian) │ │ │ time ChowV = chowForm(V,AffineChartGrass=>{1,2,3}) │ │ │ - -- used 5.08382s (cpu); 4.55589s (thread); 0s (gc) │ │ │ + -- used 5.56711s (cpu); 5.13335s (thread); 0s (gc) │ │ │ │ │ │ 4 2 2 2 2 │ │ │ o3 = x + 2x x x + x x - 2x x x + │ │ │ 1,2,4 0,2,4 1,2,4 2,3,4 0,2,4 2,3,4 1,2,3 1,2,4 1,2,5 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 2 │ │ │ x x - x x x + x x x x + │ │ │ @@ -143,19 +143,19 @@ │ │ │ 3331 0,1,2 0,1,3 0,2,3 1,2,3 0,1,4 0,2,4 1,2,4 0,3,4 1,3,4 2,3,4 0,1,5 0,2,5 1,2,5 0,3,5 1,3,5 2,3,5 0,4,5 1,4,5 2,4,5 3,4,5 │ │ │ o3 : ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- │ │ │ (x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x - x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x + x x - x x + x x , x x - x x + x x , x x - x x + x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x - x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x ) │ │ │ 2,3,5 1,4,5 1,3,5 2,4,5 1,2,5 3,4,5 2,3,4 1,4,5 1,3,4 2,4,5 1,2,4 3,4,5 2,3,5 0,4,5 0,3,5 2,4,5 0,2,5 3,4,5 1,3,5 0,4,5 0,3,5 1,4,5 0,1,5 3,4,5 1,2,5 0,4,5 0,2,5 1,4,5 0,1,5 2,4,5 2,3,4 0,4,5 0,3,4 2,4,5 0,2,4 3,4,5 1,3,4 0,4,5 0,3,4 1,4,5 0,1,4 3,4,5 1,2,4 0,4,5 0,2,4 1,4,5 0,1,4 2,4,5 1,2,3 0,4,5 0,2,3 1,4,5 0,1,3 2,4,5 0,1,2 3,4,5 2,3,4 1,3,5 1,3,4 2,3,5 1,2,3 3,4,5 1,2,5 0,3,5 0,2,5 1,3,5 0,1,5 2,3,5 2,3,4 0,3,5 0,3,4 2,3,5 0,2,3 3,4,5 1,3,4 0,3,5 0,3,4 1,3,5 0,1,3 3,4,5 1,2,4 0,3,5 0,2,4 1,3,5 0,1,4 2,3,5 0,1,2 3,4,5 1,2,3 0,3,5 0,2,3 1,3,5 0,1,3 2,3,5 2,3,4 1,2,5 1,2,4 2,3,5 1,2,3 2,4,5 1,3,4 1,2,5 1,2,4 1,3,5 1,2,3 1,4,5 0,3,4 1,2,5 0,2,4 1,3,5 0,1,4 2,3,5 0,2,3 1,4,5 0,1,3 2,4,5 0,1,2 3,4,5 2,3,4 0,2,5 0,2,4 2,3,5 0,2,3 2,4,5 1,3,4 0,2,5 0,2,4 1,3,5 0,2,3 1,4,5 0,1,2 3,4,5 0,3,4 0,2,5 0,2,4 0,3,5 0,2,3 0,4,5 1,2,4 0,2,5 0,2,4 1,2,5 0,1,2 2,4,5 1,2,3 0,2,5 0,2,3 1,2,5 0,1,2 2,3,5 2,3,4 0,1,5 0,1,4 2,3,5 0,1,3 2,4,5 0,1,2 3,4,5 1,3,4 0,1,5 0,1,4 1,3,5 0,1,3 1,4,5 0,3,4 0,1,5 0,1,4 0,3,5 0,1,3 0,4,5 1,2,4 0,1,5 0,1,4 1,2,5 0,1,2 1,4,5 0,2,4 0,1,5 0,1,4 0,2,5 0,1,2 0,4,5 1,2,3 0,1,5 0,1,3 1,2,5 0,1,2 1,3,5 0,2,3 0,1,5 0,1,3 0,2,5 0,1,2 0,3,5 1,2,4 0,3,4 0,2,4 1,3,4 0,1,4 2,3,4 1,2,3 0,3,4 0,2,3 1,3,4 0,1,3 2,3,4 1,2,3 0,2,4 0,2,3 1,2,4 0,1,2 2,3,4 1,2,3 0,1,4 0,1,3 1,2,4 0,1,2 1,3,4 0,2,3 0,1,4 0,1,3 0,2,4 0,1,2 0,3,4 │ │ │ │ │ │ i4 : -- equivalently (but faster)... │ │ │ time assert(ChowV === chowForm f) │ │ │ - -- used 1.11624s (cpu); 0.979064s (thread); 0s (gc) │ │ │ + -- used 1.19743s (cpu); 1.11522s (thread); 0s (gc) │ │ │ │ │ │ i5 : -- X-resultant of V │ │ │ time Xres = fromPluckerToStiefel dualize ChowV; │ │ │ - -- used 0.367959s (cpu); 0.220101s (thread); 0s (gc) │ │ │ + -- used 0.289111s (cpu); 0.22226s (thread); 0s (gc) │ │ │ │ │ │ i6 : -- three generic ternary quadrics │ │ │ F = genericPolynomials({2,2,2},ZZ/3331) │ │ │ │ │ │ 2 2 2 2 2 │ │ │ o6 = {a x + a x x + a x + a x x + a x x + a x , b x + b x x + b x + │ │ │ 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 │ │ │ @@ -164,12 +164,12 @@ │ │ │ b x x + b x x + b x , c x + c x x + c x + c x x + c x x + c x } │ │ │ 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : -- resultant of the three forms │ │ │ time resF = resultant F; │ │ │ - -- used 0.289565s (cpu); 0.206175s (thread); 0s (gc) │ │ │ + -- used 0.287529s (cpu); 0.210065s (thread); 0s (gc) │ │ │ │ │ │ i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring Xres)) │ │ │ │ │ │ i9 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_discriminant_lp__Ring__Element_rp.out │ │ │ @@ -4,30 +4,30 @@ │ │ │ │ │ │ 2 2 │ │ │ o2 = a*x + b*x*y + c*y │ │ │ │ │ │ o2 : ZZ[a..c][x..y] │ │ │ │ │ │ i3 : time discriminant F │ │ │ - -- used 0.00876412s (cpu); 0.00875934s (thread); 0s (gc) │ │ │ + -- used 0.0110385s (cpu); 0.0110377s (thread); 0s (gc) │ │ │ │ │ │ 2 │ │ │ o3 = - b + 4a*c │ │ │ │ │ │ o3 : ZZ[a..c] │ │ │ │ │ │ i4 : ZZ[a,b,c,d][x,y]; F = a*x^3+b*x^2*y+c*x*y^2+d*y^3 │ │ │ │ │ │ 3 2 2 3 │ │ │ o5 = a*x + b*x y + c*x*y + d*y │ │ │ │ │ │ o5 : ZZ[a..d][x..y] │ │ │ │ │ │ i6 : time discriminant F │ │ │ - -- used 0.00928698s (cpu); 0.00928758s (thread); 0s (gc) │ │ │ + -- used 0.0119987s (cpu); 0.0120007s (thread); 0s (gc) │ │ │ │ │ │ 2 2 3 3 2 2 │ │ │ o6 = - b c + 4a*c + 4b d - 18a*b*c*d + 27a d │ │ │ │ │ │ o6 : ZZ[a..d] │ │ │ │ │ │ i7 : x=symbol x; R=ZZ/331[x_0..x_3] │ │ │ @@ -59,15 +59,15 @@ │ │ │ 4 3 4 4 3 4 │ │ │ o12 = (t + t )x - t x x + t x + (t - t )x + t x x + t x │ │ │ 0 1 0 1 0 1 0 1 0 1 2 1 2 3 0 3 │ │ │ │ │ │ o12 : R' │ │ │ │ │ │ i13 : time D=discriminant pencil │ │ │ - -- used 0.521101s (cpu); 0.460767s (thread); 0s (gc) │ │ │ + -- used 0.515577s (cpu); 0.448516s (thread); 0s (gc) │ │ │ │ │ │ 108 106 2 102 6 100 8 98 10 96 12 │ │ │ o13 = - 62t + 19t t + 160t t + 91t t + 129t t + 117t t + │ │ │ 0 0 1 0 1 0 1 0 1 0 1 │ │ │ ----------------------------------------------------------------------- │ │ │ 94 14 92 16 90 18 88 20 86 22 84 24 │ │ │ 161t t + 124t t - 82t t - 21t t - 49t t - 123t t + │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_dual__Variety.out │ │ │ @@ -9,25 +9,25 @@ │ │ │ x x ) │ │ │ 0 3 │ │ │ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ 0 5 │ │ │ │ │ │ i2 : time V' = dualVariety V │ │ │ - -- used 0.196307s (cpu); 0.124873s (thread); 0s (gc) │ │ │ + -- used 0.205735s (cpu); 0.134987s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 │ │ │ o2 = ideal(x x - x x x + x x + x x - 4x x x ) │ │ │ 2 3 1 2 4 0 4 1 5 0 3 5 │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ 0 5 │ │ │ │ │ │ i3 : time V == dualVariety V' │ │ │ - -- used 0.205617s (cpu); 0.14044s (thread); 0s (gc) │ │ │ + -- used 0.228092s (cpu); 0.159805s (thread); 0s (gc) │ │ │ │ │ │ o3 = true │ │ │ │ │ │ i4 : F = first genericPolynomials({3,-1,-1},ZZ/3331) │ │ │ │ │ │ 3 2 2 3 2 2 2 │ │ │ o4 = a x + a x x + a x x + a x + a x x + a x x x + a x x + a x x + │ │ │ @@ -38,22 +38,22 @@ │ │ │ 8 1 2 9 2 │ │ │ │ │ │ ZZ │ │ │ o4 : ----[a ..a ][x ..x ] │ │ │ 3331 0 9 0 2 │ │ │ │ │ │ i5 : time discF = ideal discriminant F; │ │ │ - -- used 0.057736s (cpu); 0.0577424s (thread); 0s (gc) │ │ │ + -- used 0.0687417s (cpu); 0.0687418s (thread); 0s (gc) │ │ │ │ │ │ ZZ │ │ │ o5 : Ideal of ----[a ..a ] │ │ │ 3331 0 9 │ │ │ │ │ │ i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true); │ │ │ - -- used 0.625488s (cpu); 0.556064s (thread); 0s (gc) │ │ │ + -- used 0.740831s (cpu); 0.663289s (thread); 0s (gc) │ │ │ │ │ │ ZZ │ │ │ o6 : Ideal of ----[x ..x ] │ │ │ 3331 0 9 │ │ │ │ │ │ i7 : discF == sub(Z,vars ring discF) and Z == sub(discF,vars ring Z) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_from__Plucker__To__Stiefel.out │ │ │ @@ -6,15 +6,15 @@ │ │ │ o1 = ideal (x - x x , x x - x x , x - x x ) │ │ │ 2 1 3 1 2 0 3 1 0 2 │ │ │ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ i2 : time fromPluckerToStiefel dualize chowForm C │ │ │ - -- used 0.126592s (cpu); 0.0637976s (thread); 0s (gc) │ │ │ + -- used 0.138147s (cpu); 0.0700527s (thread); 0s (gc) │ │ │ │ │ │ 3 3 2 2 2 2 2 3 │ │ │ o2 = - x x + x x x x - x x x x + x x x - │ │ │ 0,3 1,0 0,2 0,3 1,0 1,1 0,1 0,3 1,0 1,1 0,0 0,3 1,1 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 2 2 │ │ │ x x x x + 2x x x x + x x x x x x - │ │ │ @@ -56,15 +56,15 @@ │ │ │ x x x x - 2x x x x - x x x x + x x │ │ │ 0,0 0,1 1,1 1,3 0,0 0,2 1,1 1,3 0,0 0,1 1,2 1,3 0,0 1,3 │ │ │ │ │ │ o2 : QQ[x ..x ] │ │ │ 0,0 1,3 │ │ │ │ │ │ i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1}) │ │ │ - -- used 0.137358s (cpu); 0.0611511s (thread); 0s (gc) │ │ │ + -- used 0.133466s (cpu); 0.0649393s (thread); 0s (gc) │ │ │ │ │ │ 3 2 2 │ │ │ o3 = - x x + x x x - x x x + x x + 3x x x - │ │ │ 0,3 1,2 0,2 1,2 1,3 0,2 0,3 1,2 0,2 1,3 0,3 1,2 1,3 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 3 2 │ │ │ 2x x + x + x │ │ │ @@ -85,15 +85,15 @@ │ │ │ │ │ │ o4 : QQ[a ..a ] │ │ │ 0,0 1,1 │ │ │ │ │ │ i5 : w = chowForm C; │ │ │ │ │ │ i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel(w,AffineChartGrass=>s)) │ │ │ - -- used 0.0199213s (cpu); 0.0199207s (thread); 0s (gc) │ │ │ + -- used 0.0251917s (cpu); 0.0251914s (thread); 0s (gc) │ │ │ │ │ │ 3 2 3 2 │ │ │ o6 = {ideal(- x x + x x x - x - 3x x x + 2x x + │ │ │ 0,3 1,2 0,2 1,2 1,3 0,2 0,2 0,3 1,2 0,2 1,3 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 2 3 2 │ │ │ x x x - x x + x ), ideal(x x - 2x x x x + │ │ │ @@ -130,14 +130,14 @@ │ │ │ 2 3 2 │ │ │ 2x x - x + x )} │ │ │ 0,0 1,1 1,1 1,0 │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : time apply(U,u->dim singularLocus u) │ │ │ - -- used 0.09626s (cpu); 0.0369501s (thread); 0s (gc) │ │ │ + -- used 0.109064s (cpu); 0.042567s (thread); 0s (gc) │ │ │ │ │ │ o7 = {2, 2, 2, 2, 2, 2} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_hurwitz__Form.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ + -p + -p p + 7p p + 6p p + -p p + --p ) │ │ │ 4 3 9 0 4 1 4 2 4 9 3 4 10 4 │ │ │ │ │ │ o1 : Ideal of QQ[p ..p ] │ │ │ 0 4 │ │ │ │ │ │ i2 : time hurwitzForm Q │ │ │ - -- used 0.0375527s (cpu); 0.0375568s (thread); 0s (gc) │ │ │ + -- used 0.0450159s (cpu); 0.0450171s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o2 = 11966535p + 14645610p p + 11354175p + 1666980p p + │ │ │ 0,1 0,1 0,2 0,2 0,1 1,2 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 │ │ │ 4456620p p + 1127196p + 54176850p p + 20326950p p + │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_is__Coisotropic.out │ │ │ @@ -26,15 +26,15 @@ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ o1 : -------------------------------------- │ │ │ p p - p p + p p │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ │ │ i2 : time isCoisotropic w │ │ │ - -- used 0.0967611s (cpu); 0.0287666s (thread); 0s (gc) │ │ │ + -- used 0.0871977s (cpu); 0.0257295s (thread); 0s (gc) │ │ │ │ │ │ o2 = true │ │ │ │ │ │ i3 : -- random quadric in G(1,3) │ │ │ w' = random(2,Grass(1,3)) │ │ │ │ │ │ 3 2 3 7 2 5 7 2 10 │ │ │ @@ -56,12 +56,12 @@ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ o3 : -------------------------------------- │ │ │ p p - p p + p p │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ │ │ i4 : time isCoisotropic w' │ │ │ - -- used 0.00675958s (cpu); 0.00676007s (thread); 0s (gc) │ │ │ + -- used 0.00859799s (cpu); 0.00859696s (thread); 0s (gc) │ │ │ │ │ │ o4 = false │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_is__In__Coisotropic.out │ │ │ @@ -31,12 +31,12 @@ │ │ │ 4 5 │ │ │ │ │ │ ZZ │ │ │ o3 : Ideal of -----[x ..x ] │ │ │ 33331 0 5 │ │ │ │ │ │ i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I)) │ │ │ - -- used 0.0186416s (cpu); 0.0186435s (thread); 0s (gc) │ │ │ + -- used 0.0210315s (cpu); 0.0210307s (thread); 0s (gc) │ │ │ │ │ │ o4 = true │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_macaulay__Formula.out │ │ │ @@ -13,15 +13,15 @@ │ │ │ 2 2 2 3 │ │ │ c x x x + c x x + c x x + c x x + c x } │ │ │ 4 0 1 2 7 1 2 5 0 2 8 1 2 9 2 │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : time (D,D') = macaulayFormula F │ │ │ - -- used 0.00367518s (cpu); 0.00367216s (thread); 0s (gc) │ │ │ + -- used 0.00417748s (cpu); 0.00417493s (thread); 0s (gc) │ │ │ │ │ │ o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0 0 0 0 0 0 0 0 0 0 0 │ │ │ | 0 a_0 0 a_1 a_2 0 a_3 a_4 a_5 0 0 0 0 0 0 0 0 │ │ │ | 0 0 a_0 0 a_1 a_2 0 a_3 a_4 a_5 0 0 0 0 0 0 0 │ │ │ | 0 0 0 a_0 0 0 a_1 a_2 0 0 a_3 a_4 a_5 0 0 0 0 │ │ │ | 0 0 0 0 a_0 0 0 a_1 a_2 0 0 a_3 a_4 a_5 0 0 0 │ │ │ | 0 0 0 0 0 a_0 0 0 a_1 a_2 0 0 a_3 a_4 a_5 0 0 │ │ │ @@ -78,15 +78,15 @@ │ │ │ 10 2 7 2 5 3 │ │ │ --p p + -p p + -p } │ │ │ 9 0 2 8 1 2 6 2 │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : time (D,D') = macaulayFormula F │ │ │ - -- used 0.00248359s (cpu); 0.00248394s (thread); 0s (gc) │ │ │ + -- used 0.00289059s (cpu); 0.0028891s (thread); 0s (gc) │ │ │ │ │ │ o4 = (| 9/2 9/4 3/4 7/4 7/9 7/10 0 0 0 0 0 0 0 0 0 │ │ │ | 0 9/2 0 9/4 3/4 0 7/4 7/9 7/10 0 0 0 0 0 0 │ │ │ | 0 0 9/2 0 9/4 3/4 0 7/4 7/9 7/10 0 0 0 0 0 │ │ │ | 0 0 0 9/2 0 0 9/4 3/4 0 0 7/4 7/9 7/10 0 0 │ │ │ | 0 0 0 0 9/2 0 0 9/4 3/4 0 0 7/4 7/9 7/10 0 │ │ │ | 0 0 0 0 0 9/2 0 0 9/4 3/4 0 0 7/4 7/9 7/10 │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_plucker.out │ │ │ @@ -9,29 +9,29 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ 664x ) │ │ │ 4 │ │ │ │ │ │ o3 : Ideal of P4 │ │ │ │ │ │ i4 : time p = plucker L │ │ │ - -- used 0.00445528s (cpu); 0.00445317s (thread); 0s (gc) │ │ │ + -- used 0.00939157s (cpu); 0.00939111s (thread); 0s (gc) │ │ │ │ │ │ o4 = ideal (x + 8480x , x - 6727x , x + 15777x , x + │ │ │ 2,4 3,4 1,4 3,4 0,4 3,4 2,3 │ │ │ ------------------------------------------------------------------------ │ │ │ 11656x , x - 14853x , x + 664x , x + 13522x , x + │ │ │ 3,4 1,3 3,4 0,3 3,4 1,2 3,4 0,2 │ │ │ ------------------------------------------------------------------------ │ │ │ 11804x , x + 14854x ) │ │ │ 3,4 0,1 3,4 │ │ │ │ │ │ o4 : Ideal of G'1'4 │ │ │ │ │ │ i5 : time L' = plucker p │ │ │ - -- used 0.114222s (cpu); 0.0497012s (thread); 0s (gc) │ │ │ + -- used 0.147705s (cpu); 0.0698158s (thread); 0s (gc) │ │ │ │ │ │ o5 = ideal (x + 8480x - 11656x , x - 6727x + 14853x , x + 15777x - │ │ │ 2 3 4 1 3 4 0 3 │ │ │ ------------------------------------------------------------------------ │ │ │ 664x ) │ │ │ 4 │ │ │ │ │ │ @@ -40,25 +40,25 @@ │ │ │ i6 : assert(L' == L) │ │ │ │ │ │ i7 : Y = ideal apply(5,i->random(1,G'1'4)); -- an elliptic curve │ │ │ │ │ │ o7 : Ideal of G'1'4 │ │ │ │ │ │ i8 : time W = plucker Y; -- surface swept out by the lines of Y │ │ │ - -- used 0.0312756s (cpu); 0.0312807s (thread); 0s (gc) │ │ │ + -- used 0.0394625s (cpu); 0.0394629s (thread); 0s (gc) │ │ │ │ │ │ o8 : Ideal of P4 │ │ │ │ │ │ i9 : (codim W,degree W) │ │ │ │ │ │ o9 = (2, 5) │ │ │ │ │ │ o9 : Sequence │ │ │ │ │ │ i10 : time Y' = plucker(W,1); -- variety of lines contained in W │ │ │ - -- used 0.222489s (cpu); 0.166167s (thread); 0s (gc) │ │ │ + -- used 0.258422s (cpu); 0.192273s (thread); 0s (gc) │ │ │ │ │ │ o10 : Ideal of G'1'4 │ │ │ │ │ │ i11 : assert(Y' == Y) │ │ │ │ │ │ i12 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp..._cm__Algorithm_eq_gt..._rp.out │ │ │ @@ -35,15 +35,15 @@ │ │ │ 3 2 9 7 2 9 3 1 8 4 │ │ │ -b)y*w + (-a + -b)z*w + (-a + 2b)w , 2x + -y + -z + -w} │ │ │ 4 8 8 7 4 3 5 │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : time resultant(F,Algorithm=>"Poisson2") │ │ │ - -- used 0.364823s (cpu); 0.192307s (thread); 0s (gc) │ │ │ + -- used 0.473887s (cpu); 0.256431s (thread); 0s (gc) │ │ │ │ │ │ 21002161660529014459938925799 5 2085933800619238998825958079203 4 │ │ │ o3 = - -----------------------------a - -------------------------------a b - │ │ │ 2222549728809984000000 12700284164628480000000 │ │ │ ------------------------------------------------------------------------ │ │ │ 348237304382147063838108483692249 3 2 │ │ │ ---------------------------------a b - │ │ │ @@ -56,15 +56,15 @@ │ │ │ 1146977327343523453866040839029 4 194441910898734675845094443 5 │ │ │ -------------------------------a*b - ---------------------------b │ │ │ 1119954511872000000000 895963609497600000 │ │ │ │ │ │ o3 : QQ[a..b] │ │ │ │ │ │ i4 : time resultant(F,Algorithm=>"Macaulay2") │ │ │ - -- used 0.157645s (cpu); 0.0917722s (thread); 0s (gc) │ │ │ + -- used 0.168634s (cpu); 0.100061s (thread); 0s (gc) │ │ │ │ │ │ 21002161660529014459938925799 5 2085933800619238998825958079203 4 │ │ │ o4 = - -----------------------------a - -------------------------------a b - │ │ │ 2222549728809984000000 12700284164628480000000 │ │ │ ------------------------------------------------------------------------ │ │ │ 348237304382147063838108483692249 3 2 │ │ │ ---------------------------------a b - │ │ │ @@ -77,15 +77,15 @@ │ │ │ 1146977327343523453866040839029 4 194441910898734675845094443 5 │ │ │ -------------------------------a*b - ---------------------------b │ │ │ 1119954511872000000000 895963609497600000 │ │ │ │ │ │ o4 : QQ[a..b] │ │ │ │ │ │ i5 : time resultant(F,Algorithm=>"Poisson") │ │ │ - -- used 0.381787s (cpu); 0.323386s (thread); 0s (gc) │ │ │ + -- used 0.410825s (cpu); 0.344613s (thread); 0s (gc) │ │ │ │ │ │ 21002161660529014459938925799 5 2085933800619238998825958079203 4 │ │ │ o5 = - -----------------------------a - -------------------------------a b - │ │ │ 2222549728809984000000 12700284164628480000000 │ │ │ ------------------------------------------------------------------------ │ │ │ 348237304382147063838108483692249 3 2 │ │ │ ---------------------------------a b - │ │ │ @@ -98,15 +98,15 @@ │ │ │ 1146977327343523453866040839029 4 194441910898734675845094443 5 │ │ │ -------------------------------a*b - ---------------------------b │ │ │ 1119954511872000000000 895963609497600000 │ │ │ │ │ │ o5 : QQ[a..b] │ │ │ │ │ │ i6 : time resultant(F,Algorithm=>"Macaulay") │ │ │ - -- used 0.619406s (cpu); 0.555197s (thread); 0s (gc) │ │ │ + -- used 0.673783s (cpu); 0.607181s (thread); 0s (gc) │ │ │ │ │ │ 21002161660529014459938925799 5 2085933800619238998825958079203 4 │ │ │ o6 = - -----------------------------a - -------------------------------a b - │ │ │ 2222549728809984000000 12700284164628480000000 │ │ │ ------------------------------------------------------------------------ │ │ │ 348237304382147063838108483692249 3 2 │ │ │ ---------------------------------a b - │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp__Matrix_rp.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ │ │ │ 2 2 3 2 4 │ │ │ o2 = {x + 3t*y*z - u*z , (t + 3u - 1)x - y, - t*x*y + t*x y*z + u*z } │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : time resultant F │ │ │ - -- used 0.023138s (cpu); 0.0231407s (thread); 0s (gc) │ │ │ + -- used 0.0268743s (cpu); 0.0267075s (thread); 0s (gc) │ │ │ │ │ │ 12 11 2 10 3 9 4 8 5 7 6 │ │ │ o3 = - 81t u - 1701t u - 15309t u - 76545t u - 229635t u - 413343t u │ │ │ ------------------------------------------------------------------------ │ │ │ 6 7 5 8 11 10 2 9 3 │ │ │ - 413343t u - 177147t u + 567t u + 10206t u + 76545t u + │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -64,15 +64,15 @@ │ │ │ 3 │ │ │ + c x } │ │ │ 9 2 │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : time resultant F │ │ │ - -- used 2.65652s (cpu); 1.97047s (thread); 0s (gc) │ │ │ + -- used 2.62093s (cpu); 2.03357s (thread); 0s (gc) │ │ │ │ │ │ 6 3 2 5 2 2 2 4 2 2 3 3 3 2 2 4 2 2 │ │ │ o5 = a b c - 3a a b b c + 3a a b b c - a a b c + 3a a b b c - │ │ │ 2 3 0 1 2 3 4 0 1 2 3 4 0 1 2 4 0 1 2 3 5 0 │ │ │ ------------------------------------------------------------------------ │ │ │ 3 3 2 4 2 2 2 4 2 2 2 5 2 2 6 3 2 │ │ │ 6a a b b b c + 3a a b b c + 3a a b b c - 3a a b b c + a b c - │ │ │ @@ -1690,12 +1690,12 @@ │ │ │ 2 2 2 2 │ │ │ b x x + b x x + b x , c x + c x x + c x + c x x + c x x + c x } │ │ │ 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : time # terms resultant F │ │ │ - -- used 0.530258s (cpu); 0.383961s (thread); 0s (gc) │ │ │ + -- used 0.627448s (cpu); 0.475334s (thread); 0s (gc) │ │ │ │ │ │ o7 = 21894 │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_tangential__Chow__Form.out │ │ │ @@ -8,15 +8,15 @@ │ │ │ 1 2 0 3 1 3 0 4 3 2 4 │ │ │ │ │ │ o2 : Ideal of QQ[p ..p ] │ │ │ 0 4 │ │ │ │ │ │ i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form) │ │ │ time tangentialChowForm(S,0) │ │ │ - -- used 0.0371985s (cpu); 0.0372021s (thread); 0s (gc) │ │ │ + -- used 0.0458259s (cpu); 0.0458254s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o3 = p p - p p p - p p p + p p p + p p + │ │ │ 1,3 2,3 1,2 1,3 2,4 0,3 1,3 2,4 0,2 1,4 2,4 1,2 3,4 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 │ │ │ p p - 2p p p - p p p │ │ │ @@ -26,15 +26,15 @@ │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 0,4 1,4 2,4 3,4 │ │ │ o3 : ---------------------------------------------------------------------------------------------------------------------------------------------------------------- │ │ │ (p p - p p + p p , p p - p p + p p , p p - p p + p p , p p - p p + p p , p p - p p + p p ) │ │ │ 2,3 1,4 1,3 2,4 1,2 3,4 2,3 0,4 0,3 2,4 0,2 3,4 1,3 0,4 0,3 1,4 0,1 3,4 1,2 0,4 0,2 1,4 0,1 2,4 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ │ │ i4 : -- 1-th associated hypersurface of S in G(2,4) │ │ │ time tangentialChowForm(S,1) │ │ │ - -- used 0.15512s (cpu); 0.0919453s (thread); 0s (gc) │ │ │ + -- used 0.179818s (cpu); 0.10462s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 3 2 2 │ │ │ o4 = p p + p p - 2p p + p p - │ │ │ 1,2,3 1,2,4 0,2,4 1,2,4 0,2,3 1,2,4 0,2,4 0,3,4 │ │ │ ------------------------------------------------------------------------ │ │ │ 3 3 3 │ │ │ 4p p - 4p p - 2p p + │ │ │ @@ -68,32 +68,32 @@ │ │ │ 0,1,2 0,1,3 0,2,3 1,2,3 0,1,4 0,2,4 1,2,4 0,3,4 1,3,4 2,3,4 │ │ │ o4 : ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- │ │ │ (p p - p p + p p , p p - p p + p p , p p - p p + p p , p p - p p + p p , p p - p p + p p ) │ │ │ 1,2,4 0,3,4 0,2,4 1,3,4 0,1,4 2,3,4 1,2,3 0,3,4 0,2,3 1,3,4 0,1,3 2,3,4 1,2,3 0,2,4 0,2,3 1,2,4 0,1,2 2,3,4 1,2,3 0,1,4 0,1,3 1,2,4 0,1,2 1,3,4 0,2,3 0,1,4 0,1,3 0,2,4 0,1,2 0,3,4 │ │ │ │ │ │ i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent hyperplanes to S) │ │ │ time tangentialChowForm(S,2) │ │ │ - -- used 0.0308352s (cpu); 0.0308383s (thread); 0s (gc) │ │ │ + -- used 0.0396845s (cpu); 0.0396852s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o5 = p p - p p p + p p │ │ │ 0,1,3,4 0,2,3,4 0,1,2,4 0,2,3,4 1,2,3,4 0,1,2,3 1,2,3,4 │ │ │ │ │ │ o5 : QQ[p ..p , p , p , p ] │ │ │ 0,1,2,3 0,1,2,4 0,1,3,4 0,2,3,4 1,2,3,4 │ │ │ │ │ │ i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing │ │ │ time S' = ideal dualize tangentialChowForm(S,2) │ │ │ - -- used 0.121596s (cpu); 0.0529281s (thread); 0s (gc) │ │ │ + -- used 0.137081s (cpu); 0.0640525s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o6 = ideal(p p - p p p + p p ) │ │ │ 1 2 0 1 3 0 4 │ │ │ │ │ │ o6 : Ideal of QQ[p ..p ] │ │ │ 0 4 │ │ │ │ │ │ i7 : -- we then can recover S │ │ │ time assert(dualize tangentialChowForm(S',3) == S) │ │ │ - -- used 0.18169s (cpu); 0.117706s (thread); 0s (gc) │ │ │ + -- used 0.198309s (cpu); 0.12308s (thread); 0s (gc) │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html │ │ │ @@ -91,24 +91,24 @@ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2);
│ │ │ - -- used 0.146312s (cpu); 0.0851178s (thread); 0s (gc)
    │ │ │ + -- used 0.143875s (cpu); 0.0759174s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │

    In the next example, we calculate the defining ideal of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^7$ and that of its dual variety.

    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1)
│ │ │ - -- used 0.145095s (cpu); 0.074383s (thread); 0s (gc)
│ │ │ + -- used 0.163961s (cpu); 0.0935498s (thread); 0s (gc)
│ │ │  
│ │ │                                                                             
│ │ │  o4 = (ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │                0,3 1,2    0,2 1,3   1,0 1,1    1,2 1,3   0,3 1,1    0,1 1,3 
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                              
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ @@ -135,26 +135,26 @@
│ │ │
    │ │ │

    If the option Duality is set to true, then the method applies the so-called "dual Cayley trick".

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -38,20 +38,20 @@ │ │ │ │ │ │ │ │ o2 = ideal(x x - x x ) │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ │ 0 3 │ │ │ │ i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2); │ │ │ │ - -- used 0.146312s (cpu); 0.0851178s (thread); 0s (gc) │ │ │ │ + -- used 0.143875s (cpu); 0.0759174s (thread); 0s (gc) │ │ │ │ In the next example, we calculate the defining ideal of $\mathbb │ │ │ │ {P}^1\times\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^7$ and that of its │ │ │ │ dual variety. │ │ │ │ i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1) │ │ │ │ - -- used 0.145095s (cpu); 0.074383s (thread); 0s (gc) │ │ │ │ + -- used 0.163961s (cpu); 0.0935498s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ o4 = (ideal (x x - x x , x x - x x , x x - x x , │ │ │ │ 0,3 1,2 0,2 1,3 1,0 1,1 1,2 1,3 0,3 1,1 0,1 1,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ │ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ │ @@ -73,18 +73,18 @@ │ │ │ │ 4x x x x - 2x x x x + x x )) │ │ │ │ 0,0 0,1 1,2 1,3 0,2 0,3 1,2 1,3 0,2 1,3 │ │ │ │ │ │ │ │ o4 : Sequence │ │ │ │ If the option Duality is set to true, then the method applies the so-called │ │ │ │ "dual Cayley trick". │ │ │ │ i5 : time cayleyTrick(P1xP1,1,Duality=>true); │ │ │ │ - -- used 0.257486s (cpu); 0.11514s (thread); 0s (gc) │ │ │ │ + -- used 0.28628s (cpu); 0.12188s (thread); 0s (gc) │ │ │ │ i6 : assert(oo == (P1xP1xP1,P1xP1xP1')) │ │ │ │ i7 : time cayleyTrick(P1xP1,2,Duality=>true); │ │ │ │ - -- used 0.168639s (cpu); 0.10303s (thread); 0s (gc) │ │ │ │ + -- used 0.185993s (cpu); 0.114514s (thread); 0s (gc) │ │ │ │ i8 : assert(oo == (P1xP1xP2,P1xP1xP2')) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_u_a_l_V_a_r_i_e_t_y -- projective dual variety │ │ │ │ ********** WWaayyss ttoo uussee ccaayylleeyyTTrriicckk:: ********** │ │ │ │ * cayleyTrick(Ideal,ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_a_y_l_e_y_T_r_i_c_k is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html │ │ │ @@ -95,15 +95,15 @@ │ │ │ o2 : Ideal of P3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : time cayleyTrick(P1xP1,1,Duality=>true);
│ │ │ - -- used 0.257486s (cpu); 0.11514s (thread); 0s (gc)
    │ │ │ + -- used 0.28628s (cpu); 0.12188s (thread); 0s (gc) │ │ │
    │ │ │ 
    i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
    │ │ │
    │ │ │ 
    i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ - -- used 0.168639s (cpu); 0.10303s (thread); 0s (gc)
    │ │ │ + -- used 0.185993s (cpu); 0.114514s (thread); 0s (gc) │ │ │
    │ │ │ 
    i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))
    │ │ │
    │ │ │ 
    i3 : -- Chow equations of C
│ │ │       time eqsC = chowEquations chowForm C
│ │ │ - -- used 0.135904s (cpu); 0.0637037s (thread); 0s (gc)
│ │ │ + -- used 0.135231s (cpu); 0.0709316s (thread); 0s (gc)
│ │ │  
│ │ │               2 2    2 2    2 2    4                2      2 2   2      
│ │ │  o3 = ideal (x x  + x x  + x x  + x , x x x x  + x x x  + x x , x x x  +
│ │ │               0 3    1 3    2 3    3   0 1 2 3    1 2 3    2 3   0 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │        2        3           2         2      3   3         2          2    2 2
│ │ │       x x x  + x x  - 2x x x  - 2x x x  - x x , x x  + 2x x x  - x x x  + x x 
│ │ │ @@ -167,15 +167,15 @@
│ │ │  o5 : Ideal of P3
    │ │ │
    │ │ │ 
    i6 : -- Chow equations of D
│ │ │       time eqsD = chowEquations chowForm D
│ │ │ - -- used 0.0620687s (cpu); 0.0620762s (thread); 0s (gc)
│ │ │ + -- used 0.0466663s (cpu); 0.0466682s (thread); 0s (gc)
│ │ │  
│ │ │               4      3 2     3        2 2     3      2   2   2 2      2   2 
│ │ │  o6 = ideal (x x  - x x , x x x  - x x x , x x x  - x x x , x x x  - x x x ,
│ │ │               2 3    1 3   1 2 3    0 1 3   0 2 3    0 1 3   1 2 3    0 1 3 
│ │ │       ------------------------------------------------------------------------
│ │ │            2      3 2   3        3 2   4         2         2 2       3    
│ │ │       x x x x  - x x , x x x  - x x , x x  - 4x x x x  + 3x x x , x x x  -
│ │ │ @@ -227,30 +227,30 @@
│ │ │  o9 : Ideal of P3
    │ │ │
    │ │ │ 
    i10 : -- tangential Chow forms of Q
│ │ │        time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))
│ │ │ - -- used 0.269994s (cpu); 0.136493s (thread); 0s (gc)
│ │ │ + -- used 0.290456s (cpu); 0.153597s (thread); 0s (gc)
│ │ │  
│ │ │                       2                              2
│ │ │  o10 = (x x  + x x , x    - 4x   x    + 2x   x    + x   , x     x      +
│ │ │          0 1    2 3   0,1     0,2 1,3     0,1 2,3    2,3   0,1,2 0,1,3  
│ │ │        -----------------------------------------------------------------------
│ │ │        x     x     )
│ │ │         0,2,3 1,2,3
│ │ │  
│ │ │  o10 : Sequence
    │ │ │
    │ │ │ 
    i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2))
│ │ │ - -- used 0.189432s (cpu); 0.117846s (thread); 0s (gc)
│ │ │ + -- used 0.17532s (cpu); 0.106165s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
    │ │ │
    │ │ │

    Note that chowEquations(W,0) is not the same as chowEquations W.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -28,15 +28,15 @@ │ │ │ │ 2 2 2 2 │ │ │ │ o2 = ideal (x + x + x + x , x x + x x + x x ) │ │ │ │ 0 1 2 3 0 1 1 2 2 3 │ │ │ │ │ │ │ │ o2 : Ideal of P3 │ │ │ │ i3 : -- Chow equations of C │ │ │ │ time eqsC = chowEquations chowForm C │ │ │ │ - -- used 0.135904s (cpu); 0.0637037s (thread); 0s (gc) │ │ │ │ + -- used 0.135231s (cpu); 0.0709316s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 2 2 4 2 2 2 2 │ │ │ │ o3 = ideal (x x + x x + x x + x , x x x x + x x x + x x , x x x + │ │ │ │ 0 3 1 3 2 3 3 0 1 2 3 1 2 3 2 3 0 2 3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 3 2 2 3 3 2 2 2 2 │ │ │ │ x x x + x x - 2x x x - 2x x x - x x , x x + 2x x x - x x x + x x │ │ │ │ @@ -88,15 +88,15 @@ │ │ │ │ 2 3 2 2 │ │ │ │ o5 = ideal (x - x x , x - x x x , x x - x x ) │ │ │ │ 1 0 2 2 0 1 3 1 2 0 3 │ │ │ │ │ │ │ │ o5 : Ideal of P3 │ │ │ │ i6 : -- Chow equations of D │ │ │ │ time eqsD = chowEquations chowForm D │ │ │ │ - -- used 0.0620687s (cpu); 0.0620762s (thread); 0s (gc) │ │ │ │ + -- used 0.0466663s (cpu); 0.0466682s (thread); 0s (gc) │ │ │ │ │ │ │ │ 4 3 2 3 2 2 3 2 2 2 2 2 2 │ │ │ │ o6 = ideal (x x - x x , x x x - x x x , x x x - x x x , x x x - x x x , │ │ │ │ 2 3 1 3 1 2 3 0 1 3 0 2 3 0 1 3 1 2 3 0 1 3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 3 2 3 3 2 4 2 2 2 3 │ │ │ │ x x x x - x x , x x x - x x , x x - 4x x x x + 3x x x , x x x - │ │ │ │ @@ -135,27 +135,27 @@ │ │ │ │ o9 = ideal(x x + x x ) │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o9 : Ideal of P3 │ │ │ │ i10 : -- tangential Chow forms of Q │ │ │ │ time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm │ │ │ │ (Q,1),tangentialChowForm(Q,2)) │ │ │ │ - -- used 0.269994s (cpu); 0.136493s (thread); 0s (gc) │ │ │ │ + -- used 0.290456s (cpu); 0.153597s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o10 = (x x + x x , x - 4x x + 2x x + x , x x + │ │ │ │ 0 1 2 3 0,1 0,2 1,3 0,1 2,3 2,3 0,1,2 0,1,3 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ x x ) │ │ │ │ 0,2,3 1,2,3 │ │ │ │ │ │ │ │ o10 : Sequence │ │ │ │ i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations │ │ │ │ (W2,2)) │ │ │ │ - -- used 0.189432s (cpu); 0.117846s (thread); 0s (gc) │ │ │ │ + -- used 0.17532s (cpu); 0.106165s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 = true │ │ │ │ Note that chowEquations(W,0) is not the same as chowEquations W. │ │ │ │ ********** WWaayyss ttoo uussee cchhoowwEEqquuaattiioonnss:: ********** │ │ │ │ * chowEquations(RingElement) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_h_o_w_E_q_u_a_t_i_o_n_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ 3331 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an affine chart of the Grassmannian)
│ │ │       time ChowV = chowForm(V,AffineChartGrass=>{1,2,3})
│ │ │ - -- used 5.08382s (cpu); 4.55589s (thread); 0s (gc)
│ │ │ + -- used 5.56711s (cpu); 5.13335s (thread); 0s (gc)
│ │ │  
│ │ │        4               2              2     2               2            
│ │ │  o3 = x      + 2x     x     x      + x     x      - 2x     x     x      +
│ │ │        1,2,4     0,2,4 1,2,4 2,3,4    0,2,4 2,3,4     1,2,3 1,2,4 1,2,5  
│ │ │       ------------------------------------------------------------------------
│ │ │        2     2              2                                       
│ │ │       x     x      - x     x     x      + x     x     x     x      +
│ │ │ @@ -232,22 +232,22 @@
│ │ │         2,3,5 1,4,5    1,3,5 2,4,5    1,2,5 3,4,5   2,3,4 1,4,5    1,3,4 2,4,5    1,2,4 3,4,5   2,3,5 0,4,5    0,3,5 2,4,5    0,2,5 3,4,5   1,3,5 0,4,5    0,3,5 1,4,5    0,1,5 3,4,5   1,2,5 0,4,5    0,2,5 1,4,5    0,1,5 2,4,5   2,3,4 0,4,5    0,3,4 2,4,5    0,2,4 3,4,5   1,3,4 0,4,5    0,3,4 1,4,5    0,1,4 3,4,5   1,2,4 0,4,5    0,2,4 1,4,5    0,1,4 2,4,5   1,2,3 0,4,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 1,3,5    1,3,4 2,3,5    1,2,3 3,4,5   1,2,5 0,3,5    0,2,5 1,3,5    0,1,5 2,3,5   2,3,4 0,3,5    0,3,4 2,3,5    0,2,3 3,4,5   1,3,4 0,3,5    0,3,4 1,3,5    0,1,3 3,4,5   1,2,4 0,3,5    0,2,4 1,3,5    0,1,4 2,3,5    0,1,2 3,4,5   1,2,3 0,3,5    0,2,3 1,3,5    0,1,3 2,3,5   2,3,4 1,2,5    1,2,4 2,3,5    1,2,3 2,4,5   1,3,4 1,2,5    1,2,4 1,3,5    1,2,3 1,4,5   0,3,4 1,2,5    0,2,4 1,3,5    0,1,4 2,3,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 0,2,5    0,2,4 2,3,5    0,2,3 2,4,5   1,3,4 0,2,5    0,2,4 1,3,5    0,2,3 1,4,5    0,1,2 3,4,5   0,3,4 0,2,5    0,2,4 0,3,5    0,2,3 0,4,5   1,2,4 0,2,5    0,2,4 1,2,5    0,1,2 2,4,5   1,2,3 0,2,5    0,2,3 1,2,5    0,1,2 2,3,5   2,3,4 0,1,5    0,1,4 2,3,5    0,1,3 2,4,5    0,1,2 3,4,5   1,3,4 0,1,5    0,1,4 1,3,5    0,1,3 1,4,5   0,3,4 0,1,5    0,1,4 0,3,5    0,1,3 0,4,5   1,2,4 0,1,5    0,1,4 1,2,5    0,1,2 1,4,5   0,2,4 0,1,5    0,1,4 0,2,5    0,1,2 0,4,5   1,2,3 0,1,5    0,1,3 1,2,5    0,1,2 1,3,5   0,2,3 0,1,5    0,1,3 0,2,5    0,1,2 0,3,5   1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : -- equivalently (but faster)...
│ │ │       time assert(ChowV === chowForm f)
│ │ │ - -- used 1.11624s (cpu); 0.979064s (thread); 0s (gc)
    │ │ │ + -- used 1.19743s (cpu); 1.11522s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.367959s (cpu); 0.220101s (thread); 0s (gc)
    │ │ │ + -- used 0.289111s (cpu); 0.22226s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : -- three generic ternary quadrics
│ │ │       F = genericPolynomials({2,2,2},ZZ/3331)
│ │ │  
│ │ │ @@ -262,15 +262,15 @@
│ │ │  o6 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : -- resultant of the three forms
│ │ │       time resF = resultant F;
│ │ │ - -- used 0.289565s (cpu); 0.206175s (thread); 0s (gc)
    │ │ │ + -- used 0.287529s (cpu); 0.210065s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring Xres))
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -41,15 +41,15 @@ │ │ │ │ │ │ │ │ ZZ │ │ │ │ o2 : Ideal of ----[x ..x ] │ │ │ │ 3331 0 5 │ │ │ │ i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an │ │ │ │ affine chart of the Grassmannian) │ │ │ │ time ChowV = chowForm(V,AffineChartGrass=>{1,2,3}) │ │ │ │ - -- used 5.08382s (cpu); 4.55589s (thread); 0s (gc) │ │ │ │ + -- used 5.56711s (cpu); 5.13335s (thread); 0s (gc) │ │ │ │ │ │ │ │ 4 2 2 2 2 │ │ │ │ o3 = x + 2x x x + x x - 2x x x + │ │ │ │ 1,2,4 0,2,4 1,2,4 2,3,4 0,2,4 2,3,4 1,2,3 1,2,4 1,2,5 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 2 │ │ │ │ x x - x x x + x x x x + │ │ │ │ @@ -234,33 +234,33 @@ │ │ │ │ 1,4,5 0,2,4 0,1,5 0,1,4 0,2,5 0,1,2 0,4,5 1,2,3 0,1,5 0,1,3 1,2,5 │ │ │ │ 0,1,2 1,3,5 0,2,3 0,1,5 0,1,3 0,2,5 0,1,2 0,3,5 1,2,4 0,3,4 0,2,4 │ │ │ │ 1,3,4 0,1,4 2,3,4 1,2,3 0,3,4 0,2,3 1,3,4 0,1,3 2,3,4 1,2,3 0,2,4 │ │ │ │ 0,2,3 1,2,4 0,1,2 2,3,4 1,2,3 0,1,4 0,1,3 1,2,4 0,1,2 1,3,4 0,2,3 │ │ │ │ 0,1,4 0,1,3 0,2,4 0,1,2 0,3,4 │ │ │ │ i4 : -- equivalently (but faster)... │ │ │ │ time assert(ChowV === chowForm f) │ │ │ │ - -- used 1.11624s (cpu); 0.979064s (thread); 0s (gc) │ │ │ │ + -- used 1.19743s (cpu); 1.11522s (thread); 0s (gc) │ │ │ │ i5 : -- X-resultant of V │ │ │ │ time Xres = fromPluckerToStiefel dualize ChowV; │ │ │ │ - -- used 0.367959s (cpu); 0.220101s (thread); 0s (gc) │ │ │ │ + -- used 0.289111s (cpu); 0.22226s (thread); 0s (gc) │ │ │ │ i6 : -- three generic ternary quadrics │ │ │ │ F = genericPolynomials({2,2,2},ZZ/3331) │ │ │ │ │ │ │ │ 2 2 2 2 2 │ │ │ │ o6 = {a x + a x x + a x + a x x + a x x + a x , b x + b x x + b x + │ │ │ │ 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 2 2 │ │ │ │ b x x + b x x + b x , c x + c x x + c x + c x x + c x x + c x } │ │ │ │ 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : -- resultant of the three forms │ │ │ │ time resF = resultant F; │ │ │ │ - -- used 0.289565s (cpu); 0.206175s (thread); 0s (gc) │ │ │ │ + -- used 0.287529s (cpu); 0.210065s (thread); 0s (gc) │ │ │ │ i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring │ │ │ │ Xres)) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_a_n_g_e_n_t_i_a_l_C_h_o_w_F_o_r_m -- higher Chow forms of a projective variety │ │ │ │ * _h_u_r_w_i_t_z_F_o_r_m -- Hurwitz form of a projective variety │ │ │ │ ********** WWaayyss ttoo uussee cchhoowwFFoorrmm:: ********** │ │ │ │ * chowForm(Ideal) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_discriminant_lp__Ring__Element_rp.html │ │ │ @@ -88,15 +88,15 @@ │ │ │ │ │ │ o2 : ZZ[a..c][x..y] │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time discriminant F
│ │ │ - -- used 0.00876412s (cpu); 0.00875934s (thread); 0s (gc)
│ │ │ + -- used 0.0110385s (cpu); 0.0110377s (thread); 0s (gc)
│ │ │  
│ │ │          2
│ │ │  o3 = - b  + 4a*c
│ │ │  
│ │ │  o3 : ZZ[a..c]
    │ │ │ │ │ │ │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ │ │ o5 : ZZ[a..d][x..y] │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time discriminant F
│ │ │ - -- used 0.00928698s (cpu); 0.00928758s (thread); 0s (gc)
│ │ │ + -- used 0.0119987s (cpu); 0.0120007s (thread); 0s (gc)
│ │ │  
│ │ │          2 2       3     3                   2 2
│ │ │  o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d
│ │ │  
│ │ │  o6 : ZZ[a..d]
    │ │ │ │ │ │ │ │ │ @@ -170,15 +170,15 @@ │ │ │ │ │ │ o12 : R' │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : time D=discriminant pencil
│ │ │ - -- used 0.521101s (cpu); 0.460767s (thread); 0s (gc)
│ │ │ + -- used 0.515577s (cpu); 0.448516s (thread); 0s (gc)
│ │ │  
│ │ │             108      106 2       102 6      100 8       98 10       96 12  
│ │ │  o13 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
│ │ │             0        0   1       0   1      0   1       0  1        0  1   
│ │ │        -----------------------------------------------------------------------
│ │ │            94 14       92 16      90 18      88 20      86 22       84 24  
│ │ │        161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,28 +23,28 @@
│ │ │ │  i1 : ZZ[a,b,c][x,y]; F = a*x^2+b*x*y+c*y^2
│ │ │ │  
│ │ │ │          2              2
│ │ │ │  o2 = a*x  + b*x*y + c*y
│ │ │ │  
│ │ │ │  o2 : ZZ[a..c][x..y]
│ │ │ │  i3 : time discriminant F
│ │ │ │ - -- used 0.00876412s (cpu); 0.00875934s (thread); 0s (gc)
│ │ │ │ + -- used 0.0110385s (cpu); 0.0110377s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2
│ │ │ │  o3 = - b  + 4a*c
│ │ │ │  
│ │ │ │  o3 : ZZ[a..c]
│ │ │ │  i4 : ZZ[a,b,c,d][x,y]; F = a*x^3+b*x^2*y+c*x*y^2+d*y^3
│ │ │ │  
│ │ │ │          3      2         2      3
│ │ │ │  o5 = a*x  + b*x y + c*x*y  + d*y
│ │ │ │  
│ │ │ │  o5 : ZZ[a..d][x..y]
│ │ │ │  i6 : time discriminant F
│ │ │ │ - -- used 0.00928698s (cpu); 0.00928758s (thread); 0s (gc)
│ │ │ │ + -- used 0.0119987s (cpu); 0.0120007s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2 2       3     3                   2 2
│ │ │ │  o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d
│ │ │ │  
│ │ │ │  o6 : ZZ[a..d]
│ │ │ │  The next example illustrates how computing the intersection of a pencil
│ │ │ │  generated by two degree $d$ forms $F(x_0,\ldots,x_n), G(x_0,\ldots,x_n)$ with
│ │ │ │ @@ -74,15 +74,15 @@
│ │ │ │  
│ │ │ │                  4        3      4             4        3      4
│ │ │ │  o12 = (t  + t )x  - t x x  + t x  + (t  - t )x  + t x x  + t x
│ │ │ │          0    1  0    1 0 1    0 1     0    1  2    1 2 3    0 3
│ │ │ │  
│ │ │ │  o12 : R'
│ │ │ │  i13 : time D=discriminant pencil
│ │ │ │ - -- used 0.521101s (cpu); 0.460767s (thread); 0s (gc)
│ │ │ │ + -- used 0.515577s (cpu); 0.448516s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             108      106 2       102 6      100 8       98 10       96 12
│ │ │ │  o13 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
│ │ │ │             0        0   1       0   1      0   1       0  1        0  1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            94 14       92 16      90 18      88 20      86 22       84 24
│ │ │ │        161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_dual__Variety.html
│ │ │ @@ -95,28 +95,28 @@
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   5
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time V' = dualVariety V
│ │ │ - -- used 0.196307s (cpu); 0.124873s (thread); 0s (gc)
│ │ │ + -- used 0.205735s (cpu); 0.134987s (thread); 0s (gc)
│ │ │  
│ │ │              2                 2    2
│ │ │  o2 = ideal(x x  - x x x  + x x  + x x  - 4x x x )
│ │ │              2 3    1 2 4    0 4    1 5     0 3 5
│ │ │  
│ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │                    0   5
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time V == dualVariety V'
│ │ │ - -- used 0.205617s (cpu); 0.14044s (thread); 0s (gc)
│ │ │ + -- used 0.228092s (cpu); 0.159805s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │

    In the next example, we verify that the discriminant of a generic ternary cubic form coincides with the dual variety of the 3-th Veronese embedding of the plane, which is a hypersurface of degree 12 in $\mathbb{P}^9$

    │ │ │ │ │ │ @@ -136,25 +136,25 @@ │ │ │ o4 : ----[a ..a ][x ..x ] │ │ │ 3331 0 9 0 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -31,24 +31,24 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x ) │ │ │ │ 0 3 │ │ │ │ │ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ │ 0 5 │ │ │ │ i2 : time V' = dualVariety V │ │ │ │ - -- used 0.196307s (cpu); 0.124873s (thread); 0s (gc) │ │ │ │ + -- used 0.205735s (cpu); 0.134987s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 │ │ │ │ o2 = ideal(x x - x x x + x x + x x - 4x x x ) │ │ │ │ 2 3 1 2 4 0 4 1 5 0 3 5 │ │ │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ │ 0 5 │ │ │ │ i3 : time V == dualVariety V' │ │ │ │ - -- used 0.205617s (cpu); 0.14044s (thread); 0s (gc) │ │ │ │ + -- used 0.228092s (cpu); 0.159805s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = true │ │ │ │ In the next example, we verify that the discriminant of a generic ternary cubic │ │ │ │ form coincides with the dual variety of the 3-th Veronese embedding of the │ │ │ │ plane, which is a hypersurface of degree 12 in $\mathbb{P}^9$ │ │ │ │ i4 : F = first genericPolynomials({3,-1,-1},ZZ/3331) │ │ │ │ │ │ │ │ @@ -60,21 +60,21 @@ │ │ │ │ a x x + a x │ │ │ │ 8 1 2 9 2 │ │ │ │ │ │ │ │ ZZ │ │ │ │ o4 : ----[a ..a ][x ..x ] │ │ │ │ 3331 0 9 0 2 │ │ │ │ i5 : time discF = ideal discriminant F; │ │ │ │ - -- used 0.057736s (cpu); 0.0577424s (thread); 0s (gc) │ │ │ │ + -- used 0.0687417s (cpu); 0.0687418s (thread); 0s (gc) │ │ │ │ │ │ │ │ ZZ │ │ │ │ o5 : Ideal of ----[a ..a ] │ │ │ │ 3331 0 9 │ │ │ │ i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true); │ │ │ │ - -- used 0.625488s (cpu); 0.556064s (thread); 0s (gc) │ │ │ │ + -- used 0.740831s (cpu); 0.663289s (thread); 0s (gc) │ │ │ │ │ │ │ │ ZZ │ │ │ │ o6 : Ideal of ----[x ..x ] │ │ │ │ 3331 0 9 │ │ │ │ i7 : discF == sub(Z,vars ring discF) and Z == sub(discF,vars ring Z) │ │ │ │ │ │ │ │ o7 = true │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_from__Plucker__To__Stiefel.html │ │ │ @@ -90,15 +90,15 @@ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : time discF = ideal discriminant F;
│ │ │ - -- used 0.057736s (cpu); 0.0577424s (thread); 0s (gc)
│ │ │ + -- used 0.0687417s (cpu); 0.0687418s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o5 : Ideal of ----[a ..a ]
│ │ │                3331  0   9
    │ │ │
    │ │ │ 
    i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ - -- used 0.625488s (cpu); 0.556064s (thread); 0s (gc)
│ │ │ + -- used 0.740831s (cpu); 0.663289s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o6 : Ideal of ----[x ..x ]
│ │ │                3331  0   9
    │ │ │
    │ │ │ 
    i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ - -- used 0.126592s (cpu); 0.0637976s (thread); 0s (gc)
│ │ │ + -- used 0.138147s (cpu); 0.0700527s (thread); 0s (gc)
│ │ │  
│ │ │          3   3          2   2              2       2          2   3    
│ │ │  o2 = - x   x    + x   x   x   x    - x   x   x   x    + x   x   x    -
│ │ │          0,3 1,0    0,2 0,3 1,0 1,1    0,1 0,3 1,0 1,1    0,0 0,3 1,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        2       2               2   2                                   
│ │ │       x   x   x   x    + 2x   x   x   x    + x   x   x   x   x   x    -
│ │ │ @@ -143,15 +143,15 @@
│ │ │  o2 : QQ[x   ..x   ]
│ │ │           0,0   1,3
    │ │ │
    │ │ │ 
    i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1})
│ │ │ - -- used 0.137358s (cpu); 0.0611511s (thread); 0s (gc)
│ │ │ + -- used 0.133466s (cpu); 0.0649393s (thread); 0s (gc)
│ │ │  
│ │ │              3          2                         2                        
│ │ │  o3 = - x   x    + x   x   x    - x   x   x    + x   x    + 3x   x   x    -
│ │ │          0,3 1,2    0,2 1,2 1,3    0,2 0,3 1,2    0,2 1,3     0,3 1,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │             2      3      2
│ │ │       2x   x    + x    + x
│ │ │ @@ -184,15 +184,15 @@
│ │ │
    		│ │ │ 
    i5 : w = chowForm C;
    │ │ │
    │ │ │ 
    i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel(w,AffineChartGrass=>s))
│ │ │ - -- used 0.0199213s (cpu); 0.0199207s (thread); 0s (gc)
│ │ │ + -- used 0.0251917s (cpu); 0.0251914s (thread); 0s (gc)
│ │ │  
│ │ │                     3          2          3                       2        
│ │ │  o6 = {ideal(- x   x    + x   x   x    - x    - 3x   x   x    + 2x   x    +
│ │ │                 0,3 1,2    0,2 1,2 1,3    0,2     0,2 0,3 1,2     0,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2      2            2   3               2        
│ │ │       x   x   x    - x   x    + x   ), ideal(x   x    - 2x   x   x   x    +
│ │ │ @@ -232,15 +232,15 @@
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : time apply(U,u->dim singularLocus u)
│ │ │ - -- used 0.09626s (cpu); 0.0369501s (thread); 0s (gc)
│ │ │ + -- used 0.109064s (cpu); 0.042567s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {2, 2, 2, 2, 2, 2}
│ │ │  
│ │ │  o7 : List
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -26,15 +26,15 @@ │ │ │ │ 2 2 │ │ │ │ o1 = ideal (x - x x , x x - x x , x - x x ) │ │ │ │ 2 1 3 1 2 0 3 1 0 2 │ │ │ │ │ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ │ 0 3 │ │ │ │ i2 : time fromPluckerToStiefel dualize chowForm C │ │ │ │ - -- used 0.126592s (cpu); 0.0637976s (thread); 0s (gc) │ │ │ │ + -- used 0.138147s (cpu); 0.0700527s (thread); 0s (gc) │ │ │ │ │ │ │ │ 3 3 2 2 2 2 2 3 │ │ │ │ o2 = - x x + x x x x - x x x x + x x x - │ │ │ │ 0,3 1,0 0,2 0,3 1,0 1,1 0,1 0,3 1,0 1,1 0,0 0,3 1,1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 2 2 │ │ │ │ x x x x + 2x x x x + x x x x x x - │ │ │ │ @@ -75,15 +75,15 @@ │ │ │ │ 2 2 2 2 2 2 3 3 │ │ │ │ x x x x - 2x x x x - x x x x + x x │ │ │ │ 0,0 0,1 1,1 1,3 0,0 0,2 1,1 1,3 0,0 0,1 1,2 1,3 0,0 1,3 │ │ │ │ │ │ │ │ o2 : QQ[x ..x ] │ │ │ │ 0,0 1,3 │ │ │ │ i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1}) │ │ │ │ - -- used 0.137358s (cpu); 0.0611511s (thread); 0s (gc) │ │ │ │ + -- used 0.133466s (cpu); 0.0649393s (thread); 0s (gc) │ │ │ │ │ │ │ │ 3 2 2 │ │ │ │ o3 = - x x + x x x - x x x + x x + 3x x x - │ │ │ │ 0,3 1,2 0,2 1,2 1,3 0,2 0,3 1,2 0,2 1,3 0,3 1,2 1,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 3 2 │ │ │ │ 2x x + x + x │ │ │ │ @@ -105,15 +105,15 @@ │ │ │ │ o4 : QQ[a ..a ] │ │ │ │ 0,0 1,1 │ │ │ │ As another application, we check that the singular locus of the Chow form of │ │ │ │ the twisted cubic has dimension 2 (on each standard chart). │ │ │ │ i5 : w = chowForm C; │ │ │ │ i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel │ │ │ │ (w,AffineChartGrass=>s)) │ │ │ │ - -- used 0.0199213s (cpu); 0.0199207s (thread); 0s (gc) │ │ │ │ + -- used 0.0251917s (cpu); 0.0251914s (thread); 0s (gc) │ │ │ │ │ │ │ │ 3 2 3 2 │ │ │ │ o6 = {ideal(- x x + x x x - x - 3x x x + 2x x + │ │ │ │ 0,3 1,2 0,2 1,2 1,3 0,2 0,2 0,3 1,2 0,2 1,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 2 3 2 │ │ │ │ x x x - x x + x ), ideal(x x - 2x x x x + │ │ │ │ @@ -149,15 +149,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 3 2 │ │ │ │ 2x x - x + x )} │ │ │ │ 0,0 1,1 1,1 1,0 │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : time apply(U,u->dim singularLocus u) │ │ │ │ - -- used 0.09626s (cpu); 0.0369501s (thread); 0s (gc) │ │ │ │ + -- used 0.109064s (cpu); 0.042567s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = {2, 2, 2, 2, 2, 2} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** WWaayyss ttoo uussee ffrroommPPlluucckkeerrTTooSSttiieeffeell:: ********** │ │ │ │ * fromPluckerToStiefel(Ideal) │ │ │ │ * fromPluckerToStiefel(Matrix) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_hurwitz__Form.html │ │ │ @@ -97,15 +97,15 @@ │ │ │ o1 : Ideal of QQ[p ..p ] │ │ │ 0 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time hurwitzForm Q
│ │ │ - -- used 0.0375527s (cpu); 0.0375568s (thread); 0s (gc)
│ │ │ + -- used 0.0450159s (cpu); 0.0450171s (thread); 0s (gc)
│ │ │  
│ │ │                2                                 2                      
│ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │                0,1            0,1 0,2            0,2           0,1 1,2  
│ │ │       ------------------------------------------------------------------------
│ │ │                                 2                                          
│ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │         5 2   7                       2        3 2
│ │ │ │       + -p  + -p p  + 7p p  + 6p p  + -p p  + --p )
│ │ │ │         4 3   9 0 4     1 4     2 4   9 3 4   10 4
│ │ │ │  
│ │ │ │  o1 : Ideal of QQ[p ..p ]
│ │ │ │                    0   4
│ │ │ │  i2 : time hurwitzForm Q
│ │ │ │ - -- used 0.0375527s (cpu); 0.0375568s (thread); 0s (gc)
│ │ │ │ + -- used 0.0450159s (cpu); 0.0450171s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2                                 2
│ │ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │ │                0,1            0,1 0,2            0,2           0,1 1,2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                                 2
│ │ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__Coisotropic.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time isCoisotropic w
│ │ │ - -- used 0.0967611s (cpu); 0.0287666s (thread); 0s (gc)
│ │ │ + -- used 0.0871977s (cpu); 0.0257295s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : -- random quadric in G(1,3)
│ │ │ @@ -145,15 +145,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time isCoisotropic w'
│ │ │ - -- used 0.00675958s (cpu); 0.00676007s (thread); 0s (gc)
│ │ │ + -- used 0.00859799s (cpu); 0.00859696s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -44,15 +44,15 @@ │ │ │ │ │ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ │ o1 : -------------------------------------- │ │ │ │ p p - p p + p p │ │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ i2 : time isCoisotropic w │ │ │ │ - -- used 0.0967611s (cpu); 0.0287666s (thread); 0s (gc) │ │ │ │ + -- used 0.0871977s (cpu); 0.0257295s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = true │ │ │ │ i3 : -- random quadric in G(1,3) │ │ │ │ w' = random(2,Grass(1,3)) │ │ │ │ │ │ │ │ 3 2 3 7 2 5 7 2 10 │ │ │ │ o3 = --p + -p p + -p + 5p p + -p p + -p + --p p │ │ │ │ @@ -72,15 +72,15 @@ │ │ │ │ │ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ │ o3 : -------------------------------------- │ │ │ │ p p - p p + p p │ │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ i4 : time isCoisotropic w' │ │ │ │ - -- used 0.00675958s (cpu); 0.00676007s (thread); 0s (gc) │ │ │ │ + -- used 0.00859799s (cpu); 0.00859696s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = false │ │ │ │ ********** WWaayyss ttoo uussee iissCCooiissoottrrooppiicc:: ********** │ │ │ │ * isCoisotropic(RingElement) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _i_s_C_o_i_s_o_t_r_o_p_i_c is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__In__Coisotropic.html │ │ │ @@ -122,15 +122,15 @@ │ │ │ o3 : Ideal of -----[x ..x ] │ │ │ 33331 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I))
│ │ │ - -- used 0.0186416s (cpu); 0.0186435s (thread); 0s (gc)
│ │ │ + -- used 0.0210315s (cpu); 0.0210307s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -54,15 +54,15 @@ │ │ │ │ 2380x + 9482x ) │ │ │ │ 4 5 │ │ │ │ │ │ │ │ ZZ │ │ │ │ o3 : Ideal of -----[x ..x ] │ │ │ │ 33331 0 5 │ │ │ │ i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I)) │ │ │ │ - -- used 0.0186416s (cpu); 0.0186435s (thread); 0s (gc) │ │ │ │ + -- used 0.0210315s (cpu); 0.0210307s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_a_n_g_e_n_t_i_a_l_C_h_o_w_F_o_r_m -- higher Chow forms of a projective variety │ │ │ │ * _p_l_u_c_k_e_r -- get the Plücker coordinates of a linear subspace │ │ │ │ ********** WWaayyss ttoo uussee iissIInnCCooiissoottrrooppiicc:: ********** │ │ │ │ * isInCoisotropic(Ideal,Ideal) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_macaulay__Formula.html │ │ │ @@ -92,15 +92,15 @@ │ │ │ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00367518s (cpu); 0.00367216s (thread); 0s (gc)
│ │ │ + -- used 0.00417748s (cpu); 0.00417493s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0  
│ │ │        | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0  
│ │ │        | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0  
│ │ │        | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0  
│ │ │        | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0  
│ │ │        | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0  
│ │ │ @@ -163,15 +163,15 @@
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00248359s (cpu); 0.00248394s (thread); 0s (gc)
│ │ │ + -- used 0.00289059s (cpu); 0.0028891s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0   
│ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0   
│ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0   
│ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0   
│ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0   
│ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                     2          2        2      3
│ │ │ │       c x x x  + c x x  + c x x  + c x x  + c x }
│ │ │ │        4 0 1 2    7 1 2    5 0 2    8 1 2    9 2
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : time (D,D') = macaulayFormula F
│ │ │ │ - -- used 0.00367518s (cpu); 0.00367216s (thread); 0s (gc)
│ │ │ │ + -- used 0.00417748s (cpu); 0.00417493s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0
│ │ │ │        | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0
│ │ │ │        | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0
│ │ │ │        | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0
│ │ │ │        | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0
│ │ │ │        | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0
│ │ │ │ @@ -91,15 +91,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       10   2   7   2   5 3
│ │ │ │       --p p  + -p p  + -p }
│ │ │ │        9 0 2   8 1 2   6 2
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : time (D,D') = macaulayFormula F
│ │ │ │ - -- used 0.00248359s (cpu); 0.00248394s (thread); 0s (gc)
│ │ │ │ + -- used 0.00289059s (cpu); 0.0028891s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0
│ │ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0
│ │ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0
│ │ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0
│ │ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0
│ │ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o3 : Ideal of P4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time p = plucker L
│ │ │ - -- used 0.00445528s (cpu); 0.00445317s (thread); 0s (gc)
│ │ │ + -- used 0.00939157s (cpu); 0.00939111s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = ideal (x    + 8480x   , x    - 6727x   , x    + 15777x   , x    +
│ │ │               2,4        3,4   1,4        3,4   0,4         3,4   2,3  
│ │ │       ------------------------------------------------------------------------
│ │ │       11656x   , x    - 14853x   , x    + 664x   , x    + 13522x   , x    +
│ │ │             3,4   1,3         3,4   0,3       3,4   1,2         3,4   0,2  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -114,15 +114,15 @@
│ │ │  
│ │ │  o4 : Ideal of G'1'4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time L' = plucker p
│ │ │ - -- used 0.114222s (cpu); 0.0497012s (thread); 0s (gc)
│ │ │ + -- used 0.147705s (cpu); 0.0698158s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │               2        3         4   1        3         4   0         3  
│ │ │       ------------------------------------------------------------------------
│ │ │       664x )
│ │ │           4
│ │ │  
│ │ │ @@ -143,15 +143,15 @@
│ │ │  
│ │ │  o7 : Ideal of G'1'4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ - -- used 0.0312756s (cpu); 0.0312807s (thread); 0s (gc)
│ │ │ + -- used 0.0394625s (cpu); 0.0394629s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of P4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : (codim W,degree W)
│ │ │ @@ -163,15 +163,15 @@
│ │ │            
│ │ │          
│ │ │          
    In this example, we can recover the subvariety $Y\subset\mathbb{G}(k,\mathbb{P}^n)$ by computing the Fano variety of $k$-planes contained in $W$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time Y' = plucker(W,1); -- variety of lines contained in W
│ │ │ - -- used 0.222489s (cpu); 0.166167s (thread); 0s (gc)
│ │ │ + -- used 0.258422s (cpu); 0.192273s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of G'1'4
    
│ │ │              
    
│ │ │                
    i11 : assert(Y' == Y)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,28 +28,28 @@
│ │ │ │               2        3         4   1        3         4   0         3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       664x )
│ │ │ │           4
│ │ │ │  
│ │ │ │  o3 : Ideal of P4
│ │ │ │  i4 : time p = plucker L
│ │ │ │ - -- used 0.00445528s (cpu); 0.00445317s (thread); 0s (gc)
│ │ │ │ + -- used 0.00939157s (cpu); 0.00939111s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = ideal (x    + 8480x   , x    - 6727x   , x    + 15777x   , x    +
│ │ │ │               2,4        3,4   1,4        3,4   0,4         3,4   2,3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       11656x   , x    - 14853x   , x    + 664x   , x    + 13522x   , x    +
│ │ │ │             3,4   1,3         3,4   0,3       3,4   1,2         3,4   0,2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       11804x   , x    + 14854x   )
│ │ │ │             3,4   0,1         3,4
│ │ │ │  
│ │ │ │  o4 : Ideal of G'1'4
│ │ │ │  i5 : time L' = plucker p
│ │ │ │ - -- used 0.114222s (cpu); 0.0497012s (thread); 0s (gc)
│ │ │ │ + -- used 0.147705s (cpu); 0.0698158s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │ │               2        3         4   1        3         4   0         3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       664x )
│ │ │ │           4
│ │ │ │  
│ │ │ │ @@ -60,26 +60,26 @@
│ │ │ │  $W\subset\mathbb{P}^n$ swept out by the linear spaces corresponding to points
│ │ │ │  of $Y$. As an example, we now compute a surface scroll $W\subset\mathbb{P}^4$
│ │ │ │  over an elliptic curve $Y\subset\mathbb{G}(1,\mathbb{P}^4)$.
│ │ │ │  i7 : Y = ideal apply(5,i->random(1,G'1'4)); -- an elliptic curve
│ │ │ │  
│ │ │ │  o7 : Ideal of G'1'4
│ │ │ │  i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ │ - -- used 0.0312756s (cpu); 0.0312807s (thread); 0s (gc)
│ │ │ │ + -- used 0.0394625s (cpu); 0.0394629s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of P4
│ │ │ │  i9 : (codim W,degree W)
│ │ │ │  
│ │ │ │  o9 = (2, 5)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  In this example, we can recover the subvariety $Y\subset\mathbb{G}(k,\mathbb
│ │ │ │  {P}^n)$ by computing the Fano variety of $k$-planes contained in $W$.
│ │ │ │  i10 : time Y' = plucker(W,1); -- variety of lines contained in W
│ │ │ │ - -- used 0.222489s (cpu); 0.166167s (thread); 0s (gc)
│ │ │ │ + -- used 0.258422s (cpu); 0.192273s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 : Ideal of G'1'4
│ │ │ │  i11 : assert(Y' == Y)
│ │ │ │  WWaarrnniinngg: Notice that, by default, the computation is done on a randomly chosen
│ │ │ │  affine chart on the Grassmannian. To change this behavior, you can use the
│ │ │ │  _A_f_f_i_n_e_C_h_a_r_t_G_r_a_s_s option.
│ │ │ │  ********** WWaayyss ttoo uussee pplluucckkeerr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp..._cm__Algorithm_eq_gt..._rp.html
│ │ │ @@ -113,15 +113,15 @@
│ │ │  
│ │ │  o2 : List
│ │ │              
    
│ │ │                
    i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ - -- used 0.364823s (cpu); 0.192307s (thread); 0s (gc)
│ │ │ + -- used 0.473887s (cpu); 0.256431s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o3 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ - -- used 0.157645s (cpu); 0.0917722s (thread); 0s (gc)
│ │ │ + -- used 0.168634s (cpu); 0.100061s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -161,15 +161,15 @@
│ │ │  
│ │ │  o4 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ - -- used 0.381787s (cpu); 0.323386s (thread); 0s (gc)
│ │ │ + -- used 0.410825s (cpu); 0.344613s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -185,15 +185,15 @@
│ │ │  
│ │ │  o5 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ - -- used 0.619406s (cpu); 0.555197s (thread); 0s (gc)
│ │ │ + -- used 0.673783s (cpu); 0.607181s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ ├── html2text {}
│ │ │ │ @@ -58,15 +58,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       3     2    9    7     2    9        3       1    8    4
│ │ │ │       -b)y*w  + (-a + -b)z*w  + (-a + 2b)w , 2x + -y + -z + -w}
│ │ │ │       4          8    8          7                4    3    5
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ │ - -- used 0.364823s (cpu); 0.192307s (thread); 0s (gc)
│ │ │ │ + -- used 0.473887s (cpu); 0.256431s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -78,15 +78,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o3 : QQ[a..b]
│ │ │ │  i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ │ - -- used 0.157645s (cpu); 0.0917722s (thread); 0s (gc)
│ │ │ │ + -- used 0.168634s (cpu); 0.100061s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -98,15 +98,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o4 : QQ[a..b]
│ │ │ │  i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ │ - -- used 0.381787s (cpu); 0.323386s (thread); 0s (gc)
│ │ │ │ + -- used 0.410825s (cpu); 0.344613s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -118,15 +118,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o5 : QQ[a..b]
│ │ │ │  i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ │ - -- used 0.619406s (cpu); 0.555197s (thread); 0s (gc)
│ │ │ │ + -- used 0.673783s (cpu); 0.607181s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time resultant F
│ │ │ - -- used 0.023138s (cpu); 0.0231407s (thread); 0s (gc)
│ │ │ + -- used 0.0268743s (cpu); 0.0267075s (thread); 0s (gc)
│ │ │  
│ │ │            12         11 2         10 3         9 4          8 5          7 6
│ │ │  o3 = - 81t  u - 1701t  u  - 15309t  u  - 76545t u  - 229635t u  - 413343t u 
│ │ │       ------------------------------------------------------------------------
│ │ │                6 7          5 8       11          10 2         9 3  
│ │ │       - 413343t u  - 177147t u  + 567t  u + 10206t  u  + 76545t u  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -160,15 +160,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : time resultant F
│ │ │ - -- used 2.65652s (cpu); 1.97047s (thread); 0s (gc)
│ │ │ + -- used 2.62093s (cpu); 2.03357s (thread); 0s (gc)
│ │ │  
│ │ │        6 3 2       5 2   2     2 4   2 2    3 3 3 2     2 4 2   2  
│ │ │  o5 = a b c  - 3a a b b c  + 3a a b b c  - a a b c  + 3a a b b c  -
│ │ │        2 3 0     1 2 3 4 0     1 2 3 4 0    1 2 4 0     1 2 3 5 0  
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3       2     4 2 2   2     4 2   2 2     5     2 2    6 3 2  
│ │ │       6a a b b b c  + 3a a b b c  + 3a a b b c  - 3a a b b c  + a b c  -
│ │ │ @@ -1795,15 +1795,15 @@
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time # terms resultant F
│ │ │ - -- used 0.530258s (cpu); 0.383961s (thread); 0s (gc)
│ │ │ + -- used 0.627448s (cpu); 0.475334s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 21894
    
│ │ │              
    
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -32,15 +32,15 @@ │ │ │ │ i2 : F = {x^2+3*t*y*z-u*z^2,(t+3*u-1)*x-y,-t*x*y^3+t*x^2*y*z+u*z^4} │ │ │ │ │ │ │ │ 2 2 3 2 4 │ │ │ │ o2 = {x + 3t*y*z - u*z , (t + 3u - 1)x - y, - t*x*y + t*x y*z + u*z } │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : time resultant F │ │ │ │ - -- used 0.023138s (cpu); 0.0231407s (thread); 0s (gc) │ │ │ │ + -- used 0.0268743s (cpu); 0.0267075s (thread); 0s (gc) │ │ │ │ │ │ │ │ 12 11 2 10 3 9 4 8 5 7 6 │ │ │ │ o3 = - 81t u - 1701t u - 15309t u - 76545t u - 229635t u - 413343t u │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 6 7 5 8 11 10 2 9 3 │ │ │ │ - 413343t u - 177147t u + 567t u + 10206t u + 76545t u + │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ @@ -86,15 +86,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 │ │ │ │ + c x } │ │ │ │ 9 2 │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : time resultant F │ │ │ │ - -- used 2.65652s (cpu); 1.97047s (thread); 0s (gc) │ │ │ │ + -- used 2.62093s (cpu); 2.03357s (thread); 0s (gc) │ │ │ │ │ │ │ │ 6 3 2 5 2 2 2 4 2 2 3 3 3 2 2 4 2 2 │ │ │ │ o5 = a b c - 3a a b b c + 3a a b b c - a a b c + 3a a b b c - │ │ │ │ 2 3 0 1 2 3 4 0 1 2 3 4 0 1 2 4 0 1 2 3 5 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 3 2 4 2 2 2 4 2 2 2 5 2 2 6 3 2 │ │ │ │ 6a a b b b c + 3a a b b c + 3a a b b c - 3a a b b c + a b c - │ │ │ │ @@ -1712,15 +1712,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 2 2 │ │ │ │ b x x + b x x + b x , c x + c x x + c x + c x x + c x x + c x } │ │ │ │ 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : time # terms resultant F │ │ │ │ - -- used 0.530258s (cpu); 0.383961s (thread); 0s (gc) │ │ │ │ + -- used 0.627448s (cpu); 0.475334s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = 21894 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_h_o_w_F_o_r_m -- Chow form of a projective variety │ │ │ │ * _d_i_s_c_r_i_m_i_n_a_n_t_(_R_i_n_g_E_l_e_m_e_n_t_) │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * resultant(List) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_tangential__Chow__Form.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ 0 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form)
│ │ │       time tangentialChowForm(S,0)
│ │ │ - -- used 0.0371985s (cpu); 0.0372021s (thread); 0s (gc)
│ │ │ + -- used 0.0458259s (cpu); 0.0458254s (thread); 0s (gc)
│ │ │  
│ │ │        2                                                       2        
│ │ │  o3 = p   p    - p   p   p    - p   p   p    + p   p   p    + p   p    +
│ │ │        1,3 2,3    1,2 1,3 2,4    0,3 1,3 2,4    0,2 1,4 2,4    1,2 3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │        2
│ │ │       p   p    - 2p   p   p    - p   p   p
│ │ │ @@ -123,15 +123,15 @@
│ │ │         2,3 1,4    1,3 2,4    1,2 3,4   2,3 0,4    0,3 2,4    0,2 3,4   1,3 0,4    0,3 1,4    0,1 3,4   1,2 0,4    0,2 1,4    0,1 2,4   1,2 0,3    0,2 1,3    0,1 2,3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : -- 1-th associated hypersurface of S in G(2,4)
│ │ │       time tangentialChowForm(S,1)
│ │ │ - -- used 0.15512s (cpu); 0.0919453s (thread); 0s (gc)
│ │ │ + -- used 0.179818s (cpu); 0.10462s (thread); 0s (gc)
│ │ │  
│ │ │        2     2        2     2               3        2     2      
│ │ │  o4 = p     p      + p     p      - 2p     p      + p     p      -
│ │ │        1,2,3 1,2,4    0,2,4 1,2,4     0,2,3 1,2,4    0,2,4 0,3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │               3         3               3            
│ │ │       4p     p      - 4p     p      - 2p     p      +
│ │ │ @@ -168,43 +168,43 @@
│ │ │         1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent hyperplanes to S)
│ │ │       time tangentialChowForm(S,2)
│ │ │ - -- used 0.0308352s (cpu); 0.0308383s (thread); 0s (gc)
│ │ │ + -- used 0.0396845s (cpu); 0.0396852s (thread); 0s (gc)
│ │ │  
│ │ │                2                                             2
│ │ │  o5 = p       p        - p       p       p        + p       p
│ │ │        0,1,3,4 0,2,3,4    0,1,2,4 0,2,3,4 1,2,3,4    0,1,2,3 1,2,3,4
│ │ │  
│ │ │  o5 : QQ[p       ..p       , p       , p       , p       ]
│ │ │           0,1,2,3   0,1,2,4   0,1,3,4   0,2,3,4   1,2,3,4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing
│ │ │       time S' = ideal dualize tangentialChowForm(S,2)
│ │ │ - -- used 0.121596s (cpu); 0.0529281s (thread); 0s (gc)
│ │ │ + -- used 0.137081s (cpu); 0.0640525s (thread); 0s (gc)
│ │ │  
│ │ │              2               2
│ │ │  o6 = ideal(p p  - p p p  + p p )
│ │ │              1 2    0 1 3    0 4
│ │ │  
│ │ │  o6 : Ideal of QQ[p ..p ]
│ │ │                    0   4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : -- we then can recover S
│ │ │       time assert(dualize tangentialChowForm(S',3) == S)
│ │ │ - -- used 0.18169s (cpu); 0.117706s (thread); 0s (gc)
    │ │ │ + -- used 0.198309s (cpu); 0.12308s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -120,15 +120,15 @@ │ │ │
    │ │ │

    and then call them:

    │ │ │
    │ │ │
    │ │ │ 
    i1 : fn=temporaryFileName()|".m2"
│ │ │  
│ │ │ -o1 = /tmp/M2-24089-0/0.m2
    │ │ │ +o1 = /tmp/M2-30883-0/0.m2 │ │ │
    │ │ │ 
    i2 : fn<</// square = (x) -> (stderr<<"Running"<<endl; sleep(1); x^2); ///<<endl;
    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : h=runExternalM2(fn,"square",(4));
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24089-0/1.m2" >"/tmp/M2-24089-0/1.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-30883-0/1.m2" >"/tmp/M2-30883-0/1.out" 2>&1 ))
│ │ │  Finished running.
    │ │ │
    │ │ │ 
    i7 : h
│ │ │  
│ │ │ @@ -162,29 +162,29 @@
│ │ │            
    
│ │ │            
    An abnormal program exit will have a nonzero exit code; also, the value will be null, the output file should exist, but the answer file may not exist unless the routine finished successfully.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -348,20 +348,20 @@
│ │ │          
│ │ │          
    
│ │ │                
    i10 : h=runExternalM2(fn,"justexit",());
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24089-0/2.m2" >"/tmp/M2-24089-0/2.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-30883-0/2.m2" >"/tmp/M2-30883-0/2.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    
│ │ │              
    
│ │ │                
    i11 : h
│ │ │  
│ │ │ -o11 = HashTable{"answer file" => /tmp/M2-24089-0/2.ans}
│ │ │ +o11 = HashTable{"answer file" => /tmp/M2-30883-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-24089-0/2.out
│ │ │ +                "output file" => /tmp/M2-30883-0/2.out
│ │ │                  "return code" => 6912
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 3
│ │ │ +                "time used" => 2
│ │ │                  value => null
│ │ │  
│ │ │  o11 : HashTable
    
│ │ │              
    
│ │ │ @@ -204,27 +204,27 @@
│ │ │          
    
│ │ │            
    Here, we use resource limits to limit the routine to 2 seconds of computational time, while the system is asked to use 10 seconds of computational time:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -246,40 +246,40 @@
│ │ │            
    
│ │ │            
    We can get quite a lot of detail on the resources used with the KeepStatistics command:
    
│ │ │          
│ │ │          
    
│ │ │                
    i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ -Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24089-0/3.m2" >"/tmp/M2-24089-0/3.out" 2>&1 ))
│ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-30883-0/3.m2" >"/tmp/M2-30883-0/3.out" 2>&1 ))
│ │ │  Killed
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    
│ │ │              
    
│ │ │                
    i15 : h
│ │ │  
│ │ │ -o15 = HashTable{"answer file" => /tmp/M2-24089-0/3.ans}
│ │ │ +o15 = HashTable{"answer file" => /tmp/M2-30883-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-24089-0/3.out
│ │ │ +                "output file" => /tmp/M2-30883-0/3.out
│ │ │                  "return code" => 9
│ │ │                  "statistics" => null
│ │ │                  "time used" => 2
│ │ │                  value => null
│ │ │  
│ │ │  o15 : HashTable
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -293,15 +293,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ -Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24089-0/4.m2" >"/tmp/M2-24089-0/4.out" 2>&1') >"/tmp/M2-24089-0/4.stat" 2>&1 ))
│ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-30883-0/4.m2" >"/tmp/M2-30883-0/4.out" 2>&1') >"/tmp/M2-30883-0/4.stat" 2>&1 ))
│ │ │  Finished running.
    
│ │ │              
    
│ │ │                
    i19 : h#"statistics"
│ │ │  
│ │ │ -o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24089-0/4.m2" >"/tmp/M2-24089-0/4.out" 2>&1"
│ │ │ -              User time (seconds): 6.09
│ │ │ -              System time (seconds): 0.14
│ │ │ -              Percent of CPU this job got: 90%
│ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.90
│ │ │ +o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-30883-0/4.m2" >"/tmp/M2-30883-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 5.86
│ │ │ +              System time (seconds): 0.22
│ │ │ +              Percent of CPU this job got: 120%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:05.07
│ │ │                Average shared text size (kbytes): 0
│ │ │                Average unshared data size (kbytes): 0
│ │ │                Average stack size (kbytes): 0
│ │ │                Average total size (kbytes): 0
│ │ │ -              Maximum resident set size (kbytes): 255332
│ │ │ +              Maximum resident set size (kbytes): 339296
│ │ │                Average resident set size (kbytes): 0
│ │ │                Major (requiring I/O) page faults: 0
│ │ │ -              Minor (reclaiming a frame) page faults: 8236
│ │ │ -              Voluntary context switches: 2905
│ │ │ -              Involuntary context switches: 2134
│ │ │ +              Minor (reclaiming a frame) page faults: 10987
│ │ │ +              Voluntary context switches: 12089
│ │ │ +              Involuntary context switches: 2934
│ │ │                Swaps: 0
│ │ │                File system inputs: 0
│ │ │ -              File system outputs: 0
│ │ │ +              File system outputs: 24
│ │ │                Socket messages sent: 0
│ │ │                Socket messages received: 0
│ │ │                Signals delivered: 0
│ │ │                Page size (bytes): 4096
│ │ │                Exit status: 0
    
│ │ │              
    
│ │ │                
    i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
    
│ │ │              
    
│ │ │                
    i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24089-0/6.m2" >"/tmp/M2-24089-0/6.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-30883-0/6.m2" >"/tmp/M2-30883-0/6.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -323,21 +323,21 @@
│ │ │                               2
│ │ │  o23 : R-module, quotient of R
│ │ │              
    
│ │ │                
    i24 : h=runExternalM2(fn,identity,v)
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24089-0/7.m2" >"/tmp/M2-24089-0/7.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-30883-0/7.m2" >"/tmp/M2-30883-0/7.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │ -o24 = HashTable{"answer file" => /tmp/M2-24089-0/7.ans}
│ │ │ +o24 = HashTable{"answer file" => /tmp/M2-30883-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-24089-0/7.out
│ │ │ +                "output file" => /tmp/M2-30883-0/7.out
│ │ │                  "return code" => 256
│ │ │                  "statistics" => null
│ │ │                  "time used" => 3
│ │ │                  value => null
│ │ │  
│ │ │  o24 : HashTable
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i25 : get(h#"output file")
│ │ │  
│ │ │  o25 = 
│ │ │ -      i1 : -- Script /tmp/M2-24089-0/7.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-30883-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-24089-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-30883-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-24089-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-30883-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │        stdio:4:74:(3):[2]: error: no method for binary operator ^ applied to objects:
│ │ │                    R (of class Symbol)
│ │ │              ^     2 (of class ZZ)
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -372,15 +372,15 @@
│ │ │
    		│ │ │ 
    i26 : fn<<///R=QQ[x,y];///<<endl<<flush;
    │ │ │
    │ │ │ 
    i27 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24089-0/8.m2" >"/tmp/M2-24089-0/8.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-30883-0/8.m2" >"/tmp/M2-30883-0/8.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o27 = true
    │ │ │
    │ │ │
    │ │ │ @@ -392,15 +392,15 @@ │ │ │ │ │ │ 
    i28 : v=R;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i29 : h=runExternalM2(fn,identity,v);
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24089-0/9.m2" >"/tmp/M2-24089-0/9.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-30883-0/9.m2" >"/tmp/M2-30883-0/9.out" 2>&1 ))
│ │ │  Finished running.
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i30 : h#value
│ │ │ ├── html2text {}
│ │ │ │ @@ -45,25 +45,25 @@
│ │ │ │  the output file (unless it was deleted), the name of the answer file (unless it
│ │ │ │  was deleted), any statistics recorded about the resource usage, and the value
│ │ │ │  returned by the function func. If the child process terminates abnormally, then
│ │ │ │  usually the exit code is nonzero and the value returned is _n_u_l_l.
│ │ │ │  For example, we can write a few functions to a temporary file:
│ │ │ │  i1 : fn=temporaryFileName()|".m2"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-24089-0/0.m2
│ │ │ │ +o1 = /tmp/M2-30883-0/0.m2
│ │ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-24089-0/1.out" 2>&1 ))
│ │ │ │ +M2-30883-0/1.m2" >"/tmp/M2-30883-0/1.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i7 : h
│ │ │ │  
│ │ │ │  o7 = HashTable{"answer file" => null}
│ │ │ │                 "exit code" => 0
│ │ │ │                 "output file" => null
│ │ │ │                 "return code" => 0
│ │ │ │ @@ -79,47 +79,47 @@
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  An abnormal program exit will have a nonzero exit code; also, the value will be
│ │ │ │  null, the output file should exist, but the answer file may not exist unless
│ │ │ │  the routine finished successfully.
│ │ │ │  i10 : h=runExternalM2(fn,"justexit",());
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-24089-0/2.m2" >"/tmp/M2-24089-0/2.out" 2>&1 ))
│ │ │ │ +M2-30883-0/2.m2" >"/tmp/M2-30883-0/2.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i11 : h
│ │ │ │  
│ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-24089-0/2.ans}
│ │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-30883-0/2.ans}
│ │ │ │                  "exit code" => 27
│ │ │ │ -                "output file" => /tmp/M2-24089-0/2.out
│ │ │ │ +                "output file" => /tmp/M2-30883-0/2.out
│ │ │ │                  "return code" => 6912
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 3
│ │ │ │ +                "time used" => 2
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o11 : HashTable
│ │ │ │  i12 : fileExists(h#"output file")
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : fileExists(h#"answer file")
│ │ │ │  
│ │ │ │  o13 = false
│ │ │ │  Here, we use _r_e_s_o_u_r_c_e_ _l_i_m_i_t_s to limit the routine to 2 seconds of computational
│ │ │ │  time, while the system is asked to use 10 seconds of computational time:
│ │ │ │  i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ │  Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -
│ │ │ │ -q  <"/tmp/M2-24089-0/3.m2" >"/tmp/M2-24089-0/3.out" 2>&1 ))
│ │ │ │ +q  <"/tmp/M2-30883-0/3.m2" >"/tmp/M2-30883-0/3.out" 2>&1 ))
│ │ │ │  Killed
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i15 : h
│ │ │ │  
│ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-24089-0/3.ans}
│ │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-30883-0/3.ans}
│ │ │ │                  "exit code" => 0
│ │ │ │ -                "output file" => /tmp/M2-24089-0/3.out
│ │ │ │ +                "output file" => /tmp/M2-30883-0/3.out
│ │ │ │                  "return code" => 9
│ │ │ │                  "statistics" => null
│ │ │ │                  "time used" => 2
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o15 : HashTable
│ │ │ │  i16 : if h#"output file" =!= null and fileExists(h#"output file") then get
│ │ │ │ @@ -128,110 +128,110 @@
│ │ │ │  o16 =
│ │ │ │  i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get
│ │ │ │  (h#"answer file")
│ │ │ │  We can get quite a lot of detail on the resources used with the _K_e_e_p_S_t_a_t_i_s_t_i_c_s
│ │ │ │  command:
│ │ │ │  i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ │  Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop -
│ │ │ │ --no-debug --silent  -q  <"/tmp/M2-24089-0/4.m2" >"/tmp/M2-24089-0/4.out" 2>&1')
│ │ │ │ ->"/tmp/M2-24089-0/4.stat" 2>&1 ))
│ │ │ │ +-no-debug --silent  -q  <"/tmp/M2-30883-0/4.m2" >"/tmp/M2-30883-0/4.out" 2>&1')
│ │ │ │ +>"/tmp/M2-30883-0/4.stat" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i19 : h#"statistics"
│ │ │ │  
│ │ │ │  o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug
│ │ │ │ ---silent  -q  <"/tmp/M2-24089-0/4.m2" >"/tmp/M2-24089-0/4.out" 2>&1"
│ │ │ │ -              User time (seconds): 6.09
│ │ │ │ -              System time (seconds): 0.14
│ │ │ │ -              Percent of CPU this job got: 90%
│ │ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.90
│ │ │ │ +--silent  -q  <"/tmp/M2-30883-0/4.m2" >"/tmp/M2-30883-0/4.out" 2>&1"
│ │ │ │ +              User time (seconds): 5.86
│ │ │ │ +              System time (seconds): 0.22
│ │ │ │ +              Percent of CPU this job got: 120%
│ │ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:05.07
│ │ │ │                Average shared text size (kbytes): 0
│ │ │ │                Average unshared data size (kbytes): 0
│ │ │ │                Average stack size (kbytes): 0
│ │ │ │                Average total size (kbytes): 0
│ │ │ │ -              Maximum resident set size (kbytes): 255332
│ │ │ │ +              Maximum resident set size (kbytes): 339296
│ │ │ │                Average resident set size (kbytes): 0
│ │ │ │                Major (requiring I/O) page faults: 0
│ │ │ │ -              Minor (reclaiming a frame) page faults: 8236
│ │ │ │ -              Voluntary context switches: 2905
│ │ │ │ -              Involuntary context switches: 2134
│ │ │ │ +              Minor (reclaiming a frame) page faults: 10987
│ │ │ │ +              Voluntary context switches: 12089
│ │ │ │ +              Involuntary context switches: 2934
│ │ │ │                Swaps: 0
│ │ │ │                File system inputs: 0
│ │ │ │ -              File system outputs: 0
│ │ │ │ +              File system outputs: 24
│ │ │ │                Socket messages sent: 0
│ │ │ │                Socket messages received: 0
│ │ │ │                Signals delivered: 0
│ │ │ │                Page size (bytes): 4096
│ │ │ │                Exit status: 0
│ │ │ │  We can handle most kinds of objects as return values, although _M_u_t_a_b_l_e_M_a_t_r_i_x
│ │ │ │  does not work. Here, we use the built-in _i_d_e_n_t_i_t_y function:
│ │ │ │  i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
│ │ │ │  i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-24089-0/6.m2" >"/tmp/M2-24089-0/6.out" 2>&1 ))
│ │ │ │ +M2-30883-0/6.m2" >"/tmp/M2-30883-0/6.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  Some care is required, however:
│ │ │ │  i22 : R=QQ[x,y];
│ │ │ │  i23 : v=coker random(R^2,R^{3:-1})
│ │ │ │  
│ │ │ │  o23 = cokernel | 9/2x+9/4y 7/9x+7/10y 7x+3/7y |
│ │ │ │                 | 3/4x+7/4y 7/10x+7/3y 6/7x+6y |
│ │ │ │  
│ │ │ │                               2
│ │ │ │  o23 : R-module, quotient of R
│ │ │ │  i24 : h=runExternalM2(fn,identity,v)
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-24089-0/7.m2" >"/tmp/M2-24089-0/7.out" 2>&1 ))
│ │ │ │ +M2-30883-0/7.m2" >"/tmp/M2-30883-0/7.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  
│ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-24089-0/7.ans}
│ │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-30883-0/7.ans}
│ │ │ │                  "exit code" => 1
│ │ │ │ -                "output file" => /tmp/M2-24089-0/7.out
│ │ │ │ +                "output file" => /tmp/M2-30883-0/7.out
│ │ │ │                  "return code" => 256
│ │ │ │                  "statistics" => null
│ │ │ │                  "time used" => 3
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o24 : HashTable
│ │ │ │  To view the error message:
│ │ │ │  i25 : get(h#"output file")
│ │ │ │  
│ │ │ │  o25 =
│ │ │ │ -      i1 : -- Script /tmp/M2-24089-0/7.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-30883-0/7.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-24089-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-30883-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-24089-0/7.ans",identity (cokernel
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-30883-0/7.ans",identity (cokernel
│ │ │ │  (map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/
│ │ │ │  4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ │        stdio:4:74:(3):[2]: error: no method for binary operator ^ applied to
│ │ │ │  objects:
│ │ │ │                    R (of class Symbol)
│ │ │ │              ^     2 (of class ZZ)
│ │ │ │  Keep in mind that the object you are passing must make sense in the context of
│ │ │ │  the file containing your function! For instance, here we need to define the
│ │ │ │  ring:
│ │ │ │  i26 : fn<"/tmp/M2-24089-0/8.out" 2>&1 ))
│ │ │ │ +M2-30883-0/8.m2" >"/tmp/M2-30883-0/8.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  
│ │ │ │  o27 = true
│ │ │ │  This problem can be avoided by following some _s_u_g_g_e_s_t_i_o_n_s_ _f_o_r_ _u_s_i_n_g
│ │ │ │  _R_u_n_E_x_t_e_r_n_a_l_M_2.
│ │ │ │  The objects may unavoidably lose some internal references, though:
│ │ │ │  i28 : v=R;
│ │ │ │  i29 : h=runExternalM2(fn,identity,v);
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-24089-0/9.m2" >"/tmp/M2-24089-0/9.out" 2>&1 ))
│ │ │ │ +M2-30883-0/9.m2" >"/tmp/M2-30883-0/9.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i30 : h#value
│ │ │ │  
│ │ │ │  o30 = QQ[x..y]
│ │ │ │  
│ │ │ │  o30 : PolynomialRing
│ │ │ │  i31 : v===h#value
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out
│ │ │ @@ -30,23 +30,23 @@
│ │ │                                              )
│ │ │  
│ │ │                            "variable positions" => {-1}
│ │ │  
│ │ │  o5 : InterpretedSLProgram
│ │ │  
│ │ │  i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.000238908s (cpu); 0.000234991s (thread); 0s (gc)
│ │ │ + -- used 0.000297485s (cpu); 0.000290948s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i7 : ZZ[y];
│ │ │  
│ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 5.35287s (cpu); 3.59585s (thread); 0s (gc)
│ │ │ + -- used 4.39288s (cpu); 3.15395s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html
│ │ │ @@ -109,29 +109,29 @@
│ │ │  
│ │ │  o5 : InterpretedSLProgram
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.000238908s (cpu); 0.000234991s (thread); 0s (gc)
│ │ │ + -- used 0.000297485s (cpu); 0.000290948s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : ZZ[y];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 5.35287s (cpu); 3.59585s (thread); 0s (gc)
│ │ │ + -- used 4.39288s (cpu); 3.15395s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,21 +38,21 @@
│ │ │ │                                              output nodes: 1
│ │ │ │                                              )
│ │ │ │  
│ │ │ │                            "variable positions" => {-1}
│ │ │ │  
│ │ │ │  o5 : InterpretedSLProgram
│ │ │ │  i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ │ - -- used 0.000238908s (cpu); 0.000234991s (thread); 0s (gc)
│ │ │ │ + -- used 0.000297485s (cpu); 0.000290948s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1       1
│ │ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │ │  i7 : ZZ[y];
│ │ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ │ - -- used 5.35287s (cpu); 3.59585s (thread); 0s (gc)
│ │ │ │ + -- used 4.39288s (cpu); 3.15395s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  i3 : r={5,11,3,2}
│ │ │  
│ │ │  o3 = {5, 11, 3, 2}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ - -- .00713238s elapsed
│ │ │ + -- .00772479s elapsed
│ │ │  
│ │ │         6       19       19       7       3
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │                                          
│ │ │       0       1        2        3       4
│ │ │  
│ │ │  o4 : Complex
│ │ │ @@ -41,15 +41,15 @@
│ │ │         53        53         53         53        53
│ │ │                                                  
│ │ │       -1        0          1          2         3
│ │ │  
│ │ │  o6 : Complex
│ │ │  
│ │ │  i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ - -- .00244023s elapsed
│ │ │ + -- .00318751s elapsed
│ │ │  
│ │ │  i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │                 0 => 3
│ │ │                 1 => 5
│ │ │                 2 => 2
│ │ │ @@ -85,15 +85,15 @@
│ │ │  i12 : maximalEntry complex errors
│ │ │  
│ │ │  o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .0671974s elapsed
│ │ │ + -- .0267998s elapsed
│ │ │  
│ │ │  i14 : hL === h
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : SigmaL =source U;
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Homology.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  i3 : r={4,3,3}
│ │ │  
│ │ │  o3 = {4, 3, 3}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00313593s elapsed
│ │ │ + -- .00484988s elapsed
│ │ │  
│ │ │         5       10       11       5
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o4 : Complex
│ │ │ @@ -41,25 +41,25 @@
│ │ │         53        53         53         53
│ │ │                                        
│ │ │       0         1          2          3
│ │ │  
│ │ │  o6 : Complex
│ │ │  
│ │ │  i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ - -- .000589632s elapsed
│ │ │ + -- .000763316s elapsed
│ │ │  
│ │ │  o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, )           })
│ │ │                  1 => 3             2 => (37.9214, 30.3707, 1.61954e-14)
│ │ │                  2 => 5             3 => (14.972, 8.57847, 3.90646e-15)
│ │ │                  3 => 2
│ │ │  
│ │ │  o7 : Sequence
│ │ │  
│ │ │  i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
│ │ │ - -- .00288549s elapsed
│ │ │ + -- .00167975s elapsed
│ │ │  
│ │ │  o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14)      })
│ │ │                  1 => 3             1 => (1.71747, 922.381, 2.51496e-13)
│ │ │                  2 => 5             2 => (922.381, 73.5901, 1.81323e-13)
│ │ │                  3 => 2             3 => (73.5901, , 2.82914e-13)
│ │ │  
│ │ │  o8 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_common__Entries.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i4 : r={4,3,5}
│ │ │  
│ │ │  o4 = {4, 3, 5}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ - -- .00379837s elapsed
│ │ │ + -- .00427666s elapsed
│ │ │  
│ │ │         6       10       13       8
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i4 : r={4,3,3}
│ │ │  
│ │ │  o4 = {4, 3, 3}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00544721s elapsed
│ │ │ + -- .00325643s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_project__To__Complex.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i4 : r={4,3,3}
│ │ │  
│ │ │  o4 = {4, 3, 3}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .0030229s elapsed
│ │ │ + -- .0034446s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ - -- .00713238s elapsed
│ │ │ + -- .00772479s elapsed
│ │ │  
│ │ │         6       19       19       7       3
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │                                          
│ │ │       0       1        2        3       4
│ │ │  
│ │ │  o4 : Complex
    │ │ │ @@ -145,15 +145,15 @@ │ │ │ │ │ │ o6 : Complex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ - -- .00244023s elapsed
    │ │ │ + -- .00318751s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │ @@ -207,15 +207,15 @@
│ │ │  
│ │ │  o12 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .0671974s elapsed
    │ │ │ + -- .0267998s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : hL === h
│ │ │  
│ │ │  o14 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -37,15 +37,15 @@ │ │ │ │ o2 : List │ │ │ │ i3 : r={5,11,3,2} │ │ │ │ │ │ │ │ o3 = {5, 11, 3, 2} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : elapsedTime C=randomChainComplex(h,r,Height=>4) │ │ │ │ - -- .00713238s elapsed │ │ │ │ + -- .00772479s elapsed │ │ │ │ │ │ │ │ 6 19 19 7 3 │ │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ │ │ o4 : Complex │ │ │ │ @@ -60,15 +60,15 @@ │ │ │ │ o6 = RR <-- RR <-- RR <-- RR <-- RR │ │ │ │ 53 53 53 53 53 │ │ │ │ │ │ │ │ -1 0 1 2 3 │ │ │ │ │ │ │ │ o6 : Complex │ │ │ │ i7 : elapsedTime (h,U)=SVDComplex CR; │ │ │ │ - -- .00244023s elapsed │ │ │ │ + -- .00318751s elapsed │ │ │ │ i8 : h │ │ │ │ │ │ │ │ o8 = HashTable{-1 => 1} │ │ │ │ 0 => 3 │ │ │ │ 1 => 5 │ │ │ │ 2 => 2 │ │ │ │ 3 => 1 │ │ │ │ @@ -99,15 +99,15 @@ │ │ │ │ 1)*Sigma.dd_ell*transpose U_ell); │ │ │ │ i12 : maximalEntry complex errors │ │ │ │ │ │ │ │ o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian); │ │ │ │ - -- .0671974s elapsed │ │ │ │ + -- .0267998s elapsed │ │ │ │ i14 : hL === h │ │ │ │ │ │ │ │ o14 = true │ │ │ │ i15 : SigmaL =source U; │ │ │ │ i16 : for i from min CR+1 to max CR list maximalEntry(SigmaL.dd_i -Sigma.dd_i) │ │ │ │ │ │ │ │ o16 = {1.77636e-14, 6.39488e-14, 8.52651e-14, 3.55271e-15} │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ @@ -112,15 +112,15 @@ │ │ │ │ │ │ o3 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00313593s elapsed
│ │ │ + -- .00484988s elapsed
│ │ │  
│ │ │         5       10       11       5
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o4 : Complex
    │ │ │ @@ -147,28 +147,28 @@ │ │ │ │ │ │ o6 : Complex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ - -- .000589632s elapsed
│ │ │ + -- .000763316s elapsed
│ │ │  
│ │ │  o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, )           })
│ │ │                  1 => 3             2 => (37.9214, 30.3707, 1.61954e-14)
│ │ │                  2 => 5             3 => (14.972, 8.57847, 3.90646e-15)
│ │ │                  3 => 2
│ │ │  
│ │ │  o7 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
│ │ │ - -- .00288549s elapsed
│ │ │ + -- .00167975s elapsed
│ │ │  
│ │ │  o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14)      })
│ │ │                  1 => 3             1 => (1.71747, 922.381, 2.51496e-13)
│ │ │                  2 => 5             2 => (922.381, 73.5901, 1.81323e-13)
│ │ │                  3 => 2             3 => (73.5901, , 2.82914e-13)
│ │ │  
│ │ │  o8 : Sequence
    │ │ │ ├── html2text {} │ │ │ │ @@ -40,15 +40,15 @@ │ │ │ │ o2 : List │ │ │ │ i3 : r={4,3,3} │ │ │ │ │ │ │ │ o3 = {4, 3, 3} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ │ - -- .00313593s elapsed │ │ │ │ + -- .00484988s elapsed │ │ │ │ │ │ │ │ 5 10 11 5 │ │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o4 : Complex │ │ │ │ @@ -63,24 +63,24 @@ │ │ │ │ o6 = RR <-- RR <-- RR <-- RR │ │ │ │ 53 53 53 53 │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o6 : Complex │ │ │ │ i7 : elapsedTime (h,h1)=SVDHomology CR │ │ │ │ - -- .000589632s elapsed │ │ │ │ + -- .000763316s elapsed │ │ │ │ │ │ │ │ o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, ) }) │ │ │ │ 1 => 3 2 => (37.9214, 30.3707, 1.61954e-14) │ │ │ │ 2 => 5 3 => (14.972, 8.57847, 3.90646e-15) │ │ │ │ 3 => 2 │ │ │ │ │ │ │ │ o7 : Sequence │ │ │ │ i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian) │ │ │ │ - -- .00288549s elapsed │ │ │ │ + -- .00167975s elapsed │ │ │ │ │ │ │ │ o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14) }) │ │ │ │ 1 => 3 1 => (1.71747, 922.381, 2.51496e-13) │ │ │ │ 2 => 5 2 => (922.381, 73.5901, 1.81323e-13) │ │ │ │ 3 => 2 3 => (73.5901, , 2.82914e-13) │ │ │ │ │ │ │ │ o8 : Sequence │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ @@ -115,15 +115,15 @@ │ │ │ │ │ │ o4 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ - -- .00379837s elapsed
│ │ │ + -- .00427666s elapsed
│ │ │  
│ │ │         6       10       13       8
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
    │ │ │ ├── html2text {} │ │ │ │ @@ -34,15 +34,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,5} │ │ │ │ │ │ │ │ o4 = {4, 3, 5} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true) │ │ │ │ - -- .00379837s elapsed │ │ │ │ + -- .00427666s elapsed │ │ │ │ │ │ │ │ 6 10 13 8 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ │ │ o4 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00544721s elapsed
│ │ │ + -- .00325643s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ │ - -- .00544721s elapsed │ │ │ │ + -- .00325643s elapsed │ │ │ │ │ │ │ │ 6 10 11 5 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_project__To__Complex.html │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ │ │ o4 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .0030229s elapsed
│ │ │ + -- .0034446s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ │ - -- .0030229s elapsed │ │ │ │ + -- .0034446s elapsed │ │ │ │ │ │ │ │ 6 10 11 5 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ @@ -37,33 +37,33 @@ │ │ │ o5 : Ideal of S │ │ │ │ │ │ i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5)); │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ i7 : time quotient(I^3, J^2, Strategy => Iterate); │ │ │ - -- used 0.421522s (cpu); 0.334587s (thread); 0s (gc) │ │ │ + -- used 0.384813s (cpu); 0.384569s (thread); 0s (gc) │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ │ │ i8 : time quotient(I^3, J^2, Strategy => Quotient); │ │ │ - -- used 0.598771s (cpu); 0.505197s (thread); 0s (gc) │ │ │ + -- used 0.750439s (cpu); 0.649123s (thread); 0s (gc) │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ │ │ i9 : S = ZZ/101[vars(0..4)]; │ │ │ │ │ │ i10 : I = ideal vars S; │ │ │ │ │ │ o10 : Ideal of S │ │ │ │ │ │ i11 : time quotient(I^5, I^3, Strategy => Iterate); │ │ │ - -- used 0.0212293s (cpu); 0.0212309s (thread); 0s (gc) │ │ │ + -- used 0.0271809s (cpu); 0.0271833s (thread); 0s (gc) │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ │ │ i12 : time quotient(I^5, I^3, Strategy => Quotient); │ │ │ - -- used 0.00707788s (cpu); 0.00707878s (thread); 0s (gc) │ │ │ + -- used 0.00903099s (cpu); 0.00903792s (thread); 0s (gc) │ │ │ │ │ │ o12 : Ideal of S │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ @@ -130,23 +130,23 @@ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time quotient(I^3, J^2, Strategy => Iterate);
│ │ │ - -- used 0.421522s (cpu); 0.334587s (thread); 0s (gc)
│ │ │ + -- used 0.384813s (cpu); 0.384569s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ - -- used 0.598771s (cpu); 0.505197s (thread); 0s (gc)
│ │ │ + -- used 0.750439s (cpu); 0.649123s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    Strategy => Quotient is faster in other cases:

    │ │ │ @@ -163,23 +163,23 @@ │ │ │ │ │ │ o10 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : time quotient(I^5, I^3, Strategy => Iterate);
│ │ │ - -- used 0.0212293s (cpu); 0.0212309s (thread); 0s (gc)
│ │ │ + -- used 0.0271809s (cpu); 0.0271833s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ - -- used 0.00707788s (cpu); 0.00707878s (thread); 0s (gc)
│ │ │ + -- used 0.00903099s (cpu); 0.00903792s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -56,32 +56,32 @@ │ │ │ │ i5 : I = monomialCurveIdeal(S, 1..n-1); │ │ │ │ │ │ │ │ o5 : Ideal of S │ │ │ │ i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5)); │ │ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ i7 : time quotient(I^3, J^2, Strategy => Iterate); │ │ │ │ - -- used 0.421522s (cpu); 0.334587s (thread); 0s (gc) │ │ │ │ + -- used 0.384813s (cpu); 0.384569s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ i8 : time quotient(I^3, J^2, Strategy => Quotient); │ │ │ │ - -- used 0.598771s (cpu); 0.505197s (thread); 0s (gc) │ │ │ │ + -- used 0.750439s (cpu); 0.649123s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ Strategy => Quotient is faster in other cases: │ │ │ │ i9 : S = ZZ/101[vars(0..4)]; │ │ │ │ i10 : I = ideal vars S; │ │ │ │ │ │ │ │ o10 : Ideal of S │ │ │ │ i11 : time quotient(I^5, I^3, Strategy => Iterate); │ │ │ │ - -- used 0.0212293s (cpu); 0.0212309s (thread); 0s (gc) │ │ │ │ + -- used 0.0271809s (cpu); 0.0271833s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ i12 : time quotient(I^5, I^3, Strategy => Quotient); │ │ │ │ - -- used 0.00707788s (cpu); 0.00707878s (thread); 0s (gc) │ │ │ │ + -- used 0.00903099s (cpu); 0.00903792s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of S │ │ │ │ ********** RReeffeerreenncceess ********** │ │ │ │ For further information see for example Exercise 15.41 in Eisenbud's │ │ │ │ Commutative Algebra with a View Towards Algebraic Geometry. │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd SSttrraatteeggyy:: ********** │ │ │ │ * addHook(...,Strategy=>...) -- see _a_d_d_H_o_o_k -- add a hook function to an │ │ ├── ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ @@ -40,23 +40,23 @@ │ │ │ ) │ │ │ │ │ │ o6 = f │ │ │ │ │ │ o6 : FunctionClosure │ │ │ │ │ │ i7 : for n from 2 to 10 list time f n │ │ │ - -- used 0.00491395s (cpu); 0.00490902s (thread); 0s (gc) │ │ │ - -- used 0.00666051s (cpu); 0.00666166s (thread); 0s (gc) │ │ │ - -- used 0.0100461s (cpu); 0.0100472s (thread); 0s (gc) │ │ │ - -- used 0.016907s (cpu); 0.0169085s (thread); 0s (gc) │ │ │ - -- used 0.0356775s (cpu); 0.0356803s (thread); 0s (gc) │ │ │ - -- used 0.067465s (cpu); 0.0674711s (thread); 0s (gc) │ │ │ - -- used 0.0930682s (cpu); 0.0930789s (thread); 0s (gc) │ │ │ - -- used 0.304518s (cpu); 0.186879s (thread); 0s (gc) │ │ │ - -- used 0.41398s (cpu); 0.285933s (thread); 0s (gc) │ │ │ + -- used 0.00728523s (cpu); 0.00728497s (thread); 0s (gc) │ │ │ + -- used 0.0100431s (cpu); 0.0100525s (thread); 0s (gc) │ │ │ + -- used 0.0212316s (cpu); 0.0212404s (thread); 0s (gc) │ │ │ + -- used 0.0339474s (cpu); 0.0339551s (thread); 0s (gc) │ │ │ + -- used 0.0690417s (cpu); 0.0690493s (thread); 0s (gc) │ │ │ + -- used 0.094244s (cpu); 0.0942515s (thread); 0s (gc) │ │ │ + -- used 0.102788s (cpu); 0.102797s (thread); 0s (gc) │ │ │ + -- used 0.164638s (cpu); 0.164645s (thread); 0s (gc) │ │ │ + -- used 0.412508s (cpu); 0.288768s (thread); 0s (gc) │ │ │ │ │ │ o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775, │ │ │ ------------------------------------------------------------------------ │ │ │ 289139638632755625, 520764738758073845321} │ │ │ │ │ │ o7 : List │ │ ├── ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ @@ -131,23 +131,23 @@ │ │ │ │ │ │ o6 : FunctionClosure │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : for n from 2 to 10 list time f n
│ │ │ - -- used 0.00491395s (cpu); 0.00490902s (thread); 0s (gc)
│ │ │ - -- used 0.00666051s (cpu); 0.00666166s (thread); 0s (gc)
│ │ │ - -- used 0.0100461s (cpu); 0.0100472s (thread); 0s (gc)
│ │ │ - -- used 0.016907s (cpu); 0.0169085s (thread); 0s (gc)
│ │ │ - -- used 0.0356775s (cpu); 0.0356803s (thread); 0s (gc)
│ │ │ - -- used 0.067465s (cpu); 0.0674711s (thread); 0s (gc)
│ │ │ - -- used 0.0930682s (cpu); 0.0930789s (thread); 0s (gc)
│ │ │ - -- used 0.304518s (cpu); 0.186879s (thread); 0s (gc)
│ │ │ - -- used 0.41398s (cpu); 0.285933s (thread); 0s (gc)
│ │ │ + -- used 0.00728523s (cpu); 0.00728497s (thread); 0s (gc)
│ │ │ + -- used 0.0100431s (cpu); 0.0100525s (thread); 0s (gc)
│ │ │ + -- used 0.0212316s (cpu); 0.0212404s (thread); 0s (gc)
│ │ │ + -- used 0.0339474s (cpu); 0.0339551s (thread); 0s (gc)
│ │ │ + -- used 0.0690417s (cpu); 0.0690493s (thread); 0s (gc)
│ │ │ + -- used 0.094244s (cpu); 0.0942515s (thread); 0s (gc)
│ │ │ + -- used 0.102788s (cpu); 0.102797s (thread); 0s (gc)
│ │ │ + -- used 0.164638s (cpu); 0.164645s (thread); 0s (gc)
│ │ │ + -- used 0.412508s (cpu); 0.288768s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -56,23 +56,23 @@ │ │ │ │ integral chern symmetricPower_(2*n-3) last bundles G │ │ │ │ ) │ │ │ │ │ │ │ │ o6 = f │ │ │ │ │ │ │ │ o6 : FunctionClosure │ │ │ │ i7 : for n from 2 to 10 list time f n │ │ │ │ - -- used 0.00491395s (cpu); 0.00490902s (thread); 0s (gc) │ │ │ │ - -- used 0.00666051s (cpu); 0.00666166s (thread); 0s (gc) │ │ │ │ - -- used 0.0100461s (cpu); 0.0100472s (thread); 0s (gc) │ │ │ │ - -- used 0.016907s (cpu); 0.0169085s (thread); 0s (gc) │ │ │ │ - -- used 0.0356775s (cpu); 0.0356803s (thread); 0s (gc) │ │ │ │ - -- used 0.067465s (cpu); 0.0674711s (thread); 0s (gc) │ │ │ │ - -- used 0.0930682s (cpu); 0.0930789s (thread); 0s (gc) │ │ │ │ - -- used 0.304518s (cpu); 0.186879s (thread); 0s (gc) │ │ │ │ - -- used 0.41398s (cpu); 0.285933s (thread); 0s (gc) │ │ │ │ + -- used 0.00728523s (cpu); 0.00728497s (thread); 0s (gc) │ │ │ │ + -- used 0.0100431s (cpu); 0.0100525s (thread); 0s (gc) │ │ │ │ + -- used 0.0212316s (cpu); 0.0212404s (thread); 0s (gc) │ │ │ │ + -- used 0.0339474s (cpu); 0.0339551s (thread); 0s (gc) │ │ │ │ + -- used 0.0690417s (cpu); 0.0690493s (thread); 0s (gc) │ │ │ │ + -- used 0.094244s (cpu); 0.0942515s (thread); 0s (gc) │ │ │ │ + -- used 0.102788s (cpu); 0.102797s (thread); 0s (gc) │ │ │ │ + -- used 0.164638s (cpu); 0.164645s (thread); 0s (gc) │ │ │ │ + -- used 0.412508s (cpu); 0.288768s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 289139638632755625, 520764738758073845321} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ Note: in characteristic zero, using Bertini's theorem, the numbers computed can │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/___Eor__H.out │ │ │ @@ -33,13 +33,13 @@ │ │ │ i6 : toS fe == toS fh │ │ │ │ │ │ o6 = true │ │ │ │ │ │ i7 : R = symmetricRing(QQ,12); │ │ │ │ │ │ i8 : elapsedTime jacobiTrudi({10},R,EorH => "E",Memoize => false); │ │ │ - -- .00814532s elapsed │ │ │ + -- .00439951s elapsed │ │ │ │ │ │ i9 : elapsedTime jacobiTrudi({10},R,EorH => "H",Memoize => false); │ │ │ - -- .000091682s elapsed │ │ │ + -- .000171755s elapsed │ │ │ │ │ │ i10 : │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/___Memoize.out │ │ │ @@ -1,23 +1,23 @@ │ │ │ -- -*- M2-comint -*- hash: 9113092561535051477 │ │ │ │ │ │ i1 : R = symmetricRing(QQ, 10); │ │ │ │ │ │ i2 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true); │ │ │ - -- .000426708s elapsed │ │ │ + -- .000696046s elapsed │ │ │ │ │ │ i3 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true); │ │ │ - -- .000014276s elapsed │ │ │ + -- .000032827s elapsed │ │ │ │ │ │ i4 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true); │ │ │ - -- .000378667s elapsed │ │ │ + -- .000820672s elapsed │ │ │ │ │ │ i5 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true); │ │ │ - -- .000017042s elapsed │ │ │ + -- .000031799s elapsed │ │ │ │ │ │ i6 : elapsedTime jacobiTrudi({4,3,2,1}, R); │ │ │ - -- .000011492s elapsed │ │ │ + -- .00002649s elapsed │ │ │ │ │ │ i7 : elapsedTime jacobiTrudi({4,3,2,1}, R); │ │ │ - -- .000011271s elapsed │ │ │ + -- .000024694s elapsed │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/_jacobi__Trudi.out │ │ │ @@ -50,18 +50,18 @@ │ │ │ i11 : toS fe == toS fh │ │ │ │ │ │ o11 = true │ │ │ │ │ │ i12 : R = symmetricRing(QQ,6); │ │ │ │ │ │ i13 : elapsedTime jacobiTrudi({4,3,2,1},R); │ │ │ - -- .000393305s elapsed │ │ │ + -- .000412268s elapsed │ │ │ │ │ │ i14 : elapsedTime jacobiTrudi({4,3,2,1},R); │ │ │ - -- .000013235s elapsed │ │ │ + -- .000019088s elapsed │ │ │ │ │ │ i15 : R = symmetricRing(QQ,5); │ │ │ │ │ │ i16 : S = schurRing R; │ │ │ │ │ │ i17 : jacobiTrudi({3,2,1},R) == toSymm(S_{3,2,1}) │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/_jacobi__Trudi_lp..._cm__Memoize_eq_gt..._rp.out │ │ │ @@ -3,16 +3,16 @@ │ │ │ i1 : R = symmetricRing(QQ,6); │ │ │ │ │ │ i2 : jacobiTrudi({4,3,2,1},R,Memoize => true) == jacobiTrudi({4,3,2,1},R,Memoize => false) │ │ │ │ │ │ o2 = true │ │ │ │ │ │ i3 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true); │ │ │ - -- .000437697s elapsed │ │ │ + -- .000794991s elapsed │ │ │ │ │ │ i4 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true); │ │ │ - -- .000014417s elapsed │ │ │ + -- .000021086s elapsed │ │ │ │ │ │ i5 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => false); │ │ │ - -- .000337451s elapsed │ │ │ + -- .000448389s elapsed │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/___Eor__H.html │ │ │ @@ -143,21 +143,21 @@ │ │ │ │ │ │ 
    i7 : R = symmetricRing(QQ,12);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : elapsedTime jacobiTrudi({10},R,EorH => "E",Memoize => false);
│ │ │ - -- .00814532s elapsed
    │ │ │ + -- .00439951s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : elapsedTime jacobiTrudi({10},R,EorH => "H",Memoize => false);
│ │ │ - -- .000091682s elapsed
    │ │ │ + -- .000171755s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    Functions with optional argument named EorH:

    │ │ │ ├── html2text {} │ │ │ │ @@ -46,17 +46,17 @@ │ │ │ │ o6 = true │ │ │ │ When the conjugate partition is much longer than lambda itself, the "H"-branch │ │ │ │ requires a smaller determinant and runs measurably faster. For example on │ │ │ │ lambda = (10) the conjugate is $(1^{10})$, so "H" only sets up a 1x1 │ │ │ │ determinant: │ │ │ │ i7 : R = symmetricRing(QQ,12); │ │ │ │ i8 : elapsedTime jacobiTrudi({10},R,EorH => "E",Memoize => false); │ │ │ │ - -- .00814532s elapsed │ │ │ │ + -- .00439951s elapsed │ │ │ │ i9 : elapsedTime jacobiTrudi({10},R,EorH => "H",Memoize => false); │ │ │ │ - -- .000091682s elapsed │ │ │ │ + -- .000171755s elapsed │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd EEoorrHH:: ********** │ │ │ │ * jacobiTrudi(...,EorH=>...) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _E_o_r_H is a _s_y_m_b_o_l. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:6153:0. │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/___Memoize.html │ │ │ @@ -66,59 +66,59 @@ │ │ │ │ │ │ 
    i1 : R = symmetricRing(QQ, 10);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true);
│ │ │ - -- .000426708s elapsed
    │ │ │ + -- .000696046s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true);
│ │ │ - -- .000014276s elapsed
    │ │ │ + -- .000032827s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    │ │ │

    The cache is attached to the ring R. After one partition is memoized, subsequent calls with a different partition perform the full Jacobi-Trudi determinant expansion, then cache it as well:

    │ │ │

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true);
│ │ │ - -- .000378667s elapsed
    │ │ │ + -- .000820672s elapsed │ │ │
    │ │ │ 
    i5 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true);
│ │ │ - -- .000017042s elapsed
    │ │ │ + -- .000031799s elapsed │ │ │
    │ │ │
    │ │ │

    │ │ │

    Without Memoize => true, each call recomputes the determinant from scratch; for large partitions this can be substantially more expensive than a single cached lookup.

    │ │ │

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : elapsedTime jacobiTrudi({4,3,2,1}, R);
│ │ │ - -- .000011492s elapsed
    │ │ │ + -- .00002649s elapsed │ │ │
    │ │ │ 
    i7 : elapsedTime jacobiTrudi({4,3,2,1}, R);
│ │ │ - -- .000011271s elapsed
    │ │ │ + -- .000024694s elapsed │ │ │
    │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -7,31 +7,31 @@ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ This is an optional argument for the _j_a_c_o_b_i_T_r_u_d_i function, allowing one to │ │ │ │ store its values in order to speed up computations. When Memoize => true, every │ │ │ │ computed value is cached in a hash table on the symmetric ring, so repeated │ │ │ │ calls on the same partition return the cached value in constant time. │ │ │ │ i1 : R = symmetricRing(QQ, 10); │ │ │ │ i2 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true); │ │ │ │ - -- .000426708s elapsed │ │ │ │ + -- .000696046s elapsed │ │ │ │ i3 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true); │ │ │ │ - -- .000014276s elapsed │ │ │ │ + -- .000032827s elapsed │ │ │ │ The cache is attached to the ring R. After one partition is memoized, │ │ │ │ subsequent calls with a different partition perform the full Jacobi-Trudi │ │ │ │ determinant expansion, then cache it as well: │ │ │ │ i4 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true); │ │ │ │ - -- .000378667s elapsed │ │ │ │ + -- .000820672s elapsed │ │ │ │ i5 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true); │ │ │ │ - -- .000017042s elapsed │ │ │ │ + -- .000031799s elapsed │ │ │ │ Without Memoize => true, each call recomputes the determinant from scratch; for │ │ │ │ large partitions this can be substantially more expensive than a single cached │ │ │ │ lookup. │ │ │ │ i6 : elapsedTime jacobiTrudi({4,3,2,1}, R); │ │ │ │ - -- .000011492s elapsed │ │ │ │ + -- .00002649s elapsed │ │ │ │ i7 : elapsedTime jacobiTrudi({4,3,2,1}, R); │ │ │ │ - -- .000011271s elapsed │ │ │ │ + -- .000024694s elapsed │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _j_a_c_o_b_i_T_r_u_d_i -- Jacobi-Trudi determinant │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd MMeemmooiizzee:: ********** │ │ │ │ * _j_a_c_o_b_i_T_r_u_d_i_(_._._._,_M_e_m_o_i_z_e_=_>_._._._) -- Store values of the jacobiTrudi │ │ │ │ function. │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _M_e_m_o_i_z_e is a _s_y_m_b_o_l. │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/_jacobi__Trudi.html │ │ │ @@ -186,21 +186,21 @@ │ │ │ │ │ │ 
      i12 : R = symmetricRing(QQ,6);
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i13 : elapsedTime jacobiTrudi({4,3,2,1},R);
│ │ │ - -- .000393305s elapsed
      │ │ │ + -- .000412268s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i14 : elapsedTime jacobiTrudi({4,3,2,1},R);
│ │ │ - -- .000013235s elapsed
      │ │ │ + -- .000019088s elapsed │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │

      │ │ │

      Passing a partition through toSymm applied to the corresponding Schur label reproduces the Jacobi-Trudi output:

      │ │ │

      │ │ │ ├── html2text {} │ │ │ │ @@ -67,17 +67,17 @@ │ │ │ │ │ │ │ │ o11 = true │ │ │ │ The routine caches intermediate subdeterminants on the ring via _j_a_c_o_b_i_T_r_u_d_i │ │ │ │ _(_._._._,_M_e_m_o_i_z_e_=_>_._._._), so a second call on a large partition returns almost │ │ │ │ instantly: │ │ │ │ i12 : R = symmetricRing(QQ,6); │ │ │ │ i13 : elapsedTime jacobiTrudi({4,3,2,1},R); │ │ │ │ - -- .000393305s elapsed │ │ │ │ + -- .000412268s elapsed │ │ │ │ i14 : elapsedTime jacobiTrudi({4,3,2,1},R); │ │ │ │ - -- .000013235s elapsed │ │ │ │ + -- .000019088s elapsed │ │ │ │ Passing a partition through _t_o_S_y_m_m applied to the corresponding Schur label │ │ │ │ reproduces the Jacobi-Trudi output: │ │ │ │ i15 : R = symmetricRing(QQ,5); │ │ │ │ i16 : S = schurRing R; │ │ │ │ i17 : jacobiTrudi({3,2,1},R) == toSymm(S_{3,2,1}) │ │ │ │ │ │ │ │ o17 = true │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/_jacobi__Trudi_lp..._cm__Memoize_eq_gt..._rp.html │ │ │ @@ -85,27 +85,27 @@ │ │ │ │ │ │ o2 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i3 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true);
│ │ │ - -- .000437697s elapsed
      │ │ │ + -- .000794991s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i4 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true);
│ │ │ - -- .000014417s elapsed
      │ │ │ + -- .000021086s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i5 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => false);
│ │ │ - -- .000337451s elapsed
      │ │ │ + -- .000448389s elapsed │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │
      │ │ │
      │ │ │

      Functions with optional argument named Memoize:

      │ │ │ ├── html2text {} │ │ │ │ @@ -17,19 +17,19 @@ │ │ │ │ extra memory in the ring. │ │ │ │ i1 : R = symmetricRing(QQ,6); │ │ │ │ i2 : jacobiTrudi({4,3,2,1},R,Memoize => true) == jacobiTrudi( │ │ │ │ {4,3,2,1},R,Memoize => false) │ │ │ │ │ │ │ │ o2 = true │ │ │ │ i3 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true); │ │ │ │ - -- .000437697s elapsed │ │ │ │ + -- .000794991s elapsed │ │ │ │ i4 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true); │ │ │ │ - -- .000014417s elapsed │ │ │ │ + -- .000021086s elapsed │ │ │ │ i5 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => false); │ │ │ │ - -- .000337451s elapsed │ │ │ │ + -- .000448389s elapsed │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd MMeemmooiizzee:: ********** │ │ │ │ * _j_a_c_o_b_i_T_r_u_d_i_(_._._._,_M_e_m_o_i_z_e_=_>_._._._) -- Store values of the jacobiTrudi │ │ │ │ function. │ │ │ │ ********** FFuurrtthheerr iinnffoorrmmaattiioonn ********** │ │ │ │ * Default value: _t_r_u_e │ │ │ │ * Function: _j_a_c_o_b_i_T_r_u_d_i -- Jacobi-Trudi determinant │ │ │ │ * Option key: _M_e_m_o_i_z_e -- Option to record values of the jacobiTrudi │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Component__Contained.out │ │ │ @@ -53,15 +53,15 @@ │ │ │ o9 : Ideal of R │ │ │ │ │ │ i10 : X=((W)*ideal(y)+ideal(f)); │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ │ │ i11 : time isComponentContained(X,Y) │ │ │ - -- used 3.95617s (cpu); 3.13504s (thread); 0s (gc) │ │ │ + -- used 7.31474s (cpu); 3.99554s (thread); 0s (gc) │ │ │ │ │ │ o11 = true │ │ │ │ │ │ i12 : print "we could confirm this with the computation:" │ │ │ we could confirm this with the computation: │ │ │ │ │ │ i13 : B=ideal(x)*ideal(y)*ideal(z) │ │ │ @@ -71,12 +71,12 @@ │ │ │ b*d*g, b*d*h, b*d*i, b*e*g, b*e*h, b*e*i, b*f*g, b*f*h, b*f*i, c*d*g, │ │ │ ----------------------------------------------------------------------- │ │ │ c*d*h, c*d*i, c*e*g, c*e*h, c*e*i, c*f*g, c*f*h, c*f*i) │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ │ │ i14 : time isSubset(saturate(Y,B),saturate(X,B)) │ │ │ - -- used 49.4667s (cpu); 45.7386s (thread); 0s (gc) │ │ │ + -- used 66.4694s (cpu); 60.858s (thread); 0s (gc) │ │ │ │ │ │ o14 = true │ │ │ │ │ │ i15 : │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre__Dim__X.out │ │ │ @@ -23,24 +23,24 @@ │ │ │ i5 : A = makeChowRing(R) │ │ │ │ │ │ o5 = A │ │ │ │ │ │ o5 : QuotientRing │ │ │ │ │ │ i6 : time s = segreDimX(X,Y,A) │ │ │ - -- used 0.271023s (cpu); 0.171695s (thread); 0s (gc) │ │ │ + -- used 0.4485s (cpu); 0.189487s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o6 = 2H + 4H H + 2H │ │ │ 1 1 2 2 │ │ │ │ │ │ o6 : A │ │ │ │ │ │ i7 : time segre(X,Y,A) │ │ │ - -- used 0.154172s (cpu); 0.116635s (thread); 0s (gc) │ │ │ + -- used 0.269494s (cpu); 0.135892s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ o7 = 12H H - 6H H - 6H H + 2H + 4H H + 2H │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ o7 : A │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Component__Contained.html │ │ │ @@ -167,15 +167,15 @@ │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i11 : time isComponentContained(X,Y)
│ │ │ - -- used 3.95617s (cpu); 3.13504s (thread); 0s (gc)
│ │ │ + -- used 7.31474s (cpu); 3.99554s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i12 : print "we could confirm this with the computation:"
│ │ │ @@ -194,15 +194,15 @@
│ │ │  
│ │ │  o13 : Ideal of R
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i14 : time isSubset(saturate(Y,B),saturate(X,B))
│ │ │ - -- used 49.4667s (cpu); 45.7386s (thread); 0s (gc)
│ │ │ + -- used 66.4694s (cpu); 60.858s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
      │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -68,30 +68,30 @@ │ │ │ │ i9 : Y=ideal (z_0*W_0-z_1*W_1)+ideal(f); │ │ │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ i10 : X=((W)*ideal(y)+ideal(f)); │ │ │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ i11 : time isComponentContained(X,Y) │ │ │ │ - -- used 3.95617s (cpu); 3.13504s (thread); 0s (gc) │ │ │ │ + -- used 7.31474s (cpu); 3.99554s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 = true │ │ │ │ i12 : print "we could confirm this with the computation:" │ │ │ │ we could confirm this with the computation: │ │ │ │ i13 : B=ideal(x)*ideal(y)*ideal(z) │ │ │ │ │ │ │ │ o13 = ideal (a*d*g, a*d*h, a*d*i, a*e*g, a*e*h, a*e*i, a*f*g, a*f*h, a*f*i, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ b*d*g, b*d*h, b*d*i, b*e*g, b*e*h, b*e*i, b*f*g, b*f*h, b*f*i, c*d*g, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ c*d*h, c*d*i, c*e*g, c*e*h, c*e*i, c*f*g, c*f*h, c*f*i) │ │ │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ i14 : time isSubset(saturate(Y,B),saturate(X,B)) │ │ │ │ - -- used 49.4667s (cpu); 45.7386s (thread); 0s (gc) │ │ │ │ + -- used 66.4694s (cpu); 60.858s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 = true │ │ │ │ ********** WWaayyss ttoo uussee iissCCoommppoonneennttCCoonnttaaiinneedd:: ********** │ │ │ │ * isComponentContained(Ideal,Ideal) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _i_s_C_o_m_p_o_n_e_n_t_C_o_n_t_a_i_n_e_d is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/html/_segre__Dim__X.html │ │ │ @@ -123,27 +123,27 @@ │ │ │ │ │ │ o5 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i6 : time s = segreDimX(X,Y,A)
│ │ │ - -- used 0.271023s (cpu); 0.171695s (thread); 0s (gc)
│ │ │ + -- used 0.4485s (cpu); 0.189487s (thread); 0s (gc)
│ │ │  
│ │ │         2             2
│ │ │  o6 = 2H  + 4H H  + 2H
│ │ │         1     1 2     2
│ │ │  
│ │ │  o6 : A
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i7 : time segre(X,Y,A)
│ │ │ - -- used 0.154172s (cpu); 0.116635s (thread); 0s (gc)
│ │ │ + -- used 0.269494s (cpu); 0.135892s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2     2             2
│ │ │  o7 = 12H H  - 6H H  - 6H H  + 2H  + 4H H  + 2H
│ │ │          1 2     1 2     1 2     1     1 2     2
│ │ │  
│ │ │  o7 : A
      │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -48,23 +48,23 @@ │ │ │ │ o4 : Ideal of R │ │ │ │ i5 : A = makeChowRing(R) │ │ │ │ │ │ │ │ o5 = A │ │ │ │ │ │ │ │ o5 : QuotientRing │ │ │ │ i6 : time s = segreDimX(X,Y,A) │ │ │ │ - -- used 0.271023s (cpu); 0.171695s (thread); 0s (gc) │ │ │ │ + -- used 0.4485s (cpu); 0.189487s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o6 = 2H + 4H H + 2H │ │ │ │ 1 1 2 2 │ │ │ │ │ │ │ │ o6 : A │ │ │ │ i7 : time segre(X,Y,A) │ │ │ │ - -- used 0.154172s (cpu); 0.116635s (thread); 0s (gc) │ │ │ │ + -- used 0.269494s (cpu); 0.135892s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ │ o7 = 12H H - 6H H - 6H H + 2H + 4H H + 2H │ │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ │ │ o7 : A │ │ │ │ ********** WWaayyss ttoo uussee sseeggrreeDDiimmXX:: ********** │ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/example-output/_test__Example.out │ │ │ @@ -1,6 +1,6 @@ │ │ │ -- -*- M2-comint -*- hash: 1331702921222 │ │ │ │ │ │ i1 : check SimpleDoc │ │ │ - -- capturing check(0, "SimpleDoc") -- .380049s elapsed │ │ │ + -- capturing check(0, "SimpleDoc") -- .188807s elapsed │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/html/_test__Example.html │ │ │ @@ -79,15 +79,15 @@ │ │ │
      │ │ │

      The check method executes all package tests defined this way.

      │ │ │
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │ 
      i1 : check SimpleDoc
│ │ │ - -- capturing check(0, "SimpleDoc")           -- .380049s elapsed
      │ │ │ + -- capturing check(0, "SimpleDoc") -- .188807s elapsed │ │ │
      │ │ │
      │ │ │
      │ │ │

      See also

      │ │ │
        │ │ │ ├── html2text {} │ │ │ │ @@ -10,15 +10,15 @@ │ │ │ │ The variable testExample is a _S_t_r_i_n_g containing an example of the use of _T_E_S_T │ │ │ │ to write a test case. │ │ │ │ TEST /// -* test for simpleDocFrob *- │ │ │ │ assert(simpleDocFrob(2,matrix{{1,2}}) == matrix{{1,2,0,0},{0,0,1,2}}) │ │ │ │ /// │ │ │ │ The _c_h_e_c_k method executes all package tests defined this way. │ │ │ │ i1 : check SimpleDoc │ │ │ │ - -- capturing check(0, "SimpleDoc") -- .380049s elapsed │ │ │ │ + -- capturing check(0, "SimpleDoc") -- .188807s elapsed │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _T_E_S_T -- add a test for a package │ │ │ │ * _c_h_e_c_k -- perform tests of a package │ │ │ │ * _p_a_c_k_a_g_e_T_e_m_p_l_a_t_e -- a template for a package │ │ │ │ * _d_o_c_E_x_a_m_p_l_e -- an example of a documentation string │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _t_e_s_t_E_x_a_m_p_l_e is a _s_t_r_i_n_g. │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/___Simplicial__Modules.out │ │ │ @@ -111,25 +111,25 @@ │ │ │ o7 = R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o7 : SimplicialModule │ │ │ │ │ │ i8 : elapsedTime simplicialModule(K,6) --specify top degree 6 │ │ │ - -- .123087s elapsed │ │ │ + -- .112767s elapsed │ │ │ │ │ │ 1 4 10 20 35 56 84 │ │ │ o8 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o8 : SimplicialModule │ │ │ │ │ │ i9 : elapsedTime S' = simplicialModule(K,6, Degeneracy => true) │ │ │ - -- .183923s elapsed │ │ │ + -- .175787s elapsed │ │ │ │ │ │ 1 4 10 20 35 56 84 │ │ │ o9 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o9 : SimplicialModule │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_ext__Power.out │ │ │ @@ -12,15 +12,15 @@ │ │ │ o2 = Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 │ │ │ │ │ │ o2 : Complex │ │ │ │ │ │ i3 : w3K = elapsedTime prune extPower(3, K) │ │ │ - -- 4.86653s elapsed │ │ │ + -- 6.24504s elapsed │ │ │ │ │ │ 1 18 63 91 60 15 │ │ │ o3 = Q <-- Q <-- Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 1 2 3 4 5 6 │ │ │ │ │ │ o3 : Complex │ │ │ @@ -93,15 +93,15 @@ │ │ │ 1 2 │ │ │ 2 : Q <--------------- Q : 2 │ │ │ {2} | 0 b | │ │ │ │ │ │ o7 : ComplexMap │ │ │ │ │ │ i8 : f = elapsedTime prune extPower(2, phi) │ │ │ - -- .296947s elapsed │ │ │ + -- .456911s elapsed │ │ │ │ │ │ 3 6 │ │ │ o8 = 1 : Q <------------------------- Q : 1 │ │ │ {1} | a b 0 0 0 0 | │ │ │ {1} | 0 0 0 b 0 0 | │ │ │ {2} | 0 0 0 0 ab b2 | │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_exterior__Inclusion.out │ │ │ @@ -12,15 +12,15 @@ │ │ │ o2 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o2 : Complex │ │ │ │ │ │ i3 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3 │ │ │ - -- .318475s elapsed │ │ │ + -- .217026s elapsed │ │ │ │ │ │ i4 : isWellDefined phi │ │ │ │ │ │ o4 = true │ │ │ │ │ │ i5 : isCommutative phi │ │ │ │ │ │ @@ -160,15 +160,15 @@ │ │ │ o10 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o10 : Complex │ │ │ │ │ │ i11 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3 │ │ │ - -- .30631s elapsed │ │ │ + -- .222931s elapsed │ │ │ │ │ │ i12 : isWellDefined phi │ │ │ │ │ │ o12 = true │ │ │ │ │ │ i13 : isCommutative phi │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_forget__Degeneracy_lp__Simplicial__Module_rp.out │ │ │ @@ -21,15 +21,15 @@ │ │ │ o3 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o3 : SimplicialModule │ │ │ │ │ │ i4 : elapsedTime S**S │ │ │ - -- .660129s elapsed │ │ │ + -- .401961s elapsed │ │ │ │ │ │ 1 25 225 1225 4900 15876 44100 │ │ │ o4 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o4 : SimplicialModule │ │ │ @@ -40,15 +40,15 @@ │ │ │ o5 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o5 : SimplicialModule │ │ │ │ │ │ i6 : elapsedTime fS**fS --faster when degeneracy is ignored │ │ │ - -- .449451s elapsed │ │ │ + -- .298888s elapsed │ │ │ │ │ │ 1 25 225 1225 4900 15876 44100 │ │ │ o6 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o6 : SimplicialModule │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_normalize_lp__Simplicial__Module_cm__Z__Z_rp.out │ │ │ @@ -114,35 +114,35 @@ │ │ │ o10 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ │ │ │ o10 : SimplicialModule │ │ │ │ │ │ i11 : elapsedTime prune normalize S10 │ │ │ - -- 4.64754s elapsed │ │ │ + -- 3.83116s elapsed │ │ │ │ │ │ 10 30 30 10 │ │ │ o11 = R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o11 : Complex │ │ │ │ │ │ i12 : elapsedTime prune normalize(S10, CheckSum => false) --about 3-4 times slower; becomes significant for larger ranks │ │ │ - -- 7.16268s elapsed │ │ │ + -- 7.31031s elapsed │ │ │ │ │ │ 10 30 30 10 │ │ │ o12 = R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o12 : Complex │ │ │ │ │ │ i13 : elapsedTime prune normalize(S10, 3, CheckSum => false) --MUCH FASTER! │ │ │ - -- .03868s elapsed │ │ │ + -- .0386601s elapsed │ │ │ │ │ │ 10 30 30 10 │ │ │ o13 = R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o13 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_schur__Map.out │ │ │ @@ -14,25 +14,25 @@ │ │ │ o3 = Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o3 : SimplicialModule │ │ │ │ │ │ i4 : S2S = elapsedTime schurMap({2}, S) │ │ │ - -- .113944s elapsed │ │ │ + -- .149338s elapsed │ │ │ │ │ │ 1 6 21 55 120 │ │ │ o4 = Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o4 : SimplicialModule │ │ │ │ │ │ i5 : elapsedTime schurMap({2}, S, Degeneracy => true) │ │ │ - -- .295946s elapsed │ │ │ + -- .313795s elapsed │ │ │ │ │ │ 1 6 21 55 120 │ │ │ o5 = Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o5 : SimplicialModule │ │ │ @@ -78,15 +78,15 @@ │ │ │ o10 = Q │ │ │ │ │ │ o10 : PolynomialRing │ │ │ │ │ │ i11 : K = koszulComplex vars Q; │ │ │ │ │ │ i12 : S2K = elapsedTime prune schurMap({2}, K, TopDegree => 4) │ │ │ - -- .773047s elapsed │ │ │ + -- 1.03239s elapsed │ │ │ │ │ │ 1 9 36 74 81 │ │ │ o12 = Q <-- Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o12 : Complex │ │ │ @@ -273,15 +273,15 @@ │ │ │ 1 2 │ │ │ 2 : Q <--------------- Q : 2 │ │ │ {2} | 0 b | │ │ │ │ │ │ o30 : ComplexMap │ │ │ │ │ │ i31 : f = elapsedTime prune schurMap({2}, phi) │ │ │ - -- .843938s elapsed │ │ │ + -- .955634s elapsed │ │ │ │ │ │ 1 1 │ │ │ o31 = 0 : Q <--------- Q : 0 │ │ │ | 1 | │ │ │ │ │ │ 5 9 │ │ │ 1 : Q <----------------------------------- Q : 1 │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_tensor__L__E__S.out │ │ │ @@ -62,15 +62,15 @@ │ │ │ o7 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o7 : Complex │ │ │ │ │ │ i8 : hL = elapsedTime prune tensorLES(L,4) │ │ │ - -- 1.88823s elapsed │ │ │ + -- 1.89997s elapsed │ │ │ │ │ │ 25 50 25 │ │ │ o8 = cokernel | b a | <-- cokernel | b a | <-- 0 <-- 0 <-- cokernel | a 0 b | <-- cokernel | a 0 b | <-- cokernel | b a 0 0 | <-- cokernel | b a 0 0 0 0 0 0 | <-- cokernel | 0 0 b a 0 0 0 0 | <-- cokernel | ab2 a2b b2 a2 | <-- cokernel | a2 b2 ab | <-- cokernel | ab2 a2b b2 a2 | <-- Q <-- Q <-- Q │ │ │ | 0 b a | | 0 b a | | 0 0 b a | | 0 0 b a 0 0 0 0 | | 0 0 0 0 b a 0 0 | │ │ │ 0 1 2 3 | 0 0 0 0 b a 0 0 | | ab2 a2b 0 0 0 0 b2 a2 | 9 10 11 12 13 14 │ │ │ 4 5 6 | 0 0 0 0 0 0 b a | │ │ │ 8 │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_ext__Power.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ │ │ │ o2 : Complex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i3 : w3K = elapsedTime prune extPower(3, K)
│ │ │ - -- 4.86653s elapsed
│ │ │ + -- 6.24504s elapsed
│ │ │  
│ │ │        1      18      63      91      60      15
│ │ │  o3 = Q  <-- Q   <-- Q   <-- Q   <-- Q   <-- Q
│ │ │                                               
│ │ │       1      2       3       4       5       6
│ │ │  
│ │ │  o3 : Complex
        │ │ │ @@ -203,15 +203,15 @@ │ │ │ │ │ │ o7 : ComplexMap │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i8 : f = elapsedTime prune extPower(2, phi)
│ │ │ - -- .296947s elapsed
│ │ │ + -- .456911s elapsed
│ │ │  
│ │ │            3                             6
│ │ │  o8 = 1 : Q  <------------------------- Q  : 1
│ │ │                 {1} | a b 0 0 0  0  |
│ │ │                 {1} | 0 0 0 b 0  0  |
│ │ │                 {2} | 0 0 0 0 ab b2 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │        1      2      1
│ │ │ │  o2 = Q  <-- Q  <-- Q
│ │ │ │  
│ │ │ │       0      1      2
│ │ │ │  
│ │ │ │  o2 : Complex
│ │ │ │  i3 : w3K = elapsedTime prune extPower(3, K)
│ │ │ │ - -- 4.86653s elapsed
│ │ │ │ + -- 6.24504s elapsed
│ │ │ │  
│ │ │ │        1      18      63      91      60      15
│ │ │ │  o3 = Q  <-- Q   <-- Q   <-- Q   <-- Q   <-- Q
│ │ │ │  
│ │ │ │       1      2       3       4       5       6
│ │ │ │  
│ │ │ │  o3 : Complex
│ │ │ │ @@ -115,15 +115,15 @@
│ │ │ │  
│ │ │ │            1                   2
│ │ │ │       2 : Q  <--------------- Q  : 2
│ │ │ │                 {2} | 0 b |
│ │ │ │  
│ │ │ │  o7 : ComplexMap
│ │ │ │  i8 : f = elapsedTime prune extPower(2, phi)
│ │ │ │ - -- .296947s elapsed
│ │ │ │ + -- .456911s elapsed
│ │ │ │  
│ │ │ │            3                             6
│ │ │ │  o8 = 1 : Q  <------------------------- Q  : 1
│ │ │ │                 {1} | a b 0 0 0  0  |
│ │ │ │                 {1} | 0 0 0 b 0  0  |
│ │ │ │                 {2} | 0 0 0 0 ab b2 |
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_exterior__Inclusion.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o2 : Complex
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i3 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3
│ │ │ - -- .318475s elapsed
        │ │ │ + -- .217026s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i4 : isWellDefined phi
│ │ │  
│ │ │  o4 = true
        │ │ │ @@ -278,15 +278,15 @@ │ │ │ │ │ │ o10 : Complex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i11 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3
│ │ │ - -- .30631s elapsed
        │ │ │ + -- .222931s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i12 : isWellDefined phi
│ │ │  
│ │ │  o12 = true
        │ │ │ ├── html2text {} │ │ │ │ @@ -30,15 +30,15 @@ │ │ │ │ 1 3 3 1 │ │ │ │ o2 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o2 : Complex │ │ │ │ i3 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3 │ │ │ │ - -- .318475s elapsed │ │ │ │ + -- .217026s elapsed │ │ │ │ i4 : isWellDefined phi │ │ │ │ │ │ │ │ o4 = true │ │ │ │ i5 : isCommutative phi │ │ │ │ │ │ │ │ o5 = true │ │ │ │ i6 : prune coker phi │ │ │ │ @@ -176,15 +176,15 @@ │ │ │ │ 1 3 3 1 │ │ │ │ o10 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o10 : Complex │ │ │ │ i11 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3 │ │ │ │ - -- .30631s elapsed │ │ │ │ + -- .222931s elapsed │ │ │ │ i12 : isWellDefined phi │ │ │ │ │ │ │ │ o12 = true │ │ │ │ i13 : isCommutative phi │ │ │ │ │ │ │ │ o13 = true │ │ │ │ i14 : for i to 2 list prune HH_i source phi │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_forget__Degeneracy_lp__Simplicial__Module_rp.html │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ │ │ o3 : SimplicialModule │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i4 : elapsedTime S**S
│ │ │ - -- .660129s elapsed
│ │ │ + -- .401961s elapsed
│ │ │  
│ │ │        1      25      225      1225      4900      15876      44100
│ │ │  o4 = Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q      <-- Q     <-- ...
│ │ │                                                               
│ │ │       0      1       2        3         4         5          6
│ │ │  
│ │ │  o4 : SimplicialModule
        │ │ │ @@ -134,15 +134,15 @@ │ │ │ │ │ │ o5 : SimplicialModule │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i6 : elapsedTime fS**fS --faster when degeneracy is ignored
│ │ │ - -- .449451s elapsed
│ │ │ + -- .298888s elapsed
│ │ │  
│ │ │        1      25      225      1225      4900      15876      44100
│ │ │  o6 = Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q      <-- Q     <-- ...
│ │ │                                                               
│ │ │       0      1       2        3         4         5          6
│ │ │  
│ │ │  o6 : SimplicialModule
        │ │ │ ├── html2text {} │ │ │ │ @@ -36,15 +36,15 @@ │ │ │ │ 1 5 15 35 70 126 210 │ │ │ │ o3 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o3 : SimplicialModule │ │ │ │ i4 : elapsedTime S**S │ │ │ │ - -- .660129s elapsed │ │ │ │ + -- .401961s elapsed │ │ │ │ │ │ │ │ 1 25 225 1225 4900 15876 44100 │ │ │ │ o4 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o4 : SimplicialModule │ │ │ │ @@ -53,15 +53,15 @@ │ │ │ │ 1 5 15 35 70 126 210 │ │ │ │ o5 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o5 : SimplicialModule │ │ │ │ i6 : elapsedTime fS**fS --faster when degeneracy is ignored │ │ │ │ - -- .449451s elapsed │ │ │ │ + -- .298888s elapsed │ │ │ │ │ │ │ │ 1 25 225 1225 4900 15876 44100 │ │ │ │ o6 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o6 : SimplicialModule │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_normalize_lp__Simplicial__Module_cm__Z__Z_rp.html │ │ │ @@ -237,28 +237,28 @@ │ │ │ │ │ │ o10 : SimplicialModule │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i11 : elapsedTime prune normalize S10
│ │ │ - -- 4.64754s elapsed
│ │ │ + -- 3.83116s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o11 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o11 : Complex
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i12 : elapsedTime prune normalize(S10, CheckSum => false) --about 3-4 times slower; becomes significant for larger ranks
│ │ │ - -- 7.16268s elapsed
│ │ │ + -- 7.31031s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o12 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o12 : Complex
        │ │ │ @@ -268,15 +268,15 @@ │ │ │
        │ │ │

        The user may also specify the top homological degree to compute the normalization up to. Note that this can help speed up computational time; if the user knows the normalization should have a shorter length, then they should specify this upper bound in the syntax:

        │ │ │
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
        │ │ │ 
        i13 : elapsedTime prune normalize(S10, 3, CheckSum => false) --MUCH FASTER!
│ │ │ - -- .03868s elapsed
│ │ │ + -- .0386601s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o13 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o13 : Complex
        │ │ │ ├── html2text {} │ │ │ │ @@ -156,38 +156,38 @@ │ │ │ │ │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 │ │ │ │ 9 10 │ │ │ │ │ │ │ │ o10 : SimplicialModule │ │ │ │ i11 : elapsedTime prune normalize S10 │ │ │ │ - -- 4.64754s elapsed │ │ │ │ + -- 3.83116s elapsed │ │ │ │ │ │ │ │ 10 30 30 10 │ │ │ │ o11 = R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o11 : Complex │ │ │ │ i12 : elapsedTime prune normalize(S10, CheckSum => false) --about 3-4 times │ │ │ │ slower; becomes significant for larger ranks │ │ │ │ - -- 7.16268s elapsed │ │ │ │ + -- 7.31031s elapsed │ │ │ │ │ │ │ │ 10 30 30 10 │ │ │ │ o12 = R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o12 : Complex │ │ │ │ The user may also specify the top homological degree to compute the │ │ │ │ normalization up to. Note that this can help speed up computational time; if │ │ │ │ the user knows the normalization should have a shorter length, then they should │ │ │ │ specify this upper bound in the syntax: │ │ │ │ i13 : elapsedTime prune normalize(S10, 3, CheckSum => false) --MUCH FASTER! │ │ │ │ - -- .03868s elapsed │ │ │ │ + -- .0386601s elapsed │ │ │ │ │ │ │ │ 10 30 30 10 │ │ │ │ o13 = R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o13 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_schur__Map.html │ │ │ @@ -110,28 +110,28 @@ │ │ │ │ │ │ o3 : SimplicialModule │ │ │
        │ │ │ 
        i4 : S2S = elapsedTime schurMap({2}, S)
│ │ │ - -- .113944s elapsed
│ │ │ + -- .149338s elapsed
│ │ │  
│ │ │        1      6      21      55      120
│ │ │  o4 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │                                      
│ │ │       0      1      2       3       4
│ │ │  
│ │ │  o4 : SimplicialModule
        │ │ │
        │ │ │ 
        i5 : elapsedTime schurMap({2}, S, Degeneracy => true)
│ │ │ - -- .295946s elapsed
│ │ │ + -- .313795s elapsed
│ │ │  
│ │ │        1      6      21      55      120
│ │ │  o5 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │                                      
│ │ │       0      1      2       3       4
│ │ │  
│ │ │  o5 : SimplicialModule
        │ │ │ @@ -208,15 +208,15 @@ │ │ │         		│ │ │ 
        i11 : K = koszulComplex vars Q;
        │ │ │
        │ │ │ 
        i12 : S2K = elapsedTime prune schurMap({2}, K, TopDegree => 4)
│ │ │ - -- .773047s elapsed
│ │ │ + -- 1.03239s elapsed
│ │ │  
│ │ │         1      9      36      74      81
│ │ │  o12 = Q  <-- Q  <-- Q   <-- Q   <-- Q
│ │ │                                       
│ │ │        0      1      2       3       4
│ │ │  
│ │ │  o12 : Complex
        │ │ │ @@ -470,15 +470,15 @@ │ │ │ │ │ │ o30 : ComplexMap │ │ │
        │ │ │ 
        i31 : f = elapsedTime prune schurMap({2}, phi)
│ │ │ - -- .843938s elapsed
│ │ │ + -- .955634s elapsed
│ │ │  
│ │ │             1             1
│ │ │  o31 = 0 : Q  <--------- Q  : 0
│ │ │                  | 1 |
│ │ │  
│ │ │             5                                       9
│ │ │        1 : Q  <----------------------------------- Q  : 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,24 +42,24 @@
│ │ │ │        1      3      6      10      15
│ │ │ │  o3 = Q  <-- Q  <-- Q  <-- Q   <-- Q  <-- ...
│ │ │ │  
│ │ │ │       0      1      2      3       4
│ │ │ │  
│ │ │ │  o3 : SimplicialModule
│ │ │ │  i4 : S2S = elapsedTime schurMap({2}, S)
│ │ │ │ - -- .113944s elapsed
│ │ │ │ + -- .149338s elapsed
│ │ │ │  
│ │ │ │        1      6      21      55      120
│ │ │ │  o4 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │ │  
│ │ │ │       0      1      2       3       4
│ │ │ │  
│ │ │ │  o4 : SimplicialModule
│ │ │ │  i5 : elapsedTime schurMap({2}, S, Degeneracy => true)
│ │ │ │ - -- .295946s elapsed
│ │ │ │ + -- .313795s elapsed
│ │ │ │  
│ │ │ │        1      6      21      55      120
│ │ │ │  o5 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │ │  
│ │ │ │       0      1      2       3       4
│ │ │ │  
│ │ │ │  o5 : SimplicialModule
│ │ │ │ @@ -106,15 +106,15 @@
│ │ │ │  i10 : Q = ZZ/101[a..c]
│ │ │ │  
│ │ │ │  o10 = Q
│ │ │ │  
│ │ │ │  o10 : PolynomialRing
│ │ │ │  i11 : K = koszulComplex vars Q;
│ │ │ │  i12 : S2K = elapsedTime prune schurMap({2}, K, TopDegree => 4)
│ │ │ │ - -- .773047s elapsed
│ │ │ │ + -- 1.03239s elapsed
│ │ │ │  
│ │ │ │         1      9      36      74      81
│ │ │ │  o12 = Q  <-- Q  <-- Q   <-- Q   <-- Q
│ │ │ │  
│ │ │ │        0      1      2       3       4
│ │ │ │  
│ │ │ │  o12 : Complex
│ │ │ │ @@ -299,15 +299,15 @@
│ │ │ │  
│ │ │ │             1                   2
│ │ │ │        2 : Q  <--------------- Q  : 2
│ │ │ │                  {2} | 0 b |
│ │ │ │  
│ │ │ │  o30 : ComplexMap
│ │ │ │  i31 : f = elapsedTime prune schurMap({2}, phi)
│ │ │ │ - -- .843938s elapsed
│ │ │ │ + -- .955634s elapsed
│ │ │ │  
│ │ │ │             1             1
│ │ │ │  o31 = 0 : Q  <--------- Q  : 0
│ │ │ │                  | 1 |
│ │ │ │  
│ │ │ │             5                                       9
│ │ │ │        1 : Q  <----------------------------------- Q  : 1
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_tensor__L__E__S.html
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o7 : Complex
        │ │ │
        │ │ │ 
        i8 : hL = elapsedTime prune tensorLES(L,4)
│ │ │ - -- 1.88823s elapsed
│ │ │ + -- 1.89997s elapsed
│ │ │  
│ │ │                                                                                                                                                                                                                                                                                                     25      50      25
│ │ │  o8 = cokernel | b a | <-- cokernel | b a | <-- 0 <-- 0 <-- cokernel | a 0 b | <-- cokernel | a 0 b | <-- cokernel | b a 0 0 | <-- cokernel | b a 0 0 0 0 0 0 | <-- cokernel | 0   0   b a 0 0 0  0  | <-- cokernel | ab2 a2b b2 a2 | <-- cokernel | a2 b2 ab | <-- cokernel | ab2 a2b b2 a2 | <-- Q   <-- Q   <-- Q
│ │ │                                                                      | 0 b a |              | 0 b a |              | 0 0 b a |              | 0 0 b a 0 0 0 0 |              | 0   0   0 0 b a 0  0  |                                                                                                              
│ │ │       0                    1                    2     3                                                                                     | 0 0 0 0 b a 0 0 |              | ab2 a2b 0 0 0 0 b2 a2 |     9                              10                        11                             12      13      14
│ │ │                                                             4                      5                      6                                 | 0 0 0 0 0 0 b a |      
│ │ │                                                                                                                                                                     8
│ │ │ ├── html2text {}
│ │ │ │ @@ -114,15 +114,15 @@
│ │ │ │        1      2      2      1
│ │ │ │  o7 = Q  <-- Q  <-- Q  <-- Q
│ │ │ │  
│ │ │ │       0      1      2      3
│ │ │ │  
│ │ │ │  o7 : Complex
│ │ │ │  i8 : hL = elapsedTime prune tensorLES(L,4)
│ │ │ │ - -- 1.88823s elapsed
│ │ │ │ + -- 1.89997s elapsed
│ │ │ │  
│ │ │ │  
│ │ │ │  25      50      25
│ │ │ │  o8 = cokernel | b a | <-- cokernel | b a | <-- 0 <-- 0 <-- cokernel | a 0 b |
│ │ │ │  <-- cokernel | a 0 b | <-- cokernel | b a 0 0 | <-- cokernel | b a 0 0 0 0 0 0
│ │ │ │  | <-- cokernel | 0   0   b a 0 0 0  0  | <-- cokernel | ab2 a2b b2 a2 | <-
│ │ │ │  - cokernel | a2 b2 ab | <-- cokernel | ab2 a2b b2 a2 | <-- Q   <-- Q   <-- Q
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/index.html
│ │ │ @@ -211,28 +211,28 @@
│ │ │  
│ │ │  o7 : SimplicialModule
        │ │ │
        │ │ │ 
        i8 : elapsedTime simplicialModule(K,6) --specify top degree 6
│ │ │ - -- .123087s elapsed
│ │ │ + -- .112767s elapsed
│ │ │  
│ │ │        1      4      10      20      35      56      84
│ │ │  o8 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R  <-- ...
│ │ │                                                      
│ │ │       0      1      2       3       4       5       6
│ │ │  
│ │ │  o8 : SimplicialModule
        │ │ │
        │ │ │ 
        i9 : elapsedTime S' = simplicialModule(K,6, Degeneracy => true)
│ │ │ - -- .183923s elapsed
│ │ │ + -- .175787s elapsed
│ │ │  
│ │ │        1      4      10      20      35      56      84
│ │ │  o9 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R  <-- ...
│ │ │                                                      
│ │ │       0      1      2       3       4       5       6
│ │ │  
│ │ │  o9 : SimplicialModule
        │ │ │ ├── html2text {} │ │ │ │ @@ -135,24 +135,24 @@ │ │ │ │ 1 4 10 20 │ │ │ │ o7 = R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o7 : SimplicialModule │ │ │ │ i8 : elapsedTime simplicialModule(K,6) --specify top degree 6 │ │ │ │ - -- .123087s elapsed │ │ │ │ + -- .112767s elapsed │ │ │ │ │ │ │ │ 1 4 10 20 35 56 84 │ │ │ │ o8 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o8 : SimplicialModule │ │ │ │ i9 : elapsedTime S' = simplicialModule(K,6, Degeneracy => true) │ │ │ │ - -- .183923s elapsed │ │ │ │ + -- .175787s elapsed │ │ │ │ │ │ │ │ 1 4 10 20 35 56 84 │ │ │ │ o9 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o9 : SimplicialModule │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out │ │ │ @@ -3,15 +3,15 @@ │ │ │ i1 : n = {2,3,2} │ │ │ │ │ │ o1 = {2, 3, 2} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : time degreeDeterminant n │ │ │ - -- used 8.9619e-05s (cpu); 8.491e-05s (thread); 0s (gc) │ │ │ + -- used 9.9193e-05s (cpu); 9.0656e-05s (thread); 0s (gc) │ │ │ │ │ │ o2 = 6 │ │ │ │ │ │ i3 : M = genericMultidimensionalMatrix n; │ │ │ warning: clearing value of symbol x2 to allow access to subscripted variables based on it │ │ │ : debug with expression debug 1368 or with command line option --debug 1368 │ │ │ warning: clearing value of symbol x1 to allow access to subscripted variables based on it │ │ │ @@ -19,14 +19,14 @@ │ │ │ warning: clearing value of symbol x0 to allow access to subscripted variables based on it │ │ │ : debug with expression debug 6010 or with command line option --debug 6010 │ │ │ │ │ │ o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a ..a ] │ │ │ 0,0,0 1,2,1 │ │ │ │ │ │ i4 : time degree determinant M │ │ │ - -- used 0.0343841s (cpu); 0.0336709s (thread); 0s (gc) │ │ │ + -- used 0.13651s (cpu); 0.0539424s (thread); 0s (gc) │ │ │ │ │ │ o4 = {6} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Discriminant.out │ │ │ @@ -1,13 +1,13 @@ │ │ │ -- -*- M2-comint -*- hash: 17130321902108223178 │ │ │ │ │ │ i1 : (d,n) := (2,3); │ │ │ │ │ │ i2 : time Disc = denseDiscriminant(d,n) │ │ │ - -- used 0.419312s (cpu); 0.215998s (thread); 0s (gc) │ │ │ + -- used 0.515432s (cpu); 0.264362s (thread); 0s (gc) │ │ │ │ │ │ o2 = Disc │ │ │ │ │ │ o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 2 |) │ │ │ | 0 0 0 1 1 2 0 0 1 0 | │ │ │ | 0 1 2 0 1 0 0 1 0 0 | │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out │ │ │ @@ -9,18 +9,18 @@ │ │ │ 2 │ │ │ c x x + c x + c x + c x + c ) │ │ │ 4 1 2 2 2 3 1 1 2 0 │ │ │ │ │ │ o1 : Sequence │ │ │ │ │ │ i2 : time denseResultant(f0,f1,f2); -- using Poisson formula │ │ │ - -- used 0.0888004s (cpu); 0.0887994s (thread); 0s (gc) │ │ │ + -- used 0.102734s (cpu); 0.102734s (thread); 0s (gc) │ │ │ │ │ │ i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula │ │ │ - -- used 0.305719s (cpu); 0.251533s (thread); 0s (gc) │ │ │ + -- used 0.381052s (cpu); 0.307689s (thread); 0s (gc) │ │ │ │ │ │ i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant │ │ │ - -- used 0.366337s (cpu); 0.3006s (thread); 0s (gc) │ │ │ + -- used 0.383163s (cpu); 0.329898s (thread); 0s (gc) │ │ │ │ │ │ i5 : assert(o2 == o3 and o3 == o4) │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ o1 = {{{{8, 1}, {3, 7}}, {{8, 3}, {3, 7}}}, {{{8, 8}, {5, 7}}, {{8, 5}, {2, │ │ │ ------------------------------------------------------------------------ │ │ │ 3}}}} │ │ │ │ │ │ o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ │ │ │ │ │ │ i2 : time det M │ │ │ - -- used 0.0808656s (cpu); 0.0792904s (thread); 0s (gc) │ │ │ + -- used 0.202563s (cpu); 0.113244s (thread); 0s (gc) │ │ │ │ │ │ o2 = 9698337990421512192 │ │ │ │ │ │ i3 : M = randomMultidimensionalMatrix(2,2,2,2,5) │ │ │ │ │ │ o3 = {{{{{6, 3, 6, 8, 6}, {9, 3, 7, 6, 9}}, {{6, 2, 6, 0, 2}, {6, 9, 3, 5, │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -24,13 +24,13 @@ │ │ │ 7, 4, 5}}}, {{{4, 0, 1, 4, 4}, {2, 6, 1, 1, 4}}, {{5, 4, 9, 7, 4}, {6, │ │ │ ------------------------------------------------------------------------ │ │ │ 4, 8, 4, 2}}}}} │ │ │ │ │ │ o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ │ │ │ │ │ │ i4 : time det M │ │ │ - -- used 0.519591s (cpu); 0.446362s (thread); 0s (gc) │ │ │ + -- used 0.513718s (cpu); 0.513718s (thread); 0s (gc) │ │ │ │ │ │ o4 = 912984499996938980479447727885644530753184525786986940737407301278806287 │ │ │ 9257139493926586400187927813888 │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_generic__Skew__Multidimensional__Matrix.out │ │ │ @@ -1,56 +1,56 @@ │ │ │ -- -*- M2-comint -*- hash: 6013064849487134497 │ │ │ │ │ │ i1 : genericSkewMultidimensionalMatrix(3,4) │ │ │ │ │ │ o1 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0, │ │ │ - 1 0 1 2 0 2 │ │ │ + 3 2 3 0 2 0 │ │ │ ------------------------------------------------------------------------ │ │ │ 0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a , │ │ │ - 1 0 1 3 0 3 1 │ │ │ + 3 2 3 1 2 1 3 │ │ │ ------------------------------------------------------------------------ │ │ │ 0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a , │ │ │ - 2 1 3 2 3 0 2 │ │ │ + 0 3 1 0 1 2 0 │ │ │ ------------------------------------------------------------------------ │ │ │ 0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}} │ │ │ - 0 3 2 3 │ │ │ + 2 1 0 1 │ │ │ │ │ │ o1 : 3-dimensional matrix of shape 4 x 4 x 4 over QQ[a ..a ] │ │ │ 0 3 │ │ │ │ │ │ i2 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/101) │ │ │ │ │ │ o2 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0, │ │ │ - 1 0 1 2 0 2 │ │ │ + 3 2 3 0 2 0 │ │ │ ------------------------------------------------------------------------ │ │ │ 0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a , │ │ │ - 1 0 1 3 0 3 1 │ │ │ + 3 2 3 1 2 1 3 │ │ │ ------------------------------------------------------------------------ │ │ │ 0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a , │ │ │ - 2 1 3 2 3 0 2 │ │ │ + 0 3 1 0 1 2 0 │ │ │ ------------------------------------------------------------------------ │ │ │ 0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}} │ │ │ - 0 3 2 3 │ │ │ + 2 1 0 1 │ │ │ │ │ │ ZZ │ │ │ o2 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[a ..a ] │ │ │ 101 0 3 │ │ │ │ │ │ i3 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/101,Variable=>"b") │ │ │ │ │ │ o3 = {{{0, 0, 0, 0}, {0, 0, -b , -b }, {0, b , 0, -b }, {0, b , b , 0}}, {{0, │ │ │ - 1 0 1 2 0 2 │ │ │ + 3 2 3 0 2 0 │ │ │ ------------------------------------------------------------------------ │ │ │ 0, b , b }, {0, 0, 0, 0}, {-b , 0, 0, -b }, {-b , 0, b , 0}}, {{0, -b , │ │ │ - 1 0 1 3 0 3 1 │ │ │ + 3 2 3 1 2 1 3 │ │ │ ------------------------------------------------------------------------ │ │ │ 0, b }, {b , 0, 0, b }, {0, 0, 0, 0}, {-b , -b , 0, 0}}, {{0, -b , -b , │ │ │ - 2 1 3 2 3 0 2 │ │ │ + 0 3 1 0 1 2 0 │ │ │ ------------------------------------------------------------------------ │ │ │ 0}, {b , 0, -b , 0}, {b , b , 0, 0}, {0, 0, 0, 0}}} │ │ │ - 0 3 2 3 │ │ │ + 2 1 0 1 │ │ │ │ │ │ ZZ │ │ │ o3 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[b ..b ] │ │ │ 101 0 3 │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Discriminant.out │ │ │ @@ -11,15 +11,15 @@ │ │ │ a x y z + a x y z + a x y z │ │ │ 1,1,1 1 1 1 1,2,0 1 2 0 1,2,1 1 2 1 │ │ │ │ │ │ o1 : ZZ[a ..a ][x ..x , y ..y , z ..z ] │ │ │ 0,0,0 1,2,1 0 1 0 2 0 1 │ │ │ │ │ │ i2 : time sparseDiscriminant f │ │ │ - -- used 2.65911s (cpu); 2.1603s (thread); 0s (gc) │ │ │ + -- used 2.86891s (cpu); 2.46341s (thread); 0s (gc) │ │ │ │ │ │ 2 │ │ │ o2 = a a a a a a - a a a a a - │ │ │ 0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0 0,1,0 0,2,1 1,0,0 1,0,1 1,1,0 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 2 │ │ │ a a a a + a a a a a - │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Resultant.out │ │ │ @@ -1,11 +1,11 @@ │ │ │ -- -*- M2-comint -*- hash: 16228363821945730064 │ │ │ │ │ │ i1 : time Res = sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},{1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}}) │ │ │ - -- used 0.510818s (cpu); 0.451308s (thread); 0s (gc) │ │ │ + -- used 0.634585s (cpu); 0.482453s (thread); 0s (gc) │ │ │ │ │ │ o1 = Res │ │ │ │ │ │ o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1 2 2 |, | 0 0 1 1 |}) │ │ │ | 0 0 1 1 | | 1 0 1 2 | | 0 1 0 1 | │ │ │ │ │ │ i2 : QQ[c_(1,1)..c_(3,4)][x,y]; │ │ │ @@ -18,15 +18,15 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ c x*y + c x + c y + c ) │ │ │ 3,3 3,4 3,2 3,1 │ │ │ │ │ │ o3 : Sequence │ │ │ │ │ │ i4 : time Res(f,g,h) │ │ │ - -- used 0.00961162s (cpu); 0.00961226s (thread); 0s (gc) │ │ │ + -- used 0.0119542s (cpu); 0.0119546s (thread); 0s (gc) │ │ │ │ │ │ 2 4 2 2 4 │ │ │ o4 = - c c c c c c c + c c c c c c + │ │ │ 1,2 1,3 1,4 2,1 2,2 2,3 3,1 1,2 1,3 2,1 2,2 2,4 3,1 │ │ │ ------------------------------------------------------------------------ │ │ │ 3 2 3 2 3 │ │ │ c c c c c c - 2c c c c c c c c + │ │ │ @@ -730,30 +730,30 @@ │ │ │ │ │ │ o4 : QQ[c ..c ] │ │ │ 1,1 3,4 │ │ │ │ │ │ i5 : assert(Res(f,g,h) == sparseResultant(f,g,h)) │ │ │ │ │ │ i6 : time Res = sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331); │ │ │ - -- used 0.0351859s (cpu); 0.0338381s (thread); 0s (gc) │ │ │ + -- used 0.182079s (cpu); 0.0558556s (thread); 0s (gc) │ │ │ │ │ │ o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over ZZ/3331) │ │ │ | 0 1 0 1 | │ │ │ │ │ │ i7 : ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y]; │ │ │ │ │ │ i8 : (f,g,h) = (a_0 + a_1*x + a_2*y + a_3*x*y, b_0 + b_1*x + b_2*y + b_3*x*y, c_0 + c_1*x + c_2*y + c_3*x*y) │ │ │ │ │ │ o8 = (a x*y + a x + a y + a , b x*y + b x + b y + b , c x*y + c x + c y + c ) │ │ │ 3 1 2 0 3 1 2 0 3 1 2 0 │ │ │ │ │ │ o8 : Sequence │ │ │ │ │ │ i9 : time Res(f,g,h) │ │ │ - -- used 0.00319914s (cpu); 0.0032001s (thread); 0s (gc) │ │ │ + -- used 0.00425098s (cpu); 0.00425132s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 2 2 2 │ │ │ o9 = a b b c - a a b b c - a a b b c + a a b c - a b b c c - │ │ │ 3 1 2 0 2 3 1 3 0 1 3 2 3 0 1 2 3 0 3 0 2 0 1 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 │ │ │ a a b b c c + a a b c c + a a b b c c + a b b c c - a a b b c c + │ │ │ @@ -822,15 +822,15 @@ │ │ │ 2 │ │ │ c x x + c x + c x + c x + c ) │ │ │ 4 1 2 2 2 3 1 1 2 0 │ │ │ │ │ │ o11 : Sequence │ │ │ │ │ │ i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true)); │ │ │ - -- used 0.287478s (cpu); 0.205846s (thread); 0s (gc) │ │ │ + -- used 0.229193s (cpu); 0.193292s (thread); 0s (gc) │ │ │ │ │ │ i13 : quotientRemainder(UnmixedRes,MixedRes) │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ o13 = (b c - b b c c + b b c + b c c - 2b b c c - b b c c + b c , 0) │ │ │ 5 2 4 5 2 4 2 5 4 4 2 5 2 5 2 5 2 4 4 5 2 5 │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html │ │ │ @@ -81,15 +81,15 @@ │ │ │ │ │ │ o1 : List │ │ │
        │ │ │ 
        i2 : time degreeDeterminant n
│ │ │ - -- used 8.9619e-05s (cpu); 8.491e-05s (thread); 0s (gc)
│ │ │ + -- used 9.9193e-05s (cpu); 9.0656e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 6
        │ │ │
        │ │ │ 
        i3 : M = genericMultidimensionalMatrix n;
│ │ │ @@ -103,15 +103,15 @@
│ │ │  o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a     ..a     ]
│ │ │                                                        0,0,0   1,2,1
        │ │ │
        │ │ │ 
        i4 : time degree determinant M
│ │ │ - -- used 0.0343841s (cpu); 0.0336709s (thread); 0s (gc)
│ │ │ + -- used 0.13651s (cpu); 0.0539424s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = {6}
│ │ │  
│ │ │  o4 : List
        │ │ │
        │ │ │ ├── html2text {} │ │ │ │ @@ -15,15 +15,15 @@ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : n = {2,3,2} │ │ │ │ │ │ │ │ o1 = {2, 3, 2} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : time degreeDeterminant n │ │ │ │ - -- used 8.9619e-05s (cpu); 8.491e-05s (thread); 0s (gc) │ │ │ │ + -- used 9.9193e-05s (cpu); 9.0656e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = 6 │ │ │ │ i3 : M = genericMultidimensionalMatrix n; │ │ │ │ warning: clearing value of symbol x2 to allow access to subscripted variables │ │ │ │ based on it │ │ │ │ : debug with expression debug 1368 or with command line option -- │ │ │ │ debug 1368 │ │ │ │ @@ -35,15 +35,15 @@ │ │ │ │ based on it │ │ │ │ : debug with expression debug 6010 or with command line option -- │ │ │ │ debug 6010 │ │ │ │ │ │ │ │ o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a ..a ] │ │ │ │ 0,0,0 1,2,1 │ │ │ │ i4 : time degree determinant M │ │ │ │ - -- used 0.0343841s (cpu); 0.0336709s (thread); 0s (gc) │ │ │ │ + -- used 0.13651s (cpu); 0.0539424s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = {6} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_t_e_r_m_i_n_a_n_t_(_M_u_l_t_i_d_i_m_e_n_s_i_o_n_a_l_M_a_t_r_i_x_) -- hyperdeterminant of a │ │ │ │ multidimensional matrix │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Discriminant.html │ │ │ @@ -85,15 +85,15 @@ │ │ │ │ │ │ 
        i1 : (d,n) := (2,3);
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i2 : time Disc = denseDiscriminant(d,n)
│ │ │ - -- used 0.419312s (cpu); 0.215998s (thread); 0s (gc)
│ │ │ + -- used 0.515432s (cpu); 0.264362s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = Disc
│ │ │  
│ │ │  o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 2 |)
│ │ │                                                             | 0 0 0 1 1 2 0 0 1 0 |
│ │ │                                                             | 0 1 2 0 1 0 0 1 0 0 |
        │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -18,15 +18,15 @@ │ │ │ │ * Outputs: │ │ │ │ o for (d,n), this is the same as _s_p_a_r_s_e_D_i_s_c_r_i_m_i_n_a_n_t _e_x_p_o_n_e_n_t_s_M_a_t_r_i_x │ │ │ │ ""ggeenneerriicc ppoollyynnoommiiaall ooff ddeeggrreeee dd iinn nn vvaarriiaabblleess"";; │ │ │ │ o for f, this is the same as _a_f_f_i_n_e_D_i_s_c_r_i_m_i_n_a_n_t(f). │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : (d,n) := (2,3); │ │ │ │ i2 : time Disc = denseDiscriminant(d,n) │ │ │ │ - -- used 0.419312s (cpu); 0.215998s (thread); 0s (gc) │ │ │ │ + -- used 0.515432s (cpu); 0.264362s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = Disc │ │ │ │ │ │ │ │ o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 │ │ │ │ 2 |) │ │ │ │ | 0 0 0 1 1 2 0 0 1 │ │ │ │ 0 | │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Resultant.html │ │ │ @@ -95,27 +95,27 @@ │ │ │ │ │ │ o1 : Sequence │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i2 : time denseResultant(f0,f1,f2); -- using Poisson formula
│ │ │ - -- used 0.0888004s (cpu); 0.0887994s (thread); 0s (gc)
        │ │ │ + -- used 0.102734s (cpu); 0.102734s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.305719s (cpu); 0.251533s (thread); 0s (gc)
        │ │ │ + -- used 0.381052s (cpu); 0.307689s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.366337s (cpu); 0.3006s (thread); 0s (gc)
        │ │ │ + -- used 0.383163s (cpu); 0.329898s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i5 : assert(o2 == o3 and o3 == o4)
        │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -28,20 +28,20 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 │ │ │ │ c x x + c x + c x + c x + c ) │ │ │ │ 4 1 2 2 2 3 1 1 2 0 │ │ │ │ │ │ │ │ o1 : Sequence │ │ │ │ i2 : time denseResultant(f0,f1,f2); -- using Poisson formula │ │ │ │ - -- used 0.0888004s (cpu); 0.0887994s (thread); 0s (gc) │ │ │ │ + -- used 0.102734s (cpu); 0.102734s (thread); 0s (gc) │ │ │ │ i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay │ │ │ │ formula │ │ │ │ - -- used 0.305719s (cpu); 0.251533s (thread); 0s (gc) │ │ │ │ + -- used 0.381052s (cpu); 0.307689s (thread); 0s (gc) │ │ │ │ i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant │ │ │ │ - -- used 0.366337s (cpu); 0.3006s (thread); 0s (gc) │ │ │ │ + -- used 0.383163s (cpu); 0.329898s (thread); 0s (gc) │ │ │ │ i5 : assert(o2 == o3 and o3 == o4) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_p_a_r_s_e_R_e_s_u_l_t_a_n_t -- sparse resultant (A-resultant) │ │ │ │ * _a_f_f_i_n_e_R_e_s_u_l_t_a_n_t -- affine resultant │ │ │ │ * _d_e_n_s_e_D_i_s_c_r_i_m_i_n_a_n_t -- dense discriminant (classical discriminant) │ │ │ │ * _e_x_p_o_n_e_n_t_s_M_a_t_r_i_x -- exponents in one or more polynomials │ │ │ │ * _g_e_n_e_r_i_c_L_a_u_r_e_n_t_P_o_l_y_n_o_m_i_a_l_s -- generic (Laurent) polynomials │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html │ │ │ @@ -89,15 +89,15 @@ │ │ │ │ │ │ o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i2 : time det M
│ │ │ - -- used 0.0808656s (cpu); 0.0792904s (thread); 0s (gc)
│ │ │ + -- used 0.202563s (cpu); 0.113244s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 9698337990421512192
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i3 : M = randomMultidimensionalMatrix(2,2,2,2,5)
│ │ │ @@ -114,15 +114,15 @@
│ │ │  
│ │ │  o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i4 : time det M
│ │ │ - -- used 0.519591s (cpu); 0.446362s (thread); 0s (gc)
│ │ │ + -- used 0.513718s (cpu); 0.513718s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 912984499996938980479447727885644530753184525786986940737407301278806287
│ │ │       9257139493926586400187927813888
        │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -25,15 +25,15 @@ │ │ │ │ │ │ │ │ o1 = {{{{8, 1}, {3, 7}}, {{8, 3}, {3, 7}}}, {{{8, 8}, {5, 7}}, {{8, 5}, {2, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3}}}} │ │ │ │ │ │ │ │ o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ │ │ │ │ i2 : time det M │ │ │ │ - -- used 0.0808656s (cpu); 0.0792904s (thread); 0s (gc) │ │ │ │ + -- used 0.202563s (cpu); 0.113244s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = 9698337990421512192 │ │ │ │ i3 : M = randomMultidimensionalMatrix(2,2,2,2,5) │ │ │ │ │ │ │ │ o3 = {{{{{6, 3, 6, 8, 6}, {9, 3, 7, 6, 9}}, {{6, 2, 6, 0, 2}, {6, 9, 3, 5, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 6}}}, {{{3, 5, 7, 7, 9}, {4, 5, 0, 4, 3}}, {{1, 8, 9, 1, 2}, {9, 6, 6, │ │ │ │ @@ -42,15 +42,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 7, 4, 5}}}, {{{4, 0, 1, 4, 4}, {2, 6, 1, 1, 4}}, {{5, 4, 9, 7, 4}, {6, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 4, 8, 4, 2}}}}} │ │ │ │ │ │ │ │ o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ │ │ │ │ i4 : time det M │ │ │ │ - -- used 0.519591s (cpu); 0.446362s (thread); 0s (gc) │ │ │ │ + -- used 0.513718s (cpu); 0.513718s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 912984499996938980479447727885644530753184525786986940737407301278806287 │ │ │ │ 9257139493926586400187927813888 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _M_u_l_t_i_d_i_m_e_n_s_i_o_n_a_l_M_a_t_r_i_x -- the class of all multidimensional matrices │ │ │ │ * _d_e_g_r_e_e_D_e_t_e_r_m_i_n_a_n_t -- degree of the hyperdeterminant of a generic │ │ │ │ multidimensional matrix │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_generic__Skew__Multidimensional__Matrix.html │ │ │ @@ -80,65 +80,65 @@ │ │ │

      An $n$-dimensional matrix $M$ is skew symmetric if for every permutation $s$ of the set $\{0,\ldots,n-1\}$ we have permute(M,s) == sign(s)*M.

      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │ 
      i1 : genericSkewMultidimensionalMatrix(3,4)
│ │ │  
│ │ │  o1 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0,
│ │ │ -                              1    0        1       2        0   2           
│ │ │ +                              3    2        3       0        2   0           
│ │ │       ------------------------------------------------------------------------
│ │ │       0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a ,
│ │ │ -         1   0                    1          3      0      3              1 
│ │ │ +         3   2                    3          1      2      1              3 
│ │ │       ------------------------------------------------------------------------
│ │ │       0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a ,
│ │ │ -         2     1         3                    2    3                 0    2 
│ │ │ +         0     3         1                    0    1                 2    0 
│ │ │       ------------------------------------------------------------------------
│ │ │       0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}}
│ │ │ -           0       3        2   3
│ │ │ +           2       1        0   1
│ │ │  
│ │ │  o1 : 3-dimensional matrix of shape 4 x 4 x 4 over QQ[a ..a ]
│ │ │                                                        0   3
      │ │ │
      │ │ │ 
      i2 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/101)
│ │ │  
│ │ │  o2 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0,
│ │ │ -                              1    0        1       2        0   2           
│ │ │ +                              3    2        3       0        2   0           
│ │ │       ------------------------------------------------------------------------
│ │ │       0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a ,
│ │ │ -         1   0                    1          3      0      3              1 
│ │ │ +         3   2                    3          1      2      1              3 
│ │ │       ------------------------------------------------------------------------
│ │ │       0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a ,
│ │ │ -         2     1         3                    2    3                 0    2 
│ │ │ +         0     3         1                    0    1                 2    0 
│ │ │       ------------------------------------------------------------------------
│ │ │       0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}}
│ │ │ -           0       3        2   3
│ │ │ +           2       1        0   1
│ │ │  
│ │ │                                                     ZZ
│ │ │  o2 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[a ..a ]
│ │ │                                                    101  0   3
      │ │ │
      │ │ │ 
      i3 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/101,Variable=>"b")
│ │ │  
│ │ │  o3 = {{{0, 0, 0, 0}, {0, 0, -b , -b }, {0, b , 0, -b }, {0, b , b , 0}}, {{0,
│ │ │ -                              1    0        1       2        0   2           
│ │ │ +                              3    2        3       0        2   0           
│ │ │       ------------------------------------------------------------------------
│ │ │       0, b , b }, {0, 0, 0, 0}, {-b , 0, 0, -b }, {-b , 0, b , 0}}, {{0, -b ,
│ │ │ -         1   0                    1          3      0      3              1 
│ │ │ +         3   2                    3          1      2      1              3 
│ │ │       ------------------------------------------------------------------------
│ │ │       0, b }, {b , 0, 0, b }, {0, 0, 0, 0}, {-b , -b , 0, 0}}, {{0, -b , -b ,
│ │ │ -         2     1         3                    2    3                 0    2 
│ │ │ +         0     3         1                    0    1                 2    0 
│ │ │       ------------------------------------------------------------------------
│ │ │       0}, {b , 0, -b , 0}, {b , b , 0, 0}, {0, 0, 0, 0}}}
│ │ │ -           0       3        2   3
│ │ │ +           2       1        0   1
│ │ │  
│ │ │                                                     ZZ
│ │ │  o3 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[b ..b ]
│ │ │                                                    101  0   3
      │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -17,58 +17,58 @@ │ │ │ │ $d\times\cdots\times d$ ($n$ times). │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ An $n$-dimensional matrix $M$ is skew symmetric if for every permutation $s$ of │ │ │ │ the set $\{0,\ldots,n-1\}$ we have _p_e_r_m_u_t_e(M,s) == sign(s)*M. │ │ │ │ i1 : genericSkewMultidimensionalMatrix(3,4) │ │ │ │ │ │ │ │ o1 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0, │ │ │ │ - 1 0 1 2 0 2 │ │ │ │ + 3 2 3 0 2 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a , │ │ │ │ - 1 0 1 3 0 3 1 │ │ │ │ + 3 2 3 1 2 1 3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a , │ │ │ │ - 2 1 3 2 3 0 2 │ │ │ │ + 0 3 1 0 1 2 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}} │ │ │ │ - 0 3 2 3 │ │ │ │ + 2 1 0 1 │ │ │ │ │ │ │ │ o1 : 3-dimensional matrix of shape 4 x 4 x 4 over QQ[a ..a ] │ │ │ │ 0 3 │ │ │ │ i2 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/101) │ │ │ │ │ │ │ │ o2 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0, │ │ │ │ - 1 0 1 2 0 2 │ │ │ │ + 3 2 3 0 2 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a , │ │ │ │ - 1 0 1 3 0 3 1 │ │ │ │ + 3 2 3 1 2 1 3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a , │ │ │ │ - 2 1 3 2 3 0 2 │ │ │ │ + 0 3 1 0 1 2 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}} │ │ │ │ - 0 3 2 3 │ │ │ │ + 2 1 0 1 │ │ │ │ │ │ │ │ ZZ │ │ │ │ o2 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[a ..a ] │ │ │ │ 101 0 3 │ │ │ │ i3 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/ │ │ │ │ 101,Variable=>"b") │ │ │ │ │ │ │ │ o3 = {{{0, 0, 0, 0}, {0, 0, -b , -b }, {0, b , 0, -b }, {0, b , b , 0}}, {{0, │ │ │ │ - 1 0 1 2 0 2 │ │ │ │ + 3 2 3 0 2 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0, b , b }, {0, 0, 0, 0}, {-b , 0, 0, -b }, {-b , 0, b , 0}}, {{0, -b , │ │ │ │ - 1 0 1 3 0 3 1 │ │ │ │ + 3 2 3 1 2 1 3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0, b }, {b , 0, 0, b }, {0, 0, 0, 0}, {-b , -b , 0, 0}}, {{0, -b , -b , │ │ │ │ - 2 1 3 2 3 0 2 │ │ │ │ + 0 3 1 0 1 2 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0}, {b , 0, -b , 0}, {b , b , 0, 0}, {0, 0, 0, 0}}} │ │ │ │ - 0 3 2 3 │ │ │ │ + 2 1 0 1 │ │ │ │ │ │ │ │ ZZ │ │ │ │ o3 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[b ..b ] │ │ │ │ 101 0 3 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_n_e_r_i_c_M_u_l_t_i_d_i_m_e_n_s_i_o_n_a_l_M_a_t_r_i_x -- make a generic multidimensional matrix │ │ │ │ of variables │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Discriminant.html │ │ │ @@ -95,15 +95,15 @@ │ │ │ o1 : ZZ[a ..a ][x ..x , y ..y , z ..z ] │ │ │ 0,0,0 1,2,1 0 1 0 2 0 1 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i2 : time sparseDiscriminant f
│ │ │ - -- used 2.65911s (cpu); 2.1603s (thread); 0s (gc)
│ │ │ + -- used 2.86891s (cpu); 2.46341s (thread); 0s (gc)
│ │ │  
│ │ │                                                     2                        
│ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │        0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0    0,1,0 0,2,1 1,0,0 1,0,1 1,1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │              2     2                                2            
│ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       a     x y z  + a     x y z  + a     x y z
│ │ │ │        1,1,1 1 1 1    1,2,0 1 2 0    1,2,1 1 2 1
│ │ │ │  
│ │ │ │  o1 : ZZ[a     ..a     ][x ..x , y ..y , z ..z ]
│ │ │ │           0,0,0   1,2,1   0   1   0   2   0   1
│ │ │ │  i2 : time sparseDiscriminant f
│ │ │ │ - -- used 2.65911s (cpu); 2.1603s (thread); 0s (gc)
│ │ │ │ + -- used 2.86891s (cpu); 2.46341s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                     2
│ │ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │ │        0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0    0,1,0 0,2,1 1,0,0 1,0,1 1,1,0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │              2     2                                2
│ │ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │          
      Description
      
│ │ │          
      Alternatively, one can apply the method directly to the list of Laurent polynomials $f_0,\ldots,f_n$. In this case, the matrices $A_0,\ldots,A_n$ are automatically determined by exponentsMatrix. If you want require that $A_0=\cdots=A_n$, then use the option Unmixed=>true (this could be faster). Below we consider some examples.
      
│ │ │          
      In the first example, we calculate the sparse (mixed) resultant associated to the three sets of monomials $(1,x y,x^2 y,x),(y,x^2 y^2,x^2 y,x),(1,y,x y,x)$. Then we evaluate it at the three polynomials $f = c_{(1,1)}+c_{(1,2)} x y+c_{(1,3)} x^2 y+c_{(1,4)} x, g = c_{(2,1)} y+c_{(2,2)} x^2 y^2+c_{(2,3)} x^2 y+c_{(2,4)} x, h = c_{(3,1)}+c_{(3,2)} y+c_{(3,3)} x y+c_{(3,4)} x$.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o3 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
      
│ │ │                
      i1 : time Res = sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},{1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
│ │ │ - -- used 0.510818s (cpu); 0.451308s (thread); 0s (gc)
│ │ │ + -- used 0.634585s (cpu); 0.482453s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = Res
│ │ │  
│ │ │  o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1 2 2 |, | 0 0 1 1 |})
│ │ │                                                              | 0 0 1 1 |  | 1 0 1 2 |  | 0 1 0 1 |
      
│ │ │              
      
│ │ │                
      i4 : time Res(f,g,h)
│ │ │ - -- used 0.00961162s (cpu); 0.00961226s (thread); 0s (gc)
│ │ │ + -- used 0.0119542s (cpu); 0.0119546s (thread); 0s (gc)
│ │ │  
│ │ │          2                       4      2   2               4    
│ │ │  o4 = - c   c   c   c   c   c   c    + c   c   c   c   c   c    +
│ │ │          1,2 1,3 1,4 2,1 2,2 2,3 3,1    1,2 1,3 2,1 2,2 2,4 3,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        3       2       3               2                   3        
│ │ │       c   c   c   c   c   c    - 2c   c   c   c   c   c   c   c    +
│ │ │ @@ -830,15 +830,15 @@
│ │ │            
      
│ │ │          
      In the second example, we calculate the sparse unmixed resultant associated to the set of monomials $(1,x,y,xy)$. Then we evaluate it at the three polynomials $f = a_0 + a_1 x + a_2 y + a_3 x y, g = b_0 + b_1 x + b_2 y + b_3 x y, h = c_0 + c_1 x + c_2 y + c_3 x y$. Moreover, we perform all the computation over $\mathbb{Z}/3331$.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -68,15 +68,15 @@
│ │ │ │  
│ │ │ │  o4 = | 0 1 |
│ │ │ │       | 2 3 |
│ │ │ │       | 4 |
│ │ │ │  
│ │ │ │  o4 : YoungTableau
│ │ │ │  i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ │ - -- used 0.00135189s (cpu); 0.0013499s (thread); 0s (gc)
│ │ │ │ + -- used 0.00167977s (cpu); 0.00167665s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        3 2          2 3      3     2        3 2    3   2      2   3
│ │ │ │  o5 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3 2        3 2        2 3        2 3        3   2        3 2
│ │ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │ @@ -88,15 +88,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │ │  
│ │ │ │  o5 : R
│ │ │ │  i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ │ - -- used 0.0011332s (cpu); 0.00113385s (thread); 0s (gc)
│ │ │ │ + -- used 0.00152832s (cpu); 0.00152566s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        3 2          2 3      3     2        3 2    3   2      2   3
│ │ │ │  o6 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3 2        3 2        2 3        2 3        3   2        3 2
│ │ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │ @@ -108,15 +108,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │ │  
│ │ │ │  o6 : R
│ │ │ │  i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ │ - -- used 0.00190209s (cpu); 0.00190242s (thread); 0s (gc)
│ │ │ │ + -- used 0.00276055s (cpu); 0.00276235s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )
│ │ │ │  (x )(x ))
│ │ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4
│ │ │ │  2   0
│ │ │ │  
│ │ │ │  o7 : Expression of class Product
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_representation__Multiplicity.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -116,15 +116,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  2*x_2*x_4-x_3*x_4-
│ │ │ │  2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-
│ │ │ │  3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8));
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and
│ │ │ │  sectional genus 0
│ │ │ │  i3 : time toGrass X
│ │ │ │ - -- used 3.91299s (cpu); 2.52271s (thread); 0s (gc)
│ │ │ │ + -- used 5.79131s (cpu); 3.29211s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces
│ │ │ │  of degree 2
│ │ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8
│ │ │ │  i3 : time toGrass X
│ │ │ │  warning: clearing value of symbol x to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 9868   or with command line option   --
│ │ │ │  debug 9868
│ │ │ │ - -- used 4.07004s (cpu); 2.605s (thread); 0s (gc)
│ │ │ │ + -- used 5.20248s (cpu); 3.13869s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i6 : time Res = sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ - -- used 0.0351859s (cpu); 0.0338381s (thread); 0s (gc)
│ │ │ + -- used 0.182079s (cpu); 0.0558556s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over ZZ/3331)
│ │ │                                                               | 0 1 0 1 |
      
│ │ │              
      
│ │ │ @@ -854,15 +854,15 @@
│ │ │  
│ │ │  o8 : Sequence
│ │ │              
      
│ │ │                
      i9 : time Res(f,g,h)
│ │ │ - -- used 0.00319914s (cpu); 0.0032001s (thread); 0s (gc)
│ │ │ + -- used 0.00425098s (cpu); 0.00425132s (thread); 0s (gc)
│ │ │  
│ │ │        2     2            2            2        2 2    2          
│ │ │  o9 = a b b c  - a a b b c  - a a b b c  + a a b c  - a b b c c  -
│ │ │        3 1 2 0    2 3 1 3 0    1 3 2 3 0    1 2 3 0    3 0 2 0 1  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2                       2                         
│ │ │       a a b b c c  + a a b c c  + a a b b c c  + a b b c c  - a a b b c c  +
│ │ │ @@ -943,15 +943,15 @@
│ │ │  
│ │ │  o11 : Sequence
      
│ │ │              
      
│ │ │                
      i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));
│ │ │ - -- used 0.287478s (cpu); 0.205846s (thread); 0s (gc)
      
│ │ │ + -- used 0.229193s (cpu); 0.193292s (thread); 0s (gc)
│ │ │              
      
│ │ │                
      i13 : quotientRemainder(UnmixedRes,MixedRes)
│ │ │  
│ │ │          2 2                   2    2                               2 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  In the first example, we calculate the sparse (mixed) resultant associated to
│ │ │ │  the three sets of monomials $(1,x y,x^2 y,x),(y,x^2 y^2,x^2 y,x),(1,y,x y,x)$.
│ │ │ │  Then we evaluate it at the three polynomials $f = c_{(1,1)}+c_{(1,2)} x y+c_{
│ │ │ │  (1,3)} x^2 y+c_{(1,4)} x, g = c_{(2,1)} y+c_{(2,2)} x^2 y^2+c_{(2,3)} x^2 y+c_{
│ │ │ │  (2,4)} x, h = c_{(3,1)}+c_{(3,2)} y+c_{(3,3)} x y+c_{(3,4)} x$.
│ │ │ │  i1 : time Res = sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},
│ │ │ │  {1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
│ │ │ │ - -- used 0.510818s (cpu); 0.451308s (thread); 0s (gc)
│ │ │ │ + -- used 0.634585s (cpu); 0.482453s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = Res
│ │ │ │  
│ │ │ │  o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1
│ │ │ │  2 2 |, | 0 0 1 1 |})
│ │ │ │                                                              | 0 0 1 1 |  | 1 0
│ │ │ │  1 2 |  | 0 1 0 1 |
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │         1,3       1,2       1,4     1,1   2,2        2,3       2,4     2,1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       c   x*y + c   x + c   y + c   )
│ │ │ │        3,3       3,4     3,2     3,1
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : time Res(f,g,h)
│ │ │ │ - -- used 0.00961162s (cpu); 0.00961226s (thread); 0s (gc)
│ │ │ │ + -- used 0.0119542s (cpu); 0.0119546s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2                       4      2   2               4
│ │ │ │  o4 = - c   c   c   c   c   c   c    + c   c   c   c   c   c    +
│ │ │ │          1,2 1,3 1,4 2,1 2,2 2,3 3,1    1,2 1,3 2,1 2,2 2,4 3,1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3       2       3               2                   3
│ │ │ │       c   c   c   c   c   c    - 2c   c   c   c   c   c   c   c    +
│ │ │ │ @@ -771,29 +771,29 @@
│ │ │ │  In the second example, we calculate the sparse unmixed resultant associated to
│ │ │ │  the set of monomials $(1,x,y,xy)$. Then we evaluate it at the three polynomials
│ │ │ │  $f = a_0 + a_1 x + a_2 y + a_3 x y, g = b_0 + b_1 x + b_2 y + b_3 x y, h = c_0
│ │ │ │  + c_1 x + c_2 y + c_3 x y$. Moreover, we perform all the computation over
│ │ │ │  $\mathbb{Z}/3331$.
│ │ │ │  i6 : time Res = sparseResultant(matrix{{0,0,1,1},
│ │ │ │  {0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ │ - -- used 0.0351859s (cpu); 0.0338381s (thread); 0s (gc)
│ │ │ │ + -- used 0.182079s (cpu); 0.0558556s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over
│ │ │ │  ZZ/3331)
│ │ │ │                                                               | 0 1 0 1 |
│ │ │ │  i7 : ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y];
│ │ │ │  i8 : (f,g,h) = (a_0 + a_1*x + a_2*y + a_3*x*y, b_0 + b_1*x + b_2*y + b_3*x*y,
│ │ │ │  c_0 + c_1*x + c_2*y + c_3*x*y)
│ │ │ │  
│ │ │ │  o8 = (a x*y + a x + a y + a , b x*y + b x + b y + b , c x*y + c x + c y + c )
│ │ │ │         3       1     2     0   3       1     2     0   3       1     2     0
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ │ │  i9 : time Res(f,g,h)
│ │ │ │ - -- used 0.00319914s (cpu); 0.0032001s (thread); 0s (gc)
│ │ │ │ + -- used 0.00425098s (cpu); 0.00425132s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2     2            2            2        2 2    2
│ │ │ │  o9 = a b b c  - a a b b c  - a a b b c  + a a b c  - a b b c c  -
│ │ │ │        3 1 2 0    2 3 1 3 0    1 3 2 3 0    1 2 3 0    3 0 2 0 1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                           2                       2
│ │ │ │       a a b b c c  + a a b c c  + a a b b c c  + a b b c c  - a a b b c c  +
│ │ │ │ @@ -863,15 +863,15 @@
│ │ │ │                    2
│ │ │ │        c x x  + c x  + c x  + c x  + c )
│ │ │ │         4 1 2    2 2    3 1    1 2    0
│ │ │ │  
│ │ │ │  o11 : Sequence
│ │ │ │  i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant
│ │ │ │  (f,g,h,Unmixed=>true));
│ │ │ │ - -- used 0.287478s (cpu); 0.205846s (thread); 0s (gc)
│ │ │ │ + -- used 0.229193s (cpu); 0.193292s (thread); 0s (gc)
│ │ │ │  i13 : quotientRemainder(UnmixedRes,MixedRes)
│ │ │ │  
│ │ │ │          2 2                   2    2                               2 2
│ │ │ │  o13 = (b c  - b b c c  + b b c  + b c c  - 2b b c c  - b b c c  + b c , 0)
│ │ │ │          5 2    4 5 2 4    2 5 4    4 2 5     2 5 2 5    2 4 4 5    2 5
│ │ │ │  
│ │ │ │  o13 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o4 = | 0 1 |
│ │ │       | 2 3 |
│ │ │       | 4 |
│ │ │  
│ │ │  o4 : YoungTableau
│ │ │  
│ │ │  i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.00135189s (cpu); 0.0013499s (thread); 0s (gc)
│ │ │ + -- used 0.00167977s (cpu); 0.00167665s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o5 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -46,15 +46,15 @@
│ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │  
│ │ │  o5 : R
│ │ │  
│ │ │  i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ - -- used 0.0011332s (cpu); 0.00113385s (thread); 0s (gc)
│ │ │ + -- used 0.00152832s (cpu); 0.00152566s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o6 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -67,15 +67,15 @@
│ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │  
│ │ │  o6 : R
│ │ │  
│ │ │  i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ - -- used 0.00190209s (cpu); 0.00190242s (thread); 0s (gc)
│ │ │ + -- used 0.00276055s (cpu); 0.00276235s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )(x )(x ))
│ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4   2   0
│ │ │  
│ │ │  o7 : Expression of class Product
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_representation__Multiplicity.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : tal := tally apply (H,h->conjugacyClass h);
│ │ │  
│ │ │  i4 : partis = partitions 6;
│ │ │  
│ │ │  i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity(tal,p))
│ │ │ - -- used 0.410551s (cpu); 0.308232s (thread); 0s (gc)
│ │ │ + -- used 0.44074s (cpu); 0.316292s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │                 Partition{2, 2, 2} => 1
│ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │                 Partition{3, 2, 1} => 0
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.out
│ │ │ @@ -20,15 +20,15 @@
│ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ - -- used 0.77939s (cpu); 0.519181s (thread); 0s (gc)
│ │ │ + -- used 1.00155s (cpu); 0.626697s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : seco#(new Partition from {2,2,2})
│ │ │  
│ │ │                                                        2 2 2       4 2   2     2   2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     2 2 2       4 2 2       2 2 2       2 2 2       4 2 2       2 2 2       1 2   2     1   2 2     2 2   2     1 2   2     2   2 2     1   2 2     1 2   2     2 2   2     1 2   2     1   2 2     2   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2     2   1   2   2   2 2     2   1 2     2   2   2   2   1   2   2   1 2     2   2 2     2   1 2     2   1   2   2   2   2   2   1   2   2   2     2 2   4     2 2   2     2 2   2     2 2   4     2 2   2     2 2
│ │ │  o4 = HashTable{{0, 1, 2, 3, 4, 5} => HashTable{0 => - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        }
│ │ │                                                        3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6
│ │ │                                                      2 3 2 2     4 2 3 2     2 2 2 3     4 3 2   2   2 2 3   2   2 3   2 2   2   3 2 2   2 2   3 2   4   2 3 2   2 2 2   3   4 2   2 3   2   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     1 3 2 2     1 2 3 2     2 3 2 2     1 3 2 2     2 2 3 2     1 2 3 2     1 3 2 2     2 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 3 2     2 2 2 3     4 2 2 3     2 2 2 3     2 2 2 3     4 2 2 3     2 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 3 2   2   1 2 3   2   2 3 2   2   1 3 2   2   2 2 3   2   1 2 3   2   1 3 2   2   2 3 2   2   1 3 2   2   1 2 3   2   2 2 3   2   1 2 3   2   2 3   2 2   2   3 2 2   4 3   2 2   2 3   2 2   4   3 2 2   2   3 2 2   2 3   2 2   4 3   2 2   2 3   2 2   2   3 2 2   4   3 2 2   2   3 2 2   1 2   3 2   1   2 3 2   2 2   3 2   1 2   3 2   2   2 3 2   1   2 3 2   1 2   3 2   2 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   2 2 2   3   4 2 2   3   2 2 2   3   2 2 2   3   4 2 2   3   2 2 2   3   1 2   2 3   1   2 2 3   2 2   2 3   1 2   2 3   2   2 2 3   1   2 2 3   1 2   2 3   2 2   2 3   1 2   2 3   1   2 2 3   2   2 2 3   1   2 2 3
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html
│ │ │ @@ -130,15 +130,15 @@
│ │ │  
│ │ │  o4 : YoungTableau
      
│ │ │              
      
│ │ │                
      i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.00135189s (cpu); 0.0013499s (thread); 0s (gc)
│ │ │ + -- used 0.00167977s (cpu); 0.00167665s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o5 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -154,15 +154,15 @@
│ │ │  
│ │ │  o5 : R
      
│ │ │              
      
│ │ │                
      i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ - -- used 0.0011332s (cpu); 0.00113385s (thread); 0s (gc)
│ │ │ + -- used 0.00152832s (cpu); 0.00152566s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o6 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -178,15 +178,15 @@
│ │ │  
│ │ │  o6 : R
      
│ │ │              
      
│ │ │                
      i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ - -- used 0.00190209s (cpu); 0.00190242s (thread); 0s (gc)
│ │ │ + -- used 0.00276055s (cpu); 0.00276235s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )(x )(x ))
│ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4   2   0
│ │ │  
│ │ │  o7 : Expression of class Product
      
│ │ │              
      
│ │ │                
      i4 : partis = partitions 6;
      
│ │ │              
      
│ │ │                
      i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity(tal,p))
│ │ │ - -- used 0.410551s (cpu); 0.308232s (thread); 0s (gc)
│ │ │ + -- used 0.44074s (cpu); 0.316292s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │                 Partition{2, 2, 2} => 1
│ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │                 Partition{3, 2, 1} => 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  representations of $H$ in each irreducible representation of $S_6$. We take
│ │ │ │  into account that there are multiple copies of each representation by
│ │ │ │  multiplying the values with the number of copies which is given by the
│ │ │ │  hookLengthFormula.
│ │ │ │  i4 : partis = partitions 6;
│ │ │ │  i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity
│ │ │ │  (tal,p))
│ │ │ │ - -- used 0.410551s (cpu); 0.308232s (thread); 0s (gc)
│ │ │ │ + -- used 0.44074s (cpu); 0.316292s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │ │                 Partition{2, 2, 2} => 1
│ │ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │ │                 Partition{3, 2, 1} => 0
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.html
│ │ │ @@ -114,15 +114,15 @@
│ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ - -- used 0.77939s (cpu); 0.519181s (thread); 0s (gc)
      
│ │ │ + -- used 1.00155s (cpu); 0.626697s (thread); 0s (gc)
│ │ │              
      
│ │ │                
      i4 : seco#(new Partition from {2,2,2})
│ │ │  
│ │ │                                                        2 2 2       4 2   2     2   2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     2 2 2       4 2 2       2 2 2       2 2 2       4 2 2       2 2 2       1 2   2     1   2 2     2 2   2     1 2   2     2   2 2     1   2 2     1 2   2     2 2   2     1 2   2     1   2 2     2   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2     2   1   2   2   2 2     2   1 2     2   2   2   2   1   2   2   1 2     2   2 2     2   1 2     2   1   2   2   2   2   2   1   2   2   2     2 2   4     2 2   2     2 2   2     2 2   4     2 2   2     2 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -56,15 +56,15 @@
│ │ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ │ - -- used 0.77939s (cpu); 0.519181s (thread); 0s (gc)
│ │ │ │ + -- used 1.00155s (cpu); 0.626697s (thread); 0s (gc)
│ │ │ │  i4 : seco#(new Partition from {2,2,2})
│ │ │ │  
│ │ │ │                                                        2 2 2       4 2   2     2
│ │ │ │  2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2
│ │ │ │  1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2
│ │ │ │  1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2
│ │ │ │  2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out
│ │ │ @@ -45,15 +45,15 @@
│ │ │    -- top 1, degrees: 1^1 2^3 3^3 
│ │ │    -- top 2, degrees: 2^4 3^3 
│ │ │    -- top 3, degrees: 2^3 3^4 
│ │ │    -- top 4, degrees: 2^3 3^3 4^1 
│ │ │    -- top 5, degrees: 2^3 3^3 5^1 
│ │ │    -- top 6, degrees: 2^3 3^3 6^1 
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ associated Castelnuovo successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated Castelnuovo successfully completed in 1 second (cpu: 2 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │  
│ │ │  i4 : describe X
│ │ │  
│ │ │  o4 = Complete intersection of 3 quadrics in PP^7
│ │ │       of discriminant 31 = det| 8 1 |
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Cubic__Fourfold_rp.out
│ │ │ @@ -42,15 +42,15 @@
│ │ │  -- computing the top components of (U ∩ U')\{exceptional lines} via interpolation
│ │ │    -- top 1, degrees: 1^4 2^1 
│ │ │    -- top 2, degrees: 1^3 2^2 
│ │ │    -- top 3, degrees: 1^3 2^1 3^1 
│ │ │    -- top 4, degrees: 1^3 2^1 4^1 
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ - ✦ associated K3 successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated K3 successfully completed in 1 second (cpu: 2 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : describe X
│ │ │  
│ │ │  o4 = Special cubic fourfold of discriminant 14
│ │ │       containing a rational surface of degree 4 and sectional genus 0
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -48,15 +48,15 @@
│ │ │    -- top 3, degrees: 1^1 2^4 3^2 
│ │ │    -- top 4, degrees: 1^1 2^4 4^2 
│ │ │  -- exceptional curves computed: obtained 2 line(s)
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │  -- note: invariant mismatch for standard K3 surface
│ │ │  -- computing normalization of the surface image
│ │ │ - ✦ associated K3 successfully completed in 6 seconds (cpu: 6 seconds)
│ │ │ + ✦ associated K3 successfully completed in 4 seconds (cpu: 6 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : describe X
│ │ │  
│ │ │  o4 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │       containing a surface of degree 2 and sectional genus 0
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Cubic__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -8,28 +8,28 @@
│ │ │  i2 : describe X
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 3.78689s (cpu); 2.10157s (thread); 0s (gc)
│ │ │ + -- used 3.24214s (cpu); 2.05438s (thread); 0s (gc)
│ │ │  number lines contained in the image of the cubic map and passing through a general point: 8
│ │ │  number 2-secant lines = 7
│ │ │  number 5-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
│ │ │  
│ │ │  i4 : p := point ambient X -- random point on P^5
│ │ │  
│ │ │  o4 = point of coordinates [15092, -9738, -3620, -15181, 12688, 1]
│ │ │  
│ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │  
│ │ │  i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ - -- used 0.37997s (cpu); 0.274684s (thread); 0s (gc)
│ │ │ + -- used 0.467397s (cpu); 0.337845s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, curve in PP^5
│ │ │  
│ │ │  i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree(C * surface X) == 5 and isSubset(p, C))
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │       containing a surface of degree 9 and sectional genus 2
│ │ │       cut out by 19 hypersurfaces of degree 2
│ │ │       and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 15.3871s (cpu); 8.09595s (thread); 0s (gc)
│ │ │ + -- used 20.8646s (cpu); 8.32799s (thread); 0s (gc)
│ │ │  number lines contained in the image of the quadratic map and passing through a general point: 7
│ │ │  number 1-secant lines = 6
│ │ │  number 3-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
│ │ │  
│ │ │  i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X
│ │ │ @@ -29,15 +29,15 @@
│ │ │  i5 : p := point Y -- random point on Y
│ │ │  
│ │ │  o5 = point of coordinates [14360, -1933, -494, -6471, -10457, -2246, -11879, -12725, 1]
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │  
│ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ - -- used 0.650062s (cpu); 0.407556s (thread); 0s (gc)
│ │ │ + -- used 0.800685s (cpu); 0.447939s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
│ │ │  
│ │ │  i7 : S = surface X;
│ │ │  
│ │ │  o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, curve in PP^5
│ │ │  
│ │ │  i3 : X = random({{2},{2},{2}},S);
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.374s (cpu); 0.275799s (thread); 0s (gc)
│ │ │ + -- used 0.442312s (cpu); 0.253578s (thread); 0s (gc)
│ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3 hypersurfaces of degree 2
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 32
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ │  dim GG(2,9) = 21
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Cubic__Fourfold_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = cubicFourfold V;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.705259s (cpu); 0.394869s (thread); 0s (gc)
│ │ │ + -- used 0.766728s (cpu); 0.47092s (thread); 0s (gc)
│ │ │  S: Veronese surface in PP^5
│ │ │  X: smooth cubic hypersurface in PP^5
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 27
│ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │  
│ │ │  i3 : X = gushelMukaiFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 3.94594s (cpu); 2.3306s (thread); 0s (gc)
│ │ │ + -- used 4.36832s (cpu); 3.2407s (thread); 0s (gc)
│ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │  X: GM fourfold containing S
│ │ │  Y: del Pezzo fivefold containing X
│ │ │  h^1(N_{S,Y}) = 0
│ │ │  h^0(N_{S,Y}) = 11
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  i3 : ? X
│ │ │  
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
│ │ │  
│ │ │  i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 1.2841s (cpu); 0.681116s (thread); 0s (gc)
│ │ │ + -- used 1.48062s (cpu); 0.837631s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │       source variety: PP^4
│ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees 1^2 2^5 
│ │ │       dominance: true
│ │ │       degree: 1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_polarized__K3surface.out
│ │ │ @@ -72,15 +72,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- surface was already irreducible
│ │ │  -- result: surface in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 19 seconds (cpu: 21 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 15 seconds (cpu: 20 seconds)
│ │ │  
│ │ │  o5 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -108,16 +108,16 @@
│ │ │  -- computing p2^*(H_PP^2)
│ │ │    -- obtained the curve on U: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │    -- computing image on K3 surface...
│ │ │    -- image curve: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │  -- constructing lattice polarization...
│ │ │  -- verifying self-intersection of the curve...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (8, 7, 2)
│ │ │ - ✦ polarization successfully completed in 4 seconds (cpu: 3 seconds)
│ │ │ --- total time (K3 surface + polarization): 23 seconds (cpu: 24 seconds)
│ │ │ + ✦ polarization successfully completed in 3 seconds (cpu: 5 seconds)
│ │ │ +-- total time (K3 surface + polarization): 18 seconds (cpu: 26 seconds)
│ │ │  
│ │ │  o6 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice intersection matrix on Ũ: | 14 7 |
│ │ │ @@ -205,15 +205,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- removing 1 components of degrees {3}
│ │ │  -- result: surface in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 6 seconds (cpu: 7 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 5 seconds (cpu: 9 seconds)
│ │ │  
│ │ │  o10 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -226,15 +226,15 @@
│ │ │  -- available strategies: "SpecialCurve", "MapFromW", "MapFromU", "MapFromW-Virtual", "MapFromU-Virtual"
│ │ │  -- special curves already detected on U
│ │ │    -- pushing forward curve to K3 (1/1)...
│ │ │    -- image curve: curve in PP^4 cut out by 5 hypersurfaces of degrees 2^4 3^1 
│ │ │  -- constructing lattice polarization...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (4, 5, -2)
│ │ │   ✦ polarization successfully completed in 0 seconds (cpu: 0 seconds)
│ │ │ --- total time (K3 surface + polarization): 6 seconds (cpu: 7 seconds)
│ │ │ +-- total time (K3 surface + polarization): 5 seconds (cpu: 10 seconds)
│ │ │  
│ │ │  o11 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice intersection matrix on Ũ: | 6 5  |
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : x := gens ring PP_(ZZ/33331)^8;
│ │ │  
│ │ │  i2 : X = gushelMukaiFourfold(ideal(x_6-x_7, x_5, x_3-x_4, x_1, x_0-x_4, x_2*x_7-x_4*x_8), ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8, x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8));
│ │ │  
│ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i3 : time toGrass X
│ │ │ - -- used 3.91299s (cpu); 2.52271s (thread); 0s (gc)
│ │ │ + -- used 5.79131s (cpu); 3.29211s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass_lp__Embedded__Projective__Variety_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : X = projectiveVariety ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8);
│ │ │  
│ │ │  o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8
│ │ │  
│ │ │  i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 4.07004s (cpu); 2.605s (thread); 0s (gc)
│ │ │ + -- used 5.20248s (cpu); 3.13869s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = cubicFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time f = unirationalParametrization X;
│ │ │ - -- used 0.860842s (cpu); 0.490353s (thread); 0s (gc)
│ │ │ + -- used 1.08997s (cpu); 0.652191s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │  
│ │ │  i5 : degreeSequence f
│ │ │  
│ │ │  o5 = {[10]}
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__Castelnuovo__Surface.html
│ │ │ @@ -146,15 +146,15 @@
│ │ │    -- top 1, degrees: 1^1 2^3 3^3 
│ │ │    -- top 2, degrees: 2^4 3^3 
│ │ │    -- top 3, degrees: 2^3 3^4 
│ │ │    -- top 4, degrees: 2^3 3^3 4^1 
│ │ │    -- top 5, degrees: 2^3 3^3 5^1 
│ │ │    -- top 6, degrees: 2^3 3^3 6^1 
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ associated Castelnuovo successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated Castelnuovo successfully completed in 1 second (cpu: 2 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
      
│ │ │              
      
│ │ │                
      i4 : describe X
│ │ │ ├── html2text {}
│ │ │ │ @@ -80,15 +80,15 @@
│ │ │ │    -- top 1, degrees: 1^1 2^3 3^3
│ │ │ │    -- top 2, degrees: 2^4 3^3
│ │ │ │    -- top 3, degrees: 2^3 3^4
│ │ │ │    -- top 4, degrees: 2^3 3^3 4^1
│ │ │ │    -- top 5, degrees: 2^3 3^3 5^1
│ │ │ │    -- top 6, degrees: 2^3 3^3 6^1
│ │ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ │ - ✦ associated Castelnuovo successfully completed in 2 seconds (cpu: 1 second)
│ │ │ │ + ✦ associated Castelnuovo successfully completed in 1 second (cpu: 2 seconds)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │ │  i4 : describe X
│ │ │ │  
│ │ │ │  o4 = Complete intersection of 3 quadrics in PP^7
│ │ │ │       of discriminant 31 = det| 8 1 |
│ │ │ │                               | 1 4 |
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Cubic__Fourfold_rp.html
│ │ │ @@ -144,15 +144,15 @@
│ │ │  -- computing the top components of (U ∩ U')\{exceptional lines} via interpolation
│ │ │    -- top 1, degrees: 1^4 2^1 
│ │ │    -- top 2, degrees: 1^3 2^2 
│ │ │    -- top 3, degrees: 1^3 2^1 3^1 
│ │ │    -- top 4, degrees: 1^3 2^1 4^1 
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ - ✦ associated K3 successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated K3 successfully completed in 1 second (cpu: 2 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
      
│ │ │              
      
│ │ │                
      i4 : describe X
│ │ │ ├── html2text {}
│ │ │ │ @@ -79,15 +79,15 @@
│ │ │ │  interpolation
│ │ │ │    -- top 1, degrees: 1^4 2^1
│ │ │ │    -- top 2, degrees: 1^3 2^2
│ │ │ │    -- top 3, degrees: 1^3 2^1 3^1
│ │ │ │    -- top 4, degrees: 1^3 2^1 4^1
│ │ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ │ - ✦ associated K3 successfully completed in 2 seconds (cpu: 1 second)
│ │ │ │ + ✦ associated K3 successfully completed in 1 second (cpu: 2 seconds)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : describe X
│ │ │ │  
│ │ │ │  o4 = Special cubic fourfold of discriminant 14
│ │ │ │       containing a rational surface of degree 4 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degree 2
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -150,15 +150,15 @@
│ │ │    -- top 3, degrees: 1^1 2^4 3^2 
│ │ │    -- top 4, degrees: 1^1 2^4 4^2 
│ │ │  -- exceptional curves computed: obtained 2 line(s)
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │  -- note: invariant mismatch for standard K3 surface
│ │ │  -- computing normalization of the surface image
│ │ │ - ✦ associated K3 successfully completed in 6 seconds (cpu: 6 seconds)
│ │ │ + ✦ associated K3 successfully completed in 4 seconds (cpu: 6 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
      
│ │ │              
      
│ │ │                
      i4 : describe X
│ │ │ ├── html2text {}
│ │ │ │ @@ -84,15 +84,15 @@
│ │ │ │    -- top 3, degrees: 1^1 2^4 3^2
│ │ │ │    -- top 4, degrees: 1^1 2^4 4^2
│ │ │ │  -- exceptional curves computed: obtained 2 line(s)
│ │ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ │  -- note: invariant mismatch for standard K3 surface
│ │ │ │  -- computing normalization of the surface image
│ │ │ │ - ✦ associated K3 successfully completed in 6 seconds (cpu: 6 seconds)
│ │ │ │ + ✦ associated K3 successfully completed in 4 seconds (cpu: 6 seconds)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : describe X
│ │ │ │  
│ │ │ │  o4 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │ │       containing a surface of degree 2 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degrees 1^5 2^1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Cubic__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
      
│ │ │              
      
│ │ │                
      i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 3.78689s (cpu); 2.10157s (thread); 0s (gc)
│ │ │ + -- used 3.24214s (cpu); 2.05438s (thread); 0s (gc)
│ │ │  number lines contained in the image of the cubic map and passing through a general point: 8
│ │ │  number 2-secant lines = 7
│ │ │  number 5-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
      
│ │ │              
      
│ │ │                
      i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ - -- used 0.37997s (cpu); 0.274684s (thread); 0s (gc)
│ │ │ + -- used 0.467397s (cpu); 0.337845s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, curve in PP^5
      
│ │ │              
      
│ │ │                
      i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree(C * surface X) == 5 and isSubset(p, C))
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,28 +29,28 @@
│ │ │ │  sectional genus 0
│ │ │ │  i2 : describe X
│ │ │ │  
│ │ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ │ - -- used 3.78689s (cpu); 2.10157s (thread); 0s (gc)
│ │ │ │ + -- used 3.24214s (cpu); 2.05438s (thread); 0s (gc)
│ │ │ │  number lines contained in the image of the cubic map and passing through a
│ │ │ │  general point: 8
│ │ │ │  number 2-secant lines = 7
│ │ │ │  number 5-secant conics = 1
│ │ │ │  
│ │ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
│ │ │ │  i4 : p := point ambient X -- random point on P^5
│ │ │ │  
│ │ │ │  o4 = point of coordinates [15092, -9738, -3620, -15181, 12688, 1]
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │ │  i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ │ - -- used 0.37997s (cpu); 0.274684s (thread); 0s (gc)
│ │ │ │ + -- used 0.467397s (cpu); 0.337845s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, curve in PP^5
│ │ │ │  i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree
│ │ │ │  (C * surface X) == 5 and isSubset(p, C))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_t_e_c_t_C_o_n_g_r_u_e_n_c_e_(_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_,_Z_Z_) -- detect and return a
│ │ │ │        congruence of (2e-1)-secant curves of degree e inside a del Pezzo
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Gushel__Mukai__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │              
      
│ │ │                
      i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 15.3871s (cpu); 8.09595s (thread); 0s (gc)
│ │ │ + -- used 20.8646s (cpu); 8.32799s (thread); 0s (gc)
│ │ │  number lines contained in the image of the quadratic map and passing through a general point: 7
│ │ │  number 1-secant lines = 6
│ │ │  number 3-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
      
│ │ │              
      
│ │ │                
      i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ - -- used 0.650062s (cpu); 0.407556s (thread); 0s (gc)
│ │ │ + -- used 0.800685s (cpu); 0.447939s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
      
│ │ │              
      
│ │ │                
      i7 : S = surface X;
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  o2 = Special Gushel-Mukai fourfold of discriminant 20
│ │ │ │       containing a surface of degree 9 and sectional genus 2
│ │ │ │       cut out by 19 hypersurfaces of degree 2
│ │ │ │       and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
│ │ │ │       Type: ordinary
│ │ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ │ - -- used 15.3871s (cpu); 8.09595s (thread); 0s (gc)
│ │ │ │ + -- used 20.8646s (cpu); 8.32799s (thread); 0s (gc)
│ │ │ │  number lines contained in the image of the quadratic map and passing through a
│ │ │ │  general point: 7
│ │ │ │  number 1-secant lines = 6
│ │ │ │  number 3-secant conics = 1
│ │ │ │  
│ │ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
│ │ │ │  i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X
│ │ │ │ @@ -52,15 +52,15 @@
│ │ │ │  i5 : p := point Y -- random point on Y
│ │ │ │  
│ │ │ │  o5 = point of coordinates [14360, -1933, -494, -6471, -10457, -2246, -11879, -
│ │ │ │  12725, 1]
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ │ - -- used 0.650062s (cpu); 0.407556s (thread); 0s (gc)
│ │ │ │ + -- used 0.800685s (cpu); 0.447939s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
│ │ │ │  i7 : S = surface X;
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y)
│ │ │ │  i8 : assert(dim C == 1 and degree C == 2 and dim(C*S) == 0 and degree(C*S) == 3
│ │ │ │  and isSubset(p,C) and isSubset(C,Y))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
      
│ │ │              
      
│ │ │                
      i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.374s (cpu); 0.275799s (thread); 0s (gc)
│ │ │ + -- used 0.442312s (cpu); 0.253578s (thread); 0s (gc)
│ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3 hypersurfaces of degree 2
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 32
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ │  dim GG(2,9) = 21
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │  i1 : K = ZZ/33331; S = PP_K^(1,5);
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, curve in PP^5
│ │ │ │  i3 : X = random({{2},{2},{2}},S);
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │ │  i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ │ - -- used 0.374s (cpu); 0.275799s (thread); 0s (gc)
│ │ │ │ + -- used 0.442312s (cpu); 0.253578s (thread); 0s (gc)
│ │ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3
│ │ │ │  hypersurfaces of degree 2
│ │ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │ │  h^0(N_{S,P^5}) = 32
│ │ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Cubic__Fourfold_rp.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
      
│ │ │              
      
│ │ │                
      i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.705259s (cpu); 0.394869s (thread); 0s (gc)
│ │ │ + -- used 0.766728s (cpu); 0.47092s (thread); 0s (gc)
│ │ │  S: Veronese surface in PP^5
│ │ │  X: smooth cubic hypersurface in PP^5
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 27
│ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │ │  i3 : X = cubicFourfold V;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 0.705259s (cpu); 0.394869s (thread); 0s (gc)
│ │ │ │ + -- used 0.766728s (cpu); 0.47092s (thread); 0s (gc)
│ │ │ │  S: Veronese surface in PP^5
│ │ │ │  X: smooth cubic hypersurface in PP^5
│ │ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │ │  h^0(N_{S,P^5}) = 27
│ │ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
      
│ │ │              
      
│ │ │                
      i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 3.94594s (cpu); 2.3306s (thread); 0s (gc)
│ │ │ + -- used 4.36832s (cpu); 3.2407s (thread); 0s (gc)
│ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │  X: GM fourfold containing S
│ │ │  Y: del Pezzo fivefold containing X
│ │ │  h^1(N_{S,Y}) = 0
│ │ │  h^0(N_{S,Y}) = 11
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │ │  i3 : X = gushelMukaiFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 3.94594s (cpu); 2.3306s (thread); 0s (gc)
│ │ │ │ + -- used 4.36832s (cpu); 3.2407s (thread); 0s (gc)
│ │ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │ │  X: GM fourfold containing S
│ │ │ │  Y: del Pezzo fivefold containing X
│ │ │ │  h^1(N_{S,Y}) = 0
│ │ │ │  h^0(N_{S,Y}) = 11
│ │ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
      
│ │ │              
      
│ │ │                
      i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 1.2841s (cpu); 0.681116s (thread); 0s (gc)
│ │ │ + -- used 1.48062s (cpu); 0.837631s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │       source variety: PP^4
│ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees 1^2 2^5 
│ │ │       dominance: true
│ │ │       degree: 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^9
│ │ │ │  i3 : ? X
│ │ │ │  
│ │ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │ │       1^2 2^5
│ │ │ │  i4 : time parametrizeFanoFourfold X
│ │ │ │ - -- used 1.2841s (cpu); 0.681116s (thread); 0s (gc)
│ │ │ │ + -- used 1.48062s (cpu); 0.837631s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7
│ │ │ │  hypersurfaces of degrees 1^2 2^5
│ │ │ │       dominance: true
│ │ │ │       degree: 1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_polarized__K3surface.html
│ │ │ @@ -176,15 +176,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- surface was already irreducible
│ │ │  -- result: surface in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 19 seconds (cpu: 21 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 15 seconds (cpu: 20 seconds)
│ │ │  
│ │ │  o5 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -215,16 +215,16 @@
│ │ │  -- computing p2^*(H_PP^2)
│ │ │    -- obtained the curve on U: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │    -- computing image on K3 surface...
│ │ │    -- image curve: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │  -- constructing lattice polarization...
│ │ │  -- verifying self-intersection of the curve...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (8, 7, 2)
│ │ │ - ✦ polarization successfully completed in 4 seconds (cpu: 3 seconds)
│ │ │ --- total time (K3 surface + polarization): 23 seconds (cpu: 24 seconds)
│ │ │ + ✦ polarization successfully completed in 3 seconds (cpu: 5 seconds)
│ │ │ +-- total time (K3 surface + polarization): 18 seconds (cpu: 26 seconds)
│ │ │  
│ │ │  o6 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice intersection matrix on Ũ: | 14 7 |
│ │ │ @@ -327,15 +327,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- removing 1 components of degrees {3}
│ │ │  -- result: surface in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 6 seconds (cpu: 7 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 5 seconds (cpu: 9 seconds)
│ │ │  
│ │ │  o10 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -351,15 +351,15 @@
│ │ │  -- available strategies: "SpecialCurve", "MapFromW", "MapFromU", "MapFromW-Virtual", "MapFromU-Virtual"
│ │ │  -- special curves already detected on U
│ │ │    -- pushing forward curve to K3 (1/1)...
│ │ │    -- image curve: curve in PP^4 cut out by 5 hypersurfaces of degrees 2^4 3^1 
│ │ │  -- constructing lattice polarization...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (4, 5, -2)
│ │ │   ✦ polarization successfully completed in 0 seconds (cpu: 0 seconds)
│ │ │ --- total time (K3 surface + polarization): 6 seconds (cpu: 7 seconds)
│ │ │ +-- total time (K3 surface + polarization): 5 seconds (cpu: 10 seconds)
│ │ │  
│ │ │  o11 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice intersection matrix on Ũ: | 6 5  |
│ │ │ ├── html2text {}
│ │ │ │ @@ -119,15 +119,15 @@
│ │ │ │  -- analyzing the base locus of the inverse map...
│ │ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │ │  -- projecting to PP^3 for surface decomposition
│ │ │ │    -- surface was already irreducible
│ │ │ │  -- result: surface in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ │ - ✦ underlying K3 successfully completed in 19 seconds (cpu: 21 seconds)
│ │ │ │ + ✦ underlying K3 successfully completed in 15 seconds (cpu: 20 seconds)
│ │ │ │  
│ │ │ │  o5 = Fourfold: X, cubic fourfold in C_8
│ │ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15
│ │ │ │  hypersurfaces of degree 2
│ │ │ │       No exceptional curves
│ │ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by
│ │ │ │ @@ -160,16 +160,16 @@
│ │ │ │    -- obtained the curve on U: curve in PP^8 cut out by 11 hypersurfaces of
│ │ │ │  degrees 1^3 2^8
│ │ │ │    -- computing image on K3 surface...
│ │ │ │    -- image curve: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8
│ │ │ │  -- constructing lattice polarization...
│ │ │ │  -- verifying self-intersection of the curve...
│ │ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (8, 7, 2)
│ │ │ │ - ✦ polarization successfully completed in 4 seconds (cpu: 3 seconds)
│ │ │ │ --- total time (K3 surface + polarization): 23 seconds (cpu: 24 seconds)
│ │ │ │ + ✦ polarization successfully completed in 3 seconds (cpu: 5 seconds)
│ │ │ │ +-- total time (K3 surface + polarization): 18 seconds (cpu: 26 seconds)
│ │ │ │  
│ │ │ │  o6 = Fourfold: X, cubic fourfold in C_8
│ │ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15
│ │ │ │  hypersurfaces of degree 2
│ │ │ │       No exceptional curves
│ │ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by
│ │ │ │ @@ -267,15 +267,15 @@
│ │ │ │  -- analyzing the base locus of the inverse map...
│ │ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │ │  -- projecting to PP^3 for surface decomposition
│ │ │ │    -- removing 1 components of degrees {3}
│ │ │ │  -- result: surface in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1
│ │ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ │ - ✦ underlying K3 successfully completed in 6 seconds (cpu: 7 seconds)
│ │ │ │ + ✦ underlying K3 successfully completed in 5 seconds (cpu: 9 seconds)
│ │ │ │  
│ │ │ │  o10 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │ │        Mirror fourfold: PP^4
│ │ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2
│ │ │ │  hypersurfaces of degrees 2^1 3^1
│ │ │ │        No exceptional curves
│ │ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by
│ │ │ │ @@ -292,15 +292,15 @@
│ │ │ │  Virtual", "MapFromU-Virtual"
│ │ │ │  -- special curves already detected on U
│ │ │ │    -- pushing forward curve to K3 (1/1)...
│ │ │ │    -- image curve: curve in PP^4 cut out by 5 hypersurfaces of degrees 2^4 3^1
│ │ │ │  -- constructing lattice polarization...
│ │ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (4, 5, -2)
│ │ │ │   ✦ polarization successfully completed in 0 seconds (cpu: 0 seconds)
│ │ │ │ --- total time (K3 surface + polarization): 6 seconds (cpu: 7 seconds)
│ │ │ │ +-- total time (K3 surface + polarization): 5 seconds (cpu: 10 seconds)
│ │ │ │  
│ │ │ │  o11 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │ │        Mirror fourfold: PP^4
│ │ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2
│ │ │ │  hypersurfaces of degrees 2^1 3^1
│ │ │ │        No exceptional curves
│ │ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
      
│ │ │              
      
│ │ │                
      i3 : time toGrass X
│ │ │ - -- used 3.91299s (cpu); 2.52271s (thread); 0s (gc)
│ │ │ + -- used 5.79131s (cpu); 3.29211s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
      
│ │ │              
      
│ │ │                
      i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 4.07004s (cpu); 2.605s (thread); 0s (gc)
│ │ │ + -- used 5.20248s (cpu); 3.13869s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
      
│ │ │              
      
│ │ │                
      i4 : time f = unirationalParametrization X;
│ │ │ - -- used 0.860842s (cpu); 0.490353s (thread); 0s (gc)
│ │ │ + -- used 1.08997s (cpu); 0.652191s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
      
│ │ │              
      
│ │ │                
      i5 : degreeSequence f
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │ │  i3 : X = cubicFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time f = unirationalParametrization X;
│ │ │ │ - -- used 0.860842s (cpu); 0.490353s (thread); 0s (gc)
│ │ │ │ + -- used 1.08997s (cpu); 0.652191s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │ │  i5 : degreeSequence f
│ │ │ │  
│ │ │ │  o5 = {[10]}
│ │ │ │  
│ │ │ │  o5 : List
│ │ ├── ./usr/share/doc/Macaulay2/Style/example-output/_generate__Grammar.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 3455701143666534588
│ │ │  
│ │ │  i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10637-0/0
│ │ │ +o1 = /tmp/M2-10737-0/0
│ │ │  
│ │ │  i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10637-0/0.in
│ │ │ +o2 = /tmp/M2-10737-0/0.in
│ │ │  
│ │ │  i3 : template << "@M2BANNER@" << endl << endl;
│ │ │  
│ │ │  i4 : template << "This is an example file for the generateGrammar method!";
│ │ │  
│ │ │  i5 : template << endl;
│ │ │  
│ │ │ @@ -30,15 +30,15 @@
│ │ │        String regex: @M2STRINGS@
│ │ │        List of keywords: {
│ │ │            @M2KEYWORDS@
│ │ │        }
│ │ │  
│ │ │  
│ │ │  i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ - -- generating /tmp/M2-10637-0/0
│ │ │ + -- generating /tmp/M2-10737-0/0
│ │ │  
│ │ │  i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.26.05. Do not modify this file manually.
│ │ │  
│ │ │        This is an example file for the generateGrammar method!
│ │ │        String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///"
│ │ ├── ./usr/share/doc/Macaulay2/Style/html/_generate__Grammar.html
│ │ │ @@ -87,22 +87,22 @@
│ │ │            
      The function demarkf indicates how the elements of each of the lists will be demarked in the resulting file.   The file outfile will then be generated, replacing each of these strings as indicated above.
      
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -148,15 +148,15 @@
│ │ │            @M2KEYWORDS@
│ │ │        }
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10637-0/0
      
│ │ │ +o1 = /tmp/M2-10737-0/0
│ │ │              
      
│ │ │                
      i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10637-0/0.in
      
│ │ │ +o2 = /tmp/M2-10737-0/0.in
│ │ │              
      
│ │ │                
      i3 : template << "@M2BANNER@" << endl << endl;
      
│ │ │              
      
│ │ │                
      i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ - -- generating /tmp/M2-10637-0/0
      
│ │ │ + -- generating /tmp/M2-10737-0/0
│ │ │              
      
│ │ │                
      i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.26.05. Do not modify this file manually.
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,18 +26,18 @@
│ │ │ │      * @M2CONSTANTS@, for a list of Macaulay2 symbols and packages.
│ │ │ │      * @M2STRINGS@, for a regular expression that matches Macaulay2 strings.
│ │ │ │  The function demarkf indicates how the elements of each of the lists will be
│ │ │ │  demarked in the resulting file. The file outfile will then be generated,
│ │ │ │  replacing each of these strings as indicated above.
│ │ │ │  i1 : outfile = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10637-0/0
│ │ │ │ +o1 = /tmp/M2-10737-0/0
│ │ │ │  i2 : template = outfile | ".in"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10637-0/0.in
│ │ │ │ +o2 = /tmp/M2-10737-0/0.in
│ │ │ │  i3 : template << "@M2BANNER@" << endl << endl;
│ │ │ │  i4 : template << "This is an example file for the generateGrammar method!";
│ │ │ │  i5 : template << endl;
│ │ │ │  i6 : template << "String regex: @M2STRINGS@" << endl;
│ │ │ │  i7 : template << "List of keywords: {" << endl;
│ │ │ │  i8 : template << "    @M2KEYWORDS@" << endl;
│ │ │ │  i9 : template << "}" << endl << close;
│ │ │ │ @@ -47,15 +47,15 @@
│ │ │ │  
│ │ │ │        This is an example file for the generateGrammar method!
│ │ │ │        String regex: @M2STRINGS@
│ │ │ │        List of keywords: {
│ │ │ │            @M2KEYWORDS@
│ │ │ │        }
│ │ │ │  i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ │ - -- generating /tmp/M2-10637-0/0
│ │ │ │ + -- generating /tmp/M2-10737-0/0
│ │ │ │  i12 : get outfile
│ │ │ │  
│ │ │ │  o12 = Auto-generated for Macaulay2-1.26.05. Do not modify this file manually.
│ │ │ │  
│ │ │ │        This is an example file for the generateGrammar method!
│ │ │ │        String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///"
│ │ │ │        List of keywords: {
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out
│ │ │ @@ -31,15 +31,15 @@
│ │ │  o5 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i6 : isHomogeneous P
│ │ │  
│ │ │  o6 = false
│ │ │  
│ │ │  i7 : time symbolicPower(P,4);
│ │ │ - -- used 0.385445s (cpu); 0.2032s (thread); 0s (gc)
│ │ │ + -- used 0.533912s (cpu); 0.262625s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})
│ │ │  
│ │ │               2         3         2     2
│ │ │  o8 = ideal (y  - x*z, x  - y*z, x y - z )
│ │ │ @@ -47,12 +47,12 @@
│ │ │  o8 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i9 : isHomogeneous Q
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time symbolicPower(Q,4);
│ │ │ - -- used 0.126142s (cpu); 0.0716928s (thread); 0s (gc)
│ │ │ + -- used 0.143774s (cpu); 0.0574154s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html
│ │ │ @@ -146,15 +146,15 @@
│ │ │  
│ │ │  o6 = false
      
│ │ │              
      
│ │ │                
      i7 : time symbolicPower(P,4);
│ │ │ - -- used 0.385445s (cpu); 0.2032s (thread); 0s (gc)
│ │ │ + -- used 0.533912s (cpu); 0.262625s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of QQ[x..z]
      
│ │ │              
      
│ │ │                
      i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})
│ │ │ @@ -171,15 +171,15 @@
│ │ │  
│ │ │  o9 = true
      
│ │ │              
      
│ │ │                
      i10 : time symbolicPower(Q,4);
│ │ │ - -- used 0.126142s (cpu); 0.0716928s (thread); 0s (gc)
│ │ │ + -- used 0.143774s (cpu); 0.0574154s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of QQ[x..z]
      
│ │ │              
      
│ │ │        
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -59,28 +59,28 @@
│ │ │ │  o5 = ideal (y  - x*z, x y - z , x  - y*z)
│ │ │ │  
│ │ │ │  o5 : Ideal of QQ[x..z]
│ │ │ │  i6 : isHomogeneous P
│ │ │ │  
│ │ │ │  o6 = false
│ │ │ │  i7 : time symbolicPower(P,4);
│ │ │ │ - -- used 0.385445s (cpu); 0.2032s (thread); 0s (gc)
│ │ │ │ + -- used 0.533912s (cpu); 0.262625s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of QQ[x..z]
│ │ │ │  i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})
│ │ │ │  
│ │ │ │               2         3         2     2
│ │ │ │  o8 = ideal (y  - x*z, x  - y*z, x y - z )
│ │ │ │  
│ │ │ │  o8 : Ideal of QQ[x..z]
│ │ │ │  i9 : isHomogeneous Q
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time symbolicPower(Q,4);
│ │ │ │ - -- used 0.126142s (cpu); 0.0716928s (thread); 0s (gc)
│ │ │ │ + -- used 0.143774s (cpu); 0.0574154s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 : Ideal of QQ[x..z]
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_y_m_b_P_o_w_e_r_P_r_i_m_e_P_o_s_C_h_a_r
│ │ │ │  ********** WWaayyss ttoo uussee ssyymmbboolliiccPPoowweerr:: **********
│ │ │ │      * symbolicPower(Ideal,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Window.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  o3 = 0  <-- E  <-- 0
│ │ │                      
│ │ │       -1     0      1
│ │ │  
│ │ │  o3 : Complex
│ │ │  
│ │ │  i4 : time T=tateExtension W;
│ │ │ - -- used 1.32872s (cpu); 0.768804s (thread); 0s (gc)
│ │ │ + -- used 1.12098s (cpu); 0.792973s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │  
│ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │       | 6h  3h  0 3  6  9  12 |
│ │ │       | 4h  2h  0 2  4  6  8  |
│ │ │       | 2h  h   0 1  2  3  4  |
│ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Window.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o3 : Complex
      
│ │ │              
      
│ │ │                
      i4 : time T=tateExtension W;
│ │ │ - -- used 1.32872s (cpu); 0.768804s (thread); 0s (gc)
      
│ │ │ + -- used 1.12098s (cpu); 0.792973s (thread); 0s (gc)
│ │ │              
      
│ │ │                
      i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │  
│ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │               1
│ │ │ │  o3 = 0  <-- E  <-- 0
│ │ │ │  
│ │ │ │       -1     0      1
│ │ │ │  
│ │ │ │  o3 : Complex
│ │ │ │  i4 : time T=tateExtension W;
│ │ │ │ - -- used 1.32872s (cpu); 0.768804s (thread); 0s (gc)
│ │ │ │ + -- used 1.12098s (cpu); 0.792973s (thread); 0s (gc)
│ │ │ │  i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │ │  
│ │ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │ │       | 6h  3h  0 3  6  9  12 |
│ │ │ │       | 4h  2h  0 2  4  6  8  |
│ │ │ │       | 2h  h   0 1  2  3  4  |
│ │ │ │       | 0   0   0 0  0  0  0  |
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out
│ │ │ @@ -63,20 +63,20 @@
│ │ │  o15 : Ideal of R
│ │ │  
│ │ │  i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 1.02785s (cpu); 0.700789s (thread); 0s (gc)
│ │ │ + -- used 1.27465s (cpu); 0.889554s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
│ │ │  
│ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.87751s (cpu); 2.1903s (thread); 0s (gc)
│ │ │ + -- used 3.3038s (cpu); 2.62304s (thread); 0s (gc)
│ │ │  
│ │ │  o18 : Ideal of R
│ │ │  
│ │ │  i19 : J1 == J2
│ │ │  
│ │ │  o19 = true
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out
│ │ │ @@ -7,20 +7,20 @@
│ │ │  i3 : g = map(T, S, {x^3, x^2*y, x*y^2, y^3});
│ │ │  
│ │ │  o3 : RingMap T <-- S
│ │ │  
│ │ │  i4 : R = S/(ker g);
│ │ │  
│ │ │  i5 : time isCohenMacaulay(R)
│ │ │ - -- used 0.00260629s (cpu); 0.0026015s (thread); 0s (gc)
│ │ │ + -- used 0.00351351s (cpu); 0.0035106s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00402387s (cpu); 0.00402512s (thread); 0s (gc)
│ │ │ + -- used 0.00541945s (cpu); 0.00542853s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : R = QQ[x,y,u,v]/(x*u, x*v, y*u, y*v);
│ │ │  
│ │ │  i8 : isCohenMacaulay(R)
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out
│ │ │ @@ -60,49 +60,49 @@
│ │ │  i19 : R = ZZ/5[x,y,z]/(y^2*z + x*y*z-x^3)
│ │ │  
│ │ │  o19 = R
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │  
│ │ │  i20 : time isFInjective(R)
│ │ │ - -- used 0.0243607s (cpu); 0.0243609s (thread); 0s (gc)
│ │ │ + -- used 0.0421035s (cpu); 0.0421033s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
│ │ │  
│ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 1.44732s (cpu); 1.06969s (thread); 0s (gc)
│ │ │ + -- used 1.81006s (cpu); 1.38019s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
│ │ │  
│ │ │  i23 : time isFInjective(R)
│ │ │ - -- used 0.0606528s (cpu); 0.0606579s (thread); 0s (gc)
│ │ │ + -- used 0.072186s (cpu); 0.0719428s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.0675999s (cpu); 0.0676097s (thread); 0s (gc)
│ │ │ + -- used 0.0792316s (cpu); 0.0792399s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : S = ZZ/3[xs, ys, zs, xt, yt, zt];
│ │ │  
│ │ │  i26 : EP1 = ZZ/3[x,y,z,s,t]/(x^3 + y^2*z - x*z^2);
│ │ │  
│ │ │  i27 : f = map(EP1, S, {x*s, y*s, z*s, x*t, y*t, z*t});
│ │ │  
│ │ │  o27 : RingMap EP1 <-- S
│ │ │  
│ │ │  i28 : R = S/(ker f);
│ │ │  
│ │ │  i29 : time isFInjective(R)
│ │ │ - -- used 0.778416s (cpu); 0.621243s (thread); 0s (gc)
│ │ │ + -- used 0.760386s (cpu); 0.68513s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
│ │ │  
│ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.161678s (cpu); 0.161683s (thread); 0s (gc)
│ │ │ + -- used 0.189321s (cpu); 0.189243s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out
│ │ │ @@ -80,19 +80,19 @@
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │  
│ │ │  i26 : debugLevel = 1;
│ │ │  
│ │ │  i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │ - -- used 0.043332s (cpu); 0.0433271s (thread); 0s (gc)
│ │ │ + -- used 0.0618399s (cpu); 0.0617056s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
│ │ │  
│ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.247776s (cpu); 0.203448s (thread); 0s (gc)
│ │ │ + -- used 0.253045s (cpu); 0.181362s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
│ │ │  
│ │ │  i29 : debugLevel = 0;
│ │ │  
│ │ │  i30 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out
│ │ │ @@ -81,21 +81,21 @@
│ │ │  i22 : testIdeal({3/4, 2/3, 3/5}, L)
│ │ │  
│ │ │  o22 = ideal (y, x)
│ │ │  
│ │ │  o22 : Ideal of R
│ │ │  
│ │ │  i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ - -- used 0.214265s (cpu); 0.148483s (thread); 0s (gc)
│ │ │ + -- used 0.313256s (cpu); 0.23861s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.269113s (cpu); 0.214063s (thread); 0s (gc)
│ │ │ + -- used 0.330022s (cpu); 0.254882s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html
│ │ │ @@ -231,23 +231,23 @@
│ │ │  
│ │ │  o16 : Ideal of R
      
│ │ │              
      
│ │ │                
      i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 1.02785s (cpu); 0.700789s (thread); 0s (gc)
│ │ │ + -- used 1.27465s (cpu); 0.889554s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
      
│ │ │              
      
│ │ │                
      i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.87751s (cpu); 2.1903s (thread); 0s (gc)
│ │ │ + -- used 3.3038s (cpu); 2.62304s (thread); 0s (gc)
│ │ │  
│ │ │  o18 : Ideal of R
      
│ │ │              
      
│ │ │                
      i19 : J1 == J2
│ │ │ ├── html2text {}
│ │ │ │ @@ -106,19 +106,19 @@
│ │ │ │  i15 : I2 = ideal(x^20*y^100, x + z^100);
│ │ │ │  
│ │ │ │  o15 : Ideal of R
│ │ │ │  i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │ │  
│ │ │ │  o16 : Ideal of R
│ │ │ │  i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ │ - -- used 1.02785s (cpu); 0.700789s (thread); 0s (gc)
│ │ │ │ + -- used 1.27465s (cpu); 0.889554s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 : Ideal of R
│ │ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ │ - -- used 2.87751s (cpu); 2.1903s (thread); 0s (gc)
│ │ │ │ + -- used 3.3038s (cpu); 2.62304s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o18 : Ideal of R
│ │ │ │  i19 : J1 == J2
│ │ │ │  
│ │ │ │  o19 = true
│ │ │ │  For legacy reasons, the last ideal in the list can be specified separately,
│ │ │ │  using frobeniusRoot(e, \{a_1,\ldots,a_n\}, \{I_1,\ldots,I_n\}, I). The last
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html
│ │ │ @@ -101,23 +101,23 @@
│ │ │              
      		
│ │ │                
      i4 : R = S/(ker g);
      
│ │ │              
      
│ │ │                
      i5 : time isCohenMacaulay(R)
│ │ │ - -- used 0.00260629s (cpu); 0.0026015s (thread); 0s (gc)
│ │ │ + -- used 0.00351351s (cpu); 0.0035106s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
      
│ │ │              
      
│ │ │                
      i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00402387s (cpu); 0.00402512s (thread); 0s (gc)
│ │ │ + -- used 0.00541945s (cpu); 0.00542853s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = true
      
│ │ │              
      
│ │ │          
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,19 +23,19 @@
│ │ │ │  i1 : T = ZZ/5[x,y];
│ │ │ │  i2 : S = ZZ/5[a,b,c,d];
│ │ │ │  i3 : g = map(T, S, {x^3, x^2*y, x*y^2, y^3});
│ │ │ │  
│ │ │ │  o3 : RingMap T <-- S
│ │ │ │  i4 : R = S/(ker g);
│ │ │ │  i5 : time isCohenMacaulay(R)
│ │ │ │ - -- used 0.00260629s (cpu); 0.0026015s (thread); 0s (gc)
│ │ │ │ + -- used 0.00351351s (cpu); 0.0035106s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ │ - -- used 0.00402387s (cpu); 0.00402512s (thread); 0s (gc)
│ │ │ │ + -- used 0.00541945s (cpu); 0.00542853s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : R = QQ[x,y,u,v]/(x*u, x*v, y*u, y*v);
│ │ │ │  i8 : isCohenMacaulay(R)
│ │ │ │  
│ │ │ │  o8 = false
│ │ │ │  The function isCohenMacaulay considers $R$ as a quotient of a polynomial ring,
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html
│ │ │ @@ -219,23 +219,23 @@
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i20 : time isFInjective(R)
│ │ │ - -- used 0.0243607s (cpu); 0.0243609s (thread); 0s (gc)
│ │ │ + -- used 0.0421035s (cpu); 0.0421033s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
      
│ │ │              
      
│ │ │                
      i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 1.44732s (cpu); 1.06969s (thread); 0s (gc)
│ │ │ + -- used 1.81006s (cpu); 1.38019s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
      
│ │ │              
      
│ │ │          
      
│ │ │            
      If the option AtOrigin (default value false) is set to true, isFInjective will only check $F$-injectivity at the origin. Otherwise, it will check $F$-injectivity globally. Note that checking $F$-injectivity at the origin can be slower than checking it globally. Consider the following example of a non-$F$-injective ring.
      
│ │ │ @@ -245,23 +245,23 @@
│ │ │              
│ │ │                
      i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i23 : time isFInjective(R)
│ │ │ - -- used 0.0606528s (cpu); 0.0606579s (thread); 0s (gc)
│ │ │ + -- used 0.072186s (cpu); 0.0719428s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.0675999s (cpu); 0.0676097s (thread); 0s (gc)
│ │ │ + -- used 0.0792316s (cpu); 0.0792399s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      If the option AssumeCM (default value false) is set to true, then isFInjective only checks the Frobenius action on top cohomology (which is typically much faster). Note that it can give an incorrect answer if the non-injective Frobenius occurs in a lower degree.  Consider the example of the cone over a supersingular elliptic curve times $\mathbb{P}^1$.
      
│ │ │ @@ -288,23 +288,23 @@
│ │ │              
│ │ │                
      i28 : R = S/(ker f);
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i29 : time isFInjective(R)
│ │ │ - -- used 0.778416s (cpu); 0.621243s (thread); 0s (gc)
│ │ │ + -- used 0.760386s (cpu); 0.68513s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.161678s (cpu); 0.161683s (thread); 0s (gc)
│ │ │ + -- used 0.189321s (cpu); 0.189243s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      If the option AssumedReduced is set to true (its default behavior), then the bottom local cohomology is avoided (this means the Frobenius action on the top potentially nonzero Ext is not computed).
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,52 +81,52 @@
│ │ │ │  much faster.
│ │ │ │  i19 : R = ZZ/5[x,y,z]/(y^2*z + x*y*z-x^3)
│ │ │ │  
│ │ │ │  o19 = R
│ │ │ │  
│ │ │ │  o19 : QuotientRing
│ │ │ │  i20 : time isFInjective(R)
│ │ │ │ - -- used 0.0243607s (cpu); 0.0243609s (thread); 0s (gc)
│ │ │ │ + -- used 0.0421035s (cpu); 0.0421033s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 = true
│ │ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ │ - -- used 1.44732s (cpu); 1.06969s (thread); 0s (gc)
│ │ │ │ + -- used 1.81006s (cpu); 1.38019s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  If the option AtOrigin (default value false) is set to true, isFInjective will
│ │ │ │  only check $F$-injectivity at the origin. Otherwise, it will check $F$-
│ │ │ │  injectivity globally. Note that checking $F$-injectivity at the origin can be
│ │ │ │  slower than checking it globally. Consider the following example of a non-$F$-
│ │ │ │  injective ring.
│ │ │ │  i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
│ │ │ │  i23 : time isFInjective(R)
│ │ │ │ - -- used 0.0606528s (cpu); 0.0606579s (thread); 0s (gc)
│ │ │ │ + -- used 0.072186s (cpu); 0.0719428s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = false
│ │ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ │ - -- used 0.0675999s (cpu); 0.0676097s (thread); 0s (gc)
│ │ │ │ + -- used 0.0792316s (cpu); 0.0792399s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  If the option AssumeCM (default value false) is set to true, then isFInjective
│ │ │ │  only checks the Frobenius action on top cohomology (which is typically much
│ │ │ │  faster). Note that it can give an incorrect answer if the non-injective
│ │ │ │  Frobenius occurs in a lower degree. Consider the example of the cone over a
│ │ │ │  supersingular elliptic curve times $\mathbb{P}^1$.
│ │ │ │  i25 : S = ZZ/3[xs, ys, zs, xt, yt, zt];
│ │ │ │  i26 : EP1 = ZZ/3[x,y,z,s,t]/(x^3 + y^2*z - x*z^2);
│ │ │ │  i27 : f = map(EP1, S, {x*s, y*s, z*s, x*t, y*t, z*t});
│ │ │ │  
│ │ │ │  o27 : RingMap EP1 <-- S
│ │ │ │  i28 : R = S/(ker f);
│ │ │ │  i29 : time isFInjective(R)
│ │ │ │ - -- used 0.778416s (cpu); 0.621243s (thread); 0s (gc)
│ │ │ │ + -- used 0.760386s (cpu); 0.68513s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = false
│ │ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ │ - -- used 0.161678s (cpu); 0.161683s (thread); 0s (gc)
│ │ │ │ + -- used 0.189321s (cpu); 0.189243s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = true
│ │ │ │  If the option AssumedReduced is set to true (its default behavior), then the
│ │ │ │  bottom local cohomology is avoided (this means the Frobenius action on the top
│ │ │ │  potentially nonzero Ext is not computed).
│ │ │ │  If the option AssumeNormal (default value false) is set to true, then the
│ │ │ │  bottom two local cohomology modules (or, rather, their duals) need not be
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html
│ │ │ @@ -278,23 +278,23 @@
│ │ │                
      i26 : debugLevel = 1;
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │ - -- used 0.043332s (cpu); 0.0433271s (thread); 0s (gc)
│ │ │ + -- used 0.0618399s (cpu); 0.0617056s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.247776s (cpu); 0.203448s (thread); 0s (gc)
│ │ │ + -- used 0.253045s (cpu); 0.181362s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i29 : debugLevel = 0;
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -114,19 +114,19 @@
│ │ │ │  i25 : I = minors(2, matrix {{x, y, z}, {u, v, w}});
│ │ │ │  
│ │ │ │  o25 : Ideal of S
│ │ │ │  i26 : debugLevel = 1;
│ │ │ │  i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing
│ │ │ │  DepthOfSearch will let the function search more deeply.
│ │ │ │ - -- used 0.043332s (cpu); 0.0433271s (thread); 0s (gc)
│ │ │ │ + -- used 0.0618399s (cpu); 0.0617056s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = false
│ │ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ │ - -- used 0.247776s (cpu); 0.203448s (thread); 0s (gc)
│ │ │ │ + -- used 0.253045s (cpu); 0.181362s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = true
│ │ │ │  i29 : debugLevel = 0;
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_e_s_t_I_d_e_a_l -- compute a test ideal in a Q-Gorenstein ring
│ │ │ │      * _i_s_F_R_a_t_i_o_n_a_l -- whether a ring is F-rational
│ │ │ │  ********** WWaayyss ttoo uussee iissFFRReegguullaarr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html
│ │ │ @@ -260,25 +260,25 @@
│ │ │          
      
│ │ │            
      It is often more efficient to pass a list, as opposed to finding a common denominator and passing a single element, since testIdeal can do things in a more intelligent way for such a list.
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ - -- used 0.214265s (cpu); 0.148483s (thread); 0s (gc)
│ │ │ + -- used 0.313256s (cpu); 0.23861s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
      
│ │ │              
      
│ │ │                
      i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.269113s (cpu); 0.214063s (thread); 0s (gc)
│ │ │ + -- used 0.330022s (cpu); 0.254882s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
      
│ │ │              
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -100,21 +100,21 @@
│ │ │ │  o22 = ideal (y, x)
│ │ │ │  
│ │ │ │  o22 : Ideal of R
│ │ │ │  It is often more efficient to pass a list, as opposed to finding a common
│ │ │ │  denominator and passing a single element, since testIdeal can do things in a
│ │ │ │  more intelligent way for such a list.
│ │ │ │  i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ │ - -- used 0.214265s (cpu); 0.148483s (thread); 0s (gc)
│ │ │ │ + -- used 0.313256s (cpu); 0.23861s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = ideal (y, x)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ │ - -- used 0.269113s (cpu); 0.214063s (thread); 0s (gc)
│ │ │ │ + -- used 0.330022s (cpu); 0.254882s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = ideal (y, x)
│ │ │ │  
│ │ │ │  o24 : Ideal of R
│ │ │ │  The option AssumeDomain (default value false) is used when finding a test
│ │ │ │  element. The option FrobeniusRootStrategy (default value Substitution) is
│ │ │ │  passed to internal _f_r_o_b_e_n_i_u_s_R_o_o_t calls.
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out
│ │ │ @@ -16,18 +16,16 @@
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -                                        3
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ -                                     2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +                                   3
│ │ │ +o4 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -38,16 +36,15 @@
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o4 : LineageTable
│ │ │  
│ │ │  i5 : minimize T
│ │ │  
│ │ │ -o5 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o5 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out
│ │ │ @@ -3,15 +3,14 @@
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │  o3 = LineageTable{((0, 2), 0) => null}
│ │ │ -                  ((0, 2), 1) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out
│ │ │ @@ -2,18 +2,16 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │                                       2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -24,16 +22,15 @@
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : minimize T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o4 = LineageTable{((0, 1), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out
│ │ │ @@ -2,18 +2,16 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ -                                     2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +                                   3
│ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -24,16 +22,15 @@
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out
│ │ │ @@ -6,64 +6,44 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : allowableThreads  = 4;
│ │ │  
│ │ │  i4 : H = tgb I
│ │ │  
│ │ │ -                                                                 2 12
│ │ │ -o4 = LineageTable{((((((0, 1), 2), 2), ((0, 1), 2)), 2), 2) => 9y z                           }
│ │ │ -                                                       2 16
│ │ │ -                  ((((((0, 1), 2), 2), 1), 2), 2) => 9y z
│ │ │ -                                                       2 15
│ │ │ -                  ((((((0, 1), 2), 2), 2), 2), 2) => 9y z
│ │ │ -                                                                  2 6
│ │ │ -                  ((((((0, 1), 2), 3), 2), ((0, 1), 2)), 2) => 16y z
│ │ │ -                                                            3 10
│ │ │ -                  (((((0, 1), 2), 2), ((0, 1), 2)), 2) => 9y z
│ │ │ -                                                  3 14
│ │ │ -                  (((((0, 1), 2), 2), 1), 2) => 9y z
│ │ │ -                                                  3 13
│ │ │ -                  (((((0, 1), 2), 2), 2), 2) => 9y z
│ │ │ -                                                                                  4 4
│ │ │ -                  (((((0, 1), 2), 3), 2), ((((0, 1), 2), 2), ((0, 1), 2))) => -31y z
│ │ │ -                                                             4 5
│ │ │ -                  (((((0, 1), 2), 3), 2), ((0, 1), 2)) => 13y z
│ │ │ -                                                         3 8
│ │ │ -                  (((((0, 1), 2), 3), 2), (0, 1)) => -25y z
│ │ │ -                                                  3 8
│ │ │ -                  (((((0, 1), 2), 3), 2), 2) => 9y z
│ │ │ -                                              4 13     4 9
│ │ │ -                  ((((0, 1), 2), 1), 2) => 23y z   + 6y z
│ │ │ -                                                        4 8      4 4
│ │ │ -                  ((((0, 1), 2), 2), ((0, 1), 2)) => 33y z  + 13y z
│ │ │ -                                              4 12      4 11
│ │ │ -                  ((((0, 1), 2), 2), 1) => 50y z   - 23y z
│ │ │ -                                                4 11     4 6
│ │ │ -                  ((((0, 1), 2), 2), 2) => - 26y z   + 9y z
│ │ │ -                                               4 6
│ │ │ -                  ((((0, 1), 2), 2), 3) => -13y z
│ │ │ -                                              4 6     3 16
│ │ │ -                  ((((0, 1), 2), 3), 2) => 46y z  + 9y z
│ │ │ -                                         6 4      4 6
│ │ │ -                  (((0, 1), 2), 1) => 19y z  - 30y z
│ │ │ -                                         5 4     4 7
│ │ │ -                  (((0, 1), 2), 2) => 37y z  + 9y z
│ │ │ -                                         4 14      4 11
│ │ │ -                  (((0, 1), 2), 3) => 27y z   - 16y z
│ │ │ -                                      5 5     4 4
│ │ │ -                  ((0, 1), 2) => - 24y z  + 9y z
│ │ │ -                                                                                           2 4
│ │ │ -                  ((0, 2), (((((0, 1), 2), 3), 2), ((((0, 1), 2), 2), ((0, 1), 2)))) => 41y z
│ │ │ +                                           2 5      2 4
│ │ │ +o4 = LineageTable{((0, 1), (0, 2)) => - 40y z  - 22y z }
│ │ │ +                                           5       2 4
│ │ │ +                  ((0, 1), (0, 3)) => - 46y z - 40y z
│ │ │ +                                   2 5      2 4
│ │ │ +                  ((0, 1), 2) => 7y z  + 19y z
│ │ │ +                                   2 5      2 4
│ │ │ +                  ((0, 1), 3) => 7y z  + 19y z
│ │ │ +                                    2 11      2 10
│ │ │ +                  ((0, 2), 1) => 12y z   + 47y z
│ │ │ +                                    2 10      2 9
│ │ │ +                  ((0, 3), 1) => 12y z   + 47y z
│ │ │ +                                    2 6      2 4
│ │ │ +                  ((0, 3), 2) => 23y z  + 19y z
│ │ │ +                                        2 4
│ │ │ +                  ((1, 3), (0, 2)) => -y z
│ │ │ +                                         2 4
│ │ │ +                  ((1, 3), (0, 3)) => 11y z
│ │ │ +                                    2 4
│ │ │ +                  ((2, 3), 1) => -7y z
│ │ │                                   5 2      3 4
│ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ -                                 3 5     2 4
│ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ -                               5       3 4
│ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ +                              5 3     2 4
│ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ +                              5 2      5
│ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ +                                 4 5      2 7
│ │ │ +                  (1, 2) => - 45y z  - 14y z
│ │ │ +                                 7       2 6
│ │ │ +                  (1, 3) => - 24y z - 14y z
│ │ │                                3 4     2 4
│ │ │                    (2, 3) => 7y z  - 9y z
│ │ │                                 2
│ │ │                    0 => 2x + 10y z
│ │ │                           2           3
│ │ │                    1 => 8x y + 10x*y*z
│ │ │                             3 2       3
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html
│ │ │ @@ -100,18 +100,16 @@
│ │ │            
      By default, the option is false. The basis can also be minimized after the distributed computation is finished:
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -155,15 +155,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 4.68146s (cpu); 2.95633s (thread); 0s (gc)
│ │ │ + -- used 5.58282s (cpu); 3.40705s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 252
│ │ │  
│ │ │  o5 : QQ
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i4 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -                                        3
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ -                                     2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +                                   3
│ │ │ +o4 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -125,16 +123,15 @@
│ │ │  o4 : LineageTable
      
│ │ │              
      
│ │ │                
      i5 : minimize T
│ │ │  
│ │ │ -o5 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o5 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,18 +27,16 @@
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  By default, the option is false. The basis can also be minimized after the
│ │ │ │  distributed computation is finished:
│ │ │ │  i4 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │ -                                        3
│ │ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ │ -                                     2
│ │ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ │ +                                   3
│ │ │ │ +o4 = LineageTable{((0, 2), 0) => -c      }
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │ @@ -48,16 +46,15 @@
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o4 : LineageTable
│ │ │ │  i5 : minimize T
│ │ │ │  
│ │ │ │ -o5 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ +o5 = LineageTable{((0, 2), 0) => null}
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html
│ │ │ @@ -92,15 +92,14 @@
│ │ │              
      
│ │ │                
      i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │  o3 = LineageTable{((0, 2), 0) => null}
│ │ │ -                  ((0, 2), 1) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,14 @@
│ │ │ │  Gröbner basis function _t_g_b in the expected Macaulay2 format, so that further
│ │ │ │  computation are one step easier to set up.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │  o3 = LineageTable{((0, 2), 0) => null}
│ │ │ │ -                  ((0, 2), 1) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html
│ │ │ @@ -87,18 +87,16 @@
│ │ │                
      i2 : allowableThreads= 2;
      
│ │ │              
      
│ │ │                
      i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │                                       2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -112,16 +110,15 @@
│ │ │  o3 : LineageTable
      
│ │ │              
      
│ │ │                
      i4 : minimize T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o4 = LineageTable{((0, 1), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,18 +19,16 @@
│ │ │ │  minimal generators of the ideal generated by the leading terms of the values of
│ │ │ │  H. If the values of H constitute a Gröbner basis of the ideal they generate,
│ │ │ │  this method returns a minimal Gröbner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │ -                                        3
│ │ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ │                                       2
│ │ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │ @@ -40,16 +38,15 @@
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : minimize T
│ │ │ │  
│ │ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ +o4 = LineageTable{((0, 1), 0) => null}
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html
│ │ │ @@ -87,18 +87,16 @@
│ │ │                
      i2 : allowableThreads= 2;
      
│ │ │              
      
│ │ │                
      i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ -                                     2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +                                   3
│ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -112,16 +110,15 @@
│ │ │  o3 : LineageTable
      
│ │ │              
      
│ │ │                
      i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,18 +20,16 @@
│ │ │ │  remainder on the division by the remaining values H.
│ │ │ │  If values H constitute a Gröbner basis of the ideal they generate, this method
│ │ │ │  returns a reduced Gröbner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │ │  
│ │ │ │ -                                        3
│ │ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ │ -                                     2
│ │ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ │ +                                   3
│ │ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │ @@ -41,16 +39,15 @@
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : reduce T
│ │ │ │  
│ │ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html
│ │ │ @@ -100,64 +100,44 @@
│ │ │                
      i3 : allowableThreads  = 4;
      
│ │ │              
      
│ │ │                
      i4 : H = tgb I
│ │ │  
│ │ │ -                                                                 2 12
│ │ │ -o4 = LineageTable{((((((0, 1), 2), 2), ((0, 1), 2)), 2), 2) => 9y z                           }
│ │ │ -                                                       2 16
│ │ │ -                  ((((((0, 1), 2), 2), 1), 2), 2) => 9y z
│ │ │ -                                                       2 15
│ │ │ -                  ((((((0, 1), 2), 2), 2), 2), 2) => 9y z
│ │ │ -                                                                  2 6
│ │ │ -                  ((((((0, 1), 2), 3), 2), ((0, 1), 2)), 2) => 16y z
│ │ │ -                                                            3 10
│ │ │ -                  (((((0, 1), 2), 2), ((0, 1), 2)), 2) => 9y z
│ │ │ -                                                  3 14
│ │ │ -                  (((((0, 1), 2), 2), 1), 2) => 9y z
│ │ │ -                                                  3 13
│ │ │ -                  (((((0, 1), 2), 2), 2), 2) => 9y z
│ │ │ -                                                                                  4 4
│ │ │ -                  (((((0, 1), 2), 3), 2), ((((0, 1), 2), 2), ((0, 1), 2))) => -31y z
│ │ │ -                                                             4 5
│ │ │ -                  (((((0, 1), 2), 3), 2), ((0, 1), 2)) => 13y z
│ │ │ -                                                         3 8
│ │ │ -                  (((((0, 1), 2), 3), 2), (0, 1)) => -25y z
│ │ │ -                                                  3 8
│ │ │ -                  (((((0, 1), 2), 3), 2), 2) => 9y z
│ │ │ -                                              4 13     4 9
│ │ │ -                  ((((0, 1), 2), 1), 2) => 23y z   + 6y z
│ │ │ -                                                        4 8      4 4
│ │ │ -                  ((((0, 1), 2), 2), ((0, 1), 2)) => 33y z  + 13y z
│ │ │ -                                              4 12      4 11
│ │ │ -                  ((((0, 1), 2), 2), 1) => 50y z   - 23y z
│ │ │ -                                                4 11     4 6
│ │ │ -                  ((((0, 1), 2), 2), 2) => - 26y z   + 9y z
│ │ │ -                                               4 6
│ │ │ -                  ((((0, 1), 2), 2), 3) => -13y z
│ │ │ -                                              4 6     3 16
│ │ │ -                  ((((0, 1), 2), 3), 2) => 46y z  + 9y z
│ │ │ -                                         6 4      4 6
│ │ │ -                  (((0, 1), 2), 1) => 19y z  - 30y z
│ │ │ -                                         5 4     4 7
│ │ │ -                  (((0, 1), 2), 2) => 37y z  + 9y z
│ │ │ -                                         4 14      4 11
│ │ │ -                  (((0, 1), 2), 3) => 27y z   - 16y z
│ │ │ -                                      5 5     4 4
│ │ │ -                  ((0, 1), 2) => - 24y z  + 9y z
│ │ │ -                                                                                           2 4
│ │ │ -                  ((0, 2), (((((0, 1), 2), 3), 2), ((((0, 1), 2), 2), ((0, 1), 2)))) => 41y z
│ │ │ +                                           2 5      2 4
│ │ │ +o4 = LineageTable{((0, 1), (0, 2)) => - 40y z  - 22y z }
│ │ │ +                                           5       2 4
│ │ │ +                  ((0, 1), (0, 3)) => - 46y z - 40y z
│ │ │ +                                   2 5      2 4
│ │ │ +                  ((0, 1), 2) => 7y z  + 19y z
│ │ │ +                                   2 5      2 4
│ │ │ +                  ((0, 1), 3) => 7y z  + 19y z
│ │ │ +                                    2 11      2 10
│ │ │ +                  ((0, 2), 1) => 12y z   + 47y z
│ │ │ +                                    2 10      2 9
│ │ │ +                  ((0, 3), 1) => 12y z   + 47y z
│ │ │ +                                    2 6      2 4
│ │ │ +                  ((0, 3), 2) => 23y z  + 19y z
│ │ │ +                                        2 4
│ │ │ +                  ((1, 3), (0, 2)) => -y z
│ │ │ +                                         2 4
│ │ │ +                  ((1, 3), (0, 3)) => 11y z
│ │ │ +                                    2 4
│ │ │ +                  ((2, 3), 1) => -7y z
│ │ │                                   5 2      3 4
│ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ -                                 3 5     2 4
│ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ -                               5       3 4
│ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ +                              5 3     2 4
│ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ +                              5 2      5
│ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ +                                 4 5      2 7
│ │ │ +                  (1, 2) => - 45y z  - 14y z
│ │ │ +                                 7       2 6
│ │ │ +                  (1, 3) => - 24y z - 14y z
│ │ │                                3 4     2 4
│ │ │                    (2, 3) => 7y z  - 9y z
│ │ │                                 2
│ │ │                    0 => 2x + 10y z
│ │ │                           2           3
│ │ │                    1 => 8x y + 10x*y*z
│ │ │                             3 2       3
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,69 +26,44 @@
│ │ │ │  i2 : I = ideal {2*x + 10*y^2*z, 8*x^2*y + 10*x*y*z^3, 5*x*y^3*z^2 + 9*x*z^3,
│ │ │ │  9*x*y^3*z + 10*x*y^3};
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : allowableThreads  = 4;
│ │ │ │  i4 : H = tgb I
│ │ │ │  
│ │ │ │ -                                                                 2 12
│ │ │ │ -o4 = LineageTable{((((((0, 1), 2), 2), ((0, 1), 2)), 2), 2) => 9y z
│ │ │ │ -}
│ │ │ │ -                                                       2 16
│ │ │ │ -                  ((((((0, 1), 2), 2), 1), 2), 2) => 9y z
│ │ │ │ -                                                       2 15
│ │ │ │ -                  ((((((0, 1), 2), 2), 2), 2), 2) => 9y z
│ │ │ │ -                                                                  2 6
│ │ │ │ -                  ((((((0, 1), 2), 3), 2), ((0, 1), 2)), 2) => 16y z
│ │ │ │ -                                                            3 10
│ │ │ │ -                  (((((0, 1), 2), 2), ((0, 1), 2)), 2) => 9y z
│ │ │ │ -                                                  3 14
│ │ │ │ -                  (((((0, 1), 2), 2), 1), 2) => 9y z
│ │ │ │ -                                                  3 13
│ │ │ │ -                  (((((0, 1), 2), 2), 2), 2) => 9y z
│ │ │ │ -
│ │ │ │ -4 4
│ │ │ │ -                  (((((0, 1), 2), 3), 2), ((((0, 1), 2), 2), ((0, 1), 2))) => -
│ │ │ │ -31y z
│ │ │ │ -                                                             4 5
│ │ │ │ -                  (((((0, 1), 2), 3), 2), ((0, 1), 2)) => 13y z
│ │ │ │ -                                                         3 8
│ │ │ │ -                  (((((0, 1), 2), 3), 2), (0, 1)) => -25y z
│ │ │ │ -                                                  3 8
│ │ │ │ -                  (((((0, 1), 2), 3), 2), 2) => 9y z
│ │ │ │ -                                              4 13     4 9
│ │ │ │ -                  ((((0, 1), 2), 1), 2) => 23y z   + 6y z
│ │ │ │ -                                                        4 8      4 4
│ │ │ │ -                  ((((0, 1), 2), 2), ((0, 1), 2)) => 33y z  + 13y z
│ │ │ │ -                                              4 12      4 11
│ │ │ │ -                  ((((0, 1), 2), 2), 1) => 50y z   - 23y z
│ │ │ │ -                                                4 11     4 6
│ │ │ │ -                  ((((0, 1), 2), 2), 2) => - 26y z   + 9y z
│ │ │ │ -                                               4 6
│ │ │ │ -                  ((((0, 1), 2), 2), 3) => -13y z
│ │ │ │ -                                              4 6     3 16
│ │ │ │ -                  ((((0, 1), 2), 3), 2) => 46y z  + 9y z
│ │ │ │ -                                         6 4      4 6
│ │ │ │ -                  (((0, 1), 2), 1) => 19y z  - 30y z
│ │ │ │ -                                         5 4     4 7
│ │ │ │ -                  (((0, 1), 2), 2) => 37y z  + 9y z
│ │ │ │ -                                         4 14      4 11
│ │ │ │ -                  (((0, 1), 2), 3) => 27y z   - 16y z
│ │ │ │ -                                      5 5     4 4
│ │ │ │ -                  ((0, 1), 2) => - 24y z  + 9y z
│ │ │ │ -
│ │ │ │ -2 4
│ │ │ │ -                  ((0, 2), (((((0, 1), 2), 3), 2), ((((0, 1), 2), 2), ((0, 1),
│ │ │ │ -2)))) => 41y z
│ │ │ │ +                                           2 5      2 4
│ │ │ │ +o4 = LineageTable{((0, 1), (0, 2)) => - 40y z  - 22y z }
│ │ │ │ +                                           5       2 4
│ │ │ │ +                  ((0, 1), (0, 3)) => - 46y z - 40y z
│ │ │ │ +                                   2 5      2 4
│ │ │ │ +                  ((0, 1), 2) => 7y z  + 19y z
│ │ │ │ +                                   2 5      2 4
│ │ │ │ +                  ((0, 1), 3) => 7y z  + 19y z
│ │ │ │ +                                    2 11      2 10
│ │ │ │ +                  ((0, 2), 1) => 12y z   + 47y z
│ │ │ │ +                                    2 10      2 9
│ │ │ │ +                  ((0, 3), 1) => 12y z   + 47y z
│ │ │ │ +                                    2 6      2 4
│ │ │ │ +                  ((0, 3), 2) => 23y z  + 19y z
│ │ │ │ +                                        2 4
│ │ │ │ +                  ((1, 3), (0, 2)) => -y z
│ │ │ │ +                                         2 4
│ │ │ │ +                  ((1, 3), (0, 3)) => 11y z
│ │ │ │ +                                    2 4
│ │ │ │ +                  ((2, 3), 1) => -7y z
│ │ │ │                                   5 2      3 4
│ │ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ │ -                                 3 5     2 4
│ │ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ │ -                               5       3 4
│ │ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ │ +                              5 3     2 4
│ │ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ │ +                              5 2      5
│ │ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ │ +                                 4 5      2 7
│ │ │ │ +                  (1, 2) => - 45y z  - 14y z
│ │ │ │ +                                 7       2 6
│ │ │ │ +                  (1, 3) => - 24y z - 14y z
│ │ │ │                                3 4     2 4
│ │ │ │                    (2, 3) => 7y z  - 9y z
│ │ │ │                                 2
│ │ │ │                    0 => 2x + 10y z
│ │ │ │                           2           3
│ │ │ │                    1 => 8x y + 10x*y*z
│ │ │ │                             3 2       3
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out
│ │ │ @@ -40,15 +40,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 1.23426s (cpu); 0.838012s (thread); 0s (gc)
│ │ │ + -- used 1.4641s (cpu); 0.977006s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
│ │ │  
│ │ │  i5 : time edDeg(A,ForceAmat=>true)
│ │ │  
│ │ │ @@ -56,14 +56,14 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 4.68146s (cpu); 2.95633s (thread); 0s (gc)
│ │ │ + -- used 5.58282s (cpu); 3.40705s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 252
│ │ │  
│ │ │  o5 : QQ
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html
│ │ │ @@ -136,15 +136,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 1.23426s (cpu); 0.838012s (thread); 0s (gc)
│ │ │ + -- used 1.4641s (cpu); 0.977006s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
      
│ │ │              
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -66,30 +66,30 @@
│ │ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │ │  ED Degree = 252
│ │ │ │  
│ │ │ │                         5      4      3      2
│ │ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ │ - -- used 1.23426s (cpu); 0.838012s (thread); 0s (gc)
│ │ │ │ + -- used 1.4641s (cpu); 0.977006s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 252
│ │ │ │  
│ │ │ │  o4 : QQ
│ │ │ │  i5 : time edDeg(A,ForceAmat=>true)
│ │ │ │  
│ │ │ │  The toric variety has degree = 28
│ │ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │ │  ED Degree = 252
│ │ │ │  
│ │ │ │                         5      4      3      2
│ │ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ │ - -- used 4.68146s (cpu); 2.95633s (thread); 0s (gc)
│ │ │ │ + -- used 5.58282s (cpu); 3.40705s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 252
│ │ │ │  
│ │ │ │  o5 : QQ
│ │ │ │  ********** WWaayyss ttoo uussee eeddDDeegg:: **********
│ │ │ │      * edDeg(Matrix)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out
│ │ │ @@ -17,15 +17,15 @@
│ │ │       | -1 1  2  -1 -1 1 -1 1 0 0 |
│ │ │       | 1  0  -1 0  0  0 0  0 0 0 |
│ │ │  
│ │ │                4       10
│ │ │  o2 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ - -- .101733s elapsed
│ │ │ + -- .0957823s elapsed
│ │ │  
│ │ │  i4 : select(Ts, T -> isStar T)
│ │ │  
│ │ │  o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │       1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -50,14 +50,14 @@
│ │ │  i7 : T = regularFineTriangulation A
│ │ │  
│ │ │  o7 = triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0, 1, 6, 7, 9}, {0, 2, 3, 4, 6}, {0, 2, 3, 4, 9}, {0, 2, 4, 6, 9}, {0, 3, 4, 7, 8}, {0, 3, 4, 7, 9}, {0, 3, 5, 7, 8}, {0, 4, 6, 7, 8}, {0, 4, 6, 7, 9}, {0, 5, 6, 7, 8}, {1, 2, 3, 7, 9}, {1, 2, 6, 7, 9}, {2, 3, 4, 7, 8}, {2, 3, 4, 7, 9}, {2, 3, 5, 7, 8}, {2, 4, 6, 7, 8}, {2, 4, 6, 7, 9}, {2, 5, 6, 7, 8}}
│ │ │  
│ │ │  o7 : Triangulation
│ │ │  
│ │ │  i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ - -- 1.688s elapsed
│ │ │ + -- 1.1642s elapsed
│ │ │  
│ │ │  i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/html/index.html
│ │ │ @@ -155,15 +155,15 @@
│ │ │                4       10
│ │ │  o2 : Matrix ZZ  <-- ZZ
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ - -- .101733s elapsed
      │ │ │ + -- .0957823s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i4 : select(Ts, T -> isStar T)
│ │ │  
│ │ │  o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
│ │ │ @@ -203,15 +203,15 @@
│ │ │  
│ │ │  o7 : Triangulation
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ - -- 1.688s elapsed
      │ │ │ + -- 1.1642s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
      │ │ │ ├── html2text {} │ │ │ │ @@ -54,15 +54,15 @@ │ │ │ │ | 0 0 0 1 0 0 -1 0 0 0 | │ │ │ │ | -1 1 2 -1 -1 1 -1 1 0 0 | │ │ │ │ | 1 0 -1 0 0 0 0 0 0 0 | │ │ │ │ │ │ │ │ 4 10 │ │ │ │ o2 : Matrix ZZ <-- ZZ │ │ │ │ i3 : elapsedTime Ts = allTriangulations(A, Fine => true); │ │ │ │ - -- .101733s elapsed │ │ │ │ + -- .0957823s elapsed │ │ │ │ i4 : select(Ts, T -> isStar T) │ │ │ │ │ │ │ │ o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 5, 7, 9}, {0, 3, 5, 8, 9}, {0, 4, 6, 8, 9}, {0, 5, 6, 7, 9}, {0, 5, 6, │ │ │ │ @@ -86,15 +86,15 @@ │ │ │ │ 6, 7, 9}, {0, 2, 3, 4, 6}, {0, 2, 3, 4, 9}, {0, 2, 4, 6, 9}, {0, 3, 4, 7, 8}, │ │ │ │ {0, 3, 4, 7, 9}, {0, 3, 5, 7, 8}, {0, 4, 6, 7, 8}, {0, 4, 6, 7, 9}, {0, 5, 6, │ │ │ │ 7, 8}, {1, 2, 3, 7, 9}, {1, 2, 6, 7, 9}, {2, 3, 4, 7, 8}, {2, 3, 4, 7, 9}, {2, │ │ │ │ 3, 5, 7, 8}, {2, 4, 6, 7, 8}, {2, 4, 6, 7, 9}, {2, 5, 6, 7, 8}} │ │ │ │ │ │ │ │ o7 : Triangulation │ │ │ │ i8 : elapsedTime Ts2 = generateTriangulations T; │ │ │ │ - -- 1.688s elapsed │ │ │ │ + -- 1.1642s elapsed │ │ │ │ i9 : #Ts2 == #Ts │ │ │ │ │ │ │ │ o9 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _P_o_l_y_h_e_d_r_a -- for computations with convex polyhedra, cones, and fans │ │ │ │ * _T_o_p_c_o_m -- interface to selected functions from topcom package │ │ │ │ * _R_e_f_l_e_x_i_v_e_P_o_l_y_t_o_p_e_s_D_B -- simple access to Kreuzer-Skarke database of │ │ ├── ./usr/share/doc/Macaulay2/VectorFields/dump/rawdocumentation.dump │ │ │ @@ -1,8 +1,8 @@ │ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon May 18 11:29:47 2026 │ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon May 18 11:29:46 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=10 │ │ │ Y29tbXV0YXRvcg== │ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out │ │ │ @@ -6,30 +6,30 @@ │ │ │ │ │ │ o2 = | xz yz z2 x3 | │ │ │ │ │ │ 1 4 │ │ │ o2 : Matrix S <-- S │ │ │ │ │ │ i3 : time (F,R,G,C)=localHilbertScheme(F0); │ │ │ - -- used 1.41035s (cpu); 0.72689s (thread); 0s (gc) │ │ │ + -- used 1.40893s (cpu); 0.736007s (thread); 0s (gc) │ │ │ │ │ │ i4 : T=ring first G; │ │ │ │ │ │ i5 : sum G │ │ │ │ │ │ o5 = | t_1t_16 | │ │ │ | t_9t_16 | │ │ │ | -t_4t_16 | │ │ │ | -2t_14t_16+t_15t_16 | │ │ │ │ │ │ 4 1 │ │ │ o5 : Matrix T <-- T │ │ │ │ │ │ i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false); │ │ │ - -- used 1.22449s (cpu); 0.660586s (thread); 0s (gc) │ │ │ + -- used 1.06501s (cpu); 0.527361s (thread); 0s (gc) │ │ │ │ │ │ i7 : sum G │ │ │ │ │ │ o7 = | t_1t_16 │ │ │ | 2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+ │ │ │ | -t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t │ │ │ | 2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2- │ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html │ │ │ @@ -76,15 +76,15 @@ │ │ │ │ │ │ │ │ │

      With the default setting SmartLift=>true we get very nice equations for the base space:

      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -103,15 +103,15 @@ │ │ │ │ │ │
      │ │ │ 
      i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ - -- used 1.41035s (cpu); 0.72689s (thread); 0s (gc)
      │ │ │ + -- used 1.40893s (cpu); 0.736007s (thread); 0s (gc) │ │ │
      │ │ │ 
      i4 : T=ring first G;
      │ │ │
      │ │ │

      With the setting SmartLift=>false the calculation is faster, but the equations are no longer homogeneous:

      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │ 
      i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ - -- used 1.22449s (cpu); 0.660586s (thread); 0s (gc)
      │ │ │ + -- used 1.06501s (cpu); 0.527361s (thread); 0s (gc) │ │ │
      │ │ │ 
      i7 : sum G
│ │ │  
│ │ │  o7 = | t_1t_16
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,29 +18,29 @@
│ │ │ │  o2 = | xz yz z2 x3 |
│ │ │ │  
│ │ │ │               1      4
│ │ │ │  o2 : Matrix S  <-- S
│ │ │ │  With the default setting SmartLift=>true we get very nice equations for the
│ │ │ │  base space:
│ │ │ │  i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ │ - -- used 1.41035s (cpu); 0.72689s (thread); 0s (gc)
│ │ │ │ + -- used 1.40893s (cpu); 0.736007s (thread); 0s (gc)
│ │ │ │  i4 : T=ring first G;
│ │ │ │  i5 : sum G
│ │ │ │  
│ │ │ │  o5 = | t_1t_16             |
│ │ │ │       | t_9t_16             |
│ │ │ │       | -t_4t_16            |
│ │ │ │       | -2t_14t_16+t_15t_16 |
│ │ │ │  
│ │ │ │               4      1
│ │ │ │  o5 : Matrix T  <-- T
│ │ │ │  With the setting SmartLift=>false the calculation is faster, but the equations
│ │ │ │  are no longer homogeneous:
│ │ │ │  i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ │ - -- used 1.22449s (cpu); 0.660586s (thread); 0s (gc)
│ │ │ │ + -- used 1.06501s (cpu); 0.527361s (thread); 0s (gc)
│ │ │ │  i7 : sum G
│ │ │ │  
│ │ │ │  o7 = | t_1t_16
│ │ │ │       | 2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+
│ │ │ │       | -t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t
│ │ │ │       | 2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2-
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_dualize.out
│ │ │ @@ -44,51 +44,51 @@
│ │ │  i10 : J = m^9;
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : M = J*R^1;
│ │ │  
│ │ │  i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0422547s (cpu); 0.0422579s (thread); 0s (gc)
│ │ │ + -- used 0.0525038s (cpu); 0.0525047s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
│ │ │  
│ │ │  i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.371374s (cpu); 0.371345s (thread); 0s (gc)
│ │ │ + -- used 0.492912s (cpu); 0.492755s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.535446s (cpu); 0.454421s (thread); 0s (gc)
│ │ │ + -- used 0.513252s (cpu); 0.513259s (thread); 0s (gc)
│ │ │  
│ │ │  i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00276885s (cpu); 0.00276982s (thread); 0s (gc)
│ │ │ + -- used 0.15039s (cpu); 0.0612517s (thread); 0s (gc)
│ │ │  
│ │ │  i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00222642s (cpu); 0.00222764s (thread); 0s (gc)
│ │ │ + -- used 0.00290688s (cpu); 0.00291238s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : R = ZZ/7[x,y,u,v]/ideal(x*y-u*v);
│ │ │  
│ │ │  i18 : I = ideal(x,u);
│ │ │  
│ │ │  o18 : Ideal of R
│ │ │  
│ │ │  i19 : J = I^15;
│ │ │  
│ │ │  o19 : Ideal of R
│ │ │  
│ │ │  i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0634709s (cpu); 0.0634782s (thread); 0s (gc)
│ │ │ + -- used 0.080914s (cpu); 0.0809184s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
│ │ │  
│ │ │  i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00775678s (cpu); 0.00775271s (thread); 0s (gc)
│ │ │ + -- used 0.00737026s (cpu); 0.00737614s (thread); 0s (gc)
│ │ │  
│ │ │  o21 : Ideal of R
│ │ │  
│ │ │  i22 : R = QQ[x,y]/ideal(x*y);
│ │ │  
│ │ │  i23 : J = ideal(x,y);
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexify.out
│ │ │ @@ -103,104 +103,104 @@
│ │ │  o21 : Ideal of R
│ │ │  
│ │ │  i22 : J = I^21;
│ │ │  
│ │ │  o22 : Ideal of R
│ │ │  
│ │ │  i23 : time reflexify(J);
│ │ │ - -- used 0.281734s (cpu); 0.206435s (thread); 0s (gc)
│ │ │ + -- used 0.301911s (cpu); 0.216391s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.44329s (cpu); 0.360925s (thread); 0s (gc)
│ │ │ + -- used 0.449586s (cpu); 0.374007s (thread); 0s (gc)
│ │ │  
│ │ │  i25 : R = ZZ/13[x,y,z]/ideal(x^3 + y^3-z^11*x*y);
│ │ │  
│ │ │  i26 : I = ideal(x-4*y, z);
│ │ │  
│ │ │  o26 : Ideal of R
│ │ │  
│ │ │  i27 : J = I^20;
│ │ │  
│ │ │  o27 : Ideal of R
│ │ │  
│ │ │  i28 : M = J*R^1;
│ │ │  
│ │ │  i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.0816957s (cpu); 0.0817007s (thread); 0s (gc)
│ │ │ + -- used 0.115857s (cpu); 0.115858s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o29 : Ideal of R
│ │ │  
│ │ │  i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 7.37068s (cpu); 4.72312s (thread); 0s (gc)
│ │ │ + -- used 6.33817s (cpu); 4.89547s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o30 : Ideal of R
│ │ │  
│ │ │  i31 : J1 == J2
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 6.16456s (cpu); 4.66565s (thread); 0s (gc)
│ │ │ + -- used 6.58528s (cpu); 5.12311s (thread); 0s (gc)
│ │ │  
│ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.585363s (cpu); 0.40102s (thread); 0s (gc)
│ │ │ + -- used 0.647599s (cpu); 0.433619s (thread); 0s (gc)
│ │ │  
│ │ │  i34 : R = QQ[x,y,u,v]/ideal(x*y-u*v);
│ │ │  
│ │ │  i35 : I = ideal(x,u);
│ │ │  
│ │ │  o35 : Ideal of R
│ │ │  
│ │ │  i36 : J = I^20;
│ │ │  
│ │ │  o36 : Ideal of R
│ │ │  
│ │ │  i37 : M = I^20*R^1;
│ │ │  
│ │ │  i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.686555s (cpu); 0.321328s (thread); 0s (gc)
│ │ │ + -- used 0.495374s (cpu); 0.299611s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o38 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │         19    20
│ │ │        x  u, x  )
│ │ │  
│ │ │  o38 : Ideal of R
│ │ │  
│ │ │  i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 0.0138988s (cpu); 0.0139005s (thread); 0s (gc)
│ │ │ + -- used 0.0151789s (cpu); 0.015184s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o39 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │         19    20
│ │ │        x  u, x  )
│ │ │  
│ │ │  o39 : Ideal of R
│ │ │  
│ │ │  i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 0.040219s (cpu); 0.0402241s (thread); 0s (gc)
│ │ │ + -- used 0.0576302s (cpu); 0.0576348s (thread); 0s (gc)
│ │ │  
│ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.0068728s (cpu); 0.00687423s (thread); 0s (gc)
│ │ │ + -- used 0.00736688s (cpu); 0.00737537s (thread); 0s (gc)
│ │ │  
│ │ │  i42 : R = QQ[x,y]/ideal(x*y);
│ │ │  
│ │ │  i43 : I = ideal(x,y);
│ │ │  
│ │ │  o43 : Ideal of R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexive__Power.out
│ │ │ @@ -23,44 +23,44 @@
│ │ │  i5 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │  
│ │ │  i6 : I = ideal(x-z,y-2*z);
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.0305036s (cpu); 0.0305042s (thread); 0s (gc)
│ │ │ + -- used 0.0318851s (cpu); 0.0318842s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.120736s (cpu); 0.120739s (thread); 0s (gc)
│ │ │ + -- used 0.136126s (cpu); 0.136125s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : J20a == J20b
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │  
│ │ │  i12 : I = ideal(x-z,y-2*z);
│ │ │  
│ │ │  o12 : Ideal of R
│ │ │  
│ │ │  i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0312341s (cpu); 0.0312391s (thread); 0s (gc)
│ │ │ + -- used 0.0350773s (cpu); 0.0350798s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.193455s (cpu); 0.109457s (thread); 0s (gc)
│ │ │ + -- used 0.0687027s (cpu); 0.0687086s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : J1 == J2
│ │ │  
│ │ │  o15 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_dualize.html
│ │ │ @@ -168,43 +168,43 @@
│ │ │
      		│ │ │ 
      i11 : M = J*R^1;
      │ │ │
      │ │ │ 
      i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0422547s (cpu); 0.0422579s (thread); 0s (gc)
│ │ │ + -- used 0.0525038s (cpu); 0.0525047s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
      │ │ │
      │ │ │ 
      i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.371374s (cpu); 0.371345s (thread); 0s (gc)
│ │ │ + -- used 0.492912s (cpu); 0.492755s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
      │ │ │
      │ │ │ 
      i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.535446s (cpu); 0.454421s (thread); 0s (gc)
      │ │ │ + -- used 0.513252s (cpu); 0.513259s (thread); 0s (gc) │ │ │
      │ │ │ 
      i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00276885s (cpu); 0.00276982s (thread); 0s (gc)
      │ │ │ + -- used 0.15039s (cpu); 0.0612517s (thread); 0s (gc) │ │ │
      │ │ │ 
      i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00222642s (cpu); 0.00222764s (thread); 0s (gc)
│ │ │ + -- used 0.00290688s (cpu); 0.00291238s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
      │ │ │
      │ │ │
      │ │ │

      For monomial ideals in toric rings, frequently ModuleStrategy appears faster.

      │ │ │ @@ -228,23 +228,23 @@ │ │ │ │ │ │ o19 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0634709s (cpu); 0.0634782s (thread); 0s (gc)
│ │ │ + -- used 0.080914s (cpu); 0.0809184s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00775678s (cpu); 0.00775271s (thread); 0s (gc)
│ │ │ + -- used 0.00737026s (cpu); 0.00737614s (thread); 0s (gc)
│ │ │  
│ │ │  o21 : Ideal of R
      │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │

      KnownDomain is an option for dualize. If it is false (default is true), then the computer will first check whether the ring is a domain, if it is not then it will revert to ModuleStrategy. If KnownDomain is set to true for a non-domain, then the function can return an incorrect answer.

      │ │ │ ├── html2text {} │ │ │ │ @@ -60,43 +60,43 @@ │ │ │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ i10 : J = m^9; │ │ │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ i11 : M = J*R^1; │ │ │ │ i12 : time dualize(J, Strategy=>IdealStrategy); │ │ │ │ - -- used 0.0422547s (cpu); 0.0422579s (thread); 0s (gc) │ │ │ │ + -- used 0.0525038s (cpu); 0.0525047s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of R │ │ │ │ i13 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.371374s (cpu); 0.371345s (thread); 0s (gc) │ │ │ │ + -- used 0.492912s (cpu); 0.492755s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ i14 : time dualize(M, Strategy=>IdealStrategy); │ │ │ │ - -- used 0.535446s (cpu); 0.454421s (thread); 0s (gc) │ │ │ │ + -- used 0.513252s (cpu); 0.513259s (thread); 0s (gc) │ │ │ │ i15 : time dualize(M, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00276885s (cpu); 0.00276982s (thread); 0s (gc) │ │ │ │ + -- used 0.15039s (cpu); 0.0612517s (thread); 0s (gc) │ │ │ │ i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00222642s (cpu); 0.00222764s (thread); 0s (gc) │ │ │ │ + -- used 0.00290688s (cpu); 0.00291238s (thread); 0s (gc) │ │ │ │ │ │ │ │ o16 : Ideal of R │ │ │ │ For monomial ideals in toric rings, frequently ModuleStrategy appears faster. │ │ │ │ i17 : R = ZZ/7[x,y,u,v]/ideal(x*y-u*v); │ │ │ │ i18 : I = ideal(x,u); │ │ │ │ │ │ │ │ o18 : Ideal of R │ │ │ │ i19 : J = I^15; │ │ │ │ │ │ │ │ o19 : Ideal of R │ │ │ │ i20 : time dualize(J, Strategy=>IdealStrategy); │ │ │ │ - -- used 0.0634709s (cpu); 0.0634782s (thread); 0s (gc) │ │ │ │ + -- used 0.080914s (cpu); 0.0809184s (thread); 0s (gc) │ │ │ │ │ │ │ │ o20 : Ideal of R │ │ │ │ i21 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00775678s (cpu); 0.00775271s (thread); 0s (gc) │ │ │ │ + -- used 0.00737026s (cpu); 0.00737614s (thread); 0s (gc) │ │ │ │ │ │ │ │ o21 : Ideal of R │ │ │ │ KnownDomain is an option for dualize. If it is false (default is true), then │ │ │ │ the computer will first check whether the ring is a domain, if it is not then │ │ │ │ it will revert to ModuleStrategy. If KnownDomain is set to true for a non- │ │ │ │ domain, then the function can return an incorrect answer. │ │ │ │ i22 : R = QQ[x,y]/ideal(x*y); │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexify.html │ │ │ @@ -272,23 +272,23 @@ │ │ │ │ │ │ o22 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i23 : time reflexify(J);
│ │ │ - -- used 0.281734s (cpu); 0.206435s (thread); 0s (gc)
│ │ │ + -- used 0.301911s (cpu); 0.216391s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.44329s (cpu); 0.360925s (thread); 0s (gc)
      │ │ │ + -- used 0.449586s (cpu); 0.374007s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │

      Because of this, there are two strategies for computing a reflexification (at least if the module embeds as an ideal).

      │ │ │
      │ │ │
      │ │ │ @@ -324,26 +324,26 @@ │ │ │ │ │ │ 
      i28 : M = J*R^1;
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.0816957s (cpu); 0.0817007s (thread); 0s (gc)
│ │ │ + -- used 0.115857s (cpu); 0.115858s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o29 : Ideal of R
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 7.37068s (cpu); 4.72312s (thread); 0s (gc)
│ │ │ + -- used 6.33817s (cpu); 4.89547s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o30 : Ideal of R
      │ │ │ │ │ │ │ │ │ @@ -353,21 +353,21 @@ │ │ │ │ │ │ o31 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 6.16456s (cpu); 4.66565s (thread); 0s (gc)
      │ │ │ + -- used 6.58528s (cpu); 5.12311s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.585363s (cpu); 0.40102s (thread); 0s (gc)
      │ │ │ + -- used 0.647599s (cpu); 0.433619s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │

      However, sometimes ModuleStrategy is faster, especially for Monomial ideals.

      │ │ │
      │ │ │ │ │ │ @@ -394,15 +394,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │ 
      i37 : M = I^20*R^1;
      │ │ │
      │ │ │ 
      i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.686555s (cpu); 0.321328s (thread); 0s (gc)
│ │ │ + -- used 0.495374s (cpu); 0.299611s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o38 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -411,15 +411,15 @@
│ │ │  
│ │ │  o38 : Ideal of R
      │ │ │
      │ │ │ 
      i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 0.0138988s (cpu); 0.0139005s (thread); 0s (gc)
│ │ │ + -- used 0.0151789s (cpu); 0.015184s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o39 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -428,21 +428,21 @@
│ │ │  
│ │ │  o39 : Ideal of R
      │ │ │
      │ │ │ 
      i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 0.040219s (cpu); 0.0402241s (thread); 0s (gc)
      │ │ │ + -- used 0.0576302s (cpu); 0.0576348s (thread); 0s (gc) │ │ │
      │ │ │ 
      i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.0068728s (cpu); 0.00687423s (thread); 0s (gc)
      │ │ │ + -- used 0.00736688s (cpu); 0.00737537s (thread); 0s (gc) │ │ │
      │ │ │
      │ │ │

      For ideals, if KnownDomain is false (default value is true), then the function will check whether it is a domain. If it is a domain (or assumed to be a domain), it will reflexify using a strategy which can speed up computation, if not it will compute using a sometimes slower method which is essentially reflexifying it as a module.

      │ │ │
      │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -114,19 +114,19 @@ │ │ │ │ i21 : I = ideal(x-z,y-2*z); │ │ │ │ │ │ │ │ o21 : Ideal of R │ │ │ │ i22 : J = I^21; │ │ │ │ │ │ │ │ o22 : Ideal of R │ │ │ │ i23 : time reflexify(J); │ │ │ │ - -- used 0.281734s (cpu); 0.206435s (thread); 0s (gc) │ │ │ │ + -- used 0.301911s (cpu); 0.216391s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 : Ideal of R │ │ │ │ i24 : time reflexify(J*R^1); │ │ │ │ - -- used 0.44329s (cpu); 0.360925s (thread); 0s (gc) │ │ │ │ + -- used 0.449586s (cpu); 0.374007s (thread); 0s (gc) │ │ │ │ Because of this, there are two strategies for computing a reflexification (at │ │ │ │ least if the module embeds as an ideal). │ │ │ │ IdealStrategy. In the case that $R$ is a domain, and our module is isomorphic │ │ │ │ to an ideal $I$, then one can compute the reflexification by computing colons. │ │ │ │ ModuleStrategy. This computes the reflexification simply by computing $Hom$ │ │ │ │ twice. │ │ │ │ ModuleStrategy is the default strategy for modules, IdealStrategy is the │ │ │ │ @@ -139,73 +139,73 @@ │ │ │ │ │ │ │ │ o26 : Ideal of R │ │ │ │ i27 : J = I^20; │ │ │ │ │ │ │ │ o27 : Ideal of R │ │ │ │ i28 : M = J*R^1; │ │ │ │ i29 : J1 = time reflexify( J, Strategy=>IdealStrategy ) │ │ │ │ - -- used 0.0816957s (cpu); 0.0817007s (thread); 0s (gc) │ │ │ │ + -- used 0.115857s (cpu); 0.115858s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 9 9 11 │ │ │ │ o29 = ideal (x + 5x*y + 3y , x*z - 4y*z , z + x - 4y) │ │ │ │ │ │ │ │ o29 : Ideal of R │ │ │ │ i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy ) │ │ │ │ - -- used 7.37068s (cpu); 4.72312s (thread); 0s (gc) │ │ │ │ + -- used 6.33817s (cpu); 4.89547s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 9 9 11 │ │ │ │ o30 = ideal (x + 5x*y + 3y , x*z - 4y*z , z + x - 4y) │ │ │ │ │ │ │ │ o30 : Ideal of R │ │ │ │ i31 : J1 == J2 │ │ │ │ │ │ │ │ o31 = true │ │ │ │ i32 : time reflexify( M, Strategy=>IdealStrategy ); │ │ │ │ - -- used 6.16456s (cpu); 4.66565s (thread); 0s (gc) │ │ │ │ + -- used 6.58528s (cpu); 5.12311s (thread); 0s (gc) │ │ │ │ i33 : time reflexify( M, Strategy=>ModuleStrategy ); │ │ │ │ - -- used 0.585363s (cpu); 0.40102s (thread); 0s (gc) │ │ │ │ + -- used 0.647599s (cpu); 0.433619s (thread); 0s (gc) │ │ │ │ However, sometimes ModuleStrategy is faster, especially for Monomial ideals. │ │ │ │ i34 : R = QQ[x,y,u,v]/ideal(x*y-u*v); │ │ │ │ i35 : I = ideal(x,u); │ │ │ │ │ │ │ │ o35 : Ideal of R │ │ │ │ i36 : J = I^20; │ │ │ │ │ │ │ │ o36 : Ideal of R │ │ │ │ i37 : M = I^20*R^1; │ │ │ │ i38 : time reflexify( J, Strategy=>IdealStrategy ) │ │ │ │ - -- used 0.686555s (cpu); 0.321328s (thread); 0s (gc) │ │ │ │ + -- used 0.495374s (cpu); 0.299611s (thread); 0s (gc) │ │ │ │ │ │ │ │ 20 19 2 18 3 17 4 16 5 15 6 14 7 13 8 12 │ │ │ │ o38 = ideal (u , x*u , x u , x u , x u , x u , x u , x u , x u , │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 9 11 10 10 11 9 12 8 13 7 14 6 15 5 16 4 17 3 18 2 │ │ │ │ x u , x u , x u , x u , x u , x u , x u , x u , x u , x u , │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 19 20 │ │ │ │ x u, x ) │ │ │ │ │ │ │ │ o38 : Ideal of R │ │ │ │ i39 : time reflexify( J, Strategy=>ModuleStrategy ) │ │ │ │ - -- used 0.0138988s (cpu); 0.0139005s (thread); 0s (gc) │ │ │ │ + -- used 0.0151789s (cpu); 0.015184s (thread); 0s (gc) │ │ │ │ │ │ │ │ 20 19 2 18 3 17 4 16 5 15 6 14 7 13 8 12 │ │ │ │ o39 = ideal (u , x*u , x u , x u , x u , x u , x u , x u , x u , │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 9 11 10 10 11 9 12 8 13 7 14 6 15 5 16 4 17 3 18 2 │ │ │ │ x u , x u , x u , x u , x u , x u , x u , x u , x u , x u , │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 19 20 │ │ │ │ x u, x ) │ │ │ │ │ │ │ │ o39 : Ideal of R │ │ │ │ i40 : time reflexify( M, Strategy=>IdealStrategy ); │ │ │ │ - -- used 0.040219s (cpu); 0.0402241s (thread); 0s (gc) │ │ │ │ + -- used 0.0576302s (cpu); 0.0576348s (thread); 0s (gc) │ │ │ │ i41 : time reflexify( M, Strategy=>ModuleStrategy ); │ │ │ │ - -- used 0.0068728s (cpu); 0.00687423s (thread); 0s (gc) │ │ │ │ + -- used 0.00736688s (cpu); 0.00737537s (thread); 0s (gc) │ │ │ │ For ideals, if KnownDomain is false (default value is true), then the function │ │ │ │ will check whether it is a domain. If it is a domain (or assumed to be a │ │ │ │ domain), it will reflexify using a strategy which can speed up computation, if │ │ │ │ not it will compute using a sometimes slower method which is essentially │ │ │ │ reflexifying it as a module. │ │ │ │ Consider the following example showing the importance of making the correct │ │ │ │ assumption about the ring being a domain. │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexive__Power.html │ │ │ @@ -129,30 +129,30 @@ │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.0305036s (cpu); 0.0305042s (thread); 0s (gc)
│ │ │ + -- used 0.0318851s (cpu); 0.0318842s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.120736s (cpu); 0.120739s (thread); 0s (gc)
│ │ │ + -- used 0.136126s (cpu); 0.136125s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i10 : J20a == J20b
│ │ │ @@ -176,23 +176,23 @@
│ │ │  
│ │ │  o12 : Ideal of R
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0312341s (cpu); 0.0312391s (thread); 0s (gc)
│ │ │ + -- used 0.0350773s (cpu); 0.0350798s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.193455s (cpu); 0.109457s (thread); 0s (gc)
│ │ │ + -- used 0.0687027s (cpu); 0.0687086s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i15 : J1 == J2
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,39 +40,39 @@
│ │ │ │  of the generators of $I$. Consider the example of a cone over a point on an
│ │ │ │  elliptic curve.
│ │ │ │  i5 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │ │  i6 : I = ideal(x-z,y-2*z);
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : time J20a = reflexivePower(20, I);
│ │ │ │ - -- used 0.0305036s (cpu); 0.0305042s (thread); 0s (gc)
│ │ │ │ + -- used 0.0318851s (cpu); 0.0318842s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : I20 = I^20;
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time J20b = reflexify(I20);
│ │ │ │ - -- used 0.120736s (cpu); 0.120739s (thread); 0s (gc)
│ │ │ │ + -- used 0.136126s (cpu); 0.136125s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : J20a == J20b
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  This passes the Strategy option to a reflexify call. Valid options are
│ │ │ │  IdealStrategy and ModuleStrategy.
│ │ │ │  i11 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │ │  i12 : I = ideal(x-z,y-2*z);
│ │ │ │  
│ │ │ │  o12 : Ideal of R
│ │ │ │  i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ │ - -- used 0.0312341s (cpu); 0.0312391s (thread); 0s (gc)
│ │ │ │ + -- used 0.0350773s (cpu); 0.0350798s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.193455s (cpu); 0.109457s (thread); 0s (gc)
│ │ │ │ + -- used 0.0687027s (cpu); 0.0687086s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : J1 == J2
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_f_l_e_x_i_f_y -- calculate the double dual of an ideal or module Hom(Hom(M,
│ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_map__Stratify.out
│ │ │ @@ -122,15 +122,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 2.05971s (cpu); 1.10912s (thread); 0s (gc)
│ │ │ + -- used 2.28699s (cpu); 1.3015s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o23 : List
│ │ │  
│ │ │  i24 : peek last ms
│ │ │  
│ │ │ @@ -142,15 +142,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 4.77507s (cpu); 2.67435s (thread); 0s (gc)
│ │ │ + -- used 7.38126s (cpu); 3.07829s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o25 : List
│ │ │  
│ │ │  i26 : peek last ms
│ │ │  
│ │ │ @@ -162,15 +162,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 5.618s (cpu); 3.0049s (thread); 0s (gc)
│ │ │ + -- used 8.3963s (cpu); 3.53223s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : peek last ms
│ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/html/_map__Stratify.html
│ │ │ @@ -297,15 +297,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 2.05971s (cpu); 1.10912s (thread); 0s (gc)
│ │ │ + -- used 2.28699s (cpu); 1.3015s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o23 : List
      │ │ │ │ │ │ │ │ │ │ │ │ @@ -323,15 +323,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 4.77507s (cpu); 2.67435s (thread); 0s (gc) │ │ │ + -- used 7.38126s (cpu); 3.07829s (thread); 0s (gc) │ │ │ │ │ │ o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o25 : List │ │ │ │ │ │ │ │ │ │ │ │ @@ -349,15 +349,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 5.618s (cpu); 3.0049s (thread); 0s (gc) │ │ │ + -- used 8.3963s (cpu); 3.53223s (thread); 0s (gc) │ │ │ │ │ │ o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o27 : List │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -184,15 +184,15 @@ │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ - -- used 2.05971s (cpu); 1.10912s (thread); 0s (gc) │ │ │ │ + -- used 2.28699s (cpu); 1.3015s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ │ │ o23 : List │ │ │ │ i24 : peek last ms │ │ │ │ │ │ │ │ o24 = MutableHashTable{0 => {ideal (P, M1)} } │ │ │ │ @@ -202,15 +202,15 @@ │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ - -- used 4.77507s (cpu); 2.67435s (thread); 0s (gc) │ │ │ │ + -- used 7.38126s (cpu); 3.07829s (thread); 0s (gc) │ │ │ │ │ │ │ │ o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ │ │ o25 : List │ │ │ │ i26 : peek last ms │ │ │ │ │ │ │ │ o26 = MutableHashTable{0 => {ideal (P, M1)} } │ │ │ │ @@ -220,15 +220,15 @@ │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ - -- used 5.618s (cpu); 3.0049s (thread); 0s (gc) │ │ │ │ + -- used 8.3963s (cpu); 3.53223s (thread); 0s (gc) │ │ │ │ │ │ │ │ o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ │ │ o27 : List │ │ │ │ i28 : peek last ms │ │ │ │ │ │ │ │ o28 = MutableHashTable{0 => {ideal (P, M1)} } │ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/example-output/___Installation_spand_sp__Configuration_spof_spgfan__Interface.out │ │ │ @@ -19,15 +19,15 @@ │ │ │ i4 : prefixDirectory | currentLayout#"programs" │ │ │ │ │ │ o4 = /usr/x86_64-Linux- │ │ │ Debian-forky/libexec/Macaulay2/bin/ │ │ │ │ │ │ i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true); │ │ │ -- warning: reloading gfanInterface; recreate instances of types from this package │ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16097-0/172 │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-20171-0/172 │ │ │ This is a program for computing all reduced Groebner bases of a polynomial ideal. It takes the ring and a generating set for the ideal as input. By default the enumeration is done by an almost memoryless reverse search. If the ideal is symmetric the symmetry option is useful and enumeration will be done up to symmetry using a breadth first search. The program needs a starting Groebner basis to do its computations. If the -g option is not specified it will compute one using Buchberger's algorithm. │ │ │ Options: │ │ │ -g: │ │ │ Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if it takes too much time to compute the starting (standard degree lexicographic) Groebner basis and the input is already a Groebner basis. │ │ │ │ │ │ --symmetry: │ │ │ Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup. │ │ │ @@ -38,16 +38,16 @@ │ │ │ --disableSymmetryTest: │ │ │ When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal. │ │ │ │ │ │ --parameters value: │ │ │ With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters. │ │ │ --interrupt value: │ │ │ Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans). │ │ │ -using temporary file /tmp/M2-16097-0/172 │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16097-0/174 │ │ │ +using temporary file /tmp/M2-20171-0/172 │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-20171-0/174 │ │ │ This program computes a reduced lexicographic Groebner basis of the polynomial ideal given as input. The default behavior is to use Buchberger's algorithm. The ordering of the variables is $a>b>c...$ (assuming that the ring is Q[a,b,c,...]). │ │ │ Options: │ │ │ -w: │ │ │ Compute a Groebner basis with respect to a degree lexicographic order with $a>b>c...$ instead. The degrees are given by a weight vector which is read from the input after the generating set has been read. │ │ │ │ │ │ -r: │ │ │ Use the reverse lexicographic order (or the reverse lexicographic order as a tie breaker if -w is used). The input must be homogeneous if the pure reverse lexicographic order is chosen. Ignored if -W is used. │ │ │ @@ -56,69 +56,69 @@ │ │ │ Do a Groebner walk. The input must be a minimal Groebner basis. If -W is used -w is ignored. │ │ │ │ │ │ -g: │ │ │ Do a generic Groebner walk. The input must be homogeneous and must be a minimal Groebner basis with respect to the reverse lexicographic term order. The target term order is always lexicographic. The -W option must be used. │ │ │ │ │ │ --parameters value: │ │ │ With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters. │ │ │ -using temporary file /tmp/M2-16097-0/174 │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16097-0/176 │ │ │ +using temporary file /tmp/M2-20171-0/174 │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-20171-0/176 │ │ │ This program takes a marked Groebner basis of an ideal I and a set of polynomials on its input and tests if the polynomial set is contained in I by applying the division algorithm for each element. The output is 1 for true and 0 for false. │ │ │ Options: │ │ │ --remainder: │ │ │ Tell the program to output the remainders of the divisions rather than outputting 0 or 1. │ │ │ --multiplier: │ │ │ Reads in a polynomial that will be multiplied to the polynomial to be divided before doing the division. │ │ │ -using temporary file /tmp/M2-16097-0/176 │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16097-0/178 │ │ │ +using temporary file /tmp/M2-20171-0/176 │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-20171-0/178 │ │ │ This program takes two polyhedral fans and computes their common refinement. │ │ │ Options: │ │ │ -i1 value: │ │ │ Specify the name of the first input file. │ │ │ -i2 value: │ │ │ Specify the name of the second input file. │ │ │ --stable: │ │ │ Compute the stable intersection. │ │ │ -using temporary file /tmp/M2-16097-0/178 │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16097-0/180 │ │ │ +using temporary file /tmp/M2-20171-0/178 │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-20171-0/180 │ │ │ This program takes a polyhedral fan and a vector and computes the link of the polyhedral fan around that vertex. The link will have lineality space dimension equal to the dimension of the relative open polyhedral cone of the original fan containing the vector. │ │ │ Options: │ │ │ -i value: │ │ │ Specify the name of the input file. │ │ │ --symmetry: │ │ │ Reads in a fan stored with symmetry. The generators of the symmetry group must be given on the standard input. │ │ │ │ │ │ --star: │ │ │ Computes the star instead. The star is defined as the smallest polyhedral fan containing all cones of the original fan containing the vector. │ │ │ -using temporary file /tmp/M2-16097-0/180 │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16097-0/182 │ │ │ +using temporary file /tmp/M2-20171-0/180 │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-20171-0/182 │ │ │ This program takes two polyhedral fans and computes their product. │ │ │ Options: │ │ │ -i1 value: │ │ │ Specify the name of the first input file. │ │ │ -i2 value: │ │ │ Specify the name of the second input file. │ │ │ -using temporary file /tmp/M2-16097-0/182 │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16097-0/184 │ │ │ +using temporary file /tmp/M2-20171-0/182 │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-20171-0/184 │ │ │ This program computes a Groebner cone. Three different cases are handled. The input may be a marked reduced Groebner basis in which case its Groebner cone is computed. The input may be just a marked minimal basis in which case the cone computed is not a Groebner cone in the usual sense but smaller. (These cones are described in [Fukuda, Jensen, Lauritzen, Thomas]). The third possible case is that the Groebner cone is possibly lower dimensional and given by a pair of Groebner bases as it is useful to do for tropical varieties, see option --pair. The facets of the cone can be read off in section FACETS and the equations in section IMPLIED_EQUATIONS. │ │ │ Options: │ │ │ --restrict: │ │ │ Add an inequality for each coordinate, so that the the cone is restricted to the non-negative orthant. │ │ │ --pair: │ │ │ The Groebner cone is given by a pair of compatible Groebner bases. The first basis is for the initial ideal and the second for the ideal itself. See the tropical section of the manual. │ │ │ --asfan: │ │ │ Writes the cone as a polyhedral fan with all its faces instead. In this way the extreme rays of the cone are also computed. │ │ │ --vectorinput: │ │ │ Compute a cone given list of inequalities rather than a Groebner cone. The input is an integer which specifies the dimension of the ambient space, a list of inequalities given as vectors and a list of equations. │ │ │ -using temporary file /tmp/M2-16097-0/184 │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16097-0/186 │ │ │ +using temporary file /tmp/M2-20171-0/184 │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-20171-0/186 │ │ │ This program computes the homogeneity space of a list of polynomials - as a cone. Thus generators for the homogeneity space are found in the section LINEALITY_SPACE. If you wish the homogeneity space of an ideal you should first compute a set of homogeneous generators and call the program on these. A reduced Groebner basis will always suffice for this purpose. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16097-0/186 │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16097-0/188 │ │ │ +using temporary file /tmp/M2-20171-0/186 │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-20171-0/188 │ │ │ This program homogenises a list of polynomials by introducing an extra variable. The name of the variable to be introduced is read from the input after the list of polynomials. Without the -w option the homogenisation is done with respect to total degree. │ │ │ Example: │ │ │ Input: │ │ │ Q[x,y]{y-1} │ │ │ z │ │ │ Output: │ │ │ Q[x,y,z]{y-z} │ │ │ @@ -126,30 +126,30 @@ │ │ │ -i: │ │ │ Treat input as an ideal. This will make the program compute the homogenisation of the input ideal. This is done by computing a degree Groebner basis and homogenising it. │ │ │ -w: │ │ │ Specify a homogenisation vector. The length of the vector must be the same as the number of variables in the ring. The vector is read from the input after the list of polynomials. │ │ │ │ │ │ -H: │ │ │ Let the name of the new variable be H rather than reading in a name from the input. │ │ │ -using temporary file /tmp/M2-16097-0/188 │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16097-0/190 │ │ │ +using temporary file /tmp/M2-20171-0/188 │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-20171-0/190 │ │ │ This program converts a list of polynomials to a list of their initial forms with respect to the vector given after the list. │ │ │ Options: │ │ │ --ideal: │ │ │ Treat input as an ideal. This will make the program compute the initial ideal of the ideal generated by the input polynomials. The computation is done by computing a Groebner basis with respect to the given vector. The vector must be positive or the input polynomials must be homogeneous in a positive grading. None of these conditions are checked by the program. │ │ │ │ │ │ --pair: │ │ │ Produce a pair of polynomial lists. Used together with --ideal this option will also write a compatible reduced Groebner basis for the input ideal to the output. This is useful for finding the Groebner cone of a non-monomial initial ideal. │ │ │ │ │ │ --mark: │ │ │ If the --pair option is and the --ideal option is not used this option will still make sure that the second output basis is marked consistently with the vector. │ │ │ --list: │ │ │ Read in a list of vectors instead of a single vector and produce a list of polynomial sets as output. │ │ │ -using temporary file /tmp/M2-16097-0/190 │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16097-0/192 │ │ │ +using temporary file /tmp/M2-20171-0/190 │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-20171-0/192 │ │ │ This is a program for doing interactive walks in the Groebner fan of an ideal. The input is a Groebner basis defining the starting Groebner cone of the walk. The program will list all flippable facets of the Groebner cone and ask the user to choose one. The user types in the index (number) of the facet in the list. The program will walk through the selected facet and display the new Groebner basis and a list of new facet normals for the user to choose from. Since the program reads the user's choices through the the standard input it is recommended not to redirect the standard input for this program. │ │ │ Options: │ │ │ -L: │ │ │ Latex mode. The program will try to show the current Groebner basis in a readable form by invoking LaTeX and xdvi. │ │ │ │ │ │ -x: │ │ │ Exit immediately. │ │ │ @@ -164,57 +164,57 @@ │ │ │ Tell the program to list the defining set of inequalities of the non-restricted Groebner cone as a set of vectors after having listed the current Groebner basis. │ │ │ │ │ │ -W: │ │ │ Print weight vector. This will make the program print an interior vector of the current Groebner cone and a relative interior point for each flippable facet of the current Groebner cone. │ │ │ │ │ │ --tropical: │ │ │ Traverse a tropical variety interactively. │ │ │ -using temporary file /tmp/M2-16097-0/192 │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16097-0/194 │ │ │ +using temporary file /tmp/M2-20171-0/192 │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-20171-0/194 │ │ │ This program checks if a set of marked polynomials is a Groebner basis with respect to its marking. First it is checked if the markings are consistent with respect to a positive vector. Then Buchberger's S-criterion is checked. The output is boolean value. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16097-0/194 │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16097-0/196 │ │ │ +using temporary file /tmp/M2-20171-0/194 │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-20171-0/196 │ │ │ Takes an ideal $I$ and computes the Krull dimension of R/I where R is the polynomial ring. This is done by first computing a Groebner basis. │ │ │ Options: │ │ │ -g: │ │ │ Tell the program that the input is already a reduced Groebner basis. │ │ │ -using temporary file /tmp/M2-16097-0/196 │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16097-0/198 │ │ │ +using temporary file /tmp/M2-20171-0/196 │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-20171-0/198 │ │ │ This program computes the lattice ideal of a lattice. The input is a list of generators for the lattice. │ │ │ Options: │ │ │ -t: │ │ │ Compute the toric ideal of the matrix whose rows are given on the input instead. │ │ │ --convert: │ │ │ Does not do any computation, but just converts the vectors to binomials. │ │ │ -using temporary file /tmp/M2-16097-0/198 │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16097-0/200 │ │ │ +using temporary file /tmp/M2-20171-0/198 │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-20171-0/200 │ │ │ This program converts a list of polynomials to a list of their leading terms. │ │ │ Options: │ │ │ -m: │ │ │ Do the same thing for a list of polynomial sets. That is, output the set of sets of leading terms. │ │ │ │ │ │ -using temporary file /tmp/M2-16097-0/200 │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16097-0/202 │ │ │ +using temporary file /tmp/M2-20171-0/200 │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-20171-0/202 │ │ │ This program marks a set of polynomials with respect to the vector given at the end of the input, meaning that the largest terms are moved to the front. In case of a tie the lexicographic term order with $a>b>c...$ is used to break it. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16097-0/202 │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16097-0/204 │ │ │ +using temporary file /tmp/M2-20171-0/202 │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-20171-0/204 │ │ │ This is a program for computing the normal fan of the Minkowski sum of the Newton polytopes of a list of polynomials. │ │ │ Options: │ │ │ --symmetry: │ │ │ Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup. │ │ │ │ │ │ --disableSymmetryTest: │ │ │ When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal. │ │ │ │ │ │ --nocones: │ │ │ Tell the program to not list cones in the output. │ │ │ -using temporary file /tmp/M2-16097-0/204 │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16097-0/206 │ │ │ +using temporary file /tmp/M2-20171-0/204 │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-20171-0/206 │ │ │ This program will generate the r*r minors of a d*n matrix of indeterminates. │ │ │ Options: │ │ │ -r value: │ │ │ Specify r. │ │ │ -d value: │ │ │ Specify d. │ │ │ -n value: │ │ │ @@ -229,16 +229,16 @@ │ │ │ Do nothing but produce symmetry generators for the Pluecker ideal. │ │ │ --symmetry: │ │ │ Produces a list of generators for the group of symmetries keeping the set of minors fixed. (Only without --names). │ │ │ --parametrize: │ │ │ Parametrize the set of d times n matrices of Barvinok rank less than or equal to r-1 by a list of tropical polynomials. │ │ │ --ultrametric: │ │ │ Produce tropical equations cutting out the ultrametrics. │ │ │ -using temporary file /tmp/M2-16097-0/206 │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16097-0/208 │ │ │ +using temporary file /tmp/M2-20171-0/206 │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-20171-0/208 │ │ │ This program computes the mixed volume of the Newton polytopes of a list of polynomials. The ring is specified on the input. After this follows the list of polynomials. │ │ │ Options: │ │ │ --vectorinput: │ │ │ Read in a list of point configurations instead of a polynomial ring and a list of polynomials. │ │ │ --cyclic value: │ │ │ Use cyclic-n example instead of reading input. │ │ │ --noon value: │ │ │ @@ -249,44 +249,44 @@ │ │ │ Use Katsura-n example instead of reading input. │ │ │ --gaukwa value: │ │ │ Use Gaukwa-n example instead of reading input. │ │ │ --eco value: │ │ │ Use Eco-n example instead of reading input. │ │ │ -j value: │ │ │ Number of threads │ │ │ -using temporary file /tmp/M2-16097-0/208 │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16097-0/210 │ │ │ +using temporary file /tmp/M2-20171-0/208 │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-20171-0/210 │ │ │ This program computes the union of a list of polynomial sets given as input. The polynomials must all belong to the same ring. The ring is specified on the input. After this follows the list of polynomial sets. │ │ │ Options: │ │ │ -s: │ │ │ Sort output by degree. │ │ │ │ │ │ -using temporary file /tmp/M2-16097-0/210 │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16097-0/212 │ │ │ +using temporary file /tmp/M2-20171-0/210 │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-20171-0/212 │ │ │ This program renders a Groebner fan as an xfig file. To be more precise, the input is the list of all reduced Groebner bases of an ideal. The output is a drawing of the Groebner fan intersected with a triangle. The corners of the triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top. If there are more than three variables in the ring these coordinates are extended with zeros. It is possible to shift the 1 entry cyclic with the option --shiftVariables. │ │ │ Options: │ │ │ -L: │ │ │ Make the triangle larger so that the shape of the Groebner region appears. │ │ │ --shiftVariables value: │ │ │ Shift the positions of the variables in the drawing. For example with the value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...) and top: (0,0,0,1,...). The shifting is done modulo the number of variables in the polynomial ring. The default value is 0. │ │ │ -using temporary file /tmp/M2-16097-0/212 │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16097-0/214 │ │ │ +using temporary file /tmp/M2-20171-0/212 │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-20171-0/214 │ │ │ This program renders a staircase diagram of a monomial initial ideal to an xfig file. The input is a Groebner basis of a (not necessarily monomial) polynomial ideal. The initial ideal is given by the leading terms in the Groebner basis. Using the -m option it is possible to render more than one staircase diagram. The program only works for ideals in a polynomial ring with three variables. │ │ │ Options: │ │ │ -m: │ │ │ Read multiple ideals from the input. The ideals are given as a list of lists of polynomials. For each polynomial list in the list a staircase diagram is drawn. │ │ │ │ │ │ -d value: │ │ │ Specifies the number of boxes being shown along each axis. Be sure that this number is large enough to give a correct picture of the standard monomials. The default value is 8. │ │ │ │ │ │ -w value: │ │ │ Width. Specifies the number of staircase diagrams per row in the xfig file. The default value is 5. │ │ │ │ │ │ -using temporary file /tmp/M2-16097-0/214 │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16097-0/216 │ │ │ +using temporary file /tmp/M2-20171-0/214 │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-20171-0/216 │ │ │ This program computes the resultant fan as defined in "Computing Tropical Resultants" by Jensen and Yu. The input is a polynomial ring followed by polynomials, whose coefficients are ignored. The output is the fan of coefficients such that the input system has a tropical solution. │ │ │ Options: │ │ │ --codimension: │ │ │ Compute only the codimension of the resultant fan and return. │ │ │ │ │ │ --symmetry: │ │ │ Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program DOES NOT checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry. │ │ │ @@ -299,25 +299,25 @@ │ │ │ │ │ │ --vectorinput: │ │ │ Read in a list of point configurations instead of a polynomial ring and a list of polynomials. │ │ │ │ │ │ --projection: │ │ │ Use the projection method to compute the resultant fan. This works only if the resultant fan is a hypersurface. If this option is combined with --special, then the output fan lives in the subspace of the non-specialized coordinates. │ │ │ │ │ │ -using temporary file /tmp/M2-16097-0/216 │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16097-0/218 │ │ │ +using temporary file /tmp/M2-20171-0/216 │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-20171-0/218 │ │ │ This program computes the saturation of the input ideal with the product of the variables x_1,...,x_n. The ideal does not have to be homogeneous. │ │ │ Options: │ │ │ -h: │ │ │ Tell the program that the input is a homogeneous ideal (with homogeneous generators). │ │ │ │ │ │ --noideal: │ │ │ Do not treat input as an ideal but just factor out common monomial factors of the input polynomials. │ │ │ -using temporary file /tmp/M2-16097-0/218 │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16097-0/220 │ │ │ +using temporary file /tmp/M2-20171-0/218 │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-20171-0/220 │ │ │ This program computes the secondary fan of a vector configuration. The configuration is given as an ordered list of vectors. In order to compute the secondary fan of a point configuration an additional coordinate of ones must be added. For example {(1,0),(1,1),(1,2),(1,3)}. │ │ │ Options: │ │ │ --unimodular: │ │ │ Use heuristics to search for unimodular triangulation rather than computing the complete secondary fan │ │ │ --scale value: │ │ │ Assuming that the first coordinate of each vector is 1, this option will take the polytope in the 1 plane and scale it. The point configuration will be all lattice points in that scaled polytope. The polytope must have maximal dimension. When this option is used the vector configuration must have full rank. This option may be removed in the future. │ │ │ --restrictingfan value: │ │ │ @@ -326,70 +326,70 @@ │ │ │ --symmetry: │ │ │ Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry. │ │ │ │ │ │ --nocones: │ │ │ Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used. │ │ │ --interrupt value: │ │ │ Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans). │ │ │ -using temporary file /tmp/M2-16097-0/220 │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16097-0/222 │ │ │ +using temporary file /tmp/M2-20171-0/220 │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-20171-0/222 │ │ │ This program takes a list of reduced Groebner bases for the same ideal and computes various statistics. The following information is listed: the number of bases in the input, the number of variables, the dimension of the homogeneity space, the maximal total degree of any polynomial in the input and the minimal total degree of any basis in the input, the maximal number of polynomials and terms in a basis in the input. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16097-0/222 │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16097-0/224 │ │ │ +using temporary file /tmp/M2-20171-0/222 │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-20171-0/224 │ │ │ This program changes the variable names of a polynomial ring. The input is a polynomial ring, a polynomial set in the ring and a new polynomial ring with the same coefficient field but different variable names. The output is the polynomial set written with the variable names of the second polynomial ring. │ │ │ Example: │ │ │ Input: │ │ │ Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x] │ │ │ Output: │ │ │ Q[b,a,c,x]{2*b-3*a,c+x} │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16097-0/224 │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16097-0/226 │ │ │ +using temporary file /tmp/M2-20171-0/224 │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-20171-0/226 │ │ │ This program converts ASCII math to TeX math. The data-type is specified by the options. │ │ │ Options: │ │ │ -h: │ │ │ Add a header to the output. Using this option the output will be LaTeXable right away. │ │ │ --polynomialset_: │ │ │ The data to be converted is a list of polynomials. │ │ │ --polynomialsetlist_: │ │ │ The data to be converted is a list of lists of polynomials. │ │ │ -using temporary file /tmp/M2-16097-0/226 │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16097-0/228 │ │ │ +using temporary file /tmp/M2-20171-0/226 │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-20171-0/228 │ │ │ This program takes a list of reduced Groebner bases and produces the fan of all faces of these. In this way by giving the complete list of reduced Groebner bases, the Groebner fan can be computed as a polyhedral complex. The option --restrict lets the user choose between computing the Groebner fan or the restricted Groebner fan. │ │ │ Options: │ │ │ --restrict: │ │ │ Add an inequality for each coordinate, so that the the cones are restricted to the non-negative orthant. │ │ │ --symmetry: │ │ │ Tell the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ring. The output is grouped according to these symmetries. Only one representative for each orbit is needed on the input. │ │ │ │ │ │ -using temporary file /tmp/M2-16097-0/228 │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16097-0/230 │ │ │ +using temporary file /tmp/M2-20171-0/228 │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-20171-0/230 │ │ │ This program computes a tropical basis for an ideal defining a tropical curve. Defining a tropical curve means that the Krull dimension of R/I is at most 1 + the dimension of the homogeneity space of I where R is the polynomial ring. The input is a generating set for the ideal. If the input is not homogeneous option -h must be used. │ │ │ Options: │ │ │ -h: │ │ │ Homogenise the input before computing a tropical basis and dehomogenise the output. This is needed if the input generators are not already homogeneous. │ │ │ -using temporary file /tmp/M2-16097-0/230 │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16097-0/232 │ │ │ +using temporary file /tmp/M2-20171-0/230 │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-20171-0/232 │ │ │ This program takes a marked reduced Groebner basis for a homogeneous ideal and computes the tropical variety of the ideal as a subfan of the Groebner fan. The program is slow but works for any homogeneous ideal. If you know that your ideal is prime over the complex numbers or you simply know that its tropical variety is pure and connected in codimension one then use gfan_tropicalstartingcone and gfan_tropicaltraverse instead. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16097-0/232 │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16097-0/234 │ │ │ +using temporary file /tmp/M2-20171-0/232 │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-20171-0/234 │ │ │ This program evaluates a tropical polynomial function in a given set of points. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16097-0/234 │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16097-0/236 │ │ │ +using temporary file /tmp/M2-20171-0/234 │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-20171-0/236 │ │ │ This program takes a polynomial and tropicalizes it. The output is piecewise linear function represented by a fan whose cones are the linear regions. Each ray of the fan gets the value of the tropical function assigned to it. In other words this program computes the normal fan of the Newton polytope of the input polynomial with additional information.Options: │ │ │ --exponents: │ │ │ Tell program to read a list of exponent vectors instead. │ │ │ -using temporary file /tmp/M2-16097-0/236 │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16097-0/238 │ │ │ +using temporary file /tmp/M2-20171-0/236 │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-20171-0/238 │ │ │ This program computes the tropical hypersurface defined by a principal ideal. The input is the polynomial ring followed by a set containing just a generator of the ideal.Options: │ │ │ -using temporary file /tmp/M2-16097-0/238 │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16097-0/240 │ │ │ +using temporary file /tmp/M2-20171-0/238 │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-20171-0/240 │ │ │ This program computes the set theoretical intersection of a set of tropical hypersurfaces (or to be precise, their common refinement as a fan). The input is a list of polynomials with each polynomial defining a hypersurface. Considering tropical hypersurfaces as fans, the intersection can be computed as the common refinement of these. Thus the output is a fan whose support is the intersection of the tropical hypersurfaces. │ │ │ Options: │ │ │ --tropicalbasistest: │ │ │ This option will test that the input polynomials for a tropical basis of the ideal they generate by computing the tropical prevariety of the input polynomials and then refine each cone with the Groebner fan and testing whether each cone in the refinement has an associated monomial free initial ideal. If so, then we have a tropical basis and 1 is written as output. If not, then a zero is written to the output together with a vector in the tropical prevariety but not in the variety. The actual check is done on a homogenization of the input ideal, but this does not affect the result. (This option replaces the -t option from earlier gfan versions.) │ │ │ │ │ │ --tplane: │ │ │ This option intersects the resulting fan with the plane x_0=-1, where x_0 is the first variable. To simplify the implementation the output is actually the common refinement with the non-negative half space. This means that "stuff at infinity" (where x_0=0) is not removed. │ │ │ @@ -401,16 +401,16 @@ │ │ │ Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used. │ │ │ --restrict: │ │ │ Restrict the computation to a full-dimensional cone given by a list of marked polynomials. The cone is the closure of all weight vectors choosing these marked terms. │ │ │ --stable: │ │ │ Find the stable intersection of the input polynomials using tropical intersection theory. This can be slow. Most other options are ignored. │ │ │ --parameters value: │ │ │ With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters. │ │ │ -using temporary file /tmp/M2-16097-0/240 │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16097-0/242 │ │ │ +using temporary file /tmp/M2-20171-0/240 │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-20171-0/242 │ │ │ This program is part of the Puiseux lifting algorithm implemented in Gfan and Singular. The Singular part of the implementation can be found in: │ │ │ │ │ │ Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig: │ │ │ tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, 2007 │ │ │ │ │ │ See also │ │ │ │ │ │ @@ -435,48 +435,48 @@ │ │ │ Options: │ │ │ --noMult: │ │ │ Disable the multiplicity computation. │ │ │ -n value: │ │ │ Number of variables that should have negative weight. │ │ │ -c: │ │ │ Only output a list of vectors being the possible choices. │ │ │ -using temporary file /tmp/M2-16097-0/242 │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16097-0/244 │ │ │ +using temporary file /tmp/M2-20171-0/242 │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-20171-0/244 │ │ │ This program generates tropical equations for a tropical linear space in the Speyer sense given the tropical Pluecker coordinates as input. │ │ │ Options: │ │ │ -d value: │ │ │ Specify d. │ │ │ -n value: │ │ │ Specify n. │ │ │ --trees: │ │ │ list the boundary trees (assumes d=3) │ │ │ -using temporary file /tmp/M2-16097-0/244 │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16097-0/246 │ │ │ +using temporary file /tmp/M2-20171-0/244 │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-20171-0/246 │ │ │ This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16097-0/246 │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16097-0/248 │ │ │ +using temporary file /tmp/M2-20171-0/246 │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-20171-0/248 │ │ │ This program will compute the tropical rank of matrix given as input. Tropical addition is MAXIMUM. │ │ │ Options: │ │ │ --kapranov: │ │ │ Compute Kapranov rank instead of tropical rank. │ │ │ --determinant: │ │ │ Compute the tropical determinant instead. │ │ │ -using temporary file /tmp/M2-16097-0/248 │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16097-0/250 │ │ │ +using temporary file /tmp/M2-20171-0/248 │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-20171-0/250 │ │ │ This program computes a starting pair of marked reduced Groebner bases to be used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose tropical variety is a pure d-dimensional polyhedral complex. │ │ │ Options: │ │ │ -g: │ │ │ Tell the program that the input is already a reduced Groebner basis. │ │ │ -d: │ │ │ Output dimension information to standard error. │ │ │ --stable: │ │ │ Find starting cone in the stable intersection or, equivalently, pretend that the coefficients are genereric. │ │ │ -using temporary file /tmp/M2-16097-0/250 │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16097-0/252 │ │ │ +using temporary file /tmp/M2-20171-0/250 │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-20171-0/252 │ │ │ This program computes a polyhedral fan representation of the tropical variety of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let $\omega$ be a relative interior point of $d$-dimensional Groebner cone contained in the tropical variety. The input for this program is a pair of marked reduced Groebner bases with respect to the term order represented by $\omega$, tie-broken in some way. The first one is for the initial ideal $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point for a traversal of the $d$-dimensional Groebner cones contained in the tropical variety. If the ideal is not prime but with the tropical variety still being pure $d$-dimensional the program will only compute a codimension $1$ connected component of the tropical variety. │ │ │ Options: │ │ │ --symmetry: │ │ │ Do computations up to symmetry and group the output accordingly. If this option is used the program will read in a list of generators for a symmetry group after the pair of Groebner bases have been read. Two advantages of using this option is that the output is nicely grouped and that the computation can be done faster. │ │ │ --symsigns: │ │ │ Specify for each generator of the symmetry group an element of ${-1,+1}^n$ which by its multiplication on the variables together with the permutation will keep the ideal fixed. The vectors are given as the rows of a matrix. │ │ │ --nocones: │ │ │ @@ -484,24 +484,24 @@ │ │ │ --disableSymmetryTest: │ │ │ When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal. │ │ │ │ │ │ --stable: │ │ │ Traverse the stable intersection or, equivalently, pretend that the coefficients are genereric. │ │ │ --interrupt value: │ │ │ Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans). │ │ │ -using temporary file /tmp/M2-16097-0/252 │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16097-0/254 │ │ │ +using temporary file /tmp/M2-20171-0/252 │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-20171-0/254 │ │ │ This program computes the tropical Weil divisor of piecewise linear (or tropical rational) function on a tropical k-cycle. See the Gfan manual for more information. │ │ │ Options: │ │ │ -i1 value: │ │ │ Specify the name of the Polymake input file containing the k-cycle. │ │ │ -i2 value: │ │ │ Specify the name of the Polymake input file containing the piecewise linear function. │ │ │ -using temporary file /tmp/M2-16097-0/254 │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16097-0/256 │ │ │ +using temporary file /tmp/M2-20171-0/254 │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-20171-0/256 │ │ │ This program is an experimental implementation of Groebner bases for ideals in Z[x_1,...,x_n]. │ │ │ Several operations are supported by specifying the appropriate option: │ │ │ (1) computation of the reduced Groebner basis with respect to a given vector (tiebroken lexicographically), │ │ │ (2) computation of an initial ideal, │ │ │ (3) computation of the Groebner fan, │ │ │ (4) computation of a single Groebner cone. │ │ │ Since Gfan only knows polynomial rings with coefficients being elements of a field, the ideal is specified by giving a set of polynomials in the polynomial ring Q[x_1,...,x_n]. That is, by using Q instead of Z when specifying the ring. The ideal MUST BE HOMOGENEOUS (in a positive grading) for computation of the Groebner fan. Non-homogeneous ideals are allowed for the other computations if the specified weight vectors are positive. │ │ │ @@ -521,21 +521,21 @@ │ │ │ --groebnerCone: │ │ │ Asks the program to compute a single Groebner cone containing the specified vector in its relative interior. The output is stored as a fan. The input order is: Ring ideal vector. │ │ │ -m: │ │ │ For the operations taking a vector as input, read in a list of vectors instead, and perform the operation for each vector in the list. │ │ │ -g: │ │ │ Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if the usual --groebnerFan is too slow. │ │ │ │ │ │ -using temporary file /tmp/M2-16097-0/256 │ │ │ +using temporary file /tmp/M2-20171-0/256 │ │ │ │ │ │ i6 : QQ[x,y]; │ │ │ │ │ │ i7 : gfan {x,y}; │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16097-0/258 │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-20171-0/258 │ │ │ Q[x1,x2] │ │ │ {{ │ │ │ x2, │ │ │ x1} │ │ │ } │ │ │ -using temporary file /tmp/M2-16097-0/258 │ │ │ +using temporary file /tmp/M2-20171-0/258 │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/html/___Installation_spand_sp__Configuration_spof_spgfan__Interface.html │ │ │ @@ -114,15 +114,15 @@ │ │ │

      │ │ │
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │ 
      i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true);
│ │ │   -- warning: reloading gfanInterface; recreate instances of types from this package
│ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16097-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-20171-0/172
│ │ │  This is a program for computing all reduced Groebner bases of a polynomial ideal. It takes the ring and a generating set for the ideal as input. By default the enumeration is done by an almost memoryless reverse search. If the ideal is symmetric the symmetry option is useful and enumeration will be done up to symmetry using a breadth first search. The program needs a starting Groebner basis to do its computations. If the -g option is not specified it will compute one using Buchberger's algorithm.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if it takes too much time to compute the starting (standard degree lexicographic) Groebner basis and the input is already a Groebner basis.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.
│ │ │ @@ -133,16 +133,16 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16097-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16097-0/174
│ │ │ +using temporary file /tmp/M2-20171-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-20171-0/174
│ │ │  This program computes a reduced lexicographic Groebner basis of the polynomial ideal given as input. The default behavior is to use Buchberger's algorithm. The ordering of the variables is $a>b>c...$ (assuming that the ring is Q[a,b,c,...]).
│ │ │  Options:
│ │ │  -w:
│ │ │   Compute a Groebner basis with respect to a degree lexicographic order with $a>b>c...$ instead. The degrees are given by a weight vector which is read from the input after the generating set has been read.
│ │ │  
│ │ │  -r:
│ │ │   Use the reverse lexicographic order (or the reverse lexicographic order as a tie breaker if -w is used). The input must be homogeneous if the pure reverse lexicographic order is chosen. Ignored if -W is used.
│ │ │ @@ -151,69 +151,69 @@
│ │ │   Do a Groebner walk. The input must be a minimal Groebner basis. If -W is used -w is ignored.
│ │ │  
│ │ │  -g:
│ │ │   Do a generic Groebner walk. The input must be homogeneous and must be a minimal Groebner basis with respect to the reverse lexicographic term order. The target term order is always lexicographic. The -W option must be used.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-16097-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16097-0/176
│ │ │ +using temporary file /tmp/M2-20171-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-20171-0/176
│ │ │  This program takes a marked Groebner basis of an ideal I and a set of polynomials on its input and tests if the polynomial set is contained in I by applying the division algorithm for each element. The output is 1 for true and 0 for false.
│ │ │  Options:
│ │ │  --remainder:
│ │ │   Tell the program to output the remainders of the divisions rather than outputting 0 or 1.
│ │ │  --multiplier:
│ │ │   Reads in a polynomial that will be multiplied to the polynomial to be divided before doing the division.
│ │ │ -using temporary file /tmp/M2-16097-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16097-0/178
│ │ │ +using temporary file /tmp/M2-20171-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-20171-0/178
│ │ │  This program takes two polyhedral fans and computes their common refinement.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │  --stable:
│ │ │   Compute the stable intersection.
│ │ │ -using temporary file /tmp/M2-16097-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16097-0/180
│ │ │ +using temporary file /tmp/M2-20171-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-20171-0/180
│ │ │  This program takes a polyhedral fan and a vector and computes the link of the polyhedral fan around that vertex. The link will have lineality space dimension equal to the dimension of the relative open polyhedral cone of the original fan containing the vector.
│ │ │  Options:
│ │ │  -i value:
│ │ │   Specify the name of the input file.
│ │ │  --symmetry:
│ │ │   Reads in a fan stored with symmetry. The generators of the symmetry group must be given on the standard input.
│ │ │  
│ │ │  --star:
│ │ │   Computes the star instead. The star is defined as the smallest polyhedral fan containing all cones of the original fan containing the vector.
│ │ │ -using temporary file /tmp/M2-16097-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16097-0/182
│ │ │ +using temporary file /tmp/M2-20171-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-20171-0/182
│ │ │  This program takes two polyhedral fans and computes their product.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │ -using temporary file /tmp/M2-16097-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16097-0/184
│ │ │ +using temporary file /tmp/M2-20171-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-20171-0/184
│ │ │  This program computes a Groebner cone. Three different cases are handled. The input may be a marked reduced Groebner basis in which case its Groebner cone is computed. The input may be just a marked minimal basis in which case the cone computed is not a Groebner cone in the usual sense but smaller. (These cones are described in [Fukuda, Jensen, Lauritzen, Thomas]). The third possible case is that the Groebner cone is possibly lower dimensional and given by a pair of Groebner bases as it is useful to do for tropical varieties, see option --pair. The facets of the cone can be read off in section FACETS and the equations in section IMPLIED_EQUATIONS.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cone is restricted to the non-negative orthant.
│ │ │  --pair:
│ │ │   The Groebner cone is given by a pair of compatible Groebner bases. The first basis is for the initial ideal and the second for the ideal itself. See the tropical section of the manual.
│ │ │  --asfan:
│ │ │   Writes the cone as a polyhedral fan with all its faces instead. In this way the extreme rays of the cone are also computed.
│ │ │  --vectorinput:
│ │ │   Compute a cone given list of inequalities rather than a Groebner cone. The input is an integer which specifies the dimension of the ambient space, a list of inequalities given as vectors and a list of equations.
│ │ │ -using temporary file /tmp/M2-16097-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16097-0/186
│ │ │ +using temporary file /tmp/M2-20171-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-20171-0/186
│ │ │  This program computes the homogeneity space of a list of polynomials - as a cone. Thus generators for the homogeneity space are found in the section LINEALITY_SPACE. If you wish the homogeneity space of an ideal you should first compute a set of homogeneous generators and call the program on these. A reduced Groebner basis will always suffice for this purpose.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16097-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16097-0/188
│ │ │ +using temporary file /tmp/M2-20171-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-20171-0/188
│ │ │  This program homogenises a list of polynomials by introducing an extra variable. The name of the variable to be introduced is read from the input after the list of polynomials. Without the -w option the homogenisation is done with respect to total degree.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[x,y]{y-1}
│ │ │  z
│ │ │  Output:
│ │ │  Q[x,y,z]{y-z}
│ │ │ @@ -221,30 +221,30 @@
│ │ │  -i:
│ │ │   Treat input as an ideal. This will make the program compute the homogenisation of the input ideal. This is done by computing a degree Groebner basis and homogenising it.
│ │ │  -w:
│ │ │   Specify a homogenisation vector. The length of the vector must be the same as the number of variables in the ring. The vector is read from the input after the list of polynomials.
│ │ │  
│ │ │  -H:
│ │ │   Let the name of the new variable be H rather than reading in a name from the input.
│ │ │ -using temporary file /tmp/M2-16097-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16097-0/190
│ │ │ +using temporary file /tmp/M2-20171-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-20171-0/190
│ │ │  This program converts a list of polynomials to a list of their initial forms with respect to the vector given after the list.
│ │ │  Options:
│ │ │  --ideal:
│ │ │   Treat input as an ideal. This will make the program compute the initial ideal of the ideal generated by the input polynomials. The computation is done by computing a Groebner basis with respect to the given vector. The vector must be positive or the input polynomials must be homogeneous in a positive grading. None of these conditions are checked by the program.
│ │ │  
│ │ │  --pair:
│ │ │   Produce a pair of polynomial lists. Used together with --ideal this option will also write a compatible reduced Groebner basis for the input ideal to the output. This is useful for finding the Groebner cone of a non-monomial initial ideal.
│ │ │  
│ │ │  --mark:
│ │ │   If the --pair option is and the --ideal option is not used this option will still make sure that the second output basis is marked consistently with the vector.
│ │ │  --list:
│ │ │   Read in a list of vectors instead of a single vector and produce a list of polynomial sets as output.
│ │ │ -using temporary file /tmp/M2-16097-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16097-0/192
│ │ │ +using temporary file /tmp/M2-20171-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-20171-0/192
│ │ │  This is a program for doing interactive walks in the Groebner fan of an ideal. The input is a Groebner basis defining the starting Groebner cone of the walk. The program will list all flippable facets of the Groebner cone and ask the user to choose one. The user types in the index (number) of the facet in the list. The program will walk through the selected facet and display the new Groebner basis and a list of new facet normals for the user to choose from. Since the program reads the user's choices through the the standard input it is recommended not to redirect the standard input for this program.
│ │ │  Options:
│ │ │  -L:
│ │ │   Latex mode. The program will try to show the current Groebner basis in a readable form by invoking LaTeX and xdvi.
│ │ │  
│ │ │  -x:
│ │ │   Exit immediately.
│ │ │ @@ -259,57 +259,57 @@
│ │ │   Tell the program to list the defining set of inequalities of the non-restricted Groebner cone as a set of vectors after having listed the current Groebner basis.
│ │ │  
│ │ │  -W:
│ │ │   Print weight vector. This will make the program print an interior vector of the current Groebner cone and a relative interior point for each flippable facet of the current Groebner cone.
│ │ │  
│ │ │  --tropical:
│ │ │   Traverse a tropical variety interactively.
│ │ │ -using temporary file /tmp/M2-16097-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16097-0/194
│ │ │ +using temporary file /tmp/M2-20171-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-20171-0/194
│ │ │  This program checks if a set of marked polynomials is a Groebner basis with respect to its marking. First it is checked if the markings are consistent with respect to a positive vector. Then Buchberger's S-criterion is checked. The output is boolean value.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16097-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16097-0/196
│ │ │ +using temporary file /tmp/M2-20171-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-20171-0/196
│ │ │  Takes an ideal $I$ and computes the Krull dimension of R/I where R is the polynomial ring. This is done by first computing a Groebner basis.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │ -using temporary file /tmp/M2-16097-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16097-0/198
│ │ │ +using temporary file /tmp/M2-20171-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-20171-0/198
│ │ │  This program computes the lattice ideal of a lattice. The input is a list of generators for the lattice.
│ │ │  Options:
│ │ │  -t:
│ │ │   Compute the toric ideal of the matrix whose rows are given on the input instead.
│ │ │  --convert:
│ │ │   Does not do any computation, but just converts the vectors to binomials.
│ │ │ -using temporary file /tmp/M2-16097-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16097-0/200
│ │ │ +using temporary file /tmp/M2-20171-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-20171-0/200
│ │ │  This program converts a list of polynomials to a list of their leading terms.
│ │ │  Options:
│ │ │  -m:
│ │ │   Do the same thing for a list of polynomial sets. That is, output the set of sets of leading terms.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16097-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16097-0/202
│ │ │ +using temporary file /tmp/M2-20171-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-20171-0/202
│ │ │  This program marks a set of polynomials with respect to the vector given at the end of the input, meaning that the largest terms are moved to the front. In case of a tie the lexicographic term order with $a>b>c...$ is used to break it.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16097-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16097-0/204
│ │ │ +using temporary file /tmp/M2-20171-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-20171-0/204
│ │ │  This is a program for computing the normal fan of the Minkowski sum of the Newton polytopes of a list of polynomials.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.
│ │ │  
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --nocones:
│ │ │   Tell the program to not list cones in the output.
│ │ │ -using temporary file /tmp/M2-16097-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16097-0/206
│ │ │ +using temporary file /tmp/M2-20171-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-20171-0/206
│ │ │  This program will generate the r*r minors of a d*n matrix of indeterminates.
│ │ │  Options:
│ │ │  -r value:
│ │ │   Specify r.
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │ @@ -324,16 +324,16 @@
│ │ │   Do nothing but produce symmetry generators for the Pluecker ideal.
│ │ │  --symmetry:
│ │ │   Produces a list of generators for the group of symmetries keeping the set of minors fixed. (Only without --names).
│ │ │  --parametrize:
│ │ │   Parametrize the set of d times n matrices of Barvinok rank less than or equal to r-1 by a list of tropical polynomials.
│ │ │  --ultrametric:
│ │ │   Produce tropical equations cutting out the ultrametrics.
│ │ │ -using temporary file /tmp/M2-16097-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16097-0/208
│ │ │ +using temporary file /tmp/M2-20171-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-20171-0/208
│ │ │  This program computes the mixed volume of the Newton polytopes of a list of polynomials. The ring is specified on the input. After this follows the list of polynomials.
│ │ │  Options:
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  --cyclic value:
│ │ │   Use cyclic-n example instead of reading input.
│ │ │  --noon value:
│ │ │ @@ -344,44 +344,44 @@
│ │ │   Use Katsura-n example instead of reading input.
│ │ │  --gaukwa value:
│ │ │   Use Gaukwa-n example instead of reading input.
│ │ │  --eco value:
│ │ │   Use Eco-n example instead of reading input.
│ │ │  -j value:
│ │ │   Number of threads
│ │ │ -using temporary file /tmp/M2-16097-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16097-0/210
│ │ │ +using temporary file /tmp/M2-20171-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-20171-0/210
│ │ │  This program computes the union of a list of polynomial sets given as input. The polynomials must all belong to the same ring. The ring is specified on the input. After this follows the list of polynomial sets.
│ │ │  Options:
│ │ │  -s:
│ │ │   Sort output by degree.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16097-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16097-0/212
│ │ │ +using temporary file /tmp/M2-20171-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-20171-0/212
│ │ │  This program renders a Groebner fan as an xfig file. To be more precise, the input is the list of all reduced Groebner bases of an ideal. The output is a drawing of the Groebner fan intersected with a triangle. The corners of the triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top. If there are more than three variables in the ring these coordinates are extended with zeros. It is possible to shift the 1 entry cyclic with the option --shiftVariables.
│ │ │  Options:
│ │ │  -L:
│ │ │   Make the triangle larger so that the shape of the Groebner region appears.
│ │ │  --shiftVariables value:
│ │ │   Shift the positions of the variables in the drawing. For example with the value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...) and top: (0,0,0,1,...). The shifting is done modulo the number of variables in the polynomial ring. The default value is 0.
│ │ │ -using temporary file /tmp/M2-16097-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16097-0/214
│ │ │ +using temporary file /tmp/M2-20171-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-20171-0/214
│ │ │  This program renders a staircase diagram of a monomial initial ideal to an xfig file. The input is a Groebner basis of a (not necessarily monomial) polynomial ideal. The initial ideal is given by the leading terms in the Groebner basis. Using the -m option it is possible to render more than one staircase diagram. The program only works for ideals in a polynomial ring with three variables.
│ │ │  Options:
│ │ │  -m:
│ │ │   Read multiple ideals from the input. The ideals are given as a list of lists of polynomials. For each polynomial list in the list a staircase diagram is drawn.
│ │ │  
│ │ │  -d value:
│ │ │   Specifies the number of boxes being shown along each axis. Be sure that this number is large enough to give a correct picture of the standard monomials. The default value is 8.
│ │ │  
│ │ │  -w value:
│ │ │   Width. Specifies the number of staircase diagrams per row in the xfig file. The default value is 5.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16097-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16097-0/216
│ │ │ +using temporary file /tmp/M2-20171-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-20171-0/216
│ │ │  This program computes the resultant fan as defined in "Computing Tropical Resultants" by Jensen and Yu. The input is a polynomial ring followed by polynomials, whose coefficients are ignored. The output is the fan of coefficients such that the input system has a tropical solution.
│ │ │  Options:
│ │ │  --codimension:
│ │ │   Compute only the codimension of the resultant fan and return.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program DOES NOT checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │ @@ -394,25 +394,25 @@
│ │ │  
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  
│ │ │  --projection:
│ │ │   Use the projection method to compute the resultant fan. This works only if the resultant fan is a hypersurface. If this option is combined with --special, then the output fan lives in the subspace of the non-specialized coordinates.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16097-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16097-0/218
│ │ │ +using temporary file /tmp/M2-20171-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-20171-0/218
│ │ │  This program computes the saturation of the input ideal with the product of the variables x_1,...,x_n. The ideal does not have to be homogeneous.
│ │ │  Options:
│ │ │  -h:
│ │ │   Tell the program that the input is a homogeneous ideal (with homogeneous generators).
│ │ │  
│ │ │  --noideal:
│ │ │   Do not treat input as an ideal but just factor out common monomial factors of the input polynomials.
│ │ │ -using temporary file /tmp/M2-16097-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16097-0/220
│ │ │ +using temporary file /tmp/M2-20171-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-20171-0/220
│ │ │  This program computes the secondary fan of a vector configuration. The configuration is given as an ordered list of vectors. In order to compute the secondary fan of a point configuration an additional coordinate of ones must be added. For example {(1,0),(1,1),(1,2),(1,3)}.
│ │ │  Options:
│ │ │  --unimodular:
│ │ │   Use heuristics to search for unimodular triangulation rather than computing the complete secondary fan
│ │ │  --scale value:
│ │ │   Assuming that the first coordinate of each vector is 1, this option will take the polytope in the 1 plane and scale it. The point configuration will be all lattice points in that scaled polytope. The polytope must have maximal dimension. When this option is used the vector configuration must have full rank. This option may be removed in the future.
│ │ │  --restrictingfan value:
│ │ │ @@ -421,70 +421,70 @@
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │  
│ │ │  --nocones:
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16097-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16097-0/222
│ │ │ +using temporary file /tmp/M2-20171-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-20171-0/222
│ │ │  This program takes a list of reduced Groebner bases for the same ideal and computes various statistics. The following information is listed: the number of bases in the input, the number of variables, the dimension of the homogeneity space, the maximal total degree of any polynomial in the input and the minimal total degree of any basis in the input, the maximal number of polynomials and terms in a basis in the input.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16097-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16097-0/224
│ │ │ +using temporary file /tmp/M2-20171-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-20171-0/224
│ │ │  This program changes the variable names of a polynomial ring. The input is a polynomial ring, a polynomial set in the ring and a new polynomial ring with the same coefficient field but different variable names. The output is the polynomial set written with the variable names of the second polynomial ring.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x]
│ │ │  Output:
│ │ │  Q[b,a,c,x]{2*b-3*a,c+x}
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16097-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16097-0/226
│ │ │ +using temporary file /tmp/M2-20171-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-20171-0/226
│ │ │  This program converts ASCII math to TeX math. The data-type is specified by the options.
│ │ │  Options:
│ │ │  -h:
│ │ │   Add a header to the output. Using this option the output will be LaTeXable right away.
│ │ │  --polynomialset_:
│ │ │   The data to be converted is a list of polynomials.
│ │ │  --polynomialsetlist_:
│ │ │   The data to be converted is a list of lists of polynomials.
│ │ │ -using temporary file /tmp/M2-16097-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16097-0/228
│ │ │ +using temporary file /tmp/M2-20171-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-20171-0/228
│ │ │  This program takes a list of reduced Groebner bases and produces the fan of all faces of these. In this way by giving the complete list of reduced Groebner bases, the Groebner fan can be computed as a polyhedral complex. The option --restrict lets the user choose between computing the Groebner fan or the restricted Groebner fan.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cones are restricted to the non-negative orthant.
│ │ │  --symmetry:
│ │ │   Tell the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ring. The output is grouped according to these symmetries. Only one representative for each orbit is needed on the input.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16097-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16097-0/230
│ │ │ +using temporary file /tmp/M2-20171-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-20171-0/230
│ │ │  This program computes a tropical basis for an ideal defining a tropical curve. Defining a tropical curve means that the Krull dimension of R/I is at most 1 + the dimension of the homogeneity space of I where R is the polynomial ring. The input is a generating set for the ideal. If the input is not homogeneous option -h must be used.
│ │ │  Options:
│ │ │  -h:
│ │ │   Homogenise the input before computing a tropical basis and dehomogenise the output. This is needed if the input generators are not already homogeneous.
│ │ │ -using temporary file /tmp/M2-16097-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16097-0/232
│ │ │ +using temporary file /tmp/M2-20171-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-20171-0/232
│ │ │  This program takes a marked reduced Groebner basis for a homogeneous ideal and computes the tropical variety of the ideal as a subfan of the Groebner fan. The program is slow but works for any homogeneous ideal. If you know that your ideal is prime over the complex numbers or you simply know that its tropical variety is pure and connected in codimension one then use gfan_tropicalstartingcone and gfan_tropicaltraverse instead.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16097-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16097-0/234
│ │ │ +using temporary file /tmp/M2-20171-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-20171-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16097-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16097-0/236
│ │ │ +using temporary file /tmp/M2-20171-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-20171-0/236
│ │ │  This program takes a polynomial and tropicalizes it. The output is piecewise linear function represented by a fan whose cones are the linear regions. Each ray of the fan gets the value of the tropical function assigned to it. In other words this program computes the normal fan of the Newton polytope of the input polynomial with additional information.Options:
│ │ │  --exponents:
│ │ │   Tell program to read a list of exponent vectors instead.
│ │ │ -using temporary file /tmp/M2-16097-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16097-0/238
│ │ │ +using temporary file /tmp/M2-20171-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-20171-0/238
│ │ │  This program computes the tropical hypersurface defined by a principal ideal. The input is the polynomial ring followed by a set containing just a generator of the ideal.Options:
│ │ │ -using temporary file /tmp/M2-16097-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16097-0/240
│ │ │ +using temporary file /tmp/M2-20171-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-20171-0/240
│ │ │  This program computes the set theoretical intersection of a set of tropical hypersurfaces (or to be precise, their common refinement as a fan). The input is a list of polynomials with each polynomial defining a hypersurface. Considering tropical hypersurfaces as fans, the intersection can be computed as the common refinement of these. Thus the output is a fan whose support is the intersection of the tropical hypersurfaces.
│ │ │  Options:
│ │ │  --tropicalbasistest:
│ │ │   This option will test that the input polynomials for a tropical basis of the ideal they generate by computing the tropical prevariety of the input polynomials and then refine each cone with the Groebner fan and testing whether each cone in the refinement has an associated monomial free initial ideal. If so, then we have a tropical basis and 1 is written as output. If not, then a zero is written to the output together with a vector in the tropical prevariety but not in the variety. The actual check is done on a homogenization of the input ideal, but this does not affect the result. (This option replaces the -t option from earlier gfan versions.)
│ │ │  
│ │ │  --tplane:
│ │ │   This option intersects the resulting fan with the plane x_0=-1, where x_0 is the first variable. To simplify the implementation the output is actually the common refinement with the non-negative half space. This means that "stuff at infinity" (where x_0=0) is not removed.
│ │ │ @@ -496,16 +496,16 @@
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --restrict:
│ │ │   Restrict the computation to a full-dimensional cone given by a list of marked polynomials. The cone is the closure of all weight vectors choosing these marked terms.
│ │ │  --stable:
│ │ │   Find the stable intersection of the input polynomials using tropical intersection theory. This can be slow. Most other options are ignored.
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-16097-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16097-0/242
│ │ │ +using temporary file /tmp/M2-20171-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-20171-0/242
│ │ │  This program is part of the Puiseux lifting algorithm implemented in Gfan and Singular. The Singular part of the implementation can be found in:
│ │ │  
│ │ │  Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig:
│ │ │   tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, 2007 
│ │ │  
│ │ │  See also
│ │ │  
│ │ │ @@ -530,48 +530,48 @@
│ │ │  Options:
│ │ │  --noMult:
│ │ │   Disable the multiplicity computation.
│ │ │  -n value:
│ │ │   Number of variables that should have negative weight.
│ │ │  -c:
│ │ │   Only output a list of vectors being the possible choices.
│ │ │ -using temporary file /tmp/M2-16097-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16097-0/244
│ │ │ +using temporary file /tmp/M2-20171-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-20171-0/244
│ │ │  This program generates tropical equations for a tropical linear space in the Speyer sense given the tropical Pluecker coordinates as input.
│ │ │  Options:
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │   Specify n.
│ │ │  --trees:
│ │ │   list the boundary trees (assumes d=3)
│ │ │ -using temporary file /tmp/M2-16097-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16097-0/246
│ │ │ +using temporary file /tmp/M2-20171-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-20171-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16097-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16097-0/248
│ │ │ +using temporary file /tmp/M2-20171-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-20171-0/248
│ │ │  This program will compute the tropical rank of matrix given as input. Tropical addition is MAXIMUM.
│ │ │  Options:
│ │ │  --kapranov:
│ │ │   Compute Kapranov rank instead of tropical rank.
│ │ │  --determinant:
│ │ │   Compute the tropical determinant instead.
│ │ │ -using temporary file /tmp/M2-16097-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16097-0/250
│ │ │ +using temporary file /tmp/M2-20171-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-20171-0/250
│ │ │  This program computes a starting pair of marked reduced Groebner bases to be used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose tropical variety is a pure d-dimensional polyhedral complex.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │  -d:
│ │ │   Output dimension information to standard error.
│ │ │  --stable:
│ │ │   Find starting cone in the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │ -using temporary file /tmp/M2-16097-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16097-0/252
│ │ │ +using temporary file /tmp/M2-20171-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-20171-0/252
│ │ │  This program computes a polyhedral fan representation of the tropical variety of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let $\omega$ be a relative interior point of $d$-dimensional Groebner cone contained in the tropical variety. The input for this program is a pair of marked reduced Groebner bases with respect to the term order represented by $\omega$, tie-broken in some way. The first one is for the initial ideal $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point for a traversal of the $d$-dimensional Groebner cones contained in the tropical variety. If the ideal is not prime but with the tropical variety still being pure $d$-dimensional the program will only compute a codimension $1$ connected component of the tropical variety.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Do computations up to symmetry and group the output accordingly. If this option is used the program will read in a list of generators for a symmetry group after the pair of Groebner bases have been read. Two advantages of using this option is that the output is nicely grouped and that the computation can be done faster.
│ │ │  --symsigns:
│ │ │   Specify for each generator of the symmetry group an element of ${-1,+1}^n$ which by its multiplication on the variables together with the permutation will keep the ideal fixed. The vectors are given as the rows of a matrix.
│ │ │  --nocones:
│ │ │ @@ -579,24 +579,24 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --stable:
│ │ │   Traverse the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16097-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16097-0/254
│ │ │ +using temporary file /tmp/M2-20171-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-20171-0/254
│ │ │  This program computes the tropical Weil divisor of piecewise linear (or tropical rational) function on a tropical k-cycle. See the Gfan manual for more information.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the Polymake input file containing the k-cycle.
│ │ │  -i2 value:
│ │ │   Specify the name of the Polymake input file containing the piecewise linear function.
│ │ │ -using temporary file /tmp/M2-16097-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16097-0/256
│ │ │ +using temporary file /tmp/M2-20171-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-20171-0/256
│ │ │  This program is an experimental implementation of Groebner bases for ideals in Z[x_1,...,x_n].
│ │ │  Several operations are supported by specifying the appropriate option:
│ │ │   (1) computation of the reduced Groebner basis with respect to a given vector (tiebroken lexicographically),
│ │ │   (2) computation of an initial ideal,
│ │ │   (3) computation of the Groebner fan,
│ │ │   (4) computation of a single Groebner cone.
│ │ │  Since Gfan only knows polynomial rings with coefficients being elements of a field, the ideal is specified by giving a set of polynomials in the polynomial ring Q[x_1,...,x_n]. That is, by using Q instead of Z when specifying the ring. The ideal MUST BE HOMOGENEOUS (in a positive grading) for computation of the Groebner fan. Non-homogeneous ideals are allowed for the other computations if the specified weight vectors are positive.
│ │ │ @@ -616,32 +616,32 @@
│ │ │  --groebnerCone:
│ │ │   Asks the program to compute a single Groebner cone containing the specified vector in its relative interior. The output is stored as a fan. The input order is: Ring ideal vector.
│ │ │  -m:
│ │ │   For the operations taking a vector as input, read in a list of vectors instead, and perform the operation for each vector in the list.
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if the usual --groebnerFan is too slow.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16097-0/256
      │ │ │ +using temporary file /tmp/M2-20171-0/256 │ │ │
      │ │ │ 
      i6 : QQ[x,y];
      │ │ │
      │ │ │ 
      i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16097-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-20171-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-16097-0/258
      │ │ │ +using temporary file /tmp/M2-20171-0/258 │ │ │
      │ │ │
      │ │ │

      Finally, if you want to be able to render Groebner fans and monomial staircases to .png files, you should install fig2dev. If it is installed in a non-standard location, then you may specify its path using programPaths.

      │ │ │
      │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -43,15 +43,15 @@ │ │ │ │ If you would like to see the input and output files used to communicate with │ │ │ │ gfan you can set the "keepfiles" configuration option to true. If "verbose" is │ │ │ │ set to true, gfanInterface will output the names of the temporary files used. │ │ │ │ i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, │ │ │ │ "verbose" => true}, Reload => true); │ │ │ │ -- warning: reloading gfanInterface; recreate instances of types from this │ │ │ │ package │ │ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16097-0/172 │ │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-20171-0/172 │ │ │ │ This is a program for computing all reduced Groebner bases of a polynomial │ │ │ │ ideal. It takes the ring and a generating set for the ideal as input. By │ │ │ │ default the enumeration is done by an almost memoryless reverse search. If the │ │ │ │ ideal is symmetric the symmetry option is useful and enumeration will be done │ │ │ │ up to symmetry using a breadth first search. The program needs a starting │ │ │ │ Groebner basis to do its computations. If the -g option is not specified it │ │ │ │ will compute one using Buchberger's algorithm. │ │ │ │ @@ -81,16 +81,16 @@ │ │ │ │ With this option you can specify how many variables to treat as parameters │ │ │ │ instead of variables. This makes it possible to do computations where the │ │ │ │ coefficient field is the field of rational functions in the parameters. │ │ │ │ --interrupt value: │ │ │ │ Interrupt the enumeration after a specified number of facets have been │ │ │ │ computed (works for usual symmetric traversals, but may not work in general for │ │ │ │ non-symmetric traversals or for traversals restricted to fans). │ │ │ │ -using temporary file /tmp/M2-16097-0/172 │ │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16097-0/174 │ │ │ │ +using temporary file /tmp/M2-20171-0/172 │ │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-20171-0/174 │ │ │ │ This program computes a reduced lexicographic Groebner basis of the polynomial │ │ │ │ ideal given as input. The default behavior is to use Buchberger's algorithm. │ │ │ │ The ordering of the variables is $a>b>c...$ (assuming that the ring is Q │ │ │ │ [a,b,c,...]). │ │ │ │ Options: │ │ │ │ -w: │ │ │ │ Compute a Groebner basis with respect to a degree lexicographic order with │ │ │ │ @@ -111,63 +111,63 @@ │ │ │ │ minimal Groebner basis with respect to the reverse lexicographic term order. │ │ │ │ The target term order is always lexicographic. The -W option must be used. │ │ │ │ │ │ │ │ --parameters value: │ │ │ │ With this option you can specify how many variables to treat as parameters │ │ │ │ instead of variables. This makes it possible to do computations where the │ │ │ │ coefficient field is the field of rational functions in the parameters. │ │ │ │ -using temporary file /tmp/M2-16097-0/174 │ │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16097-0/176 │ │ │ │ +using temporary file /tmp/M2-20171-0/174 │ │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-20171-0/176 │ │ │ │ This program takes a marked Groebner basis of an ideal I and a set of │ │ │ │ polynomials on its input and tests if the polynomial set is contained in I by │ │ │ │ applying the division algorithm for each element. The output is 1 for true and │ │ │ │ 0 for false. │ │ │ │ Options: │ │ │ │ --remainder: │ │ │ │ Tell the program to output the remainders of the divisions rather than │ │ │ │ outputting 0 or 1. │ │ │ │ --multiplier: │ │ │ │ Reads in a polynomial that will be multiplied to the polynomial to be divided │ │ │ │ before doing the division. │ │ │ │ -using temporary file /tmp/M2-16097-0/176 │ │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16097-0/178 │ │ │ │ +using temporary file /tmp/M2-20171-0/176 │ │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-20171-0/178 │ │ │ │ This program takes two polyhedral fans and computes their common refinement. │ │ │ │ Options: │ │ │ │ -i1 value: │ │ │ │ Specify the name of the first input file. │ │ │ │ -i2 value: │ │ │ │ Specify the name of the second input file. │ │ │ │ --stable: │ │ │ │ Compute the stable intersection. │ │ │ │ -using temporary file /tmp/M2-16097-0/178 │ │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16097-0/180 │ │ │ │ +using temporary file /tmp/M2-20171-0/178 │ │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-20171-0/180 │ │ │ │ This program takes a polyhedral fan and a vector and computes the link of the │ │ │ │ polyhedral fan around that vertex. The link will have lineality space dimension │ │ │ │ equal to the dimension of the relative open polyhedral cone of the original fan │ │ │ │ containing the vector. │ │ │ │ Options: │ │ │ │ -i value: │ │ │ │ Specify the name of the input file. │ │ │ │ --symmetry: │ │ │ │ Reads in a fan stored with symmetry. The generators of the symmetry group must │ │ │ │ be given on the standard input. │ │ │ │ │ │ │ │ --star: │ │ │ │ Computes the star instead. The star is defined as the smallest polyhedral fan │ │ │ │ containing all cones of the original fan containing the vector. │ │ │ │ -using temporary file /tmp/M2-16097-0/180 │ │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16097-0/182 │ │ │ │ +using temporary file /tmp/M2-20171-0/180 │ │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-20171-0/182 │ │ │ │ This program takes two polyhedral fans and computes their product. │ │ │ │ Options: │ │ │ │ -i1 value: │ │ │ │ Specify the name of the first input file. │ │ │ │ -i2 value: │ │ │ │ Specify the name of the second input file. │ │ │ │ -using temporary file /tmp/M2-16097-0/182 │ │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16097-0/184 │ │ │ │ +using temporary file /tmp/M2-20171-0/182 │ │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-20171-0/184 │ │ │ │ This program computes a Groebner cone. Three different cases are handled. The │ │ │ │ input may be a marked reduced Groebner basis in which case its Groebner cone is │ │ │ │ computed. The input may be just a marked minimal basis in which case the cone │ │ │ │ computed is not a Groebner cone in the usual sense but smaller. (These cones │ │ │ │ are described in [Fukuda, Jensen, Lauritzen, Thomas]). The third possible case │ │ │ │ is that the Groebner cone is possibly lower dimensional and given by a pair of │ │ │ │ Groebner bases as it is useful to do for tropical varieties, see option --pair. │ │ │ │ @@ -184,24 +184,24 @@ │ │ │ │ --asfan: │ │ │ │ Writes the cone as a polyhedral fan with all its faces instead. In this way │ │ │ │ the extreme rays of the cone are also computed. │ │ │ │ --vectorinput: │ │ │ │ Compute a cone given list of inequalities rather than a Groebner cone. The │ │ │ │ input is an integer which specifies the dimension of the ambient space, a list │ │ │ │ of inequalities given as vectors and a list of equations. │ │ │ │ -using temporary file /tmp/M2-16097-0/184 │ │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16097-0/186 │ │ │ │ +using temporary file /tmp/M2-20171-0/184 │ │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-20171-0/186 │ │ │ │ This program computes the homogeneity space of a list of polynomials - as a │ │ │ │ cone. Thus generators for the homogeneity space are found in the section │ │ │ │ LINEALITY_SPACE. If you wish the homogeneity space of an ideal you should first │ │ │ │ compute a set of homogeneous generators and call the program on these. A │ │ │ │ reduced Groebner basis will always suffice for this purpose. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16097-0/186 │ │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16097-0/188 │ │ │ │ +using temporary file /tmp/M2-20171-0/186 │ │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-20171-0/188 │ │ │ │ This program homogenises a list of polynomials by introducing an extra │ │ │ │ variable. The name of the variable to be introduced is read from the input │ │ │ │ after the list of polynomials. Without the -w option the homogenisation is done │ │ │ │ with respect to total degree. │ │ │ │ Example: │ │ │ │ Input: │ │ │ │ Q[x,y]{y-1} │ │ │ │ @@ -217,16 +217,16 @@ │ │ │ │ Specify a homogenisation vector. The length of the vector must be the same as │ │ │ │ the number of variables in the ring. The vector is read from the input after │ │ │ │ the list of polynomials. │ │ │ │ │ │ │ │ -H: │ │ │ │ Let the name of the new variable be H rather than reading in a name from the │ │ │ │ input. │ │ │ │ -using temporary file /tmp/M2-16097-0/188 │ │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16097-0/190 │ │ │ │ +using temporary file /tmp/M2-20171-0/188 │ │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-20171-0/190 │ │ │ │ This program converts a list of polynomials to a list of their initial forms │ │ │ │ with respect to the vector given after the list. │ │ │ │ Options: │ │ │ │ --ideal: │ │ │ │ Treat input as an ideal. This will make the program compute the initial ideal │ │ │ │ of the ideal generated by the input polynomials. The computation is done by │ │ │ │ computing a Groebner basis with respect to the given vector. The vector must be │ │ │ │ @@ -242,16 +242,16 @@ │ │ │ │ --mark: │ │ │ │ If the --pair option is and the --ideal option is not used this option will │ │ │ │ still make sure that the second output basis is marked consistently with the │ │ │ │ vector. │ │ │ │ --list: │ │ │ │ Read in a list of vectors instead of a single vector and produce a list of │ │ │ │ polynomial sets as output. │ │ │ │ -using temporary file /tmp/M2-16097-0/190 │ │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16097-0/192 │ │ │ │ +using temporary file /tmp/M2-20171-0/190 │ │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-20171-0/192 │ │ │ │ This is a program for doing interactive walks in the Groebner fan of an ideal. │ │ │ │ The input is a Groebner basis defining the starting Groebner cone of the walk. │ │ │ │ The program will list all flippable facets of the Groebner cone and ask the │ │ │ │ user to choose one. The user types in the index (number) of the facet in the │ │ │ │ list. The program will walk through the selected facet and display the new │ │ │ │ Groebner basis and a list of new facet normals for the user to choose from. │ │ │ │ Since the program reads the user's choices through the the standard input it is │ │ │ │ @@ -281,54 +281,54 @@ │ │ │ │ -W: │ │ │ │ Print weight vector. This will make the program print an interior vector of │ │ │ │ the current Groebner cone and a relative interior point for each flippable │ │ │ │ facet of the current Groebner cone. │ │ │ │ │ │ │ │ --tropical: │ │ │ │ Traverse a tropical variety interactively. │ │ │ │ -using temporary file /tmp/M2-16097-0/192 │ │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16097-0/194 │ │ │ │ +using temporary file /tmp/M2-20171-0/192 │ │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-20171-0/194 │ │ │ │ This program checks if a set of marked polynomials is a Groebner basis with │ │ │ │ respect to its marking. First it is checked if the markings are consistent with │ │ │ │ respect to a positive vector. Then Buchberger's S-criterion is checked. The │ │ │ │ output is boolean value. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16097-0/194 │ │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16097-0/196 │ │ │ │ +using temporary file /tmp/M2-20171-0/194 │ │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-20171-0/196 │ │ │ │ Takes an ideal $I$ and computes the Krull dimension of R/I where R is the │ │ │ │ polynomial ring. This is done by first computing a Groebner basis. │ │ │ │ Options: │ │ │ │ -g: │ │ │ │ Tell the program that the input is already a reduced Groebner basis. │ │ │ │ -using temporary file /tmp/M2-16097-0/196 │ │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16097-0/198 │ │ │ │ +using temporary file /tmp/M2-20171-0/196 │ │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-20171-0/198 │ │ │ │ This program computes the lattice ideal of a lattice. The input is a list of │ │ │ │ generators for the lattice. │ │ │ │ Options: │ │ │ │ -t: │ │ │ │ Compute the toric ideal of the matrix whose rows are given on the input │ │ │ │ instead. │ │ │ │ --convert: │ │ │ │ Does not do any computation, but just converts the vectors to binomials. │ │ │ │ -using temporary file /tmp/M2-16097-0/198 │ │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16097-0/200 │ │ │ │ +using temporary file /tmp/M2-20171-0/198 │ │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-20171-0/200 │ │ │ │ This program converts a list of polynomials to a list of their leading terms. │ │ │ │ Options: │ │ │ │ -m: │ │ │ │ Do the same thing for a list of polynomial sets. That is, output the set of │ │ │ │ sets of leading terms. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-16097-0/200 │ │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16097-0/202 │ │ │ │ +using temporary file /tmp/M2-20171-0/200 │ │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-20171-0/202 │ │ │ │ This program marks a set of polynomials with respect to the vector given at the │ │ │ │ end of the input, meaning that the largest terms are moved to the front. In │ │ │ │ case of a tie the lexicographic term order with $a>b>c...$ is used to break it. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16097-0/202 │ │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16097-0/204 │ │ │ │ +using temporary file /tmp/M2-20171-0/202 │ │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-20171-0/204 │ │ │ │ This is a program for computing the normal fan of the Minkowski sum of the │ │ │ │ Newton polytopes of a list of polynomials. │ │ │ │ Options: │ │ │ │ --symmetry: │ │ │ │ Tells the program to read in generators for a group of symmetries (subgroup of │ │ │ │ $S_n$) after having read in the ideal. The program checks that the ideal stays │ │ │ │ fixed when permuting the variables with respect to elements in the group. The │ │ │ │ @@ -338,16 +338,16 @@ │ │ │ │ --disableSymmetryTest: │ │ │ │ When using --symmetry this option will disable the check that the group read │ │ │ │ off from the input actually is a symmetry group with respect to the input │ │ │ │ ideal. │ │ │ │ │ │ │ │ --nocones: │ │ │ │ Tell the program to not list cones in the output. │ │ │ │ -using temporary file /tmp/M2-16097-0/204 │ │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16097-0/206 │ │ │ │ +using temporary file /tmp/M2-20171-0/204 │ │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-20171-0/206 │ │ │ │ This program will generate the r*r minors of a d*n matrix of indeterminates. │ │ │ │ Options: │ │ │ │ -r value: │ │ │ │ Specify r. │ │ │ │ -d value: │ │ │ │ Specify d. │ │ │ │ -n value: │ │ │ │ @@ -365,16 +365,16 @@ │ │ │ │ Produces a list of generators for the group of symmetries keeping the set of │ │ │ │ minors fixed. (Only without --names). │ │ │ │ --parametrize: │ │ │ │ Parametrize the set of d times n matrices of Barvinok rank less than or equal │ │ │ │ to r-1 by a list of tropical polynomials. │ │ │ │ --ultrametric: │ │ │ │ Produce tropical equations cutting out the ultrametrics. │ │ │ │ -using temporary file /tmp/M2-16097-0/206 │ │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16097-0/208 │ │ │ │ +using temporary file /tmp/M2-20171-0/206 │ │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-20171-0/208 │ │ │ │ This program computes the mixed volume of the Newton polytopes of a list of │ │ │ │ polynomials. The ring is specified on the input. After this follows the list of │ │ │ │ polynomials. │ │ │ │ Options: │ │ │ │ --vectorinput: │ │ │ │ Read in a list of point configurations instead of a polynomial ring and a list │ │ │ │ of polynomials. │ │ │ │ @@ -388,25 +388,25 @@ │ │ │ │ Use Katsura-n example instead of reading input. │ │ │ │ --gaukwa value: │ │ │ │ Use Gaukwa-n example instead of reading input. │ │ │ │ --eco value: │ │ │ │ Use Eco-n example instead of reading input. │ │ │ │ -j value: │ │ │ │ Number of threads │ │ │ │ -using temporary file /tmp/M2-16097-0/208 │ │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16097-0/210 │ │ │ │ +using temporary file /tmp/M2-20171-0/208 │ │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-20171-0/210 │ │ │ │ This program computes the union of a list of polynomial sets given as input. │ │ │ │ The polynomials must all belong to the same ring. The ring is specified on the │ │ │ │ input. After this follows the list of polynomial sets. │ │ │ │ Options: │ │ │ │ -s: │ │ │ │ Sort output by degree. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-16097-0/210 │ │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16097-0/212 │ │ │ │ +using temporary file /tmp/M2-20171-0/210 │ │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-20171-0/212 │ │ │ │ This program renders a Groebner fan as an xfig file. To be more precise, the │ │ │ │ input is the list of all reduced Groebner bases of an ideal. The output is a │ │ │ │ drawing of the Groebner fan intersected with a triangle. The corners of the │ │ │ │ triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top. │ │ │ │ If there are more than three variables in the ring these coordinates are │ │ │ │ extended with zeros. It is possible to shift the 1 entry cyclic with the option │ │ │ │ --shiftVariables. │ │ │ │ @@ -414,16 +414,16 @@ │ │ │ │ -L: │ │ │ │ Make the triangle larger so that the shape of the Groebner region appears. │ │ │ │ --shiftVariables value: │ │ │ │ Shift the positions of the variables in the drawing. For example with the │ │ │ │ value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...) │ │ │ │ and top: (0,0,0,1,...). The shifting is done modulo the number of variables in │ │ │ │ the polynomial ring. The default value is 0. │ │ │ │ -using temporary file /tmp/M2-16097-0/212 │ │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16097-0/214 │ │ │ │ +using temporary file /tmp/M2-20171-0/212 │ │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-20171-0/214 │ │ │ │ This program renders a staircase diagram of a monomial initial ideal to an xfig │ │ │ │ file. The input is a Groebner basis of a (not necessarily monomial) polynomial │ │ │ │ ideal. The initial ideal is given by the leading terms in the Groebner basis. │ │ │ │ Using the -m option it is possible to render more than one staircase diagram. │ │ │ │ The program only works for ideals in a polynomial ring with three variables. │ │ │ │ Options: │ │ │ │ -m: │ │ │ │ @@ -436,16 +436,16 @@ │ │ │ │ number is large enough to give a correct picture of the standard monomials. The │ │ │ │ default value is 8. │ │ │ │ │ │ │ │ -w value: │ │ │ │ Width. Specifies the number of staircase diagrams per row in the xfig file. │ │ │ │ The default value is 5. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-16097-0/214 │ │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16097-0/216 │ │ │ │ +using temporary file /tmp/M2-20171-0/214 │ │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-20171-0/216 │ │ │ │ This program computes the resultant fan as defined in "Computing Tropical │ │ │ │ Resultants" by Jensen and Yu. The input is a polynomial ring followed by │ │ │ │ polynomials, whose coefficients are ignored. The output is the fan of │ │ │ │ coefficients such that the input system has a tropical solution. │ │ │ │ Options: │ │ │ │ --codimension: │ │ │ │ Compute only the codimension of the resultant fan and return. │ │ │ │ @@ -473,28 +473,28 @@ │ │ │ │ of polynomials. │ │ │ │ │ │ │ │ --projection: │ │ │ │ Use the projection method to compute the resultant fan. This works only if the │ │ │ │ resultant fan is a hypersurface. If this option is combined with --special, │ │ │ │ then the output fan lives in the subspace of the non-specialized coordinates. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-16097-0/216 │ │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16097-0/218 │ │ │ │ +using temporary file /tmp/M2-20171-0/216 │ │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-20171-0/218 │ │ │ │ This program computes the saturation of the input ideal with the product of the │ │ │ │ variables x_1,...,x_n. The ideal does not have to be homogeneous. │ │ │ │ Options: │ │ │ │ -h: │ │ │ │ Tell the program that the input is a homogeneous ideal (with homogeneous │ │ │ │ generators). │ │ │ │ │ │ │ │ --noideal: │ │ │ │ Do not treat input as an ideal but just factor out common monomial factors of │ │ │ │ the input polynomials. │ │ │ │ -using temporary file /tmp/M2-16097-0/218 │ │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16097-0/220 │ │ │ │ +using temporary file /tmp/M2-20171-0/218 │ │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-20171-0/220 │ │ │ │ This program computes the secondary fan of a vector configuration. The │ │ │ │ configuration is given as an ordered list of vectors. In order to compute the │ │ │ │ secondary fan of a point configuration an additional coordinate of ones must be │ │ │ │ added. For example {(1,0),(1,1),(1,2),(1,3)}. │ │ │ │ Options: │ │ │ │ --unimodular: │ │ │ │ Use heuristics to search for unimodular triangulation rather than computing │ │ │ │ @@ -523,103 +523,103 @@ │ │ │ │ Tells the program not to output the CONES and MAXIMAL_CONES sections, but │ │ │ │ still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is │ │ │ │ used. │ │ │ │ --interrupt value: │ │ │ │ Interrupt the enumeration after a specified number of facets have been │ │ │ │ computed (works for usual symmetric traversals, but may not work in general for │ │ │ │ non-symmetric traversals or for traversals restricted to fans). │ │ │ │ -using temporary file /tmp/M2-16097-0/220 │ │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16097-0/222 │ │ │ │ +using temporary file /tmp/M2-20171-0/220 │ │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-20171-0/222 │ │ │ │ This program takes a list of reduced Groebner bases for the same ideal and │ │ │ │ computes various statistics. The following information is listed: the number of │ │ │ │ bases in the input, the number of variables, the dimension of the homogeneity │ │ │ │ space, the maximal total degree of any polynomial in the input and the minimal │ │ │ │ total degree of any basis in the input, the maximal number of polynomials and │ │ │ │ terms in a basis in the input. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16097-0/222 │ │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16097-0/224 │ │ │ │ +using temporary file /tmp/M2-20171-0/222 │ │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-20171-0/224 │ │ │ │ This program changes the variable names of a polynomial ring. The input is a │ │ │ │ polynomial ring, a polynomial set in the ring and a new polynomial ring with │ │ │ │ the same coefficient field but different variable names. The output is the │ │ │ │ polynomial set written with the variable names of the second polynomial ring. │ │ │ │ Example: │ │ │ │ Input: │ │ │ │ Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x] │ │ │ │ Output: │ │ │ │ Q[b,a,c,x]{2*b-3*a,c+x} │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16097-0/224 │ │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16097-0/226 │ │ │ │ +using temporary file /tmp/M2-20171-0/224 │ │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-20171-0/226 │ │ │ │ This program converts ASCII math to TeX math. The data-type is specified by the │ │ │ │ options. │ │ │ │ Options: │ │ │ │ -h: │ │ │ │ Add a header to the output. Using this option the output will be LaTeXable │ │ │ │ right away. │ │ │ │ --polynomialset_: │ │ │ │ The data to be converted is a list of polynomials. │ │ │ │ --polynomialsetlist_: │ │ │ │ The data to be converted is a list of lists of polynomials. │ │ │ │ -using temporary file /tmp/M2-16097-0/226 │ │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16097-0/228 │ │ │ │ +using temporary file /tmp/M2-20171-0/226 │ │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-20171-0/228 │ │ │ │ This program takes a list of reduced Groebner bases and produces the fan of all │ │ │ │ faces of these. In this way by giving the complete list of reduced Groebner │ │ │ │ bases, the Groebner fan can be computed as a polyhedral complex. The option -- │ │ │ │ restrict lets the user choose between computing the Groebner fan or the │ │ │ │ restricted Groebner fan. │ │ │ │ Options: │ │ │ │ --restrict: │ │ │ │ Add an inequality for each coordinate, so that the the cones are restricted to │ │ │ │ the non-negative orthant. │ │ │ │ --symmetry: │ │ │ │ Tell the program to read in generators for a group of symmetries (subgroup of │ │ │ │ $S_n$) after having read in the ring. The output is grouped according to these │ │ │ │ symmetries. Only one representative for each orbit is needed on the input. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-16097-0/228 │ │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16097-0/230 │ │ │ │ +using temporary file /tmp/M2-20171-0/228 │ │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-20171-0/230 │ │ │ │ This program computes a tropical basis for an ideal defining a tropical curve. │ │ │ │ Defining a tropical curve means that the Krull dimension of R/I is at most 1 + │ │ │ │ the dimension of the homogeneity space of I where R is the polynomial ring. The │ │ │ │ input is a generating set for the ideal. If the input is not homogeneous option │ │ │ │ -h must be used. │ │ │ │ Options: │ │ │ │ -h: │ │ │ │ Homogenise the input before computing a tropical basis and dehomogenise the │ │ │ │ output. This is needed if the input generators are not already homogeneous. │ │ │ │ -using temporary file /tmp/M2-16097-0/230 │ │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16097-0/232 │ │ │ │ +using temporary file /tmp/M2-20171-0/230 │ │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-20171-0/232 │ │ │ │ This program takes a marked reduced Groebner basis for a homogeneous ideal and │ │ │ │ computes the tropical variety of the ideal as a subfan of the Groebner fan. The │ │ │ │ program is slow but works for any homogeneous ideal. If you know that your │ │ │ │ ideal is prime over the complex numbers or you simply know that its tropical │ │ │ │ variety is pure and connected in codimension one then use │ │ │ │ gfan_tropicalstartingcone and gfan_tropicaltraverse instead. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16097-0/232 │ │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16097-0/234 │ │ │ │ +using temporary file /tmp/M2-20171-0/232 │ │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-20171-0/234 │ │ │ │ This program evaluates a tropical polynomial function in a given set of points. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16097-0/234 │ │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16097-0/236 │ │ │ │ +using temporary file /tmp/M2-20171-0/234 │ │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-20171-0/236 │ │ │ │ This program takes a polynomial and tropicalizes it. The output is piecewise │ │ │ │ linear function represented by a fan whose cones are the linear regions. Each │ │ │ │ ray of the fan gets the value of the tropical function assigned to it. In other │ │ │ │ words this program computes the normal fan of the Newton polytope of the input │ │ │ │ polynomial with additional information.Options: │ │ │ │ --exponents: │ │ │ │ Tell program to read a list of exponent vectors instead. │ │ │ │ -using temporary file /tmp/M2-16097-0/236 │ │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16097-0/238 │ │ │ │ +using temporary file /tmp/M2-20171-0/236 │ │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-20171-0/238 │ │ │ │ This program computes the tropical hypersurface defined by a principal ideal. │ │ │ │ The input is the polynomial ring followed by a set containing just a generator │ │ │ │ of the ideal.Options: │ │ │ │ -using temporary file /tmp/M2-16097-0/238 │ │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16097-0/240 │ │ │ │ +using temporary file /tmp/M2-20171-0/238 │ │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-20171-0/240 │ │ │ │ This program computes the set theoretical intersection of a set of tropical │ │ │ │ hypersurfaces (or to be precise, their common refinement as a fan). The input │ │ │ │ is a list of polynomials with each polynomial defining a hypersurface. │ │ │ │ Considering tropical hypersurfaces as fans, the intersection can be computed as │ │ │ │ the common refinement of these. Thus the output is a fan whose support is the │ │ │ │ intersection of the tropical hypersurfaces. │ │ │ │ Options: │ │ │ │ @@ -656,16 +656,16 @@ │ │ │ │ --stable: │ │ │ │ Find the stable intersection of the input polynomials using tropical │ │ │ │ intersection theory. This can be slow. Most other options are ignored. │ │ │ │ --parameters value: │ │ │ │ With this option you can specify how many variables to treat as parameters │ │ │ │ instead of variables. This makes it possible to do computations where the │ │ │ │ coefficient field is the field of rational functions in the parameters. │ │ │ │ -using temporary file /tmp/M2-16097-0/240 │ │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16097-0/242 │ │ │ │ +using temporary file /tmp/M2-20171-0/240 │ │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-20171-0/242 │ │ │ │ This program is part of the Puiseux lifting algorithm implemented in Gfan and │ │ │ │ Singular. The Singular part of the implementation can be found in: │ │ │ │ │ │ │ │ Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig: │ │ │ │ tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, │ │ │ │ 2007 │ │ │ │ │ │ │ │ @@ -693,54 +693,54 @@ │ │ │ │ Options: │ │ │ │ --noMult: │ │ │ │ Disable the multiplicity computation. │ │ │ │ -n value: │ │ │ │ Number of variables that should have negative weight. │ │ │ │ -c: │ │ │ │ Only output a list of vectors being the possible choices. │ │ │ │ -using temporary file /tmp/M2-16097-0/242 │ │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16097-0/244 │ │ │ │ +using temporary file /tmp/M2-20171-0/242 │ │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-20171-0/244 │ │ │ │ This program generates tropical equations for a tropical linear space in the │ │ │ │ Speyer sense given the tropical Pluecker coordinates as input. │ │ │ │ Options: │ │ │ │ -d value: │ │ │ │ Specify d. │ │ │ │ -n value: │ │ │ │ Specify n. │ │ │ │ --trees: │ │ │ │ list the boundary trees (assumes d=3) │ │ │ │ -using temporary file /tmp/M2-16097-0/244 │ │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16097-0/246 │ │ │ │ +using temporary file /tmp/M2-20171-0/244 │ │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-20171-0/246 │ │ │ │ This program computes the multiplicity of a tropical cone given a marked │ │ │ │ reduced Groebner basis for its initial ideal. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16097-0/246 │ │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16097-0/248 │ │ │ │ +using temporary file /tmp/M2-20171-0/246 │ │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-20171-0/248 │ │ │ │ This program will compute the tropical rank of matrix given as input. Tropical │ │ │ │ addition is MAXIMUM. │ │ │ │ Options: │ │ │ │ --kapranov: │ │ │ │ Compute Kapranov rank instead of tropical rank. │ │ │ │ --determinant: │ │ │ │ Compute the tropical determinant instead. │ │ │ │ -using temporary file /tmp/M2-16097-0/248 │ │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16097-0/250 │ │ │ │ +using temporary file /tmp/M2-20171-0/248 │ │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-20171-0/250 │ │ │ │ This program computes a starting pair of marked reduced Groebner bases to be │ │ │ │ used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose │ │ │ │ tropical variety is a pure d-dimensional polyhedral complex. │ │ │ │ Options: │ │ │ │ -g: │ │ │ │ Tell the program that the input is already a reduced Groebner basis. │ │ │ │ -d: │ │ │ │ Output dimension information to standard error. │ │ │ │ --stable: │ │ │ │ Find starting cone in the stable intersection or, equivalently, pretend that │ │ │ │ the coefficients are genereric. │ │ │ │ -using temporary file /tmp/M2-16097-0/250 │ │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16097-0/252 │ │ │ │ +using temporary file /tmp/M2-20171-0/250 │ │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-20171-0/252 │ │ │ │ This program computes a polyhedral fan representation of the tropical variety │ │ │ │ of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let │ │ │ │ $\omega$ be a relative interior point of $d$-dimensional Groebner cone │ │ │ │ contained in the tropical variety. The input for this program is a pair of │ │ │ │ marked reduced Groebner bases with respect to the term order represented by │ │ │ │ $\omega$, tie-broken in some way. The first one is for the initial ideal │ │ │ │ $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point │ │ │ │ @@ -770,27 +770,27 @@ │ │ │ │ --stable: │ │ │ │ Traverse the stable intersection or, equivalently, pretend that the │ │ │ │ coefficients are genereric. │ │ │ │ --interrupt value: │ │ │ │ Interrupt the enumeration after a specified number of facets have been │ │ │ │ computed (works for usual symmetric traversals, but may not work in general for │ │ │ │ non-symmetric traversals or for traversals restricted to fans). │ │ │ │ -using temporary file /tmp/M2-16097-0/252 │ │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16097-0/254 │ │ │ │ +using temporary file /tmp/M2-20171-0/252 │ │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-20171-0/254 │ │ │ │ This program computes the tropical Weil divisor of piecewise linear (or │ │ │ │ tropical rational) function on a tropical k-cycle. See the Gfan manual for more │ │ │ │ information. │ │ │ │ Options: │ │ │ │ -i1 value: │ │ │ │ Specify the name of the Polymake input file containing the k-cycle. │ │ │ │ -i2 value: │ │ │ │ Specify the name of the Polymake input file containing the piecewise linear │ │ │ │ function. │ │ │ │ -using temporary file /tmp/M2-16097-0/254 │ │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16097-0/256 │ │ │ │ +using temporary file /tmp/M2-20171-0/254 │ │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-20171-0/256 │ │ │ │ This program is an experimental implementation of Groebner bases for ideals in │ │ │ │ Z[x_1,...,x_n]. │ │ │ │ Several operations are supported by specifying the appropriate option: │ │ │ │ (1) computation of the reduced Groebner basis with respect to a given vector │ │ │ │ (tiebroken lexicographically), │ │ │ │ (2) computation of an initial ideal, │ │ │ │ (3) computation of the Groebner fan, │ │ │ │ @@ -825,23 +825,23 @@ │ │ │ │ For the operations taking a vector as input, read in a list of vectors │ │ │ │ instead, and perform the operation for each vector in the list. │ │ │ │ -g: │ │ │ │ Tells the program that the input is already a Groebner basis (with the initial │ │ │ │ term of each polynomial being the first ones listed). Use this option if the │ │ │ │ usual --groebnerFan is too slow. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-16097-0/256 │ │ │ │ +using temporary file /tmp/M2-20171-0/256 │ │ │ │ i6 : QQ[x,y]; │ │ │ │ i7 : gfan {x,y}; │ │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16097-0/258 │ │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-20171-0/258 │ │ │ │ Q[x1,x2] │ │ │ │ {{ │ │ │ │ x2, │ │ │ │ x1} │ │ │ │ } │ │ │ │ -using temporary file /tmp/M2-16097-0/258 │ │ │ │ +using temporary file /tmp/M2-20171-0/258 │ │ │ │ Finally, if you want to be able to render Groebner fans and monomial staircases │ │ │ │ to .png files, you should install fig2dev. If it is installed in a non-standard │ │ │ │ location, then you may specify its path using _p_r_o_g_r_a_m_P_a_t_h_s. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.05+ds/M2/Macaulay2/packages/gfanInterface.m2:2631:0. │ │ ├── ./usr/share/info/AInfinity.info.gz │ │ │ ├── AInfinity.info │ │ │ │ @@ -6123,15 +6123,15 @@ │ │ │ │ 00017ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00017ed0: 3320 3a20 656c 6170 7365 6454 696d 6520 3 : elapsedTime │ │ │ │ 00017ee0: 6275 726b 6552 6573 6f6c 7574 696f 6e28 burkeResolution( │ │ │ │ 00017ef0: 4d2c 2037 2c20 4368 6563 6b20 3d3e 2066 M, 7, Check => f │ │ │ │ 00017f00: 616c 7365 2920 2020 2020 2020 2020 2020 alse) │ │ │ │ -00017f10: 7c0a 7c20 2d2d 2031 2e38 3633 3639 7320 |.| -- 1.86369s │ │ │ │ +00017f10: 7c0a 7c20 2d2d 2031 2e36 3136 3239 7320 |.| -- 1.61629s │ │ │ │ 00017f20: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 00017f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017f50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00017f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -6166,15 +6166,15 @@ │ │ │ │ 00018150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018170: 2d2d 2d2d 2b0a 7c69 3420 3a20 656c 6170 ----+.|i4 : elap │ │ │ │ 00018180: 7365 6454 696d 6520 6275 726b 6552 6573 sedTime burkeRes │ │ │ │ 00018190: 6f6c 7574 696f 6e28 4d2c 2037 2c20 4368 olution(M, 7, Ch │ │ │ │ 000181a0: 6563 6b20 3d3e 2074 7275 6529 2020 2020 eck => true) │ │ │ │ 000181b0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2032 |.| -- 2 │ │ │ │ -000181c0: 2e32 3638 3337 7320 656c 6170 7365 6420 .26837s elapsed │ │ │ │ +000181c0: 2e30 3336 3231 7320 656c 6170 7365 6420 .03621s elapsed │ │ │ │ 000181d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000181e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000181f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00018200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/AdjunctionForSurfaces.info.gz │ │ │ ├── AdjunctionForSurfaces.info │ │ │ │ @@ -741,16 +741,16 @@ │ │ │ │ 00002e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00002e70: 7c69 3130 203a 2065 6c61 7073 6564 5469 |i10 : elapsedTi │ │ │ │ 00002e80: 6d65 2066 493d 7265 7320 4920 2020 2020 me fI=res I │ │ │ │ 00002e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00002eb0: 2020 207c 0a7c 202d 2d20 2e30 3336 3332 |.| -- .03632 │ │ │ │ -00002ec0: 3434 7320 656c 6170 7365 6420 2020 2020 44s elapsed │ │ │ │ +00002eb0: 2020 207c 0a7c 202d 2d20 2e30 3530 3132 |.| -- .05012 │ │ │ │ +00002ec0: 3231 7320 656c 6170 7365 6420 2020 2020 21s elapsed │ │ │ │ 00002ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002ef0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00002f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002f30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -1596,15 +1596,15 @@ │ │ │ │ 000063b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000063c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000063d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000063e0: 7c69 3135 203a 2065 6c61 7073 6564 5469 |i15 : elapsedTi │ │ │ │ 000063f0: 6d65 2062 6574 7469 2849 273d 7472 696d me betti(I'=trim │ │ │ │ 00006400: 206b 6572 2070 6869 2920 2020 2020 2020 ker phi) │ │ │ │ 00006410: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00006420: 2e36 3731 3637 3273 2065 6c61 7073 6564 .671672s elapsed │ │ │ │ +00006420: 2e35 3831 3138 3173 2065 6c61 7073 6564 .581181s elapsed │ │ │ │ 00006430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006450: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00006460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006480: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00006490: 0a7c 2020 2020 2020 2020 2020 2020 2030 .| 0 │ │ │ │ @@ -1651,15 +1651,15 @@ │ │ │ │ 00006720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006750: 2d2d 2d2b 0a7c 6931 3720 3a20 656c 6170 ---+.|i17 : elap │ │ │ │ 00006760: 7365 6454 696d 6520 6261 7365 5074 733d sedTime basePts= │ │ │ │ 00006770: 7072 696d 6172 7944 6563 6f6d 706f 7369 primaryDecomposi │ │ │ │ 00006780: 7469 6f6e 2069 6465 616c 2048 3b20 7c0a tion ideal H; |. │ │ │ │ -00006790: 7c20 2d2d 2036 2e34 3334 3731 7320 656c | -- 6.43471s el │ │ │ │ +00006790: 7c20 2d2d 2035 2e33 3138 3735 7320 656c | -- 5.31875s el │ │ │ │ 000067a0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 000067b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000067c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 000067d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000067e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000067f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006800: 2d2d 2d2d 2b0a 7c69 3138 203a 2074 616c ----+.|i18 : tal │ │ │ │ @@ -2608,15 +2608,15 @@ │ │ │ │ 0000a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0000a320: 3134 203a 2065 6c61 7073 6564 5469 6d65 14 : elapsedTime │ │ │ │ 0000a330: 2073 7562 2849 2c48 2920 2020 2020 2020 sub(I,H) │ │ │ │ 0000a340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a350: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -0000a360: 3132 3337 3535 7320 656c 6170 7365 6420 123755s elapsed │ │ │ │ +0000a360: 3134 3131 3034 7320 656c 6170 7365 6420 141104s elapsed │ │ │ │ 0000a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a390: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a3c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0000a3d0: 6f31 3420 3d20 6964 6561 6c20 2830 2c20 o14 = ideal (0, │ │ │ │ @@ -2648,15 +2648,15 @@ │ │ │ │ 0000a570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a5a0: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 656c -----+.|i16 : el │ │ │ │ 0000a5b0: 6170 7365 6454 696d 6520 6265 7474 6928 apsedTime betti( │ │ │ │ 0000a5c0: 4927 3d74 7269 6d20 6b65 7220 7068 6929 I'=trim ker phi) │ │ │ │ 0000a5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a5e0: 7c0a 7c20 2d2d 202e 3035 3434 3838 3373 |.| -- .0544883s │ │ │ │ +0000a5e0: 7c0a 7c20 2d2d 202e 3036 3435 3538 3673 |.| -- .0645586s │ │ │ │ 0000a5f0: 2065 6c61 7073 6564 2020 2020 2020 2020 elapsed │ │ │ │ 0000a600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a610: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000a620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a650: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ @@ -2700,16 +2700,16 @@ │ │ │ │ 0000a8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000a8e0: 0a7c 6931 3820 3a20 656c 6170 7365 6454 .|i18 : elapsedT │ │ │ │ 0000a8f0: 696d 6520 6261 7365 5074 733d 7072 696d ime basePts=prim │ │ │ │ 0000a900: 6172 7944 6563 6f6d 706f 7369 7469 6f6e aryDecomposition │ │ │ │ 0000a910: 2069 6465 616c 2048 3b20 7c0a 7c20 2d2d ideal H; |.| -- │ │ │ │ -0000a920: 2031 2e39 3433 3433 7320 656c 6170 7365 1.94343s elapse │ │ │ │ -0000a930: 6420 2020 2020 2020 2020 2020 2020 2020 d │ │ │ │ +0000a920: 2031 2e36 3235 3873 2065 6c61 7073 6564 1.6258s elapsed │ │ │ │ +0000a930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a950: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0000a960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a990: 2b0a 7c69 3139 203a 2074 616c 6c79 2061 +.|i19 : tally a │ │ │ │ 0000a9a0: 7070 6c79 2862 6173 6550 7473 2c63 2d3e pply(basePts,c-> │ │ ├── ./usr/share/info/BGG.info.gz │ │ │ ├── BGG.info │ │ │ │ @@ -4338,16 +4338,16 @@ │ │ │ │ 00010f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00010f40: 3134 203a 2074 696d 6520 6265 7474 6920 14 : time betti │ │ │ │ 00010f50: 2846 203d 2070 7572 6552 6573 6f6c 7574 (F = pureResolut │ │ │ │ 00010f60: 696f 6e28 4d2c 7b30 2c32 2c34 7d29 2920 ion(M,{0,2,4})) │ │ │ │ 00010f70: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00010f80: 302e 3434 3938 3636 7320 2863 7075 293b 0.449866s (cpu); │ │ │ │ -00010f90: 2030 2e33 3837 3331 3473 2028 7468 7265 0.387314s (thre │ │ │ │ +00010f80: 302e 3439 3339 3539 7320 2863 7075 293b 0.493959s (cpu); │ │ │ │ +00010f90: 2030 2e34 3134 3230 3873 2028 7468 7265 0.414208s (thre │ │ │ │ 00010fa0: 6164 293b 2030 7320 2867 6329 7c0a 7c20 ad); 0s (gc)|.| │ │ │ │ 00010fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010fe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00010ff0: 2020 2020 3020 3120 3220 2020 2020 2020 0 1 2 │ │ │ │ 00011000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4403,16 +4403,16 @@ │ │ │ │ 00011320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00011350: 0a7c 6931 3520 3a20 7469 6d65 2062 6574 .|i15 : time bet │ │ │ │ 00011360: 7469 2028 4620 3d20 7075 7265 5265 736f ti (F = pureReso │ │ │ │ 00011370: 6c75 7469 6f6e 2831 312c 342c 7b30 2c32 lution(11,4,{0,2 │ │ │ │ 00011380: 2c34 7d29 2920 207c 0a7c 202d 2d20 7573 ,4})) |.| -- us │ │ │ │ -00011390: 6564 2030 2e34 3933 3835 3973 2028 6370 ed 0.493859s (cp │ │ │ │ -000113a0: 7529 3b20 302e 3430 3936 3337 7320 2874 u); 0.409637s (t │ │ │ │ +00011390: 6564 2030 2e35 3835 3832 3573 2028 6370 ed 0.585825s (cp │ │ │ │ +000113a0: 7529 3b20 302e 3439 3830 3537 7320 2874 u); 0.498057s (t │ │ │ │ 000113b0: 6872 6561 6429 3b20 3073 2028 6763 297c hread); 0s (gc)| │ │ │ │ 000113c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000113d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00011400: 2020 2020 2020 2030 2031 2032 2020 2020 0 1 2 │ │ │ │ 00011410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/Benchmark.info.gz │ │ │ ├── Benchmark.info │ │ │ │ @@ -200,72 +200,77 @@ │ │ │ │ 00000c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00000c80: 3120 3a20 7275 6e42 656e 6368 6d61 726b 1 : runBenchmark │ │ │ │ 00000c90: 7320 2272 6573 3339 2220 2020 2020 2020 s "res39" │ │ │ │ 00000ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000cc0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 00000cd0: 2d20 6265 6769 6e6e 696e 6720 636f 6d70 - beginning comp │ │ │ │ -00000ce0: 7574 6174 696f 6e20 4d6f 6e20 4d61 7920 utation Mon May │ │ │ │ -00000cf0: 3138 2031 323a 3338 3a34 3120 5554 4320 18 12:38:41 UTC │ │ │ │ +00000ce0: 7574 6174 696f 6e20 5765 6420 4d61 7920 utation Wed May │ │ │ │ +00000cf0: 3230 2031 373a 3237 3a31 3520 5554 4320 20 17:27:15 UTC │ │ │ │ 00000d00: 3230 3236 2020 2020 2020 2020 2020 2020 2026 │ │ │ │ 00000d10: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 00000d20: 2d20 4c69 6e75 7820 7362 7569 6c64 2036 - Linux sbuild 6 │ │ │ │ -00000d30: 2e31 322e 3838 2b64 6562 3133 2d61 6d64 .12.88+deb13-amd │ │ │ │ -00000d40: 3634 2023 3120 534d 5020 5052 4545 4d50 64 #1 SMP PREEMP │ │ │ │ -00000d50: 545f 4459 4e41 4d49 4320 4465 6269 616e T_DYNAMIC Debian │ │ │ │ -00000d60: 2036 2e31 322e 3838 2d31 2020 7c0a 7c2d 6.12.88-1 |.|- │ │ │ │ -00000d70: 2d20 414d 4420 4550 5943 2037 3730 3250 - AMD EPYC 7702P │ │ │ │ -00000d80: 2036 342d 436f 7265 2050 726f 6365 7373 64-Core Process │ │ │ │ -00000d90: 6f72 2020 4175 7468 656e 7469 6341 4d44 or AuthenticAMD │ │ │ │ -00000da0: 2020 6370 7520 4d48 7a20 3139 3936 2e32 cpu MHz 1996.2 │ │ │ │ -00000db0: 3530 2020 2020 2020 2020 2020 7c0a 7c2d 50 |.|- │ │ │ │ +00000d30: 2e31 322e 3838 2b64 6562 3133 2d63 6c6f .12.88+deb13-clo │ │ │ │ +00000d40: 7564 2d61 6d64 3634 2023 3120 534d 5020 ud-amd64 #1 SMP │ │ │ │ +00000d50: 5052 4545 4d50 545f 4459 4e41 4d49 4320 PREEMPT_DYNAMIC │ │ │ │ +00000d60: 4465 6269 616e 2020 2020 2020 7c0a 7c2d Debian |.|- │ │ │ │ +00000d70: 2d20 496e 7465 6c20 5865 6f6e 2050 726f - Intel Xeon Pro │ │ │ │ +00000d80: 6365 7373 6f72 2028 536b 796c 616b 652c cessor (Skylake, │ │ │ │ +00000d90: 2049 4252 5329 2020 4765 6e75 696e 6549 IBRS) GenuineI │ │ │ │ +00000da0: 6e74 656c 2020 6370 7520 4d48 7a20 3230 ntel cpu MHz 20 │ │ │ │ +00000db0: 3939 2e39 3938 2020 2020 2020 7c0a 7c2d 99.998 |.|- │ │ │ │ 00000dc0: 2d20 4d61 6361 756c 6179 3220 312e 3236 - Macaulay2 1.26 │ │ │ │ 00000dd0: 2e30 352c 2063 6f6d 7069 6c65 6420 7769 .05, compiled wi │ │ │ │ 00000de0: 7468 2067 6363 2031 352e 322e 3020 2020 th gcc 15.2.0 │ │ │ │ 00000df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000e00: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 00000e10: 2d20 7265 7333 393a 2072 6573 206f 6620 - res39: res of │ │ │ │ 00000e20: 6120 6765 6e65 7269 6320 3320 6279 2039 a generic 3 by 9 │ │ │ │ 00000e30: 206d 6174 7269 7820 6f76 6572 205a 5a2f matrix over ZZ/ │ │ │ │ -00000e40: 3130 313a 202e 3133 3131 3631 2073 6563 101: .131161 sec │ │ │ │ +00000e40: 3130 313a 202e 3137 3835 3835 2073 6563 101: .178585 sec │ │ │ │ 00000e50: 6f6e 6473 2020 2020 2020 2020 7c0a 7c2d onds |.|- │ │ │ │ 00000e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00000ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c28 ------------|.|( │ │ │ │ -00000eb0: 3230 3236 2d30 352d 3135 2920 7838 365f 2026-05-15) x86_ │ │ │ │ -00000ec0: 3634 2047 4e55 2f4c 696e 7578 2020 2020 64 GNU/Linux │ │ │ │ -00000ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c36 ------------|.|6 │ │ │ │ +00000eb0: 2e31 322e 3838 2d31 2028 3230 3236 2d30 .12.88-1 (2026-0 │ │ │ │ +00000ec0: 352d 3135 2920 7838 365f 3634 2047 4e55 5-15) x86_64 GNU │ │ │ │ +00000ed0: 2f4c 696e 7578 2020 2020 2020 2020 2020 /Linux │ │ │ │ 00000ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00000ef0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00000f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00000f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00000f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00000f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00000f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 ------------+..F │ │ │ │ -00000f50: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00000f60: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00000f70: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00000f80: 202a 6e6f 7465 2072 756e 4265 6e63 686d *note runBenchm │ │ │ │ -00000f90: 6172 6b73 3a20 7275 6e42 656e 6368 6d61 arks: runBenchma │ │ │ │ -00000fa0: 726b 732c 2069 7320 6120 2a6e 6f74 6520 rks, is a *note │ │ │ │ -00000fb0: 636f 6d6d 616e 643a 0a28 4d61 6361 756c command:.(Macaul │ │ │ │ -00000fc0: 6179 3244 6f63 2943 6f6d 6d61 6e64 2c2e ay2Doc)Command,. │ │ │ │ -00000fd0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ -00000fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00000ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00001000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00001010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00001020: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -00001030: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -00001040: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -00001050: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -00001060: 6361 756c 6179 322d 312e 3236 2e30 352b caulay2-1.26.05+ │ │ │ │ -00001070: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -00001080: 7061 636b 6167 6573 2f42 656e 6368 6d61 packages/Benchma │ │ │ │ -00001090: 726b 2e0a 6d32 3a32 3937 3a30 2e0a 1f0a rk..m2:297:0.... │ │ │ │ -000010a0: 5461 6720 5461 626c 653a 0a4e 6f64 653a Tag Table:.Node: │ │ │ │ -000010b0: 2054 6f70 7f32 3334 0a4e 6f64 653a 2072 Top.234.Node: r │ │ │ │ -000010c0: 756e 4265 6e63 686d 6172 6b73 7f32 3033 unBenchmarks.203 │ │ │ │ -000010d0: 350a 1f0a 456e 6420 5461 6720 5461 626c 5...End Tag Tabl │ │ │ │ -000010e0: 650a e. │ │ │ │ +00000ef0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00000f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000f40: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00000f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00000f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00000f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00000f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00000f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 ------------+..F │ │ │ │ +00000fa0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +00000fb0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +00000fc0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +00000fd0: 202a 6e6f 7465 2072 756e 4265 6e63 686d *note runBenchm │ │ │ │ +00000fe0: 6172 6b73 3a20 7275 6e42 656e 6368 6d61 arks: runBenchma │ │ │ │ +00000ff0: 726b 732c 2069 7320 6120 2a6e 6f74 6520 rks, is a *note │ │ │ │ +00001000: 636f 6d6d 616e 643a 0a28 4d61 6361 756c command:.(Macaul │ │ │ │ +00001010: 6179 3244 6f63 2943 6f6d 6d61 6e64 2c2e ay2Doc)Command,. │ │ │ │ +00001020: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +00001030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00001040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00001050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00001060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00001070: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +00001080: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +00001090: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +000010a0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +000010b0: 6361 756c 6179 322d 312e 3236 2e30 352b caulay2-1.26.05+ │ │ │ │ +000010c0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +000010d0: 7061 636b 6167 6573 2f42 656e 6368 6d61 packages/Benchma │ │ │ │ +000010e0: 726b 2e0a 6d32 3a32 3937 3a30 2e0a 1f0a rk..m2:297:0.... │ │ │ │ +000010f0: 5461 6720 5461 626c 653a 0a4e 6f64 653a Tag Table:.Node: │ │ │ │ +00001100: 2054 6f70 7f32 3334 0a4e 6f64 653a 2072 Top.234.Node: r │ │ │ │ +00001110: 756e 4265 6e63 686d 6172 6b73 7f32 3033 unBenchmarks.203 │ │ │ │ +00001120: 350a 1f0a 456e 6420 5461 6720 5461 626c 5...End Tag Tabl │ │ │ │ +00001130: 650a e. │ │ ├── ./usr/share/info/Bertini.info.gz │ │ │ ├── Bertini.info │ │ │ │ @@ -2253,16 +2253,16 @@ │ │ │ │ 00008cc0: 616c 206e 756d 6265 720a 2020 2020 2020 al number. │ │ │ │ 00008cd0: 2020 6f72 2072 616e 646f 6d20 636f 6d70 or random comp │ │ │ │ 00008ce0: 6c65 7820 6e75 6d62 6572 0a20 2020 2020 lex number. │ │ │ │ 00008cf0: 202a 202a 6e6f 7465 2054 6f70 4469 7265 * *note TopDire │ │ │ │ 00008d00: 6374 6f72 793a 2054 6f70 4469 7265 6374 ctory: TopDirect │ │ │ │ 00008d10: 6f72 792c 203d 3e20 2e2e 2e2c 2064 6566 ory, => ..., def │ │ │ │ 00008d20: 6175 6c74 2076 616c 7565 0a20 2020 2020 ault value. │ │ │ │ -00008d30: 2020 2022 2f74 6d70 2f4d 322d 3233 3435 "/tmp/M2-2345 │ │ │ │ -00008d40: 302d 302f 3022 2c20 4f70 7469 6f6e 2074 0-0/0", Option t │ │ │ │ +00008d30: 2020 2022 2f74 6d70 2f4d 322d 3239 3930 "/tmp/M2-2990 │ │ │ │ +00008d40: 352d 302f 3022 2c20 4f70 7469 6f6e 2074 5-0/0", Option t │ │ │ │ 00008d50: 6f20 6368 616e 6765 2064 6972 6563 746f o change directo │ │ │ │ 00008d60: 7279 2066 6f72 2066 696c 6520 7374 6f72 ry for file stor │ │ │ │ 00008d70: 6167 652e 0a20 2020 2020 202a 202a 6e6f age.. * *no │ │ │ │ 00008d80: 7465 2056 6572 626f 7365 3a20 6265 7274 te Verbose: bert │ │ │ │ 00008d90: 696e 6954 7261 636b 486f 6d6f 746f 7079 iniTrackHomotopy │ │ │ │ 00008da0: 5f6c 705f 7064 5f70 645f 7064 5f63 6d56 _lp_pd_pd_pd_cmV │ │ │ │ 00008db0: 6572 626f 7365 3d3e 5f70 645f 7064 5f70 erbose=>_pd_pd_p │ │ │ │ @@ -4971,15 +4971,15 @@ │ │ │ │ 000136a0: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c ntation) => ..., │ │ │ │ 000136b0: 2064 6566 6175 6c74 2076 616c 7565 207b default value { │ │ │ │ 000136c0: 7d2c 200a 2020 2020 2020 2a20 2a6e 6f74 }, . * *not │ │ │ │ 000136d0: 6520 546f 7044 6972 6563 746f 7279 3a20 e TopDirectory: │ │ │ │ 000136e0: 546f 7044 6972 6563 746f 7279 2c20 3d3e TopDirectory, => │ │ │ │ 000136f0: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ 00013700: 6c75 650a 2020 2020 2020 2020 222f 746d lue. "/tm │ │ │ │ -00013710: 702f 4d32 2d32 3334 3530 2d30 2f30 222c p/M2-23450-0/0", │ │ │ │ +00013710: 702f 4d32 2d32 3939 3035 2d30 2f30 222c p/M2-29905-0/0", │ │ │ │ 00013720: 204f 7074 696f 6e20 746f 2063 6861 6e67 Option to chang │ │ │ │ 00013730: 6520 6469 7265 6374 6f72 7920 666f 7220 e directory for │ │ │ │ 00013740: 6669 6c65 2073 746f 7261 6765 2e0a 2020 file storage.. │ │ │ │ 00013750: 2020 2020 2a20 2a6e 6f74 6520 5665 7262 * *note Verb │ │ │ │ 00013760: 6f73 653a 2062 6572 7469 6e69 5472 6163 ose: bertiniTrac │ │ │ │ 00013770: 6b48 6f6d 6f74 6f70 795f 6c70 5f70 645f kHomotopy_lp_pd_ │ │ │ │ 00013780: 7064 5f70 645f 636d 5665 7262 6f73 653d pd_pd_cmVerbose= │ │ │ │ @@ -5472,16 +5472,16 @@ │ │ │ │ 000155f0: 6561 6c20 6e75 6d62 6572 0a20 2020 2020 eal number. │ │ │ │ 00015600: 2020 206f 7220 7261 6e64 6f6d 2063 6f6d or random com │ │ │ │ 00015610: 706c 6578 206e 756d 6265 720a 2020 2020 plex number. │ │ │ │ 00015620: 2020 2a20 2a6e 6f74 6520 546f 7044 6972 * *note TopDir │ │ │ │ 00015630: 6563 746f 7279 3a20 546f 7044 6972 6563 ectory: TopDirec │ │ │ │ 00015640: 746f 7279 2c20 3d3e 202e 2e2e 2c20 6465 tory, => ..., de │ │ │ │ 00015650: 6661 756c 7420 7661 6c75 650a 2020 2020 fault value. │ │ │ │ -00015660: 2020 2020 222f 746d 702f 4d32 2d32 3334 "/tmp/M2-234 │ │ │ │ -00015670: 3530 2d30 2f30 222c 204f 7074 696f 6e20 50-0/0", Option │ │ │ │ +00015660: 2020 2020 222f 746d 702f 4d32 2d32 3939 "/tmp/M2-299 │ │ │ │ +00015670: 3035 2d30 2f30 222c 204f 7074 696f 6e20 05-0/0", Option │ │ │ │ 00015680: 746f 2063 6861 6e67 6520 6469 7265 6374 to change direct │ │ │ │ 00015690: 6f72 7920 666f 7220 6669 6c65 2073 746f ory for file sto │ │ │ │ 000156a0: 7261 6765 2e0a 2020 2020 2020 2a20 5573 rage.. * Us │ │ │ │ 000156b0: 6552 6567 656e 6572 6174 696f 6e20 286d eRegeneration (m │ │ │ │ 000156c0: 6973 7369 6e67 2064 6f63 756d 656e 7461 issing documenta │ │ │ │ 000156d0: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465 tion) => ..., de │ │ │ │ 000156e0: 6661 756c 7420 7661 6c75 6520 2d31 2c20 fault value -1, │ │ ├── ./usr/share/info/BettiCharacters.info.gz │ │ │ ├── BettiCharacters.info │ │ │ │ @@ -16843,15 +16843,15 @@ │ │ │ │ 00041ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00041cc0: 3920 3a20 656c 6170 7365 6454 696d 6520 9 : elapsedTime │ │ │ │ 00041cd0: 6320 3d20 6368 6172 6163 7465 7220 4120 c = character A │ │ │ │ 00041ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00041d10: 2d2d 202e 3338 3336 3939 7320 656c 6170 -- .383699s elap │ │ │ │ +00041d10: 2d2d 202e 3330 3833 3238 7320 656c 6170 -- .308328s elap │ │ │ │ 00041d20: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 00041d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00041d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -18058,16 +18058,16 @@ │ │ │ │ 00046890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000468a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000468b0: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 656c ------+.|i7 : el │ │ │ │ 000468c0: 6170 7365 6454 696d 6520 633d 6368 6172 apsedTime c=char │ │ │ │ 000468d0: 6163 7465 7220 4120 2020 2020 2020 2020 acter A │ │ │ │ 000468e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000468f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046900: 2020 2020 2020 7c0a 7c20 2d2d 202e 3834 |.| -- .84 │ │ │ │ -00046910: 3231 3639 7320 656c 6170 7365 6420 2020 2169s elapsed │ │ │ │ +00046900: 2020 2020 2020 7c0a 7c20 2d2d 202e 3436 |.| -- .46 │ │ │ │ +00046910: 3235 3038 7320 656c 6170 7365 6420 2020 2508s elapsed │ │ │ │ 00046920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046950: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00046960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -19501,15 +19501,15 @@ │ │ │ │ 0004c2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c2f0: 2b0a 7c69 3230 203a 2065 6c61 7073 6564 +.|i20 : elapsed │ │ │ │ 0004c300: 5469 6d65 2061 3120 3d20 6368 6172 6163 Time a1 = charac │ │ │ │ 0004c310: 7465 7220 4131 2020 2020 2020 2020 2020 ter A1 │ │ │ │ 0004c320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004c330: 2020 7c0a 7c20 2d2d 202e 3737 3234 3233 |.| -- .772423 │ │ │ │ +0004c330: 2020 7c0a 7c20 2d2d 202e 3636 3631 3035 |.| -- .666105 │ │ │ │ 0004c340: 7320 656c 6170 7365 6420 2020 2020 2020 s elapsed │ │ │ │ 0004c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c370: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0004c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -19555,15 +19555,15 @@ │ │ │ │ 0004c620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c640: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231 ----------+.|i21 │ │ │ │ 0004c650: 203a 2065 6c61 7073 6564 5469 6d65 2061 : elapsedTime a │ │ │ │ 0004c660: 3220 3d20 6368 6172 6163 7465 7220 4132 2 = character A2 │ │ │ │ 0004c670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c680: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0004c690: 2d2d 2033 322e 3330 3932 7320 656c 6170 -- 32.3092s elap │ │ │ │ +0004c690: 2d2d 2032 362e 3233 3438 7320 656c 6170 -- 26.2348s elap │ │ │ │ 0004c6a0: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 0004c6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c6c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0004c6d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0004c6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -20066,15 +20066,15 @@ │ │ │ │ 0004e610: 4f6e 4772 6164 6564 4d6f 6475 6c65 2020 OnGradedModule │ │ │ │ 0004e620: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 0004e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004e650: 2d2b 0a7c 6933 3220 3a20 656c 6170 7365 -+.|i32 : elapse │ │ │ │ 0004e660: 6454 696d 6520 6220 3d20 6368 6172 6163 dTime b = charac │ │ │ │ 0004e670: 7465 7228 422c 3231 297c 0a7c 202d 2d20 ter(B,21)|.| -- │ │ │ │ -0004e680: 3135 2e31 3839 3473 2065 6c61 7073 6564 15.1894s elapsed │ │ │ │ +0004e680: 3132 2e30 3234 3773 2065 6c61 7073 6564 12.0247s elapsed │ │ │ │ 0004e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004e6a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0004e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004e6c0: 2020 2020 2020 2020 207c 0a7c 6f33 3220 |.|o32 │ │ │ │ 0004e6d0: 3d20 4368 6172 6163 7465 7220 6f76 6572 = Character over │ │ │ │ 0004e6e0: 206b 6b20 2020 2020 2020 2020 2020 2020 kk │ │ │ │ 0004e6f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ ├── ./usr/share/info/Bruns.info.gz │ │ │ ├── Bruns.info │ │ │ │ @@ -1095,18 +1095,18 @@ │ │ │ │ 00004460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00004480: 6932 3320 3a20 7469 6d65 206a 3d62 7275 i23 : time j=bru │ │ │ │ 00004490: 6e73 2046 2e64 645f 333b 2020 2020 2020 ns F.dd_3; │ │ │ │ 000044a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000044b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000044c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000044d0: 202d 2d20 7573 6564 2030 2e32 3638 3139 -- used 0.26819 │ │ │ │ -000044e0: 3573 2028 6370 7529 3b20 302e 3139 3738 5s (cpu); 0.1978 │ │ │ │ -000044f0: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ -00004500: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +000044d0: 202d 2d20 7573 6564 2030 2e33 3131 3934 -- used 0.31194 │ │ │ │ +000044e0: 3673 2028 6370 7529 3b20 302e 3234 3539 6s (cpu); 0.2459 │ │ │ │ +000044f0: 3632 7320 2874 6872 6561 6429 3b20 3073 62s (thread); 0s │ │ │ │ +00004500: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00004510: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00004520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004560: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00004570: 6f32 3320 3a20 4964 6561 6c20 6f66 2053 o23 : Ideal of S │ │ ├── ./usr/share/info/CellularResolutions.info.gz │ │ │ ├── CellularResolutions.info │ │ │ │ @@ -1505,25 +1505,25 @@ │ │ │ │ 00005e00: 6d65 6e73 696f 6e20 3120 7769 7468 206c mension 1 with l │ │ │ │ 00005e10: 6162 656c 7c0a 7c20 2020 2020 202d 2d2d abel|.| --- │ │ │ │ 00005e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005e60: 2d2d 2d2d 7c0a 7c20 2020 2020 2031 2c20 ----|.| 1, │ │ │ │ -00005e70: 3129 2c20 2843 656c 6c20 6f66 2064 696d 1), (Cell of dim │ │ │ │ -00005e80: 656e 7369 6f6e 2031 2077 6974 6820 6c61 ension 1 with la │ │ │ │ -00005e90: 6265 6c20 312c 202d 3129 2c20 2843 656c bel 1, -1), (Cel │ │ │ │ -00005ea0: 6c20 6f66 2064 696d 656e 7369 6f6e 2031 l of dimension 1 │ │ │ │ -00005eb0: 2020 2020 7c0a 7c20 2020 2020 202d 2d2d |.| --- │ │ │ │ +00005e70: 2d31 292c 2028 4365 6c6c 206f 6620 6469 -1), (Cell of di │ │ │ │ +00005e80: 6d65 6e73 696f 6e20 3120 7769 7468 206c mension 1 with l │ │ │ │ +00005e90: 6162 656c 2031 2c20 2d31 292c 2028 4365 abel 1, -1), (Ce │ │ │ │ +00005ea0: 6c6c 206f 6620 6469 6d65 6e73 696f 6e20 ll of dimension │ │ │ │ +00005eb0: 3120 2020 7c0a 7c20 2020 2020 202d 2d2d 1 |.| --- │ │ │ │ 00005ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005f00: 2d2d 2d2d 7c0a 7c20 2020 2020 2077 6974 ----|.| wit │ │ │ │ -00005f10: 6820 6c61 6265 6c20 312c 202d 3129 7d20 h label 1, -1)} │ │ │ │ +00005f10: 6820 6c61 6265 6c20 312c 2031 297d 2020 h label 1, 1)} │ │ │ │ 00005f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005f50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00005f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1992,41 +1992,41 @@ │ │ │ │ 00007c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00007c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00007c90: 3130 203a 2064 3120 3d20 626f 756e 6461 10 : d1 = bounda │ │ │ │ 00007ca0: 7279 4d61 705f 3120 4320 2020 2020 2020 ryMap_1 C │ │ │ │ 00007cb0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00007cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007cd0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00007ce0: 7c6f 3130 203d 207b 317d 207c 202d 7820 |o10 = {1} | -x │ │ │ │ -00007cf0: 3020 202d 7920 7c20 2020 2020 2020 2020 0 -y | │ │ │ │ +00007ce0: 7c6f 3130 203d 207b 317d 207c 202d 7920 |o10 = {1} | -y │ │ │ │ +00007cf0: 2d78 2030 2020 7c20 2020 2020 2020 2020 -x 0 | │ │ │ │ 00007d00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00007d10: 7b31 7d20 7c20 7a20 2079 2020 3020 207c {1} | z y 0 | │ │ │ │ +00007d10: 7b31 7d20 7c20 3020 207a 2020 7920 207c {1} | 0 z y | │ │ │ │ 00007d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00007d30: 7c0a 7c20 2020 2020 207b 317d 207c 2030 |.| {1} | 0 │ │ │ │ -00007d40: 2020 2d78 207a 2020 7c20 2020 2020 2020 -x z | │ │ │ │ +00007d30: 7c0a 7c20 2020 2020 207b 317d 207c 207a |.| {1} | z │ │ │ │ +00007d40: 2020 3020 202d 7820 7c20 2020 2020 2020 0 -x | │ │ │ │ 00007d50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00007d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d80: 2020 7c0a 7c6f 3130 203a 204d 6174 7269 |.|o10 : Matri │ │ │ │ 00007d90: 7820 2020 2020 2020 2020 2020 2020 2020 x │ │ │ │ 00007da0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00007db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00007dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00007dd0: 2d2d 2d2d 2b0a 7c69 3131 203a 2064 3220 ----+.|i11 : d2 │ │ │ │ 00007de0: 3d20 626f 756e 6461 7279 4d61 705f 3220 = boundaryMap_2 │ │ │ │ 00007df0: 4320 2020 2020 2020 2020 2020 207c 0a7c C |.| │ │ │ │ 00007e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e20: 2020 2020 2020 7c0a 7c6f 3131 203d 207b |.|o11 = { │ │ │ │ -00007e30: 327d 207c 202d 7920 7c20 2020 2020 2020 2} | -y | │ │ │ │ +00007e30: 327d 207c 2078 2020 7c20 2020 2020 2020 2} | x | │ │ │ │ 00007e40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00007e50: 0a7c 2020 2020 2020 7b32 7d20 7c20 7a20 .| {2} | z │ │ │ │ +00007e50: 0a7c 2020 2020 2020 7b32 7d20 7c20 2d79 .| {2} | -y │ │ │ │ 00007e60: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00007e70: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00007e80: 207b 327d 207c 2078 2020 7c20 2020 2020 {2} | x | │ │ │ │ +00007e80: 207b 327d 207c 207a 2020 7c20 2020 2020 {2} | z | │ │ │ │ 00007e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007ea0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00007eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007ec0: 2020 2020 2020 2020 2020 7c0a 7c6f 3131 |.|o11 │ │ │ │ 00007ed0: 203a 204d 6174 7269 7820 2020 2020 2020 : Matrix │ │ │ │ 00007ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007ef0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ @@ -3003,63 +3003,63 @@ │ │ │ │ 0000bba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0000bbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbf0: 2020 2020 2020 7c0a 7c6f 3820 3d20 5265 |.|o8 = Re │ │ │ │ 0000bc00: 6c61 7469 6f6e 204d 6174 7269 783a 207c lation Matrix: | │ │ │ │ -0000bc10: 2031 2030 2030 2030 2030 2031 2031 2031 1 0 0 0 0 1 1 1 │ │ │ │ -0000bc20: 2030 2030 2030 2031 2030 2031 2031 2031 0 0 0 1 0 1 1 1 │ │ │ │ -0000bc30: 2031 207c 7c0a 7c20 2020 2020 2020 2020 1 ||.| │ │ │ │ +0000bc10: 2031 2030 2030 2030 2030 2031 2031 2030 1 0 0 0 0 1 1 0 │ │ │ │ +0000bc20: 2031 2030 2030 2030 2030 2031 2031 2030 1 0 0 0 0 1 1 0 │ │ │ │ +0000bc30: 2030 207c 7c0a 7c20 2020 2020 2020 2020 0 ||.| │ │ │ │ 0000bc40: 2020 2020 2020 2020 2020 2020 207c 2030 | 0 │ │ │ │ -0000bc50: 2031 2030 2030 2030 2030 2030 2030 2031 1 0 0 0 0 0 0 1 │ │ │ │ -0000bc60: 2031 2030 2031 2030 2031 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2031 2031 207c 7c0a 0 1 0 0 1 1 ||. │ │ │ │ 0000bcf0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0000bd00: 2020 2020 2020 207c 2030 2030 2030 2030 | 0 0 0 0 │ │ │ │ -0000bd10: 2031 2030 2030 2031 2030 2030 2031 2030 1 0 0 1 0 0 1 0 │ │ │ │ -0000bd20: 2031 2030 2030 2031 2031 207c 7c0a 7c20 1 0 0 1 1 ||.| │ │ │ │ +0000bd10: 2031 2030 2030 2030 2031 2030 2031 2031 1 0 0 0 1 0 1 1 │ │ │ │ +0000bd20: 2031 2031 2031 2031 2031 207c 7c0a 7c20 1 1 1 1 1 ||.| │ │ │ │ 0000bd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bd40: 2020 2020 207c 2030 2030 2030 2030 2030 | 0 0 0 0 0 │ │ │ │ 0000bd50: 2031 2030 2030 2030 2030 2030 2030 2030 1 0 0 0 0 0 0 0 │ │ │ │ -0000bd60: 2031 2030 2031 2030 207c 7c0a 7c20 2020 1 0 1 0 ||.| │ │ │ │ +0000bd60: 2031 2030 2030 2030 207c 7c0a 7c20 2020 1 0 0 0 ||.| │ │ │ │ 0000bd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bd80: 2020 207c 2030 2030 2030 2030 2030 2030 | 0 0 0 0 0 0 │ │ │ │ 0000bd90: 2031 2030 2030 2030 2030 2030 2030 2030 1 0 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2030 2030 2030 2030 0 0 0 0 0 0 0 0 │ │ │ │ -0000bf10: 2031 2030 2030 2030 2031 207c 7c0a 7c20 1 0 0 0 1 ||.| │ │ │ │ +0000bf10: 2031 2030 2030 2031 2031 207c 7c0a 7c20 1 0 0 1 1 ||.| │ │ │ │ 0000bf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bf30: 2020 2020 207c 2030 2030 2030 2030 2030 | 0 0 0 0 0 │ │ │ │ 0000bf40: 2030 2030 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 0 │ │ │ │ 0000bf50: 2031 2030 2030 2030 207c 7c0a 7c20 2020 1 0 0 0 ||.| │ │ │ │ 0000bf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bf70: 2020 207c 2030 2030 2030 2030 2030 2030 | 0 0 0 0 0 0 │ │ │ │ 0000bf80: 2030 2030 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 0 │ │ │ │ @@ -3679,27 +3679,27 @@ │ │ │ │ 0000e5e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000e5f0: 0a7c 6f38 203d 2048 6173 6854 6162 6c65 .|o8 = HashTable │ │ │ │ 0000e600: 7b30 203d 3e20 7b78 202c 2078 2079 2c20 {0 => {x , x y, │ │ │ │ 0000e610: 7820 7920 2c20 7820 7920 2c20 782a 7920 x y , x y , x*y │ │ │ │ 0000e620: 2c20 7820 7d20 2020 2020 2020 2020 2020 , x } │ │ │ │ 0000e630: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000e640: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0000e650: 2020 2020 2020 2020 3520 3320 2020 3520 5 3 5 │ │ │ │ -0000e660: 3420 2020 3420 3220 2020 3420 3420 2020 4 4 2 4 4 │ │ │ │ -0000e670: 3520 2020 2033 2033 2020 2035 2032 2020 5 3 3 5 2 │ │ │ │ -0000e680: 2032 2034 2020 2035 2033 2020 2020 207c 2 4 5 3 | │ │ │ │ +0000e650: 2020 2020 2020 2020 3420 3420 2020 3520 4 4 5 │ │ │ │ +0000e660: 2020 2033 2033 2020 2035 2032 2020 2032 3 3 5 2 2 │ │ │ │ +0000e670: 2034 2020 2035 2033 2020 2035 2034 2020 4 5 3 5 4 │ │ │ │ +0000e680: 2035 2020 2020 3520 3220 2020 2020 207c 5 5 2 | │ │ │ │ 0000e690: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000e6a0: 2031 203d 3e20 7b78 2079 202c 2078 2079 1 => {x y , x y │ │ │ │ -0000e6b0: 202c 2078 2079 202c 2078 2079 202c 2078 , x y , x y , x │ │ │ │ -0000e6c0: 2079 2c20 7820 7920 2c20 7820 7920 2c20 y, x y , x y , │ │ │ │ -0000e6d0: 7820 7920 2c20 7820 7920 2c20 2020 207c x y , x y , | │ │ │ │ +0000e6b0: 2c20 7820 7920 2c20 7820 7920 2c20 7820 , x y , x y , x │ │ │ │ +0000e6c0: 7920 2c20 7820 7920 2c20 7820 7920 2c20 y , x y , x y , │ │ │ │ +0000e6d0: 7820 792c 2078 2079 202c 2020 2020 207c x y, x y , | │ │ │ │ 0000e6e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0000e6f0: 2020 2020 2020 2020 3520 3320 2020 3520 5 3 5 │ │ │ │ -0000e700: 3420 2020 3520 3220 2020 3520 3420 2020 4 5 2 5 4 │ │ │ │ -0000e710: 3520 3320 2020 3520 3420 2020 3520 3220 5 3 5 4 5 2 │ │ │ │ +0000e6f0: 2020 2020 2020 2020 3520 3220 2020 3520 5 2 5 │ │ │ │ +0000e700: 3420 2020 3520 3320 2020 3520 3420 2020 4 5 3 5 4 │ │ │ │ +0000e710: 3520 3220 2020 3520 3420 2020 3520 3320 5 2 5 4 5 3 │ │ │ │ 0000e720: 2020 3520 3420 2020 2020 2020 2020 207c 5 4 | │ │ │ │ 0000e730: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000e740: 2032 203d 3e20 7b78 2079 202c 2078 2079 2 => {x y , x y │ │ │ │ 0000e750: 202c 2078 2079 202c 2078 2079 202c 2078 , x y , x y , x │ │ │ │ 0000e760: 2079 202c 2078 2079 202c 2078 2079 202c y , x y , x y , │ │ │ │ 0000e770: 2078 2079 207d 2020 2020 2020 2020 207c x y } | │ │ │ │ 0000e780: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ @@ -3714,25 +3714,25 @@ │ │ │ │ 0000e810: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000e820: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 0000e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ 0000e870: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0000e880: 2020 7d20 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0000e880: 2020 207d 2020 2020 2020 2020 2020 2020 } │ │ │ │ 0000e890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e8b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000e8c0: 0a7c 2035 2034 2020 2035 2020 2020 3520 .| 5 4 5 5 │ │ │ │ -0000e8d0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0000e8c0: 0a7c 2035 2033 2020 2035 2034 2020 2034 .| 5 3 5 4 4 │ │ │ │ +0000e8d0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0000e8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e900: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000e910: 0a7c 7820 7920 2c20 7820 792c 2078 2079 .|x y , x y, x y │ │ │ │ -0000e920: 207d 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0000e910: 0a7c 7820 7920 2c20 7820 7920 2c20 7820 .|x y , x y , x │ │ │ │ +0000e920: 7920 7d20 2020 2020 2020 2020 2020 2020 y } │ │ │ │ 0000e930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e950: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000e960: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0000e970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -4695,15 +4695,15 @@ │ │ │ │ 00012560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000125a0: 2020 207c 0a7c 6f37 203d 2048 6173 6854 |.|o7 = HashT │ │ │ │ 000125b0: 6162 6c65 7b30 203d 3e20 7b43 656c 6c20 able{0 => {Cell │ │ │ │ 000125c0: 6f66 2064 696d 656e 7369 6f6e 2030 2077 of dimension 0 w │ │ │ │ -000125d0: 6974 6820 6c61 6265 6c20 7a2c 2043 656c ith label z, Cel │ │ │ │ +000125d0: 6974 6820 6c61 6265 6c20 792c 2043 656c ith label y, Cel │ │ │ │ 000125e0: 6c20 6f66 2064 696d 656e 7369 6f6e 2030 l of dimension 0 │ │ │ │ 000125f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00012600: 2020 2020 2031 203d 3e20 7b43 656c 6c20 1 => {Cell │ │ │ │ 00012610: 6f66 2064 696d 656e 7369 6f6e 2031 2077 of dimension 1 w │ │ │ │ 00012620: 6974 6820 6c61 6265 6c20 782a 797d 2020 ith label x*y} │ │ │ │ 00012630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012640: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ @@ -4720,15 +4720,15 @@ │ │ │ │ 000126f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012730: 2d2d 2d7c 0a7c 7769 7468 206c 6162 656c ---|.|with label │ │ │ │ 00012740: 2078 2c20 4365 6c6c 206f 6620 6469 6d65 x, Cell of dime │ │ │ │ 00012750: 6e73 696f 6e20 3020 7769 7468 206c 6162 nsion 0 with lab │ │ │ │ -00012760: 656c 2079 7d7d 2020 2020 2020 2020 2020 el y}} │ │ │ │ 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7369 6f6e 2030 2077 6974 dimension 0 wit │ │ │ │ -000138a0: 6820 6c61 6265 6c20 7a2c 2043 656c 6c20 h label z, Cell │ │ │ │ +000138a0: 6820 6c61 6265 6c20 792c 2043 656c 6c20 h label y, Cell │ │ │ │ 000138b0: 6f66 2064 696d 656e 7369 6f6e 2030 2077 of dimension 0 w │ │ │ │ 000138c0: 6974 6820 6c61 6265 6c20 782c 2020 2020 ith label x, │ │ │ │ 000138d0: 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d |.| -------- │ │ │ │ 000138e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000138f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013920: 7c0a 7c20 2020 2020 4365 6c6c 206f 6620 |.| Cell of │ │ │ │ 00013930: 6469 6d65 6e73 696f 6e20 3020 7769 7468 dimension 0 with │ │ │ │ -00013940: 206c 6162 656c 2079 7d20 2020 2020 2020 label y} │ │ │ │ +00013940: 206c 6162 656c 207a 7d20 2020 2020 2020 label z} │ │ │ │ 00013950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013970: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00013980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000139a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000139b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -6160,24 +6160,24 @@ │ │ │ │ 000180f0: 6f73 6574 2043 2020 2020 2020 2020 2020 oset C │ │ │ │ 00018100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018110: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00018120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018140: 7c0a 7c6f 3132 203d 2052 656c 6174 696f |.|o12 = Relatio │ │ │ │ 00018150: 6e20 4d61 7472 6978 3a20 7c20 3120 3020 n Matrix: | 1 0 │ │ │ │ -00018160: 3020 3020 3120 3020 3120 3020 3120 7c7c 0 0 1 0 1 0 1 || │ │ │ │ +00018160: 3020 3020 3020 3120 3120 3020 3120 7c7c 0 0 0 1 1 0 1 || │ │ │ │ 00018170: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00018180: 2020 2020 2020 2020 207c 2030 2031 2030 | 0 1 0 │ │ │ │ -00018190: 2030 2030 2030 2031 2031 2031 207c 7c0a 0 0 0 1 1 1 ||. │ │ │ │ +00018190: 2030 2030 2031 2030 2031 2031 207c 7c0a 0 0 1 0 1 1 ||. │ │ │ │ 000181a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000181b0: 2020 2020 2020 2020 7c20 3020 3020 3120 | 0 0 1 │ │ │ │ -000181c0: 3020 3020 3120 3020 3120 3120 7c7c 0a7c 0 0 1 0 1 1 ||.| │ │ │ │ +000181c0: 3020 3120 3020 3020 3120 3120 7c7c 0a7c 0 1 0 0 1 1 ||.| │ │ │ │ 000181d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000181e0: 2020 2020 2020 207c 2030 2030 2030 2031 | 0 0 0 1 │ │ │ │ -000181f0: 2031 2031 2030 2030 2031 207c 7c0a 7c20 1 1 0 0 1 ||.| │ │ │ │ +000181f0: 2031 2030 2031 2030 2031 207c 7c0a 7c20 1 0 1 0 1 ||.| │ │ │ │ 00018200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018210: 2020 2020 2020 7c20 3020 3020 3020 3020 | 0 0 0 0 │ │ │ │ 00018220: 3120 3020 3020 3020 3120 7c7c 0a7c 2020 1 0 0 0 1 ||.| │ │ │ │ 00018230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018240: 2020 2020 207c 2030 2030 2030 2030 2030 | 0 0 0 0 0 │ │ │ │ 00018250: 2031 2030 2030 2031 207c 7c0a 7c20 2020 1 0 0 1 ||.| │ │ │ │ 00018260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7252,23 +7252,23 @@ │ │ │ │ 0001c530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c550: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0001c560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c5a0: 7c0a 7c20 2020 2020 2020 3320 3520 2020 |.| 3 5 │ │ │ │ -0001c5b0: 2034 2035 2020 2032 2020 2020 2020 2020 4 5 2 │ │ │ │ -0001c5c0: 3220 2020 2034 2034 2020 2020 3520 3320 2 4 4 5 3 │ │ │ │ -0001c5d0: 2020 2035 2034 2020 2020 2020 2020 2020 5 4 │ │ │ │ +0001c5a0: 7c0a 7c20 2020 2020 2020 3420 3420 2020 |.| 4 4 │ │ │ │ +0001c5b0: 2035 2033 2020 2020 3520 3420 2020 3320 5 3 5 4 3 │ │ │ │ +0001c5c0: 3520 2020 2034 2035 2020 2032 2020 2020 5 4 5 2 │ │ │ │ +0001c5d0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ 0001c5e0: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ -0001c5f0: 203d 207b 7820 7920 7a2c 2078 2079 202c = {x y z, x y , │ │ │ │ -0001c600: 2078 2079 2a7a 2c20 782a 7920 7a2c 2078 x y*z, x*y z, x │ │ │ │ -0001c610: 2079 207a 2c20 7820 7920 7a2c 2078 2079 y z, x y z, x y │ │ │ │ -0001c620: 207d 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0001c5f0: 203d 207b 7820 7920 7a2c 2078 2079 207a = {x y z, x y z │ │ │ │ +0001c600: 2c20 7820 7920 2c20 7820 7920 7a2c 2078 , x y , x y z, x │ │ │ │ +0001c610: 2079 202c 2078 2079 2a7a 2c20 782a 7920 y , x y*z, x*y │ │ │ │ +0001c620: 7a7d 2020 2020 2020 2020 2020 2020 2020 z} │ │ │ │ 0001c630: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0001c640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c680: 207c 0a7c 6f35 203a 204c 6973 7420 2020 |.|o5 : List │ │ │ │ 0001c690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9402,22 +9402,22 @@ │ │ │ │ 00024b90: 6162 656c 2863 2920 2020 2020 2020 2020 abel(c) │ │ │ │ 00024ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024bb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00024bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024c00: 7c0a 7c20 2020 2020 2020 2032 2032 2020 |.| 2 2 │ │ │ │ -00024c10: 2020 2020 2032 2020 2032 2032 2020 2032 2 2 2 2 │ │ │ │ -00024c20: 2020 2032 2020 2020 2032 2020 2020 2020 2 2 │ │ │ │ +00024c00: 7c0a 7c20 2020 2020 2020 2020 2020 2032 |.| 2 │ │ │ │ +00024c10: 2020 2020 2032 2020 2020 3220 3220 2020 2 2 2 │ │ │ │ +00024c20: 3220 3220 2020 3220 2020 3220 2020 2020 2 2 2 2 │ │ │ │ 00024c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024c50: 7c0a 7c6f 3134 203d 207b 6120 6220 2c20 |.|o14 = {a b , │ │ │ │ -00024c60: 612a 622a 6320 2c20 6220 6320 2c20 6120 a*b*c , b c , a │ │ │ │ -00024c70: 622a 6320 2c20 612a 6220 637d 2020 2020 b*c , a*b c} │ │ │ │ +00024c50: 7c0a 7c6f 3134 203d 207b 612a 622a 6320 |.|o14 = {a*b*c │ │ │ │ +00024c60: 2c20 612a 6220 632c 2061 2062 202c 2062 , a*b c, a b , b │ │ │ │ +00024c70: 2063 202c 2061 2062 2a63 207d 2020 2020 c , a b*c } │ │ │ │ 00024c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024ca0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00024cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9677,15 +9677,15 @@ │ │ │ │ 00025cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025ce0: 2b0a 7c69 3130 203a 2063 656c 6c73 2831 +.|i10 : cells(1 │ │ │ │ 00025cf0: 2c43 292f 6365 6c6c 4c61 6265 6c20 2020 ,C)/cellLabel │ │ │ │ 00025d00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00025d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025d20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00025d30: 7c6f 3130 203d 207b 792a 7a2c 2078 2a79 |o10 = {y*z, x*y │ │ │ │ +00025d30: 7c6f 3130 203d 207b 782a 792c 2079 2a7a |o10 = {x*y, y*z │ │ │ │ 00025d40: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ 00025d50: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00025d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025d70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 00025d80: 3130 203a 204c 6973 7420 2020 2020 2020 10 : List │ │ │ │ 00025d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025da0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ @@ -9894,21 +9894,21 @@ │ │ │ │ 00026a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026a70: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00026a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ac0: 2020 207c 0a7c 2020 2020 2020 2034 2035 |.| 4 5 │ │ │ │ -00026ad0: 2020 2020 2032 2020 2020 3520 3420 2020 2 5 4 │ │ │ │ -00026ae0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00026ac0: 2020 207c 0a7c 2020 2020 2020 2032 2020 |.| 2 │ │ │ │ +00026ad0: 2020 2020 3420 3520 2020 3520 3420 2020 4 5 5 4 │ │ │ │ +00026ae0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00026af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00026b10: 7c6f 3520 3d20 7b78 2079 202c 2078 2a79 |o5 = {x y , x*y │ │ │ │ -00026b20: 207a 2c20 7820 7920 2c20 7820 792a 7a7d z, x y , x y*z} │ │ │ │ +00026b10: 7c6f 3520 3d20 7b78 2079 2a7a 2c20 7820 |o5 = {x y*z, x │ │ │ │ +00026b20: 7920 2c20 7820 7920 2c20 782a 7920 7a7d y , x y , x*y z} │ │ │ │ 00026b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00026b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -10856,38 +10856,38 @@ │ │ │ │ 0002a670: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0002a680: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0002a690: 2020 2020 2020 2020 2020 2020 2033 2020 3 │ │ │ │ 0002a6a0: 2020 7c0a 7c20 2020 2020 3020 3a20 5320 |.| 0 : S │ │ │ │ 0002a6b0: 203c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d <-------------- │ │ │ │ 0002a6c0: 2d2d 2d2d 2d2d 2d2d 2d20 5320 203a 2031 --------- S : 1 │ │ │ │ 0002a6d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002a6e0: 2020 7b32 7d20 7c20 2d79 3220 3020 2020 {2} | -y2 0 │ │ │ │ -0002a6f0: 2d7a 3220 7c20 2020 2020 2020 2020 7c0a -z2 | |. │ │ │ │ +0002a6e0: 2020 7b32 7d20 7c20 2d7a 3220 2d79 3220 {2} | -z2 -y2 │ │ │ │ +0002a6f0: 3020 2020 7c20 2020 2020 2020 2020 7c0a 0 | |. │ │ │ │ 0002a700: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002a710: 7b32 7d20 7c20 7832 2020 2d7a 3220 3020 {2} | x2 -z2 0 │ │ │ │ -0002a720: 2020 7c20 2020 2020 2020 2020 7c0a 7c20 | |.| │ │ │ │ +0002a710: 7b32 7d20 7c20 3020 2020 7832 2020 2d7a {2} | 0 x2 -z │ │ │ │ +0002a720: 3220 7c20 2020 2020 2020 2020 7c0a 7c20 2 | |.| │ │ │ │ 0002a730: 2020 2020 2020 2020 2020 2020 2020 7b32 {2 │ │ │ │ -0002a740: 7d20 7c20 3020 2020 7932 2020 7832 2020 } | 0 y2 x2 │ │ │ │ +0002a740: 7d20 7c20 7832 2020 3020 2020 7932 2020 } | x2 0 y2 │ │ │ │ 0002a750: 7c20 2020 2020 2020 2020 7c0a 7c20 2020 | |.| │ │ │ │ 0002a760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a780: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0002a790: 2020 2020 2033 2020 2020 2020 2020 2020 3 │ │ │ │ 0002a7a0: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ 0002a7b0: 2020 2020 2020 7c0a 7c20 2020 2020 3120 |.| 1 │ │ │ │ 0002a7c0: 3a20 5320 203c 2d2d 2d2d 2d2d 2d2d 2d2d : S <---------- │ │ │ │ 0002a7d0: 2d2d 2d2d 2d20 5320 203a 2032 2020 2020 ----- S : 2 │ │ │ │ 0002a7e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002a7f0: 2020 2020 2020 7b34 7d20 7c20 2d7a 3220 {4} | -z2 │ │ │ │ +0002a7f0: 2020 2020 2020 7b34 7d20 7c20 7932 2020 {4} | y2 │ │ │ │ 0002a800: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002a810: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002a820: 2020 2020 7b34 7d20 7c20 2d78 3220 7c20 {4} | -x2 | │ │ │ │ +0002a820: 2020 2020 7b34 7d20 7c20 2d7a 3220 7c20 {4} | -z2 | │ │ │ │ 0002a830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a840: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002a850: 2020 7b34 7d20 7c20 7932 2020 7c20 2020 {4} | y2 | │ │ │ │ +0002a850: 2020 7b34 7d20 7c20 2d78 3220 7c20 2020 {4} | -x2 | │ │ │ │ 0002a860: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0002a870: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002a880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a890: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 0002a8a0: 3520 3a20 436f 6d70 6c65 784d 6170 2020 5 : ComplexMap │ │ │ │ 0002a8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a8c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ ├── ./usr/share/info/ChainComplexExtras.info.gz │ │ │ ├── ChainComplexExtras.info │ │ │ │ @@ -4819,16 +4819,16 @@ │ │ │ │ 00012d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00012d50: 3133 203a 2074 696d 6520 6d20 3d20 6d69 13 : time m = mi │ │ │ │ 00012d60: 6e69 6d69 7a65 2028 455b 315d 293b 2020 nimize (E[1]); │ │ │ │ 00012d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012d80: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00012d90: 302e 3330 3539 3332 7320 2863 7075 293b 0.305932s (cpu); │ │ │ │ -00012da0: 2030 2e32 3235 3330 3573 2028 7468 7265 0.225305s (thre │ │ │ │ +00012d90: 302e 3233 3238 3136 7320 2863 7075 293b 0.232816s (cpu); │ │ │ │ +00012da0: 2030 2e32 3332 3831 3573 2028 7468 7265 0.232815s (thre │ │ │ │ 00012db0: 6164 293b 2030 7320 2867 6329 7c0a 2b2d ad); 0s (gc)|.+- │ │ │ │ 00012dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012df0: 2d2d 2d2d 2b0a 7c69 3134 203a 2069 7351 ----+.|i14 : isQ │ │ │ │ 00012e00: 7561 7369 4973 6f6d 6f72 7068 6973 6d20 uasiIsomorphism │ │ │ │ 00012e10: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ @@ -6579,33 +6579,33 @@ │ │ │ │ 00019b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00019b40: 3820 3a20 7469 6d65 206d 203d 2072 6573 8 : time m = res │ │ │ │ 00019b50: 6f6c 7574 696f 6e4f 6643 6861 696e 436f olutionOfChainCo │ │ │ │ 00019b60: 6d70 6c65 7820 433b 2020 2020 2020 2020 mplex C; │ │ │ │ 00019b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019b80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00019b90: 2d2d 2075 7365 6420 302e 3039 3435 3173 -- used 0.09451s │ │ │ │ -00019ba0: 2028 6370 7529 3b20 302e 3039 3439 3130 (cpu); 0.094910 │ │ │ │ +00019b90: 2d2d 2075 7365 6420 302e 3131 3939 3938 -- used 0.119998 │ │ │ │ +00019ba0: 7320 2863 7075 293b 2030 2e31 3137 3237 s (cpu); 0.11727 │ │ │ │ 00019bb0: 3973 2028 7468 7265 6164 293b 2030 7320 9s (thread); 0s │ │ │ │ 00019bc0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00019bd0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00019be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00019c30: 3920 3a20 7469 6d65 206e 203d 2063 6172 9 : time n = car │ │ │ │ 00019c40: 7461 6e45 696c 656e 6265 7267 5265 736f tanEilenbergReso │ │ │ │ 00019c50: 6c75 7469 6f6e 2043 3b20 2020 2020 2020 lution C; │ │ │ │ 00019c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00019c80: 2d2d 2075 7365 6420 302e 3131 3831 3932 -- used 0.118192 │ │ │ │ -00019c90: 7320 2863 7075 293b 2030 2e31 3138 3537 s (cpu); 0.11857 │ │ │ │ -00019ca0: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ -00019cb0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00019c80: 2d2d 2075 7365 6420 302e 3135 3138 3933 -- used 0.151893 │ │ │ │ +00019c90: 7320 2863 7075 293b 2030 2e31 3533 7320 s (cpu); 0.153s │ │ │ │ +00019ca0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +00019cb0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00019cc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00019cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00019d20: 3130 203a 2062 6574 7469 2073 6f75 7263 10 : betti sourc │ │ ├── ./usr/share/info/CharacteristicClasses.info.gz │ │ │ ├── CharacteristicClasses.info │ │ │ │ @@ -1215,16 +1215,16 @@ │ │ │ │ 00004be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004bf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ 00004c00: 2074 696d 6520 4353 4d20 5520 2020 2020 time CSM U │ │ │ │ 00004c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004c40: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00004c50: 7573 6564 2030 2e32 3636 3839 3373 2028 used 0.266893s ( │ │ │ │ -00004c60: 6370 7529 3b20 302e 3139 3438 3237 7320 cpu); 0.194827s │ │ │ │ +00004c50: 7573 6564 2030 2e32 3832 3939 3273 2028 used 0.282992s ( │ │ │ │ +00004c60: 6370 7529 3b20 302e 3139 3239 3233 7320 cpu); 0.192923s │ │ │ │ 00004c70: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00004c80: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00004c90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00004ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1300,18 +1300,18 @@ │ │ │ │ 00005130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005140: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ 00005150: 2074 696d 6520 4353 4d28 552c 4368 6563 time CSM(U,Chec │ │ │ │ 00005160: 6b53 6d6f 6f74 683d 3e66 616c 7365 2920 kSmooth=>false) │ │ │ │ 00005170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005190: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000051a0: 7573 6564 2030 2e33 3730 3134 3773 2028 used 0.370147s ( │ │ │ │ -000051b0: 6370 7529 3b20 302e 3238 3931 3534 7320 cpu); 0.289154s │ │ │ │ -000051c0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -000051d0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +000051a0: 7573 6564 2030 2e34 3030 3831 7320 2863 used 0.40081s (c │ │ │ │ +000051b0: 7075 293b 2030 2e33 3237 3737 3573 2028 pu); 0.327775s ( │ │ │ │ +000051c0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +000051d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000051e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 000051f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005230: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00005240: 2020 2037 2020 2020 2020 3620 2020 2020 7 6 │ │ │ │ @@ -4341,16 +4341,16 @@ │ │ │ │ 00010f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010f60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ 00010f70: 2074 696d 6520 4353 4d28 492c 436f 6d70 time CSM(I,Comp │ │ │ │ 00010f80: 4d65 7468 6f64 3d3e 5072 6f6a 6563 7469 Method=>Projecti │ │ │ │ 00010f90: 7665 4465 6772 6565 2920 2020 2020 2020 veDegree) │ │ │ │ 00010fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00010fb0: 2d2d 2075 7365 6420 302e 3532 3335 3773 -- used 0.52357s │ │ │ │ -00010fc0: 2028 6370 7529 3b20 302e 3332 3236 3336 (cpu); 0.322636 │ │ │ │ +00010fb0: 2d2d 2075 7365 6420 312e 3030 3834 3373 -- used 1.00843s │ │ │ │ +00010fc0: 2028 6370 7529 3b20 302e 3431 3132 3832 (cpu); 0.411282 │ │ │ │ 00010fd0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 00010fe0: 6763 2920 2020 2020 2020 2020 2020 207c gc) | │ │ │ │ 00010ff0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00011000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011030: 2020 7c0a 7c20 2020 2020 2020 3520 2020 |.| 5 │ │ │ │ @@ -4400,16 +4400,16 @@ │ │ │ │ 000112f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011310: 2d2d 2d2b 0a7c 6936 203a 2074 696d 6520 ---+.|i6 : time │ │ │ │ 00011320: 4353 4d28 492c 436f 6d70 4d65 7468 6f64 CSM(I,CompMethod │ │ │ │ 00011330: 3d3e 506e 5265 7369 6475 616c 2920 2020 =>PnResidual) │ │ │ │ 00011340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011350: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00011360: 6420 322e 3533 3535 3973 2028 6370 7529 d 2.53559s (cpu) │ │ │ │ -00011370: 3b20 322e 3232 3935 3373 2028 7468 7265 ; 2.22953s (thre │ │ │ │ +00011360: 6420 322e 3233 3730 3773 2028 6370 7529 d 2.23707s (cpu) │ │ │ │ +00011370: 3b20 322e 3035 3433 3773 2028 7468 7265 ; 2.05437s (thre │ │ │ │ 00011380: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00011390: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 000113a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000113e0: 2020 2020 2020 3520 2020 2020 2034 2020 5 4 │ │ │ │ @@ -4488,16 +4488,16 @@ │ │ │ │ 00011870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011890: 2d2d 2b0a 7c69 3130 203a 2074 696d 6520 --+.|i10 : time │ │ │ │ 000118a0: 4353 4d28 4b2c 436f 6d70 4d65 7468 6f64 CSM(K,CompMethod │ │ │ │ 000118b0: 3d3e 5072 6f6a 6563 7469 7665 4465 6772 =>ProjectiveDegr │ │ │ │ 000118c0: 6565 2920 2020 2020 2020 2020 2020 2020 ee) │ │ │ │ 000118d0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000118e0: 2030 2e32 3837 3932 3773 2028 6370 7529 0.287927s (cpu) │ │ │ │ -000118f0: 3b20 302e 3139 3934 3438 7320 2874 6872 ; 0.199448s (thr │ │ │ │ +000118e0: 2030 2e33 3136 3634 3673 2028 6370 7529 0.316646s (cpu) │ │ │ │ +000118f0: 3b20 302e 3232 3730 3534 7320 2874 6872 ; 0.227054s (thr │ │ │ │ 00011900: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00011910: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00011920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011950: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00011960: 2020 2020 2020 3320 2020 2020 3220 2020 3 2 │ │ │ │ @@ -4546,18 +4546,18 @@ │ │ │ │ 00011c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00011c40: 3131 203a 2074 696d 6520 4353 4d28 4b2c 11 : time CSM(K, │ │ │ │ 00011c50: 436f 6d70 4d65 7468 6f64 3d3e 506e 5265 CompMethod=>PnRe │ │ │ │ 00011c60: 7369 6475 616c 2920 2020 2020 2020 2020 sidual) │ │ │ │ 00011c70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011c80: 0a7c 202d 2d20 7573 6564 2030 2e30 3831 .| -- used 0.081 │ │ │ │ -00011c90: 3839 3833 7320 2863 7075 293b 2030 2e30 8983s (cpu); 0.0 │ │ │ │ -00011ca0: 3831 3930 3838 7320 2874 6872 6561 6429 819088s (thread) │ │ │ │ -00011cb0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +00011c80: 0a7c 202d 2d20 7573 6564 2030 2e30 3936 .| -- used 0.096 │ │ │ │ +00011c90: 3937 3473 2028 6370 7529 3b20 302e 3039 974s (cpu); 0.09 │ │ │ │ +00011ca0: 3638 3734 3773 2028 7468 7265 6164 293b 68747s (thread); │ │ │ │ +00011cb0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00011cc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00011cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011d00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00011d10: 3320 2020 2020 3220 2020 2020 2020 2020 3 2 │ │ │ │ 00011d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5545,16 +5545,16 @@ │ │ │ │ 00015a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015ab0: 2d2d 2b0a 7c69 3135 203a 2074 696d 6520 --+.|i15 : time │ │ │ │ 00015ac0: 6373 6d4b 3d43 534d 2841 2c4b 2920 2020 csmK=CSM(A,K) │ │ │ │ 00015ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015ae0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00015af0: 2075 7365 6420 302e 3339 3231 3239 7320 used 0.392129s │ │ │ │ -00015b00: 2863 7075 293b 2030 2e32 3834 3134 3373 (cpu); 0.284143s │ │ │ │ +00015af0: 2075 7365 6420 302e 3938 3832 3639 7320 used 0.988269s │ │ │ │ +00015b00: 2863 7075 293b 2030 2e34 3135 3738 3273 (cpu); 0.415782s │ │ │ │ 00015b10: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 00015b20: 6329 7c0a 7c20 2020 2020 2020 2020 2020 c)|.| │ │ │ │ 00015b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015b50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00015b60: 2020 2020 2032 2032 2020 2020 2032 2020 2 2 2 │ │ │ │ 00015b70: 2020 2020 2020 2032 2020 2020 3220 2020 2 2 │ │ │ │ @@ -5724,16 +5724,16 @@ │ │ │ │ 000165b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000165c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000165d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3220 ---------+.|i22 │ │ │ │ 000165e0: 3a20 7469 6d65 2043 534d 2841 2c4b 2c6d : time CSM(A,K,m │ │ │ │ 000165f0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00016600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016610: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00016620: 6564 2030 2e30 3630 3134 3732 7320 2863 ed 0.0601472s (c │ │ │ │ -00016630: 7075 293b 2030 2e30 3537 3231 3431 7320 pu); 0.0572141s │ │ │ │ +00016620: 6564 2030 2e30 3936 3839 3738 7320 2863 ed 0.0968978s (c │ │ │ │ +00016630: 7075 293b 2030 2e30 3737 3131 3936 7320 pu); 0.0771196s │ │ │ │ 00016640: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00016650: 2920 2020 207c 0a7c 2020 2020 2020 2020 ) |.| │ │ │ │ 00016660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016690: 2020 207c 0a7c 2020 2020 2020 2020 3220 |.| 2 │ │ │ │ 000166a0: 3220 2020 2020 3220 2020 2020 2020 2020 2 2 │ │ │ │ @@ -6694,16 +6694,16 @@ │ │ │ │ 0001a250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a260: 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 696d -----+.|i4 : tim │ │ │ │ 0001a270: 6520 4575 6c65 7228 492c 496e 7075 7449 e Euler(I,InputI │ │ │ │ 0001a280: 7353 6d6f 6f74 683d 3e74 7275 6529 2020 sSmooth=>true) │ │ │ │ 0001a290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a2b0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0001a2c0: 2030 2e30 3333 3537 3131 7320 2863 7075 0.0335711s (cpu │ │ │ │ -0001a2d0: 293b 2030 2e30 3332 3633 3134 7320 2874 ); 0.0326314s (t │ │ │ │ +0001a2c0: 2030 2e30 3734 3338 3733 7320 2863 7075 0.0743873s (cpu │ │ │ │ +0001a2d0: 293b 2030 2e30 3431 3735 3137 7320 2874 ); 0.0417517s (t │ │ │ │ 0001a2e0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 0001a2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a300: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0001a310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -6719,16 +6719,16 @@ │ │ │ │ 0001a3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a3f0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 696d -----+.|i5 : tim │ │ │ │ 0001a400: 6520 4575 6c65 7220 4920 2020 2020 2020 e Euler I │ │ │ │ 0001a410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a440: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0001a450: 2030 2e32 3337 3035 3473 2028 6370 7529 0.237054s (cpu) │ │ │ │ -0001a460: 3b20 302e 3134 3137 3938 7320 2874 6872 ; 0.141798s (thr │ │ │ │ +0001a450: 2030 2e32 3830 3234 3173 2028 6370 7529 0.280241s (cpu) │ │ │ │ +0001a460: 3b20 302e 3136 3736 3337 7320 2874 6872 ; 0.167637s (thr │ │ │ │ 0001a470: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 0001a480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a490: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0001a4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -6922,18 +6922,18 @@ │ │ │ │ 0001b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b0b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 ---------+.|i10 │ │ │ │ 0001b0c0: 3a20 7469 6d65 2045 756c 6572 284a 2c4d : time Euler(J,M │ │ │ │ 0001b0d0: 6574 686f 643d 3e44 6972 6563 7443 6f6d ethod=>DirectCom │ │ │ │ 0001b0e0: 706c 6574 6549 6e74 2920 2020 2020 2020 pleteInt) │ │ │ │ 0001b0f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001b100: 202d 2d20 7573 6564 2030 2e30 3732 3731 -- used 0.07271 │ │ │ │ -0001b110: 3232 7320 2863 7075 293b 2030 2e30 3731 22s (cpu); 0.071 │ │ │ │ -0001b120: 3137 3336 7320 2874 6872 6561 6429 3b20 1736s (thread); │ │ │ │ -0001b130: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +0001b100: 202d 2d20 7573 6564 2030 2e31 3736 3332 -- used 0.17632 │ │ │ │ +0001b110: 7320 2863 7075 293b 2030 2e30 3937 3334 s (cpu); 0.09734 │ │ │ │ +0001b120: 3435 7320 2874 6872 6561 6429 3b20 3073 45s (thread); 0s │ │ │ │ +0001b130: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0001b140: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001b150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b180: 2020 2020 207c 0a7c 6f31 3020 3d20 3220 |.|o10 = 2 │ │ │ │ 0001b190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -6943,18 +6943,18 @@ │ │ │ │ 0001b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 0001b210: 6931 3120 3a20 7469 6d65 2045 756c 6572 i11 : time Euler │ │ │ │ 0001b220: 284a 2c4d 6574 686f 643d 3e44 6972 6563 (J,Method=>Direc │ │ │ │ 0001b230: 7443 6f6d 706c 6574 6549 6e74 2c49 6e64 tCompleteInt,Ind │ │ │ │ 0001b240: 734f 6653 6d6f 6f74 683d 3e7b 302c 317d sOfSmooth=>{0,1} │ │ │ │ -0001b250: 297c 0a7c 202d 2d20 7573 6564 2030 2e31 )|.| -- used 0.1 │ │ │ │ -0001b260: 3832 3233 3273 2028 6370 7529 3b20 302e 82232s (cpu); 0. │ │ │ │ -0001b270: 3130 3332 3337 7320 2874 6872 6561 6429 103237s (thread) │ │ │ │ -0001b280: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +0001b250: 297c 0a7c 202d 2d20 7573 6564 2030 2e32 )|.| -- used 0.2 │ │ │ │ +0001b260: 3233 3833 3873 2028 6370 7529 3b20 302e 23838s (cpu); 0. │ │ │ │ +0001b270: 3130 3136 3773 2028 7468 7265 6164 293b 10167s (thread); │ │ │ │ +0001b280: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 0001b290: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0001b2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b2d0: 2020 2020 2020 2020 207c 0a7c 6f31 3120 |.|o11 │ │ │ │ 0001b2e0: 3d20 3220 2020 2020 2020 2020 2020 2020 = 2 │ │ │ │ 0001b2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7577,17 +7577,17 @@ │ │ │ │ 0001d980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d9b0: 2d2d 2b0a 7c69 3420 3a20 7469 6d65 2045 --+.|i4 : time E │ │ │ │ 0001d9c0: 756c 6572 4166 6669 6e65 2049 2020 2020 ulerAffine I │ │ │ │ 0001d9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d9e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001d9f0: 2d2d 2075 7365 6420 302e 3034 3736 3130 -- used 0.047610 │ │ │ │ -0001da00: 3173 2028 6370 7529 3b20 302e 3034 3732 1s (cpu); 0.0472 │ │ │ │ -0001da10: 3734 3973 2028 7468 7265 6164 293b 2030 749s (thread); 0 │ │ │ │ +0001d9f0: 2d2d 2075 7365 6420 302e 3039 3939 3730 -- used 0.099970 │ │ │ │ +0001da00: 3673 2028 6370 7529 3b20 302e 3037 3037 6s (cpu); 0.0707 │ │ │ │ +0001da10: 3435 3173 2028 7468 7265 6164 293b 2030 451s (thread); 0 │ │ │ │ 0001da20: 7320 2867 6329 7c0a 7c20 2020 2020 2020 s (gc)|.| │ │ │ │ 0001da30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001da40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001da50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001da60: 7c0a 7c6f 3420 3d20 3220 2020 2020 2020 |.|o4 = 2 │ │ │ │ 0001da70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001da80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7720,17 +7720,17 @@ │ │ │ │ 0001e270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ 0001e2a0: 203a 2074 696d 6520 4353 4d28 492c 4d65 : time CSM(I,Me │ │ │ │ 0001e2b0: 7468 6f64 3d3e 4469 7265 6374 436f 6d70 thod=>DirectComp │ │ │ │ 0001e2c0: 6c65 7449 6e74 2920 2020 2020 2020 2020 letInt) │ │ │ │ 0001e2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e2e0: 2020 7c0a 7c20 2d2d 2075 7365 6420 312e |.| -- used 1. │ │ │ │ -0001e2f0: 3434 3933 3173 2028 6370 7529 3b20 312e 44931s (cpu); 1. │ │ │ │ -0001e300: 3035 3334 3473 2028 7468 7265 6164 293b 05344s (thread); │ │ │ │ +0001e2e0: 2020 7c0a 7c20 2d2d 2075 7365 6420 342e |.| -- used 4. │ │ │ │ +0001e2f0: 3435 3539 3973 2028 6370 7529 3b20 312e 45599s (cpu); 1. │ │ │ │ +0001e300: 3236 3938 3773 2028 7468 7265 6164 293b 26987s (thread); │ │ │ │ 0001e310: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 0001e320: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0001e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e370: 7c0a 7c20 2020 2020 2020 3220 3220 2020 |.| 2 2 │ │ │ │ @@ -7783,16 +7783,16 @@ │ │ │ │ 0001e660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 0001e680: 6934 203a 2074 696d 6520 4353 4d28 492c i4 : time CSM(I, │ │ │ │ 0001e690: 4d65 7468 6f64 3d3e 4469 7265 6374 436f Method=>DirectCo │ │ │ │ 0001e6a0: 6d70 6c65 7449 6e74 2c49 6e64 734f 6653 mpletInt,IndsOfS │ │ │ │ 0001e6b0: 6d6f 6f74 683d 3e7b 312c 327d 2920 2020 mooth=>{1,2}) │ │ │ │ 0001e6c0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -0001e6d0: 312e 3535 3439 3873 2028 6370 7529 3b20 1.55498s (cpu); │ │ │ │ -0001e6e0: 312e 3136 3731 3573 2028 7468 7265 6164 1.16715s (thread │ │ │ │ +0001e6d0: 352e 3238 3035 3673 2028 6370 7529 3b20 5.28056s (cpu); │ │ │ │ +0001e6e0: 312e 3337 3432 3973 2028 7468 7265 6164 1.37429s (thread │ │ │ │ 0001e6f0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 0001e700: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0001e710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e750: 2020 7c0a 7c20 2020 2020 2020 3220 3220 |.| 2 2 │ │ │ │ @@ -7907,4328 +7907,4332 @@ │ │ │ │ 0001ee20: 6c20 7370 6565 6420 7570 2063 6f6d 7075 l speed up compu │ │ │ │ 0001ee30: 7461 7469 6f6e 7320 2869 7420 6973 2073 tations (it is s │ │ │ │ 0001ee40: 6574 2074 6f20 6661 6c73 6520 6279 0a64 et to false by.d │ │ │ │ 0001ee50: 6566 6175 6c74 292e 0a0a 2b2d 2d2d 2d2d efault)...+----- │ │ │ │ 0001ee60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ee90: 2b0a 7c69 3120 3a20 5220 3d20 5a5a 2f33 +.|i1 : R = ZZ/3 │ │ │ │ -0001eea0: 3237 3439 5b78 5f30 2e2e 785f 345d 3b20 2749[x_0..x_4]; │ │ │ │ -0001eeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eec0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001ee90: 2d2d 2b0a 7c69 3120 3a20 5220 3d20 5a5a --+.|i1 : R = ZZ │ │ │ │ +0001eea0: 2f33 3237 3439 5b78 5f30 2e2e 785f 345d /32749[x_0..x_4] │ │ │ │ +0001eeb0: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +0001eec0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0001eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ef00: 2b0a 7c69 3220 3a20 493d 6964 6561 6c28 +.|i2 : I=ideal( │ │ │ │ -0001ef10: 7261 6e64 6f6d 2832 2c52 292c 7261 6e64 random(2,R),rand │ │ │ │ -0001ef20: 6f6d 2832 2c52 292c 7261 6e64 6f6d 2831 om(2,R),random(1 │ │ │ │ -0001ef30: 2c52 2929 3b20 2020 7c0a 7c20 2020 2020 ,R)); |.| │ │ │ │ -0001ef40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ef00: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 493d ------+.|i2 : I= │ │ │ │ +0001ef10: 6964 6561 6c28 7261 6e64 6f6d 2832 2c52 ideal(random(2,R │ │ │ │ +0001ef20: 292c 7261 6e64 6f6d 2832 2c52 292c 7261 ),random(2,R),ra │ │ │ │ +0001ef30: 6e64 6f6d 2831 2c52 2929 3b20 2020 2020 ndom(1,R)); │ │ │ │ +0001ef40: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001ef50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ef60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ef70: 7c0a 7c6f 3220 3a20 4964 6561 6c20 6f66 |.|o2 : Ideal of │ │ │ │ -0001ef80: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0001ef70: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0001ef80: 3a20 4964 6561 6c20 6f66 2052 2020 2020 : Ideal of R │ │ │ │ 0001ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001efa0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0001efb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001efa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001efb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0001efc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001efd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001efe0: 2b0a 7c69 3320 3a20 7469 6d65 2043 534d +.|i3 : time CSM │ │ │ │ -0001eff0: 2049 2020 2020 2020 2020 2020 2020 2020 I │ │ │ │ +0001efe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0001eff0: 7c69 3320 3a20 7469 6d65 2043 534d 2049 |i3 : time CSM I │ │ │ │ 0001f000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f010: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -0001f020: 7365 6420 302e 3537 3938 3538 7320 2863 sed 0.579858s (c │ │ │ │ -0001f030: 7075 293b 2030 2e33 3938 3933 3173 2028 pu); 0.398931s ( │ │ │ │ -0001f040: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ -0001f050: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001f060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f020: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ +0001f030: 7365 6420 302e 3934 3135 3839 7320 2863 sed 0.941589s (c │ │ │ │ +0001f040: 7075 293b 2030 2e34 3934 3437 3573 2028 pu); 0.494475s ( │ │ │ │ +0001f050: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +0001f060: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001f070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f080: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001f090: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -0001f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f090: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001f0a0: 2020 2020 2020 3320 2020 2020 2020 2020 3 │ │ │ │ 0001f0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f0c0: 7c0a 7c6f 3320 3d20 3468 2020 2020 2020 |.|o3 = 4h │ │ │ │ -0001f0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f0d0: 2020 2020 2020 7c0a 7c6f 3320 3d20 3468 |.|o3 = 4h │ │ │ │ 0001f0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f0f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001f100: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -0001f110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f110: 7c0a 7c20 2020 2020 2020 3120 2020 2020 |.| 1 │ │ │ │ 0001f120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f130: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001f140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f140: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0001f150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f160: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001f170: 5a5a 5b68 205d 2020 2020 2020 2020 2020 ZZ[h ] │ │ │ │ -0001f180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f1a0: 7c0a 7c20 2020 2020 2020 2020 3120 2020 |.| 1 │ │ │ │ -0001f1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f1d0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ -0001f1e0: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020 ------ │ │ │ │ -0001f1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f210: 7c0a 7c20 2020 2020 2020 2035 2020 2020 |.| 5 │ │ │ │ +0001f160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f180: 2020 2020 7c0a 7c20 2020 2020 5a5a 5b68 |.| ZZ[h │ │ │ │ +0001f190: 205d 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +0001f1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f1b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001f1c0: 7c20 2020 2020 2020 2020 3120 2020 2020 | 1 │ │ │ │ +0001f1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f1f0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ +0001f200: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020 ------ │ │ │ │ +0001f210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f240: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001f250: 2020 6820 2020 2020 2020 2020 2020 2020 h │ │ │ │ -0001f260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f280: 7c0a 7c20 2020 2020 2020 2031 2020 2020 |.| 1 │ │ │ │ +0001f230: 2020 7c0a 7c20 2020 2020 2020 2035 2020 |.| 5 │ │ │ │ +0001f240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f260: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001f270: 2020 2020 2020 6820 2020 2020 2020 2020 h │ │ │ │ +0001f280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0001f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f2f0: 2b0a 7c69 3420 3a20 7469 6d65 2043 534d +.|i4 : time CSM │ │ │ │ -0001f300: 2849 2c49 6e70 7574 4973 536d 6f6f 7468 (I,InputIsSmooth │ │ │ │ -0001f310: 3d3e 7472 7565 2920 2020 2020 2020 2020 =>true) │ │ │ │ -0001f320: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -0001f330: 7365 6420 302e 3033 3135 3130 3273 2028 sed 0.0315102s ( │ │ │ │ -0001f340: 6370 7529 3b20 302e 3033 3038 3873 2028 cpu); 0.03088s ( │ │ │ │ -0001f350: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ -0001f360: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001f370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f390: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001f3a0: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +0001f2a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001f2b0: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0001f2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f2e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f310: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ +0001f320: 3a20 7469 6d65 2043 534d 2849 2c49 6e70 : time CSM(I,Inp │ │ │ │ +0001f330: 7574 4973 536d 6f6f 7468 3d3e 7472 7565 utIsSmooth=>true │ │ │ │ +0001f340: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +0001f350: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ +0001f360: 302e 3037 3632 3435 3273 2028 6370 7529 0.0762452s (cpu) │ │ │ │ +0001f370: 3b20 302e 3034 3539 3232 3173 2028 7468 ; 0.0459221s (th │ │ │ │ +0001f380: 7265 6164 293b 2030 7320 2867 6329 7c0a read); 0s (gc)|. │ │ │ │ +0001f390: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001f3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f3d0: 7c0a 7c6f 3420 3d20 3468 2020 2020 2020 |.|o4 = 4h │ │ │ │ +0001f3c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001f3d0: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0001f3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f400: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001f410: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0001f400: 2020 7c0a 7c6f 3420 3d20 3468 2020 2020 |.|o4 = 4h │ │ │ │ +0001f410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f440: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001f430: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001f440: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ 0001f450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f470: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001f480: 5a5a 5b68 205d 2020 2020 2020 2020 2020 ZZ[h ] │ │ │ │ +0001f470: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001f480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f4b0: 7c0a 7c20 2020 2020 2020 2020 3120 2020 |.| 1 │ │ │ │ +0001f4b0: 7c0a 7c20 2020 2020 5a5a 5b68 205d 2020 |.| ZZ[h ] │ │ │ │ 0001f4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f4e0: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ -0001f4f0: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020 ------ │ │ │ │ +0001f4e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001f4f0: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ 0001f500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f520: 7c0a 7c20 2020 2020 2020 2035 2020 2020 |.| 5 │ │ │ │ -0001f530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f520: 2020 2020 7c0a 7c6f 3420 3a20 2d2d 2d2d |.|o4 : ---- │ │ │ │ +0001f530: 2d2d 2020 2020 2020 2020 2020 2020 2020 -- │ │ │ │ 0001f540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f550: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001f560: 2020 6820 2020 2020 2020 2020 2020 2020 h │ │ │ │ +0001f550: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001f560: 7c20 2020 2020 2020 2035 2020 2020 2020 | 5 │ │ │ │ 0001f570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f590: 7c0a 7c20 2020 2020 2020 2031 2020 2020 |.| 1 │ │ │ │ -0001f5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f590: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001f5a0: 2020 6820 2020 2020 2020 2020 2020 2020 h │ │ │ │ 0001f5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f5c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0001f5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f600: 2b0a 0a4e 6f74 6520 7468 6174 206f 6e65 +..Note that one │ │ │ │ -0001f610: 2063 6f75 6c64 2c20 6571 7569 7661 6c65 could, equivale │ │ │ │ -0001f620: 6e74 6c79 2c20 7573 6520 7468 6520 636f ntly, use the co │ │ │ │ -0001f630: 6d6d 616e 6420 2a6e 6f74 6520 4368 6572 mmand *note Cher │ │ │ │ -0001f640: 6e3a 2043 6865 726e 2c20 696e 7374 6561 n: Chern, instea │ │ │ │ -0001f650: 640a 696e 2074 6869 7320 6361 7365 2e0a d.in this case.. │ │ │ │ -0001f660: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0001f670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f690: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ -0001f6a0: 2074 696d 6520 4368 6572 6e20 4920 2020 time Chern I │ │ │ │ -0001f6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f6d0: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -0001f6e0: 2e30 3331 3333 3833 7320 2863 7075 293b .0313383s (cpu); │ │ │ │ -0001f6f0: 2030 2e30 3330 3730 3635 7320 2874 6872 0.0307065s (thr │ │ │ │ -0001f700: 6561 6429 3b20 3073 2028 6763 297c 0a7c ead); 0s (gc)|.| │ │ │ │ -0001f710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f740: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0001f750: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +0001f5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f5d0: 2020 7c0a 7c20 2020 2020 2020 2031 2020 |.| 1 │ │ │ │ +0001f5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f600: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001f610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f640: 2d2d 2d2d 2d2d 2b0a 0a4e 6f74 6520 7468 ------+..Note th │ │ │ │ +0001f650: 6174 206f 6e65 2063 6f75 6c64 2c20 6571 at one could, eq │ │ │ │ +0001f660: 7569 7661 6c65 6e74 6c79 2c20 7573 6520 uivalently, use │ │ │ │ +0001f670: 7468 6520 636f 6d6d 616e 6420 2a6e 6f74 the command *not │ │ │ │ +0001f680: 6520 4368 6572 6e3a 2043 6865 726e 2c20 e Chern: Chern, │ │ │ │ +0001f690: 696e 7374 6561 640a 696e 2074 6869 7320 instead.in this │ │ │ │ +0001f6a0: 6361 7365 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 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================ │ │ │ │ -0001fa30: 3d3d 3d0a 0a20 202a 2022 4353 4d28 2e2e ===.. * "CSM(.. │ │ │ │ -0001fa40: 2e2c 496e 7075 7449 7353 6d6f 6f74 683d .,InputIsSmooth= │ │ │ │ -0001fa50: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e >...)" -- see *n │ │ │ │ -0001fa60: 6f74 6520 4353 4d3a 2043 534d 2c20 2d2d ote CSM: CSM, -- │ │ │ │ -0001fa70: 2054 6865 0a20 2020 2043 6865 726e 2d53 The. Chern-S │ │ │ │ -0001fa80: 6368 7761 7274 7a2d 4d61 6350 6865 7273 chwartz-MacPhers │ │ │ │ -0001fa90: 6f6e 2063 6c61 7373 0a20 202a 2045 756c on class. * Eul │ │ │ │ -0001faa0: 6572 282e 2e2e 2c49 6e70 7574 4973 536d er(...,InputIsSm │ │ │ │ -0001fab0: 6f6f 7468 3d3e 2e2e 2e29 2028 6d69 7373 ooth=>...) (miss │ │ │ │ -0001fac0: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ -0001fad0: 6e29 0a0a 466f 7220 7468 6520 7072 6f67 n)..For the prog │ │ │ │ -0001fae0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -0001faf0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -0001fb00: 626a 6563 7420 2a6e 6f74 6520 496e 7075 bject *note Inpu │ │ │ │ -0001fb10: 7449 7353 6d6f 6f74 683a 2049 6e70 7574 tIsSmooth: Input │ │ │ │ -0001fb20: 4973 536d 6f6f 7468 2c20 6973 2061 202a IsSmooth, is a * │ │ │ │ -0001fb30: 6e6f 7465 2073 796d 626f 6c3a 0a28 4d61 note symbol:.(Ma │ │ │ │ -0001fb40: 6361 756c 6179 3244 6f63 2953 796d 626f caulay2Doc)Symbo │ │ │ │ -0001fb50: 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d l,...----------- │ │ │ │ -0001fb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fba0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -0001fbb0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -0001fbc0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -0001fbd0: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -0001fbe0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -0001fbf0: 3035 2b64 732f 4d32 2f4d 6163 6175 6c61 05+ds/M2/Macaula │ │ │ │ -0001fc00: 7932 2f70 6163 6b61 6765 732f 0a43 6861 y2/packages/.Cha │ │ │ │ -0001fc10: 7261 6374 6572 6973 7469 6343 6c61 7373 racteristicClass │ │ │ │ -0001fc20: 6573 2e6d 323a 3235 3030 3a30 2e0a 1f0a es.m2:2500:0.... │ │ │ │ -0001fc30: 4669 6c65 3a20 4368 6172 6163 7465 7269 File: Characteri │ │ │ │ -0001fc40: 7374 6963 436c 6173 7365 732e 696e 666f sticClasses.info │ │ │ │ -0001fc50: 2c20 4e6f 6465 3a20 6973 4d75 6c74 6948 , Node: isMultiH │ │ │ │ -0001fc60: 6f6d 6f67 656e 656f 7573 2c20 4e65 7874 omogeneous, Next │ │ │ │ -0001fc70: 3a20 4d65 7468 6f64 2c20 5072 6576 3a20 : Method, Prev: │ │ │ │ -0001fc80: 496e 7075 7449 7353 6d6f 6f74 682c 2055 InputIsSmooth, U │ │ │ │ -0001fc90: 703a 2054 6f70 0a0a 6973 4d75 6c74 6948 p: Top..isMultiH │ │ │ │ -0001fca0: 6f6d 6f67 656e 656f 7573 202d 2d20 4368 omogeneous -- Ch │ │ │ │ -0001fcb0: 6563 6b73 2069 6620 616e 2069 6465 616c ecks if an ideal │ │ │ │ -0001fcc0: 2069 7320 686f 6d6f 6765 6e65 6f75 7320 is homogeneous │ │ │ │ -0001fcd0: 7769 7468 2072 6573 7065 6374 2074 6f20 with respect to │ │ │ │ -0001fce0: 7468 6520 6772 6164 696e 6720 6f6e 2069 the grading on i │ │ │ │ -0001fcf0: 7473 2072 696e 6720 2869 2e65 2e20 6d75 ts ring (i.e. mu │ │ │ │ -0001fd00: 6c74 692d 686f 6d6f 6765 6e65 6f75 7320 lti-homogeneous │ │ │ │ -0001fd10: 696e 2074 6865 206d 756c 7469 2d67 7261 in the multi-gra │ │ │ │ -0001fd20: 6465 6420 6361 7365 290a 2a2a 2a2a 2a2a ded case).****** │ │ │ │ -0001fd30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001fd40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001fd50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001fd60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001f980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f990: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001f9a0: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0001f9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f9d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0001f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4675 -----------+..Fu │ │ │ │ +0001fa10: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ +0001fa20: 696f 6e61 6c20 6172 6775 6d65 6e74 206e ional argument n │ │ │ │ +0001fa30: 616d 6564 2049 6e70 7574 4973 536d 6f6f amed InputIsSmoo │ │ │ │ +0001fa40: 7468 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d th:.============ │ │ │ │ +0001fa50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001fa60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001fa70: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ +0001fa80: 4353 4d28 2e2e 2e2c 496e 7075 7449 7353 CSM(...,InputIsS │ │ │ │ +0001fa90: 6d6f 6f74 683d 3e2e 2e2e 2922 202d 2d20 mooth=>...)" -- │ │ │ │ +0001faa0: 7365 6520 2a6e 6f74 6520 4353 4d3a 2043 see *note CSM: C │ │ │ │ +0001fab0: 534d 2c20 2d2d 2054 6865 0a20 2020 2043 SM, -- The. C │ │ │ │ +0001fac0: 6865 726e 2d53 6368 7761 7274 7a2d 4d61 hern-Schwartz-Ma │ │ │ │ +0001fad0: 6350 6865 7273 6f6e 2063 6c61 7373 0a20 cPherson class. │ │ │ │ +0001fae0: 202a 2045 756c 6572 282e 2e2e 2c49 6e70 * Euler(...,Inp │ │ │ │ +0001faf0: 7574 4973 536d 6f6f 7468 3d3e 2e2e 2e29 utIsSmooth=>...) │ │ │ │ +0001fb00: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ +0001fb10: 6e74 6174 696f 6e29 0a0a 466f 7220 7468 ntation)..For th │ │ │ │ +0001fb20: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +0001fb30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0001fb40: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +0001fb50: 6520 496e 7075 7449 7353 6d6f 6f74 683a e InputIsSmooth: │ │ │ │ +0001fb60: 2049 6e70 7574 4973 536d 6f6f 7468 2c20 InputIsSmooth, │ │ │ │ +0001fb70: 6973 2061 202a 6e6f 7465 2073 796d 626f is a *note symbo │ │ │ │ +0001fb80: 6c3a 0a28 4d61 6361 756c 6179 3244 6f63 l:.(Macaulay2Doc │ │ │ │ +0001fb90: 2953 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d )Symbol,...----- │ │ │ │ +0001fba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +0001fbf0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +0001fc00: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +0001fc10: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +0001fc20: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +0001fc30: 2d31 2e32 362e 3035 2b64 732f 4d32 2f4d -1.26.05+ds/M2/M │ │ │ │ +0001fc40: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +0001fc50: 732f 0a43 6861 7261 6374 6572 6973 7469 s/.Characteristi │ │ │ │ +0001fc60: 6343 6c61 7373 6573 2e6d 323a 3235 3030 cClasses.m2:2500 │ │ │ │ +0001fc70: 3a30 2e0a 1f0a 4669 6c65 3a20 4368 6172 :0....File: Char │ │ │ │ +0001fc80: 6163 7465 7269 7374 6963 436c 6173 7365 acteristicClasse │ │ │ │ +0001fc90: 732e 696e 666f 2c20 4e6f 6465 3a20 6973 s.info, Node: is │ │ │ │ +0001fca0: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573 MultiHomogeneous │ │ │ │ +0001fcb0: 2c20 4e65 7874 3a20 4d65 7468 6f64 2c20 , Next: Method, │ │ │ │ +0001fcc0: 5072 6576 3a20 496e 7075 7449 7353 6d6f Prev: InputIsSmo │ │ │ │ +0001fcd0: 6f74 682c 2055 703a 2054 6f70 0a0a 6973 oth, Up: Top..is │ │ │ │ +0001fce0: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573 MultiHomogeneous │ │ │ │ +0001fcf0: 202d 2d20 4368 6563 6b73 2069 6620 616e -- Checks if an │ │ │ │ +0001fd00: 2069 6465 616c 2069 7320 686f 6d6f 6765 ideal is homoge │ │ │ │ +0001fd10: 6e65 6f75 7320 7769 7468 2072 6573 7065 neous with respe │ │ │ │ +0001fd20: 6374 2074 6f20 7468 6520 6772 6164 696e ct to the gradin │ │ │ │ +0001fd30: 6720 6f6e 2069 7473 2072 696e 6720 2869 g on its ring (i │ │ │ │ +0001fd40: 2e65 2e20 6d75 6c74 692d 686f 6d6f 6765 .e. multi-homoge │ │ │ │ +0001fd50: 6e65 6f75 7320 696e 2074 6865 206d 756c neous in the mul │ │ │ │ +0001fd60: 7469 2d67 7261 6465 6420 6361 7365 290a ti-graded case). │ │ │ │ 0001fd70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fd80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fd90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fda0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001fdb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -0001fdc0: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -0001fdd0: 2069 734d 756c 7469 486f 6d6f 6765 6e65 isMultiHomogene │ │ │ │ -0001fde0: 6f75 7320 490a 2020 2020 2020 2020 6973 ous I. is │ │ │ │ -0001fdf0: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573 MultiHomogeneous │ │ │ │ -0001fe00: 2066 0a20 202a 2049 6e70 7574 733a 0a20 f. * Inputs:. │ │ │ │ -0001fe10: 2020 2020 202a 2049 2c20 616e 202a 6e6f * I, an *no │ │ │ │ -0001fe20: 7465 2069 6465 616c 3a20 284d 6163 6175 te ideal: (Macau │ │ │ │ -0001fe30: 6c61 7932 446f 6329 4964 6561 6c2c 2c20 lay2Doc)Ideal,, │ │ │ │ -0001fe40: 616e 2069 6465 616c 2069 6e20 6120 6772 an ideal in a gr │ │ │ │ -0001fe50: 6164 6564 206f 720a 2020 2020 2020 2020 aded or. │ │ │ │ -0001fe60: 6d75 6c74 692d 6772 6164 6564 2072 696e multi-graded rin │ │ │ │ -0001fe70: 670a 2020 2020 2020 2a20 662c 2061 202a g. * f, a * │ │ │ │ -0001fe80: 6e6f 7465 2072 696e 6720 656c 656d 656e note ring elemen │ │ │ │ -0001fe90: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ -0001fea0: 2952 696e 6745 6c65 6d65 6e74 2c2c 2061 )RingElement,, a │ │ │ │ -0001feb0: 2065 6c65 6d65 6e74 2069 6e20 610a 2020 element in a. │ │ │ │ -0001fec0: 2020 2020 2020 6772 6164 6564 206f 7220 graded or │ │ │ │ -0001fed0: 6d75 6c74 692d 6772 6164 6564 2072 696e multi-graded rin │ │ │ │ -0001fee0: 670a 2020 2a20 4f75 7470 7574 733a 0a20 g. * Outputs:. │ │ │ │ -0001fef0: 2020 2020 202a 2061 202a 6e6f 7465 2042 * a *note B │ │ │ │ -0001ff00: 6f6f 6c65 616e 2076 616c 7565 3a20 284d oolean value: (M │ │ │ │ -0001ff10: 6163 6175 6c61 7932 446f 6329 426f 6f6c acaulay2Doc)Bool │ │ │ │ -0001ff20: 6561 6e2c 2c20 0a0a 4465 7363 7269 7074 ean,, ..Descript │ │ │ │ -0001ff30: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -0001ff40: 0a54 6573 7473 2069 6620 7468 6520 696e .Tests if the in │ │ │ │ -0001ff50: 7075 7420 4964 6561 6c20 6f72 2052 696e put Ideal or Rin │ │ │ │ -0001ff60: 6745 6c65 6d65 6e74 2069 7320 486f 6d6f gElement is Homo │ │ │ │ -0001ff70: 6765 6e65 6f75 7320 7769 7468 2072 6573 geneous with res │ │ │ │ -0001ff80: 7065 6374 2074 6f20 7468 650a 6772 6164 pect to the.grad │ │ │ │ -0001ff90: 696e 6720 6f6e 2074 6865 2072 696e 672e ing on the ring. │ │ │ │ -0001ffa0: 2048 6f6d 6f67 656e 656f 7573 2069 6e70 Homogeneous inp │ │ │ │ -0001ffb0: 7574 2069 7320 7265 7175 6972 6564 2066 ut is required f │ │ │ │ -0001ffc0: 6f72 2061 6c6c 206d 6574 686f 6473 2074 or all methods t │ │ │ │ -0001ffd0: 6f20 636f 6d70 7574 650a 6368 6172 6163 o compute.charac │ │ │ │ -0001ffe0: 7465 7269 7374 6963 2063 6c61 7373 6573 teristic classes │ │ │ │ -0001fff0: 2e0a 0a54 6869 7320 6d65 7468 6f64 2077 ...This method w │ │ │ │ -00020000: 6f72 6b73 2066 6f72 2069 6465 616c 7320 orks for ideals │ │ │ │ -00020010: 696e 2074 6865 2067 7261 6465 6420 636f in the graded co │ │ │ │ -00020020: 6f72 6469 6e61 7465 2072 696e 6773 206f ordinate rings o │ │ │ │ -00020030: 6620 746f 7269 6320 7661 7269 6574 6965 f toric varietie │ │ │ │ -00020040: 732c 0a61 6e64 2068 656e 6365 2066 6f72 s,.and hence for │ │ │ │ -00020050: 2070 726f 6475 6374 7320 6f66 2070 726f products of pro │ │ │ │ -00020060: 6a65 6374 6976 6520 7370 6163 6573 2e20 jective spaces. │ │ │ │ -00020070: 5468 6573 6520 6361 6e20 6265 2063 7265 These can be cre │ │ │ │ -00020080: 6174 6564 2064 6972 6563 746c 792c 206f ated directly, o │ │ │ │ -00020090: 720a 7573 696e 6720 6d65 7468 6f64 7320 r.using methods │ │ │ │ -000200a0: 7468 6520 2a6e 6f74 6520 4d75 6c74 6950 the *note MultiP │ │ │ │ -000200b0: 726f 6a43 6f6f 7264 5269 6e67 3a20 4d75 rojCoordRing: Mu │ │ │ │ -000200c0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ -000200d0: 2c20 6d65 7468 6f64 206f 6620 7468 6973 , method of this │ │ │ │ -000200e0: 0a70 6163 6b61 6765 2c20 6f72 2077 6974 .package, or wit │ │ │ │ -000200f0: 6820 6d65 7468 6f64 7320 6672 6f6d 2074 h methods from t │ │ │ │ -00020100: 6865 204e 6f72 6d61 6c54 6f72 6963 5661 he NormalToricVa │ │ │ │ -00020110: 7269 6574 6965 7320 5061 636b 6167 652e rieties Package. │ │ │ │ -00020120: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ -00020130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020160: 2d2d 2b0a 7c69 3120 3a20 523d 4d75 6c74 --+.|i1 : R=Mult │ │ │ │ -00020170: 6950 726f 6a43 6f6f 7264 5269 6e67 287b iProjCoordRing({ │ │ │ │ -00020180: 312c 322c 317d 2920 2020 2020 2020 2020 1,2,1}) │ │ │ │ -00020190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000201a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000201b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000201c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fdb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fdc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fdd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fde0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fdf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fe00: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +0001fe10: 2020 2020 2020 2069 734d 756c 7469 486f isMultiHo │ │ │ │ +0001fe20: 6d6f 6765 6e65 6f75 7320 490a 2020 2020 mogeneous I. │ │ │ │ +0001fe30: 2020 2020 6973 4d75 6c74 6948 6f6d 6f67 isMultiHomog │ │ │ │ +0001fe40: 656e 656f 7573 2066 0a20 202a 2049 6e70 eneous f. * Inp │ │ │ │ +0001fe50: 7574 733a 0a20 2020 2020 202a 2049 2c20 uts:. * I, │ │ │ │ +0001fe60: 616e 202a 6e6f 7465 2069 6465 616c 3a20 an *note ideal: │ │ │ │ +0001fe70: 284d 6163 6175 6c61 7932 446f 6329 4964 (Macaulay2Doc)Id │ │ │ │ +0001fe80: 6561 6c2c 2c20 616e 2069 6465 616c 2069 eal,, an ideal i │ │ │ │ +0001fe90: 6e20 6120 6772 6164 6564 206f 720a 2020 n a graded or. │ │ │ │ +0001fea0: 2020 2020 2020 6d75 6c74 692d 6772 6164 multi-grad │ │ │ │ +0001feb0: 6564 2072 696e 670a 2020 2020 2020 2a20 ed ring. * │ │ │ │ +0001fec0: 662c 2061 202a 6e6f 7465 2072 696e 6720 f, a *note ring │ │ │ │ +0001fed0: 656c 656d 656e 743a 2028 4d61 6361 756c element: (Macaul │ │ │ │ +0001fee0: 6179 3244 6f63 2952 696e 6745 6c65 6d65 ay2Doc)RingEleme │ │ │ │ +0001fef0: 6e74 2c2c 2061 2065 6c65 6d65 6e74 2069 nt,, a element i │ │ │ │ +0001ff00: 6e20 610a 2020 2020 2020 2020 6772 6164 n a. grad │ │ │ │ +0001ff10: 6564 206f 7220 6d75 6c74 692d 6772 6164 ed or multi-grad │ │ │ │ +0001ff20: 6564 2072 696e 670a 2020 2a20 4f75 7470 ed ring. * Outp │ │ │ │ +0001ff30: 7574 733a 0a20 2020 2020 202a 2061 202a uts:. * a * │ │ │ │ +0001ff40: 6e6f 7465 2042 6f6f 6c65 616e 2076 616c note Boolean val │ │ │ │ +0001ff50: 7565 3a20 284d 6163 6175 6c61 7932 446f ue: (Macaulay2Do │ │ │ │ +0001ff60: 6329 426f 6f6c 6561 6e2c 2c20 0a0a 4465 c)Boolean,, ..De │ │ │ │ +0001ff70: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +0001ff80: 3d3d 3d3d 3d0a 0a54 6573 7473 2069 6620 =====..Tests if │ │ │ │ +0001ff90: 7468 6520 696e 7075 7420 4964 6561 6c20 the input Ideal │ │ │ │ +0001ffa0: 6f72 2052 696e 6745 6c65 6d65 6e74 2069 or RingElement i │ │ │ │ +0001ffb0: 7320 486f 6d6f 6765 6e65 6f75 7320 7769 s Homogeneous wi │ │ │ │ +0001ffc0: 7468 2072 6573 7065 6374 2074 6f20 7468 th respect to th │ │ │ │ +0001ffd0: 650a 6772 6164 696e 6720 6f6e 2074 6865 e.grading on the │ │ │ │ +0001ffe0: 2072 696e 672e 2048 6f6d 6f67 656e 656f ring. Homogeneo │ │ │ │ +0001fff0: 7573 2069 6e70 7574 2069 7320 7265 7175 us input is requ │ │ │ │ +00020000: 6972 6564 2066 6f72 2061 6c6c 206d 6574 ired for all met │ │ │ │ +00020010: 686f 6473 2074 6f20 636f 6d70 7574 650a hods to compute. │ │ │ │ +00020020: 6368 6172 6163 7465 7269 7374 6963 2063 characteristic c │ │ │ │ +00020030: 6c61 7373 6573 2e0a 0a54 6869 7320 6d65 lasses...This me │ │ │ │ +00020040: 7468 6f64 2077 6f72 6b73 2066 6f72 2069 thod works for i │ │ │ │ +00020050: 6465 616c 7320 696e 2074 6865 2067 7261 deals in the gra │ │ │ │ +00020060: 6465 6420 636f 6f72 6469 6e61 7465 2072 ded coordinate r │ │ │ │ +00020070: 696e 6773 206f 6620 746f 7269 6320 7661 ings of toric va │ │ │ │ +00020080: 7269 6574 6965 732c 0a61 6e64 2068 656e rieties,.and hen │ │ │ │ +00020090: 6365 2066 6f72 2070 726f 6475 6374 7320 ce for products │ │ │ │ +000200a0: 6f66 2070 726f 6a65 6374 6976 6520 7370 of projective sp │ │ │ │ +000200b0: 6163 6573 2e20 5468 6573 6520 6361 6e20 aces. These can │ │ │ │ +000200c0: 6265 2063 7265 6174 6564 2064 6972 6563 be created direc │ │ │ │ +000200d0: 746c 792c 206f 720a 7573 696e 6720 6d65 tly, or.using me │ │ │ │ +000200e0: 7468 6f64 7320 7468 6520 2a6e 6f74 6520 thods the *note │ │ │ │ +000200f0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ +00020100: 6e67 3a20 4d75 6c74 6950 726f 6a43 6f6f ng: MultiProjCoo │ │ │ │ +00020110: 7264 5269 6e67 2c20 6d65 7468 6f64 206f rdRing, method o │ │ │ │ +00020120: 6620 7468 6973 0a70 6163 6b61 6765 2c20 f this.package, │ │ │ │ +00020130: 6f72 2077 6974 6820 6d65 7468 6f64 7320 or with methods │ │ │ │ +00020140: 6672 6f6d 2074 6865 204e 6f72 6d61 6c54 from the NormalT │ │ │ │ +00020150: 6f72 6963 5661 7269 6574 6965 7320 5061 oricVarieties Pa │ │ │ │ +00020160: 636b 6167 652e 0a0a 2b2d 2d2d 2d2d 2d2d ckage...+------- │ │ │ │ +00020170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000201a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +000201b0: 523d 4d75 6c74 6950 726f 6a43 6f6f 7264 R=MultiProjCoord │ │ │ │ +000201c0: 5269 6e67 287b 312c 322c 317d 2920 2020 Ring({1,2,1}) │ │ │ │ 000201d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000201e0: 2020 2020 2020 7c0a 7c6f 3120 3d20 5220 |.|o1 = R │ │ │ │ +000201e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000201f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020220: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00020230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020220: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00020230: 3120 3d20 5220 2020 2020 2020 2020 2020 1 = R │ │ │ │ 00020240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020260: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ -00020270: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +00020260: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020270: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00020280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000202a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -000202b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000202c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000202d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000202e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -000202f0: 7c69 3220 3a20 783d 6765 6e73 2852 2920 |i2 : x=gens(R) │ │ │ │ -00020300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020330: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00020340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000202a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000202b0: 7c0a 7c6f 3120 3a20 506f 6c79 6e6f 6d69 |.|o1 : Polynomi │ │ │ │ +000202c0: 616c 5269 6e67 2020 2020 2020 2020 2020 alRing │ │ │ │ +000202d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000202e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000202f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00020300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020330: 2d2d 2d2d 2b0a 7c69 3220 3a20 783d 6765 ----+.|i2 : x=ge │ │ │ │ +00020340: 6e73 2852 2920 2020 2020 2020 2020 2020 ns(R) │ │ │ │ 00020350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020370: 2020 7c0a 7c6f 3220 3d20 7b78 202c 2078 |.|o2 = {x , x │ │ │ │ -00020380: 202c 2078 202c 2078 202c 2078 202c 2078 , x , x , x , x │ │ │ │ -00020390: 202c 2078 207d 2020 2020 2020 2020 2020 , x } │ │ │ │ +00020370: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00020380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000203a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000203b0: 2020 2020 7c0a 7c20 2020 2020 2020 3020 |.| 0 │ │ │ │ -000203c0: 2020 3120 2020 3220 2020 3320 2020 3420 1 2 3 4 │ │ │ │ -000203d0: 2020 3520 2020 3620 2020 2020 2020 2020 5 6 │ │ │ │ +000203b0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ +000203c0: 7b78 202c 2078 202c 2078 202c 2078 202c {x , x , x , x , │ │ │ │ +000203d0: 2078 202c 2078 202c 2078 207d 2020 2020 x , x , x } │ │ │ │ 000203e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000203f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00020400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000203f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00020400: 2020 2020 3020 2020 3120 2020 3220 2020 0 1 2 │ │ │ │ +00020410: 3320 2020 3420 2020 3520 2020 3620 2020 3 4 5 6 │ │ │ │ 00020420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020430: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ -00020440: 4c69 7374 2020 2020 2020 2020 2020 2020 List │ │ │ │ +00020430: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00020440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020470: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00020480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000204a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000204b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -000204c0: 3320 3a20 493d 6964 6561 6c28 785f 305e 3 : I=ideal(x_0^ │ │ │ │ -000204d0: 322a 785f 332d 785f 312a 785f 302a 785f 2*x_3-x_1*x_0*x_ │ │ │ │ -000204e0: 342c 785f 365e 3329 2020 2020 2020 2020 4,x_6^3) │ │ │ │ -000204f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00020500: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00020510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020470: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020480: 7c6f 3220 3a20 4c69 7374 2020 2020 2020 |o2 : List │ │ │ │ +00020490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000204a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000204b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000204c0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +000204d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000204e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000204f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020500: 2d2d 2b0a 7c69 3320 3a20 493d 6964 6561 --+.|i3 : I=idea │ │ │ │ +00020510: 6c28 785f 305e 322a 785f 332d 785f 312a l(x_0^2*x_3-x_1* │ │ │ │ +00020520: 785f 302a 785f 342c 785f 365e 3329 2020 x_0*x_4,x_6^3) │ │ │ │ 00020530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020540: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00020550: 3220 2020 2020 2020 2020 2020 2020 2033 2 3 │ │ │ │ +00020540: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00020550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020580: 2020 7c0a 7c6f 3320 3d20 6964 6561 6c20 |.|o3 = ideal │ │ │ │ -00020590: 2878 2078 2020 2d20 7820 7820 7820 2c20 (x x - x x x , │ │ │ │ -000205a0: 7820 2920 2020 2020 2020 2020 2020 2020 x ) │ │ │ │ +00020580: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00020590: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +000205a0: 2020 2020 2033 2020 2020 2020 2020 2020 3 │ │ │ │ 000205b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000205c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000205d0: 2020 2020 3020 3320 2020 2030 2031 2034 0 3 0 1 4 │ │ │ │ -000205e0: 2020 2036 2020 2020 2020 2020 2020 2020 6 │ │ │ │ +000205c0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ +000205d0: 6964 6561 6c20 2878 2078 2020 2d20 7820 ideal (x x - x │ │ │ │ +000205e0: 7820 7820 2c20 7820 2920 2020 2020 2020 x x , x ) │ │ │ │ 000205f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020600: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00020610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020600: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00020610: 2020 2020 2020 2020 2020 3020 3320 2020 0 3 │ │ │ │ +00020620: 2030 2031 2034 2020 2036 2020 2020 2020 0 1 4 6 │ │ │ │ 00020630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020640: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ -00020650: 4964 6561 6c20 6f66 2052 2020 2020 2020 Ideal of R │ │ │ │ +00020640: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00020650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020680: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00020690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000206a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000206b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000206c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -000206d0: 3420 3a20 6973 4d75 6c74 6948 6f6d 6f67 4 : isMultiHomog │ │ │ │ -000206e0: 656e 656f 7573 2049 2020 2020 2020 2020 eneous I │ │ │ │ -000206f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020700: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00020710: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00020720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020680: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020690: 7c6f 3320 3a20 4964 6561 6c20 6f66 2052 |o3 : Ideal of R │ │ │ │ +000206a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000206b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000206c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000206d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +000206e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000206f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020710: 2d2d 2b0a 7c69 3420 3a20 6973 4d75 6c74 --+.|i4 : isMult │ │ │ │ +00020720: 6948 6f6d 6f67 656e 656f 7573 2049 2020 iHomogeneous I │ │ │ │ 00020730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020750: 7c0a 7c6f 3420 3d20 7472 7565 2020 2020 |.|o4 = true │ │ │ │ +00020750: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00020760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020790: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -000207a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000207b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000207c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000207d0: 2d2d 2d2d 2b0a 7c69 3520 3a20 6973 4d75 ----+.|i5 : isMu │ │ │ │ -000207e0: 6c74 6948 6f6d 6f67 656e 656f 7573 2069 ltiHomogeneous i │ │ │ │ -000207f0: 6465 616c 2878 5f30 2a78 5f33 2d78 5f31 deal(x_0*x_3-x_1 │ │ │ │ -00020800: 2a78 5f30 2a78 5f34 2c78 5f36 5e33 2920 *x_0*x_4,x_6^3) │ │ │ │ -00020810: 2020 2020 2020 7c0a 7c49 6e70 7574 2074 |.|Input t │ │ │ │ -00020820: 6572 6d20 6265 6c6f 7720 6973 206e 6f74 erm below is not │ │ │ │ -00020830: 2068 6f6d 6f67 656e 656f 7573 2077 6974 homogeneous wit │ │ │ │ -00020840: 6820 7265 7370 6563 7420 746f 2074 6865 h respect to the │ │ │ │ -00020850: 2067 7261 6469 6e67 7c0a 7c2d 2078 2078 grading|.|- x x │ │ │ │ -00020860: 2078 2020 2b20 7820 7820 2020 2020 2020 x + x x │ │ │ │ -00020870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020890: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -000208a0: 3020 3120 3420 2020 2030 2033 2020 2020 0 1 4 0 3 │ │ │ │ +00020790: 2020 2020 2020 7c0a 7c6f 3420 3d20 7472 |.|o4 = tr │ │ │ │ +000207a0: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ +000207b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000207c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000207d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000207e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000207f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020810: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ +00020820: 3a20 6973 4d75 6c74 6948 6f6d 6f67 656e : isMultiHomogen │ │ │ │ +00020830: 656f 7573 2069 6465 616c 2878 5f30 2a78 eous ideal(x_0*x │ │ │ │ +00020840: 5f33 2d78 5f31 2a78 5f30 2a78 5f34 2c78 _3-x_1*x_0*x_4,x │ │ │ │ +00020850: 5f36 5e33 2920 2020 2020 2020 7c0a 7c49 _6^3) |.|I │ │ │ │ +00020860: 6e70 7574 2074 6572 6d20 6265 6c6f 7720 nput term below │ │ │ │ +00020870: 6973 206e 6f74 2068 6f6d 6f67 656e 656f is not homogeneo │ │ │ │ +00020880: 7573 2077 6974 6820 7265 7370 6563 7420 us with respect │ │ │ │ +00020890: 746f 2074 6865 2067 7261 6469 6e67 7c0a to the grading|. │ │ │ │ +000208a0: 7c2d 2078 2078 2078 2020 2b20 7820 7820 |- x x x + x x │ │ │ │ 000208b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000208c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000208d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000208e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000208f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000208d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000208e0: 7c0a 7c20 2020 3020 3120 3420 2020 2030 |.| 0 1 4 0 │ │ │ │ +000208f0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00020900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020910: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00020920: 7c6f 3520 3d20 6661 6c73 6520 2020 2020 |o5 = false │ │ │ │ +00020910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020920: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00020930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020960: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00020970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000209a0: 2d2d 2b0a 0a4e 6f74 6520 7468 6174 2066 --+..Note that f │ │ │ │ -000209b0: 6f72 2061 6e20 6964 6561 6c20 746f 2062 or an ideal to b │ │ │ │ -000209c0: 6520 6d75 6c74 692d 686f 6d6f 6765 6e65 e multi-homogene │ │ │ │ -000209d0: 6f75 7320 7468 6520 6465 6772 6565 2076 ous the degree v │ │ │ │ -000209e0: 6563 746f 7220 6f66 2061 6c6c 0a6d 6f6e ector of all.mon │ │ │ │ -000209f0: 6f6d 6961 6c73 2069 6e20 6120 6769 7665 omials in a give │ │ │ │ -00020a00: 6e20 6765 6e65 7261 746f 7220 6d75 7374 n generator must │ │ │ │ -00020a10: 2062 6520 7468 6520 7361 6d65 2e0a 0a57 be the same...W │ │ │ │ -00020a20: 6179 7320 746f 2075 7365 2069 734d 756c ays to use isMul │ │ │ │ -00020a30: 7469 486f 6d6f 6765 6e65 6f75 733a 0a3d tiHomogeneous:.= │ │ │ │ -00020a40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00020a50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00020a60: 2020 2a20 2269 734d 756c 7469 486f 6d6f * "isMultiHomo │ │ │ │ -00020a70: 6765 6e65 6f75 7328 4964 6561 6c29 220a geneous(Ideal)". │ │ │ │ -00020a80: 2020 2a20 2269 734d 756c 7469 486f 6d6f * "isMultiHomo │ │ │ │ -00020a90: 6765 6e65 6f75 7328 5269 6e67 456c 656d geneous(RingElem │ │ │ │ -00020aa0: 656e 7429 220a 0a46 6f72 2074 6865 2070 ent)"..For the p │ │ │ │ -00020ab0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -00020ac0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -00020ad0: 6520 6f62 6a65 6374 202a 6e6f 7465 2069 e object *note i │ │ │ │ -00020ae0: 734d 756c 7469 486f 6d6f 6765 6e65 6f75 sMultiHomogeneou │ │ │ │ -00020af0: 733a 2069 734d 756c 7469 486f 6d6f 6765 s: isMultiHomoge │ │ │ │ -00020b00: 6e65 6f75 732c 2069 7320 6120 2a6e 6f74 neous, is a *not │ │ │ │ -00020b10: 6520 6d65 7468 6f64 0a66 756e 6374 696f e method.functio │ │ │ │ -00020b20: 6e3a 2028 4d61 6361 756c 6179 3244 6f63 n: (Macaulay2Doc │ │ │ │ -00020b30: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ -00020b40: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ -00020b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020b90: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -00020ba0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -00020bb0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -00020bc0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -00020bd0: 6163 6175 6c61 7932 2d31 2e32 362e 3035 acaulay2-1.26.05 │ │ │ │ -00020be0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -00020bf0: 2f70 6163 6b61 6765 732f 0a43 6861 7261 /packages/.Chara │ │ │ │ -00020c00: 6374 6572 6973 7469 6343 6c61 7373 6573 cteristicClasses │ │ │ │ -00020c10: 2e6d 323a 3230 3132 3a30 2e0a 1f0a 4669 .m2:2012:0....Fi │ │ │ │ -00020c20: 6c65 3a20 4368 6172 6163 7465 7269 7374 le: Characterist │ │ │ │ -00020c30: 6963 436c 6173 7365 732e 696e 666f 2c20 icClasses.info, │ │ │ │ -00020c40: 4e6f 6465 3a20 4d65 7468 6f64 2c20 4e65 Node: Method, Ne │ │ │ │ -00020c50: 7874 3a20 4d75 6c74 6950 726f 6a43 6f6f xt: MultiProjCoo │ │ │ │ -00020c60: 7264 5269 6e67 2c20 5072 6576 3a20 6973 rdRing, Prev: is │ │ │ │ -00020c70: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573 MultiHomogeneous │ │ │ │ -00020c80: 2c20 5570 3a20 546f 700a 0a4d 6574 686f , Up: Top..Metho │ │ │ │ -00020c90: 640a 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 d.******..Descri │ │ │ │ -00020ca0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -00020cb0: 3d0a 0a54 6865 206f 7074 696f 6e20 4d65 =..The option Me │ │ │ │ -00020cc0: 7468 6f64 2069 7320 6f6e 6c79 2075 7365 thod is only use │ │ │ │ -00020cd0: 6420 6279 2074 6865 2063 6f6d 6d61 6e64 d by the command │ │ │ │ -00020ce0: 7320 2a6e 6f74 6520 4353 4d3a 2043 534d s *note CSM: CSM │ │ │ │ -00020cf0: 2c20 616e 6420 2a6e 6f74 6520 4575 6c65 , and *note Eule │ │ │ │ -00020d00: 723a 0a45 756c 6572 2c20 616e 6420 6f6e r:.Euler, and on │ │ │ │ -00020d10: 6c79 2069 6e20 636f 6d62 696e 6174 696f ly in combinatio │ │ │ │ -00020d20: 6e20 7769 7468 202a 6e6f 7465 2043 6f6d n with *note Com │ │ │ │ -00020d30: 704d 6574 686f 643a 0a43 6f6d 704d 6574 pMethod:.CompMet │ │ │ │ -00020d40: 686f 642c 3d3e 5072 6f6a 6563 7469 7665 hod,=>Projective │ │ │ │ -00020d50: 4465 6772 6565 2e20 5468 6520 4d65 7468 Degree. The Meth │ │ │ │ -00020d60: 6f64 2049 6e63 6c75 7369 6f6e 4578 636c od InclusionExcl │ │ │ │ -00020d70: 7573 696f 6e20 7769 6c6c 2061 6c77 6179 usion will alway │ │ │ │ -00020d80: 7320 6265 0a75 7365 6420 7769 7468 202a s be.used with * │ │ │ │ -00020d90: 6e6f 7465 2043 6f6d 704d 6574 686f 643a note CompMethod: │ │ │ │ -00020da0: 2043 6f6d 704d 6574 686f 642c 2050 6e52 CompMethod, PnR │ │ │ │ -00020db0: 6573 6964 7561 6c20 6f72 2062 6572 7469 esidual or berti │ │ │ │ -00020dc0: 6e69 2e20 5768 656e 2074 6865 2069 6e70 ni. When the inp │ │ │ │ -00020dd0: 7574 0a69 6465 616c 2069 7320 6120 636f ut.ideal is a co │ │ │ │ -00020de0: 6d70 6c65 7465 2069 6e74 6572 7365 6374 mplete intersect │ │ │ │ -00020df0: 696f 6e20 6f6e 6520 6d61 792c 2070 6f74 ion one may, pot │ │ │ │ -00020e00: 656e 7469 616c 6c79 2c20 7370 6565 6420 entially, speed │ │ │ │ -00020e10: 7570 2074 6865 2063 6f6d 7075 7461 7469 up the computati │ │ │ │ -00020e20: 6f6e 0a62 7920 7365 7474 696e 6720 4d65 on.by setting Me │ │ │ │ -00020e30: 7468 6f64 3d3e 2044 6972 6563 7443 6f6d thod=> DirectCom │ │ │ │ -00020e40: 706c 6574 6549 6e74 2e20 5468 6520 6f70 pleteInt. The op │ │ │ │ -00020e50: 7469 6f6e 204d 6574 686f 6420 6973 206f tion Method is o │ │ │ │ -00020e60: 6e6c 7920 7573 6564 2062 7920 7468 650a nly used by the. │ │ │ │ -00020e70: 636f 6d6d 616e 6473 202a 6e6f 7465 2043 commands *note C │ │ │ │ -00020e80: 534d 3a20 4353 4d2c 2061 6e64 202a 6e6f SM: CSM, and *no │ │ │ │ -00020e90: 7465 2045 756c 6572 3a20 4575 6c65 722c te Euler: Euler, │ │ │ │ -00020ea0: 2061 6e64 206f 6e6c 7920 696e 2063 6f6d and only in com │ │ │ │ -00020eb0: 6269 6e61 7469 6f6e 2077 6974 680a 2a6e bination with.*n │ │ │ │ -00020ec0: 6f74 6520 436f 6d70 4d65 7468 6f64 3a20 ote CompMethod: │ │ │ │ -00020ed0: 436f 6d70 4d65 7468 6f64 2c3d 3e50 726f CompMethod,=>Pro │ │ │ │ -00020ee0: 6a65 6374 6976 6544 6567 7265 652e 2054 jectiveDegree. T │ │ │ │ -00020ef0: 6865 204d 6574 686f 6420 496e 636c 7573 he Method Inclus │ │ │ │ -00020f00: 696f 6e45 7863 6c75 7369 6f6e 0a77 696c ionExclusion.wil │ │ │ │ -00020f10: 6c20 616c 7761 7973 2062 6520 7573 6564 l always be used │ │ │ │ -00020f20: 2077 6974 6820 2a6e 6f74 6520 436f 6d70 with *note Comp │ │ │ │ -00020f30: 4d65 7468 6f64 3a20 436f 6d70 4d65 7468 Method: CompMeth │ │ │ │ -00020f40: 6f64 2c20 506e 5265 7369 6475 616c 206f od, PnResidual o │ │ │ │ -00020f50: 7220 6265 7274 696e 692e 0a0a 2b2d 2d2d r bertini...+--- │ │ │ │ -00020f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020f90: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d -----+.|i1 : R = │ │ │ │ -00020fa0: 205a 5a2f 3332 3734 395b 785f 302e 2e78 ZZ/32749[x_0..x │ │ │ │ -00020fb0: 5f36 5d20 2020 2020 2020 2020 2020 2020 _6] │ │ │ │ -00020fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020fd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00020fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021000: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -00021010: 203d 2052 2020 2020 2020 2020 2020 2020 = R │ │ │ │ +00020960: 2020 2020 7c0a 7c6f 3520 3d20 6661 6c73 |.|o5 = fals │ │ │ │ +00020970: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +00020980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000209a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +000209b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000209c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000209d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000209e0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a4e 6f74 6520 --------+..Note │ │ │ │ +000209f0: 7468 6174 2066 6f72 2061 6e20 6964 6561 that for an idea │ │ │ │ +00020a00: 6c20 746f 2062 6520 6d75 6c74 692d 686f l to be multi-ho │ │ │ │ +00020a10: 6d6f 6765 6e65 6f75 7320 7468 6520 6465 mogeneous the de │ │ │ │ +00020a20: 6772 6565 2076 6563 746f 7220 6f66 2061 gree vector of a │ │ │ │ +00020a30: 6c6c 0a6d 6f6e 6f6d 6961 6c73 2069 6e20 ll.monomials in │ │ │ │ +00020a40: 6120 6769 7665 6e20 6765 6e65 7261 746f a given generato │ │ │ │ +00020a50: 7220 6d75 7374 2062 6520 7468 6520 7361 r must be the sa │ │ │ │ +00020a60: 6d65 2e0a 0a57 6179 7320 746f 2075 7365 me...Ways to use │ │ │ │ +00020a70: 2069 734d 756c 7469 486f 6d6f 6765 6e65 isMultiHomogene │ │ │ │ +00020a80: 6f75 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ous:.=========== │ │ │ │ +00020a90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00020aa0: 3d3d 3d3d 0a0a 2020 2a20 2269 734d 756c ====.. * "isMul │ │ │ │ +00020ab0: 7469 486f 6d6f 6765 6e65 6f75 7328 4964 tiHomogeneous(Id │ │ │ │ +00020ac0: 6561 6c29 220a 2020 2a20 2269 734d 756c eal)". * "isMul │ │ │ │ +00020ad0: 7469 486f 6d6f 6765 6e65 6f75 7328 5269 tiHomogeneous(Ri │ │ │ │ +00020ae0: 6e67 456c 656d 656e 7429 220a 0a46 6f72 ngElement)"..For │ │ │ │ +00020af0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +00020b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00020b10: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +00020b20: 6e6f 7465 2069 734d 756c 7469 486f 6d6f note isMultiHomo │ │ │ │ +00020b30: 6765 6e65 6f75 733a 2069 734d 756c 7469 geneous: isMulti │ │ │ │ +00020b40: 486f 6d6f 6765 6e65 6f75 732c 2069 7320 Homogeneous, is │ │ │ │ +00020b50: 6120 2a6e 6f74 6520 6d65 7468 6f64 0a66 a *note method.f │ │ │ │ +00020b60: 756e 6374 696f 6e3a 2028 4d61 6361 756c unction: (Macaul │ │ │ │ +00020b70: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +00020b80: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +00020b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020bd0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +00020be0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +00020bf0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +00020c00: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00020c10: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00020c20: 2e32 362e 3035 2b64 732f 4d32 2f4d 6163 .26.05+ds/M2/Mac │ │ │ │ +00020c30: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00020c40: 0a43 6861 7261 6374 6572 6973 7469 6343 .CharacteristicC │ │ │ │ +00020c50: 6c61 7373 6573 2e6d 323a 3230 3132 3a30 lasses.m2:2012:0 │ │ │ │ +00020c60: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163 ....File: Charac │ │ │ │ +00020c70: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ +00020c80: 696e 666f 2c20 4e6f 6465 3a20 4d65 7468 info, Node: Meth │ │ │ │ +00020c90: 6f64 2c20 4e65 7874 3a20 4d75 6c74 6950 od, Next: MultiP │ │ │ │ +00020ca0: 726f 6a43 6f6f 7264 5269 6e67 2c20 5072 rojCoordRing, Pr │ │ │ │ +00020cb0: 6576 3a20 6973 4d75 6c74 6948 6f6d 6f67 ev: isMultiHomog │ │ │ │ +00020cc0: 656e 656f 7573 2c20 5570 3a20 546f 700a eneous, Up: Top. │ │ │ │ +00020cd0: 0a4d 6574 686f 640a 2a2a 2a2a 2a2a 0a0a .Method.******.. │ │ │ │ +00020ce0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00020cf0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 7074 =======..The opt │ │ │ │ +00020d00: 696f 6e20 4d65 7468 6f64 2069 7320 6f6e ion Method is on │ │ │ │ +00020d10: 6c79 2075 7365 6420 6279 2074 6865 2063 ly used by the c │ │ │ │ +00020d20: 6f6d 6d61 6e64 7320 2a6e 6f74 6520 4353 ommands *note CS │ │ │ │ +00020d30: 4d3a 2043 534d 2c20 616e 6420 2a6e 6f74 M: CSM, and *not │ │ │ │ +00020d40: 6520 4575 6c65 723a 0a45 756c 6572 2c20 e Euler:.Euler, │ │ │ │ +00020d50: 616e 6420 6f6e 6c79 2069 6e20 636f 6d62 and only in comb │ │ │ │ +00020d60: 696e 6174 696f 6e20 7769 7468 202a 6e6f ination with *no │ │ │ │ +00020d70: 7465 2043 6f6d 704d 6574 686f 643a 0a43 te CompMethod:.C │ │ │ │ +00020d80: 6f6d 704d 6574 686f 642c 3d3e 5072 6f6a ompMethod,=>Proj │ │ │ │ +00020d90: 6563 7469 7665 4465 6772 6565 2e20 5468 ectiveDegree. Th │ │ │ │ +00020da0: 6520 4d65 7468 6f64 2049 6e63 6c75 7369 e Method Inclusi │ │ │ │ +00020db0: 6f6e 4578 636c 7573 696f 6e20 7769 6c6c onExclusion will │ │ │ │ +00020dc0: 2061 6c77 6179 7320 6265 0a75 7365 6420 always be.used │ │ │ │ +00020dd0: 7769 7468 202a 6e6f 7465 2043 6f6d 704d with *note CompM │ │ │ │ +00020de0: 6574 686f 643a 2043 6f6d 704d 6574 686f ethod: CompMetho │ │ │ │ +00020df0: 642c 2050 6e52 6573 6964 7561 6c20 6f72 d, PnResidual or │ │ │ │ +00020e00: 2062 6572 7469 6e69 2e20 5768 656e 2074 bertini. When t │ │ │ │ +00020e10: 6865 2069 6e70 7574 0a69 6465 616c 2069 he input.ideal i │ │ │ │ +00020e20: 7320 6120 636f 6d70 6c65 7465 2069 6e74 s a complete int │ │ │ │ +00020e30: 6572 7365 6374 696f 6e20 6f6e 6520 6d61 ersection one ma │ │ │ │ +00020e40: 792c 2070 6f74 656e 7469 616c 6c79 2c20 y, potentially, │ │ │ │ +00020e50: 7370 6565 6420 7570 2074 6865 2063 6f6d speed up the com │ │ │ │ +00020e60: 7075 7461 7469 6f6e 0a62 7920 7365 7474 putation.by sett │ │ │ │ +00020e70: 696e 6720 4d65 7468 6f64 3d3e 2044 6972 ing Method=> Dir │ │ │ │ +00020e80: 6563 7443 6f6d 706c 6574 6549 6e74 2e20 ectCompleteInt. │ │ │ │ +00020e90: 5468 6520 6f70 7469 6f6e 204d 6574 686f The option Metho │ │ │ │ +00020ea0: 6420 6973 206f 6e6c 7920 7573 6564 2062 d is only used b │ │ │ │ +00020eb0: 7920 7468 650a 636f 6d6d 616e 6473 202a y the.commands * │ │ │ │ +00020ec0: 6e6f 7465 2043 534d 3a20 4353 4d2c 2061 note CSM: CSM, a │ │ │ │ +00020ed0: 6e64 202a 6e6f 7465 2045 756c 6572 3a20 nd *note Euler: │ │ │ │ +00020ee0: 4575 6c65 722c 2061 6e64 206f 6e6c 7920 Euler, and only │ │ │ │ +00020ef0: 696e 2063 6f6d 6269 6e61 7469 6f6e 2077 in combination w │ │ │ │ +00020f00: 6974 680a 2a6e 6f74 6520 436f 6d70 4d65 ith.*note CompMe │ │ │ │ +00020f10: 7468 6f64 3a20 436f 6d70 4d65 7468 6f64 thod: CompMethod │ │ │ │ +00020f20: 2c3d 3e50 726f 6a65 6374 6976 6544 6567 ,=>ProjectiveDeg │ │ │ │ +00020f30: 7265 652e 2054 6865 204d 6574 686f 6420 ree. The Method │ │ │ │ +00020f40: 496e 636c 7573 696f 6e45 7863 6c75 7369 InclusionExclusi │ │ │ │ +00020f50: 6f6e 0a77 696c 6c20 616c 7761 7973 2062 on.will always b │ │ │ │ +00020f60: 6520 7573 6564 2077 6974 6820 2a6e 6f74 e used with *not │ │ │ │ +00020f70: 6520 436f 6d70 4d65 7468 6f64 3a20 436f e CompMethod: Co │ │ │ │ +00020f80: 6d70 4d65 7468 6f64 2c20 506e 5265 7369 mpMethod, PnResi │ │ │ │ +00020f90: 6475 616c 206f 7220 6265 7274 696e 692e dual or bertini. │ │ │ │ +00020fa0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +00020fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +00020fe0: 203a 2052 203d 205a 5a2f 3332 3734 395b : R = ZZ/32749[ │ │ │ │ +00020ff0: 785f 302e 2e78 5f36 5d20 2020 2020 2020 x_0..x_6] │ │ │ │ +00021000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021010: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00021020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021040: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00021050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021050: 207c 0a7c 6f31 203d 2052 2020 2020 2020 |.|o1 = R │ │ │ │ 00021060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021080: 207c 0a7c 6f31 203a 2050 6f6c 796e 6f6d |.|o1 : Polynom │ │ │ │ -00021090: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ +00021080: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00021090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000210a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000210b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -000210c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000210d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000210e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000210f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2049 -------+.|i2 : I │ │ │ │ -00021100: 3d69 6465 616c 2872 616e 646f 6d28 322c =ideal(random(2, │ │ │ │ -00021110: 5229 2c72 616e 646f 6d28 312c 5229 2c52 R),random(1,R),R │ │ │ │ -00021120: 5f30 2a52 5f31 2a52 5f36 2d52 5f30 5e33 _0*R_1*R_6-R_0^3 │ │ │ │ -00021130: 293b 7c0a 7c20 2020 2020 2020 2020 2020 );|.| │ │ │ │ -00021140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021160: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00021170: 6f32 203a 2049 6465 616c 206f 6620 5220 o2 : Ideal of R │ │ │ │ +000210b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000210c0: 2020 2020 2020 207c 0a7c 6f31 203a 2050 |.|o1 : P │ │ │ │ +000210d0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ +000210e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000210f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021100: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00021110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00021140: 6932 203a 2049 3d69 6465 616c 2872 616e i2 : I=ideal(ran │ │ │ │ +00021150: 646f 6d28 322c 5229 2c72 616e 646f 6d28 dom(2,R),random( │ │ │ │ +00021160: 312c 5229 2c52 5f30 2a52 5f31 2a52 5f36 1,R),R_0*R_1*R_6 │ │ │ │ +00021170: 2d52 5f30 5e33 293b 7c0a 7c20 2020 2020 -R_0^3);|.| │ │ │ │ 00021180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000211a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -000211b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000211c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000211d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000211e0: 2d2d 2d2b 0a7c 6933 203a 2074 696d 6520 ---+.|i3 : time │ │ │ │ -000211f0: 4353 4d20 4920 2020 2020 2020 2020 2020 CSM I │ │ │ │ -00021200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021210: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00021220: 7c20 2d2d 2075 7365 6420 312e 3135 3431 | -- used 1.1541 │ │ │ │ -00021230: 3473 2028 6370 7529 3b20 302e 3835 3931 4s (cpu); 0.8591 │ │ │ │ -00021240: 3831 7320 2874 6872 6561 6429 3b20 3073 81s (thread); 0s │ │ │ │ -00021250: 2028 6763 2920 2020 207c 0a7c 2020 2020 (gc) |.| │ │ │ │ -00021260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021290: 2020 2020 7c0a 7c20 2020 2020 2020 2035 |.| 5 │ │ │ │ -000212a0: 2020 2020 2020 3420 2020 2020 3320 2020 4 3 │ │ │ │ +000211a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000211b0: 2020 207c 0a7c 6f32 203a 2049 6465 616c |.|o2 : Ideal │ │ │ │ +000211c0: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ +000211d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000211e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000211f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00021200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021220: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ +00021230: 2074 696d 6520 4353 4d20 4920 2020 2020 time CSM I │ │ │ │ +00021240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021260: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ +00021270: 322e 3538 3839 3573 2028 6370 7529 3b20 2.58895s (cpu); │ │ │ │ +00021280: 312e 3039 3736 3773 2028 7468 7265 6164 1.09767s (thread │ │ │ │ +00021290: 293b 2030 7320 2867 6329 2020 2020 207c ); 0s (gc) | │ │ │ │ +000212a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000212b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000212c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000212d0: 0a7c 6f33 203d 2031 3268 2020 2b20 3130 .|o3 = 12h + 10 │ │ │ │ -000212e0: 6820 202b 2036 6820 2020 2020 2020 2020 h + 6h │ │ │ │ -000212f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021300: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00021310: 2020 2020 2031 2020 2020 2020 3120 2020 1 1 │ │ │ │ -00021320: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +000212c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000212d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000212e0: 2020 2020 2035 2020 2020 2020 3420 2020 5 4 │ │ │ │ +000212f0: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00021300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021310: 2020 2020 207c 0a7c 6f33 203d 2031 3268 |.|o3 = 12h │ │ │ │ +00021320: 2020 2b20 3130 6820 202b 2036 6820 2020 + 10h + 6h │ │ │ │ 00021330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021340: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00021350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021350: 7c0a 7c20 2020 2020 2020 2031 2020 2020 |.| 1 │ │ │ │ +00021360: 2020 3120 2020 2020 3120 2020 2020 2020 1 1 │ │ │ │ 00021370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021380: 7c0a 7c20 2020 2020 5a5a 5b68 205d 2020 |.| ZZ[h ] │ │ │ │ +00021380: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00021390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000213a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000213b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000213c0: 2020 2020 2020 2031 2020 2020 2020 2020 1 │ │ │ │ -000213d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000213b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000213c0: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a |.| ZZ │ │ │ │ +000213d0: 5b68 205d 2020 2020 2020 2020 2020 2020 [h ] │ │ │ │ 000213e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000213f0: 2020 2020 2020 7c0a 7c6f 3320 3a20 2d2d |.|o3 : -- │ │ │ │ -00021400: 2d2d 2d2d 2020 2020 2020 2020 2020 2020 ---- │ │ │ │ +000213f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021400: 207c 0a7c 2020 2020 2020 2020 2031 2020 |.| 1 │ │ │ │ 00021410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021430: 207c 0a7c 2020 2020 2020 2020 3720 2020 |.| 7 │ │ │ │ -00021440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021430: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00021440: 3320 3a20 2d2d 2d2d 2d2d 2020 2020 2020 3 : ------ │ │ │ │ 00021450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021460: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00021470: 2020 2020 2020 6820 2020 2020 2020 2020 h │ │ │ │ -00021480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021470: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00021480: 2020 3720 2020 2020 2020 2020 2020 2020 7 │ │ │ │ 00021490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000214a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000214b0: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +000214a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000214b0: 2020 7c0a 7c20 2020 2020 2020 6820 2020 |.| h │ │ │ │ 000214c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000214d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000214e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -000214f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00021520: 6934 203a 2074 696d 6520 4353 4d28 492c i4 : time CSM(I, │ │ │ │ -00021530: 4d65 7468 6f64 3d3e 4469 7265 6374 436f Method=>DirectCo │ │ │ │ -00021540: 6d70 6c65 7465 496e 7429 2020 2020 2020 mpleteInt) │ │ │ │ -00021550: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00021560: 7365 6420 302e 3330 3538 3538 7320 2863 sed 0.305858s (c │ │ │ │ -00021570: 7075 293b 2030 2e32 3130 3973 2028 7468 pu); 0.2109s (th │ │ │ │ -00021580: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ -00021590: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -000215a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000215b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000215c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000215d0: 7c20 2020 2020 2020 2035 2020 2020 2020 | 5 │ │ │ │ -000215e0: 3420 2020 2020 3320 2020 2020 2020 2020 4 3 │ │ │ │ +000214e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000214f0: 2020 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2920 2020 207c 0a7c 2020 2020 (gc) |.| │ │ │ │ +000215e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000215f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021600: 2020 2020 2020 2020 207c 0a7c 6f34 203d |.|o4 = │ │ │ │ -00021610: 2031 3268 2020 2b20 3130 6820 202b 2036 12h + 10h + 6 │ │ │ │ -00021620: 6820 2020 2020 2020 2020 2020 2020 2020 h │ │ │ │ +00021600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021610: 2020 2020 7c0a 7c20 2020 2020 2020 2035 |.| 5 │ │ │ │ +00021620: 2020 2020 2020 3420 2020 2020 3320 2020 4 3 │ │ │ │ 00021630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021640: 2020 2020 7c0a 7c20 2020 2020 2020 2031 |.| 1 │ │ │ │ -00021650: 2020 2020 2020 3120 2020 2020 3120 2020 1 1 │ │ │ │ -00021660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021670: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00021680: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00021690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000216a0: 2020 2020 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I │ │ │ │ -00021930: 6e64 734f 6653 6d6f 6f74 683a 0a49 6e64 ndsOfSmooth:.Ind │ │ │ │ -00021940: 734f 6653 6d6f 6f74 682c 2e0a 0a46 756e sOfSmooth,...Fun │ │ │ │ -00021950: 6374 696f 6e73 2077 6974 6820 6f70 7469 ctions with opti │ │ │ │ -00021960: 6f6e 616c 2061 7267 756d 656e 7420 6e61 onal argument na │ │ │ │ -00021970: 6d65 6420 4d65 7468 6f64 3a0a 3d3d 3d3d med Method:.==== │ │ │ │ -00021980: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00021990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000219a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -000219b0: 2243 534d 282e 2e2e 2c4d 6574 686f 643d "CSM(...,Method= │ │ │ │ -000219c0: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e >...)" -- see *n │ │ │ │ -000219d0: 6f74 6520 4353 4d3a 2043 534d 2c20 2d2d ote CSM: CSM, -- │ │ │ │ -000219e0: 2054 6865 0a20 2020 2043 6865 726e 2d53 The. Chern-S │ │ │ │ -000219f0: 6368 7761 7274 7a2d 4d61 6350 6865 7273 chwartz-MacPhers │ │ │ │ -00021a00: 6f6e 2063 6c61 7373 0a20 202a 2045 756c on class. * Eul │ │ │ │ -00021a10: 6572 282e 2e2e 2c4d 6574 686f 643d 3e2e er(...,Method=>. │ │ │ │ -00021a20: 2e2e 2920 286d 6973 7369 6e67 2064 6f63 ..) (missing doc │ │ │ │ -00021a30: 756d 656e 7461 7469 6f6e 290a 0a46 6f72 umentation)..For │ │ │ │ -00021a40: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -00021a50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00021a60: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -00021a70: 6e6f 7465 204d 6574 686f 643a 204d 6574 note Method: Met │ │ │ │ -00021a80: 686f 642c 2069 7320 6120 2a6e 6f74 6520 hod, is a *note │ │ │ │ -00021a90: 7379 6d62 6f6c 3a20 284d 6163 6175 6c61 symbol: (Macaula │ │ │ │ -00021aa0: 7932 446f 6329 5379 6d62 6f6c 2c2e 0a0a y2Doc)Symbol,... │ │ │ │ -00021ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -00021b00: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -00021b10: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -00021b20: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -00021b30: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -00021b40: 756c 6179 322d 312e 3236 2e30 352b 6473 ulay2-1.26.05+ds │ │ │ │ -00021b50: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -00021b60: 636b 6167 6573 2f0a 4368 6172 6163 7465 ckages/.Characte │ │ │ │ -00021b70: 7269 7374 6963 436c 6173 7365 732e 6d32 risticClasses.m2 │ │ │ │ -00021b80: 3a32 3433 323a 302e 0a1f 0a46 696c 653a :2432:0....File: │ │ │ │ -00021b90: 2043 6861 7261 6374 6572 6973 7469 6343 CharacteristicC │ │ │ │ -00021ba0: 6c61 7373 6573 2e69 6e66 6f2c 204e 6f64 lasses.info, Nod │ │ │ │ -00021bb0: 653a 204d 756c 7469 5072 6f6a 436f 6f72 e: MultiProjCoor │ │ │ │ -00021bc0: 6452 696e 672c 204e 6578 743a 204f 7574 dRing, Next: Out │ │ │ │ -00021bd0: 7075 742c 2050 7265 763a 204d 6574 686f put, Prev: Metho │ │ │ │ -00021be0: 642c 2055 703a 2054 6f70 0a0a 4d75 6c74 d, Up: Top..Mult │ │ │ │ -00021bf0: 6950 726f 6a43 6f6f 7264 5269 6e67 202d iProjCoordRing - │ │ │ │ -00021c00: 2d20 4120 7175 6963 6b20 7761 7920 746f - A quick way to │ │ │ │ -00021c10: 2062 7569 6c64 2074 6865 2063 6f6f 7264 build the coord │ │ │ │ -00021c20: 696e 6174 6520 7269 6e67 206f 6620 6120 inate ring of a │ │ │ │ -00021c30: 7072 6f64 7563 7420 6f66 2070 726f 6a65 product of proje │ │ │ │ -00021c40: 6374 6976 6520 7370 6163 6573 0a2a 2a2a ctive spaces.*** │ │ │ │ -00021c50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021c60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ -00021cb0: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -00021cc0: 2020 204d 756c 7469 5072 6f6a 436f 6f72 MultiProjCoor │ │ │ │ -00021cd0: 6452 696e 6720 4469 6d73 0a20 2020 2020 dRing Dims. │ │ │ │ -00021ce0: 2020 204d 756c 7469 5072 6f6a 436f 6f72 MultiProjCoor │ │ │ │ -00021cf0: 6452 696e 6720 2843 6f65 6666 5269 6e67 dRing (CoeffRing │ │ │ │ -00021d00: 2c44 696d 7329 0a20 2020 2020 2020 204d ,Dims). M │ │ │ │ -00021d10: 756c 7469 5072 6f6a 436f 6f72 6452 696e ultiProjCoordRin │ │ │ │ -00021d20: 6720 2876 6172 2c44 696d 7329 0a20 2020 g (var,Dims). │ │ │ │ -00021d30: 2020 2020 204d 756c 7469 5072 6f6a 436f MultiProjCo │ │ │ │ -00021d40: 6f72 6452 696e 6720 2843 6f65 6666 5269 ordRing (CoeffRi │ │ │ │ -00021d50: 6e67 2c76 6172 2c44 696d 7329 0a20 202a ng,var,Dims). * │ │ │ │ -00021d60: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00021d70: 2044 696d 732c 2061 202a 6e6f 7465 206c Dims, a *note l │ │ │ │ -00021d80: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ -00021d90: 6f63 294c 6973 742c 2c20 7265 7072 6573 oc)List,, repres │ │ │ │ -00021da0: 656e 7469 6e67 2074 6865 2064 696d 656e enting the dimen │ │ │ │ -00021db0: 7369 6f6e 7320 6f66 0a20 2020 2020 2020 sions of. │ │ │ │ -00021dc0: 2074 6865 2070 726f 6a65 6374 6976 6520 the projective │ │ │ │ -00021dd0: 7370 6163 6573 2c20 692e 652e 207b 6e5f spaces, i.e. {n_ │ │ │ │ -00021de0: 312c 2e2e 2e2c 6e5f 6d7d 2063 6f72 7265 1,...,n_m} corre │ │ │ │ -00021df0: 7370 6f6e 6473 2074 6f20 5c50 505e 7b6e sponds to \PP^{n │ │ │ │ -00021e00: 5f31 7d0a 2020 2020 2020 2020 782e 2e2e _1}. x... │ │ │ │ -00021e10: 2e20 7820 5c50 505e 7b6e 5f6d 7d0a 2020 . x \PP^{n_m}. │ │ │ │ -00021e20: 2020 2020 2a20 436f 6566 6652 696e 672c * CoeffRing, │ │ │ │ -00021e30: 2061 202a 6e6f 7465 2072 696e 673a 2028 a *note ring: ( │ │ │ │ -00021e40: 4d61 6361 756c 6179 3244 6f63 2952 696e Macaulay2Doc)Rin │ │ │ │ -00021e50: 672c 2c20 7468 6520 636f 6566 6669 6369 g,, the coeffici │ │ │ │ -00021e60: 656e 7420 7269 6e67 206f 660a 2020 2020 ent ring of. │ │ │ │ -00021e70: 2020 2020 7468 6520 6772 6164 6564 2070 the graded p │ │ │ │ -00021e80: 6f6c 796e 6f6d 6961 6c20 7269 6e67 2074 olynomial ring t │ │ │ │ -00021e90: 6f20 6265 2062 7569 6c74 2062 7920 7468 o be built by th │ │ │ │ -00021ea0: 6520 6d65 7468 6f64 2c20 6279 2064 6566 e method, by def │ │ │ │ -00021eb0: 6175 6c74 2074 6869 730a 2020 2020 2020 ault this. │ │ │ │ -00021ec0: 2020 6973 205c 5a5a 2f33 3237 3439 0a20 is \ZZ/32749. │ │ │ │ -00021ed0: 2020 2020 202a 2076 6172 2c20 6120 2a6e * var, a *n │ │ │ │ -00021ee0: 6f74 6520 7379 6d62 6f6c 3a20 284d 6163 ote symbol: (Mac │ │ │ │ -00021ef0: 6175 6c61 7932 446f 6329 5379 6d62 6f6c aulay2Doc)Symbol │ │ │ │ -00021f00: 2c2c 2074 6f20 6265 2075 7365 6420 666f ,, to be used fo │ │ │ │ -00021f10: 7220 7468 650a 2020 2020 2020 2020 696e r the. in │ │ │ │ -00021f20: 7465 726d 6564 6961 7465 7320 6f66 2074 termediates of t │ │ │ │ -00021f30: 6865 2067 7261 6465 6420 706f 6c79 6e6f he graded polyno │ │ │ │ -00021f40: 6d69 616c 2072 696e 6720 746f 2062 6520 mial ring to be │ │ │ │ -00021f50: 6275 696c 7420 6279 2074 6865 206d 6574 built by the met │ │ │ │ -00021f60: 686f 640a 2020 2a20 4f75 7470 7574 733a hod. * Outputs: │ │ │ │ -00021f70: 0a20 2020 2020 202a 2061 202a 6e6f 7465 . * a *note │ │ │ │ -00021f80: 2072 696e 673a 2028 4d61 6361 756c 6179 ring: (Macaulay │ │ │ │ -00021f90: 3244 6f63 2952 696e 672c 2c20 7468 6520 2Doc)Ring,, the │ │ │ │ -00021fa0: 6772 6164 6564 2063 6f6f 7264 696e 6174 graded coordinat │ │ │ │ -00021fb0: 6520 7269 6e67 206f 6620 7468 650a 2020 e ring of the. │ │ │ │ -00021fc0: 2020 2020 2020 5c50 505e 7b6e 5f31 7d20 \PP^{n_1} │ │ │ │ -00021fd0: 782e 2e2e 2e20 7820 5c50 505e 7b6e 5f6d x.... x \PP^{n_m │ │ │ │ -00021fe0: 7d20 7768 6572 6520 7b6e 5f31 2c2e 2e2e } where {n_1,... │ │ │ │ -00021ff0: 2c6e 5f6d 7d20 6973 2074 6865 2069 6e70 ,n_m} is the inp │ │ │ │ -00022000: 7574 206c 6973 7420 6f66 0a20 2020 2020 ut list of. │ │ │ │ -00022010: 2020 2064 696d 656e 7369 6f6e 730a 0a44 dimensions..D │ │ │ │ -00022020: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -00022030: 3d3d 3d3d 3d3d 0a0a 436f 6d70 7574 6573 ======..Computes │ │ │ │ -00022040: 2074 6865 2067 7261 6465 6420 636f 6f72 the graded coor │ │ │ │ -00022050: 6469 6e61 7465 2072 696e 6720 6f66 2074 dinate ring of t │ │ │ │ -00022060: 6865 205c 5050 5e7b 6e5f 317d 2078 2e2e he \PP^{n_1} x.. │ │ │ │ -00022070: 2e2e 2078 205c 5050 5e7b 6e5f 6d7d 2077 .. x \PP^{n_m} w │ │ │ │ -00022080: 6865 7265 0a7b 6e5f 312c 2e2e 2e2c 6e5f here.{n_1,...,n_ │ │ │ │ -00022090: 6d7d 2069 7320 7468 6520 696e 7075 7420 m} is the input │ │ │ │ -000220a0: 6c69 7374 206f 6620 6469 6d65 6e73 696f list of dimensio │ │ │ │ -000220b0: 6e73 2e20 5468 6973 206d 6574 686f 6420 ns. This method │ │ │ │ -000220c0: 6973 2075 7365 6420 746f 2071 7569 636b is used to quick │ │ │ │ -000220d0: 6c79 0a62 7569 6c64 2074 6865 2063 6f6f ly.build the coo │ │ │ │ -000220e0: 7264 696e 6174 6520 7269 6e67 206f 6620 rdinate ring of │ │ │ │ -000220f0: 6120 7072 6f64 7563 7420 6f66 2070 726f a product of pro │ │ │ │ -00022100: 6a65 6374 6976 6520 7370 6163 6573 2066 jective spaces f │ │ │ │ -00022110: 6f72 2075 7365 2069 6e0a 636f 6d70 7574 or use in.comput │ │ │ │ -00022120: 6174 696f 6e73 2e0a 0a2b 2d2d 2d2d 2d2d ations...+------ │ │ │ │ -00022130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022170: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053 -------+.|i1 : S │ │ │ │ -00022180: 3d4d 756c 7469 5072 6f6a 436f 6f72 6452 =MultiProjCoordR │ │ │ │ -00022190: 696e 6728 5151 2c73 796d 626f 6c20 7a2c ing(QQ,symbol z, │ │ │ │ -000221a0: 7b31 2c33 2c33 7d29 2020 2020 2020 2020 {1,3,3}) │ │ │ │ -000221b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000221c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000221d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000221e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021820: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00021830: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00021840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021860: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00021870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +000218a0: 5768 656e 2075 7369 6e67 2074 6865 2044 When using the D │ │ │ │ +000218b0: 6972 6563 7443 6f6d 706c 6574 6549 6e74 irectCompleteInt │ │ │ │ +000218c0: 206d 6574 686f 6420 6f6e 6520 6d61 7920 method one may │ │ │ │ +000218d0: 706f 7465 6e74 6961 6c6c 7920 6675 7274 potentially furt │ │ │ │ +000218e0: 6865 7220 7370 6565 6420 7570 0a63 6f6d her speed up.com │ │ │ │ +000218f0: 7075 7461 7469 6f6e 2074 696d 6520 6279 putation time by │ │ │ │ +00021900: 2073 7065 6369 6679 696e 6720 7768 6174 specifying what │ │ │ │ +00021910: 2073 7562 7365 7420 6f66 2074 6865 2067 subset of the g │ │ │ │ +00021920: 656e 6572 6174 6f72 7320 6f66 2074 6865 enerators of the │ │ │ │ +00021930: 2069 6e70 7574 2069 6465 616c 0a64 6566 input ideal.def │ │ │ │ +00021940: 696e 6520 6120 736d 6f6f 7468 2073 7562 ine a smooth sub │ │ │ │ +00021950: 7363 6865 6d65 2028 6966 2074 6869 7320 scheme (if this │ │ │ │ +00021960: 6973 206b 6e6f 776e 292c 2073 6565 202a is known), see * │ │ │ │ +00021970: 6e6f 7465 2049 6e64 734f 6653 6d6f 6f74 note IndsOfSmoot │ │ │ │ +00021980: 683a 0a49 6e64 734f 6653 6d6f 6f74 682c h:.IndsOfSmooth, │ │ │ │ +00021990: 2e0a 0a46 756e 6374 696f 6e73 2077 6974 ...Functions wit │ │ │ │ +000219a0: 6820 6f70 7469 6f6e 616c 2061 7267 756d h optional argum │ │ │ │ +000219b0: 656e 7420 6e61 6d65 6420 4d65 7468 6f64 ent named Method │ │ │ │ +000219c0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +000219d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000219e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000219f0: 0a0a 2020 2a20 2243 534d 282e 2e2e 2c4d .. * "CSM(...,M │ │ │ │ +00021a00: 6574 686f 643d 3e2e 2e2e 2922 202d 2d20 ethod=>...)" -- │ │ │ │ +00021a10: 7365 6520 2a6e 6f74 6520 4353 4d3a 2043 see *note CSM: C │ │ │ │ +00021a20: 534d 2c20 2d2d 2054 6865 0a20 2020 2043 SM, -- The. C │ │ │ │ +00021a30: 6865 726e 2d53 6368 7761 7274 7a2d 4d61 hern-Schwartz-Ma │ │ │ │ +00021a40: 6350 6865 7273 6f6e 2063 6c61 7373 0a20 cPherson class. │ │ │ │ +00021a50: 202a 2045 756c 6572 282e 2e2e 2c4d 6574 * Euler(...,Met │ │ │ │ +00021a60: 686f 643d 3e2e 2e2e 2920 286d 6973 7369 hod=>...) (missi │ │ │ │ +00021a70: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ +00021a80: 290a 0a46 6f72 2074 6865 2070 726f 6772 )..For the progr │ │ │ │ +00021a90: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +00021aa0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +00021ab0: 6a65 6374 202a 6e6f 7465 204d 6574 686f ject *note Metho │ │ │ │ +00021ac0: 643a 204d 6574 686f 642c 2069 7320 6120 d: Method, is a │ │ │ │ +00021ad0: 2a6e 6f74 6520 7379 6d62 6f6c 3a20 284d *note symbol: (M │ │ │ │ +00021ae0: 6163 6175 6c61 7932 446f 6329 5379 6d62 acaulay2Doc)Symb │ │ │ │ +00021af0: 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ol,...---------- │ │ │ │ +00021b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021b40: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +00021b50: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +00021b60: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +00021b70: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +00021b80: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +00021b90: 2e30 352b 6473 2f4d 322f 4d61 6361 756c .05+ds/M2/Macaul │ │ │ │ +00021ba0: 6179 322f 7061 636b 6167 6573 2f0a 4368 ay2/packages/.Ch │ │ │ │ +00021bb0: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ +00021bc0: 7365 732e 6d32 3a32 3433 323a 302e 0a1f ses.m2:2432:0... │ │ │ │ +00021bd0: 0a46 696c 653a 2043 6861 7261 6374 6572 .File: Character │ │ │ │ +00021be0: 6973 7469 6343 6c61 7373 6573 2e69 6e66 isticClasses.inf │ │ │ │ +00021bf0: 6f2c 204e 6f64 653a 204d 756c 7469 5072 o, Node: MultiPr │ │ │ │ +00021c00: 6f6a 436f 6f72 6452 696e 672c 204e 6578 ojCoordRing, Nex │ │ │ │ +00021c10: 743a 204f 7574 7075 742c 2050 7265 763a t: Output, Prev: │ │ │ │ +00021c20: 204d 6574 686f 642c 2055 703a 2054 6f70 Method, Up: Top │ │ │ │ +00021c30: 0a0a 4d75 6c74 6950 726f 6a43 6f6f 7264 ..MultiProjCoord │ │ │ │ +00021c40: 5269 6e67 202d 2d20 4120 7175 6963 6b20 Ring -- A quick │ │ │ │ +00021c50: 7761 7920 746f 2062 7569 6c64 2074 6865 way to build the │ │ │ │ +00021c60: 2063 6f6f 7264 696e 6174 6520 7269 6e67 coordinate ring │ │ │ │ +00021c70: 206f 6620 6120 7072 6f64 7563 7420 6f66 of a product of │ │ │ │ +00021c80: 2070 726f 6a65 6374 6976 6520 7370 6163 projective spac │ │ │ │ +00021c90: 6573 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a es.************* │ │ │ │ +00021ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021cc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021cd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021ce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021cf0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ +00021d00: 0a20 2020 2020 2020 204d 756c 7469 5072 . MultiPr │ │ │ │ +00021d10: 6f6a 436f 6f72 6452 696e 6720 4469 6d73 ojCoordRing Dims │ │ │ │ +00021d20: 0a20 2020 2020 2020 204d 756c 7469 5072 . MultiPr │ │ │ │ +00021d30: 6f6a 436f 6f72 6452 696e 6720 2843 6f65 ojCoordRing (Coe │ │ │ │ +00021d40: 6666 5269 6e67 2c44 696d 7329 0a20 2020 ffRing,Dims). │ │ │ │ +00021d50: 2020 2020 204d 756c 7469 5072 6f6a 436f MultiProjCo │ │ │ │ +00021d60: 6f72 6452 696e 6720 2876 6172 2c44 696d ordRing (var,Dim │ │ │ │ +00021d70: 7329 0a20 2020 2020 2020 204d 756c 7469 s). Multi │ │ │ │ +00021d80: 5072 6f6a 436f 6f72 6452 696e 6720 2843 ProjCoordRing (C │ │ │ │ +00021d90: 6f65 6666 5269 6e67 2c76 6172 2c44 696d oeffRing,var,Dim │ │ │ │ +00021da0: 7329 0a20 202a 2049 6e70 7574 733a 0a20 s). * Inputs:. │ │ │ │ +00021db0: 2020 2020 202a 2044 696d 732c 2061 202a * Dims, a * │ │ │ │ +00021dc0: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ +00021dd0: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ +00021de0: 7265 7072 6573 656e 7469 6e67 2074 6865 representing the │ │ │ │ +00021df0: 2064 696d 656e 7369 6f6e 7320 6f66 0a20 dimensions of. │ │ │ │ +00021e00: 2020 2020 2020 2074 6865 2070 726f 6a65 the proje │ │ │ │ +00021e10: 6374 6976 6520 7370 6163 6573 2c20 692e ctive spaces, i. │ │ │ │ +00021e20: 652e 207b 6e5f 312c 2e2e 2e2c 6e5f 6d7d e. {n_1,...,n_m} │ │ │ │ +00021e30: 2063 6f72 7265 7370 6f6e 6473 2074 6f20 corresponds to │ │ │ │ +00021e40: 5c50 505e 7b6e 5f31 7d0a 2020 2020 2020 \PP^{n_1}. │ │ │ │ +00021e50: 2020 782e 2e2e 2e20 7820 5c50 505e 7b6e x.... x \PP^{n │ │ │ │ +00021e60: 5f6d 7d0a 2020 2020 2020 2a20 436f 6566 _m}. * Coef │ │ │ │ +00021e70: 6652 696e 672c 2061 202a 6e6f 7465 2072 fRing, a *note r │ │ │ │ +00021e80: 696e 673a 2028 4d61 6361 756c 6179 3244 ing: (Macaulay2D │ │ │ │ +00021e90: 6f63 2952 696e 672c 2c20 7468 6520 636f oc)Ring,, the co │ │ │ │ +00021ea0: 6566 6669 6369 656e 7420 7269 6e67 206f efficient ring o │ │ │ │ +00021eb0: 660a 2020 2020 2020 2020 7468 6520 6772 f. the gr │ │ │ │ +00021ec0: 6164 6564 2070 6f6c 796e 6f6d 6961 6c20 aded polynomial │ │ │ │ +00021ed0: 7269 6e67 2074 6f20 6265 2062 7569 6c74 ring to be built │ │ │ │ +00021ee0: 2062 7920 7468 6520 6d65 7468 6f64 2c20 by the method, │ │ │ │ +00021ef0: 6279 2064 6566 6175 6c74 2074 6869 730a by default this. │ │ │ │ +00021f00: 2020 2020 2020 2020 6973 205c 5a5a 2f33 is \ZZ/3 │ │ │ │ +00021f10: 3237 3439 0a20 2020 2020 202a 2076 6172 2749. * var │ │ │ │ +00021f20: 2c20 6120 2a6e 6f74 6520 7379 6d62 6f6c , a *note symbol │ │ │ │ +00021f30: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00021f40: 5379 6d62 6f6c 2c2c 2074 6f20 6265 2075 Symbol,, to be u │ │ │ │ +00021f50: 7365 6420 666f 7220 7468 650a 2020 2020 sed for the. │ │ │ │ +00021f60: 2020 2020 696e 7465 726d 6564 6961 7465 intermediate │ │ │ │ +00021f70: 7320 6f66 2074 6865 2067 7261 6465 6420 s of the graded │ │ │ │ +00021f80: 706f 6c79 6e6f 6d69 616c 2072 696e 6720 polynomial ring │ │ │ │ +00021f90: 746f 2062 6520 6275 696c 7420 6279 2074 to be built by t │ │ │ │ +00021fa0: 6865 206d 6574 686f 640a 2020 2a20 4f75 he method. * Ou │ │ │ │ +00021fb0: 7470 7574 733a 0a20 2020 2020 202a 2061 tputs:. * a │ │ │ │ +00021fc0: 202a 6e6f 7465 2072 696e 673a 2028 4d61 *note ring: (Ma │ │ │ │ +00021fd0: 6361 756c 6179 3244 6f63 2952 696e 672c caulay2Doc)Ring, │ │ │ │ +00021fe0: 2c20 7468 6520 6772 6164 6564 2063 6f6f , the graded coo │ │ │ │ +00021ff0: 7264 696e 6174 6520 7269 6e67 206f 6620 rdinate ring of │ │ │ │ +00022000: 7468 650a 2020 2020 2020 2020 5c50 505e the. \PP^ │ │ │ │ +00022010: 7b6e 5f31 7d20 782e 2e2e 2e20 7820 5c50 {n_1} x.... x \P │ │ │ │ +00022020: 505e 7b6e 5f6d 7d20 7768 6572 6520 7b6e P^{n_m} where {n │ │ │ │ +00022030: 5f31 2c2e 2e2e 2c6e 5f6d 7d20 6973 2074 _1,...,n_m} is t │ │ │ │ +00022040: 6865 2069 6e70 7574 206c 6973 7420 6f66 he input list of │ │ │ │ +00022050: 0a20 2020 2020 2020 2064 696d 656e 7369 . dimensi │ │ │ │ +00022060: 6f6e 730a 0a44 6573 6372 6970 7469 6f6e ons..Description │ │ │ │ +00022070: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 436f .===========..Co │ │ │ │ +00022080: 6d70 7574 6573 2074 6865 2067 7261 6465 mputes the grade │ │ │ │ +00022090: 6420 636f 6f72 6469 6e61 7465 2072 696e d coordinate rin │ │ │ │ +000220a0: 6720 6f66 2074 6865 205c 5050 5e7b 6e5f g of the \PP^{n_ │ │ │ │ +000220b0: 317d 2078 2e2e 2e2e 2078 205c 5050 5e7b 1} x.... x \PP^{ │ │ │ │ +000220c0: 6e5f 6d7d 2077 6865 7265 0a7b 6e5f 312c n_m} where.{n_1, │ │ │ │ +000220d0: 2e2e 2e2c 6e5f 6d7d 2069 7320 7468 6520 ...,n_m} is the │ │ │ │ +000220e0: 696e 7075 7420 6c69 7374 206f 6620 6469 input list of di │ │ │ │ +000220f0: 6d65 6e73 696f 6e73 2e20 5468 6973 206d mensions. This m │ │ │ │ +00022100: 6574 686f 6420 6973 2075 7365 6420 746f ethod is used to │ │ │ │ +00022110: 2071 7569 636b 6c79 0a62 7569 6c64 2074 quickly.build t │ │ │ │ +00022120: 6865 2063 6f6f 7264 696e 6174 6520 7269 he coordinate ri │ │ │ │ +00022130: 6e67 206f 6620 6120 7072 6f64 7563 7420 ng of a product │ │ │ │ +00022140: 6f66 2070 726f 6a65 6374 6976 6520 7370 of projective sp │ │ │ │ +00022150: 6163 6573 2066 6f72 2075 7365 2069 6e0a aces for use in. │ │ │ │ +00022160: 636f 6d70 7574 6174 696f 6e73 2e0a 0a2b computations...+ │ │ │ │ +00022170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000221a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000221b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000221c0: 6931 203a 2053 3d4d 756c 7469 5072 6f6a i1 : S=MultiProj │ │ │ │ +000221d0: 436f 6f72 6452 696e 6728 5151 2c73 796d CoordRing(QQ,sym │ │ │ │ +000221e0: 626f 6c20 7a2c 7b31 2c33 2c33 7d29 2020 bol z,{1,3,3}) │ │ │ │ 000221f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022210: 2020 2020 2020 207c 0a7c 6f31 203d 2053 |.|o1 = S │ │ │ │ +00022200: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022260: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00022250: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022260: 6f31 203d 2053 2020 2020 2020 2020 2020 o1 = S │ │ │ │ 00022270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222b0: 2020 2020 2020 207c 0a7c 6f31 203a 2050 |.|o1 : P │ │ │ │ -000222c0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ +000222a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000222b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000222c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000222d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000222e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022300: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00022310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022350: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2064 -------+.|i2 : d │ │ │ │ -00022360: 6567 7265 6573 2053 2020 2020 2020 2020 egrees S │ │ │ │ -00022370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000223a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000222f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022300: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52 o1 : PolynomialR │ │ │ │ +00022310: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ +00022320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022340: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00022350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000223a0: 6932 203a 2064 6567 7265 6573 2053 2020 i2 : degrees S │ │ │ │ 000223b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000223c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000223d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000223e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000223f0: 2020 2020 2020 207c 0a7c 6f32 203d 207b |.|o2 = { │ │ │ │ -00022400: 7b31 2c20 302c 2030 7d2c 207b 312c 2030 {1, 0, 0}, {1, 0 │ │ │ │ -00022410: 2c20 307d 2c20 7b30 2c20 312c 2030 7d2c , 0}, {0, 1, 0}, │ │ │ │ -00022420: 207b 302c 2031 2c20 307d 2c20 7b30 2c20 {0, 1, 0}, {0, │ │ │ │ -00022430: 312c 2030 7d2c 207b 302c 2031 2c20 307d 1, 0}, {0, 1, 0} │ │ │ │ -00022440: 2c20 7b30 2c20 207c 0a7c 2020 2020 202d , {0, |.| - │ │ │ │ -00022450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022490: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2030 -------|.| 0 │ │ │ │ -000224a0: 2c20 317d 2c20 7b30 2c20 302c 2031 7d2c , 1}, {0, 0, 1}, │ │ │ │ -000224b0: 207b 302c 2030 2c20 317d 2c20 7b30 2c20 {0, 0, 1}, {0, │ │ │ │ -000224c0: 302c 2031 7d7d 2020 2020 2020 2020 2020 0, 1}} │ │ │ │ -000224d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000224e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000224f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000223e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000223f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022430: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022440: 6f32 203d 207b 7b31 2c20 302c 2030 7d2c o2 = {{1, 0, 0}, │ │ │ │ +00022450: 207b 312c 2030 2c20 307d 2c20 7b30 2c20 {1, 0, 0}, {0, │ │ │ │ +00022460: 312c 2030 7d2c 207b 302c 2031 2c20 307d 1, 0}, {0, 1, 0} │ │ │ │ +00022470: 2c20 7b30 2c20 312c 2030 7d2c 207b 302c , {0, 1, 0}, {0, │ │ │ │ +00022480: 2031 2c20 307d 2c20 7b30 2c20 207c 0a7c 1, 0}, {0, |.| │ │ │ │ +00022490: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d ----------- │ │ │ │ +000224a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000224b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000224c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000224d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +000224e0: 2020 2020 2030 2c20 317d 2c20 7b30 2c20 0, 1}, {0, │ │ │ │ +000224f0: 302c 2031 7d2c 207b 302c 2030 2c20 317d 0, 1}, {0, 0, 1} │ │ │ │ +00022500: 2c20 7b30 2c20 302c 2031 7d7d 2020 2020 , {0, 0, 1}} │ │ │ │ 00022510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022530: 2020 2020 2020 207c 0a7c 6f32 203a 204c |.|o2 : L │ │ │ │ -00022540: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ +00022520: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022580: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00022590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000225a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000225b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000225c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000225d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2052 -------+.|i3 : R │ │ │ │ -000225e0: 3d4d 756c 7469 5072 6f6a 436f 6f72 6452 =MultiProjCoordR │ │ │ │ -000225f0: 696e 6720 7b32 2c33 7d20 2020 2020 2020 ing {2,3} │ │ │ │ -00022600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022620: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00022630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022570: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022580: 6f32 203a 204c 6973 7420 2020 2020 2020 o2 : List │ │ │ │ +00022590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000225a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000225b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000225c0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000225d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000225e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000225f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00022620: 6933 203a 2052 3d4d 756c 7469 5072 6f6a i3 : R=MultiProj │ │ │ │ +00022630: 436f 6f72 6452 696e 6720 7b32 2c33 7d20 CoordRing {2,3} │ │ │ │ 00022640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022670: 2020 2020 2020 207c 0a7c 6f33 203d 2052 |.|o3 = R │ │ │ │ +00022660: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000226a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000226b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ 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2020 2020 │ │ │ │ +00023090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000230a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000230b0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000230c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000230d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000230e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000230f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00023110: 6938 203a 2053 6567 7265 2841 2c69 6465 i8 : Segre(A,ide │ │ │ │ +00023120: 616c 2072 616e 646f 6d28 7b31 2c31 7d2c al random({1,1}, │ │ │ │ +00023130: 5229 2920 2020 2020 2020 2020 2020 2020 R)) │ │ │ │ 00023140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023160: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00023170: 2020 3220 3320 2020 2020 3220 3220 2020 2 3 2 2 │ │ │ │ -00023180: 2020 2020 3320 2020 2020 3220 2020 2020 3 2 │ │ │ │ -00023190: 2020 2020 3220 2020 2033 2020 2020 3220 2 3 2 │ │ │ │ -000231a0: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -000231b0: 2020 2020 2020 207c 0a7c 6f38 203d 2031 |.|o8 = 1 │ │ │ │ -000231c0: 3068 2068 2020 2d20 3668 2068 2020 2d20 0h h - 6h h - │ │ │ │ -000231d0: 3468 2068 2020 2b20 3368 2068 2020 2b20 4h h + 3h h + │ │ │ │ -000231e0: 3368 2068 2020 2b20 6820 202d 2068 2020 3h h + h - h │ │ │ │ -000231f0: 2d20 3268 2068 2020 2d20 6820 202b 2068 - 2h h - h + h │ │ │ │ -00023200: 2020 2b20 6820 207c 0a7c 2020 2020 2020 + h |.| │ │ │ │ -00023210: 2020 3120 3220 2020 2020 3120 3220 2020 1 2 1 2 │ │ │ │ -00023220: 2020 3120 3220 2020 2020 3120 3220 2020 1 2 1 2 │ │ │ │ -00023230: 2020 3120 3220 2020 2032 2020 2020 3120 1 2 2 1 │ │ │ │ -00023240: 2020 2020 3120 3220 2020 2032 2020 2020 1 2 2 │ │ │ │ -00023250: 3120 2020 2032 207c 0a7c 2020 2020 2020 1 2 |.| │ │ │ │ -00023260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000232a0: 2020 2020 2020 207c 0a7c 6f38 203a 2041 |.|o8 : A │ │ │ │ +00023150: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00023160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000231a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000231b0: 2020 2020 2020 2020 3220 3320 2020 2020 2 3 │ │ │ │ +000231c0: 3220 3220 2020 2020 2020 3320 2020 2020 2 2 3 │ │ │ │ +000231d0: 3220 2020 2020 2020 2020 3220 2020 2033 2 2 3 │ │ │ │ +000231e0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +000231f0: 2032 2020 2020 2020 2020 2020 207c 0a7c 2 |.| │ │ │ │ +00023200: 6f38 203d 2031 3068 2068 2020 2d20 3668 o8 = 10h h - 6h │ │ │ │ +00023210: 2068 2020 2d20 3468 2068 2020 2b20 3368 h - 4h h + 3h │ │ │ │ +00023220: 2068 2020 2b20 3368 2068 2020 2b20 6820 h + 3h h + h │ │ │ │ +00023230: 202d 2068 2020 2d20 3268 2068 2020 2d20 - h - 2h h - │ │ │ │ +00023240: 6820 202b 2068 2020 2b20 6820 207c 0a7c h + h + h |.| │ │ │ │ +00023250: 2020 2020 2020 2020 3120 3220 2020 2020 1 2 │ │ │ │ +00023260: 3120 3220 2020 2020 3120 3220 2020 2020 1 2 1 2 │ │ │ │ +00023270: 3120 3220 2020 2020 3120 3220 2020 2032 1 2 1 2 2 │ │ │ │ +00023280: 2020 2020 3120 2020 2020 3120 3220 2020 1 1 2 │ │ │ │ +00023290: 2032 2020 2020 3120 2020 2032 207c 0a7c 2 1 2 |.| │ │ │ │ +000232a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000232b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000232c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000232d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000232e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000232f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00023300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023340: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 -------+..Ways t │ │ │ │ -00023350: 6f20 7573 6520 4d75 6c74 6950 726f 6a43 o use MultiProjC │ │ │ │ -00023360: 6f6f 7264 5269 6e67 3a0a 3d3d 3d3d 3d3d oordRing:.====== │ │ │ │ -00023370: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00023380: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ -00023390: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ -000233a0: 6e67 284c 6973 7429 220a 2020 2a20 224d ng(List)". * "M │ │ │ │ -000233b0: 756c 7469 5072 6f6a 436f 6f72 6452 696e ultiProjCoordRin │ │ │ │ -000233c0: 6728 5269 6e67 2c4c 6973 7429 220a 2020 g(Ring,List)". │ │ │ │ -000233d0: 2a20 224d 756c 7469 5072 6f6a 436f 6f72 * "MultiProjCoor │ │ │ │ -000233e0: 6452 696e 6728 5269 6e67 2c53 796d 626f dRing(Ring,Symbo │ │ │ │ -000233f0: 6c2c 4c69 7374 2922 0a20 202a 2022 4d75 l,List)". * "Mu │ │ │ │ -00023400: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ -00023410: 2853 796d 626f 6c2c 4c69 7374 2922 0a0a (Symbol,List)".. │ │ │ │ -00023420: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -00023430: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -00023440: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -00023450: 7420 2a6e 6f74 6520 4d75 6c74 6950 726f t *note MultiPro │ │ │ │ -00023460: 6a43 6f6f 7264 5269 6e67 3a20 4d75 6c74 jCoordRing: Mult │ │ │ │ -00023470: 6950 726f 6a43 6f6f 7264 5269 6e67 2c20 iProjCoordRing, │ │ │ │ -00023480: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -00023490: 640a 6675 6e63 7469 6f6e 3a20 284d 6163 d.function: (Mac │ │ │ │ -000234a0: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -000234b0: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ -000234c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000234d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000234e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000234f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -00023510: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -00023520: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -00023530: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -00023540: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -00023550: 322d 312e 3236 2e30 352b 6473 2f4d 322f 2-1.26.05+ds/M2/ │ │ │ │ -00023560: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -00023570: 6573 2f0a 4368 6172 6163 7465 7269 7374 es/.Characterist │ │ │ │ -00023580: 6963 436c 6173 7365 732e 6d32 3a32 3035 icClasses.m2:205 │ │ │ │ -00023590: 303a 302e 0a1f 0a46 696c 653a 2043 6861 0:0....File: Cha │ │ │ │ -000235a0: 7261 6374 6572 6973 7469 6343 6c61 7373 racteristicClass │ │ │ │ -000235b0: 6573 2e69 6e66 6f2c 204e 6f64 653a 204f es.info, Node: O │ │ │ │ -000235c0: 7574 7075 742c 204e 6578 743a 2070 726f utput, Next: pro │ │ │ │ -000235d0: 6261 6269 6c69 7374 6963 2061 6c67 6f72 babilistic algor │ │ │ │ -000235e0: 6974 686d 2c20 5072 6576 3a20 4d75 6c74 ithm, Prev: Mult │ │ │ │ -000235f0: 6950 726f 6a43 6f6f 7264 5269 6e67 2c20 iProjCoordRing, │ │ │ │ -00023600: 5570 3a20 546f 700a 0a4f 7574 7075 740a Up: Top..Output. │ │ │ │ -00023610: 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 7074 ******..Descript │ │ │ │ -00023620: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -00023630: 0a54 6865 206f 7074 696f 6e20 4f75 7470 .The option Outp │ │ │ │ -00023640: 7574 2069 7320 6f6e 6c79 2075 7365 6420 ut is only used │ │ │ │ -00023650: 6279 2074 6865 2063 6f6d 6d61 6e64 7320 by the commands │ │ │ │ -00023660: 2a6e 6f74 6520 4353 4d3a 2043 534d 2c2c *note CSM: CSM,, │ │ │ │ -00023670: 202a 6e6f 7465 2053 6567 7265 3a0a 5365 *note Segre:.Se │ │ │ │ -00023680: 6772 652c 2c20 2a6e 6f74 6520 4368 6572 gre,, *note Cher │ │ │ │ -00023690: 6e3a 2043 6865 726e 2c20 616e 6420 2a6e n: Chern, and *n │ │ │ │ -000236a0: 6f74 6520 4575 6c65 723a 2045 756c 6572 ote Euler: Euler │ │ │ │ -000236b0: 2c20 746f 2073 7065 6369 6679 2074 6865 , to specify the │ │ │ │ -000236c0: 2074 7970 6520 6f66 0a6f 7574 7075 7420 type of.output │ │ │ │ -000236d0: 746f 2062 6520 7265 7475 726e 6564 2074 to be returned t │ │ │ │ -000236e0: 6f20 7468 6520 7573 6564 2e20 5468 6973 o the used. This │ │ │ │ -000236f0: 206f 7074 696f 6e20 7769 6c6c 2062 6520 option will be │ │ │ │ -00023700: 6967 6e6f 7265 6420 7768 656e 2075 7365 ignored when use │ │ │ │ -00023710: 6420 7769 7468 0a2a 6e6f 7465 2043 6f6d d with.*note Com │ │ │ │ -00023720: 704d 6574 686f 643a 2043 6f6d 704d 6574 pMethod: CompMet │ │ │ │ -00023730: 686f 642c 2050 6e52 6573 6964 7561 6c20 hod, PnResidual │ │ │ │ -00023740: 6f72 2062 6572 7469 6e69 2e20 5468 6520 or bertini. The │ │ │ │ -00023750: 6f70 7469 6f6e 2077 696c 6c20 616c 736f option will also │ │ │ │ -00023760: 2062 650a 6967 6e6f 7265 2077 6865 6e20 be.ignore when │ │ │ │ -00023770: 2a6e 6f74 6520 4d65 7468 6f64 3a20 4d65 *note Method: Me │ │ │ │ -00023780: 7468 6f64 2c3d 3e44 6972 6563 7443 6f6d thod,=>DirectCom │ │ │ │ -00023790: 706c 6574 6549 6e74 2069 7320 7573 6564 pleteInt is used │ │ │ │ -000237a0: 2e20 5468 6520 6465 6661 756c 740a 6f75 . The default.ou │ │ │ │ -000237b0: 7470 7574 2066 6f72 2061 6c6c 2074 6865 tput for all the │ │ │ │ -000237c0: 7365 206d 6574 686f 6473 2069 7320 4368 se methods is Ch │ │ │ │ -000237d0: 6f77 5269 6e67 456c 656c 6d65 6e74 2077 owRingElelment w │ │ │ │ -000237e0: 6869 6368 2077 696c 6c20 7265 7475 726e hich will return │ │ │ │ -000237f0: 2061 6e20 656c 656d 656e 740a 6f66 2074 an element.of t │ │ │ │ -00023800: 6865 2061 7070 726f 7072 6961 7465 2043 he appropriate C │ │ │ │ -00023810: 686f 7720 7269 6e67 2e20 416c 6c20 6d65 how ring. All me │ │ │ │ -00023820: 7468 6f64 7320 616c 736f 2068 6176 6520 thods also have │ │ │ │ -00023830: 616e 206f 7074 696f 6e20 4861 7368 466f an option HashFo │ │ │ │ -00023840: 726d 2077 6869 6368 0a72 6574 7572 6e73 rm which.returns │ │ │ │ -00023850: 2061 6464 6974 696f 6e61 6c20 696e 666f additional info │ │ │ │ -00023860: 726d 6174 696f 6e20 636f 6d70 7574 6564 rmation computed │ │ │ │ -00023870: 2062 7920 7468 6520 6d65 7468 6f64 7320 by the methods │ │ │ │ -00023880: 6475 7269 6e67 2074 6865 6972 2073 7461 during their sta │ │ │ │ -00023890: 6e64 6172 640a 6f70 6572 6174 696f 6e2e ndard.operation. │ │ │ │ -000238a0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ -000238b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000238c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000238d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000238e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000238f0: 2b0a 7c69 3120 3a20 5220 3d20 5a5a 2f33 +.|i1 : R = ZZ/3 │ │ │ │ -00023900: 3237 3439 5b78 5f30 2e2e 785f 365d 2020 2749[x_0..x_6] │ │ │ │ -00023910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023940: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00023950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000232e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000232f0: 6f38 203a 2041 2020 2020 2020 2020 2020 o8 : A │ │ │ │ +00023300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023330: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00023340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +00023390: 5761 7973 2074 6f20 7573 6520 4d75 6c74 Ways to use Mult │ │ │ │ +000233a0: 6950 726f 6a43 6f6f 7264 5269 6e67 3a0a iProjCoordRing:. │ │ │ │ +000233b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000233c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +000233d0: 0a20 202a 2022 4d75 6c74 6950 726f 6a43 . * "MultiProjC │ │ │ │ +000233e0: 6f6f 7264 5269 6e67 284c 6973 7429 220a oordRing(List)". │ │ │ │ +000233f0: 2020 2a20 224d 756c 7469 5072 6f6a 436f * "MultiProjCo │ │ │ │ +00023400: 6f72 6452 696e 6728 5269 6e67 2c4c 6973 ordRing(Ring,Lis │ │ │ │ +00023410: 7429 220a 2020 2a20 224d 756c 7469 5072 t)". * "MultiPr │ │ │ │ +00023420: 6f6a 436f 6f72 6452 696e 6728 5269 6e67 ojCoordRing(Ring │ │ │ │ +00023430: 2c53 796d 626f 6c2c 4c69 7374 2922 0a20 ,Symbol,List)". │ │ │ │ +00023440: 202a 2022 4d75 6c74 6950 726f 6a43 6f6f * "MultiProjCoo │ │ │ │ +00023450: 7264 5269 6e67 2853 796d 626f 6c2c 4c69 rdRing(Symbol,Li │ │ │ │ +00023460: 7374 2922 0a0a 466f 7220 7468 6520 7072 st)"..For the pr │ │ │ │ +00023470: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +00023480: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +00023490: 206f 626a 6563 7420 2a6e 6f74 6520 4d75 object *note Mu │ │ │ │ +000234a0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ +000234b0: 3a20 4d75 6c74 6950 726f 6a43 6f6f 7264 : MultiProjCoord │ │ │ │ +000234c0: 5269 6e67 2c20 6973 2061 202a 6e6f 7465 Ring, is a *note │ │ │ │ +000234d0: 206d 6574 686f 640a 6675 6e63 7469 6f6e method.function │ │ │ │ +000234e0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +000234f0: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e MethodFunction,. │ │ │ │ +00023500: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +00023510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023550: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +00023560: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +00023570: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +00023580: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +00023590: 6361 756c 6179 322d 312e 3236 2e30 352b caulay2-1.26.05+ │ │ │ │ +000235a0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +000235b0: 7061 636b 6167 6573 2f0a 4368 6172 6163 packages/.Charac │ │ │ │ +000235c0: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ +000235d0: 6d32 3a32 3035 303a 302e 0a1f 0a46 696c m2:2050:0....Fil │ │ │ │ +000235e0: 653a 2043 6861 7261 6374 6572 6973 7469 e: Characteristi │ │ │ │ +000235f0: 6343 6c61 7373 6573 2e69 6e66 6f2c 204e cClasses.info, N │ │ │ │ +00023600: 6f64 653a 204f 7574 7075 742c 204e 6578 ode: Output, Nex │ │ │ │ +00023610: 743a 2070 726f 6261 6269 6c69 7374 6963 t: probabilistic │ │ │ │ +00023620: 2061 6c67 6f72 6974 686d 2c20 5072 6576 algorithm, Prev │ │ │ │ +00023630: 3a20 4d75 6c74 6950 726f 6a43 6f6f 7264 : MultiProjCoord │ │ │ │ +00023640: 5269 6e67 2c20 5570 3a20 546f 700a 0a4f Ring, Up: Top..O │ │ │ │ +00023650: 7574 7075 740a 2a2a 2a2a 2a2a 0a0a 4465 utput.******..De │ │ │ │ +00023660: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +00023670: 3d3d 3d3d 3d0a 0a54 6865 206f 7074 696f =====..The optio │ │ │ │ +00023680: 6e20 4f75 7470 7574 2069 7320 6f6e 6c79 n Output is only │ │ │ │ +00023690: 2075 7365 6420 6279 2074 6865 2063 6f6d used by the com │ │ │ │ +000236a0: 6d61 6e64 7320 2a6e 6f74 6520 4353 4d3a mands *note CSM: │ │ │ │ +000236b0: 2043 534d 2c2c 202a 6e6f 7465 2053 6567 CSM,, *note Seg │ │ │ │ +000236c0: 7265 3a0a 5365 6772 652c 2c20 2a6e 6f74 re:.Segre,, *not │ │ │ │ +000236d0: 6520 4368 6572 6e3a 2043 6865 726e 2c20 e Chern: Chern, │ │ │ │ +000236e0: 616e 6420 2a6e 6f74 6520 4575 6c65 723a and *note Euler: │ │ │ │ +000236f0: 2045 756c 6572 2c20 746f 2073 7065 6369 Euler, to speci │ │ │ │ +00023700: 6679 2074 6865 2074 7970 6520 6f66 0a6f fy the type of.o │ │ │ │ +00023710: 7574 7075 7420 746f 2062 6520 7265 7475 utput to be retu │ │ │ │ +00023720: 726e 6564 2074 6f20 7468 6520 7573 6564 rned to the used │ │ │ │ +00023730: 2e20 5468 6973 206f 7074 696f 6e20 7769 . This option wi │ │ │ │ +00023740: 6c6c 2062 6520 6967 6e6f 7265 6420 7768 ll be ignored wh │ │ │ │ +00023750: 656e 2075 7365 6420 7769 7468 0a2a 6e6f en used with.*no │ │ │ │ +00023760: 7465 2043 6f6d 704d 6574 686f 643a 2043 te CompMethod: C │ │ │ │ +00023770: 6f6d 704d 6574 686f 642c 2050 6e52 6573 ompMethod, PnRes │ │ │ │ +00023780: 6964 7561 6c20 6f72 2062 6572 7469 6e69 idual or bertini │ │ │ │ +00023790: 2e20 5468 6520 6f70 7469 6f6e 2077 696c . The option wil │ │ │ │ +000237a0: 6c20 616c 736f 2062 650a 6967 6e6f 7265 l also be.ignore │ │ │ │ +000237b0: 2077 6865 6e20 2a6e 6f74 6520 4d65 7468 when *note Meth │ │ │ │ +000237c0: 6f64 3a20 4d65 7468 6f64 2c3d 3e44 6972 od: Method,=>Dir │ │ │ │ +000237d0: 6563 7443 6f6d 706c 6574 6549 6e74 2069 ectCompleteInt i │ │ │ │ +000237e0: 7320 7573 6564 2e20 5468 6520 6465 6661 s used. The defa │ │ │ │ +000237f0: 756c 740a 6f75 7470 7574 2066 6f72 2061 ult.output for a │ │ │ │ +00023800: 6c6c 2074 6865 7365 206d 6574 686f 6473 ll these methods │ │ │ │ +00023810: 2069 7320 4368 6f77 5269 6e67 456c 656c is ChowRingElel │ │ │ │ +00023820: 6d65 6e74 2077 6869 6368 2077 696c 6c20 ment which will │ │ │ │ +00023830: 7265 7475 726e 2061 6e20 656c 656d 656e return an elemen │ │ │ │ +00023840: 740a 6f66 2074 6865 2061 7070 726f 7072 t.of the appropr │ │ │ │ +00023850: 6961 7465 2043 686f 7720 7269 6e67 2e20 iate Chow ring. │ │ │ │ +00023860: 416c 6c20 6d65 7468 6f64 7320 616c 736f All methods also │ │ │ │ +00023870: 2068 6176 6520 616e 206f 7074 696f 6e20 have an option │ │ │ │ +00023880: 4861 7368 466f 726d 2077 6869 6368 0a72 HashForm which.r │ │ │ │ +00023890: 6574 7572 6e73 2061 6464 6974 696f 6e61 eturns additiona │ │ │ │ +000238a0: 6c20 696e 666f 726d 6174 696f 6e20 636f l information co │ │ │ │ +000238b0: 6d70 7574 6564 2062 7920 7468 6520 6d65 mputed by the me │ │ │ │ +000238c0: 7468 6f64 7320 6475 7269 6e67 2074 6865 thods during the │ │ │ │ +000238d0: 6972 2073 7461 6e64 6172 640a 6f70 6572 ir standard.oper │ │ │ │ +000238e0: 6174 696f 6e2e 0a0a 2b2d 2d2d 2d2d 2d2d ation...+------- │ │ │ │ +000238f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023930: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5220 ------+.|i1 : R │ │ │ │ +00023940: 3d20 5a5a 2f33 3237 3439 5b78 5f30 2e2e = ZZ/32749[x_0.. │ │ │ │ +00023950: 785f 365d 2020 2020 2020 2020 2020 2020 x_6] │ │ │ │ 00023960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023990: 7c0a 7c6f 3120 3d20 5220 2020 2020 2020 |.|o1 = R │ │ │ │ +00023980: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000239a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000239b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000239c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000239d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000239e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000239d0: 2020 2020 2020 7c0a 7c6f 3120 3d20 5220 |.|o1 = R │ │ │ │ +000239e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000239f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a30: 7c0a 7c6f 3120 3a20 506f 6c79 6e6f 6d69 |.|o1 : Polynomi │ │ │ │ -00023a40: 616c 5269 6e67 2020 2020 2020 2020 2020 alRing │ │ │ │ +00023a20: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a80: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00023a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ad0: 2b0a 7c69 3220 3a20 413d 4368 6f77 5269 +.|i2 : A=ChowRi │ │ │ │ -00023ae0: 6e67 2852 2920 2020 2020 2020 2020 2020 ng(R) │ │ │ │ -00023af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b20: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00023a70: 2020 2020 2020 7c0a 7c6f 3120 3a20 506f |.|o1 : Po │ │ │ │ +00023a80: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020 lynomialRing │ │ │ │ +00023a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023ac0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00023ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023b10: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 413d ------+.|i2 : A= │ │ │ │ +00023b20: 4368 6f77 5269 6e67 2852 2920 2020 2020 ChowRing(R) │ │ │ │ 00023b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b70: 7c0a 7c6f 3220 3d20 4120 2020 2020 2020 |.|o2 = A │ │ │ │ +00023b60: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023bc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00023bb0: 2020 2020 2020 7c0a 7c6f 3220 3d20 4120 |.|o2 = A │ │ │ │ +00023bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023c10: 7c0a 7c6f 3220 3a20 5175 6f74 6965 6e74 |.|o2 : Quotient │ │ │ │ -00023c20: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ +00023c00: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023c60: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00023c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023cb0: 2b0a 7c69 3320 3a20 493d 6964 6561 6c28 +.|i3 : I=ideal( │ │ │ │ -00023cc0: 7261 6e64 6f6d 2832 2c52 292c 525f 302a random(2,R),R_0* │ │ │ │ -00023cd0: 525f 312a 525f 362d 525f 305e 3329 3b20 R_1*R_6-R_0^3); │ │ │ │ -00023ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023d00: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00023d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023c50: 2020 2020 2020 7c0a 7c6f 3220 3a20 5175 |.|o2 : Qu │ │ │ │ +00023c60: 6f74 6965 6e74 5269 6e67 2020 2020 2020 otientRing │ │ │ │ +00023c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023ca0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00023cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023cf0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 493d ------+.|i3 : I= │ │ │ │ +00023d00: 6964 6561 6c28 7261 6e64 6f6d 2832 2c52 ideal(random(2,R │ │ │ │ +00023d10: 292c 525f 302a 525f 312a 525f 362d 525f ),R_0*R_1*R_6-R_ │ │ │ │ +00023d20: 305e 3329 3b20 2020 2020 2020 2020 2020 0^3); │ │ │ │ 00023d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023d50: 7c0a 7c6f 3320 3a20 4964 6561 6c20 6f66 |.|o3 : Ideal of │ │ │ │ -00023d60: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +00023d40: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023da0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00023db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023df0: 2b0a 7c69 3420 3a20 6373 6d3d 4353 4d28 +.|i4 : csm=CSM( │ │ │ │ -00023e00: 412c 492c 4f75 7470 7574 3d3e 4861 7368 A,I,Output=>Hash │ │ │ │ -00023e10: 466f 726d 2920 2020 2020 2020 2020 2020 Form) │ │ │ │ -00023e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e40: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00023e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023d90: 2020 2020 2020 7c0a 7c6f 3320 3a20 4964 |.|o3 : Id │ │ │ │ +00023da0: 6561 6c20 6f66 2052 2020 2020 2020 2020 eal of R │ │ │ │ +00023db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023de0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00023df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023e30: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6373 ------+.|i4 : cs │ │ │ │ +00023e40: 6d3d 4353 4d28 412c 492c 4f75 7470 7574 m=CSM(A,I,Output │ │ │ │ +00023e50: 3d3e 4861 7368 466f 726d 2920 2020 2020 =>HashForm) │ │ │ │ 00023e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e90: 7c0a 7c6f 3420 3d20 4d75 7461 626c 6548 |.|o4 = MutableH │ │ │ │ -00023ea0: 6173 6854 6162 6c65 7b2e 2e2e 342e 2e2e ashTable{...4... │ │ │ │ -00023eb0: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +00023e80: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023ee0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00023ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023ed0: 2020 2020 2020 7c0a 7c6f 3420 3d20 4d75 |.|o4 = Mu │ │ │ │ +00023ee0: 7461 626c 6548 6173 6854 6162 6c65 7b2e tableHashTable{. │ │ │ │ +00023ef0: 2e2e 342e 2e2e 7d20 2020 2020 2020 2020 ..4...} │ │ │ │ 00023f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f30: 7c0a 7c6f 3420 3a20 4d75 7461 626c 6548 |.|o4 : MutableH │ │ │ │ -00023f40: 6173 6854 6162 6c65 2020 2020 2020 2020 ashTable │ │ │ │ +00023f20: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f80: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00023f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023fd0: 2b0a 7c69 3520 3a20 7065 656b 2063 736d +.|i5 : peek csm │ │ │ │ -00023fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024020: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00023f70: 2020 2020 2020 7c0a 7c6f 3420 3a20 4d75 |.|o4 : Mu │ │ │ │ +00023f80: 7461 626c 6548 6173 6854 6162 6c65 2020 tableHashTable │ │ │ │ +00023f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023fc0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00023fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024010: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7065 ------+.|i5 : pe │ │ │ │ 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-00025b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025ba0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025bc0: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00025bd0: 2020 3120 2020 2020 2031 2020 2020 2031 1 1 1 │ │ │ │ -00025be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025bf0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c10: 2020 2036 2020 2020 2020 3520 2020 2020 6 5 │ │ │ │ -00025c20: 2034 2020 2020 2020 3320 2020 2020 2032 4 3 2 │ │ │ │ -00025c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c40: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025c50: 2020 2020 2020 2020 2020 7b30 7d20 3d3e {0} => │ │ │ │ -00025c60: 2036 6820 202b 2031 3868 2020 2b20 3236 6h + 18h + 26 │ │ │ │ -00025c70: 6820 202b 2032 3268 2020 2b20 3130 6820 h + 22h + 10h │ │ │ │ -00025c80: 202b 2032 6820 2020 2020 2020 2020 2020 + 2h │ │ │ │ -00025c90: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025cb0: 2020 2031 2020 2020 2020 3120 2020 2020 1 1 │ │ │ │ -00025cc0: 2031 2020 2020 2020 3120 2020 2020 2031 1 1 1 │ │ │ │ -00025cd0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ -00025ce0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d00: 2020 2036 2020 2020 2020 3520 2020 2020 6 5 │ │ │ │ -00025d10: 2034 2020 2020 2020 3320 2020 2020 2032 4 3 2 │ │ │ │ -00025d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d30: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025d40: 2020 2020 2020 2020 2020 7b31 7d20 3d3e {1} => │ │ │ │ -00025d50: 2036 6820 202b 2031 3768 2020 2b20 3238 6h + 17h + 28 │ │ │ │ -00025d60: 6820 202b 2032 3768 2020 2b20 3134 6820 h + 27h + 14h │ │ │ │ -00025d70: 202b 2033 6820 2020 2020 2020 2020 2020 + 3h │ │ │ │ -00025d80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025da0: 2020 2031 2020 2020 2020 3120 2020 2020 1 1 │ │ │ │ -00025db0: 2031 2020 2020 2020 3120 2020 2020 2031 1 1 1 │ │ │ │ -00025dc0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ -00025dd0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00025de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e20: 2b0a 0a54 6865 204d 7574 6162 6c65 4861 +..The MutableHa │ │ │ │ -00025e30: 7368 5461 626c 6520 7265 7475 726e 6564 shTable returned │ │ │ │ -00025e40: 2077 6974 6820 7468 6520 6f70 7469 6f6e with the option │ │ │ │ -00025e50: 204f 7574 7075 743d 3e48 6173 6846 6f72 Output=>HashFor │ │ │ │ -00025e60: 6d20 636f 6e74 6169 6e73 0a64 6966 6665 m contains.diffe │ │ │ │ -00025e70: 7265 6e74 2069 6e66 6f72 6d61 7469 6f6e rent information │ │ │ │ -00025e80: 2064 6570 656e 6469 6e67 206f 6e20 7468 depending on th │ │ │ │ -00025e90: 6520 6d65 7468 6f64 2077 6974 6820 7768 e method with wh │ │ │ │ -00025ea0: 6963 6820 6974 2069 7320 7573 6564 2e0a ich it is used.. │ │ │ │ -00025eb0: 4164 6469 7469 6f6e 616c 6c79 2069 6620 Additionally if │ │ │ │ -00025ec0: 7468 6520 6f70 7469 6f6e 202a 6e6f 7465 the option *note │ │ │ │ -00025ed0: 2049 6e70 7574 4973 536d 6f6f 7468 3a20 InputIsSmooth: │ │ │ │ -00025ee0: 496e 7075 7449 7353 6d6f 6f74 682c 2069 InputIsSmooth, i │ │ │ │ -00025ef0: 7320 7573 6564 2074 6865 6e20 7468 650a s used then the. │ │ │ │ -00025f00: 6861 7368 2074 6162 6c65 2072 6574 7572 hash table retur │ │ │ │ -00025f10: 6e65 6420 6279 2074 6865 206d 6574 686f ned by the metho │ │ │ │ -00025f20: 6473 2045 756c 6572 2061 6e64 2043 534d ds Euler and CSM │ │ │ │ -00025f30: 2077 696c 6c20 6265 2074 6865 2073 616d will be the sam │ │ │ │ -00025f40: 6520 6173 2074 6861 740a 7265 7475 726e e as that.return │ │ │ │ -00025f50: 6564 2062 7920 4368 6572 6e2e 2057 6865 ed by Chern. Whe │ │ │ │ -00025f60: 6e20 7573 696e 6720 7468 6520 2a6e 6f74 n using the *not │ │ │ │ -00025f70: 6520 4353 4d3a 2043 534d 2c20 2063 6f6d e CSM: CSM, com │ │ │ │ -00025f80: 6d61 6e64 2069 6e20 7468 6520 6465 6661 mand in the defa │ │ │ │ -00025f90: 756c 740a 636f 6e66 6967 7572 6174 696f ult.configuratio │ │ │ │ -00025fa0: 6e73 2028 7468 6174 2069 7320 2a6e 6f74 ns (that is *not │ │ │ │ -00025fb0: 6520 4d65 7468 6f64 3a20 4d65 7468 6f64 e Method: Method │ │ │ │ -00025fc0: 2c3d 3e49 6e63 6c75 7369 6f6e 4578 636c ,=>InclusionExcl │ │ │ │ -00025fd0: 7573 696f 6e2c 202a 6e6f 7465 0a43 6f6d usion, *note.Com │ │ │ │ -00025fe0: 704d 6574 686f 643a 2043 6f6d 704d 6574 pMethod: CompMet │ │ │ │ -00025ff0: 686f 642c 3d3e 5072 6f6a 6563 7469 7665 hod,=>Projective │ │ │ │ -00026000: 4465 6772 6565 2920 7468 6572 6520 6973 Degree) there is │ │ │ │ -00026010: 2074 6865 2061 6464 6974 696f 6e61 6c20 the additional │ │ │ │ -00026020: 6f70 7469 6f6e 2074 6f0a 7365 7420 4f75 option to.set Ou │ │ │ │ -00026030: 7470 7574 3d3e 4861 7368 466f 726d 584c tput=>HashFormXL │ │ │ │ -00026040: 2e20 5468 6973 2072 6574 7572 6e73 2061 . This returns a │ │ │ │ -00026050: 6c6c 2074 6865 2075 7375 616c 2069 6e66 ll the usual inf │ │ │ │ -00026060: 6f72 6d61 7469 6f6e 2074 6861 740a 4f75 ormation that.Ou │ │ │ │ -00026070: 7470 7574 3d3e 4861 7368 466f 726d 2077 tput=>HashForm w │ │ │ │ -00026080: 6f75 6c64 2066 6f72 2074 6869 7320 636f ould for this co │ │ │ │ -00026090: 6e66 6967 7572 6174 696f 6e20 7769 7468 nfiguration with │ │ │ │ -000260a0: 2074 6865 2061 6464 6974 696f 6e20 6f66 the addition of │ │ │ │ -000260b0: 2074 6865 0a70 726f 6a65 6374 6976 6520 the.projective │ │ │ │ -000260c0: 6465 6772 6565 7320 616e 6420 5365 6772 degrees and Segr │ │ │ │ -000260d0: 6520 636c 6173 7365 7320 6f66 2073 696e e classes of sin │ │ │ │ -000260e0: 6775 6c61 7269 7479 2073 7562 7363 6865 gularity subsche │ │ │ │ -000260f0: 6d65 7320 6765 6e65 7261 7465 6420 6279 mes generated by │ │ │ │ -00026100: 2074 6865 0a68 7970 6572 7375 7266 6163 the.hypersurfac │ │ │ │ -00026110: 6573 2063 6f6e 7369 6465 7265 6420 696e es considered in │ │ │ │ -00026120: 2074 6865 2069 6e63 6c75 7369 6f6e 2f65 the inclusion/e │ │ │ │ -00026130: 7863 6c75 7369 6f6e 2070 726f 6365 6475 xclusion procedu │ │ │ │ -00026140: 7265 2c20 7468 6174 2069 7320 696e 0a66 re, that is in.f │ │ │ │ -00026150: 696e 6469 6e67 2074 6865 2043 534d 2063 inding the CSM c │ │ │ │ -00026160: 6c61 7373 206f 6620 616c 6c20 6879 7065 lass of all hype │ │ │ │ -00026170: 7273 7572 6661 6365 7320 6765 6e65 7261 rsurfaces genera │ │ │ │ -00026180: 7465 6420 6279 2074 616b 696e 6720 6120 ted by taking a │ │ │ │ -00026190: 7072 6f64 7563 7420 6f66 0a73 6f6d 6520 product of.some │ │ │ │ -000261a0: 7375 6273 6574 7320 6f66 2067 656e 6572 subsets of gener │ │ │ │ -000261b0: 6174 6f72 7320 6f66 2074 6865 2069 6e70 ators of the inp │ │ │ │ -000261c0: 7574 2069 6465 616c 2e20 4e6f 7465 2074 ut ideal. Note t │ │ │ │ -000261d0: 6861 742c 2073 696e 6365 2074 6865 2043 hat, since the C │ │ │ │ -000261e0: 534d 2063 6c61 7373 0a6f 6620 6120 7375 SM class.of a su │ │ │ │ -000261f0: 6273 6368 656d 6520 6571 7561 6c73 2074 bscheme equals t │ │ │ │ -00026200: 6865 2043 534d 2063 6c61 7373 206f 6620 he CSM class of │ │ │ │ -00026210: 6974 7320 7265 6475 6365 6420 7363 6865 its reduced sche │ │ │ │ -00026220: 6d65 2c20 6f72 2065 7175 6976 616c 656e me, or equivalen │ │ │ │ -00026230: 746c 7920 666f 720a 7573 2074 6865 2043 tly for.us the C │ │ │ │ -00026240: 534d 2063 6c61 7373 2063 6f72 7265 7370 SM class corresp │ │ │ │ -00026250: 6f6e 6469 6e67 2074 6f20 616e 2069 6465 onding to an ide │ │ │ │ -00026260: 616c 2049 2065 7175 616c 7320 7468 6520 al I equals the │ │ │ │ -00026270: 4353 4d20 636c 6173 7320 6f66 2074 6865 CSM class of the │ │ │ │ -00026280: 0a72 6164 6963 616c 206f 6620 492c 2074 .radical of I, t │ │ │ │ -00026290: 6865 6e20 696e 7465 726e 616c 6c79 2077 hen internally w │ │ │ │ -000262a0: 6520 616c 7761 7973 2077 6f72 6b20 7769 e always work wi │ │ │ │ -000262b0: 7468 2072 6164 6963 616c 2069 6465 616c th radical ideal │ │ │ │ -000262c0: 7320 2866 6f72 0a65 6666 6963 6965 6e63 s (for.efficienc │ │ │ │ -000262d0: 7920 7265 6173 6f6e 7329 2e20 4865 6e63 y reasons). Henc │ │ │ │ -000262e0: 6520 7468 6520 7072 6f6a 6563 7469 7665 e the projective │ │ │ │ -000262f0: 2064 6567 7265 6573 2061 6e64 2053 6567 degrees and Seg │ │ │ │ -00026300: 7265 2063 6c61 7373 6573 2063 6f6d 7075 re classes compu │ │ │ │ -00026310: 7465 640a 696e 7465 726e 616c 6c79 2077 ted.internally w │ │ │ │ -00026320: 696c 6c20 6265 2074 686f 7365 206f 6620 ill be those of │ │ │ │ -00026330: 7468 6520 7261 6469 6361 6c20 6f66 2061 the radical of a │ │ │ │ -00026340: 6e20 6964 6561 6c20 6465 6669 6e65 6420 n ideal defined │ │ │ │ -00026350: 6279 2061 2070 6f6c 796e 6f6d 6961 6c0a by a polynomial. │ │ │ │ -00026360: 7768 6963 6820 6973 2061 2070 726f 6475 which is a produ │ │ │ │ -00026370: 6374 206f 6620 736f 6d65 2073 7562 7365 ct of some subse │ │ │ │ -00026380: 7420 6f66 2074 6865 2067 656e 6572 6174 t of the generat │ │ │ │ -00026390: 6f72 732e 2057 6520 696c 6c75 7374 7261 ors. We illustra │ │ │ │ -000263a0: 7465 2074 6869 7320 7769 7468 2061 6e0a te this with an. │ │ │ │ -000263b0: 6578 616d 706c 6520 6265 6c6f 772e 0a0a example below... │ │ │ │ -000263c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -000263d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000263e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000263f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00026410: 7c69 3134 203a 2063 736d 584c 6861 7368 |i14 : csmXLhash │ │ │ │ -00026420: 3d43 534d 2841 2c49 2c4f 7574 7075 743d =CSM(A,I,Output= │ │ │ │ -00026430: 3e48 6173 6846 6f72 6d58 4c29 2020 2020 >HashFormXL) │ │ │ │ -00026440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026450: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00026460: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00026470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025a00: 2020 2020 2020 7c0a 7c6f 3133 203d 204d |.|o13 = M │ │ │ │ +00025a10: 7574 6162 6c65 4861 7368 5461 626c 657b utableHashTable{ │ │ │ │ +00025a20: 4575 6c65 7220 3d3e 2031 3020 2020 2020 Euler => 10 │ │ │ │ +00025a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025a40: 2020 2020 2020 2020 2020 2020 2020 207d } │ │ │ │ +00025a50: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025a70: 2020 2020 2020 2020 2020 2020 3620 2020 6 │ │ │ │ +00025a80: 2020 2035 2020 2020 2020 3420 2020 2020 5 4 │ │ │ │ +00025a90: 2033 2020 2020 2020 3220 2020 2020 2020 3 2 │ │ │ │ +00025aa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025ac0: 7b30 2c20 317d 203d 3e20 3268 2020 2b20 {0, 1} => 2h + │ │ │ │ +00025ad0: 3233 6820 202b 2033 3268 2020 2b20 3333 23h + 32h + 33 │ │ │ │ +00025ae0: 6820 202b 2031 3868 2020 2b20 3568 2020 h + 18h + 5h │ │ │ │ +00025af0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b10: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ +00025b20: 2020 2031 2020 2020 2020 3120 2020 2020 1 1 │ │ │ │ +00025b30: 2031 2020 2020 2020 3120 2020 2020 3120 1 1 1 │ │ │ │ +00025b40: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b60: 2020 2020 2020 2020 2020 3620 2020 2020 6 │ │ │ │ +00025b70: 2035 2020 2020 2020 3420 2020 2020 2033 5 4 3 │ │ │ │ +00025b80: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00025b90: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025bb0: 4353 4d20 3d3e 2031 3068 2020 2b20 3132 CSM => 10h + 12 │ │ │ │ +00025bc0: 6820 202b 2032 3268 2020 2b20 3136 6820 h + 22h + 16h │ │ │ │ +00025bd0: 202b 2036 6820 2020 2020 2020 2020 2020 + 6h │ │ │ │ +00025be0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025c00: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +00025c10: 2031 2020 2020 2020 3120 2020 2020 2031 1 1 1 │ │ │ │ +00025c20: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +00025c30: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025c50: 2020 2020 2020 2020 2036 2020 2020 2020 6 │ │ │ │ +00025c60: 3520 2020 2020 2034 2020 2020 2020 3320 5 4 3 │ │ │ │ +00025c70: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00025c80: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025ca0: 7b30 7d20 3d3e 2036 6820 202b 2031 3868 {0} => 6h + 18h │ │ │ │ +00025cb0: 2020 2b20 3236 6820 202b 2032 3268 2020 + 26h + 22h │ │ │ │ +00025cc0: 2b20 3130 6820 202b 2032 6820 2020 2020 + 10h + 2h │ │ │ │ +00025cd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025cf0: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ +00025d00: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ +00025d10: 2020 2020 2031 2020 2020 2031 2020 2020 1 1 │ │ │ │ +00025d20: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025d40: 2020 2020 2020 2020 2036 2020 2020 2020 6 │ │ │ │ +00025d50: 3520 2020 2020 2034 2020 2020 2020 3320 5 4 3 │ │ │ │ +00025d60: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00025d70: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025d90: 7b31 7d20 3d3e 2036 6820 202b 2031 3768 {1} => 6h + 17h │ │ │ │ +00025da0: 2020 2b20 3238 6820 202b 2032 3768 2020 + 28h + 27h │ │ │ │ +00025db0: 2b20 3134 6820 202b 2033 6820 2020 2020 + 14h + 3h │ │ │ │ +00025dc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025de0: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ +00025df0: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ +00025e00: 2020 2020 2031 2020 2020 2031 2020 2020 1 1 │ │ │ │ +00025e10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00025e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025e60: 2d2d 2d2d 2d2d 2b0a 0a54 6865 204d 7574 ------+..The Mut │ │ │ │ +00025e70: 6162 6c65 4861 7368 5461 626c 6520 7265 ableHashTable re │ │ │ │ +00025e80: 7475 726e 6564 2077 6974 6820 7468 6520 turned with the │ │ │ │ +00025e90: 6f70 7469 6f6e 204f 7574 7075 743d 3e48 option Output=>H │ │ │ │ +00025ea0: 6173 6846 6f72 6d20 636f 6e74 6169 6e73 ashForm contains │ │ │ │ +00025eb0: 0a64 6966 6665 7265 6e74 2069 6e66 6f72 .different infor │ │ │ │ +00025ec0: 6d61 7469 6f6e 2064 6570 656e 6469 6e67 mation depending │ │ │ │ +00025ed0: 206f 6e20 7468 6520 6d65 7468 6f64 2077 on the method w │ │ │ │ +00025ee0: 6974 6820 7768 6963 6820 6974 2069 7320 ith which it is │ │ │ │ +00025ef0: 7573 6564 2e0a 4164 6469 7469 6f6e 616c used..Additional │ │ │ │ +00025f00: 6c79 2069 6620 7468 6520 6f70 7469 6f6e ly if the option │ │ │ │ +00025f10: 202a 6e6f 7465 2049 6e70 7574 4973 536d *note InputIsSm │ │ │ │ +00025f20: 6f6f 7468 3a20 496e 7075 7449 7353 6d6f ooth: InputIsSmo │ │ │ │ +00025f30: 6f74 682c 2069 7320 7573 6564 2074 6865 oth, is used the │ │ │ │ +00025f40: 6e20 7468 650a 6861 7368 2074 6162 6c65 n the.hash table │ │ │ │ +00025f50: 2072 6574 7572 6e65 6420 6279 2074 6865 returned by the │ │ │ │ +00025f60: 206d 6574 686f 6473 2045 756c 6572 2061 methods Euler a │ │ │ │ +00025f70: 6e64 2043 534d 2077 696c 6c20 6265 2074 nd CSM will be t │ │ │ │ +00025f80: 6865 2073 616d 6520 6173 2074 6861 740a he same as that. │ │ │ │ +00025f90: 7265 7475 726e 6564 2062 7920 4368 6572 returned by Cher │ │ │ │ +00025fa0: 6e2e 2057 6865 6e20 7573 696e 6720 7468 n. When using th │ │ │ │ +00025fb0: 6520 2a6e 6f74 6520 4353 4d3a 2043 534d e *note CSM: CSM │ │ │ │ +00025fc0: 2c20 2063 6f6d 6d61 6e64 2069 6e20 7468 , command in th │ │ │ │ +00025fd0: 6520 6465 6661 756c 740a 636f 6e66 6967 e default.config │ │ │ │ +00025fe0: 7572 6174 696f 6e73 2028 7468 6174 2069 urations (that i │ │ │ │ +00025ff0: 7320 2a6e 6f74 6520 4d65 7468 6f64 3a20 s *note Method: │ │ │ │ +00026000: 4d65 7468 6f64 2c3d 3e49 6e63 6c75 7369 Method,=>Inclusi │ │ │ │ +00026010: 6f6e 4578 636c 7573 696f 6e2c 202a 6e6f onExclusion, *no │ │ │ │ +00026020: 7465 0a43 6f6d 704d 6574 686f 643a 2043 te.CompMethod: C │ │ │ │ +00026030: 6f6d 704d 6574 686f 642c 3d3e 5072 6f6a ompMethod,=>Proj │ │ │ │ +00026040: 6563 7469 7665 4465 6772 6565 2920 7468 ectiveDegree) th │ │ │ │ +00026050: 6572 6520 6973 2074 6865 2061 6464 6974 ere is the addit │ │ │ │ +00026060: 696f 6e61 6c20 6f70 7469 6f6e 2074 6f0a ional option to. │ │ │ │ +00026070: 7365 7420 4f75 7470 7574 3d3e 4861 7368 set Output=>Hash │ │ │ │ +00026080: 466f 726d 584c 2e20 5468 6973 2072 6574 FormXL. This ret │ │ │ │ +00026090: 7572 6e73 2061 6c6c 2074 6865 2075 7375 urns all the usu │ │ │ │ +000260a0: 616c 2069 6e66 6f72 6d61 7469 6f6e 2074 al information t │ │ │ │ +000260b0: 6861 740a 4f75 7470 7574 3d3e 4861 7368 hat.Output=>Hash │ │ │ │ +000260c0: 466f 726d 2077 6f75 6c64 2066 6f72 2074 Form would for t │ │ │ │ +000260d0: 6869 7320 636f 6e66 6967 7572 6174 696f his configuratio │ │ │ │ +000260e0: 6e20 7769 7468 2074 6865 2061 6464 6974 n with the addit │ │ │ │ +000260f0: 696f 6e20 6f66 2074 6865 0a70 726f 6a65 ion of the.proje │ │ │ │ +00026100: 6374 6976 6520 6465 6772 6565 7320 616e ctive degrees an │ │ │ │ +00026110: 6420 5365 6772 6520 636c 6173 7365 7320 d Segre classes │ │ │ │ +00026120: 6f66 2073 696e 6775 6c61 7269 7479 2073 of singularity s │ │ │ │ +00026130: 7562 7363 6865 6d65 7320 6765 6e65 7261 ubschemes genera │ │ │ │ +00026140: 7465 6420 6279 2074 6865 0a68 7970 6572 ted by the.hyper │ │ │ │ +00026150: 7375 7266 6163 6573 2063 6f6e 7369 6465 surfaces conside │ │ │ │ +00026160: 7265 6420 696e 2074 6865 2069 6e63 6c75 red in the inclu │ │ │ │ +00026170: 7369 6f6e 2f65 7863 6c75 7369 6f6e 2070 sion/exclusion p │ │ │ │ +00026180: 726f 6365 6475 7265 2c20 7468 6174 2069 rocedure, that i │ │ │ │ +00026190: 7320 696e 0a66 696e 6469 6e67 2074 6865 s in.finding the │ │ │ │ +000261a0: 2043 534d 2063 6c61 7373 206f 6620 616c CSM class of al │ │ │ │ +000261b0: 6c20 6879 7065 7273 7572 6661 6365 7320 l hypersurfaces │ │ │ │ +000261c0: 6765 6e65 7261 7465 6420 6279 2074 616b generated by tak │ │ │ │ +000261d0: 696e 6720 6120 7072 6f64 7563 7420 6f66 ing a product of │ │ │ │ +000261e0: 0a73 6f6d 6520 7375 6273 6574 7320 6f66 .some subsets of │ │ │ │ +000261f0: 2067 656e 6572 6174 6f72 7320 6f66 2074 generators of t │ │ │ │ +00026200: 6865 2069 6e70 7574 2069 6465 616c 2e20 he input ideal. │ │ │ │ +00026210: 4e6f 7465 2074 6861 742c 2073 696e 6365 Note that, since │ │ │ │ +00026220: 2074 6865 2043 534d 2063 6c61 7373 0a6f the CSM class.o │ │ │ │ +00026230: 6620 6120 7375 6273 6368 656d 6520 6571 f a subscheme eq │ │ │ │ +00026240: 7561 6c73 2074 6865 2043 534d 2063 6c61 uals the CSM cla │ │ │ │ +00026250: 7373 206f 6620 6974 7320 7265 6475 6365 ss of its reduce │ │ │ │ +00026260: 6420 7363 6865 6d65 2c20 6f72 2065 7175 d scheme, or equ │ │ │ │ +00026270: 6976 616c 656e 746c 7920 666f 720a 7573 ivalently for.us │ │ │ │ +00026280: 2074 6865 2043 534d 2063 6c61 7373 2063 the CSM class c │ │ │ │ +00026290: 6f72 7265 7370 6f6e 6469 6e67 2074 6f20 orresponding to │ │ │ │ +000262a0: 616e 2069 6465 616c 2049 2065 7175 616c an ideal I equal │ │ │ │ +000262b0: 7320 7468 6520 4353 4d20 636c 6173 7320 s the CSM class │ │ │ │ +000262c0: 6f66 2074 6865 0a72 6164 6963 616c 206f of the.radical o │ │ │ │ +000262d0: 6620 492c 2074 6865 6e20 696e 7465 726e f I, then intern │ │ │ │ +000262e0: 616c 6c79 2077 6520 616c 7761 7973 2077 ally we always w │ │ │ │ +000262f0: 6f72 6b20 7769 7468 2072 6164 6963 616c ork with radical │ │ │ │ +00026300: 2069 6465 616c 7320 2866 6f72 0a65 6666 ideals (for.eff │ │ │ │ +00026310: 6963 6965 6e63 7920 7265 6173 6f6e 7329 iciency reasons) │ │ │ │ +00026320: 2e20 4865 6e63 6520 7468 6520 7072 6f6a . Hence the proj │ │ │ │ +00026330: 6563 7469 7665 2064 6567 7265 6573 2061 ective degrees a │ │ │ │ +00026340: 6e64 2053 6567 7265 2063 6c61 7373 6573 nd Segre classes │ │ │ │ +00026350: 2063 6f6d 7075 7465 640a 696e 7465 726e computed.intern │ │ │ │ +00026360: 616c 6c79 2077 696c 6c20 6265 2074 686f ally will be tho │ │ │ │ +00026370: 7365 206f 6620 7468 6520 7261 6469 6361 se of the radica │ │ │ │ +00026380: 6c20 6f66 2061 6e20 6964 6561 6c20 6465 l of an ideal de │ │ │ │ +00026390: 6669 6e65 6420 6279 2061 2070 6f6c 796e fined by a polyn │ │ │ │ +000263a0: 6f6d 6961 6c0a 7768 6963 6820 6973 2061 omial.which is a │ │ │ │ +000263b0: 2070 726f 6475 6374 206f 6620 736f 6d65 product of some │ │ │ │ +000263c0: 2073 7562 7365 7420 6f66 2074 6865 2067 subset of the g │ │ │ │ +000263d0: 656e 6572 6174 6f72 732e 2057 6520 696c enerators. We il │ │ │ │ +000263e0: 6c75 7374 7261 7465 2074 6869 7320 7769 lustrate this wi │ │ │ │ +000263f0: 7468 2061 6e0a 6578 616d 706c 6520 6265 th an.example be │ │ │ │ +00026400: 6c6f 772e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d low...+--------- │ │ │ │ +00026410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026450: 2d2d 2d2d 2b0a 7c69 3134 203a 2063 736d ----+.|i14 : csm │ │ │ │ +00026460: 584c 6861 7368 3d43 534d 2841 2c49 2c4f XLhash=CSM(A,I,O │ │ │ │ +00026470: 7574 7075 743d 3e48 6173 6846 6f72 6d58 utput=>HashFormX │ │ │ │ +00026480: 4c29 2020 2020 2020 2020 2020 2020 2020 L) │ │ │ │ 00026490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000264a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000264b0: 7c6f 3134 203d 204d 7574 6162 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2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00026550: 7c6f 3134 203a 204d 7574 6162 6c65 4861 |o14 : MutableHa │ │ │ │ -00026560: 7368 5461 626c 6520 2020 2020 2020 2020 shTable │ │ │ │ +00026540: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026590: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000265a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -000265b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000265c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000265d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000265e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -000265f0: 7c69 3135 203a 2070 6565 6b20 6373 6d58 |i15 : peek csmX │ │ │ │ -00026600: 4c68 6173 6820 2020 2020 2020 2020 2020 Lhash │ │ │ │ -00026610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026630: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00026640: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00026590: 2020 2020 7c0a 7c6f 3134 203a 204d 7574 |.|o14 : Mut │ │ │ │ +000265a0: 6162 6c65 4861 7368 5461 626c 6520 2020 ableHashTable │ │ │ │ +000265b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000265c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000265d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000265e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000265f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026630: 2d2d 2d2d 2b0a 7c69 3135 203a 2070 6565 ----+.|i15 : pee │ │ │ │ +00026640: 6b20 6373 6d58 4c68 6173 6820 2020 2020 k csmXLhash │ │ │ │ 00026650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026680: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00026690: 7c6f 3135 203d 204d 7574 6162 6c65 4861 |o15 = MutableHa │ │ │ │ -000266a0: 7368 5461 626c 657b 4728 4a61 636f 6269 shTable{G(Jacobi │ │ │ │ -000266b0: 616e 297b 307d 203d 3e20 3020 2020 2020 an){0} => 0 │ │ │ │ +00026680: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000266a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000266b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000266c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000266d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000266e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000266f0: 2020 2020 2020 2020 5365 6772 6528 4a61 Segre(Ja │ │ │ │ -00026700: 636f 6269 616e 297b 307d 203d 3e20 3020 cobian){0} => 0 │ │ │ │ +000266d0: 2020 2020 7c0a 7c6f 3135 203d 204d 7574 |.|o15 = Mut │ │ │ │ +000266e0: 6162 6c65 4861 7368 5461 626c 657b 4728 ableHashTable{G( │ │ │ │ +000266f0: 4a61 636f 6269 616e 297b 307d 203d 3e20 Jacobian){0} => │ │ │ │ +00026700: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ 00026710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026720: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00026730: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00026740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026760: 2020 2020 2036 2020 2020 2020 2035 2020 6 5 │ │ │ │ -00026770: 2020 2020 2034 2020 2020 2020 2020 7c0a 4 |. │ │ │ │ -00026780: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00026790: 2020 2020 2020 2020 5365 6772 6528 4a61 Segre(Ja │ │ │ │ -000267a0: 636f 6269 616e 297b 302c 2031 7d20 3d3e cobian){0, 1} => │ │ │ │ -000267b0: 2033 3930 6820 202d 2033 3836 6820 202b 390h - 386h + │ │ │ │ -000267c0: 2031 3530 6820 202d 2020 2020 2020 7c0a 150h - |. │ │ │ │ -000267d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000267e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000267f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026800: 2020 2020 2031 2020 2020 2020 2031 2020 1 1 │ │ │ │ -00026810: 2020 2020 2031 2020 2020 2020 2020 7c0a 1 |. │ │ │ │ -00026820: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00026720: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026730: 2020 2020 2020 2020 2020 2020 2020 5365 Se │ │ │ │ +00026740: 6772 6528 4a61 636f 6269 616e 297b 307d gre(Jacobian){0} │ │ │ │ +00026750: 203d 3e20 3020 2020 2020 2020 2020 2020 => 0 │ │ │ │ +00026760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026770: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000267a0: 2020 2020 2020 2020 2020 2036 2020 2020 6 │ │ │ │ +000267b0: 2020 2035 2020 2020 2020 2034 2020 2020 5 4 │ │ │ │ +000267c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000267d0: 2020 2020 2020 2020 2020 2020 2020 5365 Se │ │ │ │ +000267e0: 6772 6528 4a61 636f 6269 616e 297b 302c gre(Jacobian){0, │ │ │ │ +000267f0: 2031 7d20 3d3e 2033 3930 6820 202d 2033 1} => 390h - 3 │ │ │ │ +00026800: 3836 6820 202b 2031 3530 6820 202d 2020 86h + 150h - │ │ │ │ +00026810: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026850: 2020 2020 3620 2020 2020 2035 2020 2020 6 5 │ │ │ │ -00026860: 2033 2020 2020 2032 2020 2020 2020 7c0a 3 2 |. │ │ │ │ -00026870: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00026880: 2020 2020 2020 2020 5365 6772 6528 4a61 Segre(Ja │ │ │ │ -00026890: 636f 6269 616e 297b 317d 203d 3e20 2d20 cobian){1} => - │ │ │ │ -000268a0: 3136 3068 2020 2b20 3332 6820 202d 2034 160h + 32h - 4 │ │ │ │ -000268b0: 6820 202b 2032 6820 2020 2020 2020 7c0a h + 2h |. │ │ │ │ -000268c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000268d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000268e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000268f0: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00026900: 2031 2020 2020 2031 2020 2020 2020 7c0a 1 1 |. │ │ │ │ -00026910: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00026840: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +00026850: 2020 2031 2020 2020 2020 2031 2020 2020 1 1 │ │ │ │ +00026860: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026890: 2020 2020 2020 2020 2020 3620 2020 2020 6 │ │ │ │ +000268a0: 2035 2020 2020 2033 2020 2020 2032 2020 5 3 2 │ │ │ │ +000268b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000268c0: 2020 2020 2020 2020 2020 2020 2020 5365 Se │ │ │ │ +000268d0: 6772 6528 4a61 636f 6269 616e 297b 317d gre(Jacobian){1} │ │ │ │ +000268e0: 203d 3e20 2d20 3136 3068 2020 2b20 3332 => - 160h + 32 │ │ │ │ +000268f0: 6820 202d 2034 6820 202b 2032 6820 2020 h - 4h + 2h │ │ │ │ +00026900: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026940: 3620 2020 2020 2035 2020 2020 2020 3420 6 5 4 │ │ │ │ -00026950: 2020 2020 2033 2020 2020 2020 2020 7c0a 3 |. │ │ │ │ -00026960: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00026970: 2020 2020 2020 2020 4728 4a61 636f 6269 G(Jacobi │ │ │ │ -00026980: 616e 297b 302c 2031 7d20 3d3e 2031 3068 an){0, 1} => 10h │ │ │ │ -00026990: 2020 2b20 3130 6820 202b 2031 3068 2020 + 10h + 10h │ │ │ │ -000269a0: 2b20 3130 6820 202b 2020 2020 2020 7c0a + 10h + |. │ │ │ │ -000269b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000269c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000269d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000269e0: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ -000269f0: 2020 2020 2031 2020 2020 2020 2020 7c0a 1 |. │ │ │ │ -00026a00: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00026930: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +00026940: 2031 2020 2020 2031 2020 2020 2031 2020 1 1 1 │ │ │ │ +00026950: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026980: 2020 2020 2020 3620 2020 2020 2035 2020 6 5 │ │ │ │ +00026990: 2020 2020 3420 2020 2020 2033 2020 2020 4 3 │ │ │ │ +000269a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000269b0: 2020 2020 2020 2020 2020 2020 2020 4728 G( │ │ │ │ +000269c0: 4a61 636f 6269 616e 297b 302c 2031 7d20 Jacobian){0, 1} │ │ │ │ +000269d0: 3d3e 2031 3068 2020 2b20 3130 6820 202b => 10h + 10h + │ │ │ │ +000269e0: 2031 3068 2020 2b20 3130 6820 202b 2020 10h + 10h + │ │ │ │ +000269f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026a20: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -00026a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026a40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00026a50: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00026a60: 2020 2020 2020 2020 4728 4a61 636f 6269 G(Jacobi │ │ │ │ -00026a70: 616e 297b 317d 203d 3e20 3268 2020 2b20 an){1} => 2h + │ │ │ │ -00026a80: 3268 2020 2b20 3120 2020 2020 2020 2020 2h + 1 │ │ │ │ -00026a90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00026aa0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00026ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ac0: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ -00026ad0: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -00026ae0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00026af0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00026a20: 2020 2020 2020 3120 2020 2020 2031 2020 1 1 │ │ │ │ +00026a30: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ +00026a40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026a70: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00026a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026a90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026aa0: 2020 2020 2020 2020 2020 2020 2020 4728 G( │ │ │ │ +00026ab0: 4a61 636f 6269 616e 297b 317d 203d 3e20 Jacobian){1} => │ │ │ │ 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2031 2020 2020 1 1 │ │ │ │ -00026bc0: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ -00026bd0: 3120 2020 2020 3120 2020 2020 2020 7c0a 1 1 |. │ │ │ │ -00026be0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00026bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c00: 2020 3620 2020 2020 2035 2020 2020 2020 6 5 │ │ │ │ -00026c10: 3420 2020 2020 2033 2020 2020 2032 2020 4 3 2 │ │ │ │ -00026c20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00026c30: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00026c40: 2020 2020 2020 2020 4353 4d20 3d3e 2031 CSM => 1 │ │ │ │ -00026c50: 3068 2020 2b20 3132 6820 202b 2032 3268 0h + 12h + 22h │ │ │ │ -00026c60: 2020 2b20 3136 6820 202b 2036 6820 2020 + 16h + 6h │ │ │ │ -00026c70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00026c80: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00026c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ca0: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ -00026cb0: 3120 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2020 7c0a |. │ │ │ │ -00027310: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00027260: 2020 2020 7c0a 7c20 2031 2020 2020 2031 |.| 1 1 │ │ │ │ +00027270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000272a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000272b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000272c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000272d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000272e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000272f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027300: 2d2d 2d2d 2b0a 7c69 3136 203a 204b 3d69 ----+.|i16 : K=i │ │ │ │ +00027310: 6465 616c 2049 5f30 2a49 5f31 3b20 2020 deal I_0*I_1; │ │ │ │ 00027320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027330: 2020 2020 2020 2020 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---------------- │ │ │ │ +00027420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027440: 2d2d 2d2d 2b0a 7c69 3137 203a 2043 534d ----+.|i17 : CSM │ │ │ │ +00027450: 2841 2c72 6164 6963 616c 204b 293d 3d43 (A,radical K)==C │ │ │ │ +00027460: 534d 2841 2c4b 2920 2020 2020 2020 2020 SM(A,K) │ │ │ │ 00027470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027490: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000274a0: 7c6f 3137 203d 2074 7275 6520 2020 2020 |o17 = true │ │ │ │ +00027490: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000274a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000274b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000274c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000274d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000274e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000274f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00027500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00027540: 7c69 3138 203a 204a 3d69 6465 616c 206a |i18 : J=ideal j │ │ │ │ -00027550: 6163 6f62 6961 6e20 7261 6469 6361 6c20 acobian radical │ │ │ │ -00027560: 4b3b 2020 2020 2020 2020 2020 2020 2020 K; │ │ │ │ -00027570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027580: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027590: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000275a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000274e0: 2020 2020 7c0a 7c6f 3137 203d 2074 7275 |.|o17 = tru │ │ │ │ +000274f0: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +00027500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027530: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00027540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027580: 2d2d 2d2d 2b0a 7c69 3138 203a 204a 3d69 ----+.|i18 : J=i │ │ │ │ +00027590: 6465 616c 206a 6163 6f62 6961 6e20 7261 deal jacobian ra │ │ │ │ +000275a0: 6469 6361 6c20 4b3b 2020 2020 2020 2020 dical K; │ │ │ │ 000275b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000275c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000275d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000275e0: 7c6f 3138 203a 2049 6465 616c 206f 6620 |o18 : Ideal of │ │ │ │ -000275f0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +000275d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000275e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000275f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027620: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027630: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00027640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00027680: 7c69 3139 203a 2073 6567 4a3d 5365 6772 |i19 : segJ=Segr │ │ │ │ -00027690: 6528 412c 4a2c 4f75 7470 7574 3d3e 4861 e(A,J,Output=>Ha │ │ │ │ -000276a0: 7368 466f 726d 2920 2020 2020 2020 2020 shForm) │ │ │ │ -000276b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000276c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000276d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000276e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027620: 2020 2020 7c0a 7c6f 3138 203a 2049 6465 |.|o18 : Ide │ │ │ │ +00027630: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ +00027640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027670: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00027680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000276a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000276b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000276c0: 2d2d 2d2d 2b0a 7c69 3139 203a 2073 6567 ----+.|i19 : seg │ │ │ │ +000276d0: 4a3d 5365 6772 6528 412c 4a2c 4f75 7470 J=Segre(A,J,Outp │ │ │ │ +000276e0: 7574 3d3e 4861 7368 466f 726d 2920 2020 ut=>HashForm) │ │ │ │ 000276f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027710: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027720: 7c6f 3139 203d 204d 7574 6162 6c65 4861 |o19 = MutableHa │ │ │ │ -00027730: 7368 5461 626c 657b 2e2e 2e34 2e2e 2e7d shTable{...4...} │ │ │ │ +00027710: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00027720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027760: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027770: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00027780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027760: 2020 2020 7c0a 7c6f 3139 203d 204d 7574 |.|o19 = Mut │ │ │ │ +00027770: 6162 6c65 4861 7368 5461 626c 657b 2e2e ableHashTable{.. │ │ │ │ +00027780: 2e34 2e2e 2e7d 2020 2020 2020 2020 2020 .4...} │ │ │ │ 00027790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000277a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000277c0: 7c6f 3139 203a 204d 7574 6162 6c65 4861 |o19 : MutableHa │ │ │ │ -000277d0: 7368 5461 626c 6520 2020 2020 2020 2020 shTable │ │ │ │ +000277b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000277c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000277d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000277e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000277f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027800: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027810: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00027820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00027860: 7c69 3230 203a 2063 736d 584c 6861 7368 |i20 : csmXLhash │ │ │ │ -00027870: 2328 2247 284a 6163 6f62 6961 6e29 227c #("G(Jacobian)"| │ │ │ │ -00027880: 746f 5374 7269 6e67 287b 302c 317d 2929 toString({0,1})) │ │ │ │ -00027890: 3d3d 7365 674a 2322 4722 2020 2020 2020 ==segJ#"G" │ │ │ │ -000278a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000278b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000278c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000278d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027800: 2020 2020 7c0a 7c6f 3139 203a 204d 7574 |.|o19 : Mut │ │ │ │ +00027810: 6162 6c65 4861 7368 5461 626c 6520 2020 ableHashTable │ │ │ │ +00027820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027850: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00027860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000278a0: 2d2d 2d2d 2b0a 7c69 3230 203a 2063 736d ----+.|i20 : csm │ │ │ │ +000278b0: 584c 6861 7368 2328 2247 284a 6163 6f62 XLhash#("G(Jacob │ │ │ │ +000278c0: 6961 6e29 227c 746f 5374 7269 6e67 287b ian)"|toString({ │ │ │ │ +000278d0: 302c 317d 2929 3d3d 7365 674a 2322 4722 0,1}))==segJ#"G" │ │ │ │ 000278e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000278f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027900: 7c6f 3230 203d 2074 7275 6520 2020 2020 |o20 = true │ │ │ │ +000278f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00027900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027940: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027950: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -000279a0: 7c69 3231 203a 2063 736d 584c 6861 7368 |i21 : csmXLhash │ │ │ │ -000279b0: 2328 2253 6567 7265 284a 6163 6f62 6961 #("Segre(Jacobia │ │ │ │ -000279c0: 6e29 227c 746f 5374 7269 6e67 287b 302c n)"|toString({0, │ │ │ │ -000279d0: 317d 2929 3d3d 7365 674a 2322 5365 6772 1}))==segJ#"Segr │ │ │ │ -000279e0: 6522 2020 2020 2020 2020 2020 2020 7c0a e" |. │ │ │ │ -000279f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00027a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a30: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027a40: 7c6f 3231 203d 2074 7275 6520 2020 2020 |o21 = true │ │ │ │ +00027940: 2020 2020 7c0a 7c6f 3230 203d 2074 7275 |.|o20 = tru │ │ │ │ +00027950: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +00027960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027990: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000279a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000279b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000279c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000279d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000279e0: 2d2d 2d2d 2b0a 7c69 3231 203a 2063 736d ----+.|i21 : csm │ │ │ │ +000279f0: 584c 6861 7368 2328 2253 6567 7265 284a XLhash#("Segre(J │ │ │ │ +00027a00: 6163 6f62 6961 6e29 227c 746f 5374 7269 acobian)"|toStri │ │ │ │ +00027a10: 6e67 287b 302c 317d 2929 3d3d 7365 674a ng({0,1}))==segJ │ │ │ │ +00027a20: 2322 5365 6772 6522 2020 2020 2020 2020 #"Segre" │ │ │ │ +00027a30: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00027a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027a90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00027ae0: 0a46 756e 6374 696f 6e73 2077 6974 6820 .Functions with │ │ │ │ -00027af0: 6f70 7469 6f6e 616c 2061 7267 756d 656e optional argumen │ │ │ │ -00027b00: 7420 6e61 6d65 6420 4f75 7470 7574 3a0a t named Output:. │ │ │ │ -00027b10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00027b20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00027b30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00027b40: 2020 2a20 2243 6865 726e 282e 2e2e 2c4f * "Chern(...,O │ │ │ │ -00027b50: 7574 7075 743d 3e2e 2e2e 2922 202d 2d20 utput=>...)" -- │ │ │ │ -00027b60: 7365 6520 2a6e 6f74 6520 4368 6572 6e3a see *note Chern: │ │ │ │ -00027b70: 2043 6865 726e 2c20 2d2d 2054 6865 2043 Chern, -- The C │ │ │ │ -00027b80: 6865 726e 2063 6c61 7373 0a20 202a 2022 hern class. * " │ │ │ │ -00027b90: 4353 4d28 2e2e 2e2c 4f75 7470 7574 3d3e CSM(...,Output=> │ │ │ │ -00027ba0: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f ...)" -- see *no │ │ │ │ -00027bb0: 7465 2043 534d 3a20 4353 4d2c 202d 2d20 te CSM: CSM, -- │ │ │ │ -00027bc0: 5468 650a 2020 2020 4368 6572 6e2d 5363 The. Chern-Sc │ │ │ │ -00027bd0: 6877 6172 747a 2d4d 6163 5068 6572 736f hwartz-MacPherso │ │ │ │ -00027be0: 6e20 636c 6173 730a 2020 2a20 2245 756c n class. * "Eul │ │ │ │ -00027bf0: 6572 282e 2e2e 2c4f 7574 7075 743d 3e2e er(...,Output=>. │ │ │ │ -00027c00: 2e2e 2922 202d 2d20 7365 6520 2a6e 6f74 ..)" -- see *not │ │ │ │ -00027c10: 6520 4575 6c65 723a 2045 756c 6572 2c20 e Euler: Euler, │ │ │ │ -00027c20: 2d2d 2054 6865 2045 756c 6572 0a20 2020 -- The Euler. │ │ │ │ -00027c30: 2043 6861 7261 6374 6572 6973 7469 630a Characteristic. │ │ │ │ -00027c40: 2020 2a20 2253 6567 7265 282e 2e2e 2c4f * "Segre(...,O │ │ │ │ -00027c50: 7574 7075 743d 3e2e 2e2e 2922 202d 2d20 utput=>...)" -- │ │ │ │ -00027c60: 7365 6520 2a6e 6f74 6520 5365 6772 653a see *note Segre: │ │ │ │ -00027c70: 2053 6567 7265 2c20 2d2d 2054 6865 2053 Segre, -- The S │ │ │ │ -00027c80: 6567 7265 2063 6c61 7373 206f 6620 610a egre class of a. │ │ │ │ -00027c90: 2020 2020 7375 6273 6368 656d 650a 0a46 subscheme..F │ │ │ │ -00027ca0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00027cb0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00027cc0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00027cd0: 202a 6e6f 7465 204f 7574 7075 743a 204f *note Output: O │ │ │ │ -00027ce0: 7574 7075 742c 2069 7320 6120 2a6e 6f74 utput, is a *not │ │ │ │ -00027cf0: 6520 7379 6d62 6f6c 3a20 284d 6163 6175 e symbol: (Macau │ │ │ │ -00027d00: 6c61 7932 446f 6329 5379 6d62 6f6c 2c2e lay2Doc)Symbol,. │ │ │ │ -00027d10: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ -00027d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d60: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -00027d70: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -00027d80: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -00027d90: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -00027da0: 6361 756c 6179 322d 312e 3236 2e30 352b caulay2-1.26.05+ │ │ │ │ -00027db0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -00027dc0: 7061 636b 6167 6573 2f0a 4368 6172 6163 packages/.Charac │ │ │ │ -00027dd0: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ -00027de0: 6d32 3a32 3436 393a 302e 0a1f 0a46 696c m2:2469:0....Fil │ │ │ │ -00027df0: 653a 2043 6861 7261 6374 6572 6973 7469 e: Characteristi │ │ │ │ -00027e00: 6343 6c61 7373 6573 2e69 6e66 6f2c 204e cClasses.info, N │ │ │ │ -00027e10: 6f64 653a 2070 726f 6261 6269 6c69 7374 ode: probabilist │ │ │ │ -00027e20: 6963 2061 6c67 6f72 6974 686d 2c20 4e65 ic algorithm, Ne │ │ │ │ -00027e30: 7874 3a20 5365 6772 652c 2050 7265 763a xt: Segre, Prev: │ │ │ │ -00027e40: 204f 7574 7075 742c 2055 703a 2054 6f70 Output, Up: Top │ │ │ │ -00027e50: 0a0a 7072 6f62 6162 696c 6973 7469 6320 ..probabilistic │ │ │ │ -00027e60: 616c 676f 7269 7468 6d0a 2a2a 2a2a 2a2a algorithm.****** │ │ │ │ -00027e70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00027e80: 2a0a 0a54 6865 2061 6c67 6f72 6974 686d *..The algorithm │ │ │ │ -00027e90: 7320 7573 6564 2066 6f72 2074 6865 2063 s used for the c │ │ │ │ -00027ea0: 6f6d 7075 7461 7469 6f6e 206f 6620 6368 omputation of ch │ │ │ │ -00027eb0: 6172 6163 7465 7269 7374 6963 2063 6c61 aracteristic cla │ │ │ │ -00027ec0: 7373 6573 2061 7265 0a70 726f 6261 6269 sses are.probabi │ │ │ │ -00027ed0: 6c69 7374 6963 2e20 5468 656f 7265 7469 listic. Theoreti │ │ │ │ -00027ee0: 6361 6c6c 792c 2074 6865 7920 6361 6c63 cally, they calc │ │ │ │ -00027ef0: 756c 6174 6520 7468 6520 636c 6173 7365 ulate the classe │ │ │ │ -00027f00: 7320 636f 7272 6563 746c 7920 666f 7220 s correctly for │ │ │ │ -00027f10: 610a 6765 6e65 7261 6c20 6368 6f69 6365 a.general choice │ │ │ │ -00027f20: 206f 6620 6365 7274 6169 6e20 706f 6c79 of certain poly │ │ │ │ -00027f30: 6e6f 6d69 616c 732e 2054 6861 7420 6973 nomials. That is │ │ │ │ -00027f40: 2c20 7468 6572 6520 6973 2061 6e20 6f70 , there is an op │ │ │ │ -00027f50: 656e 2064 656e 7365 205a 6172 6973 6b69 en dense Zariski │ │ │ │ -00027f60: 0a73 6574 2066 6f72 2077 6869 6368 2074 .set for which t │ │ │ │ -00027f70: 6865 2061 6c67 6f72 6974 686d 2079 6965 he algorithm yie │ │ │ │ -00027f80: 6c64 7320 7468 6520 636f 7272 6563 7420 lds the correct │ │ │ │ -00027f90: 636c 6173 732c 2069 2e65 2e2c 2074 6865 class, i.e., the │ │ │ │ -00027fa0: 2063 6f72 7265 6374 2063 6c61 7373 0a69 correct class.i │ │ │ │ -00027fb0: 7320 6361 6c63 756c 6174 6564 2077 6974 s calculated wit │ │ │ │ -00027fc0: 6820 7072 6f62 6162 696c 6974 7920 312e h probability 1. │ │ │ │ -00027fd0: 2048 6f77 6576 6572 2c20 7369 6e63 6520 However, since │ │ │ │ -00027fe0: 7468 6520 696d 706c 656d 656e 7461 7469 the implementati │ │ │ │ -00027ff0: 6f6e 2077 6f72 6b73 206f 7665 720a 6120 on works over.a │ │ │ │ -00028000: 6469 7363 7265 7465 2070 726f 6261 6269 discrete probabi │ │ │ │ -00028010: 6c69 7479 2073 7061 6365 2074 6865 7265 lity space there │ │ │ │ -00028020: 2069 7320 6120 7665 7279 2073 6d61 6c6c is a very small │ │ │ │ -00028030: 2c20 6275 7420 6e6f 6e2d 7a65 726f 2c20 , but non-zero, │ │ │ │ -00028040: 7072 6f62 6162 696c 6974 790a 6f66 206e probability.of n │ │ │ │ -00028050: 6f74 2063 6f6d 7075 7469 6e67 2074 6865 ot computing the │ │ │ │ -00028060: 2063 6f72 7265 6374 2063 6c61 7373 2e20 correct class. │ │ │ │ -00028070: 536b 6570 7469 6361 6c20 7573 6572 7320 Skeptical users │ │ │ │ -00028080: 7368 6f75 6c64 2072 6570 6561 7420 6361 should repeat ca │ │ │ │ -00028090: 6c63 756c 6174 696f 6e73 0a73 6576 6572 lculations.sever │ │ │ │ -000280a0: 616c 2074 696d 6573 2074 6f20 696e 6372 al times to incr │ │ │ │ -000280b0: 6561 7365 2074 6865 2070 726f 6261 6269 ease the probabi │ │ │ │ -000280c0: 6c69 7479 206f 6620 636f 6d70 7574 696e lity of computin │ │ │ │ -000280d0: 6720 7468 6520 636f 7272 6563 7420 636c g the correct cl │ │ │ │ -000280e0: 6173 732e 0a0a 496e 2074 6865 2063 6173 ass...In the cas │ │ │ │ -000280f0: 6520 6f66 2074 6865 2073 796d 626f 6c69 e of the symboli │ │ │ │ -00028100: 6320 696d 706c 656d 656e 7461 7469 6f6e c implementation │ │ │ │ -00028110: 206f 6620 7468 6520 5072 6f6a 6563 7469 of the Projecti │ │ │ │ -00028120: 7665 4465 6772 6565 206d 6574 686f 640a veDegree method. │ │ │ │ -00028130: 7072 6163 7469 6361 6c20 6578 7065 7269 practical experi │ │ │ │ -00028140: 656e 6365 2061 6e64 2061 6c67 6f72 6974 ence and algorit │ │ │ │ -00028150: 686d 2074 6573 7469 6e67 2069 6e64 6963 hm testing indic │ │ │ │ -00028160: 6174 6520 7468 6174 2061 2066 696e 6974 ate that a finit │ │ │ │ -00028170: 6520 6669 656c 6420 7769 7468 0a6f 7665 e field with.ove │ │ │ │ -00028180: 7220 3235 3030 3020 656c 656d 656e 7473 r 25000 elements │ │ │ │ -00028190: 2069 7320 6d6f 7265 2074 6861 6e20 7375 is more than su │ │ │ │ -000281a0: 6666 6963 6965 6e74 2074 6f20 6578 7065 fficient to expe │ │ │ │ -000281b0: 6374 2061 2063 6f72 7265 6374 2072 6573 ct a correct res │ │ │ │ -000281c0: 756c 7420 7769 7468 0a68 6967 6820 7072 ult with.high pr │ │ │ │ -000281d0: 6f62 6162 696c 6974 792c 2069 2e65 2e20 obability, i.e. │ │ │ │ -000281e0: 7573 696e 6720 7468 6520 6669 6e69 7465 using the finite │ │ │ │ -000281f0: 2066 6965 6c64 206b 6b3d 5a5a 2f32 3530 field kk=ZZ/250 │ │ │ │ -00028200: 3733 2074 6865 2065 7870 6572 696d 656e 73 the experimen │ │ │ │ -00028210: 7461 6c0a 6368 616e 6365 206f 6620 6661 tal.chance of fa │ │ │ │ -00028220: 696c 7572 6520 7769 7468 2074 6865 2050 ilure with the P │ │ │ │ -00028230: 726f 6a65 6374 6976 6544 6567 7265 6520 rojectiveDegree │ │ │ │ -00028240: 616c 676f 7269 7468 6d20 6f6e 2061 2076 algorithm on a v │ │ │ │ -00028250: 6172 6965 7479 206f 6620 6578 616d 706c ariety of exampl │ │ │ │ -00028260: 6573 0a77 6173 206c 6573 7320 7468 616e es.was less than │ │ │ │ -00028270: 2031 2f32 3030 302e 2055 7369 6e67 2074 1/2000. Using t │ │ │ │ -00028280: 6865 2066 696e 6974 6520 6669 656c 6420 he finite field │ │ │ │ -00028290: 6b6b 3d5a 5a2f 3332 3734 3920 7265 7375 kk=ZZ/32749 resu │ │ │ │ -000282a0: 6c74 6564 2069 6e20 6e6f 0a66 6169 6c75 lted in no.failu │ │ │ │ -000282b0: 7265 7320 696e 206f 7665 7220 3130 3030 res in over 1000 │ │ │ │ -000282c0: 3020 6174 7465 6d70 7473 206f 6620 7365 0 attempts of se │ │ │ │ -000282d0: 7665 7261 6c20 6469 6666 6572 656e 7420 veral different │ │ │ │ -000282e0: 6578 616d 706c 6573 2e0a 0a57 6520 696c examples...We il │ │ │ │ -000282f0: 6c75 7374 7261 7465 2074 6865 2070 726f lustrate the pro │ │ │ │ -00028300: 6261 6269 6c69 7374 6963 2062 6568 6176 babilistic behav │ │ │ │ -00028310: 696f 7572 2077 6974 6820 616e 2065 7861 iour with an exa │ │ │ │ -00028320: 6d70 6c65 2077 6865 7265 2074 6865 2063 mple where the c │ │ │ │ -00028330: 686f 7365 6e0a 7261 6e64 6f6d 2073 6565 hosen.random see │ │ │ │ -00028340: 6420 6c65 6164 7320 746f 2061 2077 726f d leads to a wro │ │ │ │ -00028350: 6e67 2072 6573 756c 7420 696e 2074 6865 ng result in the │ │ │ │ -00028360: 2066 6972 7374 2063 616c 6375 6c61 7469 first calculati │ │ │ │ -00028370: 6f6e 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d on...+---------- │ │ │ │ -00028380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000283a0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7365 ------+.|i1 : se │ │ │ │ -000283b0: 7452 616e 646f 6d53 6565 6420 3132 313b tRandomSeed 121; │ │ │ │ -000283c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000283d0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000283e0: 7365 7474 696e 6720 7261 6e64 6f6d 2073 setting random s │ │ │ │ -000283f0: 6565 6420 746f 2031 3231 2020 2020 2020 eed to 121 │ │ │ │ -00028400: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00028410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00028440: 0a7c 6932 203a 2052 203d 2051 515b 782c .|i2 : R = QQ[x, │ │ │ │ -00028450: 792c 7a2c 775d 2020 2020 2020 2020 2020 y,z,w] │ │ │ │ -00028460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028470: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00028480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000284a0: 2020 2020 207c 0a7c 6f32 203d 2052 2020 |.|o2 = R │ │ │ │ -000284b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027a80: 2020 2020 7c0a 7c6f 3231 203d 2074 7275 |.|o21 = tru │ │ │ │ +00027a90: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +00027aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027ad0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00027ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027b20: 2d2d 2d2d 2b0a 0a46 756e 6374 696f 6e73 ----+..Functions │ │ │ │ +00027b30: 2077 6974 6820 6f70 7469 6f6e 616c 2061 with optional a │ │ │ │ +00027b40: 7267 756d 656e 7420 6e61 6d65 6420 4f75 rgument named Ou │ │ │ │ +00027b50: 7470 7574 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d tput:.========== │ │ │ │ +00027b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00027b70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00027b80: 3d3d 3d3d 0a0a 2020 2a20 2243 6865 726e ====.. * "Chern │ │ │ │ +00027b90: 282e 2e2e 2c4f 7574 7075 743d 3e2e 2e2e (...,Output=>... │ │ │ │ +00027ba0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ +00027bb0: 4368 6572 6e3a 2043 6865 726e 2c20 2d2d Chern: Chern, -- │ │ │ │ +00027bc0: 2054 6865 2043 6865 726e 2063 6c61 7373 The Chern class │ │ │ │ +00027bd0: 0a20 202a 2022 4353 4d28 2e2e 2e2c 4f75 . * "CSM(...,Ou │ │ │ │ +00027be0: 7470 7574 3d3e 2e2e 2e29 2220 2d2d 2073 tput=>...)" -- s │ │ │ │ +00027bf0: 6565 202a 6e6f 7465 2043 534d 3a20 4353 ee *note CSM: CS │ │ │ │ +00027c00: 4d2c 202d 2d20 5468 650a 2020 2020 4368 M, -- The. Ch │ │ │ │ +00027c10: 6572 6e2d 5363 6877 6172 747a 2d4d 6163 ern-Schwartz-Mac │ │ │ │ +00027c20: 5068 6572 736f 6e20 636c 6173 730a 2020 Pherson class. │ │ │ │ +00027c30: 2a20 2245 756c 6572 282e 2e2e 2c4f 7574 * "Euler(...,Out │ │ │ │ +00027c40: 7075 743d 3e2e 2e2e 2922 202d 2d20 7365 put=>...)" -- se │ │ │ │ +00027c50: 6520 2a6e 6f74 6520 4575 6c65 723a 2045 e *note Euler: E │ │ │ │ +00027c60: 756c 6572 2c20 2d2d 2054 6865 2045 756c uler, -- The Eul │ │ │ │ +00027c70: 6572 0a20 2020 2043 6861 7261 6374 6572 er. Character │ │ │ │ +00027c80: 6973 7469 630a 2020 2a20 2253 6567 7265 istic. * "Segre │ │ │ │ +00027c90: 282e 2e2e 2c4f 7574 7075 743d 3e2e 2e2e (...,Output=>... │ │ │ │ +00027ca0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ +00027cb0: 5365 6772 653a 2053 6567 7265 2c20 2d2d Segre: Segre, -- │ │ │ │ +00027cc0: 2054 6865 2053 6567 7265 2063 6c61 7373 The Segre class │ │ │ │ +00027cd0: 206f 6620 610a 2020 2020 7375 6273 6368 of a. subsch │ │ │ │ +00027ce0: 656d 650a 0a46 6f72 2074 6865 2070 726f eme..For the pro │ │ │ │ +00027cf0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +00027d00: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +00027d10: 6f62 6a65 6374 202a 6e6f 7465 204f 7574 object *note Out │ │ │ │ +00027d20: 7075 743a 204f 7574 7075 742c 2069 7320 put: Output, is │ │ │ │ +00027d30: 6120 2a6e 6f74 6520 7379 6d62 6f6c 3a20 a *note symbol: │ │ │ │ +00027d40: 284d 6163 6175 6c61 7932 446f 6329 5379 (Macaulay2Doc)Sy │ │ │ │ +00027d50: 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d mbol,...-------- │ │ │ │ +00027d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027da0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +00027db0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +00027dc0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +00027dd0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +00027de0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00027df0: 3236 2e30 352b 6473 2f4d 322f 4d61 6361 26.05+ds/M2/Maca │ │ │ │ +00027e00: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +00027e10: 4368 6172 6163 7465 7269 7374 6963 436c CharacteristicCl │ │ │ │ +00027e20: 6173 7365 732e 6d32 3a32 3436 393a 302e asses.m2:2469:0. │ │ │ │ +00027e30: 0a1f 0a46 696c 653a 2043 6861 7261 6374 ...File: Charact │ │ │ │ +00027e40: 6572 6973 7469 6343 6c61 7373 6573 2e69 eristicClasses.i │ │ │ │ +00027e50: 6e66 6f2c 204e 6f64 653a 2070 726f 6261 nfo, Node: proba │ │ │ │ +00027e60: 6269 6c69 7374 6963 2061 6c67 6f72 6974 bilistic algorit │ │ │ │ +00027e70: 686d 2c20 4e65 7874 3a20 5365 6772 652c hm, Next: Segre, │ │ │ │ +00027e80: 2050 7265 763a 204f 7574 7075 742c 2055 Prev: Output, U │ │ │ │ +00027e90: 703a 2054 6f70 0a0a 7072 6f62 6162 696c p: Top..probabil │ │ │ │ +00027ea0: 6973 7469 6320 616c 676f 7269 7468 6d0a istic algorithm. │ │ │ │ +00027eb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00027ec0: 2a2a 2a2a 2a2a 2a0a 0a54 6865 2061 6c67 *******..The alg │ │ │ │ +00027ed0: 6f72 6974 686d 7320 7573 6564 2066 6f72 orithms used for │ │ │ │ +00027ee0: 2074 6865 2063 6f6d 7075 7461 7469 6f6e the computation │ │ │ │ +00027ef0: 206f 6620 6368 6172 6163 7465 7269 7374 of characterist │ │ │ │ +00027f00: 6963 2063 6c61 7373 6573 2061 7265 0a70 ic classes are.p │ │ │ │ +00027f10: 726f 6261 6269 6c69 7374 6963 2e20 5468 robabilistic. Th │ │ │ │ +00027f20: 656f 7265 7469 6361 6c6c 792c 2074 6865 eoretically, the │ │ │ │ +00027f30: 7920 6361 6c63 756c 6174 6520 7468 6520 y calculate the │ │ │ │ +00027f40: 636c 6173 7365 7320 636f 7272 6563 746c classes correctl │ │ │ │ +00027f50: 7920 666f 7220 610a 6765 6e65 7261 6c20 y for a.general │ │ │ │ +00027f60: 6368 6f69 6365 206f 6620 6365 7274 6169 choice of certai │ │ │ │ +00027f70: 6e20 706f 6c79 6e6f 6d69 616c 732e 2054 n polynomials. T │ │ │ │ +00027f80: 6861 7420 6973 2c20 7468 6572 6520 6973 hat is, there is │ │ │ │ +00027f90: 2061 6e20 6f70 656e 2064 656e 7365 205a an open dense Z │ │ │ │ +00027fa0: 6172 6973 6b69 0a73 6574 2066 6f72 2077 ariski.set for w │ │ │ │ +00027fb0: 6869 6368 2074 6865 2061 6c67 6f72 6974 hich the algorit │ │ │ │ +00027fc0: 686d 2079 6965 6c64 7320 7468 6520 636f hm yields the co │ │ │ │ +00027fd0: 7272 6563 7420 636c 6173 732c 2069 2e65 rrect class, i.e │ │ │ │ +00027fe0: 2e2c 2074 6865 2063 6f72 7265 6374 2063 ., the correct c │ │ │ │ +00027ff0: 6c61 7373 0a69 7320 6361 6c63 756c 6174 lass.is calculat │ │ │ │ +00028000: 6564 2077 6974 6820 7072 6f62 6162 696c ed with probabil │ │ │ │ +00028010: 6974 7920 312e 2048 6f77 6576 6572 2c20 ity 1. However, │ │ │ │ +00028020: 7369 6e63 6520 7468 6520 696d 706c 656d since the implem │ │ │ │ +00028030: 656e 7461 7469 6f6e 2077 6f72 6b73 206f entation works o │ │ │ │ +00028040: 7665 720a 6120 6469 7363 7265 7465 2070 ver.a discrete p │ │ │ │ +00028050: 726f 6261 6269 6c69 7479 2073 7061 6365 robability space │ │ │ │ +00028060: 2074 6865 7265 2069 7320 6120 7665 7279 there is a very │ │ │ │ +00028070: 2073 6d61 6c6c 2c20 6275 7420 6e6f 6e2d small, but non- │ │ │ │ +00028080: 7a65 726f 2c20 7072 6f62 6162 696c 6974 zero, probabilit │ │ │ │ +00028090: 790a 6f66 206e 6f74 2063 6f6d 7075 7469 y.of not computi │ │ │ │ +000280a0: 6e67 2074 6865 2063 6f72 7265 6374 2063 ng the correct c │ │ │ │ +000280b0: 6c61 7373 2e20 536b 6570 7469 6361 6c20 lass. Skeptical │ │ │ │ +000280c0: 7573 6572 7320 7368 6f75 6c64 2072 6570 users should rep │ │ │ │ +000280d0: 6561 7420 6361 6c63 756c 6174 696f 6e73 eat calculations │ │ │ │ +000280e0: 0a73 6576 6572 616c 2074 696d 6573 2074 .several times t │ │ │ │ +000280f0: 6f20 696e 6372 6561 7365 2074 6865 2070 o increase the p │ │ │ │ +00028100: 726f 6261 6269 6c69 7479 206f 6620 636f robability of co │ │ │ │ +00028110: 6d70 7574 696e 6720 7468 6520 636f 7272 mputing the corr │ │ │ │ +00028120: 6563 7420 636c 6173 732e 0a0a 496e 2074 ect class...In t │ │ │ │ +00028130: 6865 2063 6173 6520 6f66 2074 6865 2073 he case of the s │ │ │ │ +00028140: 796d 626f 6c69 6320 696d 706c 656d 656e ymbolic implemen │ │ │ │ +00028150: 7461 7469 6f6e 206f 6620 7468 6520 5072 tation of the Pr │ │ │ │ +00028160: 6f6a 6563 7469 7665 4465 6772 6565 206d ojectiveDegree m │ │ │ │ +00028170: 6574 686f 640a 7072 6163 7469 6361 6c20 ethod.practical │ │ │ │ +00028180: 6578 7065 7269 656e 6365 2061 6e64 2061 experience and a │ │ │ │ +00028190: 6c67 6f72 6974 686d 2074 6573 7469 6e67 lgorithm testing │ │ │ │ +000281a0: 2069 6e64 6963 6174 6520 7468 6174 2061 indicate that a │ │ │ │ +000281b0: 2066 696e 6974 6520 6669 656c 6420 7769 finite field wi │ │ │ │ +000281c0: 7468 0a6f 7665 7220 3235 3030 3020 656c th.over 25000 el │ │ │ │ +000281d0: 656d 656e 7473 2069 7320 6d6f 7265 2074 ements is more t │ │ │ │ +000281e0: 6861 6e20 7375 6666 6963 6965 6e74 2074 han sufficient t │ │ │ │ +000281f0: 6f20 6578 7065 6374 2061 2063 6f72 7265 o expect a corre │ │ │ │ +00028200: 6374 2072 6573 756c 7420 7769 7468 0a68 ct result with.h │ │ │ │ +00028210: 6967 6820 7072 6f62 6162 696c 6974 792c igh probability, │ │ │ │ +00028220: 2069 2e65 2e20 7573 696e 6720 7468 6520 i.e. using the │ │ │ │ +00028230: 6669 6e69 7465 2066 6965 6c64 206b 6b3d finite field kk= │ │ │ │ +00028240: 5a5a 2f32 3530 3733 2074 6865 2065 7870 ZZ/25073 the exp │ │ │ │ +00028250: 6572 696d 656e 7461 6c0a 6368 616e 6365 erimental.chance │ │ │ │ +00028260: 206f 6620 6661 696c 7572 6520 7769 7468 of failure with │ │ │ │ +00028270: 2074 6865 2050 726f 6a65 6374 6976 6544 the ProjectiveD │ │ │ │ +00028280: 6567 7265 6520 616c 676f 7269 7468 6d20 egree algorithm │ │ │ │ +00028290: 6f6e 2061 2076 6172 6965 7479 206f 6620 on a variety of │ │ │ │ +000282a0: 6578 616d 706c 6573 0a77 6173 206c 6573 examples.was les │ │ │ │ +000282b0: 7320 7468 616e 2031 2f32 3030 302e 2055 s than 1/2000. U │ │ │ │ +000282c0: 7369 6e67 2074 6865 2066 696e 6974 6520 sing the finite │ │ │ │ +000282d0: 6669 656c 6420 6b6b 3d5a 5a2f 3332 3734 field kk=ZZ/3274 │ │ │ │ +000282e0: 3920 7265 7375 6c74 6564 2069 6e20 6e6f 9 resulted in no │ │ │ │ +000282f0: 0a66 6169 6c75 7265 7320 696e 206f 7665 .failures in ove │ │ │ │ +00028300: 7220 3130 3030 3020 6174 7465 6d70 7473 r 10000 attempts │ │ │ │ +00028310: 206f 6620 7365 7665 7261 6c20 6469 6666 of several diff │ │ │ │ +00028320: 6572 656e 7420 6578 616d 706c 6573 2e0a erent examples.. │ │ │ │ +00028330: 0a57 6520 696c 6c75 7374 7261 7465 2074 .We illustrate t │ │ │ │ +00028340: 6865 2070 726f 6261 6269 6c69 7374 6963 he probabilistic │ │ │ │ +00028350: 2062 6568 6176 696f 7572 2077 6974 6820 behaviour with │ │ │ │ +00028360: 616e 2065 7861 6d70 6c65 2077 6865 7265 an example where │ │ │ │ +00028370: 2074 6865 2063 686f 7365 6e0a 7261 6e64 the chosen.rand │ │ │ │ +00028380: 6f6d 2073 6565 6420 6c65 6164 7320 746f om seed leads to │ │ │ │ +00028390: 2061 2077 726f 6e67 2072 6573 756c 7420 a wrong result │ │ │ │ +000283a0: 696e 2074 6865 2066 6972 7374 2063 616c in the first cal │ │ │ │ +000283b0: 6375 6c61 7469 6f6e 2e0a 0a2b 2d2d 2d2d culation...+---- │ │ │ │ +000283c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000283d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000283e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000283f0: 3120 3a20 7365 7452 616e 646f 6d53 6565 1 : setRandomSee │ │ │ │ +00028400: 6420 3132 313b 2020 2020 2020 2020 2020 d 121; │ │ │ │ +00028410: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028420: 0a7c 202d 2d20 7365 7474 696e 6720 7261 .| -- setting ra │ │ │ │ +00028430: 6e64 6f6d 2073 6565 6420 746f 2031 3231 ndom seed to 121 │ │ │ │ +00028440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028450: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00028460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028480: 2d2d 2d2d 2d2b 0a7c 6932 203a 2052 203d -----+.|i2 : R = │ │ │ │ +00028490: 2051 515b 782c 792c 7a2c 775d 2020 2020 QQ[x,y,z,w] │ │ │ │ +000284a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000284b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000284c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000284d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000284e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000284f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028500: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00028510: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ -00028520: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ -00028530: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00028540: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00028550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028570: 2d2b 0a7c 6933 203a 2049 203d 206d 696e -+.|i3 : I = min │ │ │ │ -00028580: 6f72 7328 322c 6d61 7472 6978 7b7b 782c ors(2,matrix{{x, │ │ │ │ -00028590: 792c 7a7d 2c7b 792c 7a2c 777d 7d29 2020 y,z},{y,z,w}}) │ │ │ │ -000285a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000285b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000285c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000285d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000285e0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +000284d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000284e0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +000284f0: 203d 2052 2020 2020 2020 2020 2020 2020 = R │ │ │ │ +00028500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028510: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00028520: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00028530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028550: 207c 0a7c 6f32 203a 2050 6f6c 796e 6f6d |.|o2 : Polynom │ │ │ │ +00028560: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ +00028570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028580: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00028590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000285a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000285b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2049 -------+.|i3 : I │ │ │ │ +000285c0: 203d 206d 696e 6f72 7328 322c 6d61 7472 = minors(2,matr │ │ │ │ +000285d0: 6978 7b7b 782c 792c 7a7d 2c7b 792c 7a2c ix{{x,y,z},{y,z, │ │ │ │ +000285e0: 777d 7d29 2020 2020 2020 7c0a 7c20 2020 w}}) |.| │ │ │ │ 000285f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028600: 2020 3220 2020 2020 2020 7c0a 7c6f 3320 2 |.|o3 │ │ │ │ -00028610: 3d20 6964 6561 6c20 282d 2079 2020 2b20 = ideal (- y + │ │ │ │ -00028620: 782a 7a2c 202d 2079 2a7a 202b 2078 2a77 x*z, - y*z + x*w │ │ │ │ -00028630: 2c20 2d20 7a20 202b 2079 2a77 297c 0a7c , - z + y*w)|.| │ │ │ │ -00028640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028670: 7c0a 7c6f 3320 3a20 4964 6561 6c20 6f66 |.|o3 : Ideal of │ │ │ │ -00028680: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +00028600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028610: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00028620: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +00028630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028640: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +00028650: 7c0a 7c6f 3320 3d20 6964 6561 6c20 282d |.|o3 = ideal (- │ │ │ │ +00028660: 2079 2020 2b20 782a 7a2c 202d 2079 2a7a y + x*z, - y*z │ │ │ │ +00028670: 202b 2078 2a77 2c20 2d20 7a20 202b 2079 + x*w, - z + y │ │ │ │ +00028680: 2a77 297c 0a7c 2020 2020 2020 2020 2020 *w)|.| │ │ │ │ 00028690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000286a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -000286b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000286c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000286d0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 4368 ------+.|i4 : Ch │ │ │ │ -000286e0: 6572 6e20 2849 2c43 6f6d 704d 6574 686f ern (I,CompMetho │ │ │ │ -000286f0: 643d 3e50 6e52 6573 6964 7561 6c29 2020 d=>PnResidual) │ │ │ │ -00028700: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00028710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028730: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00028740: 2020 2020 2020 3320 2020 2020 3220 2020 3 2 │ │ │ │ -00028750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028760: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028770: 0a7c 6f34 203d 2032 4820 202b 2033 4820 .|o4 = 2H + 3H │ │ │ │ -00028780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000287a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -000287b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000287c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000287d0: 2020 2020 207c 0a7c 2020 2020 205a 5a5b |.| ZZ[ │ │ │ │ -000287e0: 485d 2020 2020 2020 2020 2020 2020 2020 H] │ │ │ │ +000286a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000286b0: 2020 2020 2020 7c0a 7c6f 3320 3a20 4964 |.|o3 : Id │ │ │ │ +000286c0: 6561 6c20 6f66 2052 2020 2020 2020 2020 eal of R │ │ │ │ +000286d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000286e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000286f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00028720: 3420 3a20 4368 6572 6e20 2849 2c43 6f6d 4 : Chern (I,Com │ │ │ │ +00028730: 704d 6574 686f 643d 3e50 6e52 6573 6964 pMethod=>PnResid │ │ │ │ +00028740: 7561 6c29 2020 2020 2020 2020 2020 207c ual) | │ │ │ │ +00028750: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00028760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028780: 2020 7c0a 7c20 2020 2020 2020 3320 2020 |.| 3 │ │ │ │ +00028790: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000287a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000287b0: 2020 2020 207c 0a7c 6f34 203d 2032 4820 |.|o4 = 2H │ │ │ │ +000287c0: 202b 2033 4820 2020 2020 2020 2020 2020 + 3H │ │ │ │ +000287d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000287e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000287f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028800: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ -00028810: 2d2d 2d2d 2d20 2020 2020 2020 2020 2020 ----- │ │ │ │ -00028820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028830: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00028840: 2020 2020 2020 3420 2020 2020 2020 2020 4 │ │ │ │ -00028850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028860: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00028870: 7c20 2020 2020 2020 4820 2020 2020 2020 | H │ │ │ │ -00028880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028810: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00028820: 2020 205a 5a5b 485d 2020 2020 2020 2020 ZZ[H] │ │ │ │ +00028830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028840: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00028850: 7c6f 3420 3a20 2d2d 2d2d 2d20 2020 2020 |o4 : ----- │ │ │ │ +00028860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028880: 207c 0a7c 2020 2020 2020 2020 3420 2020 |.| 4 │ │ │ │ 00028890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000288a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -000288b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000288c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000288d0: 2d2d 2d2d 2b0a 7c69 3520 3a20 4368 6572 ----+.|i5 : Cher │ │ │ │ -000288e0: 6e20 2849 2c43 6f6d 704d 6574 686f 643d n (I,CompMethod= │ │ │ │ -000288f0: 3e50 6e52 6573 6964 7561 6c29 2020 2020 >PnResidual) │ │ │ │ -00028900: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00028910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028930: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00028940: 2020 2020 3320 2020 2020 3220 2020 2020 3 2 │ │ │ │ +000288a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000288b0: 2020 2020 7c0a 7c20 2020 2020 2020 4820 |.| H │ │ │ │ +000288c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000288d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000288e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000288f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028910: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ +00028920: 3a20 4368 6572 6e20 2849 2c43 6f6d 704d : Chern (I,CompM │ │ │ │ +00028930: 6574 686f 643d 3e50 6e52 6573 6964 7561 ethod=>PnResidua │ │ │ │ +00028940: 6c29 2020 2020 2020 2020 2020 207c 0a7c l) |.| │ │ │ │ 00028950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028960: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00028970: 6f35 203d 2032 4820 202b 2033 4820 2020 o5 = 2H + 3H │ │ │ │ -00028980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000289a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000289b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000289c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000289d0: 2020 207c 0a7c 2020 2020 205a 5a5b 485d |.| ZZ[H] │ │ │ │ -000289e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028980: 7c0a 7c20 2020 2020 2020 3320 2020 2020 |.| 3 │ │ │ │ +00028990: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000289a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000289b0: 2020 207c 0a7c 6f35 203d 2032 4820 202b |.|o5 = 2H + │ │ │ │ +000289c0: 2033 4820 2020 2020 2020 2020 2020 2020 3H │ │ │ │ +000289d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000289e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000289f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a00: 2020 2020 2020 7c0a 7c6f 3520 3a20 2d2d |.|o5 : -- │ │ │ │ -00028a10: 2d2d 2d20 2020 2020 2020 2020 2020 2020 --- │ │ │ │ -00028a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00028a40: 2020 2020 3420 2020 2020 2020 2020 2020 4 │ │ │ │ -00028a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00028a70: 2020 2020 2020 4820 2020 2020 2020 2020 H │ │ │ │ -00028a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028aa0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00028ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028ad0: 2d2d 2b0a 7c69 3620 3a20 4368 6572 6e20 --+.|i6 : Chern │ │ │ │ -00028ae0: 2849 2c43 6f6d 704d 6574 686f 643d 3e50 (I,CompMethod=>P │ │ │ │ -00028af0: 6e52 6573 6964 7561 6c29 2020 2020 2020 nResidual) │ │ │ │ -00028b00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00028b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00028b40: 2020 3320 2020 2020 3220 2020 2020 2020 3 2 │ │ │ │ +00028a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028a10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00028a20: 205a 5a5b 485d 2020 2020 2020 2020 2020 ZZ[H] │ │ │ │ +00028a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028a40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00028a50: 3520 3a20 2d2d 2d2d 2d20 2020 2020 2020 5 : ----- │ │ │ │ +00028a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028a70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028a80: 0a7c 2020 2020 2020 2020 3420 2020 2020 .| 4 │ │ │ │ +00028a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028ab0: 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│ -00028d00: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00028d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d30: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00028d40: 3320 2020 2020 3220 2020 2020 2020 2020 3 2 │ │ │ │ +00028c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028c10: 2020 2020 2020 207c 0a7c 2020 2020 205a |.| Z │ │ │ │ +00028c20: 5a5b 485d 2020 2020 2020 2020 2020 2020 Z[H] │ │ │ │ +00028c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028c40: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ +00028c50: 3a20 2d2d 2d2d 2d20 2020 2020 2020 2020 : ----- │ │ │ │ +00028c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028c70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00028c80: 2020 2020 2020 2020 3420 2020 2020 2020 4 │ │ │ │ +00028c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028cb0: 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│ │ │ -00028da0: 2020 2020 2020 3120 2020 2020 3120 2020 1 1 │ │ │ │ -00028db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028dc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028dd0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00028de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e00: 2020 7c0a 7c20 2020 2020 5a5a 5b68 205d |.| ZZ[h ] │ │ │ │ -00028e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028d70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00028d80: 2020 2020 2020 3320 2020 2020 3220 2020 3 2 │ │ │ │ +00028d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028da0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028db0: 0a7c 6f37 203d 2032 6820 202b 2033 6820 .|o7 = 2h + 3h │ │ │ │ +00028dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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-00028fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028fb0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -00028fc0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -00028fd0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -00028fe0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -00028ff0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -00029000: 2e32 362e 3035 2b64 732f 4d32 2f4d 6163 .26.05+ds/M2/Mac │ │ │ │ -00029010: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -00029020: 0a43 6861 7261 6374 6572 6973 7469 6343 .CharacteristicC │ │ │ │ -00029030: 6c61 7373 6573 2e6d 323a 3233 3738 3a30 lasses.m2:2378:0 │ │ │ │ -00029040: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163 ....File: Charac │ │ │ │ -00029050: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ -00029060: 696e 666f 2c20 4e6f 6465 3a20 5365 6772 info, Node: Segr │ │ │ │ -00029070: 652c 204e 6578 743a 2054 6f72 6963 4368 e, Next: ToricCh │ │ │ │ -00029080: 6f77 5269 6e67 2c20 5072 6576 3a20 7072 owRing, Prev: pr │ │ │ │ -00029090: 6f62 6162 696c 6973 7469 6320 616c 676f obabilistic algo │ │ │ │ -000290a0: 7269 7468 6d2c 2055 703a 2054 6f70 0a0a rithm, Up: Top.. │ │ │ │ -000290b0: 5365 6772 6520 2d2d 2054 6865 2053 6567 Segre -- The Seg │ │ │ │ -000290c0: 7265 2063 6c61 7373 206f 6620 6120 7375 re class of a su │ │ │ │ -000290d0: 6273 6368 656d 650a 2a2a 2a2a 2a2a 2a2a bscheme.******** │ │ │ │ -000290e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000290f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ -00029100: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ -00029110: 2020 2020 2053 6567 7265 2049 0a20 2020 Segre I. │ │ │ │ -00029120: 2020 2020 2053 6567 7265 2841 2c49 290a Segre(A,I). │ │ │ │ -00029130: 2020 2020 2020 2020 5365 6772 6528 582c Segre(X, │ │ │ │ -00029140: 4a29 0a20 2020 2020 2020 2053 6567 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A=\ZZ[h │ │ │ │ -00029270: 5f31 2c2e 2e2e 2c68 5f6d 5d2f 2868 5f31 _1,...,h_m]/(h_1 │ │ │ │ -00029280: 5e7b 6e5f 312b 317d 2c2e 2e2e 2c68 5f6d ^{n_1+1},...,h_m │ │ │ │ -00029290: 5e7b 6e5f 6d2b 317d 2920 7175 6f74 6965 ^{n_m+1}) quotie │ │ │ │ -000292a0: 6e74 2072 696e 670a 2020 2020 2020 2020 nt ring. │ │ │ │ -000292b0: 7265 7072 6573 656e 7469 6e67 2074 6865 representing the │ │ │ │ -000292c0: 2043 686f 7720 7269 6e67 206f 6620 5c50 Chow ring of \P │ │ │ │ -000292d0: 505e 7b6e 5f31 7d78 2e2e 2e78 5c50 505e P^{n_1}x...x\PP^ │ │ │ │ -000292e0: 7b6e 5f6d 7d2c 2074 6869 7320 7269 6e67 {n_m}, this ring │ │ │ │ -000292f0: 2073 686f 756c 640a 2020 2020 2020 2020 should. │ │ │ │ -00029300: 6265 2062 7569 6c74 2075 7369 6e67 2074 be built using t │ │ │ │ -00029310: 6865 202a 6e6f 7465 2043 686f 7752 696e he *note ChowRin │ │ │ │ -00029320: 673a 2043 686f 7752 696e 672c 2063 6f6d g: ChowRing, com │ │ │ │ -00029330: 6d61 6e64 0a20 2020 2020 202a 204a 2c20 mand. * J, │ │ │ │ -00029340: 616e 202a 6e6f 7465 2069 6465 616c 3a20 an *note ideal: │ │ │ │ -00029350: 284d 6163 6175 6c61 7932 446f 6329 4964 (Macaulay2Doc)Id │ │ │ │ -00029360: 6561 6c2c 2c20 696e 2074 6865 2067 7261 eal,, in the gra │ │ │ │ -00029370: 6465 6420 706f 6c79 6e6f 6d69 616c 2072 ded polynomial r │ │ │ │ -00029380: 696e 670a 2020 2020 2020 2020 7768 6963 ing. whic │ │ │ │ -00029390: 6820 6973 2063 6f6f 7264 696e 6174 6520 h is coordinate │ │ │ │ -000293a0: 7269 6e67 206f 6620 7468 6520 4e6f 726d ring of the Norm │ │ │ │ -000293b0: 616c 2054 6f72 6963 2056 6172 6965 7479 al Toric Variety │ │ │ │ -000293c0: 2058 0a20 2020 2020 202a 2058 2c20 6120 X. * X, a │ │ │ │ -000293d0: 2a6e 6f74 6520 6e6f 726d 616c 2074 6f72 *note normal tor │ │ │ │ -000293e0: 6963 2076 6172 6965 7479 3a0a 2020 2020 ic variety:. │ │ │ │ -000293f0: 2020 2020 284e 6f72 6d61 6c54 6f72 6963 (NormalToric │ │ │ │ -00029400: 5661 7269 6574 6965 7329 4e6f 726d 616c Varieties)Normal │ │ │ │ -00029410: 546f 7269 6356 6172 6965 7479 2c2c 2077 ToricVariety,, w │ │ │ │ -00029420: 6869 6368 2069 7320 7468 6520 616d 6269 hich is the ambi │ │ │ │ -00029430: 656e 7420 7370 6163 650a 2020 2020 2020 ent space. │ │ │ │ -00029440: 2020 7768 6963 6820 636f 6e74 6169 6e73 which contains │ │ │ │ -00029450: 2056 284a 290a 2020 2020 2020 2a20 4368 V(J). * Ch │ │ │ │ -00029460: 2c20 6120 2a6e 6f74 6520 7175 6f74 6965 , a *note quotie │ │ │ │ -00029470: 6e74 2072 696e 673a 2028 4d61 6361 756c nt ring: (Macaul │ │ │ │ -00029480: 6179 3244 6f63 2951 756f 7469 656e 7452 ay2Doc)QuotientR │ │ │ │ -00029490: 696e 672c 2c20 7468 6520 4368 6f77 2072 ing,, the Chow r │ │ │ │ -000294a0: 696e 670a 2020 2020 2020 2020 6f66 2074 ing. of t │ │ │ │ -000294b0: 6865 2074 6f72 6963 2076 6172 6965 7479 he toric variety │ │ │ │ -000294c0: 2058 2c20 4368 3d28 7269 6e67 204a 292f X, Ch=(ring J)/ │ │ │ │ -000294d0: 2853 522b 4c52 2920 7768 6572 6520 5352 (SR+LR) where SR │ │ │ │ -000294e0: 2069 7320 7468 650a 2020 2020 2020 2020 is the. │ │ │ │ -000294f0: 5374 616e 6c65 792d 5265 6973 6e65 7220 Stanley-Reisner │ │ │ │ -00029500: 6964 6561 6c20 6f66 2074 6865 2066 616e ideal of the fan │ │ │ │ -00029510: 2064 6566 696e 696e 6720 5820 616e 6420 defining X and │ │ │ │ -00029520: 4c52 2069 7320 7468 6520 6c69 6e65 6172 LR is the linear │ │ │ │ -00029530: 0a20 2020 2020 2020 2072 656c 6174 696f . relatio │ │ │ │ -00029540: 6e73 2069 6465 616c 2c20 7468 6973 2072 ns ideal, this r │ │ │ │ -00029550: 696e 6720 7368 6f75 6c64 2062 6520 6275 ing should be bu │ │ │ │ -00029560: 696c 7420 7573 696e 6720 7468 6520 2a6e ilt using the *n │ │ │ │ -00029570: 6f74 650a 2020 2020 2020 2020 546f 7269 ote. Tori │ │ │ │ -00029580: 6343 686f 7752 696e 673a 2054 6f72 6963 cChowRing: Toric │ │ │ │ -00029590: 4368 6f77 5269 6e67 2c20 636f 6d6d 616e ChowRing, comman │ │ │ │ -000295a0: 640a 2020 2a20 2a6e 6f74 6520 4f70 7469 d. * *note Opti │ │ │ │ -000295b0: 6f6e 616c 2069 6e70 7574 733a 2028 4d61 onal inputs: (Ma │ │ │ │ -000295c0: 6361 756c 6179 3244 6f63 2975 7369 6e67 caulay2Doc)using │ │ │ │ -000295d0: 2066 756e 6374 696f 6e73 2077 6974 6820 functions with │ │ │ │ -000295e0: 6f70 7469 6f6e 616c 2069 6e70 7574 732c optional inputs, │ │ │ │ -000295f0: 3a0a 2020 2020 2020 2a20 436f 6d70 4d65 :. * CompMe │ │ │ │ -00029600: 7468 6f64 2028 6d69 7373 696e 6720 646f thod (missing do │ │ │ │ -00029610: 6375 6d65 6e74 6174 696f 6e29 203d 3e20 cumentation) => │ │ │ │ -00029620: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -00029630: 7565 0a20 2020 2020 2020 2050 726f 6a65 ue. Proje │ │ │ │ -00029640: 6374 6976 6544 6567 7265 652c 2050 726f ctiveDegree, Pro │ │ │ │ -00029650: 6a65 6374 6976 6544 6567 7265 652c 2074 jectiveDegree, t │ │ │ │ -00029660: 6869 7320 616c 676f 7269 7468 6d20 6d61 his algorithm ma │ │ │ │ -00029670: 7920 6265 2075 7365 6420 666f 720a 2020 y be used for. │ │ │ │ -00029680: 2020 2020 2020 7375 6273 6368 656d 6573 subschemes │ │ │ │ -00029690: 206f 6620 616e 7920 6170 706c 6963 6162 of any applicab │ │ │ │ -000296a0: 6c65 2074 6f72 6963 2076 6172 6965 7479 le toric variety │ │ │ │ -000296b0: 2028 7468 6973 206d 6179 2062 6520 6368 (this may be ch │ │ │ │ -000296c0: 6563 6b65 6420 7573 696e 670a 2020 2020 ecked using. │ │ │ │ -000296d0: 2020 2020 7468 6520 2a6e 6f74 6520 4368 the *note Ch │ │ │ │ -000296e0: 6563 6b54 6f72 6963 5661 7269 6574 7956 eckToricVarietyV │ │ │ │ -000296f0: 616c 6964 3a20 4368 6563 6b54 6f72 6963 alid: CheckToric │ │ │ │ -00029700: 5661 7269 6574 7956 616c 6964 2c20 636f VarietyValid, co │ │ │ │ -00029710: 6d6d 616e 6429 0a20 2020 2020 202a 2043 mmand). * C │ │ │ │ -00029720: 6f6d 704d 6574 686f 6420 286d 6973 7369 ompMethod (missi │ │ │ │ -00029730: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ -00029740: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c ) => ..., defaul │ │ │ │ -00029750: 7420 7661 6c75 650a 2020 2020 2020 2020 t value. │ │ │ │ -00029760: 5072 6f6a 6563 7469 7665 4465 6772 6565 ProjectiveDegree │ │ │ │ -00029770: 2c20 506e 5265 7369 6475 616c 2c20 7468 , PnResidual, th │ │ │ │ -00029780: 6973 2061 6c67 6f72 6974 686d 206d 6179 is algorithm may │ │ │ │ -00029790: 2062 6520 7573 6564 2066 6f72 2073 7562 be used for sub │ │ │ │ -000297a0: 7363 6865 6d65 730a 2020 2020 2020 2020 schemes. │ │ │ │ -000297b0: 6f66 205c 5050 5e6e 206f 6e6c 790a 2020 of \PP^n only. │ │ │ │ -000297c0: 2020 2020 2a20 4f75 7470 7574 203d 3e20 * Output => │ │ │ │ -000297d0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -000297e0: 7565 2043 686f 7752 696e 6745 6c65 6d65 ue ChowRingEleme │ │ │ │ -000297f0: 6e74 2c20 4368 6f77 5269 6e67 456c 656d nt, ChowRingElem │ │ │ │ -00029800: 656e 742c 2072 6574 7572 6e73 0a20 2020 ent, returns. │ │ │ │ -00029810: 2020 2020 2061 2052 696e 6745 6c65 6d65 a RingEleme │ │ │ │ -00029820: 6e74 2069 6e20 7468 6520 4368 6f77 2072 nt in the Chow r │ │ │ │ -00029830: 696e 6720 6f66 2074 6865 2061 7070 726f ing of the appro │ │ │ │ -00029840: 7072 6961 7465 2061 6d62 6965 6e74 2073 priate ambient s │ │ │ │ -00029850: 7061 6365 0a20 2020 2020 202a 204f 7574 pace. * Out │ │ │ │ -00029860: 7075 7420 3d3e 202e 2e2e 2c20 6465 6661 put => ..., defa │ │ │ │ -00029870: 756c 7420 7661 6c75 6520 4368 6f77 5269 ult value ChowRi │ │ │ │ -00029880: 6e67 456c 656d 656e 742c 2048 6173 6846 ngElement, HashF │ │ │ │ -00029890: 6f72 6d2c 2048 6173 6846 6f72 6d0a 2020 orm, HashForm. │ │ │ │ -000298a0: 2020 2020 2020 7265 7475 726e 7320 6120 returns a │ │ │ │ -000298b0: 4d75 7461 626c 6548 6173 6854 6162 6c65 MutableHashTable │ │ │ │ -000298c0: 2063 6f6e 7461 696e 696e 6720 7468 6520 containing the │ │ │ │ -000298d0: 666f 6c6c 6f77 696e 6720 6b65 7973 3a20 following keys: │ │ │ │ -000298e0: 2247 2220 2874 6865 0a20 2020 2020 2020 "G" (the. │ │ │ │ -000298f0: 2070 6f6c 796e 6f6d 6961 6c20 7769 7468 polynomial with │ │ │ │ -00029900: 2063 6f65 6666 6963 6965 6e74 7320 6f66 coefficients of │ │ │ │ -00029910: 2074 6865 2068 7970 6572 706c 616e 6520 the hyperplane │ │ │ │ -00029920: 636c 6173 7365 7320 7265 7072 6573 656e classes represen │ │ │ │ -00029930: 7469 6e67 2074 6865 0a20 2020 2020 2020 ting the. │ │ │ │ -00029940: 2070 726f 6a65 6374 6976 6520 6465 6772 projective degr │ │ │ │ -00029950: 6565 7329 2c20 2247 6c69 7374 2220 2874 ees), "Glist" (t │ │ │ │ -00029960: 6865 206c 6973 7420 666f 726d 206f 6620 he list form of │ │ │ │ -00029970: 2247 2229 202c 2022 5365 6772 6522 2028 "G") , "Segre" ( │ │ │ │ -00029980: 7468 650a 2020 2020 2020 2020 746f 7461 the. tota │ │ │ │ -00029990: 6c20 5365 6772 6520 636c 6173 7320 6f66 l Segre class of │ │ │ │ -000299a0: 2074 6865 2069 6e70 7574 292c 2253 6567 the input),"Seg │ │ │ │ -000299b0: 7265 4c69 7374 2220 2874 6865 206c 6973 reList" (the lis │ │ │ │ -000299c0: 7420 666f 726d 206f 6620 2253 6567 7265 t form of "Segre │ │ │ │ -000299d0: 2229 0a20 202a 204f 7574 7075 7473 3a0a "). * Outputs:. │ │ │ │ -000299e0: 2020 2020 2020 2a20 6120 2a6e 6f74 6520 * a *note │ │ │ │ -000299f0: 7269 6e67 2065 6c65 6d65 6e74 3a20 284d ring element: (M │ │ │ │ -00029a00: 6163 6175 6c61 7932 446f 6329 5269 6e67 acaulay2Doc)Ring │ │ │ │ -00029a10: 456c 656d 656e 742c 2c20 7468 6520 7075 Element,, the pu │ │ │ │ -00029a20: 7368 666f 7277 6172 6420 6f66 0a20 2020 shforward of. │ │ │ │ -00029a30: 2020 2020 2074 6865 2074 6f74 616c 2053 the total S │ │ │ │ -00029a40: 6567 7265 2063 6c61 7373 206f 6620 7468 egre class of th │ │ │ │ -00029a50: 6520 7363 6865 6d65 2056 2064 6566 696e e scheme V defin │ │ │ │ -00029a60: 6564 2062 7920 7468 6520 696e 7075 7420 ed by the input │ │ │ │ -00029a70: 6964 6561 6c20 746f 2074 6865 0a20 2020 ideal to the. │ │ │ │ -00029a80: 2020 2020 2061 7070 726f 7072 6961 7465 appropriate │ │ │ │ -00029a90: 2043 686f 7720 7269 6e67 0a0a 4465 7363 Chow ring..Desc │ │ │ │ -00029aa0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -00029ab0: 3d3d 3d0a 0a46 6f72 2061 2073 7562 7363 ===..For a subsc │ │ │ │ -00029ac0: 6865 6d65 2056 206f 6620 616e 2061 7070 heme V of an app │ │ │ │ -00029ad0: 6c69 6361 626c 6520 746f 7269 6320 7661 licable toric va │ │ │ │ -00029ae0: 7269 6574 7920 5820 7468 6973 2063 6f6d riety X this com │ │ │ │ -00029af0: 6d61 6e64 2063 6f6d 7075 7465 7320 7468 mand computes th │ │ │ │ -00029b00: 650a 7075 7368 2d66 6f72 7761 7264 206f e.push-forward o │ │ │ │ -00029b10: 6620 7468 6520 746f 7461 6c20 5365 6772 f the total Segr │ │ │ │ -00029b20: 6520 636c 6173 7320 7328 562c 5829 206f e class s(V,X) o │ │ │ │ -00029b30: 6620 5620 696e 2058 2074 6f20 7468 6520 f V in X to the │ │ │ │ -00029b40: 4368 6f77 2072 696e 6720 6f66 2058 2e0a Chow ring of X.. │ │ │ │ -00029b50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00029b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00029b80: 0a7c 6931 203a 2073 6574 5261 6e64 6f6d .|i1 : setRandom │ │ │ │ -00029b90: 5365 6564 2037 323b 2020 2020 2020 2020 Seed 72; │ │ │ │ -00029ba0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029bb0: 0a7c 202d 2d20 7365 7474 696e 6720 7261 .| -- setting ra │ │ │ │ -00029bc0: 6e64 6f6d 2073 6565 6420 746f 2037 3220 ndom seed to 72 │ │ │ │ -00029bd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029be0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00029bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00029c10: 0a7c 6932 203a 2052 203d 205a 5a2f 3332 .|i2 : R = ZZ/32 │ │ │ │ -00029c20: 3734 395b 772c 792c 7a5d 2020 2020 2020 749[w,y,z] │ │ │ │ -00029c30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029c40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00029c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029c70: 0a7c 6f32 203d 2052 2020 2020 2020 2020 .|o2 = R │ │ │ │ -00029c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029ca0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00029cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029cc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029cd0: 0a7c 6f32 203a 2050 6f6c 796e 6f6d 6961 .|o2 : Polynomia │ │ │ │ -00029ce0: 6c52 696e 6720 2020 2020 2020 2020 2020 lRing │ │ │ │ -00029cf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029d00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00029d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00029d30: 0a7c 6933 203a 2053 6567 7265 2869 6465 .|i3 : Segre(ide │ │ │ │ -00029d40: 616c 2877 2a79 292c 436f 6d70 4d65 7468 al(w*y),CompMeth │ │ │ │ -00029d50: 6f64 3d3e 506e 5265 7369 6475 616c 297c od=>PnResidual)| │ │ │ │ -00029d60: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00029d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029d80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029d90: 0a7c 2020 2020 2020 2020 2032 2020 2020 .| 2 │ │ │ │ -00029da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029db0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029dc0: 0a7c 6f33 203d 202d 2034 4820 202b 2032 .|o3 = - 4H + 2 │ │ │ │ -00029dd0: 4820 2020 2020 2020 2020 2020 2020 2020 H │ │ │ │ -00029de0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029df0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00029e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029e20: 0a7c 2020 2020 205a 5a5b 485d 2020 2020 .| ZZ[H] │ │ │ │ -00029e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029e50: 0a7c 6f33 203a 202d 2d2d 2d2d 2020 2020 .|o3 : ----- │ │ │ │ -00029e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029e80: 0a7c 2020 2020 2020 2020 3320 2020 2020 .| 3 │ │ │ │ -00029e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ea0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029eb0: 0a7c 2020 2020 2020 2048 2020 2020 2020 .| H │ │ │ │ -00029ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ed0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029ee0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00029ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00029f10: 0a7c 6934 203a 2041 3d43 686f 7752 696e .|i4 : A=ChowRin │ │ │ │ -00029f20: 6728 5229 2020 2020 2020 2020 2020 2020 g(R) │ │ │ │ -00029f30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029f40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00029f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029f60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029f70: 0a7c 6f34 203d 2041 2020 2020 2020 2020 .|o4 = A │ │ │ │ -00029f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029f90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029fa0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00029fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029fc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029fd0: 0a7c 6f34 203a 2051 756f 7469 656e 7452 .|o4 : QuotientR │ │ │ │ -00029fe0: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -00029ff0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002a000: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0002a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002a030: 0a7c 6935 203a 2053 6567 7265 2841 2c69 .|i5 : Segre(A,i │ │ │ │ -0002a040: 6465 616c 2877 5e32 2a79 2c77 2a79 5e32 deal(w^2*y,w*y^2 │ │ │ │ -0002a050: 2929 2020 2020 2020 2020 2020 2020 207c )) | │ │ │ │ -0002a060: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002a070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a080: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002a090: 0a7c 2020 2020 2020 2020 2032 2020 2020 .| 2 │ │ │ │ -0002a0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a0b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002a0c0: 0a7c 6f35 203d 202d 2033 6820 202b 2032 .|o5 = - 3h + 2 │ │ │ │ -0002a0d0: 6820 2020 2020 2020 2020 2020 2020 2020 h │ │ │ │ -0002a0e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002a0f0: 0a7c 2020 2020 2020 2020 2031 2020 2020 .| 1 │ │ │ │ -0002a100: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -0002a110: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002a120: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002a130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a140: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002a150: 0a7c 6f35 203a 2041 2020 2020 2020 2020 .|o5 : A │ │ │ │ -0002a160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a170: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002a180: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0002a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002a1b0: 0a0a 4e6f 7720 636f 6e73 6964 6572 2061 ..Now consider a │ │ │ │ -0002a1c0: 6e20 6578 616d 706c 6520 696e 205c 5050 n example in \PP │ │ │ │ -0002a1d0: 5e32 205c 7469 6d65 7320 5c50 505e 322c ^2 \times \PP^2, │ │ │ │ -0002a1e0: 2069 6620 7765 2069 6e70 7574 2074 6865 if we input the │ │ │ │ -0002a1f0: 2043 686f 7720 7269 6e67 2041 2074 6865 Chow ring A the │ │ │ │ -0002a200: 0a6f 7574 7075 7420 7769 6c6c 2062 6520 .output will be │ │ │ │ -0002a210: 7265 7475 726e 6564 2069 6e20 7468 6520 returned in the │ │ │ │ -0002a220: 7361 6d65 2072 696e 672e 2054 6f20 656e same ring. To en │ │ │ │ -0002a230: 7375 7265 2070 726f 7065 7220 6675 6e63 sure proper func │ │ │ │ -0002a240: 7469 6f6e 206f 6620 7468 650a 6d65 7468 tion of the.meth │ │ │ │ -0002a250: 6f64 7320 7765 2062 7569 6c64 2074 6865 ods we build the │ │ │ │ -0002a260: 2043 686f 7720 7269 6e67 2075 7369 6e67 Chow ring using │ │ │ │ -0002a270: 2074 6865 202a 6e6f 7465 2043 686f 7752 the *note ChowR │ │ │ │ -0002a280: 696e 673a 2043 686f 7752 696e 672c 2063 ing: ChowRing, c │ │ │ │ -0002a290: 6f6d 6d61 6e64 2e20 5765 0a6d 6179 2061 ommand. We.may a │ │ │ │ -0002a2a0: 6c73 6f20 7265 7475 726e 2061 204d 7574 lso return a Mut │ │ │ │ -0002a2b0: 6162 6c65 4861 7368 5461 626c 652e 0a0a ableHashTable... │ │ │ │ -0002a2c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002a310: 7c69 3620 3a20 523d 4d75 6c74 6950 726f |i6 : R=MultiPro │ │ │ │ -0002a320: 6a43 6f6f 7264 5269 6e67 287b 322c 327d jCoordRing({2,2} │ │ │ │ -0002a330: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0002a340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a350: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a360: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a2d -------------+.- │ │ │ │ +00028fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +00029000: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +00029010: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +00029020: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +00029030: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +00029040: 6c61 7932 2d31 2e32 362e 3035 2b64 732f lay2-1.26.05+ds/ │ │ │ │ +00029050: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +00029060: 6b61 6765 732f 0a43 6861 7261 6374 6572 kages/.Character │ │ │ │ +00029070: 6973 7469 6343 6c61 7373 6573 2e6d 323a isticClasses.m2: │ │ │ │ +00029080: 3233 3738 3a30 2e0a 1f0a 4669 6c65 3a20 2378:0....File: │ │ │ │ +00029090: 4368 6172 6163 7465 7269 7374 6963 436c CharacteristicCl │ │ │ │ +000290a0: 6173 7365 732e 696e 666f 2c20 4e6f 6465 asses.info, Node │ │ │ │ +000290b0: 3a20 5365 6772 652c 204e 6578 743a 2054 : Segre, Next: T │ │ │ │ +000290c0: 6f72 6963 4368 6f77 5269 6e67 2c20 5072 oricChowRing, Pr │ │ │ │ +000290d0: 6576 3a20 7072 6f62 6162 696c 6973 7469 ev: probabilisti │ │ │ │ +000290e0: 6320 616c 676f 7269 7468 6d2c 2055 703a c algorithm, Up: │ │ │ │ +000290f0: 2054 6f70 0a0a 5365 6772 6520 2d2d 2054 Top..Segre -- T │ │ │ │ +00029100: 6865 2053 6567 7265 2063 6c61 7373 206f he Segre class o │ │ │ │ +00029110: 6620 6120 7375 6273 6368 656d 650a 2a2a f a subscheme.** │ │ │ │ +00029120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00029130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00029140: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ +00029150: 3a20 0a20 2020 2020 2020 2053 6567 7265 : . Segre │ │ │ │ +00029160: 2049 0a20 2020 2020 2020 2053 6567 7265 I. Segre │ │ │ │ +00029170: 2841 2c49 290a 2020 2020 2020 2020 5365 (A,I). Se │ │ │ │ +00029180: 6772 6528 582c 4a29 0a20 2020 2020 2020 gre(X,J). │ │ │ │ +00029190: 2053 6567 7265 2843 682c 582c 4a29 0a20 Segre(Ch,X,J). │ │ │ │ +000291a0: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +000291b0: 202a 2049 2c20 616e 202a 6e6f 7465 2069 * I, an *note i │ │ │ │ +000291c0: 6465 616c 3a20 284d 6163 6175 6c61 7932 deal: (Macaulay2 │ │ │ │ +000291d0: 446f 6329 4964 6561 6c2c 2c20 6120 6d75 Doc)Ideal,, a mu │ │ │ │ +000291e0: 6c74 692d 686f 6d6f 6765 6e65 6f75 7320 lti-homogeneous │ │ │ │ +000291f0: 6964 6561 6c20 696e 2061 0a20 2020 2020 ideal in a. │ │ │ │ +00029200: 2020 2067 7261 6465 6420 706f 6c79 6e6f graded polyno │ │ │ │ +00029210: 6d69 616c 2072 696e 6720 6f76 6572 2061 mial ring over a │ │ │ │ +00029220: 2066 6965 6c64 2064 6566 696e 696e 6720 field defining │ │ │ │ +00029230: 6120 636c 6f73 6564 2073 7562 7363 6865 a closed subsche │ │ │ │ +00029240: 6d65 2056 206f 660a 2020 2020 2020 2020 me V of. │ │ │ │ +00029250: 5c50 505e 7b6e 5f31 7d78 2e2e 2e78 5c50 \PP^{n_1}x...x\P │ │ │ │ +00029260: 505e 7b6e 5f6d 7d0a 2020 2020 2020 2a20 P^{n_m}. * │ │ │ │ +00029270: 412c 2061 202a 6e6f 7465 2071 756f 7469 A, a *note quoti │ │ │ │ +00029280: 656e 7420 7269 6e67 3a20 284d 6163 6175 ent ring: (Macau │ │ │ │ +00029290: 6c61 7932 446f 6329 5175 6f74 6965 6e74 lay2Doc)Quotient │ │ │ │ +000292a0: 5269 6e67 2c2c 0a20 2020 2020 2020 2041 Ring,,. A │ │ │ │ +000292b0: 3d5c 5a5a 5b68 5f31 2c2e 2e2e 2c68 5f6d =\ZZ[h_1,...,h_m │ │ │ │ +000292c0: 5d2f 2868 5f31 5e7b 6e5f 312b 317d 2c2e ]/(h_1^{n_1+1},. │ │ │ │ +000292d0: 2e2e 2c68 5f6d 5e7b 6e5f 6d2b 317d 2920 ..,h_m^{n_m+1}) │ │ │ │ +000292e0: 7175 6f74 6965 6e74 2072 696e 670a 2020 quotient ring. │ │ │ │ +000292f0: 2020 2020 2020 7265 7072 6573 656e 7469 representi │ │ │ │ +00029300: 6e67 2074 6865 2043 686f 7720 7269 6e67 ng the Chow ring │ │ │ │ +00029310: 206f 6620 5c50 505e 7b6e 5f31 7d78 2e2e of \PP^{n_1}x.. │ │ │ │ +00029320: 2e78 5c50 505e 7b6e 5f6d 7d2c 2074 6869 .x\PP^{n_m}, thi │ │ │ │ +00029330: 7320 7269 6e67 2073 686f 756c 640a 2020 s ring should. │ │ │ │ +00029340: 2020 2020 2020 6265 2062 7569 6c74 2075 be built u │ │ │ │ +00029350: 7369 6e67 2074 6865 202a 6e6f 7465 2043 sing the *note C │ │ │ │ +00029360: 686f 7752 696e 673a 2043 686f 7752 696e howRing: ChowRin │ │ │ │ +00029370: 672c 2063 6f6d 6d61 6e64 0a20 2020 2020 g, command. │ │ │ │ +00029380: 202a 204a 2c20 616e 202a 6e6f 7465 2069 * J, an *note i │ │ │ │ +00029390: 6465 616c 3a20 284d 6163 6175 6c61 7932 deal: (Macaulay2 │ │ │ │ +000293a0: 446f 6329 4964 6561 6c2c 2c20 696e 2074 Doc)Ideal,, in t │ │ │ │ +000293b0: 6865 2067 7261 6465 6420 706f 6c79 6e6f he graded polyno │ │ │ │ +000293c0: 6d69 616c 2072 696e 670a 2020 2020 2020 mial ring. │ │ │ │ +000293d0: 2020 7768 6963 6820 6973 2063 6f6f 7264 which is coord │ │ │ │ +000293e0: 696e 6174 6520 7269 6e67 206f 6620 7468 inate ring of th │ │ │ │ +000293f0: 6520 4e6f 726d 616c 2054 6f72 6963 2056 e Normal Toric V │ │ │ │ +00029400: 6172 6965 7479 2058 0a20 2020 2020 202a ariety X. * │ │ │ │ +00029410: 2058 2c20 6120 2a6e 6f74 6520 6e6f 726d X, a *note norm │ │ │ │ +00029420: 616c 2074 6f72 6963 2076 6172 6965 7479 al toric variety │ │ │ │ +00029430: 3a0a 2020 2020 2020 2020 284e 6f72 6d61 :. (Norma │ │ │ │ +00029440: 6c54 6f72 6963 5661 7269 6574 6965 7329 lToricVarieties) │ │ │ │ +00029450: 4e6f 726d 616c 546f 7269 6356 6172 6965 NormalToricVarie │ │ │ │ +00029460: 7479 2c2c 2077 6869 6368 2069 7320 7468 ty,, which is th │ │ │ │ +00029470: 6520 616d 6269 656e 7420 7370 6163 650a e ambient space. │ │ │ │ +00029480: 2020 2020 2020 2020 7768 6963 6820 636f which co │ │ │ │ +00029490: 6e74 6169 6e73 2056 284a 290a 2020 2020 ntains V(J). │ │ │ │ +000294a0: 2020 2a20 4368 2c20 6120 2a6e 6f74 6520 * Ch, a *note │ │ │ │ +000294b0: 7175 6f74 6965 6e74 2072 696e 673a 2028 quotient ring: ( │ │ │ │ +000294c0: 4d61 6361 756c 6179 3244 6f63 2951 756f Macaulay2Doc)Quo │ │ │ │ +000294d0: 7469 656e 7452 696e 672c 2c20 7468 6520 tientRing,, the │ │ │ │ +000294e0: 4368 6f77 2072 696e 670a 2020 2020 2020 Chow ring. │ │ │ │ +000294f0: 2020 6f66 2074 6865 2074 6f72 6963 2076 of the toric v │ │ │ │ +00029500: 6172 6965 7479 2058 2c20 4368 3d28 7269 ariety X, Ch=(ri │ │ │ │ +00029510: 6e67 204a 292f 2853 522b 4c52 2920 7768 ng J)/(SR+LR) wh │ │ │ │ +00029520: 6572 6520 5352 2069 7320 7468 650a 2020 ere SR is the. │ │ │ │ +00029530: 2020 2020 2020 5374 616e 6c65 792d 5265 Stanley-Re │ │ │ │ +00029540: 6973 6e65 7220 6964 6561 6c20 6f66 2074 isner ideal of t │ │ │ │ +00029550: 6865 2066 616e 2064 6566 696e 696e 6720 he fan defining │ │ │ │ +00029560: 5820 616e 6420 4c52 2069 7320 7468 6520 X and LR is the │ │ │ │ +00029570: 6c69 6e65 6172 0a20 2020 2020 2020 2072 linear. r │ │ │ │ +00029580: 656c 6174 696f 6e73 2069 6465 616c 2c20 elations ideal, │ │ │ │ +00029590: 7468 6973 2072 696e 6720 7368 6f75 6c64 this ring should │ │ │ │ +000295a0: 2062 6520 6275 696c 7420 7573 696e 6720 be built using │ │ │ │ +000295b0: 7468 6520 2a6e 6f74 650a 2020 2020 2020 the *note. │ │ │ │ +000295c0: 2020 546f 7269 6343 686f 7752 696e 673a ToricChowRing: │ │ │ │ +000295d0: 2054 6f72 6963 4368 6f77 5269 6e67 2c20 ToricChowRing, │ │ │ │ +000295e0: 636f 6d6d 616e 640a 2020 2a20 2a6e 6f74 command. * *not │ │ │ │ +000295f0: 6520 4f70 7469 6f6e 616c 2069 6e70 7574 e Optional input │ │ │ │ +00029600: 733a 2028 4d61 6361 756c 6179 3244 6f63 s: (Macaulay2Doc │ │ │ │ +00029610: 2975 7369 6e67 2066 756e 6374 696f 6e73 )using functions │ │ │ │ +00029620: 2077 6974 6820 6f70 7469 6f6e 616c 2069 with optional i │ │ │ │ +00029630: 6e70 7574 732c 3a0a 2020 2020 2020 2a20 nputs,:. * │ │ │ │ +00029640: 436f 6d70 4d65 7468 6f64 2028 6d69 7373 CompMethod (miss │ │ │ │ +00029650: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ +00029660: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175 n) => ..., defau │ │ │ │ +00029670: 6c74 2076 616c 7565 0a20 2020 2020 2020 lt value. │ │ │ │ +00029680: 2050 726f 6a65 6374 6976 6544 6567 7265 ProjectiveDegre │ │ │ │ +00029690: 652c 2050 726f 6a65 6374 6976 6544 6567 e, ProjectiveDeg │ │ │ │ +000296a0: 7265 652c 2074 6869 7320 616c 676f 7269 ree, this algori │ │ │ │ +000296b0: 7468 6d20 6d61 7920 6265 2075 7365 6420 thm may be used │ │ │ │ +000296c0: 666f 720a 2020 2020 2020 2020 7375 6273 for. subs │ │ │ │ +000296d0: 6368 656d 6573 206f 6620 616e 7920 6170 chemes of any ap │ │ │ │ +000296e0: 706c 6963 6162 6c65 2074 6f72 6963 2076 plicable toric v │ │ │ │ +000296f0: 6172 6965 7479 2028 7468 6973 206d 6179 ariety (this may │ │ │ │ +00029700: 2062 6520 6368 6563 6b65 6420 7573 696e be checked usin │ │ │ │ +00029710: 670a 2020 2020 2020 2020 7468 6520 2a6e g. the *n │ │ │ │ +00029720: 6f74 6520 4368 6563 6b54 6f72 6963 5661 ote CheckToricVa │ │ │ │ +00029730: 7269 6574 7956 616c 6964 3a20 4368 6563 rietyValid: Chec │ │ │ │ +00029740: 6b54 6f72 6963 5661 7269 6574 7956 616c kToricVarietyVal │ │ │ │ +00029750: 6964 2c20 636f 6d6d 616e 6429 0a20 2020 id, command). │ │ │ │ +00029760: 2020 202a 2043 6f6d 704d 6574 686f 6420 * CompMethod │ │ │ │ +00029770: 286d 6973 7369 6e67 2064 6f63 756d 656e (missing documen │ │ │ │ +00029780: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20 tation) => ..., │ │ │ │ +00029790: 6465 6661 756c 7420 7661 6c75 650a 2020 default value. │ │ │ │ +000297a0: 2020 2020 2020 5072 6f6a 6563 7469 7665 Projective │ │ │ │ +000297b0: 4465 6772 6565 2c20 506e 5265 7369 6475 Degree, PnResidu │ │ │ │ +000297c0: 616c 2c20 7468 6973 2061 6c67 6f72 6974 al, this algorit │ │ │ │ +000297d0: 686d 206d 6179 2062 6520 7573 6564 2066 hm may be used f │ │ │ │ +000297e0: 6f72 2073 7562 7363 6865 6d65 730a 2020 or subschemes. │ │ │ │ +000297f0: 2020 2020 2020 6f66 205c 5050 5e6e 206f of \PP^n o │ │ │ │ +00029800: 6e6c 790a 2020 2020 2020 2a20 4f75 7470 nly. * Outp │ │ │ │ +00029810: 7574 203d 3e20 2e2e 2e2c 2064 6566 6175 ut => ..., defau │ │ │ │ +00029820: 6c74 2076 616c 7565 2043 686f 7752 696e lt value ChowRin │ │ │ │ +00029830: 6745 6c65 6d65 6e74 2c20 4368 6f77 5269 gElement, ChowRi │ │ │ │ +00029840: 6e67 456c 656d 656e 742c 2072 6574 7572 ngElement, retur │ │ │ │ +00029850: 6e73 0a20 2020 2020 2020 2061 2052 696e ns. a Rin │ │ │ │ +00029860: 6745 6c65 6d65 6e74 2069 6e20 7468 6520 gElement in the │ │ │ │ +00029870: 4368 6f77 2072 696e 6720 6f66 2074 6865 Chow ring of the │ │ │ │ +00029880: 2061 7070 726f 7072 6961 7465 2061 6d62 appropriate amb │ │ │ │ +00029890: 6965 6e74 2073 7061 6365 0a20 2020 2020 ient space. │ │ │ │ +000298a0: 202a 204f 7574 7075 7420 3d3e 202e 2e2e * Output => ... │ │ │ │ +000298b0: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +000298c0: 4368 6f77 5269 6e67 456c 656d 656e 742c ChowRingElement, │ │ │ │ +000298d0: 2048 6173 6846 6f72 6d2c 2048 6173 6846 HashForm, HashF │ │ │ │ +000298e0: 6f72 6d0a 2020 2020 2020 2020 7265 7475 orm. retu │ │ │ │ +000298f0: 726e 7320 6120 4d75 7461 626c 6548 6173 rns a MutableHas │ │ │ │ +00029900: 6854 6162 6c65 2063 6f6e 7461 696e 696e hTable containin │ │ │ │ +00029910: 6720 7468 6520 666f 6c6c 6f77 696e 6720 g the following │ │ │ │ +00029920: 6b65 7973 3a20 2247 2220 2874 6865 0a20 keys: "G" (the. │ │ │ │ +00029930: 2020 2020 2020 2070 6f6c 796e 6f6d 6961 polynomia │ │ │ │ +00029940: 6c20 7769 7468 2063 6f65 6666 6963 6965 l with coefficie │ │ │ │ +00029950: 6e74 7320 6f66 2074 6865 2068 7970 6572 nts of the hyper │ │ │ │ +00029960: 706c 616e 6520 636c 6173 7365 7320 7265 plane classes re │ │ │ │ +00029970: 7072 6573 656e 7469 6e67 2074 6865 0a20 presenting the. │ │ │ │ +00029980: 2020 2020 2020 2070 726f 6a65 6374 6976 projectiv │ │ │ │ +00029990: 6520 6465 6772 6565 7329 2c20 2247 6c69 e degrees), "Gli │ │ │ │ +000299a0: 7374 2220 2874 6865 206c 6973 7420 666f st" (the list fo │ │ │ │ +000299b0: 726d 206f 6620 2247 2229 202c 2022 5365 rm of "G") , "Se │ │ │ │ +000299c0: 6772 6522 2028 7468 650a 2020 2020 2020 gre" (the. │ │ │ │ +000299d0: 2020 746f 7461 6c20 5365 6772 6520 636c total Segre cl │ │ │ │ +000299e0: 6173 7320 6f66 2074 6865 2069 6e70 7574 ass of the input │ │ │ │ +000299f0: 292c 2253 6567 7265 4c69 7374 2220 2874 ),"SegreList" (t │ │ │ │ +00029a00: 6865 206c 6973 7420 666f 726d 206f 6620 he list form of │ │ │ │ +00029a10: 2253 6567 7265 2229 0a20 202a 204f 7574 "Segre"). * Out │ │ │ │ +00029a20: 7075 7473 3a0a 2020 2020 2020 2a20 6120 puts:. * a │ │ │ │ +00029a30: 2a6e 6f74 6520 7269 6e67 2065 6c65 6d65 *note ring eleme │ │ │ │ +00029a40: 6e74 3a20 284d 6163 6175 6c61 7932 446f nt: (Macaulay2Do │ │ │ │ +00029a50: 6329 5269 6e67 456c 656d 656e 742c 2c20 c)RingElement,, │ │ │ │ +00029a60: 7468 6520 7075 7368 666f 7277 6172 6420 the pushforward │ │ │ │ +00029a70: 6f66 0a20 2020 2020 2020 2074 6865 2074 of. the t │ │ │ │ +00029a80: 6f74 616c 2053 6567 7265 2063 6c61 7373 otal Segre class │ │ │ │ +00029a90: 206f 6620 7468 6520 7363 6865 6d65 2056 of the scheme V │ │ │ │ +00029aa0: 2064 6566 696e 6564 2062 7920 7468 6520 defined by the │ │ │ │ +00029ab0: 696e 7075 7420 6964 6561 6c20 746f 2074 input ideal to t │ │ │ │ +00029ac0: 6865 0a20 2020 2020 2020 2061 7070 726f he. appro │ │ │ │ +00029ad0: 7072 6961 7465 2043 686f 7720 7269 6e67 priate Chow ring │ │ │ │ +00029ae0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +00029af0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a46 6f72 2061 =========..For a │ │ │ │ +00029b00: 2073 7562 7363 6865 6d65 2056 206f 6620 subscheme V of │ │ │ │ +00029b10: 616e 2061 7070 6c69 6361 626c 6520 746f an applicable to │ │ │ │ +00029b20: 7269 6320 7661 7269 6574 7920 5820 7468 ric variety X th │ │ │ │ +00029b30: 6973 2063 6f6d 6d61 6e64 2063 6f6d 7075 is command compu │ │ │ │ +00029b40: 7465 7320 7468 650a 7075 7368 2d66 6f72 tes the.push-for │ │ │ │ +00029b50: 7761 7264 206f 6620 7468 6520 746f 7461 ward of the tota │ │ │ │ +00029b60: 6c20 5365 6772 6520 636c 6173 7320 7328 l Segre class s( │ │ │ │ +00029b70: 562c 5829 206f 6620 5620 696e 2058 2074 V,X) of V in X t │ │ │ │ +00029b80: 6f20 7468 6520 4368 6f77 2072 696e 6720 o the Chow ring │ │ │ │ +00029b90: 6f66 2058 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d of X...+-------- │ │ │ │ +00029ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029bc0: 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 6574 -----+.|i1 : set │ │ │ │ +00029bd0: 5261 6e64 6f6d 5365 6564 2037 323b 2020 RandomSeed 72; │ │ │ │ +00029be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029bf0: 2020 2020 207c 0a7c 202d 2d20 7365 7474 |.| -- sett │ │ │ │ +00029c00: 696e 6720 7261 6e64 6f6d 2073 6565 6420 ing random seed │ │ │ │ +00029c10: 746f 2037 3220 2020 2020 2020 2020 2020 to 72 │ │ │ │ +00029c20: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00029c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029c50: 2d2d 2d2d 2d2b 0a7c 6932 203a 2052 203d -----+.|i2 : R = │ │ │ │ +00029c60: 205a 5a2f 3332 3734 395b 772c 792c 7a5d ZZ/32749[w,y,z] │ │ │ │ +00029c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029c80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029cb0: 2020 2020 207c 0a7c 6f32 203d 2052 2020 |.|o2 = R │ │ │ │ +00029cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029ce0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029d10: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c |.|o2 : Pol │ │ │ │ +00029d20: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ +00029d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029d40: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00029d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029d70: 2d2d 2d2d 2d2b 0a7c 6933 203a 2053 6567 -----+.|i3 : Seg │ │ │ │ +00029d80: 7265 2869 6465 616c 2877 2a79 292c 436f re(ideal(w*y),Co │ │ │ │ +00029d90: 6d70 4d65 7468 6f64 3d3e 506e 5265 7369 mpMethod=>PnResi │ │ │ │ +00029da0: 6475 616c 297c 0a7c 2020 2020 2020 2020 dual)|.| │ │ │ │ +00029db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029dd0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029de0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00029df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e00: 2020 2020 207c 0a7c 6f33 203d 202d 2034 |.|o3 = - 4 │ │ │ │ +00029e10: 4820 202b 2032 4820 2020 2020 2020 2020 H + 2H │ │ │ │ +00029e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e30: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e60: 2020 2020 207c 0a7c 2020 2020 205a 5a5b |.| ZZ[ │ │ │ │ +00029e70: 485d 2020 2020 2020 2020 2020 2020 2020 H] │ │ │ │ +00029e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e90: 2020 2020 207c 0a7c 6f33 203a 202d 2d2d |.|o3 : --- │ │ │ │ +00029ea0: 2d2d 2020 2020 2020 2020 2020 2020 2020 -- │ │ │ │ +00029eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029ec0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029ed0: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00029ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029ef0: 2020 2020 207c 0a7c 2020 2020 2020 2048 |.| H │ │ │ │ +00029f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f20: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00029f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029f50: 2d2d 2d2d 2d2b 0a7c 6934 203a 2041 3d43 -----+.|i4 : A=C │ │ │ │ +00029f60: 686f 7752 696e 6728 5229 2020 2020 2020 howRing(R) │ │ │ │ +00029f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029fb0: 2020 2020 207c 0a7c 6f34 203d 2041 2020 |.|o4 = A │ │ │ │ +00029fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029fe0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a010: 2020 2020 207c 0a7c 6f34 203a 2051 756f |.|o4 : Quo │ │ │ │ +0002a020: 7469 656e 7452 696e 6720 2020 2020 2020 tientRing │ │ │ │ +0002a030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a040: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a070: 2d2d 2d2d 2d2b 0a7c 6935 203a 2053 6567 -----+.|i5 : Seg │ │ │ │ +0002a080: 7265 2841 2c69 6465 616c 2877 5e32 2a79 re(A,ideal(w^2*y │ │ │ │ +0002a090: 2c77 2a79 5e32 2929 2020 2020 2020 2020 ,w*y^2)) │ │ │ │ +0002a0a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002a0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a0d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002a0e0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a100: 2020 2020 207c 0a7c 6f35 203d 202d 2033 |.|o5 = - 3 │ │ │ │ +0002a110: 6820 202b 2032 6820 2020 2020 2020 2020 h + 2h │ │ │ │ +0002a120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a130: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002a140: 2031 2020 2020 2031 2020 2020 2020 2020 1 1 │ │ │ │ +0002a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a160: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a190: 2020 2020 207c 0a7c 6f35 203a 2041 2020 |.|o5 : A │ │ │ │ +0002a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a1c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a1f0: 2d2d 2d2d 2d2b 0a0a 4e6f 7720 636f 6e73 -----+..Now cons │ │ │ │ +0002a200: 6964 6572 2061 6e20 6578 616d 706c 6520 ider an example │ │ │ │ +0002a210: 696e 205c 5050 5e32 205c 7469 6d65 7320 in \PP^2 \times │ │ │ │ +0002a220: 5c50 505e 322c 2069 6620 7765 2069 6e70 \PP^2, if we inp │ │ │ │ +0002a230: 7574 2074 6865 2043 686f 7720 7269 6e67 ut the Chow ring │ │ │ │ +0002a240: 2041 2074 6865 0a6f 7574 7075 7420 7769 A the.output wi │ │ │ │ +0002a250: 6c6c 2062 6520 7265 7475 726e 6564 2069 ll be returned i │ │ │ │ +0002a260: 6e20 7468 6520 7361 6d65 2072 696e 672e n the same ring. │ │ │ │ +0002a270: 2054 6f20 656e 7375 7265 2070 726f 7065 To ensure prope │ │ │ │ +0002a280: 7220 6675 6e63 7469 6f6e 206f 6620 7468 r function of th │ │ │ │ +0002a290: 650a 6d65 7468 6f64 7320 7765 2062 7569 e.methods we bui │ │ │ │ +0002a2a0: 6c64 2074 6865 2043 686f 7720 7269 6e67 ld the Chow ring │ │ │ │ +0002a2b0: 2075 7369 6e67 2074 6865 202a 6e6f 7465 using the *note │ │ │ │ +0002a2c0: 2043 686f 7752 696e 673a 2043 686f 7752 ChowRing: ChowR │ │ │ │ +0002a2d0: 696e 672c 2063 6f6d 6d61 6e64 2e20 5765 ing, command. We │ │ │ │ +0002a2e0: 0a6d 6179 2061 6c73 6f20 7265 7475 726e .may also return │ │ │ │ +0002a2f0: 2061 204d 7574 6162 6c65 4861 7368 5461 a MutableHashTa │ │ │ │ +0002a300: 626c 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ble...+--------- │ │ │ │ +0002a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a350: 2d2d 2d2d 2b0a 7c69 3620 3a20 523d 4d75 ----+.|i6 : R=Mu │ │ │ │ +0002a360: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ +0002a370: 287b 322c 327d 2920 2020 2020 2020 2020 ({2,2}) │ │ │ │ 0002a380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a3a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a3b0: 7c6f 3620 3d20 5220 2020 2020 2020 2020 |o6 = R │ │ │ │ +0002a3a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a3f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a400: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002a3f0: 2020 2020 7c0a 7c6f 3620 3d20 5220 2020 |.|o6 = R │ │ │ │ +0002a400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a440: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a450: 7c6f 3620 3a20 506f 6c79 6e6f 6d69 616c |o6 : Polynomial │ │ │ │ -0002a460: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ +0002a440: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a490: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a4a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002a4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002a4f0: 7c69 3720 3a20 723d 6765 6e73 2052 2020 |i7 : r=gens R │ │ │ │ -0002a500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a530: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a540: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002a490: 2020 2020 7c0a 7c6f 3620 3a20 506f 6c79 |.|o6 : Poly │ │ │ │ +0002a4a0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ +0002a4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a4e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a530: 2d2d 2d2d 2b0a 7c69 3720 3a20 723d 6765 ----+.|i7 : r=ge │ │ │ │ +0002a540: 6e73 2052 2020 2020 2020 2020 2020 2020 ns R │ │ │ │ 0002a550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a580: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a590: 7c6f 3720 3d20 7b78 202c 2078 202c 2078 |o7 = {x , x , x │ │ │ │ -0002a5a0: 202c 2078 202c 2078 202c 2078 207d 2020 , x , x , x } │ │ │ │ +0002a580: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a5d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a5e0: 7c20 2020 2020 2020 3020 2020 3120 2020 | 0 1 │ │ │ │ -0002a5f0: 3220 2020 3320 2020 3420 2020 3520 2020 2 3 4 5 │ │ │ │ +0002a5d0: 2020 2020 7c0a 7c6f 3720 3d20 7b78 202c |.|o7 = {x , │ │ │ │ +0002a5e0: 2078 202c 2078 202c 2078 202c 2078 202c x , x , x , x , │ │ │ │ +0002a5f0: 2078 207d 2020 2020 2020 2020 2020 2020 x } │ │ │ │ 0002a600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a620: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a630: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002a640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a620: 2020 2020 7c0a 7c20 2020 2020 2020 3020 |.| 0 │ │ │ │ +0002a630: 2020 3120 2020 3220 2020 3320 2020 3420 1 2 3 4 │ │ │ │ +0002a640: 2020 3520 2020 2020 2020 2020 2020 2020 5 │ │ │ │ 0002a650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a670: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a680: 7c6f 3720 3a20 4c69 7374 2020 2020 2020 |o7 : List │ │ │ │ +0002a670: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a6c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a6d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002a6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002a720: 7c69 3820 3a20 413d 4368 6f77 5269 6e67 |i8 : A=ChowRing │ │ │ │ -0002a730: 2852 2920 2020 2020 2020 2020 2020 2020 (R) │ │ │ │ -0002a740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a760: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a770: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002a6c0: 2020 2020 7c0a 7c6f 3720 3a20 4c69 7374 |.|o7 : List │ │ │ │ +0002a6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a710: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002a720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a760: 2d2d 2d2d 2b0a 7c69 3820 3a20 413d 4368 ----+.|i8 : A=Ch │ │ │ │ +0002a770: 6f77 5269 6e67 2852 2920 2020 2020 2020 owRing(R) │ │ │ │ 0002a780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a7b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a7c0: 7c6f 3820 3d20 4120 2020 2020 2020 2020 |o8 = A │ │ │ │ +0002a7b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a800: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a810: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002a800: 2020 2020 7c0a 7c6f 3820 3d20 4120 2020 |.|o8 = A │ │ │ │ +0002a810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a850: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a860: 7c6f 3820 3a20 5175 6f74 6965 6e74 5269 |o8 : QuotientRi │ │ │ │ -0002a870: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ +0002a850: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a8a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a8b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002a900: 7c69 3920 3a20 493d 6964 6561 6c28 725f |i9 : I=ideal(r_ │ │ │ │ -0002a910: 305e 322a 725f 332d 725f 342a 725f 312a 0^2*r_3-r_4*r_1* │ │ │ │ -0002a920: 725f 322c 725f 325e 322a 725f 3529 2020 r_2,r_2^2*r_5) │ │ │ │ -0002a930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a940: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a950: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002a960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a8a0: 2020 2020 7c0a 7c6f 3820 3a20 5175 6f74 |.|o8 : Quot │ │ │ │ +0002a8b0: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +0002a8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a8f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a940: 2d2d 2d2d 2b0a 7c69 3920 3a20 493d 6964 ----+.|i9 : I=id │ │ │ │ +0002a950: 6561 6c28 725f 305e 322a 725f 332d 725f eal(r_0^2*r_3-r_ │ │ │ │ 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2020 |.| │ │ │ │ +0002a9f0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +0002aa00: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002aa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002aa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aa30: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002aa40: 7c20 2020 2020 2020 2020 2020 2020 3020 | 0 │ │ │ │ -0002aa50: 3320 2020 2031 2032 2034 2020 2032 2035 3 1 2 4 2 5 │ │ │ │ +0002aa30: 2020 2020 7c0a 7c6f 3920 3d20 6964 6561 |.|o9 = idea │ │ │ │ +0002aa40: 6c20 2878 2078 2020 2d20 7820 7820 7820 l (x x - x x x │ │ │ │ +0002aa50: 2c20 7820 7820 2920 2020 2020 2020 2020 , x x ) │ │ │ │ 0002aa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002aa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aa80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002aa90: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002aaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002aa80: 2020 2020 7c0a 7c20 2020 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2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002abb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002abc0: 2d2d 2d2d 2b0a 7c69 3130 203a 2053 6567 ----+.|i10 : Seg │ │ │ │ +0002abd0: 7265 2049 2020 2020 2020 2020 2020 2020 re I │ │ │ │ 0002abe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002abf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ac00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ac10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002ac20: 7c20 2020 2020 2020 2020 3220 3220 2020 | 2 2 │ │ │ │ -0002ac30: 2020 2032 2020 2020 2020 2020 2020 3220 2 2 │ │ │ │ -0002ac40: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -0002ac50: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002ac60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002ac70: 7c6f 3130 203d 2037 3268 2068 2020 2d20 |o10 = 72h h - │ │ │ │ -0002ac80: 3234 6820 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2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ac60: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002ac70: 3220 3220 2020 2020 2032 2020 2020 2020 2 2 2 │ │ │ │ +0002ac80: 2020 2020 3220 2020 2020 3220 2020 2020 2 2 │ │ │ │ +0002ac90: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0002aca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002acb0: 2020 2020 7c0a 7c6f 3130 203d 2037 3268 |.|o10 = 72h │ │ │ │ +0002acc0: 2068 2020 2d20 3234 6820 6820 202d 2031 h - 24h h - 1 │ │ │ │ +0002acd0: 3268 2068 2020 2b20 3468 2020 2b20 3468 2h h + 4h + 4h │ │ │ │ +0002ace0: 2068 2020 2b20 6820 2020 2020 2020 2020 h + h │ │ │ │ +0002acf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ad00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002ad10: 3120 3220 2020 2020 2031 2032 2020 2020 1 2 1 2 │ │ │ │ +0002ad20: 2020 3120 3220 2020 2020 3120 2020 2020 1 2 1 │ │ │ │ +0002ad30: 3120 3220 2020 2032 2020 2020 2020 2020 1 2 2 │ │ │ │ 0002ad40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ad50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002ad60: 7c20 2020 2020 205a 5a5b 6820 2e2e 6820 | ZZ[h ..h │ │ │ │ -0002ad70: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +0002ad50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002ad60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ad70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ad80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ad90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ada0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002adb0: 7c20 2020 2020 2020 2020 2031 2020 2032 | 1 2 │ │ │ │ +0002ada0: 2020 2020 7c0a 7c20 2020 2020 205a 5a5b |.| ZZ[ │ │ │ │ +0002adb0: 6820 2e2e 6820 5d20 2020 2020 2020 2020 h ..h ] │ │ │ │ 0002adc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002add0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ade0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002adf0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002ae00: 7c6f 3130 203a 202d 2d2d 2d2d 2d2d 2d2d |o10 : --------- │ │ │ │ -0002ae10: 2d20 2020 2020 2020 2020 2020 2020 2020 - │ │ │ │ +0002adf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002ae00: 2031 2020 2032 2020 2020 2020 2020 2020 1 2 │ │ │ │ +0002ae10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ae20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ae30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ae40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002ae50: 7c20 2020 2020 2020 2020 3320 2020 3320 | 3 3 │ │ │ │ +0002ae40: 2020 2020 7c0a 7c6f 3130 203a 202d 2d2d |.|o10 : --- │ │ │ │ +0002ae50: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020 ------- │ │ │ │ 0002ae60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ae70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ae80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ae90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002aea0: 7c20 2020 2020 2020 2868 202c 2068 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2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002af80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002af90: 7c69 3131 203a 2073 313d 5365 6772 6528 |i11 : s1=Segre( │ │ │ │ -0002afa0: 412c 4929 2020 2020 2020 2020 2020 2020 A,I) │ │ │ │ -0002afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002afc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002afd0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002afe0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002af30: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002af40: 3120 2020 3220 2020 2020 2020 2020 2020 1 2 │ │ │ │ +0002af50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002af60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002af70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002af80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002afa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002afb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002afc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002afd0: 2d2d 2d2d 2b0a 7c69 3131 203a 2073 313d ----+.|i11 : s1= │ │ │ │ +0002afe0: 5365 6772 6528 412c 4929 2020 2020 2020 Segre(A,I) │ │ │ │ 0002aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b020: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b030: 7c20 2020 2020 2020 2020 3220 3220 2020 | 2 2 │ │ │ │ -0002b040: 2020 2032 2020 2020 2020 2020 2020 3220 2 2 │ │ │ │ -0002b050: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -0002b060: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002b070: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b080: 7c6f 3131 203d 2037 3268 2068 2020 2d20 |o11 = 72h h - │ │ │ │ -0002b090: 3234 6820 6820 202d 2031 3268 2068 2020 24h h - 12h h │ │ │ │ -0002b0a0: 2b20 3468 2020 2b20 3468 2068 2020 2b20 + 4h + 4h h + │ │ │ │ -0002b0b0: 6820 2020 2020 2020 2020 2020 2020 2020 h │ │ │ │ -0002b0c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b0d0: 7c20 2020 2020 2020 2020 3120 3220 2020 | 1 2 │ │ │ │ -0002b0e0: 2020 2031 2032 2020 2020 2020 3120 3220 1 2 1 2 │ │ │ │ -0002b0f0: 2020 2020 3120 2020 2020 3120 3220 2020 1 1 2 │ │ │ │ -0002b100: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002b110: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b120: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002b130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b020: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b070: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b080: 3220 3220 2020 2020 2032 2020 2020 2020 2 2 2 │ │ │ │ +0002b090: 2020 2020 3220 2020 2020 3220 2020 2020 2 2 │ │ │ │ +0002b0a0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0002b0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b0c0: 2020 2020 7c0a 7c6f 3131 203d 2037 3268 |.|o11 = 72h │ │ │ │ +0002b0d0: 2068 2020 2d20 3234 6820 6820 202d 2031 h - 24h h - 1 │ │ │ │ +0002b0e0: 3268 2068 2020 2b20 3468 2020 2b20 3468 2h h + 4h + 4h │ │ │ │ +0002b0f0: 2068 2020 2b20 6820 2020 2020 2020 2020 h + h │ │ │ │ +0002b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b110: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b120: 3120 3220 2020 2020 2031 2032 2020 2020 1 2 1 2 │ │ │ │ +0002b130: 2020 3120 3220 2020 2020 3120 2020 2020 1 2 1 │ │ │ │ +0002b140: 3120 3220 2020 2032 2020 2020 2020 2020 1 2 2 │ │ │ │ 0002b150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b160: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b170: 7c6f 3131 203a 2041 2020 2020 2020 2020 |o11 : A │ │ │ │ +0002b160: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b1b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b1c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002b210: 7c69 3132 203a 2053 6567 4861 7368 3d53 |i12 : 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│ │ │ 0002b970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b980: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b990: 7c32 3468 2068 2020 2d20 3132 6820 6820 |24h h - 12h h │ │ │ │ -0002b9a0: 2c20 3732 6820 6820 7d20 2020 2020 2020 , 72h h } │ │ │ │ +0002b980: 2020 2020 7c0a 7c20 2020 3220 2020 2020 |.| 2 │ │ │ │ +0002b990: 2020 2020 2032 2020 2020 2032 2032 2020 2 2 2 │ │ │ │ +0002b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b9d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b9e0: 7c20 2020 3120 3220 2020 2020 2031 2032 | 1 2 1 2 │ │ │ │ -0002b9f0: 2020 2020 2031 2032 2020 2020 2020 2020 1 2 │ │ │ │ +0002b9d0: 2020 2020 7c0a 7c32 3468 2068 2020 2d20 |.|24h h - │ │ │ │ +0002b9e0: 3132 6820 6820 2c20 3732 6820 6820 7d20 12h h , 72h h } │ │ │ │ +0002b9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ba00: 2020 2020 2020 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2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bbb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002bbc0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002bb10: 2020 2020 7c0a 7c20 2020 2031 2032 2020 |.| 1 2 │ │ │ │ +0002bb20: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002bb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bbb0: 2d2d 2d2d 2b0a 7c69 3134 203a 2073 313d ----+.|i14 : s1= │ │ │ │ +0002bbc0: 3d53 6567 4861 7368 2322 5365 6772 6522 =SegHash#"Segre" │ │ │ │ 0002bbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002bc10: 7c6f 3134 203d 2074 7275 6520 2020 2020 |o14 = true │ │ │ │ +0002bc00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002bc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002bc60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002bcb0: 0a49 6e20 7468 6520 6361 7365 2077 6865 .In the case whe │ │ │ │ -0002bcc0: 7265 2074 6865 2061 6d62 6965 6e74 2073 re the ambient s │ │ │ │ -0002bcd0: 7061 6365 2069 7320 6120 746f 7269 6320 pace is a toric │ │ │ │ -0002bce0: 7661 7269 6574 7920 7768 6963 6820 6973 variety which is │ │ │ │ -0002bcf0: 206e 6f74 2061 2070 726f 6475 6374 0a6f not a product.o │ │ │ │ -0002bd00: 6620 7072 6f6a 6563 7469 7665 2073 7061 f projective spa │ │ │ │ -0002bd10: 6365 7320 7765 206d 7573 7420 6c6f 6164 ces we must load │ │ │ │ -0002bd20: 2074 6865 204e 6f72 6d61 6c54 6f72 6963 the NormalToric │ │ │ │ -0002bd30: 5661 7269 6574 6965 7320 7061 636b 6167 Varieties packag │ │ │ │ -0002bd40: 6520 616e 6420 6d75 7374 0a61 6c73 6f20 e and must.also │ │ │ │ -0002bd50: 696e 7075 7420 7468 6520 746f 7269 6320 input the toric │ │ │ │ -0002bd60: 7661 7269 6574 792e 2049 6620 7468 6520 variety. If the │ │ │ │ -0002bd70: 746f 7269 6320 7661 7269 6574 7920 6973 toric variety is │ │ │ │ -0002bd80: 2061 2070 726f 6475 6374 206f 6620 7072 a product of pr │ │ │ │ -0002bd90: 6f6a 6563 7469 7665 0a73 7061 6365 2069 ojective.space i │ │ │ │ -0002bda0: 7420 6973 2072 6563 6f6d 6d65 6e64 6564 t is recommended │ │ │ │ -0002bdb0: 2074 6f20 7573 6520 7468 6520 666f 726d to use the form │ │ │ │ -0002bdc0: 2061 626f 7665 2072 6174 6865 7220 7468 above rather th │ │ │ │ -0002bdd0: 616e 2069 6e70 7574 7469 6e67 2074 6865 an inputting the │ │ │ │ -0002bde0: 2074 6f72 6963 0a76 6172 6965 7479 2066 toric.variety f │ │ │ │ -0002bdf0: 6f72 2065 6666 6963 6965 6e63 7920 7265 or efficiency re │ │ │ │ -0002be00: 6173 6f6e 732e 0a0a 2b2d 2d2d 2d2d 2d2d asons...+------- │ │ │ │ -0002be10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002be20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002be30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002be40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002be50: 2d2b 0a7c 6931 3520 3a20 6e65 6564 7350 -+.|i15 : needsP │ │ │ │ -0002be60: 6163 6b61 6765 2022 4e6f 726d 616c 546f ackage "NormalTo │ │ │ │ -0002be70: 7269 6356 6172 6965 7469 6573 2220 2020 ricVarieties" │ │ │ │ -0002be80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002be90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002bea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002beb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bc50: 2020 2020 7c0a 7c6f 3134 203d 2074 7275 |.|o14 = tru │ │ │ │ +0002bc60: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +0002bc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bca0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bcf0: 2d2d 2d2d 2b0a 0a49 6e20 7468 6520 6361 ----+..In the ca │ │ │ │ +0002bd00: 7365 2077 6865 7265 2074 6865 2061 6d62 se where the amb │ │ │ │ +0002bd10: 6965 6e74 2073 7061 6365 2069 7320 6120 ient space is a │ │ │ │ +0002bd20: 746f 7269 6320 7661 7269 6574 7920 7768 toric variety wh │ │ │ │ +0002bd30: 6963 6820 6973 206e 6f74 2061 2070 726f ich is not a pro │ │ │ │ +0002bd40: 6475 6374 0a6f 6620 7072 6f6a 6563 7469 duct.of projecti │ │ │ │ +0002bd50: 7665 2073 7061 6365 7320 7765 206d 7573 ve spaces we mus │ │ │ │ +0002bd60: 7420 6c6f 6164 2074 6865 204e 6f72 6d61 t load the Norma │ │ │ │ +0002bd70: 6c54 6f72 6963 5661 7269 6574 6965 7320 lToricVarieties │ │ │ │ +0002bd80: 7061 636b 6167 6520 616e 6420 6d75 7374 package and must │ │ │ │ +0002bd90: 0a61 6c73 6f20 696e 7075 7420 7468 6520 .also input the │ │ │ │ +0002bda0: 746f 7269 6320 7661 7269 6574 792e 2049 toric variety. I │ │ │ │ +0002bdb0: 6620 7468 6520 746f 7269 6320 7661 7269 f the toric vari │ │ │ │ +0002bdc0: 6574 7920 6973 2061 2070 726f 6475 6374 ety is a product │ │ │ │ +0002bdd0: 206f 6620 7072 6f6a 6563 7469 7665 0a73 of projective.s │ │ │ │ +0002bde0: 7061 6365 2069 7420 6973 2072 6563 6f6d pace it is recom │ │ │ │ +0002bdf0: 6d65 6e64 6564 2074 6f20 7573 6520 7468 mended to use th │ │ │ │ +0002be00: 6520 666f 726d 2061 626f 7665 2072 6174 e form above rat │ │ │ │ +0002be10: 6865 7220 7468 616e 2069 6e70 7574 7469 her than inputti │ │ │ │ +0002be20: 6e67 2074 6865 2074 6f72 6963 0a76 6172 ng the toric.var │ │ │ │ +0002be30: 6965 7479 2066 6f72 2065 6666 6963 6965 iety for efficie │ │ │ │ +0002be40: 6e63 7920 7265 6173 6f6e 732e 0a0a 2b2d ncy reasons...+- │ │ │ │ +0002be50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002be60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002be70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002be80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002be90: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 -------+.|i15 : │ │ │ │ +0002bea0: 6e65 6564 7350 6163 6b61 6765 2022 4e6f needsPackage "No │ │ │ │ +0002beb0: 726d 616c 546f 7269 6356 6172 6965 7469 rmalToricVarieti │ │ │ │ +0002bec0: 6573 2220 2020 2020 2020 2020 2020 2020 es" │ │ │ │ 0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bee0: 2020 2020 2020 207c 0a7c 6f31 3520 3d20 |.|o15 = │ │ │ │ -0002bef0: 4e6f 726d 616c 546f 7269 6356 6172 6965 NormalToricVarie │ │ │ │ -0002bf00: 7469 6573 2020 2020 2020 2020 2020 2020 ties │ │ │ │ +0002bee0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002bef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf30: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002bf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002bf30: 6f31 3520 3d20 4e6f 726d 616c 546f 7269 o15 = NormalTori │ │ │ │ +0002bf40: 6356 6172 6965 7469 6573 2020 2020 2020 cVarieties │ │ │ │ 0002bf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002bf80: 6f31 3520 3a20 5061 636b 6167 6520 2020 o15 : Package │ │ │ │ +0002bf70: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002bf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bf90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfc0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0002bfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c010: 2d2d 2d2b 0a7c 6931 3620 3a20 5268 6f20 ---+.|i16 : Rho │ │ │ │ -0002c020: 3d20 7b7b 312c 302c 307d 2c7b 302c 312c = {{1,0,0},{0,1, │ │ │ │ -0002c030: 307d 2c7b 302c 302c 317d 2c7b 2d31 2c2d 0},{0,0,1},{-1,- │ │ │ │ -0002c040: 312c 307d 2c7b 302c 302c 2d31 7d7d 2020 1,0},{0,0,-1}} │ │ │ │ -0002c050: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002c060: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c0a0: 2020 2020 2020 2020 207c 0a7c 6f31 3620 |.|o16 │ │ │ │ -0002c0b0: 3d20 7b7b 312c 2030 2c20 307d 2c20 7b30 = {{1, 0, 0}, {0 │ │ │ │ -0002c0c0: 2c20 312c 2030 7d2c 207b 302c 2030 2c20 , 1, 0}, {0, 0, │ │ │ │ -0002c0d0: 317d 2c20 7b2d 312c 202d 312c 2030 7d2c 1}, {-1, -1, 0}, │ │ │ │ -0002c0e0: 207b 302c 2030 2c20 2d31 7d7d 2020 2020 {0, 0, -1}} │ │ │ │ -0002c0f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002c100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c130: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002c140: 0a7c 6f31 3620 3a20 4c69 7374 2020 2020 .|o16 : List │ │ │ │ +0002bfc0: 2020 207c 0a7c 6f31 3520 3a20 5061 636b |.|o15 : Pack │ │ │ │ +0002bfd0: 6167 6520 2020 2020 2020 2020 2020 2020 age │ │ │ │ +0002bfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c000: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c010: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002c020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c050: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 ---------+.|i16 │ │ │ │ +0002c060: 3a20 5268 6f20 3d20 7b7b 312c 302c 307d : Rho = {{1,0,0} │ │ │ │ +0002c070: 2c7b 302c 312c 307d 2c7b 302c 302c 317d ,{0,1,0},{0,0,1} │ │ │ │ +0002c080: 2c7b 2d31 2c2d 312c 307d 2c7b 302c 302c ,{-1,-1,0},{0,0, │ │ │ │ +0002c090: 2d31 7d7d 2020 2020 2020 2020 2020 2020 -1}} │ │ │ │ +0002c0a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002c0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c0e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002c0f0: 0a7c 6f31 3620 3d20 7b7b 312c 2030 2c20 .|o16 = {{1, 0, │ │ │ │ +0002c100: 307d 2c20 7b30 2c20 312c 2030 7d2c 207b 0}, {0, 1, 0}, { │ │ │ │ +0002c110: 302c 2030 2c20 317d 2c20 7b2d 312c 202d 0, 0, 1}, {-1, - │ │ │ │ +0002c120: 312c 2030 7d2c 207b 302c 2030 2c20 2d31 1, 0}, {0, 0, -1 │ │ │ │ +0002c130: 7d7d 2020 2020 2020 2020 7c0a 7c20 2020 }} |.| │ │ │ │ +0002c140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c180: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -0002c190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c1d0: 2d2d 2d2d 2d2b 0a7c 6931 3720 3a20 5369 -----+.|i17 : Si │ │ │ │ -0002c1e0: 676d 6120 3d20 7b7b 302c 312c 327d 2c7b gma = {{0,1,2},{ │ │ │ │ -0002c1f0: 312c 322c 337d 2c7b 302c 322c 337d 2c7b 1,2,3},{0,2,3},{ │ │ │ │ -0002c200: 302c 312c 347d 2c7b 312c 332c 347d 2c7b 0,1,4},{1,3,4},{ │ │ │ │ -0002c210: 302c 332c 347d 7d20 2020 2020 2020 2020 0,3,4}} │ │ │ │ -0002c220: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002c230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c260: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0002c270: 3720 3d20 7b7b 302c 2031 2c20 327d 2c20 7 = {{0, 1, 2}, │ │ │ │ -0002c280: 7b31 2c20 322c 2033 7d2c 207b 302c 2032 {1, 2, 3}, {0, 2 │ │ │ │ -0002c290: 2c20 337d 2c20 7b30 2c20 312c 2034 7d2c , 3}, {0, 1, 4}, │ │ │ │ -0002c2a0: 207b 312c 2033 2c20 347d 2c20 7b30 2c20 {1, 3, 4}, {0, │ │ │ │ -0002c2b0: 332c 2034 7d7d 7c0a 7c20 2020 2020 2020 3, 4}}|.| │ │ │ │ -0002c2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c300: 207c 0a7c 6f31 3720 3a20 4c69 7374 2020 |.|o17 : List │ │ │ │ +0002c180: 2020 2020 207c 0a7c 6f31 3620 3a20 4c69 |.|o16 : Li │ │ │ │ +0002c190: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ +0002c1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c1d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0002c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0002c220: 3720 3a20 5369 676d 6120 3d20 7b7b 302c 7 : Sigma = {{0, │ │ │ │ +0002c230: 312c 327d 2c7b 312c 322c 337d 2c7b 302c 1,2},{1,2,3},{0, │ │ │ │ +0002c240: 322c 337d 2c7b 302c 312c 347d 2c7b 312c 2,3},{0,1,4},{1, │ │ │ │ +0002c250: 332c 347d 2c7b 302c 332c 347d 7d20 2020 3,4},{0,3,4}} │ │ │ │ +0002c260: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002c270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c2b0: 207c 0a7c 6f31 3720 3d20 7b7b 302c 2031 |.|o17 = {{0, 1 │ │ │ │ +0002c2c0: 2c20 327d 2c20 7b31 2c20 322c 2033 7d2c , 2}, {1, 2, 3}, │ │ │ │ +0002c2d0: 207b 302c 2032 2c20 337d 2c20 7b30 2c20 {0, 2, 3}, {0, │ │ │ │ +0002c2e0: 312c 2034 7d2c 207b 312c 2033 2c20 347d 1, 4}, {1, 3, 4} │ │ │ │ +0002c2f0: 2c20 7b30 2c20 332c 2034 7d7d 7c0a 7c20 , {0, 3, 4}}|.| │ │ │ │ +0002c300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c340: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0002c350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c390: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3820 3a20 -------+.|i18 : │ │ │ │ -0002c3a0: 5820 3d20 6e6f 726d 616c 546f 7269 6356 X = normalToricV │ │ │ │ -0002c3b0: 6172 6965 7479 2852 686f 2c53 6967 6d61 ariety(Rho,Sigma │ │ │ │ -0002c3c0: 2c43 6f65 6666 6963 6965 6e74 5269 6e67 ,CoefficientRing │ │ │ │ -0002c3d0: 203d 3e5a 5a2f 3332 3734 3929 2020 2020 =>ZZ/32749) │ │ │ │ -0002c3e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002c3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c420: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002c430: 6f31 3820 3d20 5820 2020 2020 2020 2020 o18 = X │ │ │ │ +0002c340: 2020 2020 2020 207c 0a7c 6f31 3720 3a20 |.|o17 : │ │ │ │ +0002c350: 4c69 7374 2020 2020 2020 2020 2020 2020 List │ │ │ │ +0002c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c390: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0002c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0002c3e0: 6931 3820 3a20 5820 3d20 6e6f 726d 616c i18 : X = normal │ │ │ │ +0002c3f0: 546f 7269 6356 6172 6965 7479 2852 686f ToricVariety(Rho │ │ │ │ +0002c400: 2c53 6967 6d61 2c43 6f65 6666 6963 6965 ,Sigma,Coefficie │ │ │ │ +0002c410: 6e74 5269 6e67 203d 3e5a 5a2f 3332 3734 ntRing =>ZZ/3274 │ │ │ │ +0002c420: 3929 2020 2020 2020 7c0a 7c20 2020 2020 9) |.| │ │ │ │ +0002c430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c470: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002c470: 2020 207c 0a7c 6f31 3820 3d20 5820 2020 |.|o18 = X │ │ │ │ 0002c480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c4c0: 2020 207c 0a7c 6f31 3820 3a20 4e6f 726d |.|o18 : Norm │ │ │ │ -0002c4d0: 616c 546f 7269 6356 6172 6965 7479 2020 alToricVariety │ │ │ │ +0002c4b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c4c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002c4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c500: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002c510: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002c520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c550: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920 ---------+.|i19 │ │ │ │ -0002c560: 3a20 4368 6563 6b54 6f72 6963 5661 7269 : CheckToricVari │ │ │ │ -0002c570: 6574 7956 616c 6964 2858 2920 2020 2020 etyValid(X) │ │ │ │ -0002c580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c5a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002c5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c500: 2020 2020 2020 2020 207c 0a7c 6f31 3820 |.|o18 │ │ │ │ +0002c510: 3a20 4e6f 726d 616c 546f 7269 6356 6172 : NormalToricVar │ │ │ │ +0002c520: 6965 7479 2020 2020 2020 2020 2020 2020 iety │ │ │ │ +0002c530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c550: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002c5a0: 0a7c 6931 3920 3a20 4368 6563 6b54 6f72 .|i19 : CheckTor │ │ │ │ +0002c5b0: 6963 5661 7269 6574 7956 616c 6964 2858 icVarietyValid(X │ │ │ │ +0002c5c0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0002c5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c5e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002c5f0: 0a7c 6f31 3920 3d20 7472 7565 2020 2020 .|o19 = true │ │ │ │ +0002c5e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002c5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c630: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -0002c640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c680: 2d2d 2d2d 2d2b 0a7c 6932 3020 3a20 523d -----+.|i20 : R= │ │ │ │ -0002c690: 7269 6e67 2858 2920 2020 2020 2020 2020 ring(X) │ │ │ │ -0002c6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c6d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002c630: 2020 2020 207c 0a7c 6f31 3920 3d20 7472 |.|o19 = tr │ │ │ │ +0002c640: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ +0002c650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c680: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0002c690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +0002c6d0: 3020 3a20 523d 7269 6e67 2858 2920 2020 0 : R=ring(X) │ │ │ │ 0002c6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c710: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0002c720: 3020 3d20 5220 2020 2020 2020 2020 2020 0 = R │ │ │ │ +0002c710: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002c720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c760: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002c760: 207c 0a7c 6f32 3020 3d20 5220 2020 2020 |.|o20 = R │ │ │ │ 0002c770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c7b0: 207c 0a7c 6f32 3020 3a20 506f 6c79 6e6f |.|o20 : Polyno │ │ │ │ -0002c7c0: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +0002c7a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002c7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c7f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0002c800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c840: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 -------+.|i21 : │ │ │ │ -0002c850: 493d 6964 6561 6c28 525f 305e 342a 525f I=ideal(R_0^4*R_ │ │ │ │ -0002c860: 312c 525f 302a 525f 332a 525f 342a 525f 1,R_0*R_3*R_4*R_ │ │ │ │ -0002c870: 322d 525f 325e 322a 525f 305e 3229 2020 2-R_2^2*R_0^2) │ │ │ │ -0002c880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c890: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002c8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c8d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002c8e0: 2020 2020 2020 2020 2020 2020 2020 3420 4 │ │ │ │ -0002c8f0: 2020 2020 2020 3220 3220 2020 2020 2020 2 2 │ │ │ │ +0002c7f0: 2020 2020 2020 207c 0a7c 6f32 3020 3a20 |.|o20 : │ │ │ │ +0002c800: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ +0002c810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c840: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0002c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0002c890: 6932 3120 3a20 493d 6964 6561 6c28 525f i21 : I=ideal(R_ │ │ │ │ +0002c8a0: 305e 342a 525f 312c 525f 302a 525f 332a 0^4*R_1,R_0*R_3* │ │ │ │ +0002c8b0: 525f 342a 525f 322d 525f 325e 322a 525f R_4*R_2-R_2^2*R_ │ │ │ │ +0002c8c0: 305e 3229 2020 2020 2020 2020 2020 2020 0^2) │ │ │ │ +0002c8d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002c8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c920: 2020 2020 2020 2020 7c0a 7c6f 3231 203d |.|o21 = │ │ │ │ -0002c930: 2069 6465 616c 2028 7820 7820 2c20 2d20 ideal (x x , - │ │ │ │ -0002c940: 7820 7820 202b 2078 2078 2078 2078 2029 x x + x x x x ) │ │ │ │ +0002c920: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002c930: 2020 2020 3420 2020 2020 2020 3220 3220 4 2 2 │ │ │ │ +0002c940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c970: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002c980: 2020 2020 3020 3120 2020 2020 3020 3220 0 1 0 2 │ │ │ │ -0002c990: 2020 2030 2032 2033 2034 2020 2020 2020 0 2 3 4 │ │ │ │ +0002c960: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c970: 7c6f 3231 203d 2069 6465 616c 2028 7820 |o21 = ideal (x │ │ │ │ +0002c980: 7820 2c20 2d20 7820 7820 202b 2078 2078 x , - x x + x x │ │ │ │ +0002c990: 2078 2078 2029 2020 2020 2020 2020 2020 x x ) │ │ │ │ 0002c9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c9b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002c9c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002c9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c9b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002c9c0: 2020 2020 2020 2020 2020 3020 3120 2020 0 1 │ │ │ │ +0002c9d0: 2020 3020 3220 2020 2030 2032 2033 2034 0 2 0 2 3 4 │ │ │ │ 0002c9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ca00: 2020 2020 2020 2020 207c 0a7c 6f32 3120 |.|o21 │ │ │ │ -0002ca10: 3a20 4964 6561 6c20 6f66 2052 2020 2020 : Ideal of R │ │ │ │ +0002ca00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002ca10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ca20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ca30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ca40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ca50: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002caa0: 0a7c 6932 3220 3a20 5365 6772 6528 582c .|i22 : Segre(X, │ │ │ │ -0002cab0: 4929 2020 2020 2020 2020 2020 2020 2020 I) │ │ │ │ -0002cac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0002caf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ca40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002ca50: 0a7c 6f32 3120 3a20 4964 6561 6c20 6f66 .|o21 : Ideal of │ │ │ │ +0002ca60: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0002ca70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ca80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ca90: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002caa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002cab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002cac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002cad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002cae0: 2d2d 2d2d 2d2b 0a7c 6932 3220 3a20 5365 -----+.|i22 : Se │ │ │ │ +0002caf0: 6772 6528 582c 4929 2020 2020 2020 2020 gre(X,I) │ │ │ │ 0002cb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cb30: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0002cb40: 2020 2032 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ +0002cb30: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002cb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cb80: 7c0a 7c6f 3232 203d 202d 2037 3278 2078 |.|o22 = - 72x x │ │ │ │ -0002cb90: 2020 2b20 3378 2020 2b20 3878 2078 2020 + 3x + 8x x │ │ │ │ -0002cba0: 2b20 7820 2020 2020 2020 2020 2020 2020 + x │ │ │ │ +0002cb70: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002cb80: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +0002cb90: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002cba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cbc0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002cbd0: 2020 2020 2020 2020 2033 2034 2020 2020 3 4 │ │ │ │ -0002cbe0: 2033 2020 2020 2033 2034 2020 2020 3320 3 3 4 3 │ │ │ │ +0002cbc0: 2020 2020 2020 7c0a 7c6f 3232 203d 202d |.|o22 = - │ │ │ │ +0002cbd0: 2037 3278 2078 2020 2b20 3378 2020 2b20 72x x + 3x + │ │ │ │ +0002cbe0: 3878 2078 2020 2b20 7820 2020 2020 2020 8x x + x │ │ │ │ 0002cbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cc10: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0002cc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cc10: 207c 0a7c 2020 2020 2020 2020 2020 2033 |.| 3 │ │ │ │ +0002cc20: 2034 2020 2020 2033 2020 2020 2033 2034 4 3 3 4 │ │ │ │ +0002cc30: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ 0002cc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cc60: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002cc70: 2020 2020 2020 2020 2020 5a5a 5b78 202e ZZ[x . │ │ │ │ -0002cc80: 2e78 205d 2020 2020 2020 2020 2020 2020 .x ] │ │ │ │ +0002cc50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002cc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002cca0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0002ccb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ccc0: 2020 2020 2020 2020 2030 2020 2034 2020 0 4 │ │ │ │ +0002ccc0: 5a5a 5b78 202e 2e78 205d 2020 2020 2020 ZZ[x ..x ] │ │ │ │ 0002ccd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ccf0: 2020 2020 2020 207c 0a7c 6f32 3220 3a20 |.|o22 : │ │ │ │ -0002cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cd20: 2d2d 2d2d 2d2d 2d2d 2d20 2020 2020 2020 --------- │ │ │ │ -0002cd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cd40: 2020 7c0a 7c20 2020 2020 2028 7820 7820 |.| (x x │ │ │ │ -0002cd50: 2c20 7820 7820 7820 2c20 7820 202d 2078 , x x x , x - x │ │ │ │ -0002cd60: 202c 2078 2020 2d20 7820 2c20 7820 202d , x - x , x - │ │ │ │ -0002cd70: 2078 2029 2020 2020 2020 2020 2020 2020 x ) │ │ │ │ -0002cd80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002cd90: 2020 2020 2020 2020 3220 3420 2020 3020 2 4 0 │ │ │ │ -0002cda0: 3120 3320 2020 3020 2020 2033 2020 2031 1 3 0 3 1 │ │ │ │ -0002cdb0: 2020 2020 3320 2020 3220 2020 2034 2020 3 2 4 │ │ │ │ +0002ccf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002cd00: 2020 2020 2020 2020 2020 2020 2020 2030 0 │ │ │ │ +0002cd10: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +0002cd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cd30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002cd40: 6f32 3220 3a20 2d2d 2d2d 2d2d 2d2d 2d2d o22 : ---------- │ │ │ │ +0002cd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002cd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d20 --------------- │ │ │ │ +0002cd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cd80: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002cd90: 2028 7820 7820 2c20 7820 7820 7820 2c20 (x x , x x x , │ │ │ │ +0002cda0: 7820 202d 2078 202c 2078 2020 2d20 7820 x - x , x - x │ │ │ │ +0002cdb0: 2c20 7820 202d 2078 2029 2020 2020 2020 , x - x ) │ │ │ │ 0002cdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cdd0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0002cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ce20: 2d2d 2d2b 0a7c 6932 3320 3a20 4368 3d54 ---+.|i23 : Ch=T │ │ │ │ -0002ce30: 6f72 6963 4368 6f77 5269 6e67 2858 2920 oricChowRing(X) │ │ │ │ -0002ce40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ce50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ce60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002ce70: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002ce80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cdd0: 2020 207c 0a7c 2020 2020 2020 2020 3220 |.| 2 │ │ │ │ +0002cde0: 3420 2020 3020 3120 3320 2020 3020 2020 4 0 1 3 0 │ │ │ │ +0002cdf0: 2033 2020 2031 2020 2020 3320 2020 3220 3 1 3 2 │ │ │ │ +0002ce00: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +0002ce10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002ce20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ce40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ce50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ce60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3320 ---------+.|i23 │ │ │ │ +0002ce70: 3a20 4368 3d54 6f72 6963 4368 6f77 5269 : Ch=ToricChowRi │ │ │ │ +0002ce80: 6e67 2858 2920 2020 2020 2020 2020 2020 ng(X) │ │ │ │ 0002ce90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ceb0: 2020 2020 2020 2020 207c 0a7c 6f32 3320 |.|o23 │ │ │ │ -0002cec0: 3d20 4368 2020 2020 2020 2020 2020 2020 = Ch │ │ │ │ +0002ceb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002cec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ced0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cf00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002cef0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002cf00: 0a7c 6f32 3320 3d20 4368 2020 2020 2020 .|o23 = Ch │ │ │ │ 0002cf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cf30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cf40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002cf50: 0a7c 6f32 3320 3a20 5175 6f74 6965 6e74 .|o23 : Quotient │ │ │ │ -0002cf60: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ +0002cf40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002cf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cf90: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -0002cfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cfe0: 2d2d 2d2d 2d2b 0a7c 6932 3420 3a20 7333 -----+.|i24 : s3 │ │ │ │ -0002cff0: 3d53 6567 7265 2843 682c 582c 4929 2020 =Segre(Ch,X,I) │ │ │ │ -0002d000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d030: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002d040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cf90: 2020 2020 207c 0a7c 6f32 3320 3a20 5175 |.|o23 : Qu │ │ │ │ +0002cfa0: 6f74 6965 6e74 5269 6e67 2020 2020 2020 otientRing │ │ │ │ +0002cfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002cfe0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0002cff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +0002d030: 3420 3a20 7333 3d53 6567 7265 2843 682c 4 : s3=Segre(Ch, │ │ │ │ +0002d040: 582c 4929 2020 2020 2020 2020 2020 2020 X,I) │ │ │ │ 0002d050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d070: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002d080: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -0002d090: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002d070: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002d080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d0c0: 2020 2020 2020 7c0a 7c6f 3234 203d 202d |.|o24 = - │ │ │ │ -0002d0d0: 2037 3278 2078 2020 2b20 3378 2020 2b20 72x x + 3x + │ │ │ │ -0002d0e0: 3878 2078 2020 2b20 7820 2020 2020 2020 8x x + x │ │ │ │ +0002d0c0: 207c 0a7c 2020 2020 2020 2020 2020 2032 |.| 2 │ │ │ │ +0002d0d0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0002d0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d110: 207c 0a7c 2020 2020 2020 2020 2020 2033 |.| 3 │ │ │ │ -0002d120: 2034 2020 2020 2033 2020 2020 2033 2034 4 3 3 4 │ │ │ │ -0002d130: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ +0002d100: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0002d110: 3234 203d 202d 2037 3278 2078 2020 2b20 24 = - 72x x + │ │ │ │ +0002d120: 3378 2020 2b20 3878 2078 2020 2b20 7820 3x + 8x x + x │ │ │ │ +0002d130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d150: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002d160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d150: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002d160: 2020 2020 2033 2034 2020 2020 2033 2020 3 4 3 │ │ │ │ +0002d170: 2020 2033 2034 2020 2020 3320 2020 2020 3 4 3 │ │ │ │ 0002d180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d1a0: 2020 2020 2020 207c 0a7c 6f32 3420 3a20 |.|o24 : │ │ │ │ -0002d1b0: 4368 2020 2020 2020 2020 2020 2020 2020 Ch │ │ │ │ +0002d1a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002d1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d1f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -0002d200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -0002d240: 416c 6c20 7468 6520 6578 616d 706c 6573 All the examples │ │ │ │ -0002d250: 2077 6572 6520 646f 6e65 2075 7369 6e67 were done using │ │ │ │ -0002d260: 2073 796d 626f 6c69 6320 636f 6d70 7574 symbolic comput │ │ │ │ -0002d270: 6174 696f 6e73 2077 6974 6820 4772 5c22 ations with Gr\" │ │ │ │ -0002d280: 6f62 6e65 7220 6261 7365 732e 0a43 6861 obner bases..Cha │ │ │ │ -0002d290: 6e67 696e 6720 7468 6520 6f70 7469 6f6e nging the option │ │ │ │ -0002d2a0: 202a 6e6f 7465 2043 6f6d 704d 6574 686f *note CompMetho │ │ │ │ -0002d2b0: 643a 2043 6f6d 704d 6574 686f 642c 2074 d: CompMethod, t │ │ │ │ -0002d2c0: 6f20 6265 7274 696e 6920 7769 6c6c 2064 o bertini will d │ │ │ │ -0002d2d0: 6f20 7468 6520 6d61 696e 0a63 6f6d 7075 o the main.compu │ │ │ │ -0002d2e0: 7461 7469 6f6e 7320 6e75 6d65 7269 6361 tations numerica │ │ │ │ -0002d2f0: 6c6c 792c 2070 726f 7669 6465 6420 4265 lly, provided Be │ │ │ │ -0002d300: 7274 696e 6920 6973 2020 2a6e 6f74 6520 rtini is *note │ │ │ │ -0002d310: 696e 7374 616c 6c65 6420 616e 6420 636f installed and co │ │ │ │ -0002d320: 6e66 6967 7572 6564 3a0a 636f 6e66 6967 nfigured:.config │ │ │ │ -0002d330: 7572 696e 6720 4265 7274 696e 692c 2e20 uring Bertini,. │ │ │ │ -0002d340: 4e6f 7465 2074 6861 7420 7468 6520 6265 Note that the be │ │ │ │ -0002d350: 7274 696e 6920 6f70 7469 6f6e 2069 7320 rtini option is │ │ │ │ -0002d360: 6f6e 6c79 2061 7661 696c 6162 6c65 2066 only available f │ │ │ │ -0002d370: 6f72 0a73 7562 7363 6865 6d65 7320 6f66 or.subschemes of │ │ │ │ -0002d380: 205c 5050 5e6e 2e0a 0a4f 6273 6572 7665 \PP^n...Observe │ │ │ │ -0002d390: 2074 6861 7420 7468 6520 616c 676f 7269 that the algori │ │ │ │ -0002d3a0: 7468 6d20 6973 2061 2070 726f 6261 6269 thm is a probabi │ │ │ │ -0002d3b0: 6c69 7374 6963 2061 6c67 6f72 6974 686d listic algorithm │ │ │ │ -0002d3c0: 2061 6e64 206d 6179 2067 6976 6520 6120 and may give a │ │ │ │ -0002d3d0: 7772 6f6e 670a 616e 7377 6572 2077 6974 wrong.answer wit │ │ │ │ -0002d3e0: 6820 6120 736d 616c 6c20 6275 7420 6e6f h a small but no │ │ │ │ -0002d3f0: 6e7a 6572 6f20 7072 6f62 6162 696c 6974 nzero probabilit │ │ │ │ -0002d400: 792e 2052 6561 6420 6d6f 7265 2075 6e64 y. Read more und │ │ │ │ -0002d410: 6572 202a 6e6f 7465 0a70 726f 6261 6269 er *note.probabi │ │ │ │ -0002d420: 6c69 7374 6963 2061 6c67 6f72 6974 686d listic algorithm │ │ │ │ -0002d430: 3a20 7072 6f62 6162 696c 6973 7469 6320 : probabilistic │ │ │ │ -0002d440: 616c 676f 7269 7468 6d2c 2e0a 0a57 6179 algorithm,...Way │ │ │ │ -0002d450: 7320 746f 2075 7365 2053 6567 7265 3a0a s to use Segre:. │ │ │ │ -0002d460: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002d470: 3d3d 0a0a 2020 2a20 2253 6567 7265 2849 ==.. * "Segre(I │ │ │ │ -0002d480: 6465 616c 2922 0a20 202a 2022 5365 6772 deal)". * "Segr │ │ │ │ -0002d490: 6528 4964 6561 6c2c 5379 6d62 6f6c 2922 e(Ideal,Symbol)" │ │ │ │ -0002d4a0: 0a20 202a 2022 5365 6772 6528 5175 6f74 . * "Segre(Quot │ │ │ │ -0002d4b0: 6965 6e74 5269 6e67 2c49 6465 616c 2922 ientRing,Ideal)" │ │ │ │ -0002d4c0: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -0002d4d0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -0002d4e0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -0002d4f0: 6563 7420 2a6e 6f74 6520 5365 6772 653a ect *note Segre: │ │ │ │ -0002d500: 2053 6567 7265 2c20 6973 2061 202a 6e6f Segre, is a *no │ │ │ │ -0002d510: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ -0002d520: 6f6e 2077 6974 6820 6f70 7469 6f6e 733a on with options: │ │ │ │ -0002d530: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ -0002d540: 6574 686f 6446 756e 6374 696f 6e57 6974 ethodFunctionWit │ │ │ │ -0002d550: 684f 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d hOptions,...---- │ │ │ │ -0002d560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -0002d5b0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -0002d5c0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -0002d5d0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -0002d5e0: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -0002d5f0: 322d 312e 3236 2e30 352b 6473 2f4d 322f 2-1.26.05+ds/M2/ │ │ │ │ -0002d600: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -0002d610: 6573 2f0a 4368 6172 6163 7465 7269 7374 es/.Characterist │ │ │ │ -0002d620: 6963 436c 6173 7365 732e 6d32 3a31 3736 icClasses.m2:176 │ │ │ │ -0002d630: 333a 302e 0a1f 0a46 696c 653a 2043 6861 3:0....File: Cha │ │ │ │ -0002d640: 7261 6374 6572 6973 7469 6343 6c61 7373 racteristicClass │ │ │ │ -0002d650: 6573 2e69 6e66 6f2c 204e 6f64 653a 2054 es.info, Node: T │ │ │ │ -0002d660: 6f72 6963 4368 6f77 5269 6e67 2c20 5072 oricChowRing, Pr │ │ │ │ -0002d670: 6576 3a20 5365 6772 652c 2055 703a 2054 ev: Segre, Up: T │ │ │ │ -0002d680: 6f70 0a0a 546f 7269 6343 686f 7752 696e op..ToricChowRin │ │ │ │ -0002d690: 6720 2d2d 2043 6f6d 7075 7465 7320 7468 g -- Computes th │ │ │ │ -0002d6a0: 6520 4368 6f77 2072 696e 6720 6f66 2061 e Chow ring of a │ │ │ │ -0002d6b0: 206e 6f72 6d61 6c20 746f 7269 6320 7661 normal toric va │ │ │ │ -0002d6c0: 7269 6574 790a 2a2a 2a2a 2a2a 2a2a 2a2a riety.********** │ │ │ │ -0002d6d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002d6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002d6f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002d700: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ -0002d710: 6765 3a20 0a20 2020 2020 2020 2054 6f72 ge: . Tor │ │ │ │ -0002d720: 6963 4368 6f77 5269 6e67 2058 0a20 202a icChowRing X. * │ │ │ │ -0002d730: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -0002d740: 2052 2c20 6120 2a6e 6f74 6520 6e6f 726d R, a *note norm │ │ │ │ -0002d750: 616c 2074 6f72 6963 2076 6172 6965 7479 al toric variety │ │ │ │ -0002d760: 3a0a 2020 2020 2020 2020 284e 6f72 6d61 :. (Norma │ │ │ │ -0002d770: 6c54 6f72 6963 5661 7269 6574 6965 7329 lToricVarieties) │ │ │ │ -0002d780: 4e6f 726d 616c 546f 7269 6356 6172 6965 NormalToricVarie │ │ │ │ -0002d790: 7479 2c2c 2041 206e 6f72 6d61 6c20 746f ty,, A normal to │ │ │ │ -0002d7a0: 7269 6320 7661 7269 6574 790a 2020 2a20 ric variety. * │ │ │ │ -0002d7b0: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -0002d7c0: 2061 202a 6e6f 7465 2071 756f 7469 656e a *note quotien │ │ │ │ -0002d7d0: 7420 7269 6e67 3a20 284d 6163 6175 6c61 t ring: (Macaula │ │ │ │ -0002d7e0: 7932 446f 6329 5175 6f74 6965 6e74 5269 y2Doc)QuotientRi │ │ │ │ -0002d7f0: 6e67 2c2c 200a 0a44 6573 6372 6970 7469 ng,, ..Descripti │ │ │ │ -0002d800: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -0002d810: 4c65 7420 5820 6265 2061 2074 6f72 6963 Let X be a toric │ │ │ │ -0002d820: 2076 6172 6965 7479 2077 6974 6820 746f variety with to │ │ │ │ -0002d830: 7461 6c20 636f 6f72 6469 6e61 7465 2072 tal coordinate r │ │ │ │ -0002d840: 696e 6720 2843 6f78 2072 696e 6729 2052 ing (Cox ring) R │ │ │ │ -0002d850: 2e20 5468 6973 206d 6574 686f 640a 636f . This method.co │ │ │ │ -0002d860: 6d70 7574 6573 2074 6865 2043 686f 7720 mputes the Chow │ │ │ │ -0002d870: 7269 6e67 2020 4368 6f77 2072 696e 6720 ring Chow ring │ │ │ │ -0002d880: 4368 3d52 2f28 5352 2b4c 5229 206f 6620 Ch=R/(SR+LR) of │ │ │ │ -0002d890: 583b 2068 6572 6520 5352 2069 7320 7468 X; here SR is th │ │ │ │ -0002d8a0: 650a 5374 616e 6c65 792d 5265 6973 6e65 e.Stanley-Reisne │ │ │ │ -0002d8b0: 7220 6964 6561 6c20 6f66 2074 6865 2063 r ideal of the c │ │ │ │ -0002d8c0: 6f72 7265 7370 6f6e 6469 6e67 2066 616e orresponding fan │ │ │ │ -0002d8d0: 2061 6e64 204c 5220 6973 2074 6865 2069 and LR is the i │ │ │ │ -0002d8e0: 6465 616c 206f 6620 6c69 6e65 6172 0a72 deal of linear.r │ │ │ │ -0002d8f0: 656c 6174 696f 6e73 2061 6d6f 756e 7420 elations amount │ │ │ │ -0002d900: 7468 6520 7261 7973 2e20 4974 2069 7320 the rays. It is │ │ │ │ -0002d910: 6e65 6564 6564 2066 6f72 2069 6e70 7574 needed for input │ │ │ │ -0002d920: 2069 6e74 6f20 7468 6520 6d65 7468 6f64 into the method │ │ │ │ -0002d930: 7320 2a6e 6f74 6520 5365 6772 653a 0a53 s *note Segre:.S │ │ │ │ -0002d940: 6567 7265 2c2c 202a 6e6f 7465 2043 6865 egre,, *note Che │ │ │ │ -0002d950: 726e 3a20 4368 6572 6e2c 2061 6e64 202a rn: Chern, and * │ │ │ │ -0002d960: 6e6f 7465 2043 534d 3a20 4353 4d2c 2069 note CSM: CSM, i │ │ │ │ -0002d970: 6e20 7468 6520 6361 7365 7320 7768 6572 n the cases wher │ │ │ │ -0002d980: 6520 6120 746f 7269 630a 7661 7269 6574 e a toric.variet │ │ │ │ -0002d990: 7920 6973 2061 6c73 6f20 696e 7075 7420 y is also input │ │ │ │ -0002d9a0: 746f 2065 6e73 7572 6520 7468 6174 2074 to ensure that t │ │ │ │ -0002d9b0: 6865 7365 206d 6574 686f 6473 2072 6574 hese methods ret │ │ │ │ -0002d9c0: 7572 6e20 7265 7375 6c74 7320 696e 2074 urn results in t │ │ │ │ -0002d9d0: 6865 2073 616d 650a 7269 6e67 2e20 5765 he same.ring. We │ │ │ │ -0002d9e0: 2067 6976 6520 616e 2065 7861 6d70 6c65 give an example │ │ │ │ -0002d9f0: 206f 6620 7468 6520 7573 6520 6f66 2074 of the use of t │ │ │ │ -0002da00: 6869 7320 6d65 7468 6f64 2074 6f20 776f his method to wo │ │ │ │ -0002da10: 726b 2077 6974 6820 656c 656d 656e 7473 rk with elements │ │ │ │ -0002da20: 206f 6620 7468 650a 4368 6f77 2072 696e of the.Chow rin │ │ │ │ -0002da30: 6720 6f66 2061 2074 6f72 6963 2076 6172 g of a toric var │ │ │ │ -0002da40: 6965 7479 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d iety..+--------- │ │ │ │ -0002da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002da60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002da90: 2d2d 2d2d 2b0a 7c69 3120 3a20 6e65 6564 ----+.|i1 : need │ │ │ │ -0002daa0: 7350 6163 6b61 6765 2022 4e6f 726d 616c sPackage "Normal │ │ │ │ -0002dab0: 546f 7269 6356 6172 6965 7469 6573 2220 ToricVarieties" │ │ │ │ -0002dac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002daf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d1e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002d1f0: 6f32 3420 3a20 4368 2020 2020 2020 2020 o24 : Ch │ │ │ │ +0002d200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d230: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0002d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d280: 2d2d 2d2b 0a0a 416c 6c20 7468 6520 6578 ---+..All the ex │ │ │ │ +0002d290: 616d 706c 6573 2077 6572 6520 646f 6e65 amples were done │ │ │ │ +0002d2a0: 2075 7369 6e67 2073 796d 626f 6c69 6320 using symbolic │ │ │ │ +0002d2b0: 636f 6d70 7574 6174 696f 6e73 2077 6974 computations wit │ │ │ │ +0002d2c0: 6820 4772 5c22 6f62 6e65 7220 6261 7365 h Gr\"obner base │ │ │ │ +0002d2d0: 732e 0a43 6861 6e67 696e 6720 7468 6520 s..Changing the │ │ │ │ +0002d2e0: 6f70 7469 6f6e 202a 6e6f 7465 2043 6f6d option *note Com │ │ │ │ +0002d2f0: 704d 6574 686f 643a 2043 6f6d 704d 6574 pMethod: CompMet │ │ │ │ +0002d300: 686f 642c 2074 6f20 6265 7274 696e 6920 hod, to bertini │ │ │ │ +0002d310: 7769 6c6c 2064 6f20 7468 6520 6d61 696e will do the main │ │ │ │ +0002d320: 0a63 6f6d 7075 7461 7469 6f6e 7320 6e75 .computations nu │ │ │ │ +0002d330: 6d65 7269 6361 6c6c 792c 2070 726f 7669 merically, provi │ │ │ │ +0002d340: 6465 6420 4265 7274 696e 6920 6973 2020 ded Bertini is │ │ │ │ +0002d350: 2a6e 6f74 6520 696e 7374 616c 6c65 6420 *note installed │ │ │ │ +0002d360: 616e 6420 636f 6e66 6967 7572 6564 3a0a and configured:. │ │ │ │ +0002d370: 636f 6e66 6967 7572 696e 6720 4265 7274 configuring Bert │ │ │ │ +0002d380: 696e 692c 2e20 4e6f 7465 2074 6861 7420 ini,. Note that │ │ │ │ +0002d390: 7468 6520 6265 7274 696e 6920 6f70 7469 the bertini opti │ │ │ │ +0002d3a0: 6f6e 2069 7320 6f6e 6c79 2061 7661 696c on is only avail │ │ │ │ +0002d3b0: 6162 6c65 2066 6f72 0a73 7562 7363 6865 able for.subsche │ │ │ │ +0002d3c0: 6d65 7320 6f66 205c 5050 5e6e 2e0a 0a4f mes of \PP^n...O │ │ │ │ +0002d3d0: 6273 6572 7665 2074 6861 7420 7468 6520 bserve that the │ │ │ │ +0002d3e0: 616c 676f 7269 7468 6d20 6973 2061 2070 algorithm is a p │ │ │ │ +0002d3f0: 726f 6261 6269 6c69 7374 6963 2061 6c67 robabilistic alg │ │ │ │ +0002d400: 6f72 6974 686d 2061 6e64 206d 6179 2067 orithm and may g │ │ │ │ +0002d410: 6976 6520 6120 7772 6f6e 670a 616e 7377 ive a wrong.answ │ │ │ │ +0002d420: 6572 2077 6974 6820 6120 736d 616c 6c20 er with a small │ │ │ │ +0002d430: 6275 7420 6e6f 6e7a 6572 6f20 7072 6f62 but nonzero prob │ │ │ │ +0002d440: 6162 696c 6974 792e 2052 6561 6420 6d6f ability. Read mo │ │ │ │ +0002d450: 7265 2075 6e64 6572 202a 6e6f 7465 0a70 re under *note.p │ │ │ │ +0002d460: 726f 6261 6269 6c69 7374 6963 2061 6c67 robabilistic alg │ │ │ │ +0002d470: 6f72 6974 686d 3a20 7072 6f62 6162 696c orithm: probabil │ │ │ │ +0002d480: 6973 7469 6320 616c 676f 7269 7468 6d2c istic algorithm, │ │ │ │ +0002d490: 2e0a 0a57 6179 7320 746f 2075 7365 2053 ...Ways to use S │ │ │ │ +0002d4a0: 6567 7265 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d egre:.========== │ │ │ │ +0002d4b0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2253 ========.. * "S │ │ │ │ +0002d4c0: 6567 7265 2849 6465 616c 2922 0a20 202a egre(Ideal)". * │ │ │ │ +0002d4d0: 2022 5365 6772 6528 4964 6561 6c2c 5379 "Segre(Ideal,Sy │ │ │ │ +0002d4e0: 6d62 6f6c 2922 0a20 202a 2022 5365 6772 mbol)". * "Segr │ │ │ │ +0002d4f0: 6528 5175 6f74 6965 6e74 5269 6e67 2c49 e(QuotientRing,I │ │ │ │ +0002d500: 6465 616c 2922 0a0a 466f 7220 7468 6520 deal)"..For the │ │ │ │ +0002d510: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ +0002d520: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +0002d530: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +0002d540: 5365 6772 653a 2053 6567 7265 2c20 6973 Segre: Segre, is │ │ │ │ +0002d550: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ +0002d560: 6675 6e63 7469 6f6e 2077 6974 6820 6f70 function with op │ │ │ │ +0002d570: 7469 6f6e 733a 0a28 4d61 6361 756c 6179 tions:.(Macaulay │ │ │ │ +0002d580: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ +0002d590: 696f 6e57 6974 684f 7074 696f 6e73 2c2e ionWithOptions,. │ │ │ │ +0002d5a0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +0002d5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d5f0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +0002d600: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +0002d610: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +0002d620: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +0002d630: 6361 756c 6179 322d 312e 3236 2e30 352b caulay2-1.26.05+ │ │ │ │ +0002d640: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +0002d650: 7061 636b 6167 6573 2f0a 4368 6172 6163 packages/.Charac │ │ │ │ +0002d660: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ +0002d670: 6d32 3a31 3736 333a 302e 0a1f 0a46 696c m2:1763:0....Fil │ │ │ │ +0002d680: 653a 2043 6861 7261 6374 6572 6973 7469 e: Characteristi │ │ │ │ +0002d690: 6343 6c61 7373 6573 2e69 6e66 6f2c 204e cClasses.info, N │ │ │ │ +0002d6a0: 6f64 653a 2054 6f72 6963 4368 6f77 5269 ode: ToricChowRi │ │ │ │ +0002d6b0: 6e67 2c20 5072 6576 3a20 5365 6772 652c ng, Prev: Segre, │ │ │ │ +0002d6c0: 2055 703a 2054 6f70 0a0a 546f 7269 6343 Up: Top..ToricC │ │ │ │ +0002d6d0: 686f 7752 696e 6720 2d2d 2043 6f6d 7075 howRing -- Compu │ │ │ │ +0002d6e0: 7465 7320 7468 6520 4368 6f77 2072 696e tes the Chow rin │ │ │ │ +0002d6f0: 6720 6f66 2061 206e 6f72 6d61 6c20 746f g of a normal to │ │ │ │ +0002d700: 7269 6320 7661 7269 6574 790a 2a2a 2a2a ric variety.**** │ │ │ │ +0002d710: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002d720: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002d730: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002d740: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ +0002d750: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ +0002d760: 2020 2054 6f72 6963 4368 6f77 5269 6e67 ToricChowRing │ │ │ │ +0002d770: 2058 0a20 202a 2049 6e70 7574 733a 0a20 X. * Inputs:. │ │ │ │ +0002d780: 2020 2020 202a 2052 2c20 6120 2a6e 6f74 * R, a *not │ │ │ │ +0002d790: 6520 6e6f 726d 616c 2074 6f72 6963 2076 e normal toric v │ │ │ │ +0002d7a0: 6172 6965 7479 3a0a 2020 2020 2020 2020 ariety:. │ │ │ │ +0002d7b0: 284e 6f72 6d61 6c54 6f72 6963 5661 7269 (NormalToricVari │ │ │ │ +0002d7c0: 6574 6965 7329 4e6f 726d 616c 546f 7269 eties)NormalTori │ │ │ │ +0002d7d0: 6356 6172 6965 7479 2c2c 2041 206e 6f72 cVariety,, A nor │ │ │ │ +0002d7e0: 6d61 6c20 746f 7269 6320 7661 7269 6574 mal toric variet │ │ │ │ +0002d7f0: 790a 2020 2a20 4f75 7470 7574 733a 0a20 y. * Outputs:. │ │ │ │ +0002d800: 2020 2020 202a 2061 202a 6e6f 7465 2071 * a *note q │ │ │ │ +0002d810: 756f 7469 656e 7420 7269 6e67 3a20 284d uotient ring: (M │ │ │ │ +0002d820: 6163 6175 6c61 7932 446f 6329 5175 6f74 acaulay2Doc)Quot │ │ │ │ +0002d830: 6965 6e74 5269 6e67 2c2c 200a 0a44 6573 ientRing,, ..Des │ │ │ │ +0002d840: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +0002d850: 3d3d 3d3d 0a0a 4c65 7420 5820 6265 2061 ====..Let X be a │ │ │ │ +0002d860: 2074 6f72 6963 2076 6172 6965 7479 2077 toric variety w │ │ │ │ +0002d870: 6974 6820 746f 7461 6c20 636f 6f72 6469 ith total coordi │ │ │ │ +0002d880: 6e61 7465 2072 696e 6720 2843 6f78 2072 nate ring (Cox r │ │ │ │ +0002d890: 696e 6729 2052 2e20 5468 6973 206d 6574 ing) R. This met │ │ │ │ +0002d8a0: 686f 640a 636f 6d70 7574 6573 2074 6865 hod.computes the │ │ │ │ +0002d8b0: 2043 686f 7720 7269 6e67 2020 4368 6f77 Chow ring Chow │ │ │ │ +0002d8c0: 2072 696e 6720 4368 3d52 2f28 5352 2b4c ring Ch=R/(SR+L │ │ │ │ +0002d8d0: 5229 206f 6620 583b 2068 6572 6520 5352 R) of X; here SR │ │ │ │ +0002d8e0: 2069 7320 7468 650a 5374 616e 6c65 792d is the.Stanley- │ │ │ │ +0002d8f0: 5265 6973 6e65 7220 6964 6561 6c20 6f66 Reisner ideal of │ │ │ │ +0002d900: 2074 6865 2063 6f72 7265 7370 6f6e 6469 the correspondi │ │ │ │ +0002d910: 6e67 2066 616e 2061 6e64 204c 5220 6973 ng fan and LR is │ │ │ │ +0002d920: 2074 6865 2069 6465 616c 206f 6620 6c69 the ideal of li │ │ │ │ +0002d930: 6e65 6172 0a72 656c 6174 696f 6e73 2061 near.relations a │ │ │ │ +0002d940: 6d6f 756e 7420 7468 6520 7261 7973 2e20 mount the rays. │ │ │ │ +0002d950: 4974 2069 7320 6e65 6564 6564 2066 6f72 It is needed for │ │ │ │ +0002d960: 2069 6e70 7574 2069 6e74 6f20 7468 6520 input into the │ │ │ │ +0002d970: 6d65 7468 6f64 7320 2a6e 6f74 6520 5365 methods *note Se │ │ │ │ +0002d980: 6772 653a 0a53 6567 7265 2c2c 202a 6e6f gre:.Segre,, *no │ │ │ │ +0002d990: 7465 2043 6865 726e 3a20 4368 6572 6e2c te Chern: Chern, │ │ │ │ +0002d9a0: 2061 6e64 202a 6e6f 7465 2043 534d 3a20 and *note CSM: │ │ │ │ +0002d9b0: 4353 4d2c 2069 6e20 7468 6520 6361 7365 CSM, in the case │ │ │ │ +0002d9c0: 7320 7768 6572 6520 6120 746f 7269 630a s where a toric. │ │ │ │ +0002d9d0: 7661 7269 6574 7920 6973 2061 6c73 6f20 variety is also │ │ │ │ +0002d9e0: 696e 7075 7420 746f 2065 6e73 7572 6520 input to ensure │ │ │ │ +0002d9f0: 7468 6174 2074 6865 7365 206d 6574 686f that these metho │ │ │ │ +0002da00: 6473 2072 6574 7572 6e20 7265 7375 6c74 ds return result │ │ │ │ +0002da10: 7320 696e 2074 6865 2073 616d 650a 7269 s in the same.ri │ │ │ │ +0002da20: 6e67 2e20 5765 2067 6976 6520 616e 2065 ng. We give an e │ │ │ │ +0002da30: 7861 6d70 6c65 206f 6620 7468 6520 7573 xample of the us │ │ │ │ +0002da40: 6520 6f66 2074 6869 7320 6d65 7468 6f64 e of this method │ │ │ │ +0002da50: 2074 6f20 776f 726b 2077 6974 6820 656c to work with el │ │ │ │ +0002da60: 656d 656e 7473 206f 6620 7468 650a 4368 ements of the.Ch │ │ │ │ +0002da70: 6f77 2072 696e 6720 6f66 2061 2074 6f72 ow ring of a tor │ │ │ │ +0002da80: 6963 2076 6172 6965 7479 0a0a 2b2d 2d2d ic variety..+--- │ │ │ │ +0002da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002daa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ +0002dae0: 3a20 6e65 6564 7350 6163 6b61 6765 2022 : needsPackage " │ │ │ │ +0002daf0: 4e6f 726d 616c 546f 7269 6356 6172 6965 NormalToricVarie │ │ │ │ +0002db00: 7469 6573 2220 2020 2020 2020 2020 2020 ties" │ │ │ │ 0002db10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db30: 2020 2020 7c0a 7c6f 3120 3d20 4e6f 726d |.|o1 = Norm │ │ │ │ -0002db40: 616c 546f 7269 6356 6172 6965 7469 6573 alToricVarieties │ │ │ │ +0002db20: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002db30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002db40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002db50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002db60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db80: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002db90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002db70: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +0002db80: 3d20 4e6f 726d 616c 546f 7269 6356 6172 = NormalToricVar │ │ │ │ +0002db90: 6965 7469 6573 2020 2020 2020 2020 2020 ieties │ │ │ │ 0002dba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dbd0: 2020 2020 7c0a 7c6f 3120 3a20 5061 636b |.|o1 : Pack │ │ │ │ -0002dbe0: 6167 6520 2020 2020 2020 2020 2020 2020 age │ │ │ │ +0002dbc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002dc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dc70: 2d2d 2d2d 2b0a 7c69 3220 3a20 5268 6f20 ----+.|i2 : Rho │ │ │ │ -0002dc80: 3d20 7b7b 312c 302c 307d 2c7b 302c 312c = {{1,0,0},{0,1, │ │ │ │ -0002dc90: 307d 2c7b 302c 302c 317d 2c7b 2d31 2c2d 0},{0,0,1},{-1,- │ │ │ │ -0002dca0: 312c 307d 2c7b 302c 302c 2d31 7d7d 2020 1,0},{0,0,-1}} │ │ │ │ -0002dcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dcc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002dcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd10: 2020 2020 7c0a 7c6f 3220 3d20 7b7b 312c |.|o2 = {{1, │ │ │ │ -0002dd20: 2030 2c20 307d 2c20 7b30 2c20 312c 2030 0, 0}, {0, 1, 0 │ │ │ │ -0002dd30: 7d2c 207b 302c 2030 2c20 317d 2c20 7b2d }, {0, 0, 1}, {- │ │ │ │ -0002dd40: 312c 202d 312c 2030 7d2c 207b 302c 2030 1, -1, 0}, {0, 0 │ │ │ │ -0002dd50: 2c20 2d31 7d7d 2020 2020 2020 2020 2020 , -1}} │ │ │ │ -0002dd60: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002dd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ddb0: 2020 2020 7c0a 7c6f 3220 3a20 4c69 7374 |.|o2 : List │ │ │ │ +0002dc10: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +0002dc20: 3a20 5061 636b 6167 6520 2020 2020 2020 : Package │ │ │ │ +0002dc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc60: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ +0002dcc0: 3a20 5268 6f20 3d20 7b7b 312c 302c 307d : Rho = {{1,0,0} │ │ │ │ +0002dcd0: 2c7b 302c 312c 307d 2c7b 302c 302c 317d ,{0,1,0},{0,0,1} │ │ │ │ +0002dce0: 2c7b 2d31 2c2d 312c 307d 2c7b 302c 302c ,{-1,-1,0},{0,0, │ │ │ │ +0002dcf0: 2d31 7d7d 2020 2020 2020 2020 2020 2020 -1}} │ │ │ │ +0002dd00: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002dd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dd50: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0002dd60: 3d20 7b7b 312c 2030 2c20 307d 2c20 7b30 = {{1, 0, 0}, {0 │ │ │ │ +0002dd70: 2c20 312c 2030 7d2c 207b 302c 2030 2c20 , 1, 0}, {0, 0, │ │ │ │ +0002dd80: 317d 2c20 7b2d 312c 202d 312c 2030 7d2c 1}, {-1, -1, 0}, │ │ │ │ +0002dd90: 207b 302c 2030 2c20 2d31 7d7d 2020 2020 {0, 0, -1}} │ │ │ │ +0002dda0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002ddb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ddc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ddd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ddf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002de00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002de10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de50: 2d2d 2d2d 2b0a 7c69 3320 3a20 5369 676d ----+.|i3 : Sigm │ │ │ │ -0002de60: 6120 3d20 7b7b 302c 312c 327d 2c7b 312c a = {{0,1,2},{1, │ │ │ │ -0002de70: 322c 337d 2c7b 302c 322c 337d 2c7b 302c 2,3},{0,2,3},{0, │ │ │ │ -0002de80: 312c 347d 2c7b 312c 332c 347d 2c7b 302c 1,4},{1,3,4},{0, │ │ │ │ -0002de90: 332c 347d 7d20 2020 2020 2020 2020 2020 3,4}} │ │ │ │ -0002dea0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002deb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ded0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002def0: 2020 2020 7c0a 7c6f 3320 3d20 7b7b 302c |.|o3 = {{0, │ │ │ │ -0002df00: 2031 2c20 327d 2c20 7b31 2c20 322c 2033 1, 2}, {1, 2, 3 │ │ │ │ -0002df10: 7d2c 207b 302c 2032 2c20 337d 2c20 7b30 }, {0, 2, 3}, {0 │ │ │ │ -0002df20: 2c20 312c 2034 7d2c 207b 312c 2033 2c20 , 1, 4}, {1, 3, │ │ │ │ -0002df30: 347d 2c20 7b30 2c20 332c 2034 7d7d 2020 4}, {0, 3, 4}} │ │ │ │ -0002df40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002df50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df90: 2020 2020 7c0a 7c6f 3320 3a20 4c69 7374 |.|o3 : List │ │ │ │ +0002ddf0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0002de00: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +0002de10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002de20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002de30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002de40: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002de50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002de60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002de70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002de80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002de90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ +0002dea0: 3a20 5369 676d 6120 3d20 7b7b 302c 312c : Sigma = {{0,1, │ │ │ │ +0002deb0: 327d 2c7b 312c 322c 337d 2c7b 302c 322c 2},{1,2,3},{0,2, │ │ │ │ +0002dec0: 337d 2c7b 302c 312c 347d 2c7b 312c 332c 3},{0,1,4},{1,3, │ │ │ │ +0002ded0: 347d 2c7b 302c 332c 347d 7d20 2020 2020 4},{0,3,4}} │ │ │ │ +0002dee0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002def0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002df00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002df10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002df30: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +0002df40: 3d20 7b7b 302c 2031 2c20 327d 2c20 7b31 = {{0, 1, 2}, {1 │ │ │ │ +0002df50: 2c20 322c 2033 7d2c 207b 302c 2032 2c20 , 2, 3}, {0, 2, │ │ │ │ +0002df60: 337d 2c20 7b30 2c20 312c 2034 7d2c 207b 3}, {0, 1, 4}, { │ │ │ │ +0002df70: 312c 2033 2c20 347d 2c20 7b30 2c20 332c 1, 3, 4}, {0, 3, │ │ │ │ +0002df80: 2034 7d7d 2020 2020 2020 7c0a 7c20 2020 4}} |.| │ │ │ │ +0002df90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dfe0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002dff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e030: 2d2d 2d2d 2b0a 7c69 3420 3a20 5820 3d20 ----+.|i4 : X = │ │ │ │ -0002e040: 6e6f 726d 616c 546f 7269 6356 6172 6965 normalToricVarie │ │ │ │ -0002e050: 7479 2852 686f 2c53 6967 6d61 2c43 6f65 ty(Rho,Sigma,Coe │ │ │ │ -0002e060: 6666 6963 6965 6e74 5269 6e67 203d 3e5a fficientRing =>Z │ │ │ │ -0002e070: 5a2f 3332 3734 3929 2020 2020 2020 2020 Z/32749) │ │ │ │ -0002e080: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002e090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e0d0: 2020 2020 7c0a 7c6f 3420 3d20 5820 2020 |.|o4 = X │ │ │ │ +0002dfd0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +0002dfe0: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +0002dff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e020: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e070: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ +0002e080: 3a20 5820 3d20 6e6f 726d 616c 546f 7269 : X = normalTori │ │ │ │ +0002e090: 6356 6172 6965 7479 2852 686f 2c53 6967 cVariety(Rho,Sig │ │ │ │ +0002e0a0: 6d61 2c43 6f65 6666 6963 6965 6e74 5269 ma,CoefficientRi │ │ │ │ +0002e0b0: 6e67 203d 3e5a 5a2f 3332 3734 3929 2020 ng =>ZZ/32749) │ │ │ │ +0002e0c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002e0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e120: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002e110: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ +0002e120: 3d20 5820 2020 2020 2020 2020 2020 2020 = X │ │ │ │ 0002e130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e170: 2020 2020 7c0a 7c6f 3420 3a20 4e6f 726d |.|o4 : Norm │ │ │ │ -0002e180: 616c 546f 7269 6356 6172 6965 7479 2020 alToricVariety │ │ │ │ +0002e160: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002e170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e1c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002e1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e210: 2d2d 2d2d 2b0a 7c69 3520 3a20 523d 7269 ----+.|i5 : R=ri │ │ │ │ -0002e220: 6e67 2058 2020 2020 2020 2020 2020 2020 ng X │ │ │ │ -0002e230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e260: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002e1b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ +0002e1c0: 3a20 4e6f 726d 616c 546f 7269 6356 6172 : NormalToricVar │ │ │ │ +0002e1d0: 6965 7479 2020 2020 2020 2020 2020 2020 iety │ │ │ │ +0002e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e200: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002e210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ +0002e260: 3a20 523d 7269 6e67 2058 2020 2020 2020 : R=ring X │ │ │ │ 0002e270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e2b0: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020 |.|o5 = R │ │ │ │ +0002e2a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002e2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e300: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002e2f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ +0002e300: 3d20 5220 2020 2020 2020 2020 2020 2020 = R │ │ │ │ 0002e310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e350: 2020 2020 7c0a 7c6f 3520 3a20 506f 6c79 |.|o5 : Poly │ │ │ │ -0002e360: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ +0002e340: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e3a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e3f0: 2d2d 2d2d 2b0a 7c69 3620 3a20 4368 3d54 ----+.|i6 : Ch=T │ │ │ │ -0002e400: 6f72 6963 4368 6f77 5269 6e67 2858 2920 oricChowRing(X) │ │ │ │ -0002e410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e440: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002e450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e390: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ +0002e3a0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +0002e3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ +0002e440: 3a20 4368 3d54 6f72 6963 4368 6f77 5269 : Ch=ToricChowRi │ │ │ │ +0002e450: 6e67 2858 2920 2020 2020 2020 2020 2020 ng(X) │ │ │ │ 0002e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e490: 2020 2020 7c0a 7c6f 3620 3d20 4368 2020 |.|o6 = Ch │ │ │ │ +0002e480: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002e4d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ +0002e4e0: 3d20 4368 2020 2020 2020 2020 2020 2020 = Ch │ │ │ │ 0002e4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e530: 2020 2020 7c0a 7c6f 3620 3a20 5175 6f74 |.|o6 : Quot │ │ │ │ -0002e540: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +0002e520: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002e530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e580: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002e590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e5d0: 2d2d 2d2d 2b0a 7c69 3720 3a20 6465 7363 ----+.|i7 : desc │ │ │ │ -0002e5e0: 7269 6265 2043 6820 2020 2020 2020 2020 ribe Ch │ │ │ │ -0002e5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e620: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002e570: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ +0002e580: 3a20 5175 6f74 6965 6e74 5269 6e67 2020 : QuotientRing │ │ │ │ +0002e590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e5c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002e5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e610: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 ----------+.|i7 │ │ │ │ +0002e620: 3a20 6465 7363 7269 6265 2043 6820 2020 : describe Ch │ │ │ │ 0002e630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e670: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002e680: 2020 2020 2020 2020 2020 2020 5a5a 5b78 ZZ[x │ │ │ │ -0002e690: 202e 2e78 205d 2020 2020 2020 2020 2020 ..x ] │ │ │ │ +0002e660: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002e670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e6c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002e6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e6e0: 3020 2020 3420 2020 2020 2020 2020 2020 0 4 │ │ │ │ +0002e6b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002e6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e6d0: 2020 5a5a 5b78 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3120 3320 2020 3020 2020 4 0 1 3 0 │ │ │ │ -0002e7d0: 2033 2020 2031 2020 2020 3320 2020 3220 3 1 3 2 │ │ │ │ -0002e7e0: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -0002e7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e800: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002e810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e850: 2d2d 2d2d 2b0a 7c69 3820 3a20 723d 6765 ----+.|i8 : r=ge │ │ │ │ -0002e860: 6e73 2052 2020 2020 2020 2020 2020 2020 ns R │ │ │ │ -0002e870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e8a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002e700: 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+0002ea50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ea60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ea70: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ea90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002eaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002eac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 ----------+.|i9 │ │ │ │ +0002ead0: 3a20 493d 6964 6561 6c28 7261 6e64 6f6d : I=ideal(random │ │ │ │ +0002eae0: 287b 312c 307d 2c52 2929 2020 2020 2020 ({1,0},R)) │ │ │ │ 0002eaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002eb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb20: 2020 2020 7c0a 7c6f 3920 3d20 6964 6561 |.|o9 = idea │ │ │ │ -0002eb30: 6c28 3130 3778 2020 2b20 3433 3736 7820 l(107x + 4376x │ │ │ │ -0002eb40: 202d 2036 3331 3678 2029 2020 2020 2020 - 6316x ) │ │ │ │ +0002eb10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002eb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002eb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb70: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002eb80: 2020 2020 2020 3020 2020 2020 2020 2031 0 1 │ │ │ │ -0002eb90: 2020 2020 2020 2020 3320 2020 2020 2020 3 │ │ │ │ +0002eb60: 2020 2020 2020 2020 2020 7c0a 7c6f 3920 |.|o9 │ │ │ │ +0002eb70: 3d20 6964 6561 6c28 3130 3778 2020 2b20 = ideal(107x + │ │ │ │ +0002eb80: 3433 3736 7820 202d 2036 3331 3678 2029 4376x - 6316x ) │ │ │ │ +0002eb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002eba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ebb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ebc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002ebd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ebb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002ebc0: 2020 2020 2020 2020 2020 2020 3020 2020 0 │ │ │ │ +0002ebd0: 2020 2020 2031 2020 2020 2020 2020 3320 1 3 │ │ │ │ 0002ebe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ebf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec10: 2020 2020 7c0a 7c6f 3920 3a20 4964 6561 |.|o9 : Idea │ │ │ │ -0002ec20: 6c20 6f66 2052 2020 2020 2020 2020 2020 l of R │ │ │ │ +0002ec00: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002ec10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ec20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ec30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ec40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002ec70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ecb0: 2d2d 2d2d 2b0a 7c69 3130 203a 204b 3d69 ----+.|i10 : K=i │ │ │ │ -0002ecc0: 6465 616c 2872 616e 646f 6d28 7b31 2c31 deal(random({1,1 │ │ │ │ -0002ecd0: 7d2c 5229 2920 2020 2020 2020 2020 2020 },R)) │ │ │ │ -0002ece0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ecf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002ed10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ec50: 2020 2020 2020 2020 2020 7c0a 7c6f 3920 |.|o9 │ │ │ │ +0002ec60: 3a20 4964 6561 6c20 6f66 2052 2020 2020 : Ideal of R │ │ │ │ +0002ec70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ec80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ec90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eca0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ecc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ecd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ece0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ecf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 ----------+.|i10 │ │ │ │ +0002ed00: 203a 204b 3d69 6465 616c 2872 616e 646f : K=ideal(rando │ │ │ │ +0002ed10: 6d28 7b31 2c31 7d2c 5229 2920 2020 2020 m({1,1},R)) │ │ │ │ 0002ed20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ed30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed50: 2020 2020 7c0a 7c6f 3130 203d 2069 6465 |.|o10 = ide │ │ │ │ -0002ed60: 616c 2833 3138 3778 2078 2020 2d20 3630 al(3187x x - 60 │ │ │ │ -0002ed70: 3533 7820 7820 202d 2031 3630 3930 7820 53x x - 16090x │ │ │ │ -0002ed80: 7820 202b 2033 3738 3378 2078 2020 2b20 x + 3783x x + │ │ │ │ -0002ed90: 3835 3730 7820 7820 202b 2038 3434 3478 8570x x + 8444x │ │ │ │ -0002eda0: 2078 2029 7c0a 7c20 2020 2020 2020 2020 x )|.| │ │ │ │ -0002edb0: 2020 2020 2020 2020 3020 3220 2020 2020 0 2 │ │ │ │ -0002edc0: 2020 2031 2032 2020 2020 2020 2020 2032 1 2 2 │ │ │ │ -0002edd0: 2033 2020 2020 2020 2020 3020 3420 2020 3 0 4 │ │ │ │ -0002ede0: 2020 2020 2031 2034 2020 2020 2020 2020 1 4 │ │ │ │ -0002edf0: 3320 3420 7c0a 7c20 2020 2020 2020 2020 3 4 |.| │ │ │ │ -0002ee00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee40: 2020 2020 7c0a 7c6f 3130 203a 2049 6465 |.|o10 : Ide │ │ │ │ -0002ee50: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ +0002ed40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002ed50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ed60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ed70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ed80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ed90: 2020 2020 2020 2020 2020 7c0a 7c6f 3130 |.|o10 │ │ │ │ +0002eda0: 203d 2069 6465 616c 2833 3138 3778 2078 = ideal(3187x x │ │ │ │ +0002edb0: 2020 2d20 3630 3533 7820 7820 202d 2031 - 6053x x - 1 │ │ │ │ +0002edc0: 3630 3930 7820 7820 202b 2033 3738 3378 6090x x + 3783x │ │ │ │ +0002edd0: 2078 2020 2b20 3835 3730 7820 7820 202b x + 8570x x + │ │ │ │ +0002ede0: 2038 3434 3478 2078 2029 7c0a 7c20 2020 8444x x )|.| │ │ │ │ +0002edf0: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ +0002ee00: 3220 2020 2020 2020 2031 2032 2020 2020 2 1 2 │ │ │ │ +0002ee10: 2020 2020 2032 2033 2020 2020 2020 2020 2 3 │ │ │ │ +0002ee20: 3020 3420 2020 2020 2020 2031 2034 2020 0 4 1 4 │ │ │ │ +0002ee30: 2020 2020 2020 3320 3420 7c0a 7c20 2020 3 4 |.| │ │ │ │ +0002ee40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ee50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ee60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ee70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee90: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eee0: 2d2d 2d2d 2b0a 7c69 3131 203a 2063 3d43 ----+.|i11 : c=C │ │ │ │ -0002eef0: 6865 726e 2843 682c 582c 4929 2020 2020 hern(Ch,X,I) │ │ │ │ -0002ef00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 2020 2020 2020 │ │ │ │ -0002f470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f480: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002f490: 3220 2020 2020 2020 3220 2020 2020 2020 2 2 │ │ │ │ +0002f470: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002f480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f4d0: 2020 2020 7c0a 7c6f 3133 203d 202d 2078 |.|o13 = - x │ │ │ │ -0002f4e0: 2078 2020 2d20 3378 2020 2d20 3478 2078 x - 3x - 4x x │ │ │ │ -0002f4f0: 2020 2b20 7820 2020 2020 2020 2020 2020 + x │ │ │ │ +0002f4c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002f4d0: 2020 2020 2020 3220 2020 2020 2020 3220 2 2 │ │ │ │ +0002f4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f520: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002f530: 3320 3420 2020 2020 3320 2020 2020 3320 3 4 3 3 │ │ │ │ -0002f540: 3420 2020 2034 2020 2020 2020 2020 2020 4 4 │ │ │ │ +0002f510: 2020 2020 2020 2020 2020 7c0a 7c6f 3133 |.|o13 │ │ │ │ +0002f520: 203d 202d 2078 2078 2020 2d20 3378 2020 = - x x - 3x │ │ │ │ +0002f530: 2d20 3478 2078 2020 2b20 7820 2020 2020 - 4x x + x │ │ │ │ +0002f540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f570: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002f580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f560: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002f570: 2020 2020 2020 3320 3420 2020 2020 3320 3 4 3 │ │ │ │ +0002f580: 2020 2020 3320 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│ -0002f760: 7820 202b 2078 2020 2b20 7820 7820 2020 x + x + x x │ │ │ │ +0002f740: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002f750: 2020 2020 2032 2020 2020 2020 3220 2020 2 2 │ │ │ │ +0002f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f7a0: 2020 2020 7c0a 7c20 2020 2020 2020 2033 |.| 3 │ │ │ │ -0002f7b0: 2034 2020 2020 3320 2020 2033 2034 2020 4 3 3 4 │ │ │ │ +0002f790: 2020 2020 2020 2020 2020 7c0a 7c6f 3134 |.|o14 │ │ │ │ +0002f7a0: 203d 2032 7820 7820 202b 2078 2020 2b20 = 2x x + x + │ │ │ │ +0002f7b0: 7820 7820 2020 2020 2020 2020 2020 2020 x x │ │ │ │ 0002f7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f7f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ 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│ -0002f8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f8e0: 2d2d 2d2d 2b0a 0a46 6f72 2074 6865 2070 ----+..For the p │ │ │ │ -0002f8f0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -0002f900: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -0002f910: 6520 6f62 6a65 6374 202a 6e6f 7465 2054 e object *note T │ │ │ │ -0002f920: 6f72 6963 4368 6f77 5269 6e67 3a20 546f oricChowRing: To │ │ │ │ -0002f930: 7269 6343 686f 7752 696e 672c 2069 7320 ricChowRing, is │ │ │ │ -0002f940: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ -0002f950: 756e 6374 696f 6e3a 0a28 4d61 6361 756c unction:.(Macaul │ │ │ │ -0002f960: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ -0002f970: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ -0002f980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f9c0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -0002f9d0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -0002f9e0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -0002f9f0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -0002fa00: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -0002fa10: 2e32 362e 3035 2b64 732f 4d32 2f4d 6163 .26.05+ds/M2/Mac │ │ │ │ -0002fa20: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -0002fa30: 0a43 6861 7261 6374 6572 6973 7469 6343 .CharacteristicC │ │ │ │ -0002fa40: 6c61 7373 6573 2e6d 323a 3139 3531 3a30 lasses.m2:1951:0 │ │ │ │ -0002fa50: 2e0a 1f0a 5461 6720 5461 626c 653a 0a4e ....Tag Table:.N │ │ │ │ -0002fa60: 6f64 653a 2054 6f70 7f32 3931 0a4e 6f64 ode: Top.291.Nod │ │ │ │ -0002fa70: 653a 2062 6572 7469 6e69 4368 6563 6b7f e: bertiniCheck. │ │ │ │ -0002fa80: 3136 3730 350a 4e6f 6465 3a20 4368 6563 16705.Node: Chec │ │ │ │ -0002fa90: 6b53 6d6f 6f74 687f 3137 3932 320a 4e6f kSmooth.17922.No │ │ │ │ -0002faa0: 6465 3a20 4368 6563 6b54 6f72 6963 5661 de: CheckToricVa │ │ │ │ -0002fab0: 7269 6574 7956 616c 6964 7f32 3237 3138 rietyValid.22718 │ │ │ │ -0002fac0: 0a4e 6f64 653a 2043 6865 726e 7f33 3337 .Node: Chern.337 │ │ │ │ -0002fad0: 3039 0a4e 6f64 653a 2043 686f 7752 696e 09.Node: ChowRin │ │ │ │ -0002fae0: 677f 3531 3231 370a 4e6f 6465 3a20 436c g.51217.Node: Cl │ │ │ │ -0002faf0: 6173 7349 6e43 686f 7752 696e 677f 3538 assInChowRing.58 │ │ │ │ -0002fb00: 3934 380a 4e6f 6465 3a20 436c 6173 7349 948.Node: ClassI │ │ │ │ -0002fb10: 6e54 6f72 6963 4368 6f77 5269 6e67 7f36 nToricChowRing.6 │ │ │ │ -0002fb20: 3133 3038 0a4e 6f64 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756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ +0002f9a0: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ +0002f9b0: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +0002f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +0002fa10: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +0002fa20: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +0002fa30: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +0002fa40: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +0002fa50: 6c61 7932 2d31 2e32 362e 3035 2b64 732f lay2-1.26.05+ds/ │ │ │ │ +0002fa60: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +0002fa70: 6b61 6765 732f 0a43 6861 7261 6374 6572 kages/.Character │ │ │ │ +0002fa80: 6973 7469 6343 6c61 7373 6573 2e6d 323a isticClasses.m2: │ │ │ │ +0002fa90: 3139 3531 3a30 2e0a 1f0a 5461 6720 5461 1951:0....Tag Ta │ │ │ │ +0002faa0: 626c 653a 0a4e 6f64 653a 2054 6f70 7f32 ble:.Node: Top.2 │ │ │ │ +0002fab0: 3931 0a4e 6f64 653a 2062 6572 7469 6e69 91.Node: bertini │ │ │ │ +0002fac0: 4368 6563 6b7f 3136 3730 350a 4e6f 6465 Check.16705.Node │ │ │ │ +0002fad0: 3a20 4368 6563 6b53 6d6f 6f74 687f 3137 : CheckSmooth.17 │ │ │ │ +0002fae0: 3932 320a 4e6f 6465 3a20 4368 6563 6b54 922.Node: CheckT │ │ │ │ +0002faf0: 6f72 6963 5661 7269 6574 7956 616c 6964 oricVarietyValid │ │ │ │ +0002fb00: 7f32 3237 3138 0a4e 6f64 653a 2043 6865 .22718.Node: Che │ │ │ │ +0002fb10: 726e 7f33 3337 3039 0a4e 6f64 653a 2043 rn.33709.Node: C │ │ │ │ +0002fb20: 686f 7752 696e 677f 3531 3231 370a 4e6f howRing.51217.No │ │ │ │ +0002fb30: 6465 3a20 436c 6173 7349 6e43 686f 7752 de: ClassInChowR │ │ │ │ +0002fb40: 696e 677f 3538 3934 380a 4e6f 6465 3a20 ing.58948.Node: │ │ │ │ +0002fb50: 436c 6173 7349 6e54 6f72 6963 4368 6f77 ClassInToricChow │ │ │ │ +0002fb60: 5269 6e67 7f36 3133 3038 0a4e 6f64 653a Ring.61308.Node: │ │ │ │ +0002fb70: 2043 6f6d 704d 6574 686f 647f 3636 3535 CompMethod.6655 │ │ │ │ +0002fb80: 310a 4e6f 6465 3a20 636f 6e66 6967 7572 1.Node: configur │ │ │ │ +0002fb90: 696e 6720 4265 7274 696e 697f 3736 3437 ing Bertini.7647 │ │ │ │ +0002fba0: 380a 4e6f 6465 3a20 4353 4d7f 3738 3131 8.Node: CSM.7811 │ │ │ │ +0002fbb0: 330a 4e6f 6465 3a20 4575 6c65 727f 3130 3.Node: Euler.10 │ │ │ │ +0002fbc0: 3138 3032 0a4e 6f64 653a 2045 756c 6572 1802.Node: Euler │ │ │ │ +0002fbd0: 4166 6669 6e65 7f31 3139 3637 390a 4e6f Affine.119679.No │ │ │ │ +0002fbe0: 6465 3a20 496e 6473 4f66 536d 6f6f 7468 de: IndsOfSmooth │ │ │ │ +0002fbf0: 7f31 3232 3139 310a 4e6f 6465 3a20 496e .122191.Node: In │ │ │ │ +0002fc00: 7075 7449 7353 6d6f 6f74 687f 3132 3631 putIsSmooth.1261 │ │ │ │ +0002fc10: 3436 0a4e 6f64 653a 2069 734d 756c 7469 46.Node: isMulti │ │ │ │ +0002fc20: 486f 6d6f 6765 6e65 6f75 737f 3133 3031 Homogeneous.1301 │ │ │ │ +0002fc30: 3634 0a4e 6f64 653a 204d 6574 686f 647f 64.Node: Method. │ │ │ │ +0002fc40: 3133 3432 3432 0a4e 6f64 653a 204d 756c 134242.Node: Mul │ │ │ │ +0002fc50: 7469 5072 6f6a 436f 6f72 6452 696e 677f tiProjCoordRing. │ │ │ │ +0002fc60: 3133 3831 3931 0a4e 6f64 653a 204f 7574 138191.Node: Out │ │ │ │ +0002fc70: 7075 747f 3134 3438 3539 0a4e 6f64 653a put.144859.Node: │ │ │ │ +0002fc80: 2070 726f 6261 6269 6c69 7374 6963 2061 probabilistic a │ │ │ │ +0002fc90: 6c67 6f72 6974 686d 7f31 3633 3337 370a lgorithm.163377. │ │ │ │ +0002fca0: 4e6f 6465 3a20 5365 6772 657f 3136 3830 Node: Segre.1680 │ │ │ │ +0002fcb0: 3732 0a4e 6f64 653a 2054 6f72 6963 4368 72.Node: ToricCh │ │ │ │ +0002fcc0: 6f77 5269 6e67 7f31 3835 3937 390a 1f0a owRing.185979... │ │ │ │ +0002fcd0: 456e 6420 5461 6720 5461 626c 650a End Tag Table. │ │ ├── ./usr/share/info/CohomCalg.info.gz │ │ │ ├── CohomCalg.info │ │ │ │ @@ -1042,16 +1042,16 @@ │ │ │ │ 00004110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00004130: 7c69 3230 203a 2065 6c61 7073 6564 5469 |i20 : elapsedTi │ │ │ │ 00004140: 6d65 2068 7665 6373 203d 2063 6f68 6f6d me hvecs = cohom │ │ │ │ 00004150: 4361 6c67 2858 2c20 4432 2920 2020 2020 Calg(X, D2) │ │ │ │ 00004160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004170: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00004180: 7c20 2d2d 2033 2e34 3437 3438 7320 656c | -- 3.44748s el │ │ │ │ -00004190: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ +00004180: 7c20 2d2d 2033 2e33 3733 3573 2065 6c61 | -- 3.3735s ela │ │ │ │ +00004190: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 000041a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000041b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000041c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000041d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000041e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000041f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1677,15 +1677,15 @@ │ │ │ │ 000068c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000068d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000068e0: 7c69 3233 203a 2065 6c61 7073 6564 5469 |i23 : elapsedTi │ │ │ │ 000068f0: 6d65 2063 6f68 6f6d 7665 6331 203d 2063 me cohomvec1 = c │ │ │ │ 00006900: 6f68 6f6d 4361 6c67 2858 5f33 202b 2058 ohomCalg(X_3 + X │ │ │ │ 00006910: 5f37 202b 2058 5f38 2920 2020 2020 2020 _7 + X_8) │ │ │ │ 00006920: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00006930: 7c20 2d2d 202e 3336 3038 3237 7320 656c | -- .360827s el │ │ │ │ +00006930: 7c20 2d2d 202e 3531 3533 3539 7320 656c | -- .515359s el │ │ │ │ 00006940: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00006950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006970: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00006980: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00006990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000069a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1712,15 +1712,15 @@ │ │ │ │ 00006af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00006b10: 7c69 3234 203a 2065 6c61 7073 6564 5469 |i24 : elapsedTi │ │ │ │ 00006b20: 6d65 2063 6f68 6f6d 7665 6332 203d 2066 me cohomvec2 = f │ │ │ │ 00006b30: 6f72 206a 2066 726f 6d20 3020 746f 2064 or j from 0 to d │ │ │ │ 00006b40: 696d 2058 206c 6973 7420 7261 6e6b 2048 im X list rank H │ │ │ │ 00006b50: 485e 6a28 582c 2020 2020 2020 2020 7c0a H^j(X, |. │ │ │ │ -00006b60: 7c20 2d2d 2031 302e 3632 3433 7320 656c | -- 10.6243s el │ │ │ │ +00006b60: 7c20 2d2d 2031 302e 3037 3635 7320 656c | -- 10.0765s el │ │ │ │ 00006b70: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00006b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006ba0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00006bb0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00006bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1797,15 +1797,15 @@ │ │ │ │ 00007040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00007050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00007060: 7c69 3237 203a 2065 6c61 7073 6564 5469 |i27 : elapsedTi │ │ │ │ 00007070: 6d65 2063 6f68 6f6d 7665 6331 203d 2063 me cohomvec1 = c │ │ │ │ 00007080: 6f68 6f6d 4361 6c67 2858 5f33 202b 2058 ohomCalg(X_3 + X │ │ │ │ 00007090: 5f37 202d 2058 5f38 2920 2020 2020 2020 _7 - X_8) │ │ │ │ 000070a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000070b0: 7c20 2d2d 202e 3333 3138 3034 7320 656c | -- .331804s el │ │ │ │ +000070b0: 7c20 2d2d 202e 3534 3434 3835 7320 656c | -- .544485s el │ │ │ │ 000070c0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 000070d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000070e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000070f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00007100: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00007110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1832,21 +1832,21 @@ │ │ │ │ 00007270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00007280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00007290: 7c69 3238 203a 2065 6c61 7073 6564 5469 |i28 : elapsedTi │ │ │ │ 000072a0: 6d65 2063 6f68 6f6d 7665 6332 203d 2065 me cohomvec2 = e │ │ │ │ 000072b0: 6c61 7073 6564 5469 6d65 2066 6f72 206a lapsedTime for j │ │ │ │ 000072c0: 2066 726f 6d20 3020 746f 2064 696d 2058 from 0 to dim X │ │ │ │ 000072d0: 206c 6973 7420 7261 6e6b 2020 2020 7c0a list rank |. │ │ │ │ -000072e0: 7c20 2d2d 202e 3535 3339 3238 7320 656c | -- .553928s el │ │ │ │ +000072e0: 7c20 2d2d 202e 3434 3531 3039 7320 656c | -- .445109s el │ │ │ │ 000072f0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00007300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007320: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00007330: 7c20 2d2d 202e 3535 3339 3673 2065 6c61 | -- .55396s ela │ │ │ │ -00007340: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ +00007330: 7c20 2d2d 202e 3434 3531 3339 7320 656c | -- .445139s el │ │ │ │ +00007340: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00007350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007370: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00007380: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00007390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000073a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000073b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/CompleteIntersectionResolutions.info.gz │ │ │ ├── CompleteIntersectionResolutions.info │ │ │ │ @@ -4343,17 +4343,17 @@ │ │ │ │ 00010f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010f80: 2d2d 2d2b 0a7c 6937 203a 2074 696d 6520 ---+.|i7 : time │ │ │ │ 00010f90: 4720 3d20 4569 7365 6e62 7564 5368 616d G = EisenbudSham │ │ │ │ 00010fa0: 6173 6828 6666 2c46 2c6c 656e 2920 2020 ash(ff,F,len) │ │ │ │ 00010fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010fd0: 2020 207c 0a7c 202d 2d20 7573 6564 2036 |.| -- used 6 │ │ │ │ -00010fe0: 2e33 3131 3134 7320 2863 7075 293b 2034 .31114s (cpu); 4 │ │ │ │ -00010ff0: 2e38 3135 3437 7320 2874 6872 6561 6429 .81547s (thread) │ │ │ │ +00010fd0: 2020 207c 0a7c 202d 2d20 7573 6564 2037 |.| -- used 7 │ │ │ │ +00010fe0: 2e35 3737 3936 7320 2863 7075 293b 2035 .57796s (cpu); 5 │ │ │ │ +00010ff0: 2e39 3832 3438 7320 2874 6872 6561 6429 .98248s (thread) │ │ │ │ 00011000: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00011010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011020: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00011030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4884,17 +4884,17 @@ │ │ │ │ 00013130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013150: 2d2b 0a7c 6932 3020 3a20 4646 203d 2074 -+.|i20 : FF = t │ │ │ │ 00013160: 696d 6520 5368 616d 6173 6828 5231 2c46 ime Shamash(R1,F │ │ │ │ 00013170: 2c34 2920 2020 2020 2020 2020 2020 2020 ,4) │ │ │ │ 00013180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013190: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -000131a0: 2e30 3637 3739 3835 7320 2863 7075 293b .0677985s (cpu); │ │ │ │ -000131b0: 2030 2e30 3637 3735 3633 7320 2874 6872 0.0677563s (thr │ │ │ │ -000131c0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +000131a0: 2e31 3031 3031 3973 2028 6370 7529 3b20 .101019s (cpu); │ │ │ │ +000131b0: 302e 3130 3130 3138 7320 2874 6872 6561 0.101018s (threa │ │ │ │ +000131c0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 000131d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000131e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000131f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013210: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00013220: 2020 3120 2020 2020 2020 3620 2020 2020 1 6 │ │ │ │ 00013230: 2020 3138 2020 2020 2020 2033 3820 2020 18 38 │ │ │ │ @@ -4925,16 +4925,16 @@ │ │ │ │ 000133c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000133d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000133e0: 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 4747 -----+.|i21 : GG │ │ │ │ 000133f0: 203d 2074 696d 6520 4569 7365 6e62 7564 = time Eisenbud │ │ │ │ 00013400: 5368 616d 6173 6828 6666 2c46 2c34 2920 Shamash(ff,F,4) │ │ │ │ 00013410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013420: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00013430: 6564 2031 2e30 3634 3431 7320 2863 7075 ed 1.06441s (cpu │ │ │ │ -00013440: 293b 2030 2e37 3937 3639 3173 2028 7468 ); 0.797691s (th │ │ │ │ +00013430: 6564 2031 2e32 3537 3539 7320 2863 7075 ed 1.25759s (cpu │ │ │ │ +00013440: 293b 2030 2e39 3430 3735 3173 2028 7468 ); 0.940751s (th │ │ │ │ 00013450: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00013460: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00013470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000134a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000134b0: 2020 2020 2f20 525c 3120 2020 2020 2f20 / R\1 / │ │ │ │ @@ -4977,24035 +4977,24031 @@ │ │ │ │ 00013700: 5468 6520 6675 6e63 7469 6f6e 2061 6c73 The function als │ │ │ │ 00013710: 6f20 6465 616c 7320 636f 7272 6563 746c o deals correctl │ │ │ │ 00013720: 7920 7769 7468 2063 6f6d 706c 6578 6573 y with complexes │ │ │ │ 00013730: 2046 2077 6865 7265 206d 696e 2046 2069 F where min F i │ │ │ │ 00013740: 7320 6e6f 7420 303a 0a0a 2b2d 2d2d 2d2d s not 0:..+----- │ │ │ │ 00013750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013780: 2b0a 7c69 3232 203a 2047 4720 3d20 7469 +.|i22 : GG = ti │ │ │ │ -00013790: 6d65 2045 6973 656e 6275 6453 6861 6d61 me EisenbudShama │ │ │ │ -000137a0: 7368 2852 312c 465b 325d 2c34 2920 2020 sh(R1,F[2],4) │ │ │ │ -000137b0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000137c0: 7365 6420 302e 3931 3533 3838 7320 2863 sed 0.915388s (c │ │ │ │ -000137d0: 7075 293b 2030 2e37 3234 3536 3673 2028 pu); 0.724566s ( │ │ │ │ -000137e0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ -000137f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00013770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00013780: 0a7c 6932 3220 3a20 4747 203d 2074 696d .|i22 : GG = tim │ │ │ │ +00013790: 6520 4569 7365 6e62 7564 5368 616d 6173 e EisenbudShamas │ │ │ │ +000137a0: 6828 5231 2c46 5b32 5d2c 3429 2020 2020 h(R1,F[2],4) │ │ │ │ +000137b0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ +000137c0: 6420 312e 3135 3936 3373 2028 6370 7529 d 1.15963s (cpu) │ │ │ │ +000137d0: 3b20 302e 3932 3432 3534 7320 2874 6872 ; 0.924254s (thr │ │ │ │ +000137e0: 6561 6429 3b20 3073 2028 6763 297c 0a7c ead); 0s (gc)|.| │ │ │ │ +000137f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013820: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00013830: 2020 2031 2020 2020 2020 2036 2020 2020 1 6 │ │ │ │ -00013840: 2020 2031 3820 2020 2020 2020 3338 2020 18 38 │ │ │ │ -00013850: 2020 2020 2036 3620 2020 2020 2020 2020 66 │ │ │ │ -00013860: 7c0a 7c6f 3232 203d 2052 3120 203c 2d2d |.|o22 = R1 <-- │ │ │ │ -00013870: 2052 3120 203c 2d2d 2052 3120 2020 3c2d R1 <-- R1 <- │ │ │ │ -00013880: 2d20 5231 2020 203c 2d2d 2052 3120 2020 - R1 <-- R1 │ │ │ │ -00013890: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00013820: 2020 2020 7c0a 7c20 2020 2020 2020 2031 |.| 1 │ │ │ │ +00013830: 2020 2020 2020 2036 2020 2020 2020 2031 6 1 │ │ │ │ +00013840: 3820 2020 2020 2020 3338 2020 2020 2020 8 38 │ │ │ │ +00013850: 2036 3620 2020 2020 2020 207c 0a7c 6f32 66 |.|o2 │ │ │ │ +00013860: 3220 3d20 5231 2020 3c2d 2d20 5231 2020 2 = R1 <-- R1 │ │ │ │ +00013870: 3c2d 2d20 5231 2020 203c 2d2d 2052 3120 <-- R1 <-- R1 │ │ │ │ +00013880: 2020 3c2d 2d20 5231 2020 2020 2020 2020 <-- R1 │ │ │ │ +00013890: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000138a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000138b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000138c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000138d0: 7c0a 7c20 2020 2020 202d 3220 2020 2020 |.| -2 │ │ │ │ -000138e0: 202d 3120 2020 2020 2030 2020 2020 2020 -1 0 │ │ │ │ -000138f0: 2020 3120 2020 2020 2020 2032 2020 2020 1 2 │ │ │ │ -00013900: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000138c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000138d0: 2020 2d32 2020 2020 2020 2d31 2020 2020 -2 -1 │ │ │ │ +000138e0: 2020 3020 2020 2020 2020 2031 2020 2020 0 1 │ │ │ │ +000138f0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00013900: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00013910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013940: 7c0a 7c6f 3232 203a 2043 6f6d 706c 6578 |.|o22 : Complex │ │ │ │ +00013930: 2020 2020 2020 207c 0a7c 6f32 3220 3a20 |.|o22 : │ │ │ │ +00013940: 436f 6d70 6c65 7820 2020 2020 2020 2020 Complex │ │ │ │ 00013950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013970: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00013960: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00013970: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00013980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000139a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000139b0: 2b0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d +..See also.==== │ │ │ │ -000139c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -000139d0: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20 makeHomotopies: │ │ │ │ -000139e0: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20 makeHomotopies, │ │ │ │ -000139f0: 2d2d 2072 6574 7572 6e73 2061 2073 7973 -- returns a sys │ │ │ │ -00013a00: 7465 6d20 6f66 2068 6967 6865 720a 2020 tem of higher. │ │ │ │ -00013a10: 2020 686f 6d6f 746f 7069 6573 0a20 202a homotopies. * │ │ │ │ -00013a20: 202a 6e6f 7465 2053 6861 6d61 7368 3a20 *note Shamash: │ │ │ │ -00013a30: 5368 616d 6173 682c 202d 2d20 436f 6d70 Shamash, -- Comp │ │ │ │ -00013a40: 7574 6573 2074 6865 2053 6861 6d61 7368 utes the Shamash │ │ │ │ -00013a50: 2043 6f6d 706c 6578 0a20 202a 202a 6e6f Complex. * *no │ │ │ │ -00013a60: 7465 2065 7870 6f3a 2065 7870 6f2c 202d te expo: expo, - │ │ │ │ -00013a70: 2d20 7265 7475 726e 7320 6120 7365 7420 - returns a set │ │ │ │ -00013a80: 636f 7272 6573 706f 6e64 696e 6720 746f corresponding to │ │ │ │ -00013a90: 2074 6865 2062 6173 6973 206f 6620 6120 the basis of a │ │ │ │ -00013aa0: 6469 7669 6465 640a 2020 2020 706f 7765 divided. powe │ │ │ │ -00013ab0: 720a 0a57 6179 7320 746f 2075 7365 2045 r..Ways to use E │ │ │ │ -00013ac0: 6973 656e 6275 6453 6861 6d61 7368 3a0a isenbudShamash:. │ │ │ │ +000139a0: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f -----+..See also │ │ │ │ +000139b0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +000139c0: 6e6f 7465 206d 616b 6548 6f6d 6f74 6f70 note makeHomotop │ │ │ │ +000139d0: 6965 733a 206d 616b 6548 6f6d 6f74 6f70 ies: makeHomotop │ │ │ │ +000139e0: 6965 732c 202d 2d20 7265 7475 726e 7320 ies, -- returns │ │ │ │ +000139f0: 6120 7379 7374 656d 206f 6620 6869 6768 a system of high │ │ │ │ +00013a00: 6572 0a20 2020 2068 6f6d 6f74 6f70 6965 er. homotopie │ │ │ │ +00013a10: 730a 2020 2a20 2a6e 6f74 6520 5368 616d s. * *note Sham │ │ │ │ +00013a20: 6173 683a 2053 6861 6d61 7368 2c20 2d2d ash: Shamash, -- │ │ │ │ +00013a30: 2043 6f6d 7075 7465 7320 7468 6520 5368 Computes the Sh │ │ │ │ +00013a40: 616d 6173 6820 436f 6d70 6c65 780a 2020 amash Complex. │ │ │ │ +00013a50: 2a20 2a6e 6f74 6520 6578 706f 3a20 6578 * *note expo: ex │ │ │ │ +00013a60: 706f 2c20 2d2d 2072 6574 7572 6e73 2061 po, -- returns a │ │ │ │ +00013a70: 2073 6574 2063 6f72 7265 7370 6f6e 6469 set correspondi │ │ │ │ +00013a80: 6e67 2074 6f20 7468 6520 6261 7369 7320 ng to the basis │ │ │ │ +00013a90: 6f66 2061 2064 6976 6964 6564 0a20 2020 of a divided. │ │ │ │ +00013aa0: 2070 6f77 6572 0a0a 5761 7973 2074 6f20 power..Ways to │ │ │ │ +00013ab0: 7573 6520 4569 7365 6e62 7564 5368 616d use EisenbudSham │ │ │ │ +00013ac0: 6173 683a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ash:.=========== │ │ │ │ 00013ad0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00013ae0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ -00013af0: 2a20 2245 6973 656e 6275 6453 6861 6d61 * "EisenbudShama │ │ │ │ -00013b00: 7368 284d 6174 7269 782c 436f 6d70 6c65 sh(Matrix,Comple │ │ │ │ -00013b10: 782c 5a5a 2922 0a20 202a 2022 4569 7365 x,ZZ)". * "Eise │ │ │ │ -00013b20: 6e62 7564 5368 616d 6173 6828 5269 6e67 nbudShamash(Ring │ │ │ │ -00013b30: 2c43 6f6d 706c 6578 2c5a 5a29 220a 0a46 ,Complex,ZZ)"..F │ │ │ │ -00013b40: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00013b50: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00013b60: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00013b70: 202a 6e6f 7465 2045 6973 656e 6275 6453 *note EisenbudS │ │ │ │ -00013b80: 6861 6d61 7368 3a20 4569 7365 6e62 7564 hamash: Eisenbud │ │ │ │ -00013b90: 5368 616d 6173 682c 2069 7320 6120 2a6e Shamash, is a *n │ │ │ │ -00013ba0: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ -00013bb0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ -00013bc0: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00013bd0: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +00013ae0: 3d0a 0a20 202a 2022 4569 7365 6e62 7564 =.. * "Eisenbud │ │ │ │ +00013af0: 5368 616d 6173 6828 4d61 7472 6978 2c43 Shamash(Matrix,C │ │ │ │ +00013b00: 6f6d 706c 6578 2c5a 5a29 220a 2020 2a20 omplex,ZZ)". * │ │ │ │ +00013b10: 2245 6973 656e 6275 6453 6861 6d61 7368 "EisenbudShamash │ │ │ │ +00013b20: 2852 696e 672c 436f 6d70 6c65 782c 5a5a (Ring,Complex,ZZ │ │ │ │ +00013b30: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ +00013b40: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +00013b50: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +00013b60: 626a 6563 7420 2a6e 6f74 6520 4569 7365 bject *note Eise │ │ │ │ +00013b70: 6e62 7564 5368 616d 6173 683a 2045 6973 nbudShamash: Eis │ │ │ │ +00013b80: 656e 6275 6453 6861 6d61 7368 2c20 6973 enbudShamash, is │ │ │ │ +00013b90: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ +00013ba0: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ +00013bb0: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ +00013bc0: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +00013bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013c20: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00013c30: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00013c40: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00013c50: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00013c60: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -00013c70: 3035 2b64 732f 4d32 2f4d 6163 6175 6c61 05+ds/M2/Macaula │ │ │ │ -00013c80: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d y2/packages/.Com │ │ │ │ -00013c90: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ -00013ca0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a nResolutions.m2: │ │ │ │ -00013cb0: 3438 3432 3a30 2e0a 1f0a 4669 6c65 3a20 4842:0....File: │ │ │ │ -00013cc0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -00013cd0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -00013ce0: 696e 666f 2c20 4e6f 6465 3a20 4569 7365 info, Node: Eise │ │ │ │ -00013cf0: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ -00013d00: 2c20 4e65 7874 3a20 6576 656e 4578 744d , Next: evenExtM │ │ │ │ -00013d10: 6f64 756c 652c 2050 7265 763a 2045 6973 odule, Prev: Eis │ │ │ │ -00013d20: 656e 6275 6453 6861 6d61 7368 2c20 5570 enbudShamash, Up │ │ │ │ -00013d30: 3a20 546f 700a 0a45 6973 656e 6275 6453 : Top..EisenbudS │ │ │ │ -00013d40: 6861 6d61 7368 546f 7461 6c20 2d2d 2050 hamashTotal -- P │ │ │ │ -00013d50: 7265 6375 7273 6f72 2063 6f6d 706c 6578 recursor complex │ │ │ │ -00013d60: 206f 6620 746f 7461 6c20 4578 740a 2a2a of total Ext.** │ │ │ │ +00013c10: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +00013c20: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +00013c30: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +00013c40: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +00013c50: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +00013c60: 312e 3236 2e30 352b 6473 2f4d 322f 4d61 1.26.05+ds/M2/Ma │ │ │ │ +00013c70: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +00013c80: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ +00013c90: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ +00013ca0: 732e 6d32 3a34 3834 323a 302e 0a1f 0a46 s.m2:4842:0....F │ │ │ │ +00013cb0: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ +00013cc0: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00013cd0: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ +00013ce0: 2045 6973 656e 6275 6453 6861 6d61 7368 EisenbudShamash │ │ │ │ +00013cf0: 546f 7461 6c2c 204e 6578 743a 2065 7665 Total, Next: eve │ │ │ │ +00013d00: 6e45 7874 4d6f 6475 6c65 2c20 5072 6576 nExtModule, Prev │ │ │ │ +00013d10: 3a20 4569 7365 6e62 7564 5368 616d 6173 : EisenbudShamas │ │ │ │ +00013d20: 682c 2055 703a 2054 6f70 0a0a 4569 7365 h, Up: Top..Eise │ │ │ │ +00013d30: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ +00013d40: 202d 2d20 5072 6563 7572 736f 7220 636f -- Precursor co │ │ │ │ +00013d50: 6d70 6c65 7820 6f66 2074 6f74 616c 2045 mplex of total E │ │ │ │ +00013d60: 7874 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a xt.************* │ │ │ │ 00013d70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00013d80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00013d90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00013da0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -00013db0: 200a 2020 2020 2020 2020 2864 302c 6431 . (d0,d1 │ │ │ │ -00013dc0: 2920 3d20 2045 6973 656e 6275 6453 6861 ) = EisenbudSha │ │ │ │ -00013dd0: 6d61 7368 546f 7461 6c20 4d0a 2020 2a20 mashTotal M. * │ │ │ │ -00013de0: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -00013df0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c M, a *note modul │ │ │ │ -00013e00: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -00013e10: 294d 6f64 756c 652c 2c20 6f76 6572 2061 )Module,, over a │ │ │ │ -00013e20: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -00013e30: 6563 7469 6f6e 0a20 202a 202a 6e6f 7465 ection. * *note │ │ │ │ -00013e40: 204f 7074 696f 6e61 6c20 696e 7075 7473 Optional inputs │ │ │ │ -00013e50: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00013e60: 7573 696e 6720 6675 6e63 7469 6f6e 7320 using functions │ │ │ │ -00013e70: 7769 7468 206f 7074 696f 6e61 6c20 696e with optional in │ │ │ │ -00013e80: 7075 7473 2c3a 0a20 2020 2020 202a 2043 puts,:. * C │ │ │ │ -00013e90: 6865 636b 203d 3e20 2e2e 2e2c 2064 6566 heck => ..., def │ │ │ │ -00013ea0: 6175 6c74 2076 616c 7565 2066 616c 7365 ault value false │ │ │ │ -00013eb0: 0a20 2020 2020 202a 2047 7261 6469 6e67 . * Grading │ │ │ │ -00013ec0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ -00013ed0: 2076 616c 7565 2032 0a20 2020 2020 202a value 2. * │ │ │ │ -00013ee0: 2056 6172 6961 626c 6573 203d 3e20 2e2e Variables => .. │ │ │ │ -00013ef0: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -00013f00: 2073 0a20 202a 204f 7574 7075 7473 3a0a s. * Outputs:. │ │ │ │ -00013f10: 2020 2020 2020 2a20 6430 2c20 6120 2a6e * d0, a *n │ │ │ │ -00013f20: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ -00013f30: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ -00013f40: 2c2c 206d 6170 206f 6620 6672 6565 206d ,, map of free m │ │ │ │ -00013f50: 6f64 756c 6573 206f 7665 7220 616e 0a20 odules over an. │ │ │ │ -00013f60: 2020 2020 2020 2065 6e6c 6172 6765 6420 enlarged │ │ │ │ -00013f70: 7269 6e67 0a20 2020 2020 202a 2064 312c ring. * d1, │ │ │ │ -00013f80: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ -00013f90: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ -00013fa0: 6174 7269 782c 2c20 6d61 7020 6f66 2066 atrix,, map of f │ │ │ │ -00013fb0: 7265 6520 6d6f 6475 6c65 7320 6f76 6572 ree modules over │ │ │ │ -00013fc0: 2061 6e0a 2020 2020 2020 2020 656e 6c61 an. enla │ │ │ │ -00013fd0: 7267 6564 2072 696e 670a 0a44 6573 6372 rged ring..Descr │ │ │ │ -00013fe0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -00013ff0: 3d3d 0a0a 4173 7375 6d65 2074 6861 7420 ==..Assume that │ │ │ │ -00014000: 4d20 6973 2064 6566 696e 6564 206f 7665 M is defined ove │ │ │ │ -00014010: 7220 6120 7269 6e67 206f 6620 7468 6520 r a ring of the │ │ │ │ -00014020: 666f 726d 2052 6261 7220 3d20 522f 2866 form Rbar = R/(f │ │ │ │ -00014030: 5f30 2e2e 665f 7b63 2d31 7d29 2c20 610a _0..f_{c-1}), a. │ │ │ │ -00014040: 636f 6d70 6c65 7465 2069 6e74 6572 7365 complete interse │ │ │ │ -00014050: 6374 696f 6e2c 2061 6e64 2074 6861 7420 ction, and that │ │ │ │ -00014060: 4d20 6861 7320 6120 6669 6e69 7465 2066 M has a finite f │ │ │ │ -00014070: 7265 6520 7265 736f 6c75 7469 6f6e 2047 ree resolution G │ │ │ │ -00014080: 206f 7665 7220 522e 2049 6e0a 7468 6973 over R. In.this │ │ │ │ -00014090: 2063 6173 6520 4d20 6861 7320 6120 6672 case M has a fr │ │ │ │ -000140a0: 6565 2072 6573 6f6c 7574 696f 6e20 4620 ee resolution F │ │ │ │ -000140b0: 6f76 6572 2052 6261 7220 7768 6f73 6520 over Rbar whose │ │ │ │ -000140c0: 6475 616c 2c20 465e 2a20 6973 2061 2066 dual, F^* is a f │ │ │ │ -000140d0: 696e 6974 656c 790a 6765 6e65 7261 7465 initely.generate │ │ │ │ -000140e0: 642c 205a 2d67 7261 6465 6420 6672 6565 d, Z-graded free │ │ │ │ -000140f0: 206d 6f64 756c 6520 6f76 6572 2061 2072 module over a r │ │ │ │ -00014100: 696e 6720 5362 6172 5c63 6f6e 6720 6b6b ing Sbar\cong kk │ │ │ │ -00014110: 5b73 5f30 2e2e 735f 7b63 2d31 7d2c 6765 [s_0..s_{c-1},ge │ │ │ │ -00014120: 6e73 0a52 6261 725d 2c20 7768 6572 6520 ns.Rbar], where │ │ │ │ -00014130: 7468 6520 6465 6772 6565 7320 6f66 2074 the degrees of t │ │ │ │ -00014140: 6865 2073 5f69 2061 7265 207b 2d32 2c20 he s_i are {-2, │ │ │ │ -00014150: 2d64 6567 7265 6520 665f 697d 2e20 5468 -degree f_i}. Th │ │ │ │ -00014160: 6973 2072 6573 6f6c 7574 696f 6e20 6973 is resolution is │ │ │ │ -00014170: 0a69 7320 636f 6e73 7472 7563 7465 6420 .is constructed │ │ │ │ -00014180: 6672 6f6d 2074 6865 2064 7561 6c20 6f66 from the dual of │ │ │ │ -00014190: 2047 2c20 746f 6765 7468 6572 2077 6974 G, together wit │ │ │ │ -000141a0: 6820 7468 6520 6475 616c 7320 6f66 2074 h the duals of t │ │ │ │ -000141b0: 6865 2068 6967 6865 720a 686f 6d6f 746f he higher.homoto │ │ │ │ -000141c0: 7069 6573 206f 6e20 4720 6465 6669 6e65 pies on G define │ │ │ │ -000141d0: 6420 6279 2045 6973 656e 6275 642e 0a0a d by Eisenbud... │ │ │ │ -000141e0: 5468 6520 6675 6e63 7469 6f6e 2072 6574 The function ret │ │ │ │ -000141f0: 7572 6e73 2074 6865 2064 6966 6665 7265 urns the differe │ │ │ │ -00014200: 6e74 6961 6c73 2064 303a 465e 2a5f 7b65 ntials d0:F^*_{e │ │ │ │ -00014210: 7665 6e7d 205c 746f 2046 5e2a 5f7b 6f64 ven} \to F^*_{od │ │ │ │ -00014220: 647d 2061 6e64 0a64 313a 465e 2a5f 7b6f d} and.d1:F^*_{o │ │ │ │ -00014230: 6464 7d5c 746f 2046 5e2a 5f7b 6576 656e dd}\to F^*_{even │ │ │ │ -00014240: 7d2e 0a0a 5468 6520 6d61 7073 2064 302c }...The maps d0, │ │ │ │ -00014250: 6431 2066 6f72 6d20 6120 6d61 7472 6978 d1 form a matrix │ │ │ │ -00014260: 2066 6163 746f 7269 7a61 7469 6f6e 206f factorization o │ │ │ │ -00014270: 6620 7375 6d28 632c 2069 2d3e 735f 692a f sum(c, i->s_i* │ │ │ │ -00014280: 665f 6929 2e20 5468 6520 6861 7665 2074 f_i). The have t │ │ │ │ -00014290: 6865 0a70 726f 7065 7274 7920 7468 6174 he.property that │ │ │ │ -000142a0: 2066 6f72 2061 6e79 2052 6261 7220 6d6f for any Rbar mo │ │ │ │ -000142b0: 6475 6c65 204e 2c0a 0a48 485f 3120 636f dule N,..HH_1 co │ │ │ │ -000142c0: 6d70 6c65 7820 5c7b 6430 2a2a 4e2c 2064 mplex \{d0**N, d │ │ │ │ -000142d0: 312a 2a4e 5c7d 203d 2045 7874 5e7b 6576 1**N\} = Ext^{ev │ │ │ │ -000142e0: 656e 7d5f 7b52 6261 727d 284d 2c4e 290a en}_{Rbar}(M,N). │ │ │ │ -000142f0: 0a53 5e7b 7b31 2c30 7d7d 2a2a 4848 5f31 .S^{{1,0}}**HH_1 │ │ │ │ -00014300: 2063 6f6d 706c 6578 205c 7b53 5e7b 7b2d complex \{S^{{- │ │ │ │ -00014310: 322c 307d 7d2a 2a64 312a 2a4e 2c20 6430 2,0}}**d1**N, d0 │ │ │ │ -00014320: 2a2a 4e5c 7d20 3d20 4578 745e 7b6f 6464 **N\} = Ext^{odd │ │ │ │ -00014330: 7d5f 7b52 6261 727d 284d 2c4e 290a 0a54 }_{Rbar}(M,N)..T │ │ │ │ -00014340: 6869 7320 6973 2065 6e63 6f64 6564 2069 his is encoded i │ │ │ │ -00014350: 6e20 7468 6520 7363 7269 7074 206e 6577 n the script new │ │ │ │ -00014360: 4578 740a 0a4f 7074 696f 6e20 6465 6661 Ext..Option defa │ │ │ │ -00014370: 756c 7473 3a20 4368 6563 6b3d 3e66 616c ults: Check=>fal │ │ │ │ -00014380: 7365 2056 6172 6961 626c 6573 3d3e 6765 se Variables=>ge │ │ │ │ -00014390: 7453 796d 626f 6c20 2273 222c 2047 7261 tSymbol "s", Gra │ │ │ │ -000143a0: 6469 6e67 203d 3e32 7d0a 0a49 6620 4772 ding =>2}..If Gr │ │ │ │ -000143b0: 6164 696e 6720 3d3e 312c 2074 6865 6e20 ading =>1, then │ │ │ │ -000143c0: 6120 7369 6e67 6c79 2067 7261 6465 6420 a singly graded │ │ │ │ -000143d0: 7265 7375 6c74 2069 7320 7265 7475 726e result is return │ │ │ │ -000143e0: 6564 2028 6a75 7374 2066 6f72 6765 7474 ed (just forgett │ │ │ │ -000143f0: 696e 6720 7468 650a 686f 6d6f 6c6f 6769 ing the.homologi │ │ │ │ -00014400: 6361 6c20 6772 6164 696e 672e 290a 0a0a cal grading.)... │ │ │ │ -00014410: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00013d90: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +00013da0: 7361 6765 3a20 0a20 2020 2020 2020 2028 sage: . ( │ │ │ │ +00013db0: 6430 2c64 3129 203d 2020 4569 7365 6e62 d0,d1) = Eisenb │ │ │ │ +00013dc0: 7564 5368 616d 6173 6854 6f74 616c 204d udShamashTotal M │ │ │ │ +00013dd0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ +00013de0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520 * M, a *note │ │ │ │ +00013df0: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61 module: (Macaula │ │ │ │ +00013e00: 7932 446f 6329 4d6f 6475 6c65 2c2c 206f y2Doc)Module,, o │ │ │ │ +00013e10: 7665 7220 6120 636f 6d70 6c65 7465 2069 ver a complete i │ │ │ │ +00013e20: 6e74 6572 7365 6374 696f 6e0a 2020 2a20 ntersection. * │ │ │ │ +00013e30: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069 *note Optional i │ │ │ │ +00013e40: 6e70 7574 733a 2028 4d61 6361 756c 6179 nputs: (Macaulay │ │ │ │ +00013e50: 3244 6f63 2975 7369 6e67 2066 756e 6374 2Doc)using funct │ │ │ │ +00013e60: 696f 6e73 2077 6974 6820 6f70 7469 6f6e ions with option │ │ │ │ +00013e70: 616c 2069 6e70 7574 732c 3a0a 2020 2020 al inputs,:. │ │ │ │ +00013e80: 2020 2a20 4368 6563 6b20 3d3e 202e 2e2e * Check => ... │ │ │ │ +00013e90: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +00013ea0: 6661 6c73 650a 2020 2020 2020 2a20 4772 false. * Gr │ │ │ │ +00013eb0: 6164 696e 6720 3d3e 202e 2e2e 2c20 6465 ading => ..., de │ │ │ │ +00013ec0: 6661 756c 7420 7661 6c75 6520 320a 2020 fault value 2. │ │ │ │ +00013ed0: 2020 2020 2a20 5661 7269 6162 6c65 7320 * Variables │ │ │ │ +00013ee0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ +00013ef0: 7661 6c75 6520 730a 2020 2a20 4f75 7470 value s. * Outp │ │ │ │ +00013f00: 7574 733a 0a20 2020 2020 202a 2064 302c uts:. * d0, │ │ │ │ +00013f10: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ +00013f20: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +00013f30: 6174 7269 782c 2c20 6d61 7020 6f66 2066 atrix,, map of f │ │ │ │ +00013f40: 7265 6520 6d6f 6475 6c65 7320 6f76 6572 ree modules over │ │ │ │ +00013f50: 2061 6e0a 2020 2020 2020 2020 656e 6c61 an. enla │ │ │ │ +00013f60: 7267 6564 2072 696e 670a 2020 2020 2020 rged ring. │ │ │ │ +00013f70: 2a20 6431 2c20 6120 2a6e 6f74 6520 6d61 * d1, a *note ma │ │ │ │ +00013f80: 7472 6978 3a20 284d 6163 6175 6c61 7932 trix: (Macaulay2 │ │ │ │ +00013f90: 446f 6329 4d61 7472 6978 2c2c 206d 6170 Doc)Matrix,, map │ │ │ │ +00013fa0: 206f 6620 6672 6565 206d 6f64 756c 6573 of free modules │ │ │ │ +00013fb0: 206f 7665 7220 616e 0a20 2020 2020 2020 over an. │ │ │ │ +00013fc0: 2065 6e6c 6172 6765 6420 7269 6e67 0a0a enlarged ring.. │ │ │ │ +00013fd0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00013fe0: 3d3d 3d3d 3d3d 3d0a 0a41 7373 756d 6520 =======..Assume │ │ │ │ +00013ff0: 7468 6174 204d 2069 7320 6465 6669 6e65 that M is define │ │ │ │ +00014000: 6420 6f76 6572 2061 2072 696e 6720 6f66 d over a ring of │ │ │ │ +00014010: 2074 6865 2066 6f72 6d20 5262 6172 203d the form Rbar = │ │ │ │ +00014020: 2052 2f28 665f 302e 2e66 5f7b 632d 317d R/(f_0..f_{c-1} │ │ │ │ +00014030: 292c 2061 0a63 6f6d 706c 6574 6520 696e ), a.complete in │ │ │ │ +00014040: 7465 7273 6563 7469 6f6e 2c20 616e 6420 tersection, and │ │ │ │ +00014050: 7468 6174 204d 2068 6173 2061 2066 696e that M has a fin │ │ │ │ +00014060: 6974 6520 6672 6565 2072 6573 6f6c 7574 ite free resolut │ │ │ │ +00014070: 696f 6e20 4720 6f76 6572 2052 2e20 496e ion G over R. In │ │ │ │ +00014080: 0a74 6869 7320 6361 7365 204d 2068 6173 .this case M has │ │ │ │ +00014090: 2061 2066 7265 6520 7265 736f 6c75 7469 a free resoluti │ │ │ │ +000140a0: 6f6e 2046 206f 7665 7220 5262 6172 2077 on F over Rbar w │ │ │ │ +000140b0: 686f 7365 2064 7561 6c2c 2046 5e2a 2069 hose dual, F^* i │ │ │ │ +000140c0: 7320 6120 6669 6e69 7465 6c79 0a67 656e s a finitely.gen │ │ │ │ +000140d0: 6572 6174 6564 2c20 5a2d 6772 6164 6564 erated, Z-graded │ │ │ │ +000140e0: 2066 7265 6520 6d6f 6475 6c65 206f 7665 free module ove │ │ │ │ +000140f0: 7220 6120 7269 6e67 2053 6261 725c 636f r a ring Sbar\co │ │ │ │ +00014100: 6e67 206b 6b5b 735f 302e 2e73 5f7b 632d ng kk[s_0..s_{c- │ │ │ │ +00014110: 317d 2c67 656e 730a 5262 6172 5d2c 2077 1},gens.Rbar], w │ │ │ │ +00014120: 6865 7265 2074 6865 2064 6567 7265 6573 here the degrees │ │ │ │ +00014130: 206f 6620 7468 6520 735f 6920 6172 6520 of the s_i are │ │ │ │ +00014140: 7b2d 322c 202d 6465 6772 6565 2066 5f69 {-2, -degree f_i │ │ │ │ +00014150: 7d2e 2054 6869 7320 7265 736f 6c75 7469 }. This resoluti │ │ │ │ +00014160: 6f6e 2069 730a 6973 2063 6f6e 7374 7275 on is.is constru │ │ │ │ +00014170: 6374 6564 2066 726f 6d20 7468 6520 6475 cted from the du │ │ │ │ +00014180: 616c 206f 6620 472c 2074 6f67 6574 6865 al of G, togethe │ │ │ │ +00014190: 7220 7769 7468 2074 6865 2064 7561 6c73 r with the duals │ │ │ │ +000141a0: 206f 6620 7468 6520 6869 6768 6572 0a68 of the higher.h │ │ │ │ +000141b0: 6f6d 6f74 6f70 6965 7320 6f6e 2047 2064 omotopies on G d │ │ │ │ +000141c0: 6566 696e 6564 2062 7920 4569 7365 6e62 efined by Eisenb │ │ │ │ +000141d0: 7564 2e0a 0a54 6865 2066 756e 6374 696f ud...The functio │ │ │ │ +000141e0: 6e20 7265 7475 726e 7320 7468 6520 6469 n returns the di │ │ │ │ +000141f0: 6666 6572 656e 7469 616c 7320 6430 3a46 fferentials d0:F │ │ │ │ +00014200: 5e2a 5f7b 6576 656e 7d20 5c74 6f20 465e ^*_{even} \to F^ │ │ │ │ +00014210: 2a5f 7b6f 6464 7d20 616e 640a 6431 3a46 *_{odd} and.d1:F │ │ │ │ +00014220: 5e2a 5f7b 6f64 647d 5c74 6f20 465e 2a5f ^*_{odd}\to F^*_ │ │ │ │ +00014230: 7b65 7665 6e7d 2e0a 0a54 6865 206d 6170 {even}...The map │ │ │ │ +00014240: 7320 6430 2c64 3120 666f 726d 2061 206d s d0,d1 form a m │ │ │ │ +00014250: 6174 7269 7820 6661 6374 6f72 697a 6174 atrix factorizat │ │ │ │ +00014260: 696f 6e20 6f66 2073 756d 2863 2c20 692d ion of sum(c, i- │ │ │ │ +00014270: 3e73 5f69 2a66 5f69 292e 2054 6865 2068 >s_i*f_i). The h │ │ │ │ +00014280: 6176 6520 7468 650a 7072 6f70 6572 7479 ave the.property │ │ │ │ +00014290: 2074 6861 7420 666f 7220 616e 7920 5262 that for any Rb │ │ │ │ +000142a0: 6172 206d 6f64 756c 6520 4e2c 0a0a 4848 ar module N,..HH │ │ │ │ +000142b0: 5f31 2063 6f6d 706c 6578 205c 7b64 302a _1 complex \{d0* │ │ │ │ +000142c0: 2a4e 2c20 6431 2a2a 4e5c 7d20 3d20 4578 *N, d1**N\} = Ex │ │ │ │ +000142d0: 745e 7b65 7665 6e7d 5f7b 5262 6172 7d28 t^{even}_{Rbar}( │ │ │ │ +000142e0: 4d2c 4e29 0a0a 535e 7b7b 312c 307d 7d2a M,N)..S^{{1,0}}* │ │ │ │ +000142f0: 2a48 485f 3120 636f 6d70 6c65 7820 5c7b *HH_1 complex \{ │ │ │ │ +00014300: 535e 7b7b 2d32 2c30 7d7d 2a2a 6431 2a2a S^{{-2,0}}**d1** │ │ │ │ +00014310: 4e2c 2064 302a 2a4e 5c7d 203d 2045 7874 N, d0**N\} = Ext │ │ │ │ +00014320: 5e7b 6f64 647d 5f7b 5262 6172 7d28 4d2c ^{odd}_{Rbar}(M, │ │ │ │ +00014330: 4e29 0a0a 5468 6973 2069 7320 656e 636f N)..This is enco │ │ │ │ +00014340: 6465 6420 696e 2074 6865 2073 6372 6970 ded in the scrip │ │ │ │ +00014350: 7420 6e65 7745 7874 0a0a 4f70 7469 6f6e t newExt..Option │ │ │ │ +00014360: 2064 6566 6175 6c74 733a 2043 6865 636b defaults: Check │ │ │ │ +00014370: 3d3e 6661 6c73 6520 5661 7269 6162 6c65 =>false Variable │ │ │ │ +00014380: 733d 3e67 6574 5379 6d62 6f6c 2022 7322 s=>getSymbol "s" │ │ │ │ +00014390: 2c20 4772 6164 696e 6720 3d3e 327d 0a0a , Grading =>2}.. │ │ │ │ +000143a0: 4966 2047 7261 6469 6e67 203d 3e31 2c20 If Grading =>1, │ │ │ │ +000143b0: 7468 656e 2061 2073 696e 676c 7920 6772 then a singly gr │ │ │ │ +000143c0: 6164 6564 2072 6573 756c 7420 6973 2072 aded result is r │ │ │ │ +000143d0: 6574 7572 6e65 6420 286a 7573 7420 666f eturned (just fo │ │ │ │ +000143e0: 7267 6574 7469 6e67 2074 6865 0a68 6f6d rgetting the.hom │ │ │ │ +000143f0: 6f6c 6f67 6963 616c 2067 7261 6469 6e67 ological grading │ │ │ │ +00014400: 2e29 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d .)....+--------- │ │ │ │ +00014410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00014460: 0a7c 6931 203a 206e 203d 2033 2020 2020 .|i1 : n = 3 │ │ │ │ +00014450: 2d2d 2d2d 2b0a 7c69 3120 3a20 6e20 3d20 ----+.|i1 : n = │ │ │ │ +00014460: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00014470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000144a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000144b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000144a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000144b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000144f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014500: 0a7c 6f31 203d 2033 2020 2020 2020 2020 .|o1 = 3 │ │ │ │ +000144f0: 2020 2020 7c0a 7c6f 3120 3d20 3320 2020 |.|o1 = 3 │ │ │ │ +00014500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014540: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014550: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014540: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000145a0: 0a7c 6932 203a 2063 203d 2032 2020 2020 .|i2 : c = 2 │ │ │ │ +00014590: 2d2d 2d2d 2b0a 7c69 3220 3a20 6320 3d20 ----+.|i2 : c = │ │ │ │ +000145a0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 000145b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000145c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000145d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000145e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000145f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000145e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000145f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014630: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014640: 0a7c 6f32 203d 2032 2020 2020 2020 2020 .|o2 = 2 │ │ │ │ +00014630: 2020 2020 7c0a 7c6f 3220 3d20 3220 2020 |.|o2 = 2 │ │ │ │ +00014640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014680: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014690: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014680: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000146a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000146b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000146c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000146d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000146e0: 0a7c 6933 203a 206b 6b20 3d20 5a5a 2f31 .|i3 : kk = ZZ/1 │ │ │ │ -000146f0: 3031 2020 2020 2020 2020 2020 2020 2020 01 │ │ │ │ +000146d0: 2d2d 2d2d 2b0a 7c69 3320 3a20 6b6b 203d ----+.|i3 : kk = │ │ │ │ +000146e0: 205a 5a2f 3130 3120 2020 2020 2020 2020 ZZ/101 │ │ │ │ +000146f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014720: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014730: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014720: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014770: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014780: 0a7c 6f33 203d 206b 6b20 2020 2020 2020 .|o3 = kk │ │ │ │ +00014770: 2020 2020 7c0a 7c6f 3320 3d20 6b6b 2020 |.|o3 = kk │ │ │ │ +00014780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000147a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000147b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000147c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000147d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000147c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000147d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000147e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000147f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014810: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014820: 0a7c 6f33 203a 2051 756f 7469 656e 7452 .|o3 : QuotientR │ │ │ │ -00014830: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ +00014810: 2020 2020 7c0a 7c6f 3320 3a20 5175 6f74 |.|o3 : Quot │ │ │ │ +00014820: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +00014830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014860: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014870: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014860: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000148a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000148b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000148c0: 0a7c 6934 203a 2052 203d 206b 6b5b 785f .|i4 : R = kk[x_ │ │ │ │ -000148d0: 302e 2e78 5f28 6e2d 3129 5d20 2020 2020 0..x_(n-1)] │ │ │ │ +000148b0: 2d2d 2d2d 2b0a 7c69 3420 3a20 5220 3d20 ----+.|i4 : R = │ │ │ │ +000148c0: 6b6b 5b78 5f30 2e2e 785f 286e 2d31 295d kk[x_0..x_(n-1)] │ │ │ │ +000148d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000148e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000148f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014900: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014910: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014900: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014950: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014960: 0a7c 6f34 203d 2052 2020 2020 2020 2020 .|o4 = R │ │ │ │ +00014950: 2020 2020 7c0a 7c6f 3420 3d20 5220 2020 |.|o4 = R │ │ │ │ +00014960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000149a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000149b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000149a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000149b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000149f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014a00: 0a7c 6f34 203a 2050 6f6c 796e 6f6d 6961 .|o4 : Polynomia │ │ │ │ -00014a10: 6c52 696e 6720 2020 2020 2020 2020 2020 lRing │ │ │ │ +000149f0: 2020 2020 7c0a 7c6f 3420 3a20 506f 6c79 |.|o4 : Poly │ │ │ │ +00014a00: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ +00014a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014a40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014a50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014a40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00014aa0: 0a7c 6935 203a 2049 203d 2069 6465 616c .|i5 : I = ideal │ │ │ │ -00014ab0: 2878 5f30 5e32 2c20 785f 325e 3329 2020 (x_0^2, x_2^3) │ │ │ │ +00014a90: 2d2d 2d2d 2b0a 7c69 3520 3a20 4920 3d20 ----+.|i5 : I = │ │ │ │ +00014aa0: 6964 6561 6c28 785f 305e 322c 2078 5f32 ideal(x_0^2, x_2 │ │ │ │ +00014ab0: 5e33 2920 2020 2020 2020 2020 2020 2020 ^3) │ │ │ │ 00014ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014ae0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014af0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014ae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014b30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014b40: 0a7c 2020 2020 2020 2020 2020 2020 2032 .| 2 │ │ │ │ -00014b50: 2020 2033 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00014b30: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014b40: 2020 2020 3220 2020 3320 2020 2020 2020 2 3 │ │ │ │ +00014b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014b80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014b90: 0a7c 6f35 203d 2069 6465 616c 2028 7820 .|o5 = ideal (x │ │ │ │ -00014ba0: 2c20 7820 2920 2020 2020 2020 2020 2020 , x ) │ │ │ │ +00014b80: 2020 2020 7c0a 7c6f 3520 3d20 6964 6561 |.|o5 = idea │ │ │ │ +00014b90: 6c20 2878 202c 2078 2029 2020 2020 2020 l (x , x ) │ │ │ │ +00014ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014bd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014be0: 0a7c 2020 2020 2020 2020 2020 2020 2030 .| 0 │ │ │ │ -00014bf0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00014bd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014be0: 2020 2020 3020 2020 3220 2020 2020 2020 0 2 │ │ │ │ +00014bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014c20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014c30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014c20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014c70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014c80: 0a7c 6f35 203a 2049 6465 616c 206f 6620 .|o5 : Ideal of │ │ │ │ -00014c90: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +00014c70: 2020 2020 7c0a 7c6f 3520 3a20 4964 6561 |.|o5 : Idea │ │ │ │ +00014c80: 6c20 6f66 2052 2020 2020 2020 2020 2020 l of R │ │ │ │ +00014c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014cc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014cd0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014cc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00014d20: 0a7c 6936 203a 2066 6620 3d20 6765 6e73 .|i6 : ff = gens │ │ │ │ -00014d30: 2049 2020 2020 2020 2020 2020 2020 2020 I │ │ │ │ +00014d10: 2d2d 2d2d 2b0a 7c69 3620 3a20 6666 203d ----+.|i6 : ff = │ │ │ │ +00014d20: 2067 656e 7320 4920 2020 2020 2020 2020 gens I │ │ │ │ +00014d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014d60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014d70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014d60: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014db0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014dc0: 0a7c 6f36 203d 207c 2078 5f30 5e32 2078 .|o6 = | x_0^2 x │ │ │ │ -00014dd0: 5f32 5e33 207c 2020 2020 2020 2020 2020 _2^3 | │ │ │ │ +00014db0: 2020 2020 7c0a 7c6f 3620 3d20 7c20 785f |.|o6 = | x_ │ │ │ │ +00014dc0: 305e 3220 785f 325e 3320 7c20 2020 2020 0^2 x_2^3 | │ │ │ │ +00014dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014e00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014e10: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014e00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014e50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014e60: 0a7c 2020 2020 2020 2020 2020 2020 2031 .| 1 │ │ │ │ -00014e70: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +00014e50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014e60: 2020 2020 3120 2020 2020 2032 2020 2020 1 2 │ │ │ │ +00014e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014ea0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014eb0: 0a7c 6f36 203a 204d 6174 7269 7820 5220 .|o6 : Matrix R │ │ │ │ -00014ec0: 203c 2d2d 2052 2020 2020 2020 2020 2020 <-- R │ │ │ │ +00014ea0: 2020 2020 7c0a 7c6f 3620 3a20 4d61 7472 |.|o6 : Matr │ │ │ │ +00014eb0: 6978 2052 2020 3c2d 2d20 5220 2020 2020 ix R <-- R │ │ │ │ +00014ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014ef0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014f00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014ef0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00014f50: 0a7c 6937 203a 2052 6261 7220 3d20 522f .|i7 : Rbar = R/ │ │ │ │ -00014f60: 4920 2020 2020 2020 2020 2020 2020 2020 I │ │ │ │ +00014f40: 2d2d 2d2d 2b0a 7c69 3720 3a20 5262 6172 ----+.|i7 : Rbar │ │ │ │ +00014f50: 203d 2052 2f49 2020 2020 2020 2020 2020 = R/I │ │ │ │ +00014f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014f90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014fa0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014f90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014fe0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014ff0: 0a7c 6f37 203d 2052 6261 7220 2020 2020 .|o7 = Rbar │ │ │ │ +00014fe0: 2020 2020 7c0a 7c6f 3720 3d20 5262 6172 |.|o7 = Rbar │ │ │ │ +00014ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015030: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015040: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015030: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015080: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015090: 0a7c 6f37 203a 2051 756f 7469 656e 7452 .|o7 : QuotientR │ │ │ │ -000150a0: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ +00015080: 2020 2020 7c0a 7c6f 3720 3a20 5175 6f74 |.|o7 : Quot │ │ │ │ +00015090: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +000150a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000150b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000150c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000150d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000150e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000150d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000150e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000150f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00015130: 0a7c 6938 203a 2062 6172 203d 206d 6170 .|i8 : bar = map │ │ │ │ -00015140: 2852 6261 722c 2052 2920 2020 2020 2020 (Rbar, R) │ │ │ │ +00015120: 2d2d 2d2d 2b0a 7c69 3820 3a20 6261 7220 ----+.|i8 : bar │ │ │ │ +00015130: 3d20 6d61 7028 5262 6172 2c20 5229 2020 = map(Rbar, R) │ │ │ │ +00015140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015170: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015180: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015170: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000151a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000151b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000151c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000151d0: 0a7c 6f38 203d 206d 6170 2028 5262 6172 .|o8 = map (Rbar │ │ │ │ -000151e0: 2c20 522c 207b 7820 2c20 7820 2c20 7820 , R, {x , x , x │ │ │ │ -000151f0: 7d29 2020 2020 2020 2020 2020 2020 2020 }) │ │ │ │ +000151c0: 2020 2020 7c0a 7c6f 3820 3d20 6d61 7020 |.|o8 = map │ │ │ │ +000151d0: 2852 6261 722c 2052 2c20 7b78 202c 2078 (Rbar, R, {x , x │ │ │ │ +000151e0: 202c 2078 207d 2920 2020 2020 2020 2020 , x }) │ │ │ │ +000151f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015210: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015220: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00015230: 2020 2020 2020 2030 2020 2031 2020 2032 0 1 2 │ │ │ │ +00015210: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015220: 2020 2020 2020 2020 2020 2020 3020 2020 0 │ │ │ │ +00015230: 3120 2020 3220 2020 2020 2020 2020 2020 1 2 │ │ │ │ 00015240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015260: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015270: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015260: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000152a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000152b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000152c0: 0a7c 6f38 203a 2052 696e 674d 6170 2052 .|o8 : RingMap R │ │ │ │ -000152d0: 6261 7220 3c2d 2d20 5220 2020 2020 2020 bar <-- R │ │ │ │ +000152b0: 2020 2020 7c0a 7c6f 3820 3a20 5269 6e67 |.|o8 : Ring │ │ │ │ +000152c0: 4d61 7020 5262 6172 203c 2d2d 2052 2020 Map Rbar <-- R │ │ │ │ +000152d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000152e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000152f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015300: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015310: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00015300: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00015310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00015360: 0a7c 6939 203a 204d 6261 7220 3d20 7072 .|i9 : Mbar = pr │ │ │ │ -00015370: 756e 6520 636f 6b65 7220 7261 6e64 6f6d une coker random │ │ │ │ -00015380: 2852 6261 725e 312c 2052 6261 725e 7b2d (Rbar^1, Rbar^{- │ │ │ │ -00015390: 327d 2920 2020 2020 2020 2020 2020 2020 2}) │ │ │ │ -000153a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000153b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015350: 2d2d 2d2d 2b0a 7c69 3920 3a20 4d62 6172 ----+.|i9 : Mbar │ │ │ │ +00015360: 203d 2070 7275 6e65 2063 6f6b 6572 2072 = prune coker r │ │ │ │ +00015370: 616e 646f 6d28 5262 6172 5e31 2c20 5262 andom(Rbar^1, Rb │ │ │ │ +00015380: 6172 5e7b 2d32 7d29 2020 2020 2020 2020 ar^{-2}) │ │ │ │ +00015390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000153a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000153b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000153c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000153d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000153e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000153f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015400: 0a7c 6f39 203d 2063 6f6b 6572 6e65 6c20 .|o9 = cokernel │ │ │ │ -00015410: 7c20 785f 3078 5f31 2b32 3478 5f31 5e32 | x_0x_1+24x_1^2 │ │ │ │ -00015420: 2b34 3978 5f30 785f 322b 3378 5f31 785f +49x_0x_2+3x_1x_ │ │ │ │ -00015430: 322b 3578 5f32 5e32 207c 2020 2020 2020 2+5x_2^2 | │ │ │ │ -00015440: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015450: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000153f0: 2020 2020 7c0a 7c6f 3920 3d20 636f 6b65 |.|o9 = coke │ │ │ │ +00015400: 726e 656c 207c 2078 5f30 785f 312b 3234 rnel | x_0x_1+24 │ │ │ │ +00015410: 785f 315e 322b 3439 785f 3078 5f32 2b33 x_1^2+49x_0x_2+3 │ │ │ │ +00015420: 785f 3178 5f32 2b35 785f 325e 3220 7c20 x_1x_2+5x_2^2 | │ │ │ │ +00015430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015440: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015490: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000154a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000154b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000154c0: 2020 2020 3120 2020 2020 2020 2020 2020 1 │ │ │ │ +00015490: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000154a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000154b0: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ +000154c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000154d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000154e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000154f0: 0a7c 6f39 203a 2052 6261 722d 6d6f 6475 .|o9 : Rbar-modu │ │ │ │ -00015500: 6c65 2c20 7175 6f74 6965 6e74 206f 6620 le, quotient of │ │ │ │ -00015510: 5262 6172 2020 2020 2020 2020 2020 2020 Rbar │ │ │ │ +000154e0: 2020 2020 7c0a 7c6f 3920 3a20 5262 6172 |.|o9 : Rbar │ │ │ │ +000154f0: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ +00015500: 7420 6f66 2052 6261 7220 2020 2020 2020 t of Rbar │ │ │ │ +00015510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015530: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015540: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00015530: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00015540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00015590: 0a7c 6931 3020 3a20 2864 302c 6431 2920 .|i10 : (d0,d1) │ │ │ │ -000155a0: 3d20 4569 7365 6e62 7564 5368 616d 6173 = EisenbudShamas │ │ │ │ -000155b0: 6854 6f74 616c 284d 6261 722c 4772 6164 hTotal(Mbar,Grad │ │ │ │ -000155c0: 696e 6720 3d3e 3129 2020 2020 2020 2020 ing =>1) │ │ │ │ -000155d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000155e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015580: 2d2d 2d2d 2b0a 7c69 3130 203a 2028 6430 ----+.|i10 : (d0 │ │ │ │ +00015590: 2c64 3129 203d 2045 6973 656e 6275 6453 ,d1) = EisenbudS │ │ │ │ +000155a0: 6861 6d61 7368 546f 7461 6c28 4d62 6172 hamashTotal(Mbar │ │ │ │ +000155b0: 2c47 7261 6469 6e67 203d 3e31 2920 2020 ,Grading =>1) │ │ │ │ +000155c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000155d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000155e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000155f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015620: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015630: 0a7c 6f31 3020 3d20 287b 2d32 7d20 7c20 .|o10 = ({-2} | │ │ │ │ -00015640: 785f 305e 3220 2020 2020 2020 2020 2020 x_0^2 │ │ │ │ -00015650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015660: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ -00015670: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015680: 0a7c 2020 2020 2020 207b 2d32 7d20 7c20 .| {-2} | │ │ │ │ -00015690: 785f 3078 5f31 2b32 3478 5f31 5e32 2b34 x_0x_1+24x_1^2+4 │ │ │ │ -000156a0: 3978 5f30 785f 322b 3378 5f31 785f 322b 9x_0x_2+3x_1x_2+ │ │ │ │ -000156b0: 3578 5f32 5e32 2033 3073 5f30 2020 2020 5x_2^2 30s_0 │ │ │ │ -000156c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000156d0: 0a7c 2020 2020 2020 207b 2d33 7d20 7c20 .| {-3} | │ │ │ │ -000156e0: 785f 325e 3320 2020 2020 2020 2020 2020 x_2^3 │ │ │ │ -000156f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015700: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ -00015710: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015720: 0a7c 2020 2020 2020 207b 2d37 7d20 7c20 .| {-7} | │ │ │ │ -00015730: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ -00015740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015750: 2020 2020 2020 2078 5f32 5e33 2020 2020 x_2^3 │ │ │ │ -00015760: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015770: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ +00015620: 2020 2020 7c0a 7c6f 3130 203d 2028 7b2d |.|o10 = ({- │ │ │ │ +00015630: 327d 207c 2078 5f30 5e32 2020 2020 2020 2} | x_0^2 │ │ │ │ +00015640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015650: 2020 2020 2020 2020 2020 2020 3020 2020 0 │ │ │ │ +00015660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015670: 2020 2020 7c0a 7c20 2020 2020 2020 7b2d |.| {- │ │ │ │ +00015680: 327d 207c 2078 5f30 785f 312b 3234 785f 2} | x_0x_1+24x_ │ │ │ │ +00015690: 315e 322b 3439 785f 3078 5f32 2b33 785f 1^2+49x_0x_2+3x_ │ │ │ │ +000156a0: 3178 5f32 2b35 785f 325e 3220 3330 735f 1x_2+5x_2^2 30s_ │ │ │ │ +000156b0: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +000156c0: 2020 2020 7c0a 7c20 2020 2020 2020 7b2d |.| {- │ │ │ │ +000156d0: 337d 207c 2078 5f32 5e33 2020 2020 2020 3} | x_2^3 │ │ │ │ +000156e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000156f0: 2020 2020 2020 2020 2020 2020 3020 2020 0 │ │ │ │ +00015700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015710: 2020 2020 7c0a 7c20 2020 2020 2020 7b2d |.| {- │ │ │ │ +00015720: 377d 207c 2030 2020 2020 2020 2020 2020 7} | 0 │ │ │ │ +00015730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015740: 2020 2020 2020 2020 2020 2020 785f 325e x_2^ │ │ │ │ +00015750: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00015760: 2020 2020 7c0a 7c20 2020 2020 202d 2d2d |.| --- │ │ │ │ +00015770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000157a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000157b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -000157c0: 0a7c 2020 2020 2020 2d73 5f31 2020 2020 .| -s_1 │ │ │ │ +000157b0: 2d2d 2d2d 7c0a 7c20 2020 2020 202d 735f ----|.| -s_ │ │ │ │ +000157c0: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 000157d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000157e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000157f0: 2020 3020 2020 2020 2020 207c 2c20 7b30 0 |, {0 │ │ │ │ -00015800: 7d20 207c 2020 2020 2020 2020 2020 207c } | | │ │ │ │ -00015810: 0a7c 2020 2020 2020 3020 2020 2020 2020 .| 0 │ │ │ │ +000157e0: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ +000157f0: 7c2c 207b 307d 2020 7c20 2020 2020 2020 |, {0} | │ │ │ │ +00015800: 2020 2020 7c0a 7c20 2020 2020 2030 2020 |.| 0 │ │ │ │ +00015810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015840: 2020 2d73 5f31 2020 2020 207c 2020 7b2d -s_1 | {- │ │ │ │ -00015850: 347d 207c 2020 2020 2020 2020 2020 207c 4} | | │ │ │ │ -00015860: 0a7c 2020 2020 2020 735f 3020 2020 2020 .| s_0 │ │ │ │ +00015830: 2020 2020 2020 202d 735f 3120 2020 2020 -s_1 │ │ │ │ +00015840: 7c20 207b 2d34 7d20 7c20 2020 2020 2020 | {-4} | │ │ │ │ +00015850: 2020 2020 7c0a 7c20 2020 2020 2073 5f30 |.| s_0 │ │ │ │ +00015860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015890: 2020 3020 2020 2020 2020 207c 2020 7b2d 0 | {- │ │ │ │ -000158a0: 357d 207c 2020 2020 2020 2020 2020 207c 5} | | │ │ │ │ -000158b0: 0a7c 2020 2020 2020 3337 785f 3078 5f31 .| 37x_0x_1 │ │ │ │ -000158c0: 2d32 3178 5f31 5e32 2d35 785f 3078 5f32 -21x_1^2-5x_0x_2 │ │ │ │ -000158d0: 2b31 3078 5f31 785f 322d 3137 785f 325e +10x_1x_2-17x_2^ │ │ │ │ -000158e0: 3220 2d33 3778 5f30 5e32 207c 2020 7b2d 2 -37x_0^2 | {- │ │ │ │ -000158f0: 357d 207c 2020 2020 2020 2020 2020 207c 5} | | │ │ │ │ -00015900: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ +00015880: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ +00015890: 7c20 207b 2d35 7d20 7c20 2020 2020 2020 | {-5} | │ │ │ │ +000158a0: 2020 2020 7c0a 7c20 2020 2020 2033 3778 |.| 37x │ │ │ │ +000158b0: 5f30 785f 312d 3231 785f 315e 322d 3578 _0x_1-21x_1^2-5x │ │ │ │ +000158c0: 5f30 785f 322b 3130 785f 3178 5f32 2d31 _0x_2+10x_1x_2-1 │ │ │ │ +000158d0: 3778 5f32 5e32 202d 3337 785f 305e 3220 7x_2^2 -37x_0^2 │ │ │ │ +000158e0: 7c20 207b 2d35 7d20 7c20 2020 2020 2020 | 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| │ │ │ │ -00016580: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016570: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000165a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000165b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000165c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000165d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000165e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000165f0: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -00016600: 2020 2020 2020 2020 2020 2020 2034 2020 4 │ │ │ │ -00016610: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016620: 0a7c 6f31 3220 3a20 4d61 7472 6978 2028 .|o12 : Matrix ( │ │ │ │ -00016630: 6b6b 5b73 202e 2e73 202c 2078 202e 2e78 kk[s ..s , x ..x │ │ │ │ -00016640: 205d 2920 203c 2d2d 2028 6b6b 5b73 202e ]) <-- (kk[s . │ │ │ │ -00016650: 2e73 202c 2078 202e 2e78 205d 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2020 2020 2020 2020 │ │ │ │ 00016c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016c50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016c60: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00016c70: 3020 2020 3220 2020 2020 2020 2020 2020 0 2 │ │ │ │ +00016c50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016c60: 2020 2020 2030 2020 2032 2020 2020 2020 0 2 │ │ │ │ +00016c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ca0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016cb0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016ca0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016cf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016d00: 0a7c 6f31 3520 3a20 4964 6561 6c20 6f66 .|o15 : Ideal of │ │ │ │ -00016d10: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +00016cf0: 2020 2020 7c0a 7c6f 3135 203a 2049 6465 |.|o15 : Ide │ │ │ │ +00016d00: 616c 206f 6620 5320 2020 2020 2020 2020 al of S │ │ │ │ +00016d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016d40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016d50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00016d40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00016d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00016da0: 0a7c 6931 3620 3a20 5362 6172 203d 2053 .|i16 : Sbar = S │ │ │ │ -00016db0: 2f49 5320 2020 2020 2020 2020 2020 2020 /IS │ │ │ │ +00016d90: 2d2d 2d2d 2b0a 7c69 3136 203a 2053 6261 ----+.|i16 : Sba │ │ │ │ +00016da0: 7220 3d20 532f 4953 2020 2020 2020 2020 r = S/IS │ │ │ │ +00016db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016de0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016df0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016de0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016e40: 0a7c 6f31 3620 3d20 5362 6172 2020 2020 .|o16 = Sbar │ │ │ │ +00016e30: 2020 2020 7c0a 7c6f 3136 203d 2053 6261 |.|o16 = Sba │ │ │ │ +00016e40: 7220 2020 2020 2020 2020 2020 2020 2020 r │ │ │ │ 00016e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016e90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016e80: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ed0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016ee0: 0a7c 6f31 3620 3a20 5175 6f74 6965 6e74 .|o16 : Quotient │ │ │ │ -00016ef0: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ +00016ed0: 2020 2020 7c0a 7c6f 3136 203a 2051 756f |.|o16 : Quo │ │ │ │ +00016ee0: 7469 656e 7452 696e 6720 2020 2020 2020 tientRing │ │ │ │ +00016ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016f20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016f30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00016f20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00016f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00016f80: 0a7c 6931 3720 3a20 534d 6261 7220 3d20 .|i17 : SMbar = │ │ │ │ -00016f90: 5362 6172 2a2a 4d62 6172 2020 2020 2020 Sbar**Mbar │ │ │ │ +00016f70: 2d2d 2d2d 2b0a 7c69 3137 203a 2053 4d62 ----+.|i17 : SMb │ │ │ │ +00016f80: 6172 203d 2053 6261 722a 2a4d 6261 7220 ar = Sbar**Mbar │ │ │ │ +00016f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016fc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016fd0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016fc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017010: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017020: 0a7c 6f31 3720 3d20 636f 6b65 726e 656c .|o17 = cokernel │ │ │ │ -00017030: 207c 2078 5f30 785f 312b 3234 785f 315e | x_0x_1+24x_1^ │ │ │ │ -00017040: 322b 3439 785f 3078 5f32 2b33 785f 3178 2+49x_0x_2+3x_1x │ │ │ │ -00017050: 5f32 2b35 785f 325e 3220 7c20 2020 2020 _2+5x_2^2 | │ │ │ │ -00017060: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017070: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00017010: 2020 2020 7c0a 7c6f 3137 203d 2063 6f6b |.|o17 = cok │ │ │ │ +00017020: 6572 6e65 6c20 7c20 785f 3078 5f31 2b32 ernel | x_0x_1+2 │ │ │ │ +00017030: 3478 5f31 5e32 2b34 3978 5f30 785f 322b 4x_1^2+49x_0x_2+ │ │ │ │ +00017040: 3378 5f31 785f 322b 3578 5f32 5e32 207c 3x_1x_2+5x_2^2 | │ │ │ │ +00017050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017060: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00017070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000170a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000170b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000170c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000170d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000170e0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +000170b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000170c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000170d0: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +000170e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000170f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017100: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017110: 0a7c 6f31 3720 3a20 5362 6172 2d6d 6f64 .|o17 : Sbar-mod │ │ │ │ -00017120: 756c 652c 2071 756f 7469 656e 7420 6f66 ule, quotient of │ │ │ │ -00017130: 2053 6261 7220 2020 2020 2020 2020 2020 Sbar │ │ │ │ +00017100: 2020 2020 7c0a 7c6f 3137 203a 2053 6261 |.|o17 : Sba │ │ │ │ +00017110: 722d 6d6f 6475 6c65 2c20 7175 6f74 6965 r-module, quotie │ │ │ │ +00017120: 6e74 206f 6620 5362 6172 2020 2020 2020 nt of Sbar │ │ │ │ +00017130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017150: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017160: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00017150: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00017160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000171a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000171b0: 0a0a 486f 6d28 6430 2c53 6261 7229 2061 ..Hom(d0,Sbar) a │ │ │ │ -000171c0: 6e64 2048 6f6d 2864 312c 5362 6172 2920 nd Hom(d1,Sbar) │ │ │ │ -000171d0: 746f 6765 7468 6572 2066 6f72 6d20 7468 together form th │ │ │ │ -000171e0: 6520 7265 736f 6c75 7469 6f6e 206f 6620 e resolution of │ │ │ │ -000171f0: 4d62 6172 3b20 7468 7573 2074 6865 0a68 Mbar; thus the.h │ │ │ │ -00017200: 6f6d 6f6c 6f67 7920 6f66 206f 6e65 2063 omology of one c │ │ │ │ -00017210: 6f6d 706f 7369 7469 6f6e 2069 7320 302c omposition is 0, │ │ │ │ -00017220: 2077 6869 6c65 2074 6865 206f 7468 6572 while the other │ │ │ │ -00017230: 2069 7320 4d62 6172 0a0a 2b2d 2d2d 2d2d is Mbar..+----- │ │ │ │ +000171a0: 2d2d 2d2d 2b0a 0a48 6f6d 2864 302c 5362 ----+..Hom(d0,Sb │ │ │ │ +000171b0: 6172 2920 616e 6420 486f 6d28 6431 2c53 ar) and Hom(d1,S │ │ │ │ +000171c0: 6261 7229 2074 6f67 6574 6865 7220 666f bar) together fo │ │ │ │ +000171d0: 726d 2074 6865 2072 6573 6f6c 7574 696f rm the resolutio │ │ │ │ +000171e0: 6e20 6f66 204d 6261 723b 2074 6875 7320 n of Mbar; thus │ │ │ │ +000171f0: 7468 650a 686f 6d6f 6c6f 6779 206f 6620 the.homology of │ │ │ │ +00017200: 6f6e 6520 636f 6d70 6f73 6974 696f 6e20 one composition │ │ │ │ +00017210: 6973 2030 2c20 7768 696c 6520 7468 6520 is 0, while the │ │ │ │ +00017220: 6f74 6865 7220 6973 204d 6261 720a 0a2b other is Mbar..+ │ │ │ │ +00017230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017280: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a --------+.|i18 : │ │ │ │ -00017290: 2070 7275 6e65 2048 485f 3120 636f 6d70 prune HH_1 comp │ │ │ │ -000172a0: 6c65 787b 6475 616c 2028 5362 6172 2a2a lex{dual (Sbar** │ │ │ │ -000172b0: 6430 292c 2064 7561 6c28 5362 6172 2a2a d0), dual(Sbar** │ │ │ │ -000172c0: 6431 297d 203d 3d20 3020 2020 2020 2020 d1)} == 0 │ │ │ │ -000172d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00017270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00017280: 6931 3820 3a20 7072 756e 6520 4848 5f31 i18 : prune HH_1 │ │ │ │ +00017290: 2063 6f6d 706c 6578 7b64 7561 6c20 2853 complex{dual (S │ │ │ │ +000172a0: 6261 722a 2a64 3029 2c20 6475 616c 2853 bar**d0), dual(S │ │ │ │ +000172b0: 6261 722a 2a64 3129 7d20 3d3d 2030 2020 bar**d1)} == 0 │ │ │ │ +000172c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000172d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000172e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000172f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017320: 2020 2020 2020 2020 7c0a 7c6f 3138 203d |.|o18 = │ │ │ │ -00017330: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ +00017310: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017320: 6f31 3820 3d20 7472 7565 2020 2020 2020 o18 = true │ │ │ │ +00017330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017370: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00017360: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00017370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000173a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000173b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000173c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a --------+.|i19 : │ │ │ │ -000173d0: 204d 6261 7227 203d 2053 6261 725e 312f Mbar' = Sbar^1/ │ │ │ │ -000173e0: 2853 6261 725f 302c 2053 6261 725f 3129 (Sbar_0, Sbar_1) │ │ │ │ -000173f0: 2a2a 534d 6261 7220 2020 2020 2020 2020 **SMbar │ │ │ │ -00017400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017410: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000173b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000173c0: 6931 3920 3a20 4d62 6172 2720 3d20 5362 i19 : Mbar' = Sb │ │ │ │ +000173d0: 6172 5e31 2f28 5362 6172 5f30 2c20 5362 ar^1/(Sbar_0, Sb │ │ │ │ +000173e0: 6172 5f31 292a 2a53 4d62 6172 2020 2020 ar_1)**SMbar │ │ │ │ +000173f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017400: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017460: 2020 2020 2020 2020 7c0a 7c6f 3139 203d |.|o19 = │ │ │ │ -00017470: 2063 6f6b 6572 6e65 6c20 7c20 785f 3078 cokernel | x_0x │ │ │ │ -00017480: 5f31 2b32 3478 5f31 5e32 2b34 3978 5f30 _1+24x_1^2+49x_0 │ │ │ │ -00017490: 785f 322b 3378 5f31 785f 322b 3578 5f32 x_2+3x_1x_2+5x_2 │ │ │ │ -000174a0: 5e32 2073 5f30 2073 5f31 207c 2020 2020 ^2 s_0 s_1 | │ │ │ │ -000174b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00017450: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017460: 6f31 3920 3d20 636f 6b65 726e 656c 207c o19 = cokernel | │ │ │ │ +00017470: 2078 5f30 785f 312b 3234 785f 315e 322b x_0x_1+24x_1^2+ │ │ │ │ +00017480: 3439 785f 3078 5f32 2b33 785f 3178 5f32 49x_0x_2+3x_1x_2 │ │ │ │ +00017490: 2b35 785f 325e 3220 735f 3020 735f 3120 +5x_2^2 s_0 s_1 │ │ │ │ +000174a0: 7c20 2020 2020 2020 2020 2020 207c 0a7c | |.| │ │ │ │ +000174b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000174c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000174d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000174e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000174f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017500: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000174f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017520: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ +00017520: 2020 2031 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 00017530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017550: 2020 2020 2020 2020 7c0a 7c6f 3139 203a |.|o19 : │ │ │ │ -00017560: 2053 6261 722d 6d6f 6475 6c65 2c20 7175 Sbar-module, qu │ │ │ │ -00017570: 6f74 6965 6e74 206f 6620 5362 6172 2020 otient of Sbar │ │ │ │ +00017540: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017550: 6f31 3920 3a20 5362 6172 2d6d 6f64 756c o19 : Sbar-modul │ │ │ │ +00017560: 652c 2071 756f 7469 656e 7420 6f66 2053 e, quotient of S │ │ │ │ +00017570: 6261 7220 2020 2020 2020 2020 2020 2020 bar │ │ │ │ 00017580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000175a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00017590: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000175a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000175b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000175c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000175d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000175e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000175f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a --------+.|i20 : │ │ │ │ -00017600: 2069 6465 616c 2070 7265 7365 6e74 6174 ideal presentat │ │ │ │ -00017610: 696f 6e20 7072 756e 6520 4848 5f31 2063 ion prune HH_1 c │ │ │ │ -00017620: 6f6d 706c 6578 7b64 7561 6c20 2853 6261 omplex{dual (Sba │ │ │ │ -00017630: 722a 2a64 3129 2c20 6475 616c 2853 6261 r**d1), dual(Sba │ │ │ │ -00017640: 722a 2a64 3029 7d20 7c0a 7c20 2020 2020 r**d0)} |.| │ │ │ │ +000175e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000175f0: 6932 3020 3a20 6964 6561 6c20 7072 6573 i20 : ideal pres │ │ │ │ +00017600: 656e 7461 7469 6f6e 2070 7275 6e65 2048 entation prune H │ │ │ │ +00017610: 485f 3120 636f 6d70 6c65 787b 6475 616c H_1 complex{dual │ │ │ │ +00017620: 2028 5362 6172 2a2a 6431 292c 2064 7561 (Sbar**d1), dua │ │ │ │ +00017630: 6c28 5362 6172 2a2a 6430 297d 207c 0a7c l(Sbar**d0)} |.| │ │ │ │ +00017640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017690: 2020 2020 2020 2020 7c0a 7c6f 3230 203d |.|o20 = │ │ │ │ -000176a0: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ +00017680: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017690: 6f32 3020 3d20 7472 7565 2020 2020 2020 o20 = true │ │ │ │ +000176a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000176b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000176c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000176d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000176e0: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d |.|----- │ │ │ │ +000176d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000176e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000176f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017730: 2d2d 2d2d 2d2d 2d2d 7c0a 7c3d 3d20 6964 --------|.|== id │ │ │ │ -00017740: 6561 6c20 7072 6573 656e 7461 7469 6f6e eal presentation │ │ │ │ -00017750: 204d 6261 7227 2020 2020 2020 2020 2020 Mbar' │ │ │ │ +00017720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +00017730: 3d3d 2069 6465 616c 2070 7265 7365 6e74 == ideal present │ │ │ │ +00017740: 6174 696f 6e20 4d62 6172 2720 2020 2020 ation Mbar' │ │ │ │ +00017750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017780: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00017770: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00017780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000177a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000177b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000177c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000177d0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -000177e0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -000177f0: 2a20 2a6e 6f74 6520 4578 743a 2028 4d61 * *note Ext: (Ma │ │ │ │ -00017800: 6361 756c 6179 3244 6f63 2945 7874 2c20 caulay2Doc)Ext, │ │ │ │ -00017810: 2d2d 2063 6f6d 7075 7465 2061 6e20 4578 -- compute an Ex │ │ │ │ -00017820: 7420 6d6f 6475 6c65 0a20 202a 202a 6e6f t module. * *no │ │ │ │ -00017830: 7465 206e 6577 4578 743a 206e 6577 4578 te newExt: newEx │ │ │ │ -00017840: 742c 202d 2d20 476c 6f62 616c 2045 7874 t, -- Global Ext │ │ │ │ -00017850: 2066 6f72 206d 6f64 756c 6573 206f 7665 for modules ove │ │ │ │ -00017860: 7220 6120 636f 6d70 6c65 7465 0a20 2020 r a complete. │ │ │ │ -00017870: 2049 6e74 6572 7365 6374 696f 6e0a 2020 Intersection. │ │ │ │ -00017880: 2a20 2a6e 6f74 6520 6d61 6b65 486f 6d6f * *note makeHomo │ │ │ │ -00017890: 746f 7069 6573 3a20 6d61 6b65 486f 6d6f topies: makeHomo │ │ │ │ -000178a0: 746f 7069 6573 2c20 2d2d 2072 6574 7572 topies, -- retur │ │ │ │ -000178b0: 6e73 2061 2073 7973 7465 6d20 6f66 2068 ns a system of h │ │ │ │ -000178c0: 6967 6865 720a 2020 2020 686f 6d6f 746f igher. homoto │ │ │ │ -000178d0: 7069 6573 0a0a 5761 7973 2074 6f20 7573 pies..Ways to us │ │ │ │ -000178e0: 6520 4569 7365 6e62 7564 5368 616d 6173 e EisenbudShamas │ │ │ │ -000178f0: 6854 6f74 616c 3a0a 3d3d 3d3d 3d3d 3d3d hTotal:.======== │ │ │ │ -00017900: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00017910: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ -00017920: 4569 7365 6e62 7564 5368 616d 6173 6854 EisenbudShamashT │ │ │ │ -00017930: 6f74 616c 284d 6f64 756c 6529 220a 0a46 otal(Module)"..F │ │ │ │ -00017940: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00017950: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00017960: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00017970: 202a 6e6f 7465 2045 6973 656e 6275 6453 *note EisenbudS │ │ │ │ -00017980: 6861 6d61 7368 546f 7461 6c3a 2045 6973 hamashTotal: Eis │ │ │ │ -00017990: 656e 6275 6453 6861 6d61 7368 546f 7461 enbudShamashTota │ │ │ │ -000179a0: 6c2c 2069 7320 6120 2a6e 6f74 6520 6d65 l, is a *note me │ │ │ │ -000179b0: 7468 6f64 0a66 756e 6374 696f 6e20 7769 thod.function wi │ │ │ │ -000179c0: 7468 206f 7074 696f 6e73 3a20 284d 6163 th options: (Mac │ │ │ │ -000179d0: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -000179e0: 4675 6e63 7469 6f6e 5769 7468 4f70 7469 FunctionWithOpti │ │ │ │ -000179f0: 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ons,...--------- │ │ │ │ +000177c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +000177d0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +000177e0: 3d0a 0a20 202a 202a 6e6f 7465 2045 7874 =.. * *note Ext │ │ │ │ +000177f0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00017800: 4578 742c 202d 2d20 636f 6d70 7574 6520 Ext, -- compute │ │ │ │ +00017810: 616e 2045 7874 206d 6f64 756c 650a 2020 an Ext module. │ │ │ │ +00017820: 2a20 2a6e 6f74 6520 6e65 7745 7874 3a20 * *note newExt: │ │ │ │ +00017830: 6e65 7745 7874 2c20 2d2d 2047 6c6f 6261 newExt, -- Globa │ │ │ │ +00017840: 6c20 4578 7420 666f 7220 6d6f 6475 6c65 l Ext for module │ │ │ │ +00017850: 7320 6f76 6572 2061 2063 6f6d 706c 6574 s over a complet │ │ │ │ +00017860: 650a 2020 2020 496e 7465 7273 6563 7469 e. Intersecti │ │ │ │ +00017870: 6f6e 0a20 202a 202a 6e6f 7465 206d 616b on. * *note mak │ │ │ │ +00017880: 6548 6f6d 6f74 6f70 6965 733a 206d 616b eHomotopies: mak │ │ │ │ +00017890: 6548 6f6d 6f74 6f70 6965 732c 202d 2d20 eHomotopies, -- │ │ │ │ +000178a0: 7265 7475 726e 7320 6120 7379 7374 656d returns a system │ │ │ │ +000178b0: 206f 6620 6869 6768 6572 0a20 2020 2068 of higher. h │ │ │ │ +000178c0: 6f6d 6f74 6f70 6965 730a 0a57 6179 7320 omotopies..Ways │ │ │ │ +000178d0: 746f 2075 7365 2045 6973 656e 6275 6453 to use EisenbudS │ │ │ │ +000178e0: 6861 6d61 7368 546f 7461 6c3a 0a3d 3d3d hamashTotal:.=== │ │ │ │ +000178f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00017900: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +00017910: 2020 2a20 2245 6973 656e 6275 6453 6861 * "EisenbudSha │ │ │ │ +00017920: 6d61 7368 546f 7461 6c28 4d6f 6475 6c65 mashTotal(Module │ │ │ │ +00017930: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ +00017940: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +00017950: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +00017960: 626a 6563 7420 2a6e 6f74 6520 4569 7365 bject *note Eise │ │ │ │ +00017970: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ +00017980: 3a20 4569 7365 6e62 7564 5368 616d 6173 : EisenbudShamas │ │ │ │ +00017990: 6854 6f74 616c 2c20 6973 2061 202a 6e6f hTotal, is a *no │ │ │ │ +000179a0: 7465 206d 6574 686f 640a 6675 6e63 7469 te method.functi │ │ │ │ +000179b0: 6f6e 2077 6974 6820 6f70 7469 6f6e 733a on with options: │ │ │ │ +000179c0: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +000179d0: 6574 686f 6446 756e 6374 696f 6e57 6974 ethodFunctionWit │ │ │ │ +000179e0: 684f 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d hOptions,...---- │ │ │ │ +000179f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017a40: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -00017a50: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -00017a60: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00017a70: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00017a80: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00017a90: 362e 3035 2b64 732f 4d32 2f4d 6163 6175 6.05+ds/M2/Macau │ │ │ │ -00017aa0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ -00017ab0: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -00017ac0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d ionResolutions.m │ │ │ │ -00017ad0: 323a 3531 3639 3a30 2e0a 1f0a 4669 6c65 2:5169:0....File │ │ │ │ -00017ae0: 3a20 436f 6d70 6c65 7465 496e 7465 7273 : CompleteInters │ │ │ │ -00017af0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -00017b00: 732e 696e 666f 2c20 4e6f 6465 3a20 6576 s.info, Node: ev │ │ │ │ -00017b10: 656e 4578 744d 6f64 756c 652c 204e 6578 enExtModule, Nex │ │ │ │ -00017b20: 743a 2065 7870 6f2c 2050 7265 763a 2045 t: expo, Prev: E │ │ │ │ -00017b30: 6973 656e 6275 6453 6861 6d61 7368 546f isenbudShamashTo │ │ │ │ -00017b40: 7461 6c2c 2055 703a 2054 6f70 0a0a 6576 tal, Up: Top..ev │ │ │ │ -00017b50: 656e 4578 744d 6f64 756c 6520 2d2d 2065 enExtModule -- e │ │ │ │ -00017b60: 7665 6e20 7061 7274 206f 6620 4578 745e ven part of Ext^ │ │ │ │ -00017b70: 2a28 4d2c 6b29 206f 7665 7220 6120 636f *(M,k) over a co │ │ │ │ -00017b80: 6d70 6c65 7465 2069 6e74 6572 7365 6374 mplete intersect │ │ │ │ -00017b90: 696f 6e20 6173 206d 6f64 756c 6520 6f76 ion as module ov │ │ │ │ -00017ba0: 6572 2043 4920 6f70 6572 6174 6f72 2072 er CI operator r │ │ │ │ -00017bb0: 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ing.************ │ │ │ │ +00017a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +00017a40: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +00017a50: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +00017a60: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +00017a70: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00017a80: 322d 312e 3236 2e30 352b 6473 2f4d 322f 2-1.26.05+ds/M2/ │ │ │ │ +00017a90: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +00017aa0: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ +00017ab0: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +00017ac0: 6f6e 732e 6d32 3a35 3136 393a 302e 0a1f ons.m2:5169:0... │ │ │ │ +00017ad0: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ +00017ae0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +00017af0: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ +00017b00: 653a 2065 7665 6e45 7874 4d6f 6475 6c65 e: evenExtModule │ │ │ │ +00017b10: 2c20 4e65 7874 3a20 6578 706f 2c20 5072 , Next: expo, Pr │ │ │ │ +00017b20: 6576 3a20 4569 7365 6e62 7564 5368 616d ev: EisenbudSham │ │ │ │ +00017b30: 6173 6854 6f74 616c 2c20 5570 3a20 546f ashTotal, Up: To │ │ │ │ +00017b40: 700a 0a65 7665 6e45 7874 4d6f 6475 6c65 p..evenExtModule │ │ │ │ +00017b50: 202d 2d20 6576 656e 2070 6172 7420 6f66 -- even part of │ │ │ │ +00017b60: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572 Ext^*(M,k) over │ │ │ │ +00017b70: 2061 2063 6f6d 706c 6574 6520 696e 7465 a complete inte │ │ │ │ +00017b80: 7273 6563 7469 6f6e 2061 7320 6d6f 6475 rsection as modu │ │ │ │ +00017b90: 6c65 206f 7665 7220 4349 206f 7065 7261 le over CI opera │ │ │ │ +00017ba0: 746f 7220 7269 6e67 0a2a 2a2a 2a2a 2a2a tor ring.******* │ │ │ │ +00017bb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017bc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017bd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017be0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017bf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00017c00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00017c10: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -00017c20: 7361 6765 3a20 0a20 2020 2020 2020 2045 sage: . E │ │ │ │ -00017c30: 203d 2065 7665 6e45 7874 4d6f 6475 6c65 = evenExtModule │ │ │ │ -00017c40: 204d 0a20 202a 2049 6e70 7574 733a 0a20 M. * Inputs:. │ │ │ │ -00017c50: 2020 2020 202a 204d 2c20 6120 2a6e 6f74 * M, a *not │ │ │ │ -00017c60: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ -00017c70: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ -00017c80: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ -00017c90: 2069 6e74 6572 7365 6374 696f 6e0a 2020 intersection. │ │ │ │ -00017ca0: 2020 2020 2020 7269 6e67 0a20 202a 202a ring. * * │ │ │ │ -00017cb0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e note Optional in │ │ │ │ -00017cc0: 7075 7473 3a20 284d 6163 6175 6c61 7932 puts: (Macaulay2 │ │ │ │ -00017cd0: 446f 6329 7573 696e 6720 6675 6e63 7469 Doc)using functi │ │ │ │ -00017ce0: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ -00017cf0: 6c20 696e 7075 7473 2c3a 0a20 2020 2020 l inputs,:. │ │ │ │ -00017d00: 202a 204f 7574 5269 6e67 203d 3e20 2e2e * OutRing => .. │ │ │ │ -00017d10: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -00017d20: 2030 0a20 202a 204f 7574 7075 7473 3a0a 0. * Outputs:. │ │ │ │ -00017d30: 2020 2020 2020 2a20 452c 2061 202a 6e6f * E, a *no │ │ │ │ -00017d40: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ -00017d50: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ -00017d60: 2c20 6f76 6572 2061 2070 6f6c 796e 6f6d , over a polynom │ │ │ │ -00017d70: 6961 6c20 7269 6e67 2077 6974 680a 2020 ial ring with. │ │ │ │ -00017d80: 2020 2020 2020 6765 6e73 2069 6e20 6465 gens in de │ │ │ │ -00017d90: 6772 6565 2031 0a0a 4465 7363 7269 7074 gree 1..Descript │ │ │ │ -00017da0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -00017db0: 0a45 7874 7261 6374 7320 7468 6520 6576 .Extracts the ev │ │ │ │ -00017dc0: 656e 2064 6567 7265 6520 7061 7274 2066 en degree part f │ │ │ │ -00017dd0: 726f 6d20 4578 744d 6f64 756c 6520 4d20 rom ExtModule M │ │ │ │ -00017de0: 4966 2074 6865 206f 7074 696f 6e61 6c20 If the optional │ │ │ │ -00017df0: 6172 6775 6d65 6e74 204f 7574 5269 6e67 argument OutRing │ │ │ │ -00017e00: 0a3d 3e20 5420 6973 2067 6976 656e 2c20 .=> T is given, │ │ │ │ -00017e10: 616e 6420 636c 6173 7320 5420 3d3d 3d20 and class T === │ │ │ │ -00017e20: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c20 PolynomialRing, │ │ │ │ -00017e30: 7468 656e 2074 6865 206f 7574 7075 7420 then the output │ │ │ │ -00017e40: 7769 6c6c 2062 6520 6120 6d6f 6475 6c65 will be a module │ │ │ │ -00017e50: 0a6f 7665 7220 542e 0a0a 2b2d 2d2d 2d2d .over T...+----- │ │ │ │ +00017c00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +00017c10: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +00017c20: 2020 2020 4520 3d20 6576 656e 4578 744d E = evenExtM │ │ │ │ +00017c30: 6f64 756c 6520 4d0a 2020 2a20 496e 7075 odule M. * Inpu │ │ │ │ +00017c40: 7473 3a0a 2020 2020 2020 2a20 4d2c 2061 ts:. * M, a │ │ │ │ +00017c50: 202a 6e6f 7465 206d 6f64 756c 653a 2028 *note module: ( │ │ │ │ +00017c60: 4d61 6361 756c 6179 3244 6f63 294d 6f64 Macaulay2Doc)Mod │ │ │ │ +00017c70: 756c 652c 2c20 6f76 6572 2061 2063 6f6d ule,, over a com │ │ │ │ +00017c80: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ +00017c90: 6f6e 0a20 2020 2020 2020 2072 696e 670a on. ring. │ │ │ │ +00017ca0: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e * *note Option │ │ │ │ +00017cb0: 616c 2069 6e70 7574 733a 2028 4d61 6361 al inputs: (Maca │ │ │ │ +00017cc0: 756c 6179 3244 6f63 2975 7369 6e67 2066 ulay2Doc)using f │ │ │ │ +00017cd0: 756e 6374 696f 6e73 2077 6974 6820 6f70 unctions with op │ │ │ │ +00017ce0: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a tional inputs,:. │ │ │ │ +00017cf0: 2020 2020 2020 2a20 4f75 7452 696e 6720 * OutRing │ │ │ │ +00017d00: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ +00017d10: 7661 6c75 6520 300a 2020 2a20 4f75 7470 value 0. * Outp │ │ │ │ +00017d20: 7574 733a 0a20 2020 2020 202a 2045 2c20 uts:. * E, │ │ │ │ +00017d30: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20 a *note module: │ │ │ │ +00017d40: 284d 6163 6175 6c61 7932 446f 6329 4d6f (Macaulay2Doc)Mo │ │ │ │ +00017d50: 6475 6c65 2c2c 206f 7665 7220 6120 706f dule,, over a po │ │ │ │ +00017d60: 6c79 6e6f 6d69 616c 2072 696e 6720 7769 lynomial ring wi │ │ │ │ +00017d70: 7468 0a20 2020 2020 2020 2067 656e 7320 th. gens │ │ │ │ +00017d80: 696e 2064 6567 7265 6520 310a 0a44 6573 in degree 1..Des │ │ │ │ +00017d90: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +00017da0: 3d3d 3d3d 0a0a 4578 7472 6163 7473 2074 ====..Extracts t │ │ │ │ +00017db0: 6865 2065 7665 6e20 6465 6772 6565 2070 he even degree p │ │ │ │ +00017dc0: 6172 7420 6672 6f6d 2045 7874 4d6f 6475 art from ExtModu │ │ │ │ +00017dd0: 6c65 204d 2049 6620 7468 6520 6f70 7469 le M If the opti │ │ │ │ +00017de0: 6f6e 616c 2061 7267 756d 656e 7420 4f75 onal argument Ou │ │ │ │ +00017df0: 7452 696e 670a 3d3e 2054 2069 7320 6769 tRing.=> T is gi │ │ │ │ +00017e00: 7665 6e2c 2061 6e64 2063 6c61 7373 2054 ven, and class T │ │ │ │ +00017e10: 203d 3d3d 2050 6f6c 796e 6f6d 6961 6c52 === PolynomialR │ │ │ │ +00017e20: 696e 672c 2074 6865 6e20 7468 6520 6f75 ing, then the ou │ │ │ │ +00017e30: 7470 7574 2077 696c 6c20 6265 2061 206d tput will be a m │ │ │ │ +00017e40: 6f64 756c 650a 6f76 6572 2054 2e0a 0a2b odule.over T...+ │ │ │ │ +00017e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017e90: 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b 3d20 ----+.|i1 : kk= │ │ │ │ -00017ea0: 5a5a 2f31 3031 2020 2020 2020 2020 2020 ZZ/101 │ │ │ │ +00017e80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ +00017e90: 206b 6b3d 205a 5a2f 3130 3120 2020 2020 kk= ZZ/101 │ │ │ │ +00017ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017ed0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00017ec0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00017ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00017f10: 3120 3d20 6b6b 2020 2020 2020 2020 2020 1 = kk │ │ │ │ +00017f00: 207c 0a7c 6f31 203d 206b 6b20 2020 2020 |.|o1 = kk │ │ │ │ +00017f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00017f30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f80: 2020 2020 7c0a 7c6f 3120 3a20 5175 6f74 |.|o1 : Quot │ │ │ │ -00017f90: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +00017f70: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ +00017f80: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +00017f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017fc0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00017fb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00017fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00018000: 3220 3a20 5320 3d20 6b6b 5b78 2c79 2c7a 2 : S = kk[x,y,z │ │ │ │ -00018010: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ -00018020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018030: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00017ff0: 2d2b 0a7c 6932 203a 2053 203d 206b 6b5b -+.|i2 : S = kk[ │ │ │ │ +00018000: 782c 792c 7a5d 2020 2020 2020 2020 2020 x,y,z] │ │ │ │ +00018010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018020: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018070: 2020 2020 7c0a 7c6f 3220 3d20 5320 2020 |.|o2 = S │ │ │ │ +00018060: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ +00018070: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 00018080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000180a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000180b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000180a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000180b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000180c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000180d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000180e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -000180f0: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269 2 : PolynomialRi │ │ │ │ -00018100: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ -00018110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018120: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000180e0: 207c 0a7c 6f32 203a 2050 6f6c 796e 6f6d |.|o2 : Polynom │ │ │ │ +000180f0: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ +00018100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018110: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00018120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018160: 2d2d 2d2d 2b0a 7c69 3320 3a20 4932 203d ----+.|i3 : I2 = │ │ │ │ -00018170: 2069 6465 616c 2278 332c 797a 2220 2020 ideal"x3,yz" │ │ │ │ +00018150: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ +00018160: 2049 3220 3d20 6964 6561 6c22 7833 2c79 I2 = ideal"x3,y │ │ │ │ +00018170: 7a22 2020 2020 2020 2020 2020 2020 2020 z" │ │ │ │ 00018180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000181a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018190: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000181a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000181b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000181c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000181d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000181e0: 2020 2020 2020 2020 2020 2020 3320 2020 3 │ │ │ │ +000181d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000181e0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 000181f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018210: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -00018220: 6964 6561 6c20 2878 202c 2079 2a7a 2920 ideal (x , y*z) │ │ │ │ +00018200: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018210: 6f33 203d 2069 6465 616c 2028 7820 2c20 o3 = ideal (x , │ │ │ │ +00018220: 792a 7a29 2020 2020 2020 2020 2020 2020 y*z) │ │ │ │ 00018230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018250: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00018240: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018290: 7c0a 7c6f 3320 3a20 4964 6561 6c20 6f66 |.|o3 : Ideal of │ │ │ │ -000182a0: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +00018280: 2020 2020 207c 0a7c 6f33 203a 2049 6465 |.|o3 : Ide │ │ │ │ +00018290: 616c 206f 6620 5320 2020 2020 2020 2020 al of S │ │ │ │ +000182a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000182b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000182c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000182c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 000182d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000182e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000182f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018300: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ -00018310: 5232 203d 2053 2f49 3220 2020 2020 2020 R2 = S/I2 │ │ │ │ +000182f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00018300: 6934 203a 2052 3220 3d20 532f 4932 2020 i4 : R2 = S/I2 │ │ │ │ +00018310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018340: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00018330: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018380: 7c0a 7c6f 3420 3d20 5232 2020 2020 2020 |.|o4 = R2 │ │ │ │ +00018370: 2020 2020 207c 0a7c 6f34 203d 2052 3220 |.|o4 = R2 │ │ │ │ +00018380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000183a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000183b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000183b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000183c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000183d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000183e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000183f0: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ -00018400: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +000183e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000183f0: 6f34 203a 2051 756f 7469 656e 7452 696e o4 : QuotientRin │ │ │ │ +00018400: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ 00018410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018430: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00018420: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00018430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018470: 2b0a 7c69 3520 3a20 4d32 203d 2052 325e +.|i5 : M2 = R2^ │ │ │ │ -00018480: 312f 6964 6561 6c22 7832 2c79 2c7a 2220 1/ideal"x2,y,z" │ │ │ │ +00018460: 2d2d 2d2d 2d2b 0a7c 6935 203a 204d 3220 -----+.|i5 : M2 │ │ │ │ +00018470: 3d20 5232 5e31 2f69 6465 616c 2278 322c = R2^1/ideal"x2, │ │ │ │ +00018480: 792c 7a22 2020 2020 2020 2020 2020 2020 y,z" │ │ │ │ 00018490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000184a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000184a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000184b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000184c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000184d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000184e0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ -000184f0: 636f 6b65 726e 656c 207c 2078 3220 7920 cokernel | x2 y │ │ │ │ -00018500: 7a20 7c20 2020 2020 2020 2020 2020 2020 z | │ │ │ │ -00018510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018520: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000184d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000184e0: 6f35 203d 2063 6f6b 6572 6e65 6c20 7c20 o5 = cokernel | │ │ │ │ +000184f0: 7832 2079 207a 207c 2020 2020 2020 2020 x2 y z | │ │ │ │ +00018500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018510: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018560: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00018570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018580: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -00018590: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -000185a0: 3520 3a20 5232 2d6d 6f64 756c 652c 2071 5 : R2-module, q │ │ │ │ -000185b0: 756f 7469 656e 7420 6f66 2052 3220 2020 uotient of R2 │ │ │ │ -000185c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000185d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00018550: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018570: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +00018580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018590: 207c 0a7c 6f35 203a 2052 322d 6d6f 6475 |.|o5 : R2-modu │ │ │ │ +000185a0: 6c65 2c20 7175 6f74 6965 6e74 206f 6620 le, quotient of │ │ │ │ +000185b0: 5232 2020 2020 2020 2020 2020 2020 2020 R2 │ │ │ │ +000185c0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000185d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000185e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000185f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018610: 2d2d 2d2d 2b0a 7c69 3620 3a20 6265 7474 ----+.|i6 : bett │ │ │ │ -00018620: 6920 6672 6565 5265 736f 6c75 7469 6f6e i freeResolution │ │ │ │ -00018630: 2028 4d32 2c20 4c65 6e67 7468 4c69 6d69 (M2, LengthLimi │ │ │ │ -00018640: 7420 3d3e 3130 2920 2020 2020 2020 2020 t =>10) │ │ │ │ -00018650: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018600: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ +00018610: 2062 6574 7469 2066 7265 6552 6573 6f6c betti freeResol │ │ │ │ +00018620: 7574 696f 6e20 284d 322c 204c 656e 6774 ution (M2, Lengt │ │ │ │ +00018630: 684c 696d 6974 203d 3e31 3029 2020 2020 hLimit =>10) │ │ │ │ +00018640: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018680: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00018690: 2020 2020 2020 2020 2020 2030 2031 2032 0 1 2 │ │ │ │ -000186a0: 2033 2034 2020 3520 2036 2020 3720 2038 3 4 5 6 7 8 │ │ │ │ -000186b0: 2020 3920 3130 2020 2020 2020 2020 2020 9 10 │ │ │ │ -000186c0: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ -000186d0: 746f 7461 6c3a 2031 2033 2035 2037 2039 total: 1 3 5 7 9 │ │ │ │ -000186e0: 2031 3120 3133 2031 3520 3137 2031 3920 11 13 15 17 19 │ │ │ │ -000186f0: 3231 2020 2020 2020 2020 2020 2020 2020 21 │ │ │ │ -00018700: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00018710: 303a 2031 2032 2032 2032 2032 2020 3220 0: 1 2 2 2 2 2 │ │ │ │ -00018720: 2032 2020 3220 2032 2020 3220 2032 2020 2 2 2 2 2 │ │ │ │ -00018730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018740: 7c0a 7c20 2020 2020 2020 2020 313a 202e |.| 1: . │ │ │ │ -00018750: 2031 2033 2034 2034 2020 3420 2034 2020 1 3 4 4 4 4 │ │ │ │ -00018760: 3420 2034 2020 3420 2034 2020 2020 2020 4 4 4 4 │ │ │ │ -00018770: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00018780: 2020 2020 2020 2020 323a 202e 202e 202e 2: . . . │ │ │ │ -00018790: 2031 2033 2020 3420 2034 2020 3420 2034 1 3 4 4 4 4 │ │ │ │ -000187a0: 2020 3420 2034 2020 2020 2020 2020 2020 4 4 │ │ │ │ -000187b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000187c0: 2020 2020 333a 202e 202e 202e 202e 202e 3: . . . . . │ │ │ │ -000187d0: 2020 3120 2033 2020 3420 2034 2020 3420 1 3 4 4 4 │ │ │ │ -000187e0: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -000187f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00018800: 343a 202e 202e 202e 202e 202e 2020 2e20 4: . . . . . . │ │ │ │ -00018810: 202e 2020 3120 2033 2020 3420 2034 2020 . 1 3 4 4 │ │ │ │ -00018820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018830: 7c0a 7c20 2020 2020 2020 2020 353a 202e |.| 5: . │ │ │ │ -00018840: 202e 202e 202e 202e 2020 2e20 202e 2020 . . . . . . │ │ │ │ -00018850: 2e20 202e 2020 3120 2033 2020 2020 2020 . . 1 3 │ │ │ │ -00018860: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00018680: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018690: 3020 3120 3220 3320 3420 2035 2020 3620 0 1 2 3 4 5 6 │ │ │ │ +000186a0: 2037 2020 3820 2039 2031 3020 2020 2020 7 8 9 10 │ │ │ │ +000186b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000186c0: 6f36 203d 2074 6f74 616c 3a20 3120 3320 o6 = total: 1 3 │ │ │ │ +000186d0: 3520 3720 3920 3131 2031 3320 3135 2031 5 7 9 11 13 15 1 │ │ │ │ +000186e0: 3720 3139 2032 3120 2020 2020 2020 2020 7 19 21 │ │ │ │ +000186f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018700: 2020 2020 2030 3a20 3120 3220 3220 3220 0: 1 2 2 2 │ │ │ │ +00018710: 3220 2032 2020 3220 2032 2020 3220 2032 2 2 2 2 2 2 │ │ │ │ +00018720: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00018730: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018740: 2031 3a20 2e20 3120 3320 3420 3420 2034 1: . 1 3 4 4 4 │ │ │ │ +00018750: 2020 3420 2034 2020 3420 2034 2020 3420 4 4 4 4 4 │ │ │ │ +00018760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018770: 207c 0a7c 2020 2020 2020 2020 2032 3a20 |.| 2: │ │ │ │ +00018780: 2e20 2e20 2e20 3120 3320 2034 2020 3420 . . . 1 3 4 4 │ │ │ │ +00018790: 2034 2020 3420 2034 2020 3420 2020 2020 4 4 4 4 │ │ │ │ +000187a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000187b0: 2020 2020 2020 2020 2033 3a20 2e20 2e20 3: . . │ │ │ │ +000187c0: 2e20 2e20 2e20 2031 2020 3320 2034 2020 . . . 1 3 4 │ │ │ │ +000187d0: 3420 2034 2020 3420 2020 2020 2020 2020 4 4 4 │ │ │ │ +000187e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000187f0: 2020 2020 2034 3a20 2e20 2e20 2e20 2e20 4: . . . . │ │ │ │ +00018800: 2e20 202e 2020 2e20 2031 2020 3320 2034 . . . 1 3 4 │ │ │ │ +00018810: 2020 3420 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +00018820: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018830: 2035 3a20 2e20 2e20 2e20 2e20 2e20 202e 5: . . . . . . │ │ │ │ +00018840: 2020 2e20 202e 2020 2e20 2031 2020 3320 . . . 1 3 │ │ │ │ +00018850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018860: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00018870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000188a0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20 |.|o6 : │ │ │ │ -000188b0: 4265 7474 6954 616c 6c79 2020 2020 2020 BettiTally │ │ │ │ +00018890: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000188a0: 6f36 203a 2042 6574 7469 5461 6c6c 7920 o6 : BettiTally │ │ │ │ +000188b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000188c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000188d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000188e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000188d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000188e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000188f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018920: 2b0a 7c69 3720 3a20 4520 3d20 4578 744d +.|i7 : E = ExtM │ │ │ │ -00018930: 6f64 756c 6520 4d32 2020 2020 2020 2020 odule M2 │ │ │ │ +00018910: 2d2d 2d2d 2d2b 0a7c 6937 203a 2045 203d -----+.|i7 : E = │ │ │ │ +00018920: 2045 7874 4d6f 6475 6c65 204d 3220 2020 ExtModule M2 │ │ │ │ +00018930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018950: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00018950: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00018960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018990: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000189a0: 2020 2020 2020 2020 2020 2020 3820 2020 8 │ │ │ │ +00018980: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000189a0: 2038 2020 2020 2020 2020 2020 2020 2020 8 │ │ │ │ 000189b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000189c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000189d0: 2020 2020 7c0a 7c6f 3720 3d20 286b 6b5b |.|o7 = (kk[ │ │ │ │ -000189e0: 5820 2e2e 5820 5d29 2020 2020 2020 2020 X ..X ]) │ │ │ │ +000189c0: 2020 2020 2020 2020 207c 0a7c 6f37 203d |.|o7 = │ │ │ │ +000189d0: 2028 6b6b 5b58 202e 2e58 205d 2920 2020 (kk[X ..X ]) │ │ │ │ +000189e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000189f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a10: 7c0a 7c20 2020 2020 2020 2020 2030 2020 |.| 0 │ │ │ │ -00018a20: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00018a00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018a10: 2020 3020 2020 3120 2020 2020 2020 2020 0 1 │ │ │ │ +00018a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00018a40: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00018a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a80: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20 |.|o7 : │ │ │ │ -00018a90: 6b6b 5b58 202e 2e58 205d 2d6d 6f64 756c kk[X ..X ]-modul │ │ │ │ -00018aa0: 652c 2066 7265 652c 2064 6567 7265 6573 e, free, degrees │ │ │ │ -00018ab0: 207b 302e 2e31 2c20 323a 312c 2033 3a32 {0..1, 2:1, 3:2 │ │ │ │ -00018ac0: 2c20 337d 7c0a 7c20 2020 2020 2020 2020 , 3}|.| │ │ │ │ -00018ad0: 3020 2020 3120 2020 2020 2020 2020 2020 0 1 │ │ │ │ +00018a70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018a80: 6f37 203a 206b 6b5b 5820 2e2e 5820 5d2d o7 : kk[X ..X ]- │ │ │ │ +00018a90: 6d6f 6475 6c65 2c20 6672 6565 2c20 6465 module, free, de │ │ │ │ +00018aa0: 6772 6565 7320 7b30 2e2e 312c 2032 3a31 grees {0..1, 2:1 │ │ │ │ +00018ab0: 2c20 333a 322c 2033 7d7c 0a7c 2020 2020 , 3:2, 3}|.| │ │ │ │ +00018ac0: 2020 2020 2030 2020 2031 2020 2020 2020 0 1 │ │ │ │ +00018ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018b00: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00018af0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00018b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00018b40: 3820 3a20 6170 706c 7928 746f 4c69 7374 8 : apply(toList │ │ │ │ -00018b50: 2830 2e2e 3130 292c 2069 2d3e 6869 6c62 (0..10), i->hilb │ │ │ │ -00018b60: 6572 7446 756e 6374 696f 6e28 692c 2045 ertFunction(i, E │ │ │ │ -00018b70: 2929 2020 2020 2020 7c0a 7c20 2020 2020 )) |.| │ │ │ │ +00018b30: 2d2b 0a7c 6938 203a 2061 7070 6c79 2874 -+.|i8 : apply(t │ │ │ │ +00018b40: 6f4c 6973 7428 302e 2e31 3029 2c20 692d oList(0..10), i- │ │ │ │ +00018b50: 3e68 696c 6265 7274 4675 6e63 7469 6f6e >hilbertFunction │ │ │ │ +00018b60: 2869 2c20 4529 2920 2020 2020 207c 0a7c (i, E)) |.| │ │ │ │ +00018b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018bb0: 2020 2020 7c0a 7c6f 3820 3d20 7b31 2c20 |.|o8 = {1, │ │ │ │ -00018bc0: 332c 2035 2c20 372c 2039 2c20 3131 2c20 3, 5, 7, 9, 11, │ │ │ │ -00018bd0: 3133 2c20 3135 2c20 3137 2c20 3139 2c20 13, 15, 17, 19, │ │ │ │ -00018be0: 3231 7d20 2020 2020 2020 2020 2020 2020 21} │ │ │ │ -00018bf0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018ba0: 2020 2020 2020 2020 207c 0a7c 6f38 203d |.|o8 = │ │ │ │ +00018bb0: 207b 312c 2033 2c20 352c 2037 2c20 392c {1, 3, 5, 7, 9, │ │ │ │ +00018bc0: 2031 312c 2031 332c 2031 352c 2031 372c 11, 13, 15, 17, │ │ │ │ +00018bd0: 2031 392c 2032 317d 2020 2020 2020 2020 19, 21} │ │ │ │ +00018be0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018c20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00018c30: 3820 3a20 4c69 7374 2020 2020 2020 2020 8 : List │ │ │ │ +00018c20: 207c 0a7c 6f38 203a 204c 6973 7420 2020 |.|o8 : List │ │ │ │ +00018c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018c60: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00018c50: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00018c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018ca0: 2d2d 2d2d 2b0a 7c69 3920 3a20 4565 7665 ----+.|i9 : Eeve │ │ │ │ -00018cb0: 6e20 3d20 6576 656e 4578 744d 6f64 756c n = evenExtModul │ │ │ │ -00018cc0: 6520 4d32 2020 2020 2020 2020 2020 2020 e M2 │ │ │ │ -00018cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018ce0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018c90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a ---------+.|i9 : │ │ │ │ +00018ca0: 2045 6576 656e 203d 2065 7665 6e45 7874 Eeven = evenExt │ │ │ │ +00018cb0: 4d6f 6475 6c65 204d 3220 2020 2020 2020 Module M2 │ │ │ │ +00018cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00018d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d30: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -00018d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d50: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20 |.|o9 = │ │ │ │ -00018d60: 286b 6b5b 5820 2e2e 5820 5d29 2020 2020 (kk[X ..X ]) │ │ │ │ +00018d10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018d20: 2020 2020 2034 2020 2020 2020 2020 2020 4 │ │ │ │ +00018d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018d40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018d50: 6f39 203d 2028 6b6b 5b58 202e 2e58 205d o9 = (kk[X ..X ] │ │ │ │ +00018d60: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00018d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00018da0: 2030 2020 2031 2020 2020 2020 2020 2020 0 1 │ │ │ │ +00018d80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018d90: 2020 2020 2020 3020 2020 3120 2020 2020 0 1 │ │ │ │ +00018da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018dd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018dc0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018e00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00018e10: 3920 3a20 6b6b 5b58 202e 2e58 205d 2d6d 9 : kk[X ..X ]-m │ │ │ │ -00018e20: 6f64 756c 652c 2066 7265 652c 2064 6567 odule, free, deg │ │ │ │ -00018e30: 7265 6573 207b 302e 2e31 2c20 323a 317d rees {0..1, 2:1} │ │ │ │ -00018e40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00018e50: 2020 2020 3020 2020 3120 2020 2020 2020 0 1 │ │ │ │ +00018e00: 207c 0a7c 6f39 203a 206b 6b5b 5820 2e2e |.|o9 : kk[X .. │ │ │ │ +00018e10: 5820 5d2d 6d6f 6475 6c65 2c20 6672 6565 X ]-module, free │ │ │ │ +00018e20: 2c20 6465 6772 6565 7320 7b30 2e2e 312c , degrees {0..1, │ │ │ │ +00018e30: 2032 3a31 7d20 2020 2020 2020 207c 0a7c 2:1} |.| │ │ │ │ +00018e40: 2020 2020 2020 2020 2030 2020 2031 2020 0 1 │ │ │ │ +00018e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018e80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00018e70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00018e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018ec0: 2b0a 7c69 3130 203a 2061 7070 6c79 2874 +.|i10 : apply(t │ │ │ │ -00018ed0: 6f4c 6973 7428 302e 2e35 292c 2069 2d3e oList(0..5), i-> │ │ │ │ -00018ee0: 6869 6c62 6572 7446 756e 6374 696f 6e28 hilbertFunction( │ │ │ │ -00018ef0: 692c 2045 6576 656e 2929 2020 7c0a 7c20 i, Eeven)) |.| │ │ │ │ +00018eb0: 2d2d 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+00018f40: 3133 2c20 3137 2c20 3231 7d20 2020 2020 13, 17, 21} │ │ │ │ +00018f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018f60: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018fb0: 7c0a 7c6f 3130 203a 204c 6973 7420 2020 |.|o10 : List │ │ │ │ +00018fa0: 2020 2020 207c 0a7c 6f31 3020 3a20 4c69 |.|o10 : Li │ │ │ │ +00018fb0: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ 00018fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018fe0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00018fe0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00018ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019020: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -00019030: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -00019040: 2a20 2a6e 6f74 6520 4578 744d 6f64 756c * *note ExtModul │ │ │ │ -00019050: 653a 2045 7874 4d6f 6475 6c65 2c20 2d2d e: ExtModule, -- │ │ │ │ -00019060: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572 Ext^*(M,k) over │ │ │ │ -00019070: 2061 2063 6f6d 706c 6574 6520 696e 7465 a complete inte │ │ │ │ -00019080: 7273 6563 7469 6f6e 2061 730a 2020 2020 rsection as. │ │ │ │ -00019090: 6d6f 6475 6c65 206f 7665 7220 4349 206f module over CI o │ │ │ │ -000190a0: 7065 7261 746f 7220 7269 6e67 0a20 202a perator ring. * │ │ │ │ -000190b0: 202a 6e6f 7465 206f 6464 4578 744d 6f64 *note oddExtMod │ │ │ │ -000190c0: 756c 653a 206f 6464 4578 744d 6f64 756c ule: oddExtModul │ │ │ │ -000190d0: 652c 202d 2d20 6f64 6420 7061 7274 206f e, -- odd part o │ │ │ │ -000190e0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665 f Ext^*(M,k) ove │ │ │ │ -000190f0: 7220 6120 636f 6d70 6c65 7465 0a20 2020 r a complete. │ │ │ │ -00019100: 2069 6e74 6572 7365 6374 696f 6e20 6173 intersection as │ │ │ │ -00019110: 206d 6f64 756c 6520 6f76 6572 2043 4920 module over CI │ │ │ │ -00019120: 6f70 6572 6174 6f72 2072 696e 670a 2020 operator ring. │ │ │ │ -00019130: 2a20 2a6e 6f74 6520 4f75 7452 696e 673a * *note OutRing: │ │ │ │ -00019140: 204f 7574 5269 6e67 2c20 2d2d 204f 7074 OutRing, -- Opt │ │ │ │ -00019150: 696f 6e20 616c 6c6f 7769 6e67 2073 7065 ion allowing spe │ │ │ │ -00019160: 6369 6669 6361 7469 6f6e 206f 6620 7468 cification of th │ │ │ │ -00019170: 6520 7269 6e67 206f 7665 720a 2020 2020 e ring over. │ │ │ │ -00019180: 7768 6963 6820 7468 6520 6f75 7470 7574 which the output │ │ │ │ -00019190: 2069 7320 6465 6669 6e65 640a 0a57 6179 is defined..Way │ │ │ │ -000191a0: 7320 746f 2075 7365 2065 7665 6e45 7874 s to use evenExt │ │ │ │ -000191b0: 4d6f 6475 6c65 3a0a 3d3d 3d3d 3d3d 3d3d Module:.======== │ │ │ │ -000191c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000191d0: 3d3d 0a0a 2020 2a20 2265 7665 6e45 7874 ==.. * "evenExt │ │ │ │ -000191e0: 4d6f 6475 6c65 284d 6f64 756c 6529 220a Module(Module)". │ │ │ │ -000191f0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ -00019200: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ -00019210: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ -00019220: 6374 202a 6e6f 7465 2065 7665 6e45 7874 ct *note evenExt │ │ │ │ -00019230: 4d6f 6475 6c65 3a20 6576 656e 4578 744d Module: evenExtM │ │ │ │ -00019240: 6f64 756c 652c 2069 7320 6120 2a6e 6f74 odule, is a *not │ │ │ │ -00019250: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ -00019260: 6e20 7769 7468 0a6f 7074 696f 6e73 3a20 n with.options: │ │ │ │ -00019270: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -00019280: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ -00019290: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ +00019010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +00019020: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +00019030: 3d0a 0a20 202a 202a 6e6f 7465 2045 7874 =.. * *note Ext │ │ │ │ +00019040: 4d6f 6475 6c65 3a20 4578 744d 6f64 756c Module: ExtModul │ │ │ │ +00019050: 652c 202d 2d20 4578 745e 2a28 4d2c 6b29 e, -- Ext^*(M,k) │ │ │ │ +00019060: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ +00019070: 2069 6e74 6572 7365 6374 696f 6e20 6173 intersection as │ │ │ │ +00019080: 0a20 2020 206d 6f64 756c 6520 6f76 6572 . module over │ │ │ │ +00019090: 2043 4920 6f70 6572 6174 6f72 2072 696e CI operator rin │ │ │ │ +000190a0: 670a 2020 2a20 2a6e 6f74 6520 6f64 6445 g. * *note oddE │ │ │ │ +000190b0: 7874 4d6f 6475 6c65 3a20 6f64 6445 7874 xtModule: oddExt │ │ │ │ +000190c0: 4d6f 6475 6c65 2c20 2d2d 206f 6464 2070 Module, -- odd p │ │ │ │ +000190d0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b art of Ext^*(M,k │ │ │ │ +000190e0: 2920 6f76 6572 2061 2063 6f6d 706c 6574 ) over a complet │ │ │ │ +000190f0: 650a 2020 2020 696e 7465 7273 6563 7469 e. intersecti │ │ │ │ +00019100: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665 on as module ove │ │ │ │ +00019110: 7220 4349 206f 7065 7261 746f 7220 7269 r CI operator ri │ │ │ │ +00019120: 6e67 0a20 202a 202a 6e6f 7465 204f 7574 ng. * *note Out │ │ │ │ +00019130: 5269 6e67 3a20 4f75 7452 696e 672c 202d Ring: OutRing, - │ │ │ │ +00019140: 2d20 4f70 7469 6f6e 2061 6c6c 6f77 696e - Option allowin │ │ │ │ +00019150: 6720 7370 6563 6966 6963 6174 696f 6e20 g specification │ │ │ │ +00019160: 6f66 2074 6865 2072 696e 6720 6f76 6572 of the ring over │ │ │ │ +00019170: 0a20 2020 2077 6869 6368 2074 6865 206f . which the o │ │ │ │ +00019180: 7574 7075 7420 6973 2064 6566 696e 6564 utput is defined │ │ │ │ +00019190: 0a0a 5761 7973 2074 6f20 7573 6520 6576 ..Ways to use ev │ │ │ │ +000191a0: 656e 4578 744d 6f64 756c 653a 0a3d 3d3d enExtModule:.=== │ │ │ │ +000191b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000191c0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6576 =======.. * "ev │ │ │ │ +000191d0: 656e 4578 744d 6f64 756c 6528 4d6f 6475 enExtModule(Modu │ │ │ │ +000191e0: 6c65 2922 0a0a 466f 7220 7468 6520 7072 le)"..For the pr │ │ │ │ +000191f0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +00019200: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +00019210: 206f 626a 6563 7420 2a6e 6f74 6520 6576 object *note ev │ │ │ │ +00019220: 656e 4578 744d 6f64 756c 653a 2065 7665 enExtModule: eve │ │ │ │ +00019230: 6e45 7874 4d6f 6475 6c65 2c20 6973 2061 nExtModule, is a │ │ │ │ +00019240: 202a 6e6f 7465 206d 6574 686f 6420 6675 *note method fu │ │ │ │ +00019250: 6e63 7469 6f6e 2077 6974 680a 6f70 7469 nction with.opti │ │ │ │ +00019260: 6f6e 733a 2028 4d61 6361 756c 6179 3244 ons: (Macaulay2D │ │ │ │ +00019270: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ +00019280: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a nWithOptions,... │ │ │ │ +00019290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000192a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000192b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000192c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000192d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000192e0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -000192f0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -00019300: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -00019310: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -00019320: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -00019330: 2d31 2e32 362e 3035 2b64 732f 4d32 2f4d -1.26.05+ds/M2/M │ │ │ │ -00019340: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00019350: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ -00019360: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -00019370: 6e73 2e6d 323a 3336 3433 3a30 2e0a 1f0a ns.m2:3643:0.... │ │ │ │ -00019380: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ -00019390: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -000193a0: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ -000193b0: 3a20 6578 706f 2c20 4e65 7874 3a20 6578 : expo, Next: ex │ │ │ │ -000193c0: 7465 7269 6f72 4578 744d 6f64 756c 652c teriorExtModule, │ │ │ │ -000193d0: 2050 7265 763a 2065 7665 6e45 7874 4d6f Prev: evenExtMo │ │ │ │ -000193e0: 6475 6c65 2c20 5570 3a20 546f 700a 0a65 dule, Up: Top..e │ │ │ │ -000193f0: 7870 6f20 2d2d 2072 6574 7572 6e73 2061 xpo -- returns a │ │ │ │ -00019400: 2073 6574 2063 6f72 7265 7370 6f6e 6469 set correspondi │ │ │ │ -00019410: 6e67 2074 6f20 7468 6520 6261 7369 7320 ng to the basis │ │ │ │ -00019420: 6f66 2061 2064 6976 6964 6564 2070 6f77 of a divided pow │ │ │ │ -00019430: 6572 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a er.************* │ │ │ │ +000192d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +000192e0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +000192f0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +00019300: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +00019310: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +00019320: 756c 6179 322d 312e 3236 2e30 352b 6473 ulay2-1.26.05+ds │ │ │ │ +00019330: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +00019340: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ +00019350: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +00019360: 6c75 7469 6f6e 732e 6d32 3a33 3634 333a lutions.m2:3643: │ │ │ │ +00019370: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ +00019380: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +00019390: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ +000193a0: 204e 6f64 653a 2065 7870 6f2c 204e 6578 Node: expo, Nex │ │ │ │ +000193b0: 743a 2065 7874 6572 696f 7245 7874 4d6f t: exteriorExtMo │ │ │ │ +000193c0: 6475 6c65 2c20 5072 6576 3a20 6576 656e dule, Prev: even │ │ │ │ +000193d0: 4578 744d 6f64 756c 652c 2055 703a 2054 ExtModule, Up: T │ │ │ │ +000193e0: 6f70 0a0a 6578 706f 202d 2d20 7265 7475 op..expo -- retu │ │ │ │ +000193f0: 726e 7320 6120 7365 7420 636f 7272 6573 rns a set corres │ │ │ │ +00019400: 706f 6e64 696e 6720 746f 2074 6865 2062 ponding to the b │ │ │ │ +00019410: 6173 6973 206f 6620 6120 6469 7669 6465 asis of a divide │ │ │ │ +00019420: 6420 706f 7765 720a 2a2a 2a2a 2a2a 2a2a d power.******** │ │ │ │ +00019430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00019440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00019450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00019460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00019470: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -00019480: 653a 200a 2020 2020 2020 2020 4220 3d20 e: . B = │ │ │ │ -00019490: 6578 706f 2863 2c4e 290a 2020 2020 2020 expo(c,N). │ │ │ │ -000194a0: 2020 4220 3d20 6578 706f 2863 2c4c 290a B = expo(c,L). │ │ │ │ -000194b0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -000194c0: 2020 2a20 4e2c 2061 6e20 2a6e 6f74 6520 * N, an *note │ │ │ │ -000194d0: 696e 7465 6765 723a 2028 4d61 6361 756c integer: (Macaul │ │ │ │ -000194e0: 6179 3244 6f63 295a 5a2c 2c20 0a20 2020 ay2Doc)ZZ,, . │ │ │ │ -000194f0: 2020 202a 2063 2c20 616e 202a 6e6f 7465 * c, an *note │ │ │ │ -00019500: 2069 6e74 6567 6572 3a20 284d 6163 6175 integer: (Macau │ │ │ │ -00019510: 6c61 7932 446f 6329 5a5a 2c2c 200a 2020 lay2Doc)ZZ,, . │ │ │ │ -00019520: 2020 2020 2a20 4c2c 2061 202a 6e6f 7465 * L, a *note │ │ │ │ -00019530: 206c 6973 743a 2028 4d61 6361 756c 6179 list: (Macaulay │ │ │ │ -00019540: 3244 6f63 294c 6973 742c 2c20 6f66 2063 2Doc)List,, of c │ │ │ │ -00019550: 206e 6f6e 2d6e 6567 6174 6976 6520 696e non-negative in │ │ │ │ -00019560: 7465 6765 7273 0a20 202a 204f 7574 7075 tegers. * Outpu │ │ │ │ -00019570: 7473 3a0a 2020 2020 2020 2a20 422c 2061 ts:. * B, a │ │ │ │ -00019580: 202a 6e6f 7465 206c 6973 743a 2028 4d61 *note list: (Ma │ │ │ │ -00019590: 6361 756c 6179 3244 6f63 294c 6973 742c caulay2Doc)List, │ │ │ │ -000195a0: 2c20 7061 7274 6974 696f 6e73 2077 6974 , partitions wit │ │ │ │ -000195b0: 6820 6320 6e6f 6e2d 6e65 6761 7469 7665 h c non-negative │ │ │ │ -000195c0: 0a20 2020 2020 2020 2070 6172 7473 0a0a . parts.. │ │ │ │ -000195d0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -000195e0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 2066 6f72 =======..The for │ │ │ │ -000195f0: 6d20 6578 706f 2863 2c4e 2920 7265 7475 m expo(c,N) retu │ │ │ │ -00019600: 726e 7320 7061 7274 6974 696f 6e73 206f rns partitions o │ │ │ │ -00019610: 6620 4e20 7769 7468 2063 206e 6f6e 2d6e f N with c non-n │ │ │ │ -00019620: 6567 6174 6976 6520 7061 7274 732e 2054 egative parts. T │ │ │ │ -00019630: 6865 2066 6f72 6d0a 6578 706f 2863 2c20 he form.expo(c, │ │ │ │ -00019640: 4c29 2072 6574 7572 6e73 2070 6172 7469 L) returns parti │ │ │ │ -00019650: 7469 6f6e 7320 7769 7468 206e 6f6e 2d6e tions with non-n │ │ │ │ -00019660: 6567 6174 6976 6520 7061 7274 7320 7468 egative parts th │ │ │ │ -00019670: 6174 2061 7265 2063 6f6d 706f 6e65 6e74 at are component │ │ │ │ -00019680: 7769 7365 203c 3d0a 4c20 2861 6e64 2061 wise <=.L (and a │ │ │ │ -00019690: 6e79 2073 756d 203c 3d20 7375 6d20 4c29 ny sum <= sum L) │ │ │ │ -000196a0: 2e0a 0a54 6865 206c 6973 7420 6578 706f ...The list expo │ │ │ │ -000196b0: 2863 2c4e 2920 206d 6179 2062 6520 7468 (c,N) may be th │ │ │ │ -000196c0: 6f75 6768 7420 6f66 2061 7320 7468 6520 ought of as the │ │ │ │ -000196d0: 6c69 7374 206f 6620 6578 706f 6e65 6e74 list of exponent │ │ │ │ -000196e0: 2076 6563 746f 7273 206f 6620 7468 650a vectors of the. │ │ │ │ -000196f0: 6d6f 6e6f 6d69 616c 7320 6f66 2064 6567 monomials of deg │ │ │ │ -00019700: 7265 6520 4e20 696e 2063 2076 6172 6961 ree N in c varia │ │ │ │ -00019710: 626c 6573 2e20 5468 6973 2069 7320 7573 bles. This is us │ │ │ │ -00019720: 6564 2069 6e20 7468 6520 636f 6e73 7472 ed in the constr │ │ │ │ -00019730: 7563 7469 6f6e 206f 6620 7468 650a 4569 uction of the.Ei │ │ │ │ -00019740: 7365 6e62 7564 2d53 6861 6d61 7368 2072 senbud-Shamash r │ │ │ │ -00019750: 6573 6f6c 7574 696f 6e2e 0a0a 5468 6520 esolution...The │ │ │ │ -00019760: 6c69 7374 2065 7870 6f28 632c 204c 292c list expo(c, L), │ │ │ │ -00019770: 206f 6e20 7468 6520 6f74 6865 7220 6861 on the other ha │ │ │ │ -00019780: 6e64 2c20 6d61 7920 6265 2074 686f 7567 nd, may be thoug │ │ │ │ -00019790: 6874 206f 6620 6173 2074 6865 206c 6973 ht of as the lis │ │ │ │ -000197a0: 7420 6f66 0a64 6976 6973 6f72 7320 6f66 t of.divisors of │ │ │ │ -000197b0: 2065 5e4c 203d 2065 5f30 5e7b 4c5f 307d e^L = e_0^{L_0} │ │ │ │ -000197c0: 202e 2e2e 2065 5f63 5e7b 4c5f 637d 2e20 ... e_c^{L_c}. │ │ │ │ -000197d0: 5468 6973 2069 7320 7573 6564 2069 6e20 This is used in │ │ │ │ -000197e0: 7468 6520 636f 6e73 7472 7563 7469 6f6e the construction │ │ │ │ -000197f0: 206f 660a 7468 6520 6869 6768 6572 2068 of.the higher h │ │ │ │ -00019800: 6f6d 6f74 6f70 6965 7320 6f6e 2061 2063 omotopies on a c │ │ │ │ -00019810: 6f6d 706c 6578 2e0a 0a2b 2d2d 2d2d 2d2d omplex...+------ │ │ │ │ +00019460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00019470: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +00019480: 2042 203d 2065 7870 6f28 632c 4e29 0a20 B = expo(c,N). │ │ │ │ +00019490: 2020 2020 2020 2042 203d 2065 7870 6f28 B = expo( │ │ │ │ +000194a0: 632c 4c29 0a20 202a 2049 6e70 7574 733a c,L). * Inputs: │ │ │ │ +000194b0: 0a20 2020 2020 202a 204e 2c20 616e 202a . * N, an * │ │ │ │ +000194c0: 6e6f 7465 2069 6e74 6567 6572 3a20 284d note integer: (M │ │ │ │ +000194d0: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c acaulay2Doc)ZZ,, │ │ │ │ +000194e0: 200a 2020 2020 2020 2a20 632c 2061 6e20 . * c, an │ │ │ │ +000194f0: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ +00019500: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ +00019510: 2c20 0a20 2020 2020 202a 204c 2c20 6120 , . * L, a │ │ │ │ +00019520: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ +00019530: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ +00019540: 206f 6620 6320 6e6f 6e2d 6e65 6761 7469 of c non-negati │ │ │ │ +00019550: 7665 2069 6e74 6567 6572 730a 2020 2a20 ve integers. * │ │ │ │ +00019560: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ +00019570: 2042 2c20 6120 2a6e 6f74 6520 6c69 7374 B, a *note list │ │ │ │ +00019580: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00019590: 4c69 7374 2c2c 2070 6172 7469 7469 6f6e List,, partition │ │ │ │ +000195a0: 7320 7769 7468 2063 206e 6f6e 2d6e 6567 s with c non-neg │ │ │ │ +000195b0: 6174 6976 650a 2020 2020 2020 2020 7061 ative. pa │ │ │ │ +000195c0: 7274 730a 0a44 6573 6372 6970 7469 6f6e rts..Description │ │ │ │ +000195d0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 .===========..Th │ │ │ │ +000195e0: 6520 666f 726d 2065 7870 6f28 632c 4e29 e form expo(c,N) │ │ │ │ +000195f0: 2072 6574 7572 6e73 2070 6172 7469 7469 returns partiti │ │ │ │ +00019600: 6f6e 7320 6f66 204e 2077 6974 6820 6320 ons of N with c │ │ │ │ +00019610: 6e6f 6e2d 6e65 6761 7469 7665 2070 6172 non-negative par │ │ │ │ +00019620: 7473 2e20 5468 6520 666f 726d 0a65 7870 ts. The form.exp │ │ │ │ +00019630: 6f28 632c 204c 2920 7265 7475 726e 7320 o(c, L) returns │ │ │ │ +00019640: 7061 7274 6974 696f 6e73 2077 6974 6820 partitions with │ │ │ │ +00019650: 6e6f 6e2d 6e65 6761 7469 7665 2070 6172 non-negative par │ │ │ │ +00019660: 7473 2074 6861 7420 6172 6520 636f 6d70 ts that are comp │ │ │ │ +00019670: 6f6e 656e 7477 6973 6520 3c3d 0a4c 2028 onentwise <=.L ( │ │ │ │ +00019680: 616e 6420 616e 7920 7375 6d20 3c3d 2073 and any sum <= s │ │ │ │ +00019690: 756d 204c 292e 0a0a 5468 6520 6c69 7374 um L)...The list │ │ │ │ +000196a0: 2065 7870 6f28 632c 4e29 2020 6d61 7920 expo(c,N) may │ │ │ │ +000196b0: 6265 2074 686f 7567 6874 206f 6620 6173 be thought of as │ │ │ │ +000196c0: 2074 6865 206c 6973 7420 6f66 2065 7870 the list of exp │ │ │ │ +000196d0: 6f6e 656e 7420 7665 6374 6f72 7320 6f66 onent vectors of │ │ │ │ +000196e0: 2074 6865 0a6d 6f6e 6f6d 6961 6c73 206f the.monomials o │ │ │ │ +000196f0: 6620 6465 6772 6565 204e 2069 6e20 6320 f degree N in c │ │ │ │ +00019700: 7661 7269 6162 6c65 732e 2054 6869 7320 variables. This │ │ │ │ +00019710: 6973 2075 7365 6420 696e 2074 6865 2063 is used in the c │ │ │ │ +00019720: 6f6e 7374 7275 6374 696f 6e20 6f66 2074 onstruction of t │ │ │ │ +00019730: 6865 0a45 6973 656e 6275 642d 5368 616d he.Eisenbud-Sham │ │ │ │ +00019740: 6173 6820 7265 736f 6c75 7469 6f6e 2e0a ash resolution.. │ │ │ │ +00019750: 0a54 6865 206c 6973 7420 6578 706f 2863 .The list expo(c │ │ │ │ +00019760: 2c20 4c29 2c20 6f6e 2074 6865 206f 7468 , L), on the oth │ │ │ │ +00019770: 6572 2068 616e 642c 206d 6179 2062 6520 er hand, may be │ │ │ │ +00019780: 7468 6f75 6768 7420 6f66 2061 7320 7468 thought of as th │ │ │ │ +00019790: 6520 6c69 7374 206f 660a 6469 7669 736f e list of.diviso │ │ │ │ +000197a0: 7273 206f 6620 655e 4c20 3d20 655f 305e rs of e^L = e_0^ │ │ │ │ +000197b0: 7b4c 5f30 7d20 2e2e 2e20 655f 635e 7b4c {L_0} ... e_c^{L │ │ │ │ +000197c0: 5f63 7d2e 2054 6869 7320 6973 2075 7365 _c}. This is use │ │ │ │ +000197d0: 6420 696e 2074 6865 2063 6f6e 7374 7275 d in the constru │ │ │ │ +000197e0: 6374 696f 6e20 6f66 0a74 6865 2068 6967 ction of.the hig │ │ │ │ +000197f0: 6865 7220 686f 6d6f 746f 7069 6573 206f her homotopies o │ │ │ │ +00019800: 6e20 6120 636f 6d70 6c65 782e 0a0a 2b2d n a complex...+- │ │ │ │ +00019810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019860: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2065 -------+.|i1 : e │ │ │ │ -00019870: 7870 6f28 332c 3529 2020 2020 2020 2020 xpo(3,5) │ │ │ │ +00019850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00019860: 3120 3a20 6578 706f 2833 2c35 2920 2020 1 : expo(3,5) │ │ │ │ +00019870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000198a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000198b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000198a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000198b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000198c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000198d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000198e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000198f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019900: 2020 2020 2020 207c 0a7c 6f31 203d 207b |.|o1 = { │ │ │ │ -00019910: 7b35 2c20 302c 2030 7d2c 207b 342c 2031 {5, 0, 0}, {4, 1 │ │ │ │ -00019920: 2c20 307d 2c20 7b34 2c20 302c 2031 7d2c , 0}, {4, 0, 1}, │ │ │ │ -00019930: 207b 332c 2032 2c20 307d 2c20 7b33 2c20 {3, 2, 0}, {3, │ │ │ │ -00019940: 312c 2031 7d2c 207b 332c 2030 2c20 327d 1, 1}, {3, 0, 2} │ │ │ │ -00019950: 2c20 7b32 2c20 207c 0a7c 2020 2020 202d , {2, |.| - │ │ │ │ +000198f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00019900: 3120 3d20 7b7b 352c 2030 2c20 307d 2c20 1 = {{5, 0, 0}, │ │ │ │ +00019910: 7b34 2c20 312c 2030 7d2c 207b 342c 2030 {4, 1, 0}, {4, 0 │ │ │ │ +00019920: 2c20 317d 2c20 7b33 2c20 322c 2030 7d2c , 1}, {3, 2, 0}, │ │ │ │ +00019930: 207b 332c 2031 2c20 317d 2c20 7b33 2c20 {3, 1, 1}, {3, │ │ │ │ +00019940: 302c 2032 7d2c 207b 322c 2020 7c0a 7c20 0, 2}, {2, |.| │ │ │ │ +00019950: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00019960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000199a0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2033 -------|.| 3 │ │ │ │ -000199b0: 2c20 307d 2c20 7b32 2c20 322c 2031 7d2c , 0}, {2, 2, 1}, │ │ │ │ -000199c0: 207b 322c 2031 2c20 327d 2c20 7b32 2c20 {2, 1, 2}, {2, │ │ │ │ -000199d0: 302c 2033 7d2c 207b 312c 2034 2c20 307d 0, 3}, {1, 4, 0} │ │ │ │ -000199e0: 2c20 7b31 2c20 332c 2031 7d2c 207b 312c , {1, 3, 1}, {1, │ │ │ │ -000199f0: 2032 2c20 327d 2c7c 0a7c 2020 2020 202d 2, 2},|.| - │ │ │ │ +00019990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +000199a0: 2020 2020 332c 2030 7d2c 207b 322c 2032 3, 0}, {2, 2 │ │ │ │ +000199b0: 2c20 317d 2c20 7b32 2c20 312c 2032 7d2c , 1}, {2, 1, 2}, │ │ │ │ +000199c0: 207b 322c 2030 2c20 337d 2c20 7b31 2c20 {2, 0, 3}, {1, │ │ │ │ +000199d0: 342c 2030 7d2c 207b 312c 2033 2c20 317d 4, 0}, {1, 3, 1} │ │ │ │ +000199e0: 2c20 7b31 2c20 322c 2032 7d2c 7c0a 7c20 , {1, 2, 2},|.| │ │ │ │ +000199f0: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00019a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019a40: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 207b -------|.| { │ │ │ │ -00019a50: 312c 2031 2c20 337d 2c20 7b31 2c20 302c 1, 1, 3}, {1, 0, │ │ │ │ -00019a60: 2034 7d2c 207b 302c 2035 2c20 307d 2c20 4}, {0, 5, 0}, │ │ │ │ -00019a70: 7b30 2c20 342c 2031 7d2c 207b 302c 2033 {0, 4, 1}, {0, 3 │ │ │ │ -00019a80: 2c20 327d 2c20 7b30 2c20 322c 2033 7d2c , 2}, {0, 2, 3}, │ │ │ │ -00019a90: 207b 302c 2031 2c7c 0a7c 2020 2020 202d {0, 1,|.| - │ │ │ │ +00019a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00019a40: 2020 2020 7b31 2c20 312c 2033 7d2c 207b {1, 1, 3}, { │ │ │ │ +00019a50: 312c 2030 2c20 347d 2c20 7b30 2c20 352c 1, 0, 4}, {0, 5, │ │ │ │ +00019a60: 2030 7d2c 207b 302c 2034 2c20 317d 2c20 0}, {0, 4, 1}, │ │ │ │ +00019a70: 7b30 2c20 332c 2032 7d2c 207b 302c 2032 {0, 3, 2}, {0, 2 │ │ │ │ +00019a80: 2c20 337d 2c20 7b30 2c20 312c 7c0a 7c20 , 3}, {0, 1,|.| │ │ │ │ +00019a90: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00019aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019ae0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2034 -------|.| 4 │ │ │ │ -00019af0: 7d2c 207b 302c 2030 2c20 357d 7d20 2020 }, {0, 0, 5}} │ │ │ │ +00019ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00019ae0: 2020 2020 347d 2c20 7b30 2c20 302c 2035 4}, {0, 0, 5 │ │ │ │ +00019af0: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ 00019b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019b30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00019b20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00019b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019b80: 2020 2020 2020 207c 0a7c 6f31 203a 204c |.|o1 : L │ │ │ │ -00019b90: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ +00019b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00019b80: 3120 3a20 4c69 7374 2020 2020 2020 2020 1 : List │ │ │ │ +00019b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019bd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00019bc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00019bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019c20: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2065 -------+.|i2 : e │ │ │ │ -00019c30: 7870 6f28 332c 207b 332c 322c 317d 2920 xpo(3, {3,2,1}) │ │ │ │ +00019c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00019c20: 3220 3a20 6578 706f 2833 2c20 7b33 2c32 2 : expo(3, {3,2 │ │ │ │ +00019c30: 2c31 7d29 2020 2020 2020 2020 2020 2020 ,1}) │ │ │ │ 00019c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019c70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00019c60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00019c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019cc0: 2020 2020 2020 207c 0a7c 6f32 203d 207b |.|o2 = { │ │ │ │ -00019cd0: 7b30 2c20 302c 2030 7d2c 207b 312c 2030 {0, 0, 0}, {1, 0 │ │ │ │ -00019ce0: 2c20 307d 2c20 7b30 2c20 312c 2030 7d2c , 0}, {0, 1, 0}, │ │ │ │ -00019cf0: 207b 302c 2030 2c20 317d 2c20 7b32 2c20 {0, 0, 1}, {2, │ │ │ │ -00019d00: 302c 2030 7d2c 207b 312c 2031 2c20 307d 0, 0}, {1, 1, 0} │ │ │ │ -00019d10: 2c20 7b31 2c20 207c 0a7c 2020 2020 202d , {1, |.| - │ │ │ │ +00019cb0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00019cc0: 3220 3d20 7b7b 302c 2030 2c20 307d 2c20 2 = {{0, 0, 0}, │ │ │ │ +00019cd0: 7b31 2c20 302c 2030 7d2c 207b 302c 2031 {1, 0, 0}, {0, 1 │ │ │ │ +00019ce0: 2c20 307d 2c20 7b30 2c20 302c 2031 7d2c , 0}, {0, 0, 1}, │ │ │ │ +00019cf0: 207b 322c 2030 2c20 307d 2c20 7b31 2c20 {2, 0, 0}, {1, │ │ │ │ +00019d00: 312c 2030 7d2c 207b 312c 2020 7c0a 7c20 1, 0}, {1, |.| │ │ │ │ +00019d10: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00019d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019d60: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2030 -------|.| 0 │ │ │ │ -00019d70: 2c20 317d 2c20 7b30 2c20 322c 2030 7d2c , 1}, {0, 2, 0}, │ │ │ │ -00019d80: 207b 302c 2031 2c20 317d 2c20 7b33 2c20 {0, 1, 1}, {3, │ │ │ │ -00019d90: 302c 2030 7d2c 207b 322c 2031 2c20 307d 0, 0}, {2, 1, 0} │ │ │ │ -00019da0: 2c20 7b32 2c20 302c 2031 7d2c 207b 312c , {2, 0, 1}, {1, │ │ │ │ -00019db0: 2032 2c20 307d 2c7c 0a7c 2020 2020 202d 2, 0},|.| - │ │ │ │ +00019d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00019d60: 2020 2020 302c 2031 7d2c 207b 302c 2032 0, 1}, {0, 2 │ │ │ │ +00019d70: 2c20 307d 2c20 7b30 2c20 312c 2031 7d2c , 0}, {0, 1, 1}, │ │ │ │ +00019d80: 207b 332c 2030 2c20 307d 2c20 7b32 2c20 {3, 0, 0}, {2, │ │ │ │ +00019d90: 312c 2030 7d2c 207b 322c 2030 2c20 317d 1, 0}, {2, 0, 1} │ │ │ │ +00019da0: 2c20 7b31 2c20 322c 2030 7d2c 7c0a 7c20 , {1, 2, 0},|.| │ │ │ │ +00019db0: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00019dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019e00: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 207b -------|.| { │ │ │ │ -00019e10: 312c 2031 2c20 317d 2c20 7b30 2c20 322c 1, 1, 1}, {0, 2, │ │ │ │ -00019e20: 2031 7d2c 207b 332c 2031 2c20 307d 2c20 1}, {3, 1, 0}, │ │ │ │ -00019e30: 7b33 2c20 302c 2031 7d2c 207b 322c 2032 {3, 0, 1}, {2, 2 │ │ │ │ -00019e40: 2c20 307d 2c20 7b32 2c20 312c 2031 7d2c , 0}, {2, 1, 1}, │ │ │ │ -00019e50: 207b 312c 2032 2c7c 0a7c 2020 2020 202d {1, 2,|.| - │ │ │ │ +00019df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00019e00: 2020 2020 7b31 2c20 312c 2031 7d2c 207b {1, 1, 1}, { │ │ │ │ +00019e10: 302c 2032 2c20 317d 2c20 7b33 2c20 312c 0, 2, 1}, {3, 1, │ │ │ │ +00019e20: 2030 7d2c 207b 332c 2030 2c20 317d 2c20 0}, {3, 0, 1}, │ │ │ │ +00019e30: 7b32 2c20 322c 2030 7d2c 207b 322c 2031 {2, 2, 0}, {2, 1 │ │ │ │ +00019e40: 2c20 317d 2c20 7b31 2c20 322c 7c0a 7c20 , 1}, {1, 2,|.| │ │ │ │ +00019e50: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00019e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019ea0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2031 -------|.| 1 │ │ │ │ -00019eb0: 7d2c 207b 332c 2032 2c20 307d 2c20 7b33 }, {3, 2, 0}, {3 │ │ │ │ -00019ec0: 2c20 312c 2031 7d2c 207b 322c 2032 2c20 , 1, 1}, {2, 2, │ │ │ │ -00019ed0: 317d 7d20 2020 2020 2020 2020 2020 2020 1}} │ │ │ │ -00019ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019ef0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00019e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00019ea0: 2020 2020 317d 2c20 7b33 2c20 322c 2030 1}, {3, 2, 0 │ │ │ │ +00019eb0: 7d2c 207b 332c 2031 2c20 317d 2c20 7b32 }, {3, 1, 1}, {2 │ │ │ │ +00019ec0: 2c20 322c 2031 7d7d 2020 2020 2020 2020 , 2, 1}} │ │ │ │ +00019ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019ee0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00019ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019f40: 2020 2020 2020 207c 0a7c 6f32 203a 204c |.|o2 : L │ │ │ │ -00019f50: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ +00019f30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00019f40: 3220 3a20 4c69 7374 2020 2020 2020 2020 2 : List │ │ │ │ +00019f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019f90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00019f80: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00019f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019fe0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ -00019ff0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -0001a000: 202a 6e6f 7465 2045 6973 656e 6275 6453 *note EisenbudS │ │ │ │ -0001a010: 6861 6d61 7368 3a20 4569 7365 6e62 7564 hamash: Eisenbud │ │ │ │ -0001a020: 5368 616d 6173 682c 202d 2d20 436f 6d70 Shamash, -- Comp │ │ │ │ -0001a030: 7574 6573 2074 6865 2045 6973 656e 6275 utes the Eisenbu │ │ │ │ -0001a040: 642d 5368 616d 6173 680a 2020 2020 436f d-Shamash. Co │ │ │ │ -0001a050: 6d70 6c65 780a 2020 2a20 2a6e 6f74 6520 mplex. * *note │ │ │ │ -0001a060: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20 makeHomotopies: │ │ │ │ -0001a070: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20 makeHomotopies, │ │ │ │ -0001a080: 2d2d 2072 6574 7572 6e73 2061 2073 7973 -- returns a sys │ │ │ │ -0001a090: 7465 6d20 6f66 2068 6967 6865 720a 2020 tem of higher. │ │ │ │ -0001a0a0: 2020 686f 6d6f 746f 7069 6573 0a0a 5761 homotopies..Wa │ │ │ │ -0001a0b0: 7973 2074 6f20 7573 6520 6578 706f 3a0a ys to use expo:. │ │ │ │ -0001a0c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001a0d0: 3d0a 0a20 202a 2022 6578 706f 285a 5a2c =.. * "expo(ZZ, │ │ │ │ -0001a0e0: 4c69 7374 2922 0a20 202a 2022 6578 706f List)". * "expo │ │ │ │ -0001a0f0: 285a 5a2c 5a5a 2922 0a0a 466f 7220 7468 (ZZ,ZZ)"..For th │ │ │ │ -0001a100: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -0001a110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -0001a120: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -0001a130: 6520 6578 706f 3a20 6578 706f 2c20 6973 e expo: expo, is │ │ │ │ -0001a140: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ -0001a150: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ -0001a160: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ -0001a170: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +00019fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 ------------+..S │ │ │ │ +00019fe0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +00019ff0: 0a0a 2020 2a20 2a6e 6f74 6520 4569 7365 .. * *note Eise │ │ │ │ +0001a000: 6e62 7564 5368 616d 6173 683a 2045 6973 nbudShamash: Eis │ │ │ │ +0001a010: 656e 6275 6453 6861 6d61 7368 2c20 2d2d enbudShamash, -- │ │ │ │ +0001a020: 2043 6f6d 7075 7465 7320 7468 6520 4569 Computes the Ei │ │ │ │ +0001a030: 7365 6e62 7564 2d53 6861 6d61 7368 0a20 senbud-Shamash. │ │ │ │ +0001a040: 2020 2043 6f6d 706c 6578 0a20 202a 202a Complex. * * │ │ │ │ +0001a050: 6e6f 7465 206d 616b 6548 6f6d 6f74 6f70 note makeHomotop │ │ │ │ +0001a060: 6965 733a 206d 616b 6548 6f6d 6f74 6f70 ies: makeHomotop │ │ │ │ +0001a070: 6965 732c 202d 2d20 7265 7475 726e 7320 ies, -- returns │ │ │ │ +0001a080: 6120 7379 7374 656d 206f 6620 6869 6768 a system of high │ │ │ │ +0001a090: 6572 0a20 2020 2068 6f6d 6f74 6f70 6965 er. homotopie │ │ │ │ +0001a0a0: 730a 0a57 6179 7320 746f 2075 7365 2065 s..Ways to use e │ │ │ │ +0001a0b0: 7870 6f3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d xpo:.=========== │ │ │ │ +0001a0c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2265 7870 ======.. * "exp │ │ │ │ +0001a0d0: 6f28 5a5a 2c4c 6973 7429 220a 2020 2a20 o(ZZ,List)". * │ │ │ │ +0001a0e0: 2265 7870 6f28 5a5a 2c5a 5a29 220a 0a46 "expo(ZZ,ZZ)"..F │ │ │ │ +0001a0f0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +0001a100: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +0001a110: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +0001a120: 202a 6e6f 7465 2065 7870 6f3a 2065 7870 *note expo: exp │ │ │ │ +0001a130: 6f2c 2069 7320 6120 2a6e 6f74 6520 6d65 o, is a *note me │ │ │ │ +0001a140: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ +0001a150: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ +0001a160: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +0001a170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a1c0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -0001a1d0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -0001a1e0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -0001a1f0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -0001a200: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -0001a210: 312e 3236 2e30 352b 6473 2f4d 322f 4d61 1.26.05+ds/M2/Ma │ │ │ │ -0001a220: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -0001a230: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ -0001a240: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -0001a250: 732e 6d32 3a35 3038 363a 302e 0a1f 0a46 s.m2:5086:0....F │ │ │ │ -0001a260: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ -0001a270: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ -0001a280: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ -0001a290: 2065 7874 6572 696f 7245 7874 4d6f 6475 exteriorExtModu │ │ │ │ -0001a2a0: 6c65 2c20 4e65 7874 3a20 6578 7465 7269 le, Next: exteri │ │ │ │ -0001a2b0: 6f72 486f 6d6f 6c6f 6779 4d6f 6475 6c65 orHomologyModule │ │ │ │ -0001a2c0: 2c20 5072 6576 3a20 6578 706f 2c20 5570 , Prev: expo, Up │ │ │ │ -0001a2d0: 3a20 546f 700a 0a65 7874 6572 696f 7245 : Top..exteriorE │ │ │ │ -0001a2e0: 7874 4d6f 6475 6c65 202d 2d20 4578 7428 xtModule -- Ext( │ │ │ │ -0001a2f0: 4d2c 6b29 206f 7220 4578 7428 4d2c 4e29 M,k) or Ext(M,N) │ │ │ │ -0001a300: 2061 7320 6120 6d6f 6475 6c65 206f 7665 as a module ove │ │ │ │ -0001a310: 7220 616e 2065 7874 6572 696f 7220 616c r an exterior al │ │ │ │ -0001a320: 6765 6272 610a 2a2a 2a2a 2a2a 2a2a 2a2a gebra.********** │ │ │ │ +0001a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +0001a1c0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +0001a1d0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +0001a1e0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +0001a1f0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +0001a200: 6c61 7932 2d31 2e32 362e 3035 2b64 732f lay2-1.26.05+ds/ │ │ │ │ +0001a210: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +0001a220: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ +0001a230: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +0001a240: 7574 696f 6e73 2e6d 323a 3530 3836 3a30 utions.m2:5086:0 │ │ │ │ +0001a250: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ +0001a260: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ +0001a270: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ +0001a280: 4e6f 6465 3a20 6578 7465 7269 6f72 4578 Node: exteriorEx │ │ │ │ +0001a290: 744d 6f64 756c 652c 204e 6578 743a 2065 tModule, Next: e │ │ │ │ +0001a2a0: 7874 6572 696f 7248 6f6d 6f6c 6f67 794d xteriorHomologyM │ │ │ │ +0001a2b0: 6f64 756c 652c 2050 7265 763a 2065 7870 odule, Prev: exp │ │ │ │ +0001a2c0: 6f2c 2055 703a 2054 6f70 0a0a 6578 7465 o, Up: Top..exte │ │ │ │ +0001a2d0: 7269 6f72 4578 744d 6f64 756c 6520 2d2d riorExtModule -- │ │ │ │ +0001a2e0: 2045 7874 284d 2c6b 2920 6f72 2045 7874 Ext(M,k) or Ext │ │ │ │ +0001a2f0: 284d 2c4e 2920 6173 2061 206d 6f64 756c (M,N) as a modul │ │ │ │ +0001a300: 6520 6f76 6572 2061 6e20 6578 7465 7269 e over an exteri │ │ │ │ +0001a310: 6f72 2061 6c67 6562 7261 0a2a 2a2a 2a2a or algebra.***** │ │ │ │ +0001a320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001a330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001a340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001a350: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001a360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001a370: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -0001a380: 200a 2020 2020 2020 2020 4520 3d20 6578 . E = ex │ │ │ │ -0001a390: 7465 7269 6f72 4578 744d 6f64 756c 6528 teriorExtModule( │ │ │ │ -0001a3a0: 662c 4d29 0a20 202a 2049 6e70 7574 733a f,M). * Inputs: │ │ │ │ -0001a3b0: 0a20 2020 2020 202a 2066 2c20 6120 2a6e . * f, a *n │ │ │ │ -0001a3c0: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ -0001a3d0: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ -0001a3e0: 2c2c 2031 2078 2063 2c20 656e 7472 6965 ,, 1 x c, entrie │ │ │ │ -0001a3f0: 7320 6d75 7374 2062 650a 2020 2020 2020 s must be. │ │ │ │ -0001a400: 2020 686f 6d6f 746f 7069 6320 746f 2030 homotopic to 0 │ │ │ │ -0001a410: 206f 6e20 460a 2020 2020 2020 2a20 4d2c on F. * M, │ │ │ │ -0001a420: 2061 202a 6e6f 7465 206d 6f64 756c 653a a *note module: │ │ │ │ -0001a430: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ -0001a440: 6f64 756c 652c 2c20 616e 6e69 6869 6c61 odule,, annihila │ │ │ │ -0001a450: 7465 6420 6279 2074 6865 2065 6c65 6d65 ted by the eleme │ │ │ │ -0001a460: 6e74 730a 2020 2020 2020 2020 6f66 2066 nts. of f │ │ │ │ -0001a470: 660a 2020 2020 2020 2a20 4e2c 2061 202a f. * N, a * │ │ │ │ -0001a480: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -0001a490: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -0001a4a0: 652c 2c20 616e 6e69 6869 6c61 7465 6420 e,, annihilated │ │ │ │ -0001a4b0: 6279 2074 6865 2065 6c65 6d65 6e74 730a by the elements. │ │ │ │ -0001a4c0: 2020 2020 2020 2020 6f66 2066 660a 2020 of ff. │ │ │ │ -0001a4d0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ -0001a4e0: 202a 2045 2c20 6120 2a6e 6f74 6520 6d6f * E, a *note mo │ │ │ │ -0001a4f0: 6475 6c65 3a20 284d 6163 6175 6c61 7932 dule: (Macaulay2 │ │ │ │ -0001a500: 446f 6329 4d6f 6475 6c65 2c2c 204d 6f64 Doc)Module,, Mod │ │ │ │ -0001a510: 756c 6520 6f76 6572 2061 6e20 6578 7465 ule over an exte │ │ │ │ -0001a520: 7269 6f72 0a20 2020 2020 2020 2061 6c67 rior. alg │ │ │ │ -0001a530: 6562 7261 2077 6974 6820 7661 7269 6162 ebra with variab │ │ │ │ -0001a540: 6c65 7320 636f 7272 6573 706f 6e64 696e les correspondin │ │ │ │ -0001a550: 6720 746f 2065 6c65 6d65 6e74 7320 6f66 g to elements of │ │ │ │ -0001a560: 2066 0a0a 4465 7363 7269 7074 696f 6e0a f..Description. │ │ │ │ -0001a570: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620 ===========..If │ │ │ │ -0001a580: 4d2c 4e20 6172 6520 532d 6d6f 6475 6c65 M,N are S-module │ │ │ │ -0001a590: 7320 616e 6e69 6869 6c61 7465 6420 6279 s annihilated by │ │ │ │ -0001a5a0: 2074 6865 2065 6c65 6d65 6e74 7320 6f66 the elements of │ │ │ │ -0001a5b0: 2074 6865 206d 6174 7269 7820 6666 203d the matrix ff = │ │ │ │ -0001a5c0: 2028 665f 312e 2e66 5f63 292c 0a61 6e64 (f_1..f_c),.and │ │ │ │ -0001a5d0: 206b 2069 7320 7468 6520 7265 7369 6475 k is the residu │ │ │ │ -0001a5e0: 6520 6669 656c 6420 6f66 2053 2c20 7468 e field of S, th │ │ │ │ -0001a5f0: 656e 2074 6865 2073 6372 6970 7420 6578 en the script ex │ │ │ │ -0001a600: 7465 7269 6f72 4578 744d 6f64 756c 6528 teriorExtModule( │ │ │ │ -0001a610: 662c 4d29 2072 6574 7572 6e73 0a45 7874 f,M) returns.Ext │ │ │ │ -0001a620: 5f53 284d 2c20 6b29 2061 7320 6120 6d6f _S(M, k) as a mo │ │ │ │ -0001a630: 6475 6c65 206f 7665 7220 616e 2065 7874 dule over an ext │ │ │ │ -0001a640: 6572 696f 7220 616c 6765 6272 6120 4520 erior algebra E │ │ │ │ -0001a650: 3d20 6b3c 655f 312c 2e2e 2e2c 655f 633e = k │ │ │ │ -0001a660: 2c20 7768 6572 6520 7468 650a 655f 6920 , where the.e_i │ │ │ │ -0001a670: 6861 7665 2064 6567 7265 6520 312e 2049 have degree 1. I │ │ │ │ -0001a680: 7420 6973 2063 6f6d 7075 7465 6420 6173 t is computed as │ │ │ │ -0001a690: 2074 6865 2045 2d64 7561 6c20 6f66 2065 the E-dual of e │ │ │ │ -0001a6a0: 7874 6572 696f 7254 6f72 4d6f 6475 6c65 xteriorTorModule │ │ │ │ -0001a6b0: 2e0a 0a54 6865 2073 6372 6970 7420 6578 ...The script ex │ │ │ │ -0001a6c0: 7465 7269 6f72 546f 724d 6f64 756c 6528 teriorTorModule( │ │ │ │ -0001a6d0: 662c 4d2c 4e29 2072 6574 7572 6e73 2045 f,M,N) returns E │ │ │ │ -0001a6e0: 7874 5f53 284d 2c4e 2920 6173 2061 206d xt_S(M,N) as a m │ │ │ │ -0001a6f0: 6f64 756c 6520 6f76 6572 2061 0a62 6967 odule over a.big │ │ │ │ -0001a700: 7261 6465 6420 7269 6e67 2053 4520 3d20 raded ring SE = │ │ │ │ -0001a710: 533c 655f 312c 2e2e 2c65 5f63 3e2c 2077 S, w │ │ │ │ -0001a720: 6865 7265 2074 6865 2065 5f69 2068 6176 here the e_i hav │ │ │ │ -0001a730: 6520 6465 6772 6565 7320 7b64 5f69 2c31 e degrees {d_i,1 │ │ │ │ -0001a740: 7d2c 2077 6865 7265 2064 5f69 0a69 7320 }, where d_i.is │ │ │ │ -0001a750: 7468 6520 6465 6772 6565 206f 6620 665f the degree of f_ │ │ │ │ -0001a760: 692e 2054 6865 206d 6f64 756c 6520 7374 i. The module st │ │ │ │ -0001a770: 7275 6374 7572 652c 2069 6e20 6569 7468 ructure, in eith │ │ │ │ -0001a780: 6572 2063 6173 652c 2069 7320 6465 6669 er case, is defi │ │ │ │ -0001a790: 6e65 6420 6279 2074 6865 0a68 6f6d 6f74 ned by the.homot │ │ │ │ -0001a7a0: 6f70 6965 7320 666f 7220 7468 6520 665f opies for the f_ │ │ │ │ -0001a7b0: 6920 6f6e 2074 6865 2072 6573 6f6c 7574 i on the resolut │ │ │ │ -0001a7c0: 696f 6e20 6f66 204d 2c20 636f 6d70 7574 ion of M, comput │ │ │ │ -0001a7d0: 6564 2062 7920 7468 6520 7363 7269 7074 ed by the script │ │ │ │ -0001a7e0: 0a6d 616b 6548 6f6d 6f74 6f70 6965 7331 .makeHomotopies1 │ │ │ │ -0001a7f0: 2e54 6865 2073 6372 6970 7420 6361 6c6c .The script call │ │ │ │ -0001a800: 7320 6d61 6b65 4d6f 6475 6c65 2074 6f20 s makeModule to │ │ │ │ -0001a810: 636f 6d70 7574 6520 6120 286e 6f6e 2d6d compute a (non-m │ │ │ │ -0001a820: 696e 696d 616c 290a 7072 6573 656e 7461 inimal).presenta │ │ │ │ -0001a830: 7469 6f6e 206f 6620 7468 6973 206d 6f64 tion of this mod │ │ │ │ -0001a840: 756c 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ule...+--------- │ │ │ │ +0001a360: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +0001a370: 7361 6765 3a20 0a20 2020 2020 2020 2045 sage: . E │ │ │ │ +0001a380: 203d 2065 7874 6572 696f 7245 7874 4d6f = exteriorExtMo │ │ │ │ +0001a390: 6475 6c65 2866 2c4d 290a 2020 2a20 496e dule(f,M). * In │ │ │ │ +0001a3a0: 7075 7473 3a0a 2020 2020 2020 2a20 662c puts:. * f, │ │ │ │ +0001a3b0: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ +0001a3c0: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +0001a3d0: 6174 7269 782c 2c20 3120 7820 632c 2065 atrix,, 1 x c, e │ │ │ │ +0001a3e0: 6e74 7269 6573 206d 7573 7420 6265 0a20 ntries must be. │ │ │ │ +0001a3f0: 2020 2020 2020 2068 6f6d 6f74 6f70 6963 homotopic │ │ │ │ +0001a400: 2074 6f20 3020 6f6e 2046 0a20 2020 2020 to 0 on F. │ │ │ │ +0001a410: 202a 204d 2c20 6120 2a6e 6f74 6520 6d6f * M, a *note mo │ │ │ │ +0001a420: 6475 6c65 3a20 284d 6163 6175 6c61 7932 dule: (Macaulay2 │ │ │ │ +0001a430: 446f 6329 4d6f 6475 6c65 2c2c 2061 6e6e Doc)Module,, ann │ │ │ │ +0001a440: 6968 696c 6174 6564 2062 7920 7468 6520 ihilated by the │ │ │ │ +0001a450: 656c 656d 656e 7473 0a20 2020 2020 2020 elements. │ │ │ │ +0001a460: 206f 6620 6666 0a20 2020 2020 202a 204e of ff. * N │ │ │ │ +0001a470: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +0001a480: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0001a490: 4d6f 6475 6c65 2c2c 2061 6e6e 6968 696c Module,, annihil │ │ │ │ +0001a4a0: 6174 6564 2062 7920 7468 6520 656c 656d ated by the elem │ │ │ │ +0001a4b0: 656e 7473 0a20 2020 2020 2020 206f 6620 ents. of │ │ │ │ +0001a4c0: 6666 0a20 202a 204f 7574 7075 7473 3a0a ff. * Outputs:. │ │ │ │ +0001a4d0: 2020 2020 2020 2a20 452c 2061 202a 6e6f * E, a *no │ │ │ │ +0001a4e0: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ +0001a4f0: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ +0001a500: 2c20 4d6f 6475 6c65 206f 7665 7220 616e , Module over an │ │ │ │ +0001a510: 2065 7874 6572 696f 720a 2020 2020 2020 exterior. │ │ │ │ +0001a520: 2020 616c 6765 6272 6120 7769 7468 2076 algebra with v │ │ │ │ +0001a530: 6172 6961 626c 6573 2063 6f72 7265 7370 ariables corresp │ │ │ │ +0001a540: 6f6e 6469 6e67 2074 6f20 656c 656d 656e onding to elemen │ │ │ │ +0001a550: 7473 206f 6620 660a 0a44 6573 6372 6970 ts of f..Descrip │ │ │ │ +0001a560: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +0001a570: 0a0a 4966 204d 2c4e 2061 7265 2053 2d6d ..If M,N are S-m │ │ │ │ +0001a580: 6f64 756c 6573 2061 6e6e 6968 696c 6174 odules annihilat │ │ │ │ +0001a590: 6564 2062 7920 7468 6520 656c 656d 656e ed by the elemen │ │ │ │ +0001a5a0: 7473 206f 6620 7468 6520 6d61 7472 6978 ts of the matrix │ │ │ │ +0001a5b0: 2066 6620 3d20 2866 5f31 2e2e 665f 6329 ff = (f_1..f_c) │ │ │ │ +0001a5c0: 2c0a 616e 6420 6b20 6973 2074 6865 2072 ,.and k is the r │ │ │ │ +0001a5d0: 6573 6964 7565 2066 6965 6c64 206f 6620 esidue field of │ │ │ │ +0001a5e0: 532c 2074 6865 6e20 7468 6520 7363 7269 S, then the scri │ │ │ │ +0001a5f0: 7074 2065 7874 6572 696f 7245 7874 4d6f pt exteriorExtMo │ │ │ │ +0001a600: 6475 6c65 2866 2c4d 2920 7265 7475 726e dule(f,M) return │ │ │ │ +0001a610: 730a 4578 745f 5328 4d2c 206b 2920 6173 s.Ext_S(M, k) as │ │ │ │ +0001a620: 2061 206d 6f64 756c 6520 6f76 6572 2061 a module over a │ │ │ │ +0001a630: 6e20 6578 7465 7269 6f72 2061 6c67 6562 n exterior algeb │ │ │ │ +0001a640: 7261 2045 203d 206b 3c65 5f31 2c2e 2e2e ra E = k, where the │ │ │ │ +0001a660: 0a65 5f69 2068 6176 6520 6465 6772 6565 .e_i have degree │ │ │ │ +0001a670: 2031 2e20 4974 2069 7320 636f 6d70 7574 1. It is comput │ │ │ │ +0001a680: 6564 2061 7320 7468 6520 452d 6475 616c ed as the E-dual │ │ │ │ +0001a690: 206f 6620 6578 7465 7269 6f72 546f 724d of exteriorTorM │ │ │ │ +0001a6a0: 6f64 756c 652e 0a0a 5468 6520 7363 7269 odule...The scri │ │ │ │ +0001a6b0: 7074 2065 7874 6572 696f 7254 6f72 4d6f pt exteriorTorMo │ │ │ │ +0001a6c0: 6475 6c65 2866 2c4d 2c4e 2920 7265 7475 dule(f,M,N) retu │ │ │ │ +0001a6d0: 726e 7320 4578 745f 5328 4d2c 4e29 2061 rns Ext_S(M,N) a │ │ │ │ +0001a6e0: 7320 6120 6d6f 6475 6c65 206f 7665 7220 s a module over │ │ │ │ +0001a6f0: 610a 6269 6772 6164 6564 2072 696e 6720 a.bigraded ring │ │ │ │ +0001a700: 5345 203d 2053 3c65 5f31 2c2e 2e2c 655f SE = S, where the e_ │ │ │ │ +0001a720: 6920 6861 7665 2064 6567 7265 6573 207b i have degrees { │ │ │ │ +0001a730: 645f 692c 317d 2c20 7768 6572 6520 645f d_i,1}, where d_ │ │ │ │ +0001a740: 690a 6973 2074 6865 2064 6567 7265 6520 i.is the degree │ │ │ │ +0001a750: 6f66 2066 5f69 2e20 5468 6520 6d6f 6475 of f_i. The modu │ │ │ │ +0001a760: 6c65 2073 7472 7563 7475 7265 2c20 696e le structure, in │ │ │ │ +0001a770: 2065 6974 6865 7220 6361 7365 2c20 6973 either case, is │ │ │ │ +0001a780: 2064 6566 696e 6564 2062 7920 7468 650a defined by the. │ │ │ │ +0001a790: 686f 6d6f 746f 7069 6573 2066 6f72 2074 homotopies for t │ │ │ │ +0001a7a0: 6865 2066 5f69 206f 6e20 7468 6520 7265 he f_i on the re │ │ │ │ +0001a7b0: 736f 6c75 7469 6f6e 206f 6620 4d2c 2063 solution of M, c │ │ │ │ +0001a7c0: 6f6d 7075 7465 6420 6279 2074 6865 2073 omputed by the s │ │ │ │ +0001a7d0: 6372 6970 740a 6d61 6b65 486f 6d6f 746f cript.makeHomoto │ │ │ │ +0001a7e0: 7069 6573 312e 5468 6520 7363 7269 7074 pies1.The script │ │ │ │ +0001a7f0: 2063 616c 6c73 206d 616b 654d 6f64 756c calls makeModul │ │ │ │ +0001a800: 6520 746f 2063 6f6d 7075 7465 2061 2028 e to compute a ( │ │ │ │ +0001a810: 6e6f 6e2d 6d69 6e69 6d61 6c29 0a70 7265 non-minimal).pre │ │ │ │ +0001a820: 7365 6e74 6174 696f 6e20 6f66 2074 6869 sentation of thi │ │ │ │ +0001a830: 7320 6d6f 6475 6c65 2e0a 0a2b 2d2d 2d2d s module...+---- │ │ │ │ +0001a840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a880: 2d2d 2b0a 7c69 3120 3a20 6b6b 203d 205a --+.|i1 : kk = Z │ │ │ │ -0001a890: 5a2f 3130 3120 2020 2020 2020 2020 2020 Z/101 │ │ │ │ +0001a870: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b -------+.|i1 : k │ │ │ │ +0001a880: 6b20 3d20 5a5a 2f31 3031 2020 2020 2020 k = ZZ/101 │ │ │ │ +0001a890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001a8b0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001a8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001a900: 7c6f 3120 3d20 6b6b 2020 2020 2020 2020 |o1 = kk │ │ │ │ +0001a8f0: 2020 207c 0a7c 6f31 203d 206b 6b20 2020 |.|o1 = kk │ │ │ │ +0001a900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a930: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001a930: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001a940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a970: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ -0001a980: 3a20 5175 6f74 6965 6e74 5269 6e67 2020 : QuotientRing │ │ │ │ +0001a960: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a970: 0a7c 6f31 203a 2051 756f 7469 656e 7452 .|o1 : QuotientR │ │ │ │ +0001a980: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ 0001a990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a9b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001a9a0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0001a9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a9f0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 ------+.|i2 : S │ │ │ │ -0001aa00: 3d20 6b6b 5b61 2c62 2c63 5d20 2020 2020 = kk[a,b,c] │ │ │ │ +0001a9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +0001a9f0: 203a 2053 203d 206b 6b5b 612c 622c 635d : S = kk[a,b,c] │ │ │ │ +0001aa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aa30: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001aa20: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001aa30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aa70: 2020 7c0a 7c6f 3220 3d20 5320 2020 2020 |.|o2 = S │ │ │ │ +0001aa60: 2020 2020 2020 207c 0a7c 6f32 203d 2053 |.|o2 = S │ │ │ │ +0001aa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aab0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001aaa0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001aab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aae0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001aaf0: 7c6f 3220 3a20 506f 6c79 6e6f 6d69 616c |o2 : Polynomial │ │ │ │ -0001ab00: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ +0001aae0: 2020 207c 0a7c 6f32 203a 2050 6f6c 796e |.|o2 : Polyn │ │ │ │ +0001aaf0: 6f6d 6961 6c52 696e 6720 2020 2020 2020 omialRing │ │ │ │ +0001ab00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ab10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ab20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001ab20: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001ab30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ab40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ -0001ab70: 3a20 6620 3d20 6d61 7472 6978 2261 342c : f = matrix"a4, │ │ │ │ -0001ab80: 6234 2c63 3422 2020 2020 2020 2020 2020 b4,c4" │ │ │ │ -0001ab90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aba0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001ab60: 0a7c 6933 203a 2066 203d 206d 6174 7269 .|i3 : f = matri │ │ │ │ +0001ab70: 7822 6134 2c62 342c 6334 2220 2020 2020 x"a4,b4,c4" │ │ │ │ +0001ab80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ab90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001aba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001abb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001abc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001abd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001abe0: 2020 2020 2020 7c0a 7c6f 3320 3d20 7c20 |.|o3 = | │ │ │ │ -0001abf0: 6134 2062 3420 6334 207c 2020 2020 2020 a4 b4 c4 | │ │ │ │ +0001abd0: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +0001abe0: 203d 207c 2061 3420 6234 2063 3420 7c20 = | a4 b4 c4 | │ │ │ │ +0001abf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ac00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ac10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ac20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001ac10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001ac20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ac30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ac40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ac50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ac60: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001ac70: 2020 3120 2020 2020 2033 2020 2020 2020 1 3 │ │ │ │ +0001ac50: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001ac60: 2020 2020 2020 2031 2020 2020 2020 3320 1 3 │ │ │ │ +0001ac70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ac80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ac90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aca0: 7c0a 7c6f 3320 3a20 4d61 7472 6978 2053 |.|o3 : Matrix S │ │ │ │ -0001acb0: 2020 3c2d 2d20 5320 2020 2020 2020 2020 <-- S │ │ │ │ +0001ac90: 2020 2020 207c 0a7c 6f33 203a 204d 6174 |.|o3 : Mat │ │ │ │ +0001aca0: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ +0001acb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001acc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001acd0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001ace0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001acd0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001ace0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001acf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ad00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ad10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001ad20: 3420 3a20 5220 3d20 532f 6964 6561 6c20 4 : R = S/ideal │ │ │ │ -0001ad30: 6620 2020 2020 2020 2020 2020 2020 2020 f │ │ │ │ -0001ad40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ad50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001ad10: 2d2b 0a7c 6934 203a 2052 203d 2053 2f69 -+.|i4 : R = S/i │ │ │ │ +0001ad20: 6465 616c 2066 2020 2020 2020 2020 2020 deal f │ │ │ │ +0001ad30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ad40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001ad50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001ad60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ad70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ad80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ad90: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ -0001ada0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0001ad80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001ad90: 6f34 203d 2052 2020 2020 2020 2020 2020 o4 = R │ │ │ │ +0001ada0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001adb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001adc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001add0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001adc0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001add0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ade0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001adf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae10: 2020 2020 7c0a 7c6f 3420 3a20 5175 6f74 |.|o4 : Quot │ │ │ │ -0001ae20: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +0001ae00: 2020 2020 2020 2020 207c 0a7c 6f34 203a |.|o4 : │ │ │ │ +0001ae10: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +0001ae20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ae30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001ae40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0001ae50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ae90: 2b0a 7c69 3520 3a20 7020 3d20 6d61 7028 +.|i5 : p = map( │ │ │ │ -0001aea0: 522c 5329 2020 2020 2020 2020 2020 2020 R,S) │ │ │ │ +0001ae80: 2d2d 2d2d 2d2b 0a7c 6935 203a 2070 203d -----+.|i5 : p = │ │ │ │ +0001ae90: 206d 6170 2852 2c53 2920 2020 2020 2020 map(R,S) │ │ │ │ +0001aea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aec0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001aed0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001aec0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001aed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001af00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001af10: 3520 3d20 6d61 7020 2852 2c20 532c 207b 5 = map (R, S, { │ │ │ │ -0001af20: 612c 2062 2c20 637d 2920 2020 2020 2020 a, b, c}) │ │ │ │ -0001af30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001af40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001af00: 207c 0a7c 6f35 203d 206d 6170 2028 522c |.|o5 = map (R, │ │ │ │ +0001af10: 2053 2c20 7b61 2c20 622c 2063 7d29 2020 S, {a, b, c}) │ │ │ │ +0001af20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001af30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001af40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001af50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001af60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001af70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001af80: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20 |.|o5 : │ │ │ │ -0001af90: 5269 6e67 4d61 7020 5220 3c2d 2d20 5320 RingMap R <-- S │ │ │ │ +0001af70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001af80: 6f35 203a 2052 696e 674d 6170 2052 203c o5 : RingMap R < │ │ │ │ +0001af90: 2d2d 2053 2020 2020 2020 2020 2020 2020 -- S │ │ │ │ 0001afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001afc0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001afb0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001afc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001afd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001afe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001aff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b000: 2d2d 2d2d 2b0a 7c69 3620 3a20 4d20 3d20 ----+.|i6 : M = │ │ │ │ -0001b010: 636f 6b65 7220 6d61 7028 525e 322c 2052 coker map(R^2, R │ │ │ │ -0001b020: 5e7b 333a 2d31 7d2c 207b 7b61 2c62 2c63 ^{3:-1}, {{a,b,c │ │ │ │ -0001b030: 7d2c 7b62 2c63 2c61 7d7d 2920 2020 2020 },{b,c,a}}) │ │ │ │ -0001b040: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001aff0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ +0001b000: 204d 203d 2063 6f6b 6572 206d 6170 2852 M = coker map(R │ │ │ │ +0001b010: 5e32 2c20 525e 7b33 3a2d 317d 2c20 7b7b ^2, R^{3:-1}, {{ │ │ │ │ +0001b020: 612c 622c 637d 2c7b 622c 632c 617d 7d29 a,b,c},{b,c,a}}) │ │ │ │ +0001b030: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b080: 7c0a 7c6f 3620 3d20 636f 6b65 726e 656c |.|o6 = cokernel │ │ │ │ -0001b090: 207c 2061 2062 2063 207c 2020 2020 2020 | a b c | │ │ │ │ +0001b070: 2020 2020 207c 0a7c 6f36 203d 2063 6f6b |.|o6 = cok │ │ │ │ +0001b080: 6572 6e65 6c20 7c20 6120 6220 6320 7c20 ernel | a b c | │ │ │ │ +0001b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001b0c0: 7c20 2020 2020 2020 2020 2020 2020 207c | | │ │ │ │ -0001b0d0: 2062 2063 2061 207c 2020 2020 2020 2020 b c a | │ │ │ │ +0001b0b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b0c0: 2020 2020 7c20 6220 6320 6120 7c20 2020 | b c a | │ │ │ │ +0001b0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b0f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001b0f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b130: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b150: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -0001b160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b170: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20 |.|o6 : │ │ │ │ -0001b180: 522d 6d6f 6475 6c65 2c20 7175 6f74 6965 R-module, quotie │ │ │ │ -0001b190: 6e74 206f 6620 5220 2020 2020 2020 2020 nt of R │ │ │ │ -0001b1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b1b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001b120: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001b130: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001b140: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +0001b150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b160: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001b170: 6f36 203a 2052 2d6d 6f64 756c 652c 2071 o6 : R-module, q │ │ │ │ +0001b180: 756f 7469 656e 7420 6f66 2052 2020 2020 uotient of R │ │ │ │ +0001b190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b1a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b1f0: 2d2d 2d2d 2b0a 7c69 3720 3a20 6265 7474 ----+.|i7 : bett │ │ │ │ -0001b200: 6920 2846 4620 3d66 7265 6552 6573 6f6c i (FF =freeResol │ │ │ │ -0001b210: 7574 696f 6e28 204d 2c20 4c65 6e67 7468 ution( M, Length │ │ │ │ -0001b220: 4c69 6d69 7420 3d3e 3629 2920 2020 2020 Limit =>6)) │ │ │ │ -0001b230: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b1e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a ---------+.|i7 : │ │ │ │ +0001b1f0: 2062 6574 7469 2028 4646 203d 6672 6565 betti (FF =free │ │ │ │ +0001b200: 5265 736f 6c75 7469 6f6e 2820 4d2c 204c Resolution( M, L │ │ │ │ +0001b210: 656e 6774 684c 696d 6974 203d 3e36 2929 engthLimit =>6)) │ │ │ │ +0001b220: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001b230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b270: 7c0a 7c20 2020 2020 2020 2020 2020 2030 |.| 0 │ │ │ │ -0001b280: 2031 2032 2033 2034 2020 3520 2036 2020 1 2 3 4 5 6 │ │ │ │ +0001b260: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001b270: 2020 2020 3020 3120 3220 3320 3420 2035 0 1 2 3 4 5 │ │ │ │ +0001b280: 2020 3620 2020 2020 2020 2020 2020 2020 6 │ │ │ │ 0001b290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b2a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001b2b0: 7c6f 3720 3d20 746f 7461 6c3a 2032 2033 |o7 = total: 2 3 │ │ │ │ -0001b2c0: 2034 2036 2039 2031 3320 3138 2020 2020 4 6 9 13 18 │ │ │ │ +0001b2a0: 2020 207c 0a7c 6f37 203d 2074 6f74 616c |.|o7 = total │ │ │ │ +0001b2b0: 3a20 3220 3320 3420 3620 3920 3133 2031 : 2 3 4 6 9 13 1 │ │ │ │ +0001b2c0: 3820 2020 2020 2020 2020 2020 2020 2020 8 │ │ │ │ 0001b2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b2e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001b2f0: 2020 2020 2020 2020 303a 2032 2033 202e 0: 2 3 . │ │ │ │ -0001b300: 202e 202e 2020 2e20 202e 2020 2020 2020 . . . . │ │ │ │ -0001b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b320: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001b330: 2020 2020 2020 313a 202e 202e 2031 202e 1: . . 1 . │ │ │ │ -0001b340: 202e 2020 2e20 202e 2020 2020 2020 2020 . . . │ │ │ │ -0001b350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b360: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001b370: 2020 2020 323a 202e 202e 2033 2033 202e 2: . . 3 3 . │ │ │ │ -0001b380: 2020 2e20 202e 2020 2020 2020 2020 2020 . . │ │ │ │ -0001b390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b3a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001b3b0: 2020 333a 202e 202e 202e 2033 2033 2020 3: . . . 3 3 │ │ │ │ -0001b3c0: 2e20 202e 2020 2020 2020 2020 2020 2020 . . │ │ │ │ -0001b3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b3e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001b3f0: 343a 202e 202e 202e 202e 2033 2020 3320 4: . . . . 3 3 │ │ │ │ -0001b400: 202e 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -0001b410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b420: 2020 7c0a 7c20 2020 2020 2020 2020 353a |.| 5: │ │ │ │ -0001b430: 202e 202e 202e 202e 2033 2020 3920 2036 . . . . 3 9 6 │ │ │ │ +0001b2e0: 207c 0a7c 2020 2020 2020 2020 2030 3a20 |.| 0: │ │ │ │ +0001b2f0: 3220 3320 2e20 2e20 2e20 202e 2020 2e20 2 3 . . . . . │ │ │ │ +0001b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b310: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001b320: 0a7c 2020 2020 2020 2020 2031 3a20 2e20 .| 1: . │ │ │ │ +0001b330: 2e20 3120 2e20 2e20 202e 2020 2e20 2020 . 1 . . . . │ │ │ │ +0001b340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b350: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001b360: 2020 2020 2020 2020 2032 3a20 2e20 2e20 2: . . │ │ │ │ +0001b370: 3320 3320 2e20 202e 2020 2e20 2020 2020 3 3 . . . │ │ │ │ +0001b380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b390: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001b3a0: 2020 2020 2020 2033 3a20 2e20 2e20 2e20 3: . . . │ │ │ │ +0001b3b0: 3320 3320 202e 2020 2e20 2020 2020 2020 3 3 . . │ │ │ │ +0001b3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b3d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001b3e0: 2020 2020 2034 3a20 2e20 2e20 2e20 2e20 4: . . . . │ │ │ │ +0001b3f0: 3320 2033 2020 2e20 2020 2020 2020 2020 3 3 . │ │ │ │ +0001b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b410: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001b420: 2020 2035 3a20 2e20 2e20 2e20 2e20 3320 5: . . . . 3 │ │ │ │ +0001b430: 2039 2020 3620 2020 2020 2020 2020 2020 9 6 │ │ │ │ 0001b440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b460: 7c0a 7c20 2020 2020 2020 2020 363a 202e |.| 6: . │ │ │ │ -0001b470: 202e 202e 202e 202e 2020 2e20 2033 2020 . . . . . 3 │ │ │ │ +0001b450: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001b460: 2036 3a20 2e20 2e20 2e20 2e20 2e20 202e 6: . . . . . . │ │ │ │ +0001b470: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0001b480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b490: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001b4a0: 7c20 2020 2020 2020 2020 373a 202e 202e | 7: . . │ │ │ │ -0001b4b0: 202e 202e 202e 2020 3120 2039 2020 2020 . . . 1 9 │ │ │ │ +0001b490: 2020 207c 0a7c 2020 2020 2020 2020 2037 |.| 7 │ │ │ │ +0001b4a0: 3a20 2e20 2e20 2e20 2e20 2e20 2031 2020 : . . . . . 1 │ │ │ │ +0001b4b0: 3920 2020 2020 2020 2020 2020 2020 2020 9 │ │ │ │ 0001b4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b4d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001b4d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001b4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b510: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ -0001b520: 3a20 4265 7474 6954 616c 6c79 2020 2020 : BettiTally │ │ │ │ +0001b500: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001b510: 0a7c 6f37 203a 2042 6574 7469 5461 6c6c .|o7 : BettiTall │ │ │ │ +0001b520: 7920 2020 2020 2020 2020 2020 2020 2020 y │ │ │ │ 0001b530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b550: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001b540: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0001b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b590: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 4d53 ------+.|i8 : MS │ │ │ │ -0001b5a0: 203d 2070 7275 6e65 2070 7573 6846 6f72 = prune pushFor │ │ │ │ -0001b5b0: 7761 7264 2870 2c20 636f 6b65 7220 4646 ward(p, coker FF │ │ │ │ -0001b5c0: 2e64 645f 3629 3b20 2020 2020 2020 2020 .dd_6); │ │ │ │ -0001b5d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001b580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ +0001b590: 203a 204d 5320 3d20 7072 756e 6520 7075 : MS = prune pu │ │ │ │ +0001b5a0: 7368 466f 7277 6172 6428 702c 2063 6f6b shForward(p, cok │ │ │ │ +0001b5b0: 6572 2046 462e 6464 5f36 293b 2020 2020 er FF.dd_6); │ │ │ │ +0001b5c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001b5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b610: 2d2d 2b0a 7c69 3920 3a20 7265 7346 6c64 --+.|i9 : resFld │ │ │ │ -0001b620: 203a 3d20 7075 7368 466f 7277 6172 6428 := pushForward( │ │ │ │ -0001b630: 702c 2063 6f6b 6572 2076 6172 7320 5229 p, coker vars R) │ │ │ │ -0001b640: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ -0001b650: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001b600: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2072 -------+.|i9 : r │ │ │ │ +0001b610: 6573 466c 6420 3a3d 2070 7573 6846 6f72 esFld := pushFor │ │ │ │ +0001b620: 7761 7264 2870 2c20 636f 6b65 7220 7661 ward(p, coker va │ │ │ │ +0001b630: 7273 2052 293b 2020 2020 2020 2020 2020 rs R); │ │ │ │ +0001b640: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001b650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001b690: 7c69 3130 203a 2054 203d 2065 7874 6572 |i10 : T = exter │ │ │ │ -0001b6a0: 696f 7254 6f72 4d6f 6475 6c65 2866 2c4d iorTorModule(f,M │ │ │ │ -0001b6b0: 5329 3b20 2020 2020 2020 2020 2020 2020 S); │ │ │ │ -0001b6c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001b680: 2d2d 2d2b 0a7c 6931 3020 3a20 5420 3d20 ---+.|i10 : T = │ │ │ │ +0001b690: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ +0001b6a0: 6528 662c 4d53 293b 2020 2020 2020 2020 e(f,MS); │ │ │ │ +0001b6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b6c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001b6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b700: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 ----------+.|i11 │ │ │ │ -0001b710: 203a 2045 203d 2065 7874 6572 696f 7245 : E = exteriorE │ │ │ │ -0001b720: 7874 4d6f 6475 6c65 2866 2c4d 5329 3b20 xtModule(f,MS); │ │ │ │ -0001b730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b740: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001b6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001b700: 0a7c 6931 3120 3a20 4520 3d20 6578 7465 .|i11 : E = exte │ │ │ │ +0001b710: 7269 6f72 4578 744d 6f64 756c 6528 662c riorExtModule(f, │ │ │ │ +0001b720: 4d53 293b 2020 2020 2020 2020 2020 2020 MS); │ │ │ │ +0001b730: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0001b740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b780: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2068 ------+.|i12 : h │ │ │ │ -0001b790: 6628 2d34 2e2e 302c 4529 2020 2020 2020 f(-4..0,E) │ │ │ │ +0001b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0001b780: 3220 3a20 6866 282d 342e 2e30 2c45 2920 2 : hf(-4..0,E) │ │ │ │ +0001b790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b7c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001b7b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001b7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b800: 2020 7c0a 7c6f 3132 203d 207b 302c 2039 |.|o12 = {0, 9 │ │ │ │ -0001b810: 2c20 3239 2c20 3333 2c20 3133 7d20 2020 , 29, 33, 13} │ │ │ │ +0001b7f0: 2020 2020 2020 207c 0a7c 6f31 3220 3d20 |.|o12 = │ │ │ │ +0001b800: 7b30 2c20 392c 2032 392c 2033 332c 2031 {0, 9, 29, 33, 1 │ │ │ │ +0001b810: 337d 2020 2020 2020 2020 2020 2020 2020 3} │ │ │ │ 0001b820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b840: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b830: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001b840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b870: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001b880: 7c6f 3132 203a 204c 6973 7420 2020 2020 |o12 : List │ │ │ │ +0001b870: 2020 207c 0a7c 6f31 3220 3a20 4c69 7374 |.|o12 : List │ │ │ │ +0001b880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b8b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001b8b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001b8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 ----------+.|i13 │ │ │ │ -0001b900: 203a 2062 6574 7469 2066 7265 6552 6573 : betti freeRes │ │ │ │ -0001b910: 6f6c 7574 696f 6e20 4d53 2020 2020 2020 olution MS │ │ │ │ -0001b920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b930: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001b8f0: 0a7c 6931 3320 3a20 6265 7474 6920 6672 .|i13 : betti fr │ │ │ │ +0001b900: 6565 5265 736f 6c75 7469 6f6e 204d 5320 eeResolution MS │ │ │ │ +0001b910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b920: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001b930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b970: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001b980: 2020 2020 2020 2030 2020 3120 2032 2033 0 1 2 3 │ │ │ │ +0001b960: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001b970: 2020 2020 2020 2020 2020 2020 3020 2031 0 1 │ │ │ │ +0001b980: 2020 3220 3320 2020 2020 2020 2020 2020 2 3 │ │ │ │ 0001b990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b9b0: 2020 2020 7c0a 7c6f 3133 203d 2074 6f74 |.|o13 = tot │ │ │ │ -0001b9c0: 616c 3a20 3133 2033 3320 3239 2039 2020 al: 13 33 29 9 │ │ │ │ +0001b9a0: 2020 2020 2020 2020 207c 0a7c 6f31 3320 |.|o13 │ │ │ │ +0001b9b0: 3d20 746f 7461 6c3a 2031 3320 3333 2032 = total: 13 33 2 │ │ │ │ +0001b9c0: 3920 3920 2020 2020 2020 2020 2020 2020 9 9 │ │ │ │ 0001b9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b9f0: 2020 7c0a 7c20 2020 2020 2020 2020 2039 |.| 9 │ │ │ │ -0001ba00: 3a20 2033 2020 2e20 202e 202e 2020 2020 : 3 . . . │ │ │ │ +0001b9e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001b9f0: 2020 2020 393a 2020 3320 202e 2020 2e20 9: 3 . . │ │ │ │ +0001ba00: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ 0001ba10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba30: 7c0a 7c20 2020 2020 2020 2020 3130 3a20 |.| 10: │ │ │ │ -0001ba40: 2039 2020 3620 202e 202e 2020 2020 2020 9 6 . . │ │ │ │ +0001ba20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001ba30: 2031 303a 2020 3920 2036 2020 2e20 2e20 10: 9 6 . . │ │ │ │ +0001ba40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ba50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001ba70: 7c20 2020 2020 2020 2020 3131 3a20 202e | 11: . │ │ │ │ -0001ba80: 2020 3320 202e 202e 2020 2020 2020 2020 3 . . │ │ │ │ +0001ba60: 2020 207c 0a7c 2020 2020 2020 2020 2031 |.| 1 │ │ │ │ +0001ba70: 313a 2020 2e20 2033 2020 2e20 2e20 2020 1: . 3 . . │ │ │ │ +0001ba80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ba90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001baa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001bab0: 2020 2020 2020 2020 3132 3a20 2031 2031 12: 1 1 │ │ │ │ -0001bac0: 3520 202e 202e 2020 2020 2020 2020 2020 5 . . │ │ │ │ -0001bad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001baf0: 2020 2020 2020 3133 3a20 202e 2020 3920 13: . 9 │ │ │ │ -0001bb00: 2038 202e 2020 2020 2020 2020 2020 2020 8 . │ │ │ │ -0001bb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bb20: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001bb30: 2020 2020 3134 3a20 202e 2020 2e20 2036 14: . . 6 │ │ │ │ -0001bb40: 202e 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -0001bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bb60: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001bb70: 2020 3135 3a20 202e 2020 2e20 3132 202e 15: . . 12 . │ │ │ │ +0001baa0: 207c 0a7c 2020 2020 2020 2020 2031 323a |.| 12: │ │ │ │ +0001bab0: 2020 3120 3135 2020 2e20 2e20 2020 2020 1 15 . . │ │ │ │ +0001bac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bad0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001bae0: 0a7c 2020 2020 2020 2020 2031 333a 2020 .| 13: │ │ │ │ +0001baf0: 2e20 2039 2020 3820 2e20 2020 2020 2020 . 9 8 . │ │ │ │ +0001bb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bb10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001bb20: 2020 2020 2020 2020 2031 343a 2020 2e20 14: . │ │ │ │ +0001bb30: 202e 2020 3620 2e20 2020 2020 2020 2020 . 6 . │ │ │ │ +0001bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bb50: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001bb60: 2020 2020 2020 2031 353a 2020 2e20 202e 15: . . │ │ │ │ +0001bb70: 2031 3220 2e20 2020 2020 2020 2020 2020 12 . │ │ │ │ 0001bb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bba0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001bbb0: 3136 3a20 202e 2020 2e20 2033 2033 2020 16: . . 3 3 │ │ │ │ +0001bb90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001bba0: 2020 2020 2031 363a 2020 2e20 202e 2020 16: . . │ │ │ │ +0001bbb0: 3320 3320 2020 2020 2020 2020 2020 2020 3 3 │ │ │ │ 0001bbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bbe0: 2020 7c0a 7c20 2020 2020 2020 2020 3137 |.| 17 │ │ │ │ -0001bbf0: 3a20 202e 2020 2e20 202e 2033 2020 2020 : . . . 3 │ │ │ │ +0001bbd0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001bbe0: 2020 2031 373a 2020 2e20 202e 2020 2e20 17: . . . │ │ │ │ +0001bbf0: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0001bc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bc20: 7c0a 7c20 2020 2020 2020 2020 3138 3a20 |.| 18: │ │ │ │ -0001bc30: 202e 2020 2e20 202e 2033 2020 2020 2020 . . . 3 │ │ │ │ +0001bc10: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001bc20: 2031 383a 2020 2e20 202e 2020 2e20 3320 18: . . . 3 │ │ │ │ +0001bc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bc50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001bc60: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001bc50: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001bc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bc90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001bca0: 3133 203a 2042 6574 7469 5461 6c6c 7920 13 : BettiTally │ │ │ │ +0001bc90: 207c 0a7c 6f31 3320 3a20 4265 7474 6954 |.|o13 : BettiT │ │ │ │ +0001bca0: 616c 6c79 2020 2020 2020 2020 2020 2020 ally │ │ │ │ 0001bcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bcd0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0001bcc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001bcd0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0001bce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001bcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bd10: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a --------+.|i14 : │ │ │ │ -0001bd20: 2062 6574 7469 2066 7265 6552 6573 6f6c betti freeResol │ │ │ │ -0001bd30: 7574 696f 6e20 2850 4520 3d20 7072 756e ution (PE = prun │ │ │ │ -0001bd40: 6520 452c 204c 656e 6774 684c 696d 6974 e E, LengthLimit │ │ │ │ -0001bd50: 203d 3e20 3629 7c0a 7c20 2020 2020 2020 => 6)|.| │ │ │ │ +0001bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0001bd10: 6931 3420 3a20 6265 7474 6920 6672 6565 i14 : betti free │ │ │ │ +0001bd20: 5265 736f 6c75 7469 6f6e 2028 5045 203d Resolution (PE = │ │ │ │ +0001bd30: 2070 7275 6e65 2045 2c20 4c65 6e67 7468 prune E, Length │ │ │ │ +0001bd40: 4c69 6d69 7420 3d3e 2036 297c 0a7c 2020 Limit => 6)|.| │ │ │ │ +0001bd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bd90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001bda0: 2020 2020 2030 2020 3120 2032 2020 3320 0 1 2 3 │ │ │ │ -0001bdb0: 2034 2020 2035 2020 2036 2020 2020 2020 4 5 6 │ │ │ │ -0001bdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bdd0: 2020 7c0a 7c6f 3134 203d 2074 6f74 616c |.|o14 = total │ │ │ │ -0001bde0: 3a20 3136 2031 3320 3235 2034 3920 3831 : 16 13 25 49 81 │ │ │ │ -0001bdf0: 2031 3231 2031 3639 2020 2020 2020 2020 121 169 │ │ │ │ -0001be00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001be10: 7c0a 7c20 2020 2020 2020 2020 2d33 3a20 |.| -3: │ │ │ │ -0001be20: 2039 2020 3420 2033 2020 3320 2033 2020 9 4 3 3 3 │ │ │ │ -0001be30: 2033 2020 2033 2020 2020 2020 2020 2020 3 3 │ │ │ │ -0001be40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001be50: 7c20 2020 2020 2020 2020 2d32 3a20 2036 | -2: 6 │ │ │ │ -0001be60: 2020 3320 202e 2020 2e20 202e 2020 202e 3 . . . . │ │ │ │ -0001be70: 2020 202e 2020 2020 2020 2020 2020 2020 . │ │ │ │ -0001be80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001be90: 2020 2020 2020 2020 2d31 3a20 202e 2020 -1: . │ │ │ │ -0001bea0: 2e20 2037 2031 3820 3333 2020 3532 2020 . 7 18 33 52 │ │ │ │ -0001beb0: 3735 2020 2020 2020 2020 2020 2020 2020 75 │ │ │ │ -0001bec0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001bed0: 2020 2020 2020 2030 3a20 2031 2020 3620 0: 1 6 │ │ │ │ -0001bee0: 3135 2032 3820 3435 2020 3636 2020 3931 15 28 45 66 91 │ │ │ │ -0001bef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf00: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001bd80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001bd90: 2020 2020 2020 2020 2020 3020 2031 2020 0 1 │ │ │ │ +0001bda0: 3220 2033 2020 3420 2020 3520 2020 3620 2 3 4 5 6 │ │ │ │ +0001bdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bdc0: 2020 2020 2020 207c 0a7c 6f31 3420 3d20 |.|o14 = │ │ │ │ +0001bdd0: 746f 7461 6c3a 2031 3620 3133 2032 3520 total: 16 13 25 │ │ │ │ +0001bde0: 3439 2038 3120 3132 3120 3136 3920 2020 49 81 121 169 │ │ │ │ +0001bdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001be00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001be10: 202d 333a 2020 3920 2034 2020 3320 2033 -3: 9 4 3 3 │ │ │ │ +0001be20: 2020 3320 2020 3320 2020 3320 2020 2020 3 3 3 │ │ │ │ +0001be30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001be40: 2020 207c 0a7c 2020 2020 2020 2020 202d |.| - │ │ │ │ +0001be50: 323a 2020 3620 2033 2020 2e20 202e 2020 2: 6 3 . . │ │ │ │ +0001be60: 2e20 2020 2e20 2020 2e20 2020 2020 2020 . . . │ │ │ │ +0001be70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001be80: 207c 0a7c 2020 2020 2020 2020 202d 313a |.| -1: │ │ │ │ +0001be90: 2020 2e20 202e 2020 3720 3138 2033 3320 . . 7 18 33 │ │ │ │ +0001bea0: 2035 3220 2037 3520 2020 2020 2020 2020 52 75 │ │ │ │ +0001beb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001bec0: 0a7c 2020 2020 2020 2020 2020 303a 2020 .| 0: │ │ │ │ +0001bed0: 3120 2036 2031 3520 3238 2034 3520 2036 1 6 15 28 45 6 │ │ │ │ +0001bee0: 3620 2039 3120 2020 2020 2020 2020 2020 6 91 │ │ │ │ +0001bef0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001bf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf40: 2020 2020 2020 7c0a 7c6f 3134 203a 2042 |.|o14 : B │ │ │ │ -0001bf50: 6574 7469 5461 6c6c 7920 2020 2020 2020 ettiTally │ │ │ │ +0001bf30: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0001bf40: 3420 3a20 4265 7474 6954 616c 6c79 2020 4 : BettiTally │ │ │ │ +0001bf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001bf70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001bf80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001bf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001bfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bfc0: 2d2d 2b0a 7c69 3135 203a 2062 6574 7469 --+.|i15 : betti │ │ │ │ -0001bfd0: 2066 7265 6552 6573 6f6c 7574 696f 6e20 freeResolution │ │ │ │ -0001bfe0: 2850 5420 3d20 7072 756e 6520 542c 204c (PT = prune T, L │ │ │ │ -0001bff0: 656e 6774 684c 696d 6974 203d 3e20 3629 engthLimit => 6) │ │ │ │ -0001c000: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001bfb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 -------+.|i15 : │ │ │ │ +0001bfc0: 6265 7474 6920 6672 6565 5265 736f 6c75 betti freeResolu │ │ │ │ +0001bfd0: 7469 6f6e 2028 5054 203d 2070 7275 6e65 tion (PT = prune │ │ │ │ +0001bfe0: 2054 2c20 4c65 6e67 7468 4c69 6d69 7420 T, LengthLimit │ │ │ │ +0001bff0: 3d3e 2036 297c 0a7c 2020 2020 2020 2020 => 6)|.| │ │ │ │ +0001c000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c030: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001c040: 7c20 2020 2020 2020 2020 2020 2020 2030 | 0 │ │ │ │ -0001c050: 2020 3120 2032 2020 2033 2020 2034 2020 1 2 3 4 │ │ │ │ -0001c060: 2035 2020 2036 2020 2020 2020 2020 2020 5 6 │ │ │ │ -0001c070: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001c080: 3135 203d 2074 6f74 616c 3a20 3331 2035 15 = total: 31 5 │ │ │ │ -0001c090: 3520 3837 2031 3237 2031 3735 2032 3331 5 87 127 175 231 │ │ │ │ -0001c0a0: 2032 3935 2020 2020 2020 2020 2020 2020 295 │ │ │ │ -0001c0b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001c0c0: 2020 2020 2020 2030 3a20 3133 2032 3420 0: 13 24 │ │ │ │ -0001c0d0: 3339 2020 3538 2020 3831 2031 3038 2031 39 58 81 108 1 │ │ │ │ -0001c0e0: 3339 2020 2020 2020 2020 2020 2020 2020 39 │ │ │ │ -0001c0f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001c100: 2020 2020 2031 3a20 3138 2033 3120 3438 1: 18 31 48 │ │ │ │ -0001c110: 2020 3639 2020 3934 2031 3233 2031 3536 69 94 123 156 │ │ │ │ -0001c120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c130: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001c030: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c040: 2020 2020 3020 2031 2020 3220 2020 3320 0 1 2 3 │ │ │ │ +0001c050: 2020 3420 2020 3520 2020 3620 2020 2020 4 5 6 │ │ │ │ +0001c060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c070: 207c 0a7c 6f31 3520 3d20 746f 7461 6c3a |.|o15 = total: │ │ │ │ +0001c080: 2033 3120 3535 2038 3720 3132 3720 3137 31 55 87 127 17 │ │ │ │ +0001c090: 3520 3233 3120 3239 3520 2020 2020 2020 5 231 295 │ │ │ │ +0001c0a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001c0b0: 0a7c 2020 2020 2020 2020 2020 303a 2031 .| 0: 1 │ │ │ │ +0001c0c0: 3320 3234 2033 3920 2035 3820 2038 3120 3 24 39 58 81 │ │ │ │ +0001c0d0: 3130 3820 3133 3920 2020 2020 2020 2020 108 139 │ │ │ │ +0001c0e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001c0f0: 2020 2020 2020 2020 2020 313a 2031 3820 1: 18 │ │ │ │ +0001c100: 3331 2034 3820 2036 3920 2039 3420 3132 31 48 69 94 12 │ │ │ │ +0001c110: 3320 3135 3620 2020 2020 2020 2020 2020 3 156 │ │ │ │ +0001c120: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001c130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c170: 2020 2020 7c0a 7c6f 3135 203a 2042 6574 |.|o15 : Bet │ │ │ │ -0001c180: 7469 5461 6c6c 7920 2020 2020 2020 2020 tiTally │ │ │ │ +0001c160: 2020 2020 2020 2020 207c 0a7c 6f31 3520 |.|o15 │ │ │ │ +0001c170: 3a20 4265 7474 6954 616c 6c79 2020 2020 : BettiTally │ │ │ │ +0001c180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c1b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001c1a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0001c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c1f0: 2b0a 7c69 3136 203a 2045 3120 3d20 7072 +.|i16 : E1 = pr │ │ │ │ -0001c200: 756e 6520 6578 7465 7269 6f72 4578 744d une exteriorExtM │ │ │ │ -0001c210: 6f64 756c 6528 662c 204d 532c 2072 6573 odule(f, MS, res │ │ │ │ -0001c220: 466c 6429 3b20 2020 2020 2020 2020 7c0a Fld); |. │ │ │ │ -0001c230: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001c1e0: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 4531 -----+.|i16 : E1 │ │ │ │ +0001c1f0: 203d 2070 7275 6e65 2065 7874 6572 696f = prune exterio │ │ │ │ +0001c200: 7245 7874 4d6f 6475 6c65 2866 2c20 4d53 rExtModule(f, MS │ │ │ │ +0001c210: 2c20 7265 7346 6c64 293b 2020 2020 2020 , resFld); │ │ │ │ +0001c220: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001c230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001c270: 3137 203a 2072 696e 6720 4531 2020 2020 17 : ring E1 │ │ │ │ +0001c260: 2d2b 0a7c 6931 3720 3a20 7269 6e67 2045 -+.|i17 : ring E │ │ │ │ +0001c270: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 0001c280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c2a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001c290: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001c2a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001c2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c2e0: 2020 2020 2020 2020 7c0a 7c6f 3137 203d |.|o17 = │ │ │ │ -0001c2f0: 206b 6b5b 5820 2e2e 5820 2c20 6520 2e2e kk[X ..X , e .. │ │ │ │ -0001c300: 6520 5d20 2020 2020 2020 2020 2020 2020 e ] │ │ │ │ -0001c310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c320: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001c330: 2020 2030 2020 2032 2020 2030 2020 2032 0 2 0 2 │ │ │ │ +0001c2d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001c2e0: 6f31 3720 3d20 6b6b 5b58 202e 2e58 202c o17 = kk[X ..X , │ │ │ │ +0001c2f0: 2065 202e 2e65 205d 2020 2020 2020 2020 e ..e ] │ │ │ │ +0001c300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c310: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001c320: 2020 2020 2020 2020 3020 2020 3220 2020 0 2 │ │ │ │ +0001c330: 3020 2020 3220 2020 2020 2020 2020 2020 0 2 │ │ │ │ 0001c340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c360: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001c350: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c3a0: 2020 7c0a 7c6f 3137 203a 2050 6f6c 796e |.|o17 : Polyn │ │ │ │ -0001c3b0: 6f6d 6961 6c52 696e 672c 2033 2073 6b65 omialRing, 3 ske │ │ │ │ -0001c3c0: 7720 636f 6d6d 7574 6174 6976 6520 7661 w commutative va │ │ │ │ -0001c3d0: 7269 6162 6c65 2873 2920 2020 2020 2020 riable(s) │ │ │ │ -0001c3e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001c390: 2020 2020 2020 207c 0a7c 6f31 3720 3a20 |.|o17 : │ │ │ │ +0001c3a0: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c20 PolynomialRing, │ │ │ │ +0001c3b0: 3320 736b 6577 2063 6f6d 6d75 7461 7469 3 skew commutati │ │ │ │ +0001c3c0: 7665 2076 6172 6961 626c 6528 7329 2020 ve variable(s) │ │ │ │ +0001c3d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001c3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001c420: 7c69 3138 203a 2065 7852 696e 6720 3d20 |i18 : exRing = │ │ │ │ -0001c430: 6b6b 5b65 5f30 2c65 5f31 2c65 5f32 2c20 kk[e_0,e_1,e_2, │ │ │ │ -0001c440: 536b 6577 436f 6d6d 7574 6174 6976 6520 SkewCommutative │ │ │ │ -0001c450: 3d3e 7472 7565 5d20 2020 2020 7c0a 7c20 =>true] |.| │ │ │ │ +0001c410: 2d2d 2d2b 0a7c 6931 3820 3a20 6578 5269 ---+.|i18 : exRi │ │ │ │ +0001c420: 6e67 203d 206b 6b5b 655f 302c 655f 312c ng = kk[e_0,e_1, │ │ │ │ +0001c430: 655f 322c 2053 6b65 7743 6f6d 6d75 7461 e_2, SkewCommuta │ │ │ │ +0001c440: 7469 7665 203d 3e74 7275 655d 2020 2020 tive =>true] │ │ │ │ +0001c450: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001c460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c490: 2020 2020 2020 2020 2020 7c0a 7c6f 3138 |.|o18 │ │ │ │ -0001c4a0: 203d 2065 7852 696e 6720 2020 2020 2020 = exRing │ │ │ │ +0001c480: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001c490: 0a7c 6f31 3820 3d20 6578 5269 6e67 2020 .|o18 = exRing │ │ │ │ +0001c4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c4d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001c4c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001c4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c510: 2020 2020 2020 7c0a 7c6f 3138 203a 2050 |.|o18 : P │ │ │ │ -0001c520: 6f6c 796e 6f6d 6961 6c52 696e 672c 2033 olynomialRing, 3 │ │ │ │ -0001c530: 2073 6b65 7720 636f 6d6d 7574 6174 6976 skew commutativ │ │ │ │ -0001c540: 6520 7661 7269 6162 6c65 2873 2920 2020 e variable(s) │ │ │ │ -0001c550: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001c500: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0001c510: 3820 3a20 506f 6c79 6e6f 6d69 616c 5269 8 : PolynomialRi │ │ │ │ +0001c520: 6e67 2c20 3320 736b 6577 2063 6f6d 6d75 ng, 3 skew commu │ │ │ │ +0001c530: 7461 7469 7665 2076 6172 6961 626c 6528 tative variable( │ │ │ │ +0001c540: 7329 2020 2020 2020 207c 0a2b 2d2d 2d2d s) |.+---- │ │ │ │ +0001c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c590: 2d2d 2b0a 0a57 6520 6361 6e20 616c 736f --+..We can also │ │ │ │ -0001c5a0: 2063 6f6e 7374 7275 6374 2074 6865 2065 construct the e │ │ │ │ -0001c5b0: 7874 6572 696f 7245 7874 4d6f 6475 6c65 xteriorExtModule │ │ │ │ -0001c5c0: 2061 7320 6120 6269 6772 6164 6564 206d as a bigraded m │ │ │ │ -0001c5d0: 6f64 756c 652c 206f 7665 7220 6120 7269 odule, over a ri │ │ │ │ -0001c5e0: 6e67 0a53 4520 7468 6174 2068 6173 2062 ng.SE that has b │ │ │ │ -0001c5f0: 6f74 6820 706f 6c79 6e6f 6d69 616c 2076 oth polynomial v │ │ │ │ -0001c600: 6172 6961 626c 6573 206c 696b 6520 5320 ariables like S │ │ │ │ -0001c610: 616e 6420 6578 7465 7269 6f72 2076 6172 and exterior var │ │ │ │ -0001c620: 6961 626c 6573 206c 696b 6520 452e 2054 iables like E. T │ │ │ │ -0001c630: 6865 0a70 6f6c 796e 6f6d 6961 6c20 7661 he.polynomial va │ │ │ │ -0001c640: 7269 6162 6c65 7320 6861 7665 2064 6567 riables have deg │ │ │ │ -0001c650: 7265 6573 207b 312c 307d 2e20 5468 6520 rees {1,0}. The │ │ │ │ -0001c660: 6578 7465 7269 6f72 2076 6172 6961 626c exterior variabl │ │ │ │ -0001c670: 6573 2068 6176 6520 6465 6772 6565 730a es have degrees. │ │ │ │ -0001c680: 7b64 6567 2066 665f 692c 2031 7d2e 0a0a {deg ff_i, 1}... │ │ │ │ -0001c690: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001c580: 2d2d 2d2d 2d2d 2d2b 0a0a 5765 2063 616e -------+..We can │ │ │ │ +0001c590: 2061 6c73 6f20 636f 6e73 7472 7563 7420 also construct │ │ │ │ +0001c5a0: 7468 6520 6578 7465 7269 6f72 4578 744d the exteriorExtM │ │ │ │ +0001c5b0: 6f64 756c 6520 6173 2061 2062 6967 7261 odule as a bigra │ │ │ │ +0001c5c0: 6465 6420 6d6f 6475 6c65 2c20 6f76 6572 ded module, over │ │ │ │ +0001c5d0: 2061 2072 696e 670a 5345 2074 6861 7420 a ring.SE that │ │ │ │ +0001c5e0: 6861 7320 626f 7468 2070 6f6c 796e 6f6d has both polynom │ │ │ │ +0001c5f0: 6961 6c20 7661 7269 6162 6c65 7320 6c69 ial variables li │ │ │ │ +0001c600: 6b65 2053 2061 6e64 2065 7874 6572 696f ke S and exterio │ │ │ │ +0001c610: 7220 7661 7269 6162 6c65 7320 6c69 6b65 r variables like │ │ │ │ +0001c620: 2045 2e20 5468 650a 706f 6c79 6e6f 6d69 E. The.polynomi │ │ │ │ +0001c630: 616c 2076 6172 6961 626c 6573 2068 6176 al variables hav │ │ │ │ +0001c640: 6520 6465 6772 6565 7320 7b31 2c30 7d2e e degrees {1,0}. │ │ │ │ +0001c650: 2054 6865 2065 7874 6572 696f 7220 7661 The exterior va │ │ │ │ +0001c660: 7269 6162 6c65 7320 6861 7665 2064 6567 riables have deg │ │ │ │ +0001c670: 7265 6573 0a7b 6465 6720 6666 5f69 2c20 rees.{deg ff_i, │ │ │ │ +0001c680: 317d 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 1}...+---------- │ │ │ │ +0001c690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c6c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20 -------+.|i19 : │ │ │ │ -0001c6d0: 4531 203d 2070 7275 6e65 2065 7874 6572 E1 = prune exter │ │ │ │ -0001c6e0: 696f 7245 7874 4d6f 6475 6c65 2866 2c20 iorExtModule(f, │ │ │ │ -0001c6f0: 4d53 2c20 7265 7346 6c64 293b 2020 2020 MS, resFld); │ │ │ │ -0001c700: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0001c6c0: 3139 203a 2045 3120 3d20 7072 756e 6520 19 : E1 = prune │ │ │ │ +0001c6d0: 6578 7465 7269 6f72 4578 744d 6f64 756c exteriorExtModul │ │ │ │ +0001c6e0: 6528 662c 204d 532c 2072 6573 466c 6429 e(f, MS, resFld) │ │ │ │ +0001c6f0: 3b20 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d ; |.+-------- │ │ │ │ +0001c700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c730: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020 ---------+.|i20 │ │ │ │ -0001c740: 3a20 7269 6e67 2045 3120 2020 2020 2020 : ring E1 │ │ │ │ +0001c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0001c730: 7c69 3230 203a 2072 696e 6720 4531 2020 |i20 : ring E1 │ │ │ │ +0001c740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c770: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c760: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001c770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c7a0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0001c7b0: 3020 3d20 6b6b 5b58 202e 2e58 202c 2065 0 = kk[X ..X , e │ │ │ │ -0001c7c0: 202e 2e65 205d 2020 2020 2020 2020 2020 ..e ] │ │ │ │ -0001c7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c7e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001c7f0: 2030 2020 2032 2020 2030 2020 2032 2020 0 2 0 2 │ │ │ │ +0001c7a0: 7c0a 7c6f 3230 203d 206b 6b5b 5820 2e2e |.|o20 = kk[X .. │ │ │ │ +0001c7b0: 5820 2c20 6520 2e2e 6520 5d20 2020 2020 X , e ..e ] │ │ │ │ +0001c7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c7d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001c7e0: 2020 2020 2020 3020 2020 3220 2020 3020 0 2 0 │ │ │ │ +0001c7f0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0001c800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c810: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001c810: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001c820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c850: 2020 2020 2020 7c0a 7c6f 3230 203a 2050 |.|o20 : P │ │ │ │ -0001c860: 6f6c 796e 6f6d 6961 6c52 696e 672c 2033 olynomialRing, 3 │ │ │ │ -0001c870: 2073 6b65 7720 636f 6d6d 7574 6174 6976 skew commutativ │ │ │ │ -0001c880: 6520 7661 7269 6162 6c65 2873 2920 207c e variable(s) | │ │ │ │ -0001c890: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0001c840: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +0001c850: 3020 3a20 506f 6c79 6e6f 6d69 616c 5269 0 : PolynomialRi │ │ │ │ +0001c860: 6e67 2c20 3320 736b 6577 2063 6f6d 6d75 ng, 3 skew commu │ │ │ │ +0001c870: 7461 7469 7665 2076 6172 6961 626c 6528 tative variable( │ │ │ │ +0001c880: 7329 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d s) |.+--------- │ │ │ │ +0001c890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c8c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a --------+.|i21 : │ │ │ │ -0001c8d0: 2065 7852 696e 6720 3d20 6b6b 5b65 5f30 exRing = kk[e_0 │ │ │ │ -0001c8e0: 2c65 5f31 2c65 5f32 2c20 536b 6577 436f ,e_1,e_2, SkewCo │ │ │ │ -0001c8f0: 6d6d 7574 6174 6976 6520 3d3e 7472 7565 mmutative =>true │ │ │ │ -0001c900: 5d7c 0a7c 2020 2020 2020 2020 2020 2020 ]|.| │ │ │ │ +0001c8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0001c8c0: 6932 3120 3a20 6578 5269 6e67 203d 206b i21 : exRing = k │ │ │ │ +0001c8d0: 6b5b 655f 302c 655f 312c 655f 322c 2053 k[e_0,e_1,e_2, S │ │ │ │ +0001c8e0: 6b65 7743 6f6d 6d75 7461 7469 7665 203d kewCommutative = │ │ │ │ +0001c8f0: 3e74 7275 655d 7c0a 7c20 2020 2020 2020 >true]|.| │ │ │ │ +0001c900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c930: 2020 2020 2020 2020 2020 7c0a 7c6f 3231 |.|o21 │ │ │ │ -0001c940: 203d 2065 7852 696e 6720 2020 2020 2020 = exRing │ │ │ │ +0001c920: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001c930: 0a7c 6f32 3120 3d20 6578 5269 6e67 2020 .|o21 = exRing │ │ │ │ +0001c940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c970: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c960: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001c970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c9a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001c9b0: 3231 203a 2050 6f6c 796e 6f6d 6961 6c52 21 : PolynomialR │ │ │ │ -0001c9c0: 696e 672c 2033 2073 6b65 7720 636f 6d6d ing, 3 skew comm │ │ │ │ -0001c9d0: 7574 6174 6976 6520 7661 7269 6162 6c65 utative variable │ │ │ │ -0001c9e0: 2873 2920 207c 0a2b 2d2d 2d2d 2d2d 2d2d (s) |.+-------- │ │ │ │ +0001c9a0: 207c 0a7c 6f32 3120 3a20 506f 6c79 6e6f |.|o21 : Polyno │ │ │ │ +0001c9b0: 6d69 616c 5269 6e67 2c20 3320 736b 6577 mialRing, 3 skew │ │ │ │ +0001c9c0: 2063 6f6d 6d75 7461 7469 7665 2076 6172 commutative var │ │ │ │ +0001c9d0: 6961 626c 6528 7329 2020 7c0a 2b2d 2d2d iable(s) |.+--- │ │ │ │ +0001c9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ca00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ca10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001ca20: 0a54 6f20 7365 6520 7468 6174 2074 6869 .To see that thi │ │ │ │ -0001ca30: 7320 6973 2072 6561 6c6c 7920 7468 6520 s is really the │ │ │ │ -0001ca40: 7361 6d65 206d 6f64 756c 652c 2077 6974 same module, wit │ │ │ │ -0001ca50: 6820 6120 6d6f 7265 2063 6f6d 706c 6578 h a more complex │ │ │ │ -0001ca60: 2067 7261 6469 6e67 2c20 7765 2063 616e grading, we can │ │ │ │ -0001ca70: 0a62 7269 6e67 2069 7420 6f76 6572 2074 .bring it over t │ │ │ │ -0001ca80: 6f20 6120 7075 7265 2065 7874 6572 696f o a pure exterio │ │ │ │ -0001ca90: 7220 616c 6765 6272 612e 204e 6f74 6520 r algebra. Note │ │ │ │ -0001caa0: 7468 6174 2074 6865 206e 6563 6573 7361 that the necessa │ │ │ │ -0001cab0: 7279 206d 6170 206f 6620 7269 6e67 730a ry map of rings. │ │ │ │ -0001cac0: 6d75 7374 2063 6f6e 7461 696e 2061 2044 must contain a D │ │ │ │ -0001cad0: 6567 7265 654d 6170 206f 7074 696f 6e2e egreeMap option. │ │ │ │ -0001cae0: 2049 6e20 6765 6e65 7261 6c20 7765 2063 In general we c │ │ │ │ -0001caf0: 6f75 6c64 206f 6e6c 7920 7461 6b65 2074 ould only take t │ │ │ │ -0001cb00: 6865 2064 6567 7265 6573 206f 660a 7468 he degrees of.th │ │ │ │ -0001cb10: 6520 6765 6e65 7261 746f 7273 206f 6620 e generators of │ │ │ │ -0001cb20: 7468 6520 6578 7465 7269 6f72 2061 6c67 the exterior alg │ │ │ │ -0001cb30: 6562 7261 2074 6f20 6265 2074 6865 2067 ebra to be the g │ │ │ │ -0001cb40: 6364 206f 6620 2074 6865 2064 6567 2066 cd of the deg f │ │ │ │ -0001cb50: 665f 6920 3b20 696e 2074 6865 0a65 7861 f_i ; in the.exa │ │ │ │ -0001cb60: 6d70 6c65 2061 626f 7665 2074 6869 7320 mple above this │ │ │ │ -0001cb70: 6973 2031 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d is 1...+-------- │ │ │ │ +0001ca10: 2d2d 2d2b 0a0a 546f 2073 6565 2074 6861 ---+..To see tha │ │ │ │ +0001ca20: 7420 7468 6973 2069 7320 7265 616c 6c79 t this is really │ │ │ │ +0001ca30: 2074 6865 2073 616d 6520 6d6f 6475 6c65 the same module │ │ │ │ +0001ca40: 2c20 7769 7468 2061 206d 6f72 6520 636f , with a more co │ │ │ │ +0001ca50: 6d70 6c65 7820 6772 6164 696e 672c 2077 mplex grading, w │ │ │ │ +0001ca60: 6520 6361 6e0a 6272 696e 6720 6974 206f e can.bring it o │ │ │ │ +0001ca70: 7665 7220 746f 2061 2070 7572 6520 6578 ver to a pure ex │ │ │ │ +0001ca80: 7465 7269 6f72 2061 6c67 6562 7261 2e20 terior algebra. │ │ │ │ +0001ca90: 4e6f 7465 2074 6861 7420 7468 6520 6e65 Note that the ne │ │ │ │ +0001caa0: 6365 7373 6172 7920 6d61 7020 6f66 2072 cessary map of r │ │ │ │ +0001cab0: 696e 6773 0a6d 7573 7420 636f 6e74 6169 ings.must contai │ │ │ │ +0001cac0: 6e20 6120 4465 6772 6565 4d61 7020 6f70 n a DegreeMap op │ │ │ │ +0001cad0: 7469 6f6e 2e20 496e 2067 656e 6572 616c tion. In general │ │ │ │ +0001cae0: 2077 6520 636f 756c 6420 6f6e 6c79 2074 we could only t │ │ │ │ +0001caf0: 616b 6520 7468 6520 6465 6772 6565 7320 ake the degrees │ │ │ │ +0001cb00: 6f66 0a74 6865 2067 656e 6572 6174 6f72 of.the generator │ │ │ │ +0001cb10: 7320 6f66 2074 6865 2065 7874 6572 696f s of the exterio │ │ │ │ +0001cb20: 7220 616c 6765 6272 6120 746f 2062 6520 r algebra to be │ │ │ │ +0001cb30: 7468 6520 6763 6420 6f66 2020 7468 6520 the gcd of the │ │ │ │ +0001cb40: 6465 6720 6666 5f69 203b 2069 6e20 7468 deg ff_i ; in th │ │ │ │ +0001cb50: 650a 6578 616d 706c 6520 6162 6f76 6520 e.example above │ │ │ │ +0001cb60: 7468 6973 2069 7320 312e 0a0a 2b2d 2d2d this is 1...+--- │ │ │ │ +0001cb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cbc0: 2d2d 2b0a 7c69 3232 203a 2071 203d 206d --+.|i22 : q = m │ │ │ │ -0001cbd0: 6170 2865 7852 696e 672c 2072 696e 6720 ap(exRing, ring │ │ │ │ -0001cbe0: 4531 2c20 7b33 3a30 2c65 5f30 2c65 5f31 E1, {3:0,e_0,e_1 │ │ │ │ -0001cbf0: 2c65 5f32 7d2c 2044 6567 7265 654d 6170 ,e_2}, DegreeMap │ │ │ │ -0001cc00: 203d 3e20 6420 2d3e 207b 645f 317d 297c => d -> {d_1})| │ │ │ │ -0001cc10: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001cbb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3220 3a20 -------+.|i22 : │ │ │ │ +0001cbc0: 7120 3d20 6d61 7028 6578 5269 6e67 2c20 q = map(exRing, │ │ │ │ +0001cbd0: 7269 6e67 2045 312c 207b 333a 302c 655f ring E1, {3:0,e_ │ │ │ │ +0001cbe0: 302c 655f 312c 655f 327d 2c20 4465 6772 0,e_1,e_2}, Degr │ │ │ │ +0001cbf0: 6565 4d61 7020 3d3e 2064 202d 3e20 7b64 eeMap => d -> {d │ │ │ │ +0001cc00: 5f31 7d29 7c0a 7c20 2020 2020 2020 2020 _1})|.| │ │ │ │ +0001cc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cc50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001cc60: 3232 203d 206d 6170 2028 6578 5269 6e67 22 = map (exRing │ │ │ │ -0001cc70: 2c20 6b6b 5b58 202e 2e58 202c 2065 202e , kk[X ..X , e . │ │ │ │ -0001cc80: 2e65 205d 2c20 7b30 2c20 302c 2030 2c20 .e ], {0, 0, 0, │ │ │ │ -0001cc90: 6520 2c20 6520 2c20 6520 7d29 2020 2020 e , e , e }) │ │ │ │ -0001cca0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0001ccb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ccc0: 2020 2030 2020 2032 2020 2030 2020 2032 0 2 0 2 │ │ │ │ -0001ccd0: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ -0001cce0: 2020 3120 2020 3220 2020 2020 2020 2020 1 2 │ │ │ │ -0001ccf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001cc50: 207c 0a7c 6f32 3220 3d20 6d61 7020 2865 |.|o22 = map (e │ │ │ │ +0001cc60: 7852 696e 672c 206b 6b5b 5820 2e2e 5820 xRing, kk[X ..X │ │ │ │ +0001cc70: 2c20 6520 2e2e 6520 5d2c 207b 302c 2030 , e ..e ], {0, 0 │ │ │ │ +0001cc80: 2c20 302c 2065 202c 2065 202c 2065 207d , 0, e , e , e } │ │ │ │ +0001cc90: 2920 2020 2020 2020 2020 2020 2020 7c0a ) |. │ │ │ │ +0001cca0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001ccb0: 2020 2020 2020 2020 3020 2020 3220 2020 0 2 │ │ │ │ +0001ccc0: 3020 2020 3220 2020 2020 2020 2020 2020 0 2 │ │ │ │ +0001ccd0: 2020 2030 2020 2031 2020 2032 2020 2020 0 1 2 │ │ │ │ +0001cce0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001ccf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cd40: 2020 207c 0a7c 6f32 3220 3a20 5269 6e67 |.|o22 : Ring │ │ │ │ -0001cd50: 4d61 7020 6578 5269 6e67 203c 2d2d 206b Map exRing <-- k │ │ │ │ -0001cd60: 6b5b 5820 2e2e 5820 2c20 6520 2e2e 6520 k[X ..X , e ..e │ │ │ │ -0001cd70: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ -0001cd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cd90: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cdb0: 3020 2020 3220 2020 3020 2020 3220 2020 0 2 0 2 │ │ │ │ +0001cd30: 2020 2020 2020 2020 7c0a 7c6f 3232 203a |.|o22 : │ │ │ │ +0001cd40: 2052 696e 674d 6170 2065 7852 696e 6720 RingMap exRing │ │ │ │ +0001cd50: 3c2d 2d20 6b6b 5b58 202e 2e58 202c 2065 <-- kk[X ..X , e │ │ │ │ +0001cd60: 202e 2e65 205d 2020 2020 2020 2020 2020 ..e ] │ │ │ │ +0001cd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cd80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001cd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cda0: 2020 2020 2030 2020 2032 2020 2030 2020 0 2 0 │ │ │ │ +0001cdb0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0001cdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cdd0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0001cdd0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0001cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ce20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233 ----------+.|i23 │ │ │ │ -0001ce30: 203a 2045 3220 3d20 636f 6b65 7220 7120 : E2 = coker q │ │ │ │ -0001ce40: 7072 6573 656e 7461 7469 6f6e 2045 313b presentation E1; │ │ │ │ +0001ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001ce20: 0a7c 6932 3320 3a20 4532 203d 2063 6f6b .|i23 : E2 = cok │ │ │ │ +0001ce30: 6572 2071 2070 7265 7365 6e74 6174 696f er q presentatio │ │ │ │ +0001ce40: 6e20 4531 3b20 2020 2020 2020 2020 2020 n E1; │ │ │ │ 0001ce50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ce60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ce70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0001ce60: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001ce70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ce80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ce90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ceb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cec0: 2d2d 2d2d 2b0a 7c69 3234 203a 2068 6628 ----+.|i24 : hf( │ │ │ │ -0001ced0: 2d35 2e2e 352c 4532 2920 3d3d 2068 6628 -5..5,E2) == hf( │ │ │ │ -0001cee0: 2d35 2e2e 352c 4529 2020 2020 2020 2020 -5..5,E) │ │ │ │ +0001ceb0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3420 ---------+.|i24 │ │ │ │ +0001cec0: 3a20 6866 282d 352e 2e35 2c45 3229 203d : hf(-5..5,E2) = │ │ │ │ +0001ced0: 3d20 6866 282d 352e 2e35 2c45 2920 2020 = hf(-5..5,E) │ │ │ │ +0001cee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cf10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001cf00: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001cf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cf50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001cf60: 7c6f 3234 203d 2074 7275 6520 2020 2020 |o24 = true │ │ │ │ +0001cf50: 2020 207c 0a7c 6f32 3420 3d20 7472 7565 |.|o24 = true │ │ │ │ +0001cf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cfa0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001cfa0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0001cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cff0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -0001d000: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -0001d010: 2a20 2a6e 6f74 6520 6578 7465 7269 6f72 * *note exterior │ │ │ │ -0001d020: 546f 724d 6f64 756c 653a 2065 7874 6572 TorModule: exter │ │ │ │ -0001d030: 696f 7254 6f72 4d6f 6475 6c65 2c20 2d2d iorTorModule, -- │ │ │ │ -0001d040: 2054 6f72 2061 7320 6120 6d6f 6475 6c65 Tor as a module │ │ │ │ -0001d050: 206f 7665 7220 616e 0a20 2020 2065 7874 over an. ext │ │ │ │ -0001d060: 6572 696f 7220 616c 6765 6272 6120 6f72 erior algebra or │ │ │ │ -0001d070: 2062 6967 7261 6465 6420 616c 6765 6272 bigraded algebr │ │ │ │ -0001d080: 610a 2020 2a20 2a6e 6f74 6520 6d61 6b65 a. * *note make │ │ │ │ -0001d090: 4d6f 6475 6c65 3a20 6d61 6b65 4d6f 6475 Module: makeModu │ │ │ │ -0001d0a0: 6c65 2c20 2d2d 206d 616b 6573 2061 204d le, -- makes a M │ │ │ │ -0001d0b0: 6f64 756c 6520 6f75 7420 6f66 2061 2063 odule out of a c │ │ │ │ -0001d0c0: 6f6c 6c65 6374 696f 6e20 6f66 0a20 2020 ollection of. │ │ │ │ -0001d0d0: 206d 6f64 756c 6573 2061 6e64 206d 6170 modules and map │ │ │ │ -0001d0e0: 730a 0a57 6179 7320 746f 2075 7365 2065 s..Ways to use e │ │ │ │ -0001d0f0: 7874 6572 696f 7245 7874 4d6f 6475 6c65 xteriorExtModule │ │ │ │ -0001d100: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -0001d110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001d120: 0a0a 2020 2a20 2265 7874 6572 696f 7245 .. * "exteriorE │ │ │ │ -0001d130: 7874 4d6f 6475 6c65 284d 6174 7269 782c xtModule(Matrix, │ │ │ │ -0001d140: 4d6f 6475 6c65 2922 0a20 202a 2022 6578 Module)". * "ex │ │ │ │ -0001d150: 7465 7269 6f72 4578 744d 6f64 756c 6528 teriorExtModule( │ │ │ │ -0001d160: 4d61 7472 6978 2c4d 6f64 756c 652c 4d6f Matrix,Module,Mo │ │ │ │ -0001d170: 6475 6c65 2922 0a0a 466f 7220 7468 6520 dule)"..For the │ │ │ │ -0001d180: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -0001d190: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -0001d1a0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -0001d1b0: 6578 7465 7269 6f72 4578 744d 6f64 756c exteriorExtModul │ │ │ │ -0001d1c0: 653a 2065 7874 6572 696f 7245 7874 4d6f e: exteriorExtMo │ │ │ │ -0001d1d0: 6475 6c65 2c20 6973 2061 202a 6e6f 7465 dule, is a *note │ │ │ │ -0001d1e0: 206d 6574 686f 640a 6675 6e63 7469 6f6e method.function │ │ │ │ -0001d1f0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -0001d200: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e MethodFunction,. │ │ │ │ -0001d210: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +0001cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +0001cff0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +0001d000: 3d0a 0a20 202a 202a 6e6f 7465 2065 7874 =.. * *note ext │ │ │ │ +0001d010: 6572 696f 7254 6f72 4d6f 6475 6c65 3a20 eriorTorModule: │ │ │ │ +0001d020: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ +0001d030: 652c 202d 2d20 546f 7220 6173 2061 206d e, -- Tor as a m │ │ │ │ +0001d040: 6f64 756c 6520 6f76 6572 2061 6e0a 2020 odule over an. │ │ │ │ +0001d050: 2020 6578 7465 7269 6f72 2061 6c67 6562 exterior algeb │ │ │ │ +0001d060: 7261 206f 7220 6269 6772 6164 6564 2061 ra or bigraded a │ │ │ │ +0001d070: 6c67 6562 7261 0a20 202a 202a 6e6f 7465 lgebra. * *note │ │ │ │ +0001d080: 206d 616b 654d 6f64 756c 653a 206d 616b makeModule: mak │ │ │ │ +0001d090: 654d 6f64 756c 652c 202d 2d20 6d61 6b65 eModule, -- make │ │ │ │ +0001d0a0: 7320 6120 4d6f 6475 6c65 206f 7574 206f s a Module out o │ │ │ │ +0001d0b0: 6620 6120 636f 6c6c 6563 7469 6f6e 206f f a collection o │ │ │ │ +0001d0c0: 660a 2020 2020 6d6f 6475 6c65 7320 616e f. modules an │ │ │ │ +0001d0d0: 6420 6d61 7073 0a0a 5761 7973 2074 6f20 d maps..Ways to │ │ │ │ +0001d0e0: 7573 6520 6578 7465 7269 6f72 4578 744d use exteriorExtM │ │ │ │ +0001d0f0: 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d 3d3d odule:.========= │ │ │ │ +0001d100: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001d110: 3d3d 3d3d 3d0a 0a20 202a 2022 6578 7465 =====.. * "exte │ │ │ │ +0001d120: 7269 6f72 4578 744d 6f64 756c 6528 4d61 riorExtModule(Ma │ │ │ │ +0001d130: 7472 6978 2c4d 6f64 756c 6529 220a 2020 trix,Module)". │ │ │ │ +0001d140: 2a20 2265 7874 6572 696f 7245 7874 4d6f * "exteriorExtMo │ │ │ │ +0001d150: 6475 6c65 284d 6174 7269 782c 4d6f 6475 dule(Matrix,Modu │ │ │ │ +0001d160: 6c65 2c4d 6f64 756c 6529 220a 0a46 6f72 le,Module)"..For │ │ │ │ +0001d170: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +0001d180: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001d190: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +0001d1a0: 6e6f 7465 2065 7874 6572 696f 7245 7874 note exteriorExt │ │ │ │ +0001d1b0: 4d6f 6475 6c65 3a20 6578 7465 7269 6f72 Module: exterior │ │ │ │ +0001d1c0: 4578 744d 6f64 756c 652c 2069 7320 6120 ExtModule, is a │ │ │ │ +0001d1d0: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e *note method.fun │ │ │ │ +0001d1e0: 6374 696f 6e3a 2028 4d61 6361 756c 6179 ction: (Macaulay │ │ │ │ +0001d1f0: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ +0001d200: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ion,...--------- │ │ │ │ +0001d210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d260: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -0001d270: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -0001d280: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -0001d290: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -0001d2a0: 6361 756c 6179 322d 312e 3236 2e30 352b caulay2-1.26.05+ │ │ │ │ -0001d2b0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -0001d2c0: 7061 636b 6167 6573 2f0a 436f 6d70 6c65 packages/.Comple │ │ │ │ -0001d2d0: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ -0001d2e0: 736f 6c75 7469 6f6e 732e 6d32 3a34 3236 solutions.m2:426 │ │ │ │ -0001d2f0: 333a 302e 0a1f 0a46 696c 653a 2043 6f6d 3:0....File: Com │ │ │ │ -0001d300: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ -0001d310: 6e52 6573 6f6c 7574 696f 6e73 2e69 6e66 nResolutions.inf │ │ │ │ -0001d320: 6f2c 204e 6f64 653a 2065 7874 6572 696f o, Node: exterio │ │ │ │ -0001d330: 7248 6f6d 6f6c 6f67 794d 6f64 756c 652c rHomologyModule, │ │ │ │ -0001d340: 204e 6578 743a 2065 7874 6572 696f 7254 Next: exteriorT │ │ │ │ -0001d350: 6f72 4d6f 6475 6c65 2c20 5072 6576 3a20 orModule, Prev: │ │ │ │ -0001d360: 6578 7465 7269 6f72 4578 744d 6f64 756c exteriorExtModul │ │ │ │ -0001d370: 652c 2055 703a 2054 6f70 0a0a 6578 7465 e, Up: Top..exte │ │ │ │ -0001d380: 7269 6f72 486f 6d6f 6c6f 6779 4d6f 6475 riorHomologyModu │ │ │ │ -0001d390: 6c65 202d 2d20 4d61 6b65 2074 6865 2068 le -- Make the h │ │ │ │ -0001d3a0: 6f6d 6f6c 6f67 7920 6f66 2061 2063 6f6d omology of a com │ │ │ │ -0001d3b0: 706c 6578 2069 6e74 6f20 6120 6d6f 6475 plex into a modu │ │ │ │ -0001d3c0: 6c65 206f 7665 7220 616e 2065 7874 6572 le over an exter │ │ │ │ -0001d3d0: 696f 7220 616c 6765 6272 610a 2a2a 2a2a ior algebra.**** │ │ │ │ +0001d250: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +0001d260: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +0001d270: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +0001d280: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +0001d290: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +0001d2a0: 362e 3035 2b64 732f 4d32 2f4d 6163 6175 6.05+ds/M2/Macau │ │ │ │ +0001d2b0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ +0001d2c0: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ +0001d2d0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d ionResolutions.m │ │ │ │ +0001d2e0: 323a 3432 3633 3a30 2e0a 1f0a 4669 6c65 2:4263:0....File │ │ │ │ +0001d2f0: 3a20 436f 6d70 6c65 7465 496e 7465 7273 : CompleteInters │ │ │ │ +0001d300: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ +0001d310: 732e 696e 666f 2c20 4e6f 6465 3a20 6578 s.info, Node: ex │ │ │ │ +0001d320: 7465 7269 6f72 486f 6d6f 6c6f 6779 4d6f teriorHomologyMo │ │ │ │ +0001d330: 6475 6c65 2c20 4e65 7874 3a20 6578 7465 dule, Next: exte │ │ │ │ +0001d340: 7269 6f72 546f 724d 6f64 756c 652c 2050 riorTorModule, P │ │ │ │ +0001d350: 7265 763a 2065 7874 6572 696f 7245 7874 rev: exteriorExt │ │ │ │ +0001d360: 4d6f 6475 6c65 2c20 5570 3a20 546f 700a Module, Up: Top. │ │ │ │ +0001d370: 0a65 7874 6572 696f 7248 6f6d 6f6c 6f67 .exteriorHomolog │ │ │ │ +0001d380: 794d 6f64 756c 6520 2d2d 204d 616b 6520 yModule -- Make │ │ │ │ +0001d390: 7468 6520 686f 6d6f 6c6f 6779 206f 6620 the homology of │ │ │ │ +0001d3a0: 6120 636f 6d70 6c65 7820 696e 746f 2061 a complex into a │ │ │ │ +0001d3b0: 206d 6f64 756c 6520 6f76 6572 2061 6e20 module over an │ │ │ │ +0001d3c0: 6578 7465 7269 6f72 2061 6c67 6562 7261 exterior algebra │ │ │ │ +0001d3d0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ 0001d3e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001d3f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001d400: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001d410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001d420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001d430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -0001d440: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -0001d450: 204d 203d 2065 7874 6572 696f 7248 6f6d M = exteriorHom │ │ │ │ -0001d460: 6f6c 6f67 794d 6f64 756c 6528 6666 2c20 ologyModule(ff, │ │ │ │ -0001d470: 4329 0a20 202a 2049 6e70 7574 733a 0a20 C). * Inputs:. │ │ │ │ -0001d480: 2020 2020 202a 2066 662c 2061 202a 6e6f * ff, a *no │ │ │ │ -0001d490: 7465 206d 6174 7269 783a 2028 4d61 6361 te matrix: (Maca │ │ │ │ -0001d4a0: 756c 6179 3244 6f63 294d 6174 7269 782c ulay2Doc)Matrix, │ │ │ │ -0001d4b0: 2c20 4d61 7472 6978 206f 6620 656c 656d , Matrix of elem │ │ │ │ -0001d4c0: 656e 7473 2074 6861 7420 6172 650a 2020 ents that are. │ │ │ │ -0001d4d0: 2020 2020 2020 686f 6d6f 746f 7069 6320 homotopic │ │ │ │ -0001d4e0: 746f 2030 206f 6e20 430a 2020 2020 2020 to 0 on C. │ │ │ │ -0001d4f0: 2a20 432c 2061 202a 6e6f 7465 2063 6f6d * C, a *note com │ │ │ │ -0001d500: 706c 6578 3a20 2843 6f6d 706c 6578 6573 plex: (Complexes │ │ │ │ -0001d510: 2943 6f6d 706c 6578 2c2c 200a 2020 2a20 )Complex,, . * │ │ │ │ -0001d520: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -0001d530: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -0001d540: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -0001d550: 6329 4d6f 6475 6c65 2c2c 200a 0a44 6573 c)Module,, ..Des │ │ │ │ -0001d560: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -0001d570: 3d3d 3d3d 0a0a 4173 7375 6d69 6e67 2074 ====..Assuming t │ │ │ │ -0001d580: 6861 7420 7468 6520 656c 656d 656e 7473 hat the elements │ │ │ │ -0001d590: 206f 6620 7468 6520 3178 6320 6d61 7472 of the 1xc matr │ │ │ │ -0001d5a0: 6978 2066 6620 6172 6520 6e75 6c6c 2d68 ix ff are null-h │ │ │ │ -0001d5b0: 6f6d 6f74 6f70 6963 206f 6e20 432c 2074 omotopic on C, t │ │ │ │ -0001d5c0: 6865 0a73 6372 6970 7420 7265 7475 726e he.script return │ │ │ │ -0001d5d0: 7320 7468 6520 6469 7265 6374 2073 756d s the direct sum │ │ │ │ -0001d5e0: 206f 6620 7468 6520 686f 6d6f 6c6f 6779 of the homology │ │ │ │ -0001d5f0: 206f 6620 4320 6173 2061 206d 6f64 756c of C as a modul │ │ │ │ -0001d600: 6520 6f76 6572 2061 206e 6577 2072 696e e over a new rin │ │ │ │ -0001d610: 672c 0a63 6f6e 7369 7374 696e 6720 6f66 g,.consisting of │ │ │ │ -0001d620: 2072 696e 6720 4320 7769 7468 2063 2065 ring C with c e │ │ │ │ -0001d630: 7874 6572 696f 7220 7661 7269 6162 6c65 xterior variable │ │ │ │ -0001d640: 7320 6164 6a6f 696e 6564 2e20 5468 6520 s adjoined. The │ │ │ │ -0001d650: 7363 7269 7074 2069 7320 7468 6520 6d61 script is the ma │ │ │ │ -0001d660: 696e 0a63 6f6d 706f 6e65 6e74 206f 6620 in.component of │ │ │ │ -0001d670: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ -0001d680: 650a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d e..See also.==== │ │ │ │ -0001d690: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -0001d6a0: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ -0001d6b0: 653a 2065 7874 6572 696f 7254 6f72 4d6f e: exteriorTorMo │ │ │ │ -0001d6c0: 6475 6c65 2c20 2d2d 2054 6f72 2061 7320 dule, -- Tor as │ │ │ │ -0001d6d0: 6120 6d6f 6475 6c65 206f 7665 7220 616e a module over an │ │ │ │ -0001d6e0: 0a20 2020 2065 7874 6572 696f 7220 616c . exterior al │ │ │ │ -0001d6f0: 6765 6272 6120 6f72 2062 6967 7261 6465 gebra or bigrade │ │ │ │ -0001d700: 6420 616c 6765 6272 610a 2020 2a20 2a6e d algebra. * *n │ │ │ │ -0001d710: 6f74 6520 6d61 6b65 486f 6d6f 746f 7069 ote makeHomotopi │ │ │ │ -0001d720: 6573 4f6e 486f 6d6f 6c6f 6779 3a20 6d61 esOnHomology: ma │ │ │ │ -0001d730: 6b65 486f 6d6f 746f 7069 6573 4f6e 486f keHomotopiesOnHo │ │ │ │ -0001d740: 6d6f 6c6f 6779 2c20 2d2d 2048 6f6d 6f6c mology, -- Homol │ │ │ │ -0001d750: 6f67 7920 6f66 2061 0a20 2020 2063 6f6d ogy of a. com │ │ │ │ -0001d760: 706c 6578 2061 7320 6578 7465 7269 6f72 plex as exterior │ │ │ │ -0001d770: 206d 6f64 756c 650a 0a57 6179 7320 746f module..Ways to │ │ │ │ -0001d780: 2075 7365 2065 7874 6572 696f 7248 6f6d use exteriorHom │ │ │ │ -0001d790: 6f6c 6f67 794d 6f64 756c 653a 0a3d 3d3d ologyModule:.=== │ │ │ │ +0001d430: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +0001d440: 2020 2020 2020 4d20 3d20 6578 7465 7269 M = exteri │ │ │ │ +0001d450: 6f72 486f 6d6f 6c6f 6779 4d6f 6475 6c65 orHomologyModule │ │ │ │ +0001d460: 2866 662c 2043 290a 2020 2a20 496e 7075 (ff, C). * Inpu │ │ │ │ +0001d470: 7473 3a0a 2020 2020 2020 2a20 6666 2c20 ts:. * ff, │ │ │ │ +0001d480: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ +0001d490: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ +0001d4a0: 7472 6978 2c2c 204d 6174 7269 7820 6f66 trix,, Matrix of │ │ │ │ +0001d4b0: 2065 6c65 6d65 6e74 7320 7468 6174 2061 elements that a │ │ │ │ +0001d4c0: 7265 0a20 2020 2020 2020 2068 6f6d 6f74 re. homot │ │ │ │ +0001d4d0: 6f70 6963 2074 6f20 3020 6f6e 2043 0a20 opic to 0 on C. │ │ │ │ +0001d4e0: 2020 2020 202a 2043 2c20 6120 2a6e 6f74 * C, a *not │ │ │ │ +0001d4f0: 6520 636f 6d70 6c65 783a 2028 436f 6d70 e complex: (Comp │ │ │ │ +0001d500: 6c65 7865 7329 436f 6d70 6c65 782c 2c20 lexes)Complex,, │ │ │ │ +0001d510: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +0001d520: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +0001d530: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +0001d540: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +0001d550: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +0001d560: 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 7373 756d =========..Assum │ │ │ │ +0001d570: 696e 6720 7468 6174 2074 6865 2065 6c65 ing that the ele │ │ │ │ +0001d580: 6d65 6e74 7320 6f66 2074 6865 2031 7863 ments of the 1xc │ │ │ │ +0001d590: 206d 6174 7269 7820 6666 2061 7265 206e matrix ff are n │ │ │ │ +0001d5a0: 756c 6c2d 686f 6d6f 746f 7069 6320 6f6e ull-homotopic on │ │ │ │ +0001d5b0: 2043 2c20 7468 650a 7363 7269 7074 2072 C, the.script r │ │ │ │ +0001d5c0: 6574 7572 6e73 2074 6865 2064 6972 6563 eturns the direc │ │ │ │ +0001d5d0: 7420 7375 6d20 6f66 2074 6865 2068 6f6d t sum of the hom │ │ │ │ +0001d5e0: 6f6c 6f67 7920 6f66 2043 2061 7320 6120 ology of C as a │ │ │ │ +0001d5f0: 6d6f 6475 6c65 206f 7665 7220 6120 6e65 module over a ne │ │ │ │ +0001d600: 7720 7269 6e67 2c0a 636f 6e73 6973 7469 w ring,.consisti │ │ │ │ +0001d610: 6e67 206f 6620 7269 6e67 2043 2077 6974 ng of ring C wit │ │ │ │ +0001d620: 6820 6320 6578 7465 7269 6f72 2076 6172 h c exterior var │ │ │ │ +0001d630: 6961 626c 6573 2061 646a 6f69 6e65 642e iables adjoined. │ │ │ │ +0001d640: 2054 6865 2073 6372 6970 7420 6973 2074 The script is t │ │ │ │ +0001d650: 6865 206d 6169 6e0a 636f 6d70 6f6e 656e he main.componen │ │ │ │ +0001d660: 7420 6f66 2065 7874 6572 696f 7254 6f72 t of exteriorTor │ │ │ │ +0001d670: 4d6f 6475 6c65 0a0a 5365 6520 616c 736f Module..See also │ │ │ │ +0001d680: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +0001d690: 6e6f 7465 2065 7874 6572 696f 7254 6f72 note exteriorTor │ │ │ │ +0001d6a0: 4d6f 6475 6c65 3a20 6578 7465 7269 6f72 Module: exterior │ │ │ │ +0001d6b0: 546f 724d 6f64 756c 652c 202d 2d20 546f TorModule, -- To │ │ │ │ +0001d6c0: 7220 6173 2061 206d 6f64 756c 6520 6f76 r as a module ov │ │ │ │ +0001d6d0: 6572 2061 6e0a 2020 2020 6578 7465 7269 er an. exteri │ │ │ │ +0001d6e0: 6f72 2061 6c67 6562 7261 206f 7220 6269 or algebra or bi │ │ │ │ +0001d6f0: 6772 6164 6564 2061 6c67 6562 7261 0a20 graded algebra. │ │ │ │ +0001d700: 202a 202a 6e6f 7465 206d 616b 6548 6f6d * *note makeHom │ │ │ │ +0001d710: 6f74 6f70 6965 734f 6e48 6f6d 6f6c 6f67 otopiesOnHomolog │ │ │ │ +0001d720: 793a 206d 616b 6548 6f6d 6f74 6f70 6965 y: makeHomotopie │ │ │ │ +0001d730: 734f 6e48 6f6d 6f6c 6f67 792c 202d 2d20 sOnHomology, -- │ │ │ │ +0001d740: 486f 6d6f 6c6f 6779 206f 6620 610a 2020 Homology of a. │ │ │ │ +0001d750: 2020 636f 6d70 6c65 7820 6173 2065 7874 complex as ext │ │ │ │ +0001d760: 6572 696f 7220 6d6f 6475 6c65 0a0a 5761 erior module..Wa │ │ │ │ +0001d770: 7973 2074 6f20 7573 6520 6578 7465 7269 ys to use exteri │ │ │ │ +0001d780: 6f72 486f 6d6f 6c6f 6779 4d6f 6475 6c65 orHomologyModule │ │ │ │ +0001d790: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ 0001d7a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001d7b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001d7c0: 0a0a 2020 2a20 2265 7874 6572 696f 7248 .. * "exteriorH │ │ │ │ -0001d7d0: 6f6d 6f6c 6f67 794d 6f64 756c 6528 4d61 omologyModule(Ma │ │ │ │ -0001d7e0: 7472 6978 2c43 6f6d 706c 6578 2922 0a0a trix,Complex)".. │ │ │ │ -0001d7f0: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -0001d800: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -0001d810: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -0001d820: 7420 2a6e 6f74 6520 6578 7465 7269 6f72 t *note exterior │ │ │ │ -0001d830: 486f 6d6f 6c6f 6779 4d6f 6475 6c65 3a20 HomologyModule: │ │ │ │ -0001d840: 6578 7465 7269 6f72 486f 6d6f 6c6f 6779 exteriorHomology │ │ │ │ -0001d850: 4d6f 6475 6c65 2c20 6973 2061 202a 6e6f Module, is a *no │ │ │ │ -0001d860: 7465 0a6d 6574 686f 6420 6675 6e63 7469 te.method functi │ │ │ │ -0001d870: 6f6e 3a20 284d 6163 6175 6c61 7932 446f on: (Macaulay2Do │ │ │ │ -0001d880: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ -0001d890: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ +0001d7b0: 3d3d 3d3d 3d0a 0a20 202a 2022 6578 7465 =====.. * "exte │ │ │ │ +0001d7c0: 7269 6f72 486f 6d6f 6c6f 6779 4d6f 6475 riorHomologyModu │ │ │ │ +0001d7d0: 6c65 284d 6174 7269 782c 436f 6d70 6c65 le(Matrix,Comple │ │ │ │ +0001d7e0: 7829 220a 0a46 6f72 2074 6865 2070 726f x)"..For the pro │ │ │ │ +0001d7f0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +0001d800: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +0001d810: 6f62 6a65 6374 202a 6e6f 7465 2065 7874 object *note ext │ │ │ │ +0001d820: 6572 696f 7248 6f6d 6f6c 6f67 794d 6f64 eriorHomologyMod │ │ │ │ +0001d830: 756c 653a 2065 7874 6572 696f 7248 6f6d ule: exteriorHom │ │ │ │ +0001d840: 6f6c 6f67 794d 6f64 756c 652c 2069 7320 ologyModule, is │ │ │ │ +0001d850: 6120 2a6e 6f74 650a 6d65 7468 6f64 2066 a *note.method f │ │ │ │ +0001d860: 756e 6374 696f 6e3a 2028 4d61 6361 756c unction: (Macaul │ │ │ │ +0001d870: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +0001d880: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +0001d890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d8e0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -0001d8f0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -0001d900: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -0001d910: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -0001d920: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -0001d930: 352b 6473 2f4d 322f 4d61 6361 756c 6179 5+ds/M2/Macaulay │ │ │ │ -0001d940: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ -0001d950: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -0001d960: 5265 736f 6c75 7469 6f6e 732e 6d32 3a32 Resolutions.m2:2 │ │ │ │ -0001d970: 3738 353a 302e 0a1f 0a46 696c 653a 2043 785:0....File: C │ │ │ │ -0001d980: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -0001d990: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ -0001d9a0: 6e66 6f2c 204e 6f64 653a 2065 7874 6572 nfo, Node: exter │ │ │ │ -0001d9b0: 696f 7254 6f72 4d6f 6475 6c65 2c20 4e65 iorTorModule, Ne │ │ │ │ -0001d9c0: 7874 3a20 6578 7449 734f 6e65 506f 6c79 xt: extIsOnePoly │ │ │ │ -0001d9d0: 6e6f 6d69 616c 2c20 5072 6576 3a20 6578 nomial, Prev: ex │ │ │ │ -0001d9e0: 7465 7269 6f72 486f 6d6f 6c6f 6779 4d6f teriorHomologyMo │ │ │ │ -0001d9f0: 6475 6c65 2c20 5570 3a20 546f 700a 0a65 dule, Up: Top..e │ │ │ │ -0001da00: 7874 6572 696f 7254 6f72 4d6f 6475 6c65 xteriorTorModule │ │ │ │ -0001da10: 202d 2d20 546f 7220 6173 2061 206d 6f64 -- Tor as a mod │ │ │ │ -0001da20: 756c 6520 6f76 6572 2061 6e20 6578 7465 ule over an exte │ │ │ │ -0001da30: 7269 6f72 2061 6c67 6562 7261 206f 7220 rior algebra or │ │ │ │ -0001da40: 6269 6772 6164 6564 2061 6c67 6562 7261 bigraded algebra │ │ │ │ -0001da50: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ +0001d8d0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +0001d8e0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0001d8f0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0001d900: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0001d910: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0001d920: 2e32 362e 3035 2b64 732f 4d32 2f4d 6163 .26.05+ds/M2/Mac │ │ │ │ +0001d930: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +0001d940: 0a43 6f6d 706c 6574 6549 6e74 6572 7365 .CompleteInterse │ │ │ │ +0001d950: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ +0001d960: 2e6d 323a 3237 3835 3a30 2e0a 1f0a 4669 .m2:2785:0....Fi │ │ │ │ +0001d970: 6c65 3a20 436f 6d70 6c65 7465 496e 7465 le: CompleteInte │ │ │ │ +0001d980: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +0001d990: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20 ons.info, Node: │ │ │ │ +0001d9a0: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ +0001d9b0: 652c 204e 6578 743a 2065 7874 4973 4f6e e, Next: extIsOn │ │ │ │ +0001d9c0: 6550 6f6c 796e 6f6d 6961 6c2c 2050 7265 ePolynomial, Pre │ │ │ │ +0001d9d0: 763a 2065 7874 6572 696f 7248 6f6d 6f6c v: exteriorHomol │ │ │ │ +0001d9e0: 6f67 794d 6f64 756c 652c 2055 703a 2054 ogyModule, Up: T │ │ │ │ +0001d9f0: 6f70 0a0a 6578 7465 7269 6f72 546f 724d op..exteriorTorM │ │ │ │ +0001da00: 6f64 756c 6520 2d2d 2054 6f72 2061 7320 odule -- Tor as │ │ │ │ +0001da10: 6120 6d6f 6475 6c65 206f 7665 7220 616e a module over an │ │ │ │ +0001da20: 2065 7874 6572 696f 7220 616c 6765 6272 exterior algebr │ │ │ │ +0001da30: 6120 6f72 2062 6967 7261 6465 6420 616c a or bigraded al │ │ │ │ +0001da40: 6765 6272 610a 2a2a 2a2a 2a2a 2a2a 2a2a gebra.********** │ │ │ │ +0001da50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001da60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001da70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001da80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001da90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001daa0: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ -0001dab0: 2020 2020 2020 2020 5420 3d20 6578 7465 T = exte │ │ │ │ -0001dac0: 7269 6f72 546f 724d 6f64 756c 6528 662c riorTorModule(f, │ │ │ │ -0001dad0: 4629 0a20 2020 2020 2020 2054 203d 2065 F). T = e │ │ │ │ -0001dae0: 7874 6572 696f 7254 6f72 4d6f 6475 6c65 xteriorTorModule │ │ │ │ -0001daf0: 2866 2c4d 2c4e 290a 2020 2a20 496e 7075 (f,M,N). * Inpu │ │ │ │ -0001db00: 7473 3a0a 2020 2020 2020 2a20 662c 2061 ts:. * f, a │ │ │ │ -0001db10: 202a 6e6f 7465 206d 6174 7269 783a 2028 *note matrix: ( │ │ │ │ -0001db20: 4d61 6361 756c 6179 3244 6f63 294d 6174 Macaulay2Doc)Mat │ │ │ │ -0001db30: 7269 782c 2c20 3120 7820 632c 2065 6e74 rix,, 1 x c, ent │ │ │ │ -0001db40: 7269 6573 206d 7573 7420 6265 0a20 2020 ries must be. │ │ │ │ -0001db50: 2020 2020 2068 6f6d 6f74 6f70 6963 2074 homotopic t │ │ │ │ -0001db60: 6f20 3020 6f6e 2046 0a20 2020 2020 202a o 0 on F. * │ │ │ │ -0001db70: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -0001db80: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -0001db90: 6329 4d6f 6475 6c65 2c2c 2053 2d6d 6f64 c)Module,, S-mod │ │ │ │ -0001dba0: 756c 6520 616e 6e69 6869 6c61 7465 6420 ule annihilated │ │ │ │ -0001dbb0: 6279 2069 6465 616c 0a20 2020 2020 2020 by ideal. │ │ │ │ -0001dbc0: 2066 0a20 2020 2020 202a 204e 2c20 6120 f. * N, a │ │ │ │ -0001dbd0: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ -0001dbe0: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ -0001dbf0: 6c65 2c2c 2053 2d6d 6f64 756c 6520 616e le,, S-module an │ │ │ │ -0001dc00: 6e69 6869 6c61 7465 6420 6279 2069 6465 nihilated by ide │ │ │ │ -0001dc10: 616c 0a20 2020 2020 2020 2066 0a20 202a al. f. * │ │ │ │ -0001dc20: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -0001dc30: 2a20 542c 2061 202a 6e6f 7465 206d 6f64 * T, a *note mod │ │ │ │ -0001dc40: 756c 653a 2028 4d61 6361 756c 6179 3244 ule: (Macaulay2D │ │ │ │ -0001dc50: 6f63 294d 6f64 756c 652c 2c20 546f 725e oc)Module,, Tor^ │ │ │ │ -0001dc60: 5328 4d2c 4e29 2061 7320 6120 4d6f 6475 S(M,N) as a Modu │ │ │ │ -0001dc70: 6c65 206f 7665 720a 2020 2020 2020 2020 le over. │ │ │ │ -0001dc80: 616e 2065 7874 6572 696f 7220 616c 6765 an exterior alge │ │ │ │ -0001dc90: 6272 610a 0a44 6573 6372 6970 7469 6f6e bra..Description │ │ │ │ -0001dca0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966 .===========..If │ │ │ │ -0001dcb0: 204d 2c4e 2061 7265 2053 2d6d 6f64 756c M,N are S-modul │ │ │ │ -0001dcc0: 6573 2061 6e6e 6968 696c 6174 6564 2062 es annihilated b │ │ │ │ -0001dcd0: 7920 7468 6520 656c 656d 656e 7473 206f y the elements o │ │ │ │ -0001dce0: 6620 7468 6520 6d61 7472 6978 2066 6620 f the matrix ff │ │ │ │ -0001dcf0: 3d20 2866 5f31 2e2e 665f 6329 2c0a 616e = (f_1..f_c),.an │ │ │ │ -0001dd00: 6420 6b20 6973 2074 6865 2072 6573 6964 d k is the resid │ │ │ │ -0001dd10: 7565 2066 6965 6c64 206f 6620 532c 2074 ue field of S, t │ │ │ │ -0001dd20: 6865 6e20 7468 6520 7363 7269 7074 2065 hen the script e │ │ │ │ -0001dd30: 7874 6572 696f 7254 6f72 4d6f 6475 6c65 xteriorTorModule │ │ │ │ -0001dd40: 2866 2c4d 2920 7265 7475 726e 730a 546f (f,M) returns.To │ │ │ │ -0001dd50: 725e 5328 4d2c 206b 2920 6173 2061 206d r^S(M, k) as a m │ │ │ │ -0001dd60: 6f64 756c 6520 6f76 6572 2061 6e20 6578 odule over an ex │ │ │ │ -0001dd70: 7465 7269 6f72 2061 6c67 6562 7261 206b terior algebra k │ │ │ │ -0001dd80: 3c65 5f31 2c2e 2e2e 2c65 5f63 3e2c 2077 , w │ │ │ │ -0001dd90: 6865 7265 2074 6865 2065 5f69 0a68 6176 here the e_i.hav │ │ │ │ -0001dda0: 6520 6465 6772 6565 2031 2c20 7768 696c e degree 1, whil │ │ │ │ -0001ddb0: 6520 6578 7465 7269 6f72 546f 724d 6f64 e exteriorTorMod │ │ │ │ -0001ddc0: 756c 6528 662c 4d2c 4e29 2072 6574 7572 ule(f,M,N) retur │ │ │ │ -0001ddd0: 6e73 2054 6f72 5e53 284d 2c4e 2920 6173 ns Tor^S(M,N) as │ │ │ │ -0001dde0: 2061 206d 6f64 756c 650a 6f76 6572 2061 a module.over a │ │ │ │ -0001ddf0: 2062 6967 7261 6465 6420 7269 6e67 2053 bigraded ring S │ │ │ │ -0001de00: 4520 3d20 533c 655f 312c 2e2e 2c65 5f63 E = S, where the e_i │ │ │ │ -0001de20: 2068 6176 6520 6465 6772 6565 7320 7b64 have degrees {d │ │ │ │ -0001de30: 5f69 2c31 7d2c 0a77 6865 7265 2064 5f69 _i,1},.where d_i │ │ │ │ -0001de40: 2069 7320 7468 6520 6465 6772 6565 206f is the degree o │ │ │ │ -0001de50: 6620 665f 692e 2054 6865 206d 6f64 756c f f_i. The modul │ │ │ │ -0001de60: 6520 7374 7275 6374 7572 652c 2069 6e20 e structure, in │ │ │ │ -0001de70: 6569 7468 6572 2063 6173 652c 2069 730a either case, is. │ │ │ │ -0001de80: 6465 6669 6e65 6420 6279 2074 6865 2068 defined by the h │ │ │ │ -0001de90: 6f6d 6f74 6f70 6965 7320 666f 7220 7468 omotopies for th │ │ │ │ -0001dea0: 6520 665f 6920 6f6e 2074 6865 2072 6573 e f_i on the res │ │ │ │ -0001deb0: 6f6c 7574 696f 6e20 6f66 204d 2c20 636f olution of M, co │ │ │ │ -0001dec0: 6d70 7574 6564 2062 7920 7468 650a 7363 mputed by the.sc │ │ │ │ -0001ded0: 7269 7074 206d 616b 6548 6f6d 6f74 6f70 ript makeHomotop │ │ │ │ -0001dee0: 6965 7331 2e0a 0a54 6865 2073 6372 6970 ies1...The scrip │ │ │ │ -0001def0: 7473 2063 616c 6c20 6d61 6b65 4d6f 6475 ts call makeModu │ │ │ │ -0001df00: 6c65 2074 6f20 636f 6d70 7574 6520 6120 le to compute a │ │ │ │ -0001df10: 286e 6f6e 2d6d 696e 696d 616c 2920 7072 (non-minimal) pr │ │ │ │ -0001df20: 6573 656e 7461 7469 6f6e 206f 6620 7468 esentation of th │ │ │ │ -0001df30: 6973 0a6d 6f64 756c 652e 0a0a 4672 6f6d is.module...From │ │ │ │ -0001df40: 2074 6865 2064 6573 6372 6970 7469 6f6e the description │ │ │ │ -0001df50: 2062 7920 6d61 7472 6978 2066 6163 746f by matrix facto │ │ │ │ -0001df60: 7269 7a61 7469 6f6e 7320 616e 6420 7468 rizations and th │ │ │ │ -0001df70: 6520 7061 7065 7220 2254 6f72 2061 7320 e paper "Tor as │ │ │ │ -0001df80: 6120 6d6f 6475 6c65 0a6f 7665 7220 616e a module.over an │ │ │ │ -0001df90: 2065 7874 6572 696f 7220 616c 6765 6272 exterior algebr │ │ │ │ -0001dfa0: 6122 206f 6620 4569 7365 6e62 7564 2c20 a" of Eisenbud, │ │ │ │ -0001dfb0: 5065 6576 6120 616e 6420 5363 6872 6579 Peeva and Schrey │ │ │ │ -0001dfc0: 6572 2069 7420 666f 6c6c 6f77 7320 7468 er it follows th │ │ │ │ -0001dfd0: 6174 2077 6865 6e0a 4d20 6973 2061 2068 at when.M is a h │ │ │ │ -0001dfe0: 6967 6820 7379 7a79 6779 2061 6e64 2046 igh syzygy and F │ │ │ │ -0001dff0: 2069 7320 6974 7320 7265 736f 6c75 7469 is its resoluti │ │ │ │ -0001e000: 6f6e 2c20 7468 656e 2074 6865 2070 7265 on, then the pre │ │ │ │ -0001e010: 7365 6e74 6174 696f 6e20 6f66 0a54 6f72 sentation of.Tor │ │ │ │ -0001e020: 284d 2c53 5e31 2f6d 6d29 2061 6c77 6179 (M,S^1/mm) alway │ │ │ │ -0001e030: 7320 6861 7320 6765 6e65 7261 746f 7273 s has generators │ │ │ │ -0001e040: 2069 6e20 6465 6772 6565 7320 302c 312c in degrees 0,1, │ │ │ │ -0001e050: 2063 6f72 7265 7370 6f6e 6469 6e67 2074 corresponding t │ │ │ │ -0001e060: 6f20 7468 650a 7461 7267 6574 7320 616e o the.targets an │ │ │ │ -0001e070: 6420 736f 7572 6365 7320 6f66 2074 6865 d sources of the │ │ │ │ -0001e080: 2073 7461 636b 206f 6620 6d61 7073 2042 stack of maps B │ │ │ │ -0001e090: 2869 292c 2061 6e64 2074 6861 7420 7468 (i), and that th │ │ │ │ -0001e0a0: 6520 7265 736f 6c75 7469 6f6e 2069 730a e resolution is. │ │ │ │ -0001e0b0: 636f 6d70 6f6e 656e 7477 6973 6520 6c69 componentwise li │ │ │ │ -0001e0c0: 6e65 6172 2069 6e20 6120 7375 6974 6162 near in a suitab │ │ │ │ -0001e0d0: 6c65 2073 656e 7365 2e20 496e 2074 6865 le sense. In the │ │ │ │ -0001e0e0: 2066 6f6c 6c6f 7769 6e67 2065 7861 6d70 following examp │ │ │ │ -0001e0f0: 6c65 2c20 7468 6573 6520 6661 6374 730a le, these facts. │ │ │ │ -0001e100: 6172 6520 7665 7269 6669 6564 2e20 5468 are verified. Th │ │ │ │ -0001e110: 6520 546f 7220 6d6f 6475 6c65 2064 6f65 e Tor module doe │ │ │ │ -0001e120: 7320 4e4f 5420 7370 6c69 7420 696e 746f s NOT split into │ │ │ │ -0001e130: 2074 6865 2064 6972 6563 7420 7375 6d20 the direct sum │ │ │ │ -0001e140: 6f66 2074 6865 0a73 7562 6d6f 6475 6c65 of the.submodule │ │ │ │ -0001e150: 7320 6765 6e65 7261 7465 6420 696e 2064 s generated in d │ │ │ │ -0001e160: 6567 7265 6573 2030 2061 6e64 2031 2c20 egrees 0 and 1, │ │ │ │ -0001e170: 686f 7765 7665 722e 0a0a 0a0a 2b2d 2d2d however.....+--- │ │ │ │ +0001da90: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ +0001daa0: 6765 3a20 0a20 2020 2020 2020 2054 203d ge: . T = │ │ │ │ +0001dab0: 2065 7874 6572 696f 7254 6f72 4d6f 6475 exteriorTorModu │ │ │ │ +0001dac0: 6c65 2866 2c46 290a 2020 2020 2020 2020 le(f,F). │ │ │ │ +0001dad0: 5420 3d20 6578 7465 7269 6f72 546f 724d T = exteriorTorM │ │ │ │ +0001dae0: 6f64 756c 6528 662c 4d2c 4e29 0a20 202a odule(f,M,N). * │ │ │ │ +0001daf0: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +0001db00: 2066 2c20 6120 2a6e 6f74 6520 6d61 7472 f, a *note matr │ │ │ │ +0001db10: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ +0001db20: 6329 4d61 7472 6978 2c2c 2031 2078 2063 c)Matrix,, 1 x c │ │ │ │ +0001db30: 2c20 656e 7472 6965 7320 6d75 7374 2062 , entries must b │ │ │ │ +0001db40: 650a 2020 2020 2020 2020 686f 6d6f 746f e. homoto │ │ │ │ +0001db50: 7069 6320 746f 2030 206f 6e20 460a 2020 pic to 0 on F. │ │ │ │ +0001db60: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +0001db70: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +0001db80: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +0001db90: 532d 6d6f 6475 6c65 2061 6e6e 6968 696c S-module annihil │ │ │ │ +0001dba0: 6174 6564 2062 7920 6964 6561 6c0a 2020 ated by ideal. │ │ │ │ +0001dbb0: 2020 2020 2020 660a 2020 2020 2020 2a20 f. * │ │ │ │ +0001dbc0: 4e2c 2061 202a 6e6f 7465 206d 6f64 756c N, a *note modul │ │ │ │ +0001dbd0: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ +0001dbe0: 294d 6f64 756c 652c 2c20 532d 6d6f 6475 )Module,, S-modu │ │ │ │ +0001dbf0: 6c65 2061 6e6e 6968 696c 6174 6564 2062 le annihilated b │ │ │ │ +0001dc00: 7920 6964 6561 6c0a 2020 2020 2020 2020 y ideal. │ │ │ │ +0001dc10: 660a 2020 2a20 4f75 7470 7574 733a 0a20 f. * Outputs:. │ │ │ │ +0001dc20: 2020 2020 202a 2054 2c20 6120 2a6e 6f74 * T, a *not │ │ │ │ +0001dc30: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ +0001dc40: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ +0001dc50: 2054 6f72 5e53 284d 2c4e 2920 6173 2061 Tor^S(M,N) as a │ │ │ │ +0001dc60: 204d 6f64 756c 6520 6f76 6572 0a20 2020 Module over. │ │ │ │ +0001dc70: 2020 2020 2061 6e20 6578 7465 7269 6f72 an exterior │ │ │ │ +0001dc80: 2061 6c67 6562 7261 0a0a 4465 7363 7269 algebra..Descri │ │ │ │ +0001dc90: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ +0001dca0: 3d0a 0a49 6620 4d2c 4e20 6172 6520 532d =..If M,N are S- │ │ │ │ +0001dcb0: 6d6f 6475 6c65 7320 616e 6e69 6869 6c61 modules annihila │ │ │ │ +0001dcc0: 7465 6420 6279 2074 6865 2065 6c65 6d65 ted by the eleme │ │ │ │ +0001dcd0: 6e74 7320 6f66 2074 6865 206d 6174 7269 nts of the matri │ │ │ │ +0001dce0: 7820 6666 203d 2028 665f 312e 2e66 5f63 x ff = (f_1..f_c │ │ │ │ +0001dcf0: 292c 0a61 6e64 206b 2069 7320 7468 6520 ),.and k is the │ │ │ │ +0001dd00: 7265 7369 6475 6520 6669 656c 6420 6f66 residue field of │ │ │ │ +0001dd10: 2053 2c20 7468 656e 2074 6865 2073 6372 S, then the scr │ │ │ │ +0001dd20: 6970 7420 6578 7465 7269 6f72 546f 724d ipt exteriorTorM │ │ │ │ +0001dd30: 6f64 756c 6528 662c 4d29 2072 6574 7572 odule(f,M) retur │ │ │ │ +0001dd40: 6e73 0a54 6f72 5e53 284d 2c20 6b29 2061 ns.Tor^S(M, k) a │ │ │ │ +0001dd50: 7320 6120 6d6f 6475 6c65 206f 7665 7220 s a module over │ │ │ │ +0001dd60: 616e 2065 7874 6572 696f 7220 616c 6765 an exterior alge │ │ │ │ +0001dd70: 6272 6120 6b3c 655f 312c 2e2e 2e2c 655f bra k, where the e_ │ │ │ │ +0001dd90: 690a 6861 7665 2064 6567 7265 6520 312c i.have degree 1, │ │ │ │ +0001dda0: 2077 6869 6c65 2065 7874 6572 696f 7254 while exteriorT │ │ │ │ +0001ddb0: 6f72 4d6f 6475 6c65 2866 2c4d 2c4e 2920 orModule(f,M,N) │ │ │ │ +0001ddc0: 7265 7475 726e 7320 546f 725e 5328 4d2c returns Tor^S(M, │ │ │ │ +0001ddd0: 4e29 2061 7320 6120 6d6f 6475 6c65 0a6f N) as a module.o │ │ │ │ +0001dde0: 7665 7220 6120 6269 6772 6164 6564 2072 ver a bigraded r │ │ │ │ +0001ddf0: 696e 6720 5345 203d 2053 3c65 5f31 2c2e ing SE = S, where th │ │ │ │ +0001de10: 6520 655f 6920 6861 7665 2064 6567 7265 e e_i have degre │ │ │ │ +0001de20: 6573 207b 645f 692c 317d 2c0a 7768 6572 es {d_i,1},.wher │ │ │ │ +0001de30: 6520 645f 6920 6973 2074 6865 2064 6567 e d_i is the deg │ │ │ │ +0001de40: 7265 6520 6f66 2066 5f69 2e20 5468 6520 ree of f_i. The │ │ │ │ +0001de50: 6d6f 6475 6c65 2073 7472 7563 7475 7265 module structure │ │ │ │ +0001de60: 2c20 696e 2065 6974 6865 7220 6361 7365 , in either case │ │ │ │ +0001de70: 2c20 6973 0a64 6566 696e 6564 2062 7920 , is.defined by │ │ │ │ +0001de80: 7468 6520 686f 6d6f 746f 7069 6573 2066 the homotopies f │ │ │ │ +0001de90: 6f72 2074 6865 2066 5f69 206f 6e20 7468 or the f_i on th │ │ │ │ +0001dea0: 6520 7265 736f 6c75 7469 6f6e 206f 6620 e resolution of │ │ │ │ +0001deb0: 4d2c 2063 6f6d 7075 7465 6420 6279 2074 M, computed by t │ │ │ │ +0001dec0: 6865 0a73 6372 6970 7420 6d61 6b65 486f he.script makeHo │ │ │ │ +0001ded0: 6d6f 746f 7069 6573 312e 0a0a 5468 6520 motopies1...The │ │ │ │ +0001dee0: 7363 7269 7074 7320 6361 6c6c 206d 616b scripts call mak │ │ │ │ +0001def0: 654d 6f64 756c 6520 746f 2063 6f6d 7075 eModule to compu │ │ │ │ +0001df00: 7465 2061 2028 6e6f 6e2d 6d69 6e69 6d61 te a (non-minima │ │ │ │ +0001df10: 6c29 2070 7265 7365 6e74 6174 696f 6e20 l) presentation │ │ │ │ +0001df20: 6f66 2074 6869 730a 6d6f 6475 6c65 2e0a of this.module.. │ │ │ │ +0001df30: 0a46 726f 6d20 7468 6520 6465 7363 7269 .From the descri │ │ │ │ +0001df40: 7074 696f 6e20 6279 206d 6174 7269 7820 ption by matrix │ │ │ │ +0001df50: 6661 6374 6f72 697a 6174 696f 6e73 2061 factorizations a │ │ │ │ +0001df60: 6e64 2074 6865 2070 6170 6572 2022 546f nd the paper "To │ │ │ │ +0001df70: 7220 6173 2061 206d 6f64 756c 650a 6f76 r as a module.ov │ │ │ │ +0001df80: 6572 2061 6e20 6578 7465 7269 6f72 2061 er an exterior a │ │ │ │ +0001df90: 6c67 6562 7261 2220 6f66 2045 6973 656e lgebra" of Eisen │ │ │ │ +0001dfa0: 6275 642c 2050 6565 7661 2061 6e64 2053 bud, Peeva and S │ │ │ │ +0001dfb0: 6368 7265 7965 7220 6974 2066 6f6c 6c6f chreyer it follo │ │ │ │ +0001dfc0: 7773 2074 6861 7420 7768 656e 0a4d 2069 ws that when.M i │ │ │ │ +0001dfd0: 7320 6120 6869 6768 2073 797a 7967 7920 s a high syzygy │ │ │ │ +0001dfe0: 616e 6420 4620 6973 2069 7473 2072 6573 and F is its res │ │ │ │ +0001dff0: 6f6c 7574 696f 6e2c 2074 6865 6e20 7468 olution, then th │ │ │ │ +0001e000: 6520 7072 6573 656e 7461 7469 6f6e 206f e presentation o │ │ │ │ +0001e010: 660a 546f 7228 4d2c 535e 312f 6d6d 2920 f.Tor(M,S^1/mm) │ │ │ │ +0001e020: 616c 7761 7973 2068 6173 2067 656e 6572 always has gener │ │ │ │ +0001e030: 6174 6f72 7320 696e 2064 6567 7265 6573 ators in degrees │ │ │ │ +0001e040: 2030 2c31 2c20 636f 7272 6573 706f 6e64 0,1, correspond │ │ │ │ +0001e050: 696e 6720 746f 2074 6865 0a74 6172 6765 ing to the.targe │ │ │ │ +0001e060: 7473 2061 6e64 2073 6f75 7263 6573 206f ts and sources o │ │ │ │ +0001e070: 6620 7468 6520 7374 6163 6b20 6f66 206d f the stack of m │ │ │ │ +0001e080: 6170 7320 4228 6929 2c20 616e 6420 7468 aps B(i), and th │ │ │ │ +0001e090: 6174 2074 6865 2072 6573 6f6c 7574 696f at the resolutio │ │ │ │ +0001e0a0: 6e20 6973 0a63 6f6d 706f 6e65 6e74 7769 n is.componentwi │ │ │ │ +0001e0b0: 7365 206c 696e 6561 7220 696e 2061 2073 se linear in a s │ │ │ │ +0001e0c0: 7569 7461 626c 6520 7365 6e73 652e 2049 uitable sense. I │ │ │ │ +0001e0d0: 6e20 7468 6520 666f 6c6c 6f77 696e 6720 n the following │ │ │ │ +0001e0e0: 6578 616d 706c 652c 2074 6865 7365 2066 example, these f │ │ │ │ +0001e0f0: 6163 7473 0a61 7265 2076 6572 6966 6965 acts.are verifie │ │ │ │ +0001e100: 642e 2054 6865 2054 6f72 206d 6f64 756c d. The Tor modul │ │ │ │ +0001e110: 6520 646f 6573 204e 4f54 2073 706c 6974 e does NOT split │ │ │ │ +0001e120: 2069 6e74 6f20 7468 6520 6469 7265 6374 into the direct │ │ │ │ +0001e130: 2073 756d 206f 6620 7468 650a 7375 626d sum of the.subm │ │ │ │ +0001e140: 6f64 756c 6573 2067 656e 6572 6174 6564 odules generated │ │ │ │ +0001e150: 2069 6e20 6465 6772 6565 7320 3020 616e in degrees 0 an │ │ │ │ +0001e160: 6420 312c 2068 6f77 6576 6572 2e0a 0a0a d 1, however.... │ │ │ │ +0001e170: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0001e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e1c0: 2b0a 7c69 3120 3a20 6b6b 203d 205a 5a2f +.|i1 : kk = ZZ/ │ │ │ │ -0001e1d0: 3130 3120 2020 2020 2020 2020 2020 2020 101 │ │ │ │ +0001e1b0: 2d2d 2d2d 2d2b 0a7c 6931 203a 206b 6b20 -----+.|i1 : kk │ │ │ │ +0001e1c0: 3d20 5a5a 2f31 3031 2020 2020 2020 2020 = ZZ/101 │ │ │ │ +0001e1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e200: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001e1f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001e200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e240: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001e250: 3120 3d20 6b6b 2020 2020 2020 2020 2020 1 = kk │ │ │ │ +0001e240: 207c 0a7c 6f31 203d 206b 6b20 2020 2020 |.|o1 = kk │ │ │ │ +0001e250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e290: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001e280: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001e290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e2d0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ -0001e2e0: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +0001e2c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001e2d0: 6f31 203a 2051 756f 7469 656e 7452 696e o1 : QuotientRin │ │ │ │ +0001e2e0: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ 0001e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e310: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001e320: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001e310: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001e320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e360: 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 3d20 ----+.|i2 : S = │ │ │ │ -0001e370: 6b6b 5b61 2c62 2c63 5d20 2020 2020 2020 kk[a,b,c] │ │ │ │ +0001e350: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ +0001e360: 2053 203d 206b 6b5b 612c 622c 635d 2020 S = kk[a,b,c] │ │ │ │ +0001e370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e3a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001e390: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001e3a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001e3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e3f0: 7c0a 7c6f 3220 3d20 5320 2020 2020 2020 |.|o2 = S │ │ │ │ +0001e3e0: 2020 2020 207c 0a7c 6f32 203d 2053 2020 |.|o2 = S │ │ │ │ +0001e3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e430: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001e420: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e470: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001e480: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269 2 : PolynomialRi │ │ │ │ -0001e490: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ +0001e470: 207c 0a7c 6f32 203a 2050 6f6c 796e 6f6d |.|o2 : Polynom │ │ │ │ +0001e480: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ +0001e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e4c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001e4b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0001e4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e500: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ -0001e510: 6620 3d20 6d61 7472 6978 2261 342c 6234 f = matrix"a4,b4 │ │ │ │ -0001e520: 2c63 3422 2020 2020 2020 2020 2020 2020 ,c4" │ │ │ │ +0001e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0001e500: 6933 203a 2066 203d 206d 6174 7269 7822 i3 : f = matrix" │ │ │ │ +0001e510: 6134 2c62 342c 6334 2220 2020 2020 2020 a4,b4,c4" │ │ │ │ +0001e520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e540: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001e550: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001e540: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001e550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e590: 2020 2020 7c0a 7c6f 3320 3d20 7c20 6134 |.|o3 = | a4 │ │ │ │ -0001e5a0: 2062 3420 6334 207c 2020 2020 2020 2020 b4 c4 | │ │ │ │ +0001e580: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ +0001e590: 207c 2061 3420 6234 2063 3420 7c20 2020 | a4 b4 c4 | │ │ │ │ +0001e5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e5d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001e5c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001e5d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001e5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e620: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001e630: 3120 2020 2020 2033 2020 2020 2020 2020 1 3 │ │ │ │ +0001e610: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001e620: 2020 2020 2031 2020 2020 2020 3320 2020 1 3 │ │ │ │ +0001e630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e660: 2020 2020 2020 7c0a 7c6f 3320 3a20 4d61 |.|o3 : Ma │ │ │ │ -0001e670: 7472 6978 2053 2020 3c2d 2d20 5320 2020 trix S <-- S │ │ │ │ +0001e650: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +0001e660: 203a 204d 6174 7269 7820 5320 203c 2d2d : Matrix S <-- │ │ │ │ +0001e670: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 0001e680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e6a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001e6a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001e6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e6f0: 2d2d 2b0a 7c69 3420 3a20 5220 3d20 532f --+.|i4 : R = S/ │ │ │ │ -0001e700: 6964 6561 6c20 6620 2020 2020 2020 2020 ideal f │ │ │ │ +0001e6e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 -------+.|i4 : R │ │ │ │ +0001e6f0: 203d 2053 2f69 6465 616c 2066 2020 2020 = S/ideal f │ │ │ │ +0001e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e730: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001e720: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001e730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e770: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001e780: 7c6f 3420 3d20 5220 2020 2020 2020 2020 |o4 = R │ │ │ │ +0001e770: 2020 207c 0a7c 6f34 203d 2052 2020 2020 |.|o4 = R │ │ │ │ +0001e780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e7c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001e7b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001e7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e800: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ -0001e810: 3a20 5175 6f74 6965 6e74 5269 6e67 2020 : QuotientRing │ │ │ │ +0001e7f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001e800: 0a7c 6f34 203a 2051 756f 7469 656e 7452 .|o4 : QuotientR │ │ │ │ +0001e810: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ 0001e820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e850: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001e840: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e890: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7020 ------+.|i5 : p │ │ │ │ -0001e8a0: 3d20 6d61 7028 522c 5329 2020 2020 2020 = map(R,S) │ │ │ │ +0001e880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ +0001e890: 203a 2070 203d 206d 6170 2852 2c53 2920 : p = map(R,S) │ │ │ │ +0001e8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e8d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001e8d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001e8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e920: 2020 7c0a 7c6f 3520 3d20 6d61 7020 2852 |.|o5 = map (R │ │ │ │ -0001e930: 2c20 532c 207b 612c 2062 2c20 637d 2920 , S, {a, b, c}) │ │ │ │ +0001e910: 2020 2020 2020 207c 0a7c 6f35 203d 206d |.|o5 = m │ │ │ │ +0001e920: 6170 2028 522c 2053 2c20 7b61 2c20 622c ap (R, S, {a, b, │ │ │ │ +0001e930: 2063 7d29 2020 2020 2020 2020 2020 2020 c}) │ │ │ │ 0001e940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e960: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001e950: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001e960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e9a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001e9b0: 7c6f 3520 3a20 5269 6e67 4d61 7020 5220 |o5 : RingMap R │ │ │ │ -0001e9c0: 3c2d 2d20 5320 2020 2020 2020 2020 2020 <-- S │ │ │ │ +0001e9a0: 2020 207c 0a7c 6f35 203a 2052 696e 674d |.|o5 : RingM │ │ │ │ +0001e9b0: 6170 2052 203c 2d2d 2053 2020 2020 2020 ap R <-- S │ │ │ │ +0001e9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e9f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001e9e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ea00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ -0001ea40: 3a20 4d20 3d20 636f 6b65 7220 6d61 7028 : M = coker map( │ │ │ │ -0001ea50: 525e 322c 2052 5e7b 333a 2d31 7d2c 207b R^2, R^{3:-1}, { │ │ │ │ -0001ea60: 7b61 2c62 2c63 7d2c 7b62 2c63 2c61 7d7d {a,b,c},{b,c,a}} │ │ │ │ -0001ea70: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0001ea80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001ea30: 0a7c 6936 203a 204d 203d 2063 6f6b 6572 .|i6 : M = coker │ │ │ │ +0001ea40: 206d 6170 2852 5e32 2c20 525e 7b33 3a2d map(R^2, R^{3:- │ │ │ │ +0001ea50: 317d 2c20 7b7b 612c 622c 637d 2c7b 622c 1}, {{a,b,c},{b, │ │ │ │ +0001ea60: 632c 617d 7d29 2020 2020 2020 2020 2020 c,a}}) │ │ │ │ +0001ea70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001ea80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eac0: 2020 2020 2020 7c0a 7c6f 3620 3d20 636f |.|o6 = co │ │ │ │ -0001ead0: 6b65 726e 656c 207c 2061 2062 2063 207c kernel | a b c | │ │ │ │ +0001eab0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ +0001eac0: 203d 2063 6f6b 6572 6e65 6c20 7c20 6120 = cokernel | a │ │ │ │ +0001ead0: 6220 6320 7c20 2020 2020 2020 2020 2020 b c | │ │ │ │ 0001eae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001eb10: 2020 2020 2020 2020 2020 2020 207c 2062 | b │ │ │ │ -0001eb20: 2063 2061 207c 2020 2020 2020 2020 2020 c a | │ │ │ │ +0001eb00: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001eb10: 2020 7c20 6220 6320 6120 7c20 2020 2020 | b c a | │ │ │ │ +0001eb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb50: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001eb40: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001eb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb90: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001eba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ebb0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0001eb80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001eb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001eba0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +0001ebb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ebd0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001ebe0: 7c6f 3620 3a20 522d 6d6f 6475 6c65 2c20 |o6 : R-module, │ │ │ │ -0001ebf0: 7175 6f74 6965 6e74 206f 6620 5220 2020 quotient of R │ │ │ │ +0001ebd0: 2020 207c 0a7c 6f36 203a 2052 2d6d 6f64 |.|o6 : R-mod │ │ │ │ +0001ebe0: 756c 652c 2071 756f 7469 656e 7420 6f66 ule, quotient of │ │ │ │ +0001ebf0: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 0001ec00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ec10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ec20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001ec10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001ec20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ec30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ec40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ec50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ec60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 ----------+.|i7 │ │ │ │ -0001ec70: 3a20 6265 7474 6920 2846 4620 3d66 7265 : betti (FF =fre │ │ │ │ -0001ec80: 6552 6573 6f6c 7574 696f 6e28 204d 2c20 eResolution( M, │ │ │ │ -0001ec90: 4c65 6e67 7468 4c69 6d69 7420 3d3e 3629 LengthLimit =>6) │ │ │ │ -0001eca0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0001ecb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001ec50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001ec60: 0a7c 6937 203a 2062 6574 7469 2028 4646 .|i7 : betti (FF │ │ │ │ +0001ec70: 203d 6672 6565 5265 736f 6c75 7469 6f6e =freeResolution │ │ │ │ +0001ec80: 2820 4d2c 204c 656e 6774 684c 696d 6974 ( M, LengthLimit │ │ │ │ +0001ec90: 203d 3e36 2929 2020 2020 2020 2020 2020 =>6)) │ │ │ │ +0001eca0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001ecb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ecc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ecd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ece0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ecf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001ed00: 2020 2020 2030 2031 2032 2033 2034 2020 0 1 2 3 4 │ │ │ │ -0001ed10: 3520 2036 2020 2020 2020 2020 2020 2020 5 6 │ │ │ │ +0001ece0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001ecf0: 2020 2020 2020 2020 2020 3020 3120 3220 0 1 2 │ │ │ │ +0001ed00: 3320 3420 2035 2020 3620 2020 2020 2020 3 4 5 6 │ │ │ │ +0001ed10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ed20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ed30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001ed40: 3720 3d20 746f 7461 6c3a 2032 2033 2034 7 = total: 2 3 4 │ │ │ │ -0001ed50: 2036 2039 2031 3320 3138 2020 2020 2020 6 9 13 18 │ │ │ │ +0001ed30: 207c 0a7c 6f37 203d 2074 6f74 616c 3a20 |.|o7 = total: │ │ │ │ +0001ed40: 3220 3320 3420 3620 3920 3133 2031 3820 2 3 4 6 9 13 18 │ │ │ │ +0001ed50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ed60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ed70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ed80: 2020 7c0a 7c20 2020 2020 2020 2020 303a |.| 0: │ │ │ │ -0001ed90: 2032 2033 202e 202e 202e 2020 2e20 202e 2 3 . . . . . │ │ │ │ +0001ed70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001ed80: 2020 2030 3a20 3220 3320 2e20 2e20 2e20 0: 2 3 . . . │ │ │ │ +0001ed90: 202e 2020 2e20 2020 2020 2020 2020 2020 . . │ │ │ │ 0001eda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001edb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001edc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001edd0: 2020 2020 313a 202e 202e 2031 202e 202e 1: . . 1 . . │ │ │ │ -0001ede0: 2020 2e20 202e 2020 2020 2020 2020 2020 . . │ │ │ │ +0001edb0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001edc0: 2020 2020 2020 2020 2031 3a20 2e20 2e20 1: . . │ │ │ │ +0001edd0: 3120 2e20 2e20 202e 2020 2e20 2020 2020 1 . . . . │ │ │ │ +0001ede0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001edf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ee00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001ee10: 7c20 2020 2020 2020 2020 323a 202e 202e | 2: . . │ │ │ │ -0001ee20: 2033 2033 202e 2020 2e20 202e 2020 2020 3 3 . . . │ │ │ │ +0001ee00: 2020 207c 0a7c 2020 2020 2020 2020 2032 |.| 2 │ │ │ │ +0001ee10: 3a20 2e20 2e20 3320 3320 2e20 202e 2020 : . . 3 3 . . │ │ │ │ +0001ee20: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ 0001ee30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ee40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ee50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001ee60: 333a 202e 202e 202e 2033 2033 2020 2e20 3: . . . 3 3 . │ │ │ │ -0001ee70: 202e 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -0001ee80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ee90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001eea0: 2020 2020 2020 343a 202e 202e 202e 202e 4: . . . . │ │ │ │ -0001eeb0: 2033 2020 3320 202e 2020 2020 2020 2020 3 3 . │ │ │ │ +0001ee40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001ee50: 2020 2020 2033 3a20 2e20 2e20 2e20 3320 3: . . . 3 │ │ │ │ +0001ee60: 3320 202e 2020 2e20 2020 2020 2020 2020 3 . . │ │ │ │ +0001ee70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ee80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001ee90: 0a7c 2020 2020 2020 2020 2034 3a20 2e20 .| 4: . │ │ │ │ +0001eea0: 2e20 2e20 2e20 3320 2033 2020 2e20 2020 . . . 3 3 . │ │ │ │ +0001eeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eee0: 7c0a 7c20 2020 2020 2020 2020 353a 202e |.| 5: . │ │ │ │ -0001eef0: 202e 202e 202e 2033 2020 3920 2036 2020 . . . 3 9 6 │ │ │ │ +0001eed0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001eee0: 2035 3a20 2e20 2e20 2e20 2e20 3320 2039 5: . . . . 3 9 │ │ │ │ +0001eef0: 2020 3620 2020 2020 2020 2020 2020 2020 6 │ │ │ │ 0001ef00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ef10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ef20: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001ef30: 2020 363a 202e 202e 202e 202e 202e 2020 6: . . . . . │ │ │ │ -0001ef40: 2e20 2033 2020 2020 2020 2020 2020 2020 . 3 │ │ │ │ +0001ef10: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001ef20: 2020 2020 2020 2036 3a20 2e20 2e20 2e20 6: . . . │ │ │ │ +0001ef30: 2e20 2e20 202e 2020 3320 2020 2020 2020 . . . 3 │ │ │ │ +0001ef40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ef50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ef60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001ef70: 2020 2020 2020 2020 373a 202e 202e 202e 7: . . . │ │ │ │ -0001ef80: 202e 202e 2020 3120 2039 2020 2020 2020 . . 1 9 │ │ │ │ +0001ef60: 207c 0a7c 2020 2020 2020 2020 2037 3a20 |.| 7: │ │ │ │ +0001ef70: 2e20 2e20 2e20 2e20 2e20 2031 2020 3920 . . . . . 1 9 │ │ │ │ +0001ef80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001efa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001efb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001efa0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001efb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001efc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001efd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001efe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eff0: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20 |.|o7 : │ │ │ │ -0001f000: 4265 7474 6954 616c 6c79 2020 2020 2020 BettiTally │ │ │ │ +0001efe0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001eff0: 6f37 203a 2042 6574 7469 5461 6c6c 7920 o7 : BettiTally │ │ │ │ +0001f000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f030: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001f040: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001f030: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001f040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f080: 2d2d 2d2d 2b0a 7c69 3820 3a20 4d53 203d ----+.|i8 : MS = │ │ │ │ -0001f090: 2070 7275 6e65 2070 7573 6846 6f72 7761 prune pushForwa │ │ │ │ -0001f0a0: 7264 2870 2c20 636f 6b65 7220 4646 2e64 rd(p, coker FF.d │ │ │ │ -0001f0b0: 645f 3629 3b20 2020 2020 2020 2020 2020 d_6); │ │ │ │ -0001f0c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0001f070: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a ---------+.|i8 : │ │ │ │ +0001f080: 204d 5320 3d20 7072 756e 6520 7075 7368 MS = prune push │ │ │ │ +0001f090: 466f 7277 6172 6428 702c 2063 6f6b 6572 Forward(p, coker │ │ │ │ +0001f0a0: 2046 462e 6464 5f36 293b 2020 2020 2020 FF.dd_6); │ │ │ │ +0001f0b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001f0c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0001f0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f110: 2b0a 7c69 3920 3a20 5420 3d20 6578 7465 +.|i9 : T = exte │ │ │ │ -0001f120: 7269 6f72 546f 724d 6f64 756c 6528 662c riorTorModule(f, │ │ │ │ -0001f130: 4d53 293b 2020 2020 2020 2020 2020 2020 MS); │ │ │ │ -0001f140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f150: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001f100: 2d2d 2d2d 2d2b 0a7c 6939 203a 2054 203d -----+.|i9 : T = │ │ │ │ +0001f110: 2065 7874 6572 696f 7254 6f72 4d6f 6475 exteriorTorModu │ │ │ │ +0001f120: 6c65 2866 2c4d 5329 3b20 2020 2020 2020 le(f,MS); │ │ │ │ +0001f130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f140: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001f150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001f1a0: 3130 203a 2062 6574 7469 2054 2020 2020 10 : betti T │ │ │ │ +0001f190: 2d2b 0a7c 6931 3020 3a20 6265 7474 6920 -+.|i10 : betti │ │ │ │ +0001f1a0: 5420 2020 2020 2020 2020 2020 2020 2020 T │ │ │ │ 0001f1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f1e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001f1d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001f1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f220: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001f230: 2020 2020 2020 2020 2030 2020 2031 2020 0 1 │ │ │ │ +0001f210: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001f220: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ +0001f230: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 0001f240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f260: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001f270: 7c6f 3130 203d 2074 6f74 616c 3a20 3834 |o10 = total: 84 │ │ │ │ -0001f280: 2032 3532 2020 2020 2020 2020 2020 2020 252 │ │ │ │ +0001f260: 2020 207c 0a7c 6f31 3020 3d20 746f 7461 |.|o10 = tota │ │ │ │ +0001f270: 6c3a 2038 3420 3235 3220 2020 2020 2020 l: 84 252 │ │ │ │ +0001f280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001f2c0: 2030 3a20 3133 2020 3339 2020 2020 2020 0: 13 39 │ │ │ │ +0001f2a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001f2b0: 2020 2020 2020 303a 2031 3320 2033 3920 0: 13 39 │ │ │ │ +0001f2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001f300: 2020 2020 2020 2031 3a20 3333 2020 3939 1: 33 99 │ │ │ │ +0001f2e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001f2f0: 0a7c 2020 2020 2020 2020 2020 313a 2033 .| 1: 3 │ │ │ │ +0001f300: 3320 2039 3920 2020 2020 2020 2020 2020 3 99 │ │ │ │ 0001f310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f340: 7c0a 7c20 2020 2020 2020 2020 2032 3a20 |.| 2: │ │ │ │ -0001f350: 3239 2020 3837 2020 2020 2020 2020 2020 29 87 │ │ │ │ +0001f330: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001f340: 2020 323a 2032 3920 2038 3720 2020 2020 2: 29 87 │ │ │ │ +0001f350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f380: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001f390: 2020 2033 3a20 2039 2020 3237 2020 2020 3: 9 27 │ │ │ │ +0001f370: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001f380: 2020 2020 2020 2020 333a 2020 3920 2032 3: 9 2 │ │ │ │ +0001f390: 3720 2020 2020 2020 2020 2020 2020 2020 7 │ │ │ │ 0001f3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f3c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001f3c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001f3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f410: 2020 7c0a 7c6f 3130 203a 2042 6574 7469 |.|o10 : Betti │ │ │ │ -0001f420: 5461 6c6c 7920 2020 2020 2020 2020 2020 Tally │ │ │ │ +0001f400: 2020 2020 2020 207c 0a7c 6f31 3020 3a20 |.|o10 : │ │ │ │ +0001f410: 4265 7474 6954 616c 6c79 2020 2020 2020 BettiTally │ │ │ │ +0001f420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f450: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001f440: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0001f450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001f4a0: 7c69 3131 203a 2062 6574 7469 2066 7265 |i11 : betti fre │ │ │ │ -0001f4b0: 6552 6573 6f6c 7574 696f 6e20 2850 5420 eResolution (PT │ │ │ │ -0001f4c0: 3d20 7072 756e 6520 542c 204c 656e 6774 = prune T, Lengt │ │ │ │ -0001f4d0: 684c 696d 6974 203d 3e20 3429 2020 2020 hLimit => 4) │ │ │ │ -0001f4e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001f490: 2d2d 2d2b 0a7c 6931 3120 3a20 6265 7474 ---+.|i11 : bett │ │ │ │ +0001f4a0: 6920 6672 6565 5265 736f 6c75 7469 6f6e i freeResolution │ │ │ │ +0001f4b0: 2028 5054 203d 2070 7275 6e65 2054 2c20 (PT = prune T, │ │ │ │ +0001f4c0: 4c65 6e67 7468 4c69 6d69 7420 3d3e 2034 LengthLimit => 4 │ │ │ │ +0001f4d0: 2920 2020 2020 2020 207c 0a7c 2020 2020 ) |.| │ │ │ │ +0001f4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f520: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001f530: 2020 2020 2020 2020 2020 2030 2020 3120 0 1 │ │ │ │ -0001f540: 2032 2020 2033 2020 2034 2020 2020 2020 2 3 4 │ │ │ │ +0001f510: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001f520: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001f530: 3020 2031 2020 3220 2020 3320 2020 3420 0 1 2 3 4 │ │ │ │ +0001f540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f570: 7c0a 7c6f 3131 203d 2074 6f74 616c 3a20 |.|o11 = total: │ │ │ │ -0001f580: 3331 2035 3520 3837 2031 3237 2031 3735 31 55 87 127 175 │ │ │ │ +0001f560: 2020 2020 207c 0a7c 6f31 3120 3d20 746f |.|o11 = to │ │ │ │ +0001f570: 7461 6c3a 2033 3120 3535 2038 3720 3132 tal: 31 55 87 12 │ │ │ │ +0001f580: 3720 3137 3520 2020 2020 2020 2020 2020 7 175 │ │ │ │ 0001f590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f5b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001f5c0: 2020 2030 3a20 3133 2032 3420 3339 2020 0: 13 24 39 │ │ │ │ -0001f5d0: 3538 2020 3831 2020 2020 2020 2020 2020 58 81 │ │ │ │ +0001f5a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001f5b0: 2020 2020 2020 2020 303a 2031 3320 3234 0: 13 24 │ │ │ │ +0001f5c0: 2033 3920 2035 3820 2038 3120 2020 2020 39 58 81 │ │ │ │ +0001f5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f5f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001f600: 2020 2020 2020 2020 2031 3a20 3138 2033 1: 18 3 │ │ │ │ -0001f610: 3120 3438 2020 3639 2020 3934 2020 2020 1 48 69 94 │ │ │ │ +0001f5f0: 207c 0a7c 2020 2020 2020 2020 2020 313a |.| 1: │ │ │ │ +0001f600: 2031 3820 3331 2034 3820 2036 3920 2039 18 31 48 69 9 │ │ │ │ +0001f610: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ 0001f620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f640: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001f630: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001f640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f680: 2020 2020 2020 2020 7c0a 7c6f 3131 203a |.|o11 : │ │ │ │ -0001f690: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +0001f670: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001f680: 6f31 3120 3a20 4265 7474 6954 616c 6c79 o11 : BettiTally │ │ │ │ +0001f690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f6c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001f6d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001f6c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001f6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f710: 2d2d 2d2d 2b0a 7c69 3132 203a 2061 6e6e ----+.|i12 : ann │ │ │ │ -0001f720: 2050 5420 2020 2020 2020 2020 2020 2020 PT │ │ │ │ +0001f700: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 ---------+.|i12 │ │ │ │ +0001f710: 3a20 616e 6e20 5054 2020 2020 2020 2020 : ann PT │ │ │ │ +0001f720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f750: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001f740: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001f750: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f7a0: 7c0a 7c6f 3132 203d 2069 6465 616c 2865 |.|o12 = ideal(e │ │ │ │ -0001f7b0: 2065 2065 2029 2020 2020 2020 2020 2020 e e ) │ │ │ │ +0001f790: 2020 2020 207c 0a7c 6f31 3220 3d20 6964 |.|o12 = id │ │ │ │ +0001f7a0: 6561 6c28 6520 6520 6520 2920 2020 2020 eal(e e e ) │ │ │ │ +0001f7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f7e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001f7f0: 2020 2020 2020 3020 3120 3220 2020 2020 0 1 2 │ │ │ │ +0001f7d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001f7e0: 2020 2020 2020 2020 2020 2030 2031 2032 0 1 2 │ │ │ │ +0001f7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f820: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001f820: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001f830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f870: 2020 7c0a 7c6f 3132 203a 2049 6465 616c |.|o12 : Ideal │ │ │ │ -0001f880: 206f 6620 6b6b 5b65 202e 2e65 205d 2020 of kk[e ..e ] │ │ │ │ +0001f860: 2020 2020 2020 207c 0a7c 6f31 3220 3a20 |.|o12 : │ │ │ │ +0001f870: 4964 6561 6c20 6f66 206b 6b5b 6520 2e2e Ideal of kk[e .. │ │ │ │ +0001f880: 6520 5d20 2020 2020 2020 2020 2020 2020 e ] │ │ │ │ 0001f890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f8b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001f8c0: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ -0001f8d0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0001f8a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001f8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f8c0: 2020 2030 2020 2032 2020 2020 2020 2020 0 2 │ │ │ │ +0001f8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f8f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001f900: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001f8f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f940: 2d2d 2d2d 2b0a 7c69 3133 203a 2050 5430 ----+.|i13 : PT0 │ │ │ │ -0001f950: 203d 2069 6d61 6765 2028 696e 6475 6365 = image (induce │ │ │ │ -0001f960: 644d 6170 2850 542c 636f 7665 7220 5054 dMap(PT,cover PT │ │ │ │ -0001f970: 292a 2028 2863 6f76 6572 2050 5429 5f7b )* ((cover PT)_{ │ │ │ │ -0001f980: 302e 2e31 327d 2929 3b20 7c0a 2b2d 2d2d 0..12})); |.+--- │ │ │ │ +0001f930: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 ---------+.|i13 │ │ │ │ +0001f940: 3a20 5054 3020 3d20 696d 6167 6520 2869 : PT0 = image (i │ │ │ │ +0001f950: 6e64 7563 6564 4d61 7028 5054 2c63 6f76 nducedMap(PT,cov │ │ │ │ +0001f960: 6572 2050 5429 2a20 2828 636f 7665 7220 er PT)* ((cover │ │ │ │ +0001f970: 5054 295f 7b30 2e2e 3132 7d29 293b 207c PT)_{0..12})); | │ │ │ │ +0001f980: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0001f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f9d0: 2b0a 7c69 3134 203a 2050 5431 203d 2069 +.|i14 : PT1 = i │ │ │ │ -0001f9e0: 6d61 6765 2028 696e 6475 6365 644d 6170 mage (inducedMap │ │ │ │ -0001f9f0: 2850 542c 636f 7665 7220 5054 292a 2028 (PT,cover PT)* ( │ │ │ │ -0001fa00: 2863 6f76 6572 2050 5429 5f7b 3133 2e2e (cover PT)_{13.. │ │ │ │ -0001fa10: 3330 7d29 293b 7c0a 2b2d 2d2d 2d2d 2d2d 30}));|.+------- │ │ │ │ +0001f9c0: 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 5054 -----+.|i14 : PT │ │ │ │ +0001f9d0: 3120 3d20 696d 6167 6520 2869 6e64 7563 1 = image (induc │ │ │ │ +0001f9e0: 6564 4d61 7028 5054 2c63 6f76 6572 2050 edMap(PT,cover P │ │ │ │ +0001f9f0: 5429 2a20 2828 636f 7665 7220 5054 295f T)* ((cover PT)_ │ │ │ │ +0001fa00: 7b31 332e 2e33 307d 2929 3b7c 0a2b 2d2d {13..30}));|.+-- │ │ │ │ +0001fa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fa50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001fa60: 3135 203a 2062 6574 7469 2066 7265 6552 15 : betti freeR │ │ │ │ -0001fa70: 6573 6f6c 7574 696f 6e28 7072 756e 6520 esolution(prune │ │ │ │ -0001fa80: 5054 302c 204c 656e 6774 684c 696d 6974 PT0, LengthLimit │ │ │ │ -0001fa90: 203d 3e20 3429 2020 2020 2020 2020 2020 => 4) │ │ │ │ -0001faa0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001fa50: 2d2b 0a7c 6931 3520 3a20 6265 7474 6920 -+.|i15 : betti │ │ │ │ +0001fa60: 6672 6565 5265 736f 6c75 7469 6f6e 2870 freeResolution(p │ │ │ │ +0001fa70: 7275 6e65 2050 5430 2c20 4c65 6e67 7468 rune PT0, Length │ │ │ │ +0001fa80: 4c69 6d69 7420 3d3e 2034 2920 2020 2020 Limit => 4) │ │ │ │ +0001fa90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001faa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fae0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001faf0: 2020 2020 2020 2020 2030 2020 3120 2032 0 1 2 │ │ │ │ -0001fb00: 2020 3320 2034 2020 2020 2020 2020 2020 3 4 │ │ │ │ +0001fad0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001fae0: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ +0001faf0: 2031 2020 3220 2033 2020 3420 2020 2020 1 2 3 4 │ │ │ │ +0001fb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fb20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001fb30: 7c6f 3135 203d 2074 6f74 616c 3a20 3133 |o15 = total: 13 │ │ │ │ -0001fb40: 2032 3420 3339 2035 3820 3831 2020 2020 24 39 58 81 │ │ │ │ +0001fb20: 2020 207c 0a7c 6f31 3520 3d20 746f 7461 |.|o15 = tota │ │ │ │ +0001fb30: 6c3a 2031 3320 3234 2033 3920 3538 2038 l: 13 24 39 58 8 │ │ │ │ +0001fb40: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 0001fb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fb70: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001fb80: 2030 3a20 3133 2032 3420 3339 2035 3820 0: 13 24 39 58 │ │ │ │ -0001fb90: 3831 2020 2020 2020 2020 2020 2020 2020 81 │ │ │ │ -0001fba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fbb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001fb60: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001fb70: 2020 2020 2020 303a 2031 3320 3234 2033 0: 13 24 3 │ │ │ │ +0001fb80: 3920 3538 2038 3120 2020 2020 2020 2020 9 58 81 │ │ │ │ +0001fb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fba0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001fbb0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001fbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc00: 7c0a 7c6f 3135 203a 2042 6574 7469 5461 |.|o15 : BettiTa │ │ │ │ -0001fc10: 6c6c 7920 2020 2020 2020 2020 2020 2020 lly │ │ │ │ +0001fbf0: 2020 2020 207c 0a7c 6f31 3520 3a20 4265 |.|o15 : Be │ │ │ │ +0001fc00: 7474 6954 616c 6c79 2020 2020 2020 2020 ttiTally │ │ │ │ +0001fc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001fc30: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001fc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001fc90: 3136 203a 2062 6574 7469 2066 7265 6552 16 : betti freeR │ │ │ │ -0001fca0: 6573 6f6c 7574 696f 6e28 7072 756e 6520 esolution(prune │ │ │ │ -0001fcb0: 5054 312c 204c 656e 6774 684c 696d 6974 PT1, LengthLimit │ │ │ │ -0001fcc0: 203d 3e20 3429 2020 2020 2020 2020 2020 => 4) │ │ │ │ -0001fcd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001fc80: 2d2b 0a7c 6931 3620 3a20 6265 7474 6920 -+.|i16 : betti │ │ │ │ +0001fc90: 6672 6565 5265 736f 6c75 7469 6f6e 2870 freeResolution(p │ │ │ │ +0001fca0: 7275 6e65 2050 5431 2c20 4c65 6e67 7468 rune PT1, Length │ │ │ │ +0001fcb0: 4c69 6d69 7420 3d3e 2034 2920 2020 2020 Limit => 4) │ │ │ │ +0001fcc0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001fcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fd10: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001fd20: 2020 2020 2020 2020 2030 2020 3120 2032 0 1 2 │ │ │ │ -0001fd30: 2020 3320 2034 2020 2020 2020 2020 2020 3 4 │ │ │ │ +0001fd00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001fd10: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ +0001fd20: 2031 2020 3220 2033 2020 3420 2020 2020 1 2 3 4 │ │ │ │ +0001fd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fd50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001fd60: 7c6f 3136 203d 2074 6f74 616c 3a20 3138 |o16 = total: 18 │ │ │ │ -0001fd70: 2032 3820 3339 2035 3120 3634 2020 2020 28 39 51 64 │ │ │ │ +0001fd50: 2020 207c 0a7c 6f31 3620 3d20 746f 7461 |.|o16 = tota │ │ │ │ +0001fd60: 6c3a 2031 3820 3238 2033 3920 3531 2036 l: 18 28 39 51 6 │ │ │ │ +0001fd70: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ 0001fd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fda0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001fdb0: 2031 3a20 3138 2032 3820 3339 2035 3120 1: 18 28 39 51 │ │ │ │ -0001fdc0: 3634 2020 2020 2020 2020 2020 2020 2020 64 │ │ │ │ -0001fdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fde0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001fd90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001fda0: 2020 2020 2020 313a 2031 3820 3238 2033 1: 18 28 3 │ │ │ │ +0001fdb0: 3920 3531 2036 3420 2020 2020 2020 2020 9 51 64 │ │ │ │ +0001fdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fdd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001fde0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001fdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fe00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fe10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fe20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fe30: 7c0a 7c6f 3136 203a 2042 6574 7469 5461 |.|o16 : BettiTa │ │ │ │ -0001fe40: 6c6c 7920 2020 2020 2020 2020 2020 2020 lly │ │ │ │ +0001fe20: 2020 2020 207c 0a7c 6f31 3620 3a20 4265 |.|o16 : Be │ │ │ │ +0001fe30: 7474 6954 616c 6c79 2020 2020 2020 2020 ttiTally │ │ │ │ +0001fe40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fe50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fe60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fe70: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001fe60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001fe70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fe90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001feb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001fec0: 3137 203a 2062 6574 7469 2066 7265 6552 17 : betti freeR │ │ │ │ -0001fed0: 6573 6f6c 7574 696f 6e28 7072 756e 6520 esolution(prune │ │ │ │ -0001fee0: 5054 2c20 4c65 6e67 7468 4c69 6d69 7420 PT, LengthLimit │ │ │ │ -0001fef0: 3d3e 2034 2920 2020 2020 2020 2020 2020 => 4) │ │ │ │ -0001ff00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001feb0: 2d2b 0a7c 6931 3720 3a20 6265 7474 6920 -+.|i17 : betti │ │ │ │ +0001fec0: 6672 6565 5265 736f 6c75 7469 6f6e 2870 freeResolution(p │ │ │ │ +0001fed0: 7275 6e65 2050 542c 204c 656e 6774 684c rune PT, LengthL │ │ │ │ +0001fee0: 696d 6974 203d 3e20 3429 2020 2020 2020 imit => 4) │ │ │ │ +0001fef0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001ff00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ff10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ff20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ff30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ff40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001ff50: 2020 2020 2020 2020 2030 2020 3120 2032 0 1 2 │ │ │ │ -0001ff60: 2020 2033 2020 2034 2020 2020 2020 2020 3 4 │ │ │ │ +0001ff30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001ff40: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ +0001ff50: 2031 2020 3220 2020 3320 2020 3420 2020 1 2 3 4 │ │ │ │ +0001ff60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ff70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ff80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001ff90: 7c6f 3137 203d 2074 6f74 616c 3a20 3331 |o17 = total: 31 │ │ │ │ -0001ffa0: 2035 3520 3837 2031 3237 2031 3735 2020 55 87 127 175 │ │ │ │ +0001ff80: 2020 207c 0a7c 6f31 3720 3d20 746f 7461 |.|o17 = tota │ │ │ │ +0001ff90: 6c3a 2033 3120 3535 2038 3720 3132 3720 l: 31 55 87 127 │ │ │ │ +0001ffa0: 3137 3520 2020 2020 2020 2020 2020 2020 175 │ │ │ │ 0001ffb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ffc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ffd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001ffe0: 2030 3a20 3133 2032 3420 3339 2020 3538 0: 13 24 39 58 │ │ │ │ -0001fff0: 2020 3831 2020 2020 2020 2020 2020 2020 81 │ │ │ │ -00020000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020010: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00020020: 2020 2020 2020 2031 3a20 3138 2033 3120 1: 18 31 │ │ │ │ -00020030: 3438 2020 3639 2020 3934 2020 2020 2020 48 69 94 │ │ │ │ +0001ffc0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001ffd0: 2020 2020 2020 303a 2031 3320 3234 2033 0: 13 24 3 │ │ │ │ +0001ffe0: 3920 2035 3820 2038 3120 2020 2020 2020 9 58 81 │ │ │ │ +0001fff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020000: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00020010: 0a7c 2020 2020 2020 2020 2020 313a 2031 .| 1: 1 │ │ │ │ +00020020: 3820 3331 2034 3820 2036 3920 2039 3420 8 31 48 69 94 │ │ │ │ +00020030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020060: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00020050: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00020060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000200a0: 2020 2020 2020 7c0a 7c6f 3137 203a 2042 |.|o17 : B │ │ │ │ -000200b0: 6574 7469 5461 6c6c 7920 2020 2020 2020 ettiTally │ │ │ │ +00020090: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +000200a0: 3720 3a20 4265 7474 6954 616c 6c79 2020 7 : BettiTally │ │ │ │ +000200b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000200c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000200d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000200e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000200e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 000200f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020130: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d --+..See also.== │ │ │ │ -00020140: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ -00020150: 6520 6d61 6b65 4d6f 6475 6c65 3a20 6d61 e makeModule: ma │ │ │ │ -00020160: 6b65 4d6f 6475 6c65 2c20 2d2d 206d 616b keModule, -- mak │ │ │ │ -00020170: 6573 2061 204d 6f64 756c 6520 6f75 7420 es a Module out │ │ │ │ -00020180: 6f66 2061 2063 6f6c 6c65 6374 696f 6e20 of a collection │ │ │ │ -00020190: 6f66 0a20 2020 206d 6f64 756c 6573 2061 of. modules a │ │ │ │ -000201a0: 6e64 206d 6170 730a 0a57 6179 7320 746f nd maps..Ways to │ │ │ │ -000201b0: 2075 7365 2065 7874 6572 696f 7254 6f72 use exteriorTor │ │ │ │ -000201c0: 4d6f 6475 6c65 3a0a 3d3d 3d3d 3d3d 3d3d Module:.======== │ │ │ │ -000201d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000201e0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2265 7874 ======.. * "ext │ │ │ │ -000201f0: 6572 696f 7254 6f72 4d6f 6475 6c65 284d eriorTorModule(M │ │ │ │ -00020200: 6174 7269 782c 4d6f 6475 6c65 2922 0a20 atrix,Module)". │ │ │ │ -00020210: 202a 2022 6578 7465 7269 6f72 546f 724d * "exteriorTorM │ │ │ │ -00020220: 6f64 756c 6528 4d61 7472 6978 2c4d 6f64 odule(Matrix,Mod │ │ │ │ -00020230: 756c 652c 4d6f 6475 6c65 2922 0a0a 466f ule,Module)"..Fo │ │ │ │ -00020240: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -00020250: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -00020260: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -00020270: 2a6e 6f74 6520 6578 7465 7269 6f72 546f *note exteriorTo │ │ │ │ -00020280: 724d 6f64 756c 653a 2065 7874 6572 696f rModule: exterio │ │ │ │ -00020290: 7254 6f72 4d6f 6475 6c65 2c20 6973 2061 rTorModule, is a │ │ │ │ -000202a0: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ -000202b0: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61 nction: (Macaula │ │ │ │ -000202c0: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ -000202d0: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ +00020120: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ +00020130: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ +00020140: 202a 6e6f 7465 206d 616b 654d 6f64 756c *note makeModul │ │ │ │ +00020150: 653a 206d 616b 654d 6f64 756c 652c 202d e: makeModule, - │ │ │ │ +00020160: 2d20 6d61 6b65 7320 6120 4d6f 6475 6c65 - makes a Module │ │ │ │ +00020170: 206f 7574 206f 6620 6120 636f 6c6c 6563 out of a collec │ │ │ │ +00020180: 7469 6f6e 206f 660a 2020 2020 6d6f 6475 tion of. modu │ │ │ │ +00020190: 6c65 7320 616e 6420 6d61 7073 0a0a 5761 les and maps..Wa │ │ │ │ +000201a0: 7973 2074 6f20 7573 6520 6578 7465 7269 ys to use exteri │ │ │ │ +000201b0: 6f72 546f 724d 6f64 756c 653a 0a3d 3d3d orTorModule:.=== │ │ │ │ +000201c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000201d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +000201e0: 2022 6578 7465 7269 6f72 546f 724d 6f64 "exteriorTorMod │ │ │ │ +000201f0: 756c 6528 4d61 7472 6978 2c4d 6f64 756c ule(Matrix,Modul │ │ │ │ +00020200: 6529 220a 2020 2a20 2265 7874 6572 696f e)". * "exterio │ │ │ │ +00020210: 7254 6f72 4d6f 6475 6c65 284d 6174 7269 rTorModule(Matri │ │ │ │ +00020220: 782c 4d6f 6475 6c65 2c4d 6f64 756c 6529 x,Module,Module) │ │ │ │ +00020230: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ +00020240: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +00020250: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +00020260: 6a65 6374 202a 6e6f 7465 2065 7874 6572 ject *note exter │ │ │ │ +00020270: 696f 7254 6f72 4d6f 6475 6c65 3a20 6578 iorTorModule: ex │ │ │ │ +00020280: 7465 7269 6f72 546f 724d 6f64 756c 652c teriorTorModule, │ │ │ │ +00020290: 2069 7320 6120 2a6e 6f74 6520 6d65 7468 is a *note meth │ │ │ │ +000202a0: 6f64 0a66 756e 6374 696f 6e3a 2028 4d61 od.function: (Ma │ │ │ │ +000202b0: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ +000202c0: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d dFunction,...--- │ │ │ │ +000202d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000202e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000202f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020320: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -00020330: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -00020340: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -00020350: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -00020360: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -00020370: 3236 2e30 352b 6473 2f4d 322f 4d61 6361 26.05+ds/M2/Maca │ │ │ │ -00020380: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ -00020390: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -000203a0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -000203b0: 6d32 3a34 3138 323a 302e 0a1f 0a46 696c m2:4182:0....Fil │ │ │ │ -000203c0: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ -000203d0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -000203e0: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2065 ns.info, Node: e │ │ │ │ -000203f0: 7874 4973 4f6e 6550 6f6c 796e 6f6d 6961 xtIsOnePolynomia │ │ │ │ -00020400: 6c2c 204e 6578 743a 2045 7874 4d6f 6475 l, Next: ExtModu │ │ │ │ -00020410: 6c65 2c20 5072 6576 3a20 6578 7465 7269 le, Prev: exteri │ │ │ │ -00020420: 6f72 546f 724d 6f64 756c 652c 2055 703a orTorModule, Up: │ │ │ │ -00020430: 2054 6f70 0a0a 6578 7449 734f 6e65 506f Top..extIsOnePo │ │ │ │ -00020440: 6c79 6e6f 6d69 616c 202d 2d20 6368 6563 lynomial -- chec │ │ │ │ -00020450: 6b20 7768 6574 6865 7220 7468 6520 4869 k whether the Hi │ │ │ │ -00020460: 6c62 6572 7420 6675 6e63 7469 6f6e 206f lbert function o │ │ │ │ -00020470: 6620 4578 7428 4d2c 6b29 2069 7320 6f6e f Ext(M,k) is on │ │ │ │ -00020480: 6520 706f 6c79 6e6f 6d69 616c 0a2a 2a2a e polynomial.*** │ │ │ │ +00020310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +00020320: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +00020330: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +00020340: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +00020350: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +00020360: 7932 2d31 2e32 362e 3035 2b64 732f 4d32 y2-1.26.05+ds/M2 │ │ │ │ +00020370: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +00020380: 6765 732f 0a43 6f6d 706c 6574 6549 6e74 ges/.CompleteInt │ │ │ │ +00020390: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +000203a0: 696f 6e73 2e6d 323a 3431 3832 3a30 2e0a ions.m2:4182:0.. │ │ │ │ +000203b0: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465 ..File: Complete │ │ │ │ +000203c0: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +000203d0: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f lutions.info, No │ │ │ │ +000203e0: 6465 3a20 6578 7449 734f 6e65 506f 6c79 de: extIsOnePoly │ │ │ │ +000203f0: 6e6f 6d69 616c 2c20 4e65 7874 3a20 4578 nomial, Next: Ex │ │ │ │ +00020400: 744d 6f64 756c 652c 2050 7265 763a 2065 tModule, Prev: e │ │ │ │ +00020410: 7874 6572 696f 7254 6f72 4d6f 6475 6c65 xteriorTorModule │ │ │ │ +00020420: 2c20 5570 3a20 546f 700a 0a65 7874 4973 , Up: Top..extIs │ │ │ │ +00020430: 4f6e 6550 6f6c 796e 6f6d 6961 6c20 2d2d OnePolynomial -- │ │ │ │ +00020440: 2063 6865 636b 2077 6865 7468 6572 2074 check whether t │ │ │ │ +00020450: 6865 2048 696c 6265 7274 2066 756e 6374 he Hilbert funct │ │ │ │ +00020460: 696f 6e20 6f66 2045 7874 284d 2c6b 2920 ion of Ext(M,k) │ │ │ │ +00020470: 6973 206f 6e65 2070 6f6c 796e 6f6d 6961 is one polynomia │ │ │ │ +00020480: 6c0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a l.************** │ │ │ │ 00020490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000204a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000204b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000204c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000204d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000204e0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -000204f0: 0a20 2020 2020 2020 2028 702c 7429 203d . (p,t) = │ │ │ │ -00020500: 2065 7874 4973 4f6e 6550 6f6c 796e 6f6d extIsOnePolynom │ │ │ │ -00020510: 6961 6c20 4d0a 2020 2a20 496e 7075 7473 ial M. * Inputs │ │ │ │ -00020520: 3a0a 2020 2020 2020 2a20 4d2c 2061 202a :. * M, a * │ │ │ │ -00020530: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -00020540: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -00020550: 652c 2c20 6d6f 6475 6c65 206f 7665 7220 e,, module over │ │ │ │ -00020560: 6120 636f 6d70 6c65 7465 0a20 2020 2020 a complete. │ │ │ │ -00020570: 2020 2069 6e74 6572 7365 6374 696f 6e0a intersection. │ │ │ │ -00020580: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -00020590: 2020 202a 2070 2c20 6120 2a6e 6f74 6520 * p, a *note │ │ │ │ -000205a0: 7269 6e67 2065 6c65 6d65 6e74 3a20 284d ring element: (M │ │ │ │ -000205b0: 6163 6175 6c61 7932 446f 6329 5269 6e67 acaulay2Doc)Ring │ │ │ │ -000205c0: 456c 656d 656e 742c 2c20 7028 7a29 3d70 Element,, p(z)=p │ │ │ │ -000205d0: 6528 7a2f 3229 2c0a 2020 2020 2020 2020 e(z/2),. │ │ │ │ -000205e0: 7768 6572 6520 7065 2069 7320 7468 6520 where pe is the │ │ │ │ -000205f0: 4869 6c62 6572 7420 706f 6c79 206f 6620 Hilbert poly of │ │ │ │ -00020600: 4578 745e 7b65 7665 6e7d 284d 2c6b 290a Ext^{even}(M,k). │ │ │ │ -00020610: 2020 2020 2020 2a20 742c 2061 202a 6e6f * t, a *no │ │ │ │ -00020620: 7465 2042 6f6f 6c65 616e 2076 616c 7565 te Boolean value │ │ │ │ -00020630: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00020640: 426f 6f6c 6561 6e2c 2c20 7472 7565 2069 Boolean,, true i │ │ │ │ -00020650: 6620 7468 6520 6576 656e 2061 6e64 0a20 f the even and. │ │ │ │ -00020660: 2020 2020 2020 206f 6464 2070 6f6c 796e odd polyn │ │ │ │ -00020670: 6f6d 6961 6c73 206d 6174 6368 2074 6f20 omials match to │ │ │ │ -00020680: 666f 726d 206f 6e65 2070 6f6c 796e 6f6d form one polynom │ │ │ │ -00020690: 6961 6c0a 0a44 6573 6372 6970 7469 6f6e ial..Description │ │ │ │ -000206a0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 436f .===========..Co │ │ │ │ -000206b0: 6d70 7574 6573 2074 6865 2048 696c 6265 mputes the Hilbe │ │ │ │ -000206c0: 7274 2070 6f6c 796e 6f6d 6961 6c73 2070 rt polynomials p │ │ │ │ -000206d0: 6528 7a29 2c20 706f 287a 2920 6f66 2065 e(z), po(z) of e │ │ │ │ -000206e0: 7665 6e45 7874 4d6f 6475 6c65 2061 6e64 venExtModule and │ │ │ │ -000206f0: 0a6f 6464 4578 744d 6f64 756c 652e 2049 .oddExtModule. I │ │ │ │ -00020700: 7420 7265 7475 726e 7320 7065 287a 2f32 t returns pe(z/2 │ │ │ │ -00020710: 292c 2061 6e64 2063 6f6d 7061 7265 7320 ), and compares │ │ │ │ -00020720: 746f 2073 6565 2077 6865 7468 6572 2074 to see whether t │ │ │ │ -00020730: 6869 7320 6973 2065 7175 616c 2074 6f0a his is equal to. │ │ │ │ -00020740: 706f 287a 2f32 2d31 2f32 292e 2041 7672 po(z/2-1/2). Avr │ │ │ │ -00020750: 616d 6f76 2c20 5365 6365 6c65 616e 7520 amov, Seceleanu │ │ │ │ -00020760: 616e 6420 5a68 656e 6720 6861 7665 2070 and Zheng have p │ │ │ │ -00020770: 726f 7665 6e20 7468 6174 2069 6620 7468 roven that if th │ │ │ │ -00020780: 6520 6964 6561 6c20 6f66 0a71 7561 6472 e ideal of.quadr │ │ │ │ -00020790: 6174 6963 206c 6561 6469 6e67 2066 6f72 atic leading for │ │ │ │ -000207a0: 6d73 206f 6620 6120 636f 6d70 6c65 7465 ms of a complete │ │ │ │ -000207b0: 2069 6e74 6572 7365 6374 696f 6e20 6f66 intersection of │ │ │ │ -000207c0: 2063 6f64 696d 656e 7369 6f6e 2063 2067 codimension c g │ │ │ │ -000207d0: 656e 6572 6174 6520 616e 0a69 6465 616c enerate an.ideal │ │ │ │ -000207e0: 206f 6620 636f 6469 6d65 6e73 696f 6e20 of codimension │ │ │ │ -000207f0: 6174 206c 6561 7374 2063 2d31 2c20 7468 at least c-1, th │ │ │ │ -00020800: 656e 2074 6865 2042 6574 7469 206e 756d en the Betti num │ │ │ │ -00020810: 6265 7273 206f 6620 616e 7920 6d6f 6475 bers of any modu │ │ │ │ -00020820: 6c65 2067 726f 772c 0a65 7665 6e74 7561 le grow,.eventua │ │ │ │ -00020830: 6c6c 792c 2061 7320 6120 7369 6e67 6c65 lly, as a single │ │ │ │ -00020840: 2070 6f6c 796e 6f6d 6961 6c20 2869 6e73 polynomial (ins │ │ │ │ -00020850: 7465 6164 206f 6620 7265 7175 6972 696e tead of requirin │ │ │ │ -00020860: 6720 7365 7061 7261 7465 2070 6f6c 796e g separate polyn │ │ │ │ -00020870: 6f6d 6961 6c73 0a66 6f72 2065 7665 6e20 omials.for even │ │ │ │ -00020880: 616e 6420 6f64 6420 7465 726d 732e 2920 and odd terms.) │ │ │ │ -00020890: 5468 6973 2073 6372 6970 7420 6368 6563 This script chec │ │ │ │ -000208a0: 6b73 2074 6865 2072 6573 756c 7420 696e ks the result in │ │ │ │ -000208b0: 2074 6865 2068 6f6d 6f67 656e 656f 7573 the homogeneous │ │ │ │ -000208c0: 2063 6173 650a 2869 6e20 7768 6963 6820 case.(in which │ │ │ │ -000208d0: 6361 7365 2074 6865 2063 6f6e 6469 7469 case the conditi │ │ │ │ -000208e0: 6f6e 2069 7320 6e65 6365 7373 6172 7920 on is necessary │ │ │ │ -000208f0: 616e 6420 7375 6666 6963 6965 6e74 2e29 and sufficient.) │ │ │ │ -00020900: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +000204d0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ +000204e0: 6167 653a 200a 2020 2020 2020 2020 2870 age: . (p │ │ │ │ +000204f0: 2c74 2920 3d20 6578 7449 734f 6e65 506f ,t) = extIsOnePo │ │ │ │ +00020500: 6c79 6e6f 6d69 616c 204d 0a20 202a 2049 lynomial M. * I │ │ │ │ +00020510: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ +00020520: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +00020530: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00020540: 4d6f 6475 6c65 2c2c 206d 6f64 756c 6520 Module,, module │ │ │ │ +00020550: 6f76 6572 2061 2063 6f6d 706c 6574 650a over a complete. │ │ │ │ +00020560: 2020 2020 2020 2020 696e 7465 7273 6563 intersec │ │ │ │ +00020570: 7469 6f6e 0a20 202a 204f 7574 7075 7473 tion. * Outputs │ │ │ │ +00020580: 3a0a 2020 2020 2020 2a20 702c 2061 202a :. * p, a * │ │ │ │ +00020590: 6e6f 7465 2072 696e 6720 656c 656d 656e note ring elemen │ │ │ │ +000205a0: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ +000205b0: 2952 696e 6745 6c65 6d65 6e74 2c2c 2070 )RingElement,, p │ │ │ │ +000205c0: 287a 293d 7065 287a 2f32 292c 0a20 2020 (z)=pe(z/2),. │ │ │ │ +000205d0: 2020 2020 2077 6865 7265 2070 6520 6973 where pe is │ │ │ │ +000205e0: 2074 6865 2048 696c 6265 7274 2070 6f6c the Hilbert pol │ │ │ │ +000205f0: 7920 6f66 2045 7874 5e7b 6576 656e 7d28 y of Ext^{even}( │ │ │ │ +00020600: 4d2c 6b29 0a20 2020 2020 202a 2074 2c20 M,k). * t, │ │ │ │ +00020610: 6120 2a6e 6f74 6520 426f 6f6c 6561 6e20 a *note Boolean │ │ │ │ +00020620: 7661 6c75 653a 2028 4d61 6361 756c 6179 value: (Macaulay │ │ │ │ +00020630: 3244 6f63 2942 6f6f 6c65 616e 2c2c 2074 2Doc)Boolean,, t │ │ │ │ +00020640: 7275 6520 6966 2074 6865 2065 7665 6e20 rue if the even │ │ │ │ +00020650: 616e 640a 2020 2020 2020 2020 6f64 6420 and. odd │ │ │ │ +00020660: 706f 6c79 6e6f 6d69 616c 7320 6d61 7463 polynomials matc │ │ │ │ +00020670: 6820 746f 2066 6f72 6d20 6f6e 6520 706f h to form one po │ │ │ │ +00020680: 6c79 6e6f 6d69 616c 0a0a 4465 7363 7269 lynomial..Descri │ │ │ │ +00020690: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ +000206a0: 3d0a 0a43 6f6d 7075 7465 7320 7468 6520 =..Computes the │ │ │ │ +000206b0: 4869 6c62 6572 7420 706f 6c79 6e6f 6d69 Hilbert polynomi │ │ │ │ +000206c0: 616c 7320 7065 287a 292c 2070 6f28 7a29 als pe(z), po(z) │ │ │ │ +000206d0: 206f 6620 6576 656e 4578 744d 6f64 756c of evenExtModul │ │ │ │ +000206e0: 6520 616e 640a 6f64 6445 7874 4d6f 6475 e and.oddExtModu │ │ │ │ +000206f0: 6c65 2e20 4974 2072 6574 7572 6e73 2070 le. It returns p │ │ │ │ +00020700: 6528 7a2f 3229 2c20 616e 6420 636f 6d70 e(z/2), and comp │ │ │ │ +00020710: 6172 6573 2074 6f20 7365 6520 7768 6574 ares to see whet │ │ │ │ +00020720: 6865 7220 7468 6973 2069 7320 6571 7561 her this is equa │ │ │ │ +00020730: 6c20 746f 0a70 6f28 7a2f 322d 312f 3229 l to.po(z/2-1/2) │ │ │ │ +00020740: 2e20 4176 7261 6d6f 762c 2053 6563 656c . Avramov, Secel │ │ │ │ +00020750: 6561 6e75 2061 6e64 205a 6865 6e67 2068 eanu and Zheng h │ │ │ │ +00020760: 6176 6520 7072 6f76 656e 2074 6861 7420 ave proven that │ │ │ │ +00020770: 6966 2074 6865 2069 6465 616c 206f 660a if the ideal of. │ │ │ │ +00020780: 7175 6164 7261 7469 6320 6c65 6164 696e quadratic leadin │ │ │ │ +00020790: 6720 666f 726d 7320 6f66 2061 2063 6f6d g forms of a com │ │ │ │ +000207a0: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ +000207b0: 6f6e 206f 6620 636f 6469 6d65 6e73 696f on of codimensio │ │ │ │ +000207c0: 6e20 6320 6765 6e65 7261 7465 2061 6e0a n c generate an. │ │ │ │ +000207d0: 6964 6561 6c20 6f66 2063 6f64 696d 656e ideal of codimen │ │ │ │ +000207e0: 7369 6f6e 2061 7420 6c65 6173 7420 632d sion at least c- │ │ │ │ +000207f0: 312c 2074 6865 6e20 7468 6520 4265 7474 1, then the Bett │ │ │ │ +00020800: 6920 6e75 6d62 6572 7320 6f66 2061 6e79 i numbers of any │ │ │ │ +00020810: 206d 6f64 756c 6520 6772 6f77 2c0a 6576 module grow,.ev │ │ │ │ +00020820: 656e 7475 616c 6c79 2c20 6173 2061 2073 entually, as a s │ │ │ │ +00020830: 696e 676c 6520 706f 6c79 6e6f 6d69 616c ingle polynomial │ │ │ │ +00020840: 2028 696e 7374 6561 6420 6f66 2072 6571 (instead of req │ │ │ │ +00020850: 7569 7269 6e67 2073 6570 6172 6174 6520 uiring separate │ │ │ │ +00020860: 706f 6c79 6e6f 6d69 616c 730a 666f 7220 polynomials.for │ │ │ │ +00020870: 6576 656e 2061 6e64 206f 6464 2074 6572 even and odd ter │ │ │ │ +00020880: 6d73 2e29 2054 6869 7320 7363 7269 7074 ms.) This script │ │ │ │ +00020890: 2063 6865 636b 7320 7468 6520 7265 7375 checks the resu │ │ │ │ +000208a0: 6c74 2069 6e20 7468 6520 686f 6d6f 6765 lt in the homoge │ │ │ │ +000208b0: 6e65 6f75 7320 6361 7365 0a28 696e 2077 neous case.(in w │ │ │ │ +000208c0: 6869 6368 2063 6173 6520 7468 6520 636f hich case the co │ │ │ │ +000208d0: 6e64 6974 696f 6e20 6973 206e 6563 6573 ndition is neces │ │ │ │ +000208e0: 7361 7279 2061 6e64 2073 7566 6669 6369 sary and suffici │ │ │ │ +000208f0: 656e 742e 290a 0a2b 2d2d 2d2d 2d2d 2d2d ent.)..+-------- │ │ │ │ +00020900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00020940: 203a 2052 313d 5a5a 2f31 3031 5b61 2c62 : R1=ZZ/101[a,b │ │ │ │ -00020950: 2c63 5d2f 6964 6561 6c28 615e 322c 625e ,c]/ideal(a^2,b^ │ │ │ │ -00020960: 322c 635e 3529 2020 2020 2020 2020 2020 2,c^5) │ │ │ │ -00020970: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00020930: 2b0a 7c69 3120 3a20 5231 3d5a 5a2f 3130 +.|i1 : R1=ZZ/10 │ │ │ │ +00020940: 315b 612c 622c 635d 2f69 6465 616c 2861 1[a,b,c]/ideal(a │ │ │ │ +00020950: 5e32 2c62 5e32 2c63 5e35 2920 2020 2020 ^2,b^2,c^5) │ │ │ │ +00020960: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00020970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000209a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000209b0: 207c 0a7c 6f31 203d 2052 3120 2020 2020 |.|o1 = R1 │ │ │ │ +000209a0: 2020 2020 2020 7c0a 7c6f 3120 3d20 5231 |.|o1 = R1 │ │ │ │ +000209b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000209c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000209d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000209e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000209e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000209f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020a20: 2020 2020 2020 207c 0a7c 6f31 203a 2051 |.|o1 : Q │ │ │ │ -00020a30: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +00020a10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00020a20: 3120 3a20 5175 6f74 6965 6e74 5269 6e67 1 : QuotientRing │ │ │ │ +00020a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020a60: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00020a50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00020a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00020aa0: 6932 203a 2052 323d 5a5a 2f31 3031 5b61 i2 : R2=ZZ/101[a │ │ │ │ -00020ab0: 2c62 2c63 5d2f 6964 6561 6c28 615e 332c ,b,c]/ideal(a^3, │ │ │ │ -00020ac0: 625e 3329 2020 2020 2020 2020 2020 2020 b^3) │ │ │ │ -00020ad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00020a90: 2d2d 2b0a 7c69 3220 3a20 5232 3d5a 5a2f --+.|i2 : R2=ZZ/ │ │ │ │ +00020aa0: 3130 315b 612c 622c 635d 2f69 6465 616c 101[a,b,c]/ideal │ │ │ │ +00020ab0: 2861 5e33 2c62 5e33 2920 2020 2020 2020 (a^3,b^3) │ │ │ │ +00020ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00020ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020b10: 2020 207c 0a7c 6f32 203d 2052 3220 2020 |.|o2 = R2 │ │ │ │ +00020b00: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ +00020b10: 5232 2020 2020 2020 2020 2020 2020 2020 R2 │ │ │ │ 00020b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020b40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00020b50: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00020b40: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00020b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020b80: 2020 2020 2020 2020 207c 0a7c 6f32 203a |.|o2 : │ │ │ │ -00020b90: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +00020b70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020b80: 7c6f 3220 3a20 5175 6f74 6965 6e74 5269 |o2 : QuotientRi │ │ │ │ +00020b90: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ 00020ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020bc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00020bb0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00020bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00020c00: 0a7c 6933 203a 2065 7874 4973 4f6e 6550 .|i3 : extIsOneP │ │ │ │ -00020c10: 6f6c 796e 6f6d 6961 6c20 636f 6b65 7220 olynomial coker │ │ │ │ -00020c20: 7261 6e64 6f6d 2852 315e 7b30 2c31 7d2c random(R1^{0,1}, │ │ │ │ -00020c30: 5231 5e7b 333a 2d31 7d29 7c0a 7c20 2020 R1^{3:-1})|.| │ │ │ │ +00020bf0: 2d2d 2d2d 2b0a 7c69 3320 3a20 6578 7449 ----+.|i3 : extI │ │ │ │ +00020c00: 734f 6e65 506f 6c79 6e6f 6d69 616c 2063 sOnePolynomial c │ │ │ │ +00020c10: 6f6b 6572 2072 616e 646f 6d28 5231 5e7b oker random(R1^{ │ │ │ │ +00020c20: 302c 317d 2c52 315e 7b33 3a2d 317d 297c 0,1},R1^{3:-1})| │ │ │ │ +00020c30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00020c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020c70: 2020 2020 207c 0a7c 2020 2020 2020 3120 |.| 1 │ │ │ │ -00020c80: 3220 2020 3120 2020 2020 2020 2020 2020 2 1 │ │ │ │ +00020c60: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00020c70: 2020 2031 2032 2020 2031 2020 2020 2020 1 2 1 │ │ │ │ +00020c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020cb0: 7c0a 7c6f 3320 3d20 282d 7a20 202d 202d |.|o3 = (-z - - │ │ │ │ -00020cc0: 7a20 2b20 332c 2074 7275 6529 2020 2020 z + 3, true) │ │ │ │ +00020ca0: 2020 2020 207c 0a7c 6f33 203d 2028 2d7a |.|o3 = (-z │ │ │ │ +00020cb0: 2020 2d20 2d7a 202b 2033 2c20 7472 7565 - -z + 3, true │ │ │ │ +00020cc0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00020cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020ce0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00020cf0: 2020 2020 3220 2020 2020 3220 2020 2020 2 2 │ │ │ │ +00020ce0: 7c0a 7c20 2020 2020 2032 2020 2020 2032 |.| 2 2 │ │ │ │ +00020cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020d20: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00020d10: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00020d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020d60: 207c 0a7c 6f33 203a 2053 6571 7565 6e63 |.|o3 : Sequenc │ │ │ │ -00020d70: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +00020d50: 2020 2020 2020 7c0a 7c6f 3320 3a20 5365 |.|o3 : Se │ │ │ │ +00020d60: 7175 656e 6365 2020 2020 2020 2020 2020 quence │ │ │ │ +00020d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020d90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00020d90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00020da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020dd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2065 -------+.|i4 : e │ │ │ │ -00020de0: 7874 4973 4f6e 6550 6f6c 796e 6f6d 6961 xtIsOnePolynomia │ │ │ │ -00020df0: 6c20 636f 6b65 7220 7261 6e64 6f6d 2852 l coker random(R │ │ │ │ -00020e00: 325e 7b30 2c31 7d2c 5232 5e7b 333a 2d31 2^{0,1},R2^{3:-1 │ │ │ │ -00020e10: 7d29 7c0a 7c20 2020 2020 2020 2020 2020 })|.| │ │ │ │ +00020dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00020dd0: 3420 3a20 6578 7449 734f 6e65 506f 6c79 4 : extIsOnePoly │ │ │ │ +00020de0: 6e6f 6d69 616c 2063 6f6b 6572 2072 616e nomial coker ran │ │ │ │ +00020df0: 646f 6d28 5232 5e7b 302c 317d 2c52 325e dom(R2^{0,1},R2^ │ │ │ │ +00020e00: 7b33 3a2d 317d 297c 0a7c 2020 2020 2020 {3:-1})|.| │ │ │ │ +00020e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020e40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00020e50: 6f34 203d 2028 337a 202d 2032 2c20 6661 o4 = (3z - 2, fa │ │ │ │ -00020e60: 6c73 6529 2020 2020 2020 2020 2020 2020 lse) │ │ │ │ -00020e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020e80: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00020e40: 2020 7c0a 7c6f 3420 3d20 2833 7a20 2d20 |.|o4 = (3z - │ │ │ │ +00020e50: 322c 2066 616c 7365 2920 2020 2020 2020 2, false) │ │ │ │ +00020e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020e70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00020e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020ec0: 2020 207c 0a7c 6f34 203a 2053 6571 7565 |.|o4 : Seque │ │ │ │ -00020ed0: 6e63 6520 2020 2020 2020 2020 2020 2020 nce │ │ │ │ +00020eb0: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ +00020ec0: 5365 7175 656e 6365 2020 2020 2020 2020 Sequence │ │ │ │ +00020ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020ef0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00020f00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00020ef0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00020f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020f30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 ---------+..See │ │ │ │ -00020f40: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 also.========.. │ │ │ │ -00020f50: 202a 202a 6e6f 7465 2065 7665 6e45 7874 * *note evenExt │ │ │ │ -00020f60: 4d6f 6475 6c65 3a20 6576 656e 4578 744d Module: evenExtM │ │ │ │ -00020f70: 6f64 756c 652c 202d 2d20 6576 656e 2070 odule, -- even p │ │ │ │ -00020f80: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b art of Ext^*(M,k │ │ │ │ -00020f90: 2920 6f76 6572 2061 0a20 2020 2063 6f6d ) over a. com │ │ │ │ -00020fa0: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ -00020fb0: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665 on as module ove │ │ │ │ -00020fc0: 7220 4349 206f 7065 7261 746f 7220 7269 r CI operator ri │ │ │ │ -00020fd0: 6e67 0a20 202a 202a 6e6f 7465 206f 6464 ng. * *note odd │ │ │ │ -00020fe0: 4578 744d 6f64 756c 653a 206f 6464 4578 ExtModule: oddEx │ │ │ │ -00020ff0: 744d 6f64 756c 652c 202d 2d20 6f64 6420 tModule, -- odd │ │ │ │ -00021000: 7061 7274 206f 6620 4578 745e 2a28 4d2c part of Ext^*(M, │ │ │ │ -00021010: 6b29 206f 7665 7220 6120 636f 6d70 6c65 k) over a comple │ │ │ │ -00021020: 7465 0a20 2020 2069 6e74 6572 7365 6374 te. intersect │ │ │ │ -00021030: 696f 6e20 6173 206d 6f64 756c 6520 6f76 ion as module ov │ │ │ │ -00021040: 6572 2043 4920 6f70 6572 6174 6f72 2072 er CI operator r │ │ │ │ -00021050: 696e 670a 0a57 6179 7320 746f 2075 7365 ing..Ways to use │ │ │ │ -00021060: 2065 7874 4973 4f6e 6550 6f6c 796e 6f6d extIsOnePolynom │ │ │ │ -00021070: 6961 6c3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ial:.=========== │ │ │ │ -00021080: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00021090: 3d3d 3d3d 0a0a 2020 2a20 2265 7874 4973 ====.. * "extIs │ │ │ │ -000210a0: 4f6e 6550 6f6c 796e 6f6d 6961 6c28 4d6f OnePolynomial(Mo │ │ │ │ -000210b0: 6475 6c65 2922 0a0a 466f 7220 7468 6520 dule)"..For the │ │ │ │ -000210c0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -000210d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -000210e0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -000210f0: 6578 7449 734f 6e65 506f 6c79 6e6f 6d69 extIsOnePolynomi │ │ │ │ -00021100: 616c 3a20 6578 7449 734f 6e65 506f 6c79 al: extIsOnePoly │ │ │ │ -00021110: 6e6f 6d69 616c 2c20 6973 2061 202a 6e6f nomial, is a *no │ │ │ │ -00021120: 7465 206d 6574 686f 640a 6675 6e63 7469 te method.functi │ │ │ │ -00021130: 6f6e 3a20 284d 6163 6175 6c61 7932 446f on: (Macaulay2Do │ │ │ │ -00021140: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ -00021150: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ +00020f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00020f30: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ +00020f40: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6576 ==.. * *note ev │ │ │ │ +00020f50: 656e 4578 744d 6f64 756c 653a 2065 7665 enExtModule: eve │ │ │ │ +00020f60: 6e45 7874 4d6f 6475 6c65 2c20 2d2d 2065 nExtModule, -- e │ │ │ │ +00020f70: 7665 6e20 7061 7274 206f 6620 4578 745e ven part of Ext^ │ │ │ │ +00020f80: 2a28 4d2c 6b29 206f 7665 7220 610a 2020 *(M,k) over a. │ │ │ │ +00020f90: 2020 636f 6d70 6c65 7465 2069 6e74 6572 complete inter │ │ │ │ +00020fa0: 7365 6374 696f 6e20 6173 206d 6f64 756c section as modul │ │ │ │ +00020fb0: 6520 6f76 6572 2043 4920 6f70 6572 6174 e over CI operat │ │ │ │ +00020fc0: 6f72 2072 696e 670a 2020 2a20 2a6e 6f74 or ring. * *not │ │ │ │ +00020fd0: 6520 6f64 6445 7874 4d6f 6475 6c65 3a20 e oddExtModule: │ │ │ │ +00020fe0: 6f64 6445 7874 4d6f 6475 6c65 2c20 2d2d oddExtModule, -- │ │ │ │ +00020ff0: 206f 6464 2070 6172 7420 6f66 2045 7874 odd part of Ext │ │ │ │ +00021000: 5e2a 284d 2c6b 2920 6f76 6572 2061 2063 ^*(M,k) over a c │ │ │ │ +00021010: 6f6d 706c 6574 650a 2020 2020 696e 7465 omplete. inte │ │ │ │ +00021020: 7273 6563 7469 6f6e 2061 7320 6d6f 6475 rsection as modu │ │ │ │ +00021030: 6c65 206f 7665 7220 4349 206f 7065 7261 le over CI opera │ │ │ │ +00021040: 746f 7220 7269 6e67 0a0a 5761 7973 2074 tor ring..Ways t │ │ │ │ +00021050: 6f20 7573 6520 6578 7449 734f 6e65 506f o use extIsOnePo │ │ │ │ +00021060: 6c79 6e6f 6d69 616c 3a0a 3d3d 3d3d 3d3d lynomial:.====== │ │ │ │ +00021070: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00021080: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ +00021090: 6578 7449 734f 6e65 506f 6c79 6e6f 6d69 extIsOnePolynomi │ │ │ │ +000210a0: 616c 284d 6f64 756c 6529 220a 0a46 6f72 al(Module)"..For │ │ │ │ +000210b0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +000210c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000210d0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +000210e0: 6e6f 7465 2065 7874 4973 4f6e 6550 6f6c note extIsOnePol │ │ │ │ +000210f0: 796e 6f6d 6961 6c3a 2065 7874 4973 4f6e ynomial: extIsOn │ │ │ │ +00021100: 6550 6f6c 796e 6f6d 6961 6c2c 2069 7320 ePolynomial, is │ │ │ │ +00021110: 6120 2a6e 6f74 6520 6d65 7468 6f64 0a66 a *note method.f │ │ │ │ +00021120: 756e 6374 696f 6e3a 2028 4d61 6361 756c unction: (Macaul │ │ │ │ +00021130: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +00021140: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +00021150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000211a0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -000211b0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -000211c0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -000211d0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -000211e0: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -000211f0: 352b 6473 2f4d 322f 4d61 6361 756c 6179 5+ds/M2/Macaulay │ │ │ │ -00021200: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ -00021210: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -00021220: 5265 736f 6c75 7469 6f6e 732e 6d32 3a34 Resolutions.m2:4 │ │ │ │ -00021230: 3933 313a 302e 0a1f 0a46 696c 653a 2043 931:0....File: C │ │ │ │ -00021240: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -00021250: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ -00021260: 6e66 6f2c 204e 6f64 653a 2045 7874 4d6f nfo, Node: ExtMo │ │ │ │ -00021270: 6475 6c65 2c20 4e65 7874 3a20 4578 744d dule, Next: ExtM │ │ │ │ -00021280: 6f64 756c 6544 6174 612c 2050 7265 763a oduleData, Prev: │ │ │ │ -00021290: 2065 7874 4973 4f6e 6550 6f6c 796e 6f6d extIsOnePolynom │ │ │ │ -000212a0: 6961 6c2c 2055 703a 2054 6f70 0a0a 4578 ial, Up: Top..Ex │ │ │ │ -000212b0: 744d 6f64 756c 6520 2d2d 2045 7874 5e2a tModule -- Ext^* │ │ │ │ -000212c0: 284d 2c6b 2920 6f76 6572 2061 2063 6f6d (M,k) over a com │ │ │ │ -000212d0: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ -000212e0: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665 on as module ove │ │ │ │ -000212f0: 7220 4349 206f 7065 7261 746f 7220 7269 r CI operator ri │ │ │ │ -00021300: 6e67 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ng.************* │ │ │ │ +00021190: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +000211a0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +000211b0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +000211c0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +000211d0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +000211e0: 2e32 362e 3035 2b64 732f 4d32 2f4d 6163 .26.05+ds/M2/Mac │ │ │ │ +000211f0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00021200: 0a43 6f6d 706c 6574 6549 6e74 6572 7365 .CompleteInterse │ │ │ │ +00021210: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ +00021220: 2e6d 323a 3439 3331 3a30 2e0a 1f0a 4669 .m2:4931:0....Fi │ │ │ │ +00021230: 6c65 3a20 436f 6d70 6c65 7465 496e 7465 le: CompleteInte │ │ │ │ +00021240: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +00021250: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20 ons.info, Node: │ │ │ │ +00021260: 4578 744d 6f64 756c 652c 204e 6578 743a ExtModule, Next: │ │ │ │ +00021270: 2045 7874 4d6f 6475 6c65 4461 7461 2c20 ExtModuleData, │ │ │ │ +00021280: 5072 6576 3a20 6578 7449 734f 6e65 506f Prev: extIsOnePo │ │ │ │ +00021290: 6c79 6e6f 6d69 616c 2c20 5570 3a20 546f lynomial, Up: To │ │ │ │ +000212a0: 700a 0a45 7874 4d6f 6475 6c65 202d 2d20 p..ExtModule -- │ │ │ │ +000212b0: 4578 745e 2a28 4d2c 6b29 206f 7665 7220 Ext^*(M,k) over │ │ │ │ +000212c0: 6120 636f 6d70 6c65 7465 2069 6e74 6572 a complete inter │ │ │ │ +000212d0: 7365 6374 696f 6e20 6173 206d 6f64 756c section as modul │ │ │ │ +000212e0: 6520 6f76 6572 2043 4920 6f70 6572 6174 e over CI operat │ │ │ │ +000212f0: 6f72 2072 696e 670a 2a2a 2a2a 2a2a 2a2a or ring.******** │ │ │ │ +00021300: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00021310: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00021320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00021330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021350: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ -00021360: 6765 3a20 0a20 2020 2020 2020 2045 203d ge: . E = │ │ │ │ -00021370: 2045 7874 4d6f 6475 6c65 204d 0a20 202a ExtModule M. * │ │ │ │ -00021380: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00021390: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -000213a0: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -000213b0: 6329 4d6f 6475 6c65 2c2c 206f 7665 7220 c)Module,, over │ │ │ │ -000213c0: 6120 636f 6d70 6c65 7465 2069 6e74 6572 a complete inter │ │ │ │ -000213d0: 7365 6374 696f 6e0a 2020 2020 2020 2020 section. │ │ │ │ -000213e0: 7269 6e67 0a20 202a 204f 7574 7075 7473 ring. * Outputs │ │ │ │ -000213f0: 3a0a 2020 2020 2020 2a20 452c 2061 202a :. * E, a * │ │ │ │ -00021400: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -00021410: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -00021420: 652c 2c20 6f76 6572 2061 2070 6f6c 796e e,, over a polyn │ │ │ │ -00021430: 6f6d 6961 6c20 7269 6e67 2077 6974 680a omial ring with. │ │ │ │ -00021440: 2020 2020 2020 2020 6765 6e73 2069 6e20 gens in │ │ │ │ -00021450: 6576 656e 2064 6567 7265 650a 0a44 6573 even degree..Des │ │ │ │ -00021460: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -00021470: 3d3d 3d3d 0a0a 5573 6573 2063 6f64 6520 ====..Uses code │ │ │ │ -00021480: 6f66 2041 7672 616d 6f76 2d47 7261 7973 of Avramov-Grays │ │ │ │ -00021490: 6f6e 2064 6573 6372 6962 6564 2069 6e20 on described in │ │ │ │ -000214a0: 4d61 6361 756c 6179 3220 626f 6f6b 0a0a Macaulay2 book.. │ │ │ │ -000214b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00021340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ +00021350: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ +00021360: 2020 4520 3d20 4578 744d 6f64 756c 6520 E = ExtModule │ │ │ │ +00021370: 4d0a 2020 2a20 496e 7075 7473 3a0a 2020 M. * Inputs:. │ │ │ │ +00021380: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +00021390: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +000213a0: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +000213b0: 6f76 6572 2061 2063 6f6d 706c 6574 6520 over a complete │ │ │ │ +000213c0: 696e 7465 7273 6563 7469 6f6e 0a20 2020 intersection. │ │ │ │ +000213d0: 2020 2020 2072 696e 670a 2020 2a20 4f75 ring. * Ou │ │ │ │ +000213e0: 7470 7574 733a 0a20 2020 2020 202a 2045 tputs:. * E │ │ │ │ +000213f0: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +00021400: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00021410: 4d6f 6475 6c65 2c2c 206f 7665 7220 6120 Module,, over a │ │ │ │ +00021420: 706f 6c79 6e6f 6d69 616c 2072 696e 6720 polynomial ring │ │ │ │ +00021430: 7769 7468 0a20 2020 2020 2020 2067 656e with. gen │ │ │ │ +00021440: 7320 696e 2065 7665 6e20 6465 6772 6565 s in even degree │ │ │ │ +00021450: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +00021460: 3d3d 3d3d 3d3d 3d3d 3d0a 0a55 7365 7320 =========..Uses │ │ │ │ +00021470: 636f 6465 206f 6620 4176 7261 6d6f 762d code of Avramov- │ │ │ │ +00021480: 4772 6179 736f 6e20 6465 7363 7269 6265 Grayson describe │ │ │ │ +00021490: 6420 696e 204d 6163 6175 6c61 7932 2062 d in Macaulay2 b │ │ │ │ +000214a0: 6f6f 6b0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ook..+---------- │ │ │ │ +000214b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000214c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000214d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000214e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -000214f0: 203a 206b 6b3d 205a 5a2f 3130 3120 2020 : kk= ZZ/101 │ │ │ │ +000214e0: 2b0a 7c69 3120 3a20 6b6b 3d20 5a5a 2f31 +.|i1 : kk= ZZ/1 │ │ │ │ +000214f0: 3031 2020 2020 2020 2020 2020 2020 2020 01 │ │ │ │ 00021500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021520: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00021510: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00021520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021560: 2020 2020 207c 0a7c 6f31 203d 206b 6b20 |.|o1 = kk │ │ │ │ +00021550: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +00021560: 3d20 6b6b 2020 2020 2020 2020 2020 2020 = kk │ │ │ │ 00021570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000215a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00021590: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000215a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000215b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000215c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000215d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000215e0: 0a7c 6f31 203a 2051 756f 7469 656e 7452 .|o1 : QuotientR │ │ │ │ -000215f0: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ +000215d0: 2020 2020 7c0a 7c6f 3120 3a20 5175 6f74 |.|o1 : Quot │ │ │ │ +000215e0: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +000215f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021610: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00021610: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00021620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021650: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -00021660: 2053 203d 206b 6b5b 782c 792c 7a5d 2020 S = kk[x,y,z] │ │ │ │ +00021640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00021650: 7c69 3220 3a20 5320 3d20 6b6b 5b78 2c79 |i2 : S = kk[x,y │ │ │ │ +00021660: 2c7a 5d20 2020 2020 2020 2020 2020 2020 ,z] │ │ │ │ 00021670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021690: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00021680: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00021690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000216a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000216b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000216c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000216d0: 2020 207c 0a7c 6f32 203d 2053 2020 2020 |.|o2 = S │ │ │ │ +000216c0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ +000216d0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 000216e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000216f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021710: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00021700: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00021710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021740: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00021750: 6f32 203a 2050 6f6c 796e 6f6d 6961 6c52 o2 : PolynomialR │ │ │ │ -00021760: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -00021770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021780: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00021740: 2020 7c0a 7c6f 3220 3a20 506f 6c79 6e6f |.|o2 : Polyno │ │ │ │ +00021750: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +00021760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021770: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00021780: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00021790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000217a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000217b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000217c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2049 -------+.|i3 : I │ │ │ │ -000217d0: 3120 3d20 6964 6561 6c20 2278 3379 2220 1 = ideal "x3y" │ │ │ │ +000217b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000217c0: 3320 3a20 4931 203d 2069 6465 616c 2022 3 : I1 = ideal " │ │ │ │ +000217d0: 7833 7922 2020 2020 2020 2020 2020 2020 x3y" │ │ │ │ 000217e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000217f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021800: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000217f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00021800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021840: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00021850: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00021830: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00021840: 2020 2020 2033 2020 2020 2020 2020 2020 3 │ │ │ │ +00021850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021870: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00021880: 7c6f 3320 3d20 6964 6561 6c28 7820 7929 |o3 = ideal(x y) │ │ │ │ +00021870: 2020 207c 0a7c 6f33 203d 2069 6465 616c |.|o3 = ideal │ │ │ │ +00021880: 2878 2079 2920 2020 2020 2020 2020 2020 (x y) │ │ │ │ 00021890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000218a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000218b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000218b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000218c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000218d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000218e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000218f0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ -00021900: 4964 6561 6c20 6f66 2053 2020 2020 2020 Ideal of S │ │ │ │ +000218e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000218f0: 6f33 203a 2049 6465 616c 206f 6620 5320 o3 : Ideal of S │ │ │ │ +00021900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021930: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00021920: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00021930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021970: 2d2d 2b0a 7c69 3420 3a20 5231 203d 2053 --+.|i4 : R1 = S │ │ │ │ -00021980: 2f49 3120 2020 2020 2020 2020 2020 2020 /I1 │ │ │ │ +00021960: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 -------+.|i4 : R │ │ │ │ +00021970: 3120 3d20 532f 4931 2020 2020 2020 2020 1 = S/I1 │ │ │ │ +00021980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000219a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000219b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000219a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000219b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000219c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000219d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000219e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -000219f0: 3420 3d20 5231 2020 2020 2020 2020 2020 4 = R1 │ │ │ │ +000219e0: 207c 0a7c 6f34 203d 2052 3120 2020 2020 |.|o4 = R1 │ │ │ │ +000219f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021a20: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00021a10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021a20: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00021a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021a60: 2020 2020 2020 7c0a 7c6f 3420 3a20 5175 |.|o4 : Qu │ │ │ │ -00021a70: 6f74 6965 6e74 5269 6e67 2020 2020 2020 otientRing │ │ │ │ +00021a50: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ +00021a60: 203a 2051 756f 7469 656e 7452 696e 6720 : QuotientRing │ │ │ │ +00021a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021aa0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00021a90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00021aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021ae0: 2b0a 7c69 3520 3a20 4d31 203d 2052 315e +.|i5 : M1 = R1^ │ │ │ │ -00021af0: 312f 6964 6561 6c28 785e 3229 2020 2020 1/ideal(x^2) │ │ │ │ +00021ad0: 2d2d 2d2d 2d2b 0a7c 6935 203a 204d 3120 -----+.|i5 : M1 │ │ │ │ +00021ae0: 3d20 5231 5e31 2f69 6465 616c 2878 5e32 = R1^1/ideal(x^2 │ │ │ │ +00021af0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00021b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021b10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00021b10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00021b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021b30: 2020 2020 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312c 2031 2c20 {1, 1, 1, 1, 1, │ │ │ │ +00022250: 312c 2031 2c20 312c 2031 2c20 312c 2031 1, 1, 1, 1, 1, 1 │ │ │ │ +00022260: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +00022270: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00022280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000222a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000222c0: 6f38 203a 204c 6973 7420 2020 2020 2020 o8 : List │ │ │ │ +000222b0: 2020 7c0a 7c6f 3820 3a20 4c69 7374 2020 |.|o8 : List │ │ │ │ +000222c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000222d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +000222e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000222f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 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2020 2020 2020 2020 2020 │ │ │ │ 00022390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000223a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000223b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000223c0: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +000223a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000223b0: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +000223c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000223d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000223e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000223f0: 7c6f 3920 3d20 286b 6b5b 5820 5d29 2020 |o9 = (kk[X ]) │ │ │ │ +000223e0: 2020 207c 0a7c 6f39 203d 2028 6b6b 5b58 |.|o9 = (kk[X │ │ │ │ +000223f0: 205d 2920 2020 2020 2020 2020 2020 2020 ]) │ │ │ │ 00022400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022420: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00022430: 2020 2020 2020 2020 3020 2020 2020 2020 0 │ │ │ │ +00022420: 7c0a 7c20 2020 2020 2020 2020 2030 2020 |.| 0 │ │ │ │ +00022430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022460: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00022450: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000224a0: 2020 2020 207c 0a7c 6f39 203a 206b 6b5b |.|o9 : kk[ │ │ │ │ -000224b0: 5820 5d2d 6d6f 6475 6c65 2c20 6672 6565 X ]-module, free │ │ │ │ +00022490: 2020 2020 2020 2020 2020 7c0a 7c6f 3920 |.|o9 │ │ │ │ +000224a0: 3a20 6b6b 5b58 205d 2d6d 6f64 756c 652c : kk[X ]-module, │ │ │ │ +000224b0: 2066 7265 6520 2020 2020 2020 2020 2020 free │ │ │ │ 000224c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000224d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000224e0: 2020 7c0a 7c20 2020 2020 2020 2020 3020 |.| 0 │ │ │ │ +000224d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000224e0: 2020 2030 2020 2020 2020 2020 2020 2020 0 │ │ │ │ 000224f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022510: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00022520: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00022510: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00022520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00022560: 3130 203a 2061 7070 6c79 2874 6f4c 6973 10 : apply(toLis │ │ │ │ -00022570: 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2020 2020 │ │ │ │ -00022610: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000225c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +000225d0: 3020 3d20 7b31 2c20 312c 2031 2c20 312c 0 = {1, 1, 1, 1, │ │ │ │ +000225e0: 2031 2c20 317d 2020 2020 2020 2020 2020 1, 1} │ │ │ │ +000225f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022600: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00022610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022650: 7c0a 7c6f 3130 203a 204c 6973 7420 2020 |.|o10 : List │ │ │ │ +00022640: 2020 2020 207c 0a7c 6f31 3020 3a20 4c69 |.|o10 : Li │ │ │ │ +00022650: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ 00022660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022680: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00022680: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00022690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000226a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000226b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000226c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 ----------+.|i11 │ │ │ │ -000226d0: 203a 2045 6f64 6420 3d20 6f64 6445 7874 : Eodd = oddExt │ │ │ │ -000226e0: 4d6f 6475 6c65 284d 3129 2020 2020 2020 Module(M1) │ │ │ │ -000226f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022700: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000226b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +000226c0: 0a7c 6931 3120 3a20 456f 6464 203d 206f .|i11 : Eodd = o │ │ │ │ +000226d0: 6464 4578 744d 6f64 756c 6528 4d31 2920 ddExtModule(M1) │ │ │ │ +000226e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000226f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00022700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022740: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00022750: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +00022730: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00022740: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +00022750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022780: 207c 0a7c 6f31 3120 3d20 286b 6b5b 5820 |.|o11 = (kk[X │ │ │ │ -00022790: 5d29 2020 2020 2020 2020 2020 2020 2020 ]) │ │ │ │ +00022770: 2020 2020 2020 7c0a 7c6f 3131 203d 2028 |.|o11 = ( │ │ │ │ +00022780: 6b6b 5b58 205d 2920 2020 2020 2020 2020 kk[X ]) │ │ │ │ +00022790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000227a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000227b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000227c0: 7c20 2020 2020 2020 2020 2020 3020 2020 | 0 │ │ │ │ +000227b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000227c0: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ 000227d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000227e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000227f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000227f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00022800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022830: 2020 2020 2020 2020 7c0a 7c6f 3131 203a |.|o11 : │ │ │ │ -00022840: 206b 6b5b 5820 5d2d 6d6f 6475 6c65 2c20 kk[X ]-module, │ │ │ │ -00022850: 6672 6565 2020 2020 2020 2020 2020 2020 free │ │ │ │ -00022860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022870: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00022880: 2020 3020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +00022820: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022830: 6f31 3120 3a20 6b6b 5b58 205d 2d6d 6f64 o11 : kk[X ]-mod │ │ │ │ +00022840: 756c 652c 2066 7265 6520 2020 2020 2020 ule, free │ │ │ │ +00022850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022860: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00022870: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ +00022880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000228a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000228b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000228a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000228b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000228c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000228d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000228e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000228f0: 0a7c 6931 3220 3a20 6170 706c 7928 746f .|i12 : apply(to │ │ │ │ -00022900: 4c69 7374 2830 2e2e 3529 2c20 692d 3e68 List(0..5), i->h │ │ │ │ -00022910: 696c 6265 7274 4675 6e63 7469 6f6e 2869 ilbertFunction(i │ │ │ │ -00022920: 2c20 456f 6464 2929 2020 2020 7c0a 7c20 , Eodd)) |.| │ │ │ │ +000228e0: 2d2d 2d2d 2b0a 7c69 3132 203a 2061 7070 ----+.|i12 : app │ │ │ │ +000228f0: 6c79 2874 6f4c 6973 7428 302e 2e35 292c ly(toList(0..5), │ │ │ │ +00022900: 2069 2d3e 6869 6c62 6572 7446 756e 6374 i->hilbertFunct │ │ │ │ +00022910: 696f 6e28 692c 2045 6f64 6429 2920 2020 ion(i, Eodd)) │ │ │ │ +00022920: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00022930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022960: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ -00022970: 3d20 7b31 2c20 312c 2031 2c20 312c 2031 = {1, 1, 1, 1, 1 │ │ │ │ -00022980: 2c20 317d 2020 2020 2020 2020 2020 2020 , 1} │ │ │ │ -00022990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00022950: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00022960: 7c6f 3132 203d 207b 312c 2031 2c20 312c |o12 = {1, 1, 1, │ │ │ │ +00022970: 2031 2c20 312c 2031 7d20 2020 2020 2020 1, 1, 1} │ │ │ │ +00022980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022990: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000229a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000229b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000229c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229e0: 2020 207c 0a7c 6f31 3220 3a20 4c69 7374 |.|o12 : List │ │ │ │ +000229d0: 2020 2020 2020 2020 7c0a 7c6f 3132 203a |.|o12 : │ │ │ │ +000229e0: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ 000229f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022a20: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00022a10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00022a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00022a60: 6931 3320 3a20 7573 6520 5320 2020 2020 i13 : use S │ │ │ │ +00022a50: 2d2d 2b0a 7c69 3133 203a 2075 7365 2053 --+.|i13 : use S │ │ │ │ +00022a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022a90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00022a80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00022a90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00022aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ad0: 2020 2020 2020 207c 0a7c 6f31 3320 3d20 |.|o13 = │ │ │ │ -00022ae0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +00022ac0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00022ad0: 3133 203d 2053 2020 2020 2020 2020 2020 13 = S │ │ │ │ +00022ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022b10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00022b00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00022b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022b50: 207c 0a7c 6f31 3320 3a20 506f 6c79 6e6f |.|o13 : Polyno │ │ │ │ -00022b60: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +00022b40: 2020 2020 2020 7c0a 7c6f 3133 203a 2050 |.|o13 : P │ │ │ │ +00022b50: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ +00022b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022b80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00022b90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00022b80: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00022b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00022bd0: 3420 3a20 4932 203d 2069 6465 616c 2278 4 : I2 = ideal"x │ │ │ │ -00022be0: 332c 797a 2220 2020 2020 2020 2020 2020 3,yz" │ │ │ │ -00022bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c00: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00022bc0: 2b0a 7c69 3134 203a 2049 3220 3d20 6964 +.|i14 : I2 = id │ │ │ │ +00022bd0: 6561 6c22 7833 2c79 7a22 2020 2020 2020 eal"x3,yz" │ │ │ │ +00022be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c40: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00022c50: 2020 2020 2020 3320 2020 2020 2020 2020 3 │ │ │ │ +00022c30: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00022c40: 2020 2020 2020 2020 2020 2033 2020 2020 3 │ │ │ │ +00022c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c80: 2020 7c0a 7c6f 3134 203d 2069 6465 616c |.|o14 = ideal │ │ │ │ -00022c90: 2028 7820 2c20 792a 7a29 2020 2020 2020 (x , y*z) │ │ │ │ +00022c70: 2020 2020 2020 207c 0a7c 6f31 3420 3d20 |.|o14 = │ │ │ │ +00022c80: 6964 6561 6c20 2878 202c 2079 2a7a 2920 ideal (x , y*z) │ │ │ │ +00022c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022cb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00022cc0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00022cb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00022cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022cf0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00022d00: 3134 203a 2049 6465 616c 206f 6620 5320 14 : Ideal of S │ │ │ │ +00022cf0: 207c 0a7c 6f31 3420 3a20 4964 6561 6c20 |.|o14 : Ideal │ │ │ │ +00022d00: 6f66 2053 2020 2020 2020 2020 2020 2020 of S │ │ │ │ 00022d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00022d20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00022d30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00022d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022d70: 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a 2052 ------+.|i15 : R │ │ │ │ -00022d80: 3220 3d20 532f 4932 2020 2020 2020 2020 2 = S/I2 │ │ │ │ +00022d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +00022d70: 3520 3a20 5232 203d 2053 2f49 3220 2020 5 : R2 = S/I2 │ │ │ │ +00022d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022db0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00022da0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00022db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022df0: 7c0a 7c6f 3135 203d 2052 3220 2020 2020 |.|o15 = R2 │ │ │ │ +00022de0: 2020 2020 207c 0a7c 6f31 3520 3d20 5232 |.|o15 = R2 │ │ │ │ +00022df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022e20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022e20: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00022e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022e60: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ -00022e70: 203a 2051 756f 7469 656e 7452 696e 6720 : QuotientRing │ │ │ │ +00022e50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00022e60: 0a7c 6f31 3520 3a20 5175 6f74 6965 6e74 .|o15 : Quotient │ │ │ │ +00022e70: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ 00022e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ea0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00022e90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00022ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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2020 2020 2020 │ │ │ │ 00022f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022f50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00022f60: 7c6f 3136 203d 2063 6f6b 6572 6e65 6c20 |o16 = cokernel │ │ │ │ -00022f70: 7c20 7832 2079 207a 207c 2020 2020 2020 | x2 y z | │ │ │ │ +00022f50: 2020 207c 0a7c 6f31 3620 3d20 636f 6b65 |.|o16 = coke │ │ │ │ +00022f60: 726e 656c 207c 2078 3220 7920 7a20 7c20 rnel | x2 y z | │ │ │ │ +00022f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022f90: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00022f90: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00022fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022fd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00022fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ff0: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ -00023000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023010: 2020 2020 207c 0a7c 6f31 3620 3a20 5232 |.|o16 : R2 │ │ │ │ -00023020: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ -00023030: 7420 6f66 2052 3220 2020 2020 2020 2020 t of R2 │ │ │ │ -00023040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023050: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00022fc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022fe0: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ +00022ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023000: 2020 2020 2020 2020 2020 7c0a 7c6f 3136 |.|o16 │ │ │ │ +00023010: 203a 2052 322d 6d6f 6475 6c65 2c20 7175 : R2-module, qu │ │ │ │ +00023020: 6f74 6965 6e74 206f 6620 5232 2020 2020 otient of R2 │ │ │ │ +00023030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023040: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00023050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00023090: 0a7c 6931 3720 3a20 6265 7474 6920 6672 .|i17 : betti fr │ │ │ │ -000230a0: 6565 5265 736f 6c75 7469 6f6e 2028 4d32 eeResolution (M2 │ │ │ │ -000230b0: 2c20 4c65 6e67 7468 4c69 6d69 7420 3d3e , LengthLimit => │ │ │ │ -000230c0: 3130 2920 2020 2020 2020 2020 7c0a 7c20 10) |.| │ │ │ │ +00023080: 2d2d 2d2d 2b0a 7c69 3137 203a 2062 6574 ----+.|i17 : bet │ │ │ │ +00023090: 7469 2066 7265 6552 6573 6f6c 7574 696f ti freeResolutio │ │ │ │ +000230a0: 6e20 284d 322c 204c 656e 6774 684c 696d n (M2, LengthLim │ │ │ │ +000230b0: 6974 203d 3e31 3029 2020 2020 2020 2020 it =>10) │ │ │ │ +000230c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000230d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000230e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000230f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023100: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00023110: 2020 2020 2020 2020 2030 2031 2032 2033 0 1 2 3 │ │ │ │ -00023120: 2034 2020 3520 2036 2020 3720 2038 2020 4 5 6 7 8 │ │ │ │ -00023130: 3920 3130 2020 2020 2020 2020 2020 2020 9 10 │ │ │ │ -00023140: 2020 2020 2020 7c0a 7c6f 3137 203d 2074 |.|o17 = t │ │ │ │ -00023150: 6f74 616c 3a20 3120 3320 3520 3720 3920 otal: 1 3 5 7 9 │ │ │ │ -00023160: 3131 2031 3320 3135 2031 3720 3139 2032 11 13 15 17 19 2 │ │ │ │ -00023170: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -00023180: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00023190: 303a 2031 2032 2032 2032 2032 2020 3220 0: 1 2 2 2 2 2 │ │ │ │ -000231a0: 2032 2020 3220 2032 2020 3220 2032 2020 2 2 2 2 2 │ │ │ │ -000231b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000231c0: 7c0a 7c20 2020 2020 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3220 2032 2020 3220 2032 2 2 2 2 2 2 │ │ │ │ +000231a0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000231b0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000231c0: 2020 313a 202e 2031 2033 2034 2034 2020 1: . 1 3 4 4 │ │ │ │ +000231d0: 3420 2034 2020 3420 2034 2020 3420 2034 4 4 4 4 4 4 │ │ │ │ +000231e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000231f0: 2020 7c0a 7c20 2020 2020 2020 2020 2032 |.| 2 │ │ │ │ +00023200: 3a20 2e20 2e20 2e20 3120 3320 2034 2020 : . . . 1 3 4 │ │ │ │ +00023210: 3420 2034 2020 3420 2034 2020 3420 2020 4 4 4 4 4 │ │ │ │ +00023220: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00023230: 0a7c 2020 2020 2020 2020 2020 333a 202e .| 3: . │ │ │ │ +00023240: 202e 202e 202e 202e 2020 3120 2033 2020 . . . . 1 3 │ │ │ │ +00023250: 3420 2034 2020 3420 2034 2020 2020 2020 4 4 4 4 │ │ │ │ +00023260: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00023270: 2020 2020 2020 2020 2034 3a20 2e20 2e20 4: . . │ │ │ │ +00023280: 2e20 2e20 2e20 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2020 │ │ │ │ 00023350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023360: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00023360: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00023370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000233a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a --------+.|i18 : │ │ │ │ -000233b0: 2045 203d 2045 7874 4d6f 6475 6c65 204d E = ExtModule M │ │ │ │ -000233c0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000233d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000233e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00023390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000233a0: 6931 3820 3a20 4520 3d20 4578 744d 6f64 i18 : E = ExtMod │ │ │ │ +000233b0: 756c 6520 4d32 2020 2020 2020 2020 2020 ule M2 │ │ │ │ +000233c0: 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+000237c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000237d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000237e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000237f0: 2020 7c0a 7c6f 3230 203d 2028 6b6b 5b58 |.|o20 = (kk[X │ │ │ │ -00023800: 202e 2e58 205d 2920 2020 2020 2020 2020 ..X ]) │ │ │ │ +000237e0: 2020 2020 2020 207c 0a7c 6f32 3020 3d20 |.|o20 = │ │ │ │ +000237f0: 286b 6b5b 5820 2e2e 5820 5d29 2020 2020 (kk[X ..X ]) │ │ │ │ +00023800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023820: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00023830: 0a7c 2020 2020 2020 2020 2020 2030 2020 .| 0 │ │ │ │ -00023840: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00023820: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00023830: 2020 3020 2020 3120 2020 2020 2020 2020 0 1 │ │ │ │ +00023840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023860: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00023860: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00023870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000238a0: 2020 2020 2020 2020 207c 0a7c 6f32 3020 |.|o20 │ │ │ │ -000238b0: 3a20 6b6b 5b58 202e 2e58 205d 2d6d 6f64 : kk[X ..X ]-mod │ │ │ │ -000238c0: 756c 652c 2066 7265 652c 2064 6567 7265 ule, free, degre │ │ │ │ -000238d0: 6573 207b 302e 2e31 2c20 323a 317d 2020 es {0..1, 2:1} │ │ │ │ -000238e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -000238f0: 2020 2030 2020 2031 2020 2020 2020 2020 0 1 │ │ │ │ +00023890: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000238a0: 7c6f 3230 203a 206b 6b5b 5820 2e2e 5820 |o20 : kk[X ..X │ │ │ │ +000238b0: 5d2d 6d6f 6475 6c65 2c20 6672 6565 2c20 ]-module, free, │ │ │ │ +000238c0: 6465 6772 6565 7320 7b30 2e2e 312c 2032 degrees {0..1, 2 │ │ │ │ +000238d0: 3a31 7d20 2020 2020 2020 207c 0a7c 2020 :1} |.| │ │ │ │ +000238e0: 2020 2020 2020 2020 3020 2020 3120 2020 0 1 │ │ │ │ +000238f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023920: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00023910: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00023920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023960: 2b0a 7c69 3231 203a 2061 7070 6c79 2874 +.|i21 : apply(t │ │ │ │ -00023970: 6f4c 6973 7428 302e 2e35 292c 2069 2d3e oList(0..5), i-> │ │ │ │ -00023980: 6869 6c62 6572 7446 756e 6374 696f 6e28 hilbertFunction( │ │ │ │ -00023990: 692c 2045 6576 656e 2929 2020 207c 0a7c i, Eeven)) |.| │ │ │ │ +00023950: 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 6170 -----+.|i21 : ap │ │ │ │ +00023960: 706c 7928 746f 4c69 7374 2830 2e2e 3529 ply(toList(0..5) │ │ │ │ +00023970: 2c20 692d 3e68 696c 6265 7274 4675 6e63 , i->hilbertFunc │ │ │ │ +00023980: 7469 6f6e 2869 2c20 4565 7665 6e29 2920 tion(i, Eeven)) │ │ │ │ +00023990: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000239a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000239b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000239c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000239d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3231 |.|o21 │ │ │ │ -000239e0: 203d 207b 312c 2035 2c20 392c 2031 332c = {1, 5, 9, 13, │ │ │ │ -000239f0: 2031 372c 2032 317d 2020 2020 2020 2020 17, 21} │ │ │ │ -00023a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a10: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000239c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000239d0: 0a7c 6f32 3120 3d20 7b31 2c20 352c 2039 .|o21 = {1, 5, 9 │ │ │ │ +000239e0: 2c20 3133 2c20 3137 2c20 3231 7d20 2020 , 13, 17, 21} │ │ │ │ +000239f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023a00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00023a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a50: 2020 2020 7c0a 7c6f 3231 203a 204c 6973 |.|o21 : Lis │ │ │ │ -00023a60: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00023a40: 2020 2020 2020 2020 207c 0a7c 6f32 3120 |.|o21 │ │ │ │ +00023a50: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +00023a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00023a80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00023a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00023ad0: 7c69 3232 203a 2045 6f64 6420 3d20 6f64 |i22 : Eodd = od │ │ │ │ -00023ae0: 6445 7874 4d6f 6475 6c65 204d 3220 2020 dExtModule M2 │ │ │ │ +00023ac0: 2d2d 2d2b 0a7c 6932 3220 3a20 456f 6464 ---+.|i22 : Eodd │ │ │ │ +00023ad0: 203d 206f 6464 4578 744d 6f64 756c 6520 = oddExtModule │ │ │ │ +00023ae0: 4d32 2020 2020 2020 2020 2020 2020 2020 M2 │ │ │ │ 00023af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b00: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00023b00: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00023b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00023b50: 2020 2020 2020 2020 2020 2020 2034 2020 4 │ │ │ │ +00023b30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00023b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023b50: 2020 3420 2020 2020 2020 2020 2020 2020 4 │ │ │ │ 00023b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b80: 2020 2020 207c 0a7c 6f32 3220 3d20 286b |.|o22 = (k │ │ │ │ -00023b90: 6b5b 5820 2e2e 5820 5d29 2020 2020 2020 k[X ..X ]) │ │ │ │ +00023b70: 2020 2020 2020 2020 2020 7c0a 7c6f 3232 |.|o22 │ │ │ │ +00023b80: 203d 2028 6b6b 5b58 202e 2e58 205d 2920 = (kk[X ..X ]) │ │ │ │ +00023b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023bc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00023bd0: 3020 2020 3120 2020 2020 2020 2020 2020 0 1 │ │ │ │ +00023bb0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00023bc0: 2020 2020 2030 2020 2031 2020 2020 2020 0 1 │ │ │ │ +00023bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023bf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00023c00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00023bf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00023c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023c30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00023c40: 3232 203a 206b 6b5b 5820 2e2e 5820 5d2d 22 : kk[X ..X ]- │ │ │ │ -00023c50: 6d6f 6475 6c65 2c20 6672 6565 2c20 6465 module, free, de │ │ │ │ -00023c60: 6772 6565 7320 7b33 3a30 2c20 317d 2020 grees {3:0, 1} │ │ │ │ -00023c70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00023c80: 2020 2020 2020 3020 2020 3120 2020 2020 0 1 │ │ │ │ +00023c30: 207c 0a7c 6f32 3220 3a20 6b6b 5b58 202e |.|o22 : kk[X . │ │ │ │ +00023c40: 2e58 205d 2d6d 6f64 756c 652c 2066 7265 .X ]-module, fre │ │ │ │ +00023c50: 652c 2064 6567 7265 6573 207b 333a 302c e, degrees {3:0, │ │ │ │ +00023c60: 2031 7d20 2020 2020 2020 2020 2020 7c0a 1} |. │ │ │ │ +00023c70: 7c20 2020 2020 2020 2020 2030 2020 2031 | 0 1 │ │ │ │ +00023c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023cb0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00023ca0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00023cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023cf0: 2d2d 2d2b 0a7c 6932 3320 3a20 6170 706c ---+.|i23 : appl │ │ │ │ -00023d00: 7928 746f 4c69 7374 2830 2e2e 3529 2c20 y(toList(0..5), │ │ │ │ -00023d10: 692d 3e68 696c 6265 7274 4675 6e63 7469 i->hilbertFuncti │ │ │ │ -00023d20: 6f6e 2869 2c20 456f 6464 2929 2020 2020 on(i, Eodd)) │ │ │ │ -00023d30: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00023ce0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a --------+.|i23 : │ │ │ │ +00023cf0: 2061 7070 6c79 2874 6f4c 6973 7428 302e apply(toList(0. │ │ │ │ +00023d00: 2e35 292c 2069 2d3e 6869 6c62 6572 7446 .5), i->hilbertF │ │ │ │ +00023d10: 756e 6374 696f 6e28 692c 2045 6f64 6429 unction(i, Eodd) │ │ │ │ +00023d20: 2920 2020 207c 0a7c 2020 2020 2020 2020 ) |.| │ │ │ │ +00023d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023d60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023d70: 6f32 3320 3d20 7b33 2c20 372c 2031 312c o23 = {3, 7, 11, │ │ │ │ -00023d80: 2031 352c 2031 392c 2032 337d 2020 2020 15, 19, 23} │ │ │ │ -00023d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023da0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00023d60: 2020 7c0a 7c6f 3233 203d 207b 332c 2037 |.|o23 = {3, 7 │ │ │ │ +00023d70: 2c20 3131 2c20 3135 2c20 3139 2c20 3233 , 11, 15, 19, 23 │ │ │ │ +00023d80: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +00023d90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00023da0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00023db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023de0: 2020 2020 2020 207c 0a7c 6f32 3320 3a20 |.|o23 : │ │ │ │ -00023df0: 4c69 7374 2020 2020 2020 2020 2020 2020 List │ │ │ │ +00023dd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00023de0: 3233 203a 204c 6973 7420 2020 2020 2020 23 : List │ │ │ │ +00023df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00023e10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00023e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023e60: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d -+..See also.=== │ │ │ │ -00023e70: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -00023e80: 2065 7665 6e45 7874 4d6f 6475 6c65 3a20 evenExtModule: │ │ │ │ -00023e90: 6576 656e 4578 744d 6f64 756c 652c 202d evenExtModule, - │ │ │ │ -00023ea0: 2d20 6576 656e 2070 6172 7420 6f66 2045 - even part of E │ │ │ │ -00023eb0: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061 xt^*(M,k) over a │ │ │ │ -00023ec0: 0a20 2020 2063 6f6d 706c 6574 6520 696e . complete in │ │ │ │ -00023ed0: 7465 7273 6563 7469 6f6e 2061 7320 6d6f tersection as mo │ │ │ │ -00023ee0: 6475 6c65 206f 7665 7220 4349 206f 7065 dule over CI ope │ │ │ │ -00023ef0: 7261 746f 7220 7269 6e67 0a20 202a 202a rator ring. * * │ │ │ │ -00023f00: 6e6f 7465 206f 6464 4578 744d 6f64 756c note oddExtModul │ │ │ │ -00023f10: 653a 206f 6464 4578 744d 6f64 756c 652c e: oddExtModule, │ │ │ │ -00023f20: 202d 2d20 6f64 6420 7061 7274 206f 6620 -- odd part of │ │ │ │ -00023f30: 4578 745e 2a28 4d2c 6b29 206f 7665 7220 Ext^*(M,k) over │ │ │ │ -00023f40: 6120 636f 6d70 6c65 7465 0a20 2020 2069 a complete. i │ │ │ │ -00023f50: 6e74 6572 7365 6374 696f 6e20 6173 206d ntersection as m │ │ │ │ -00023f60: 6f64 756c 6520 6f76 6572 2043 4920 6f70 odule over CI op │ │ │ │ -00023f70: 6572 6174 6f72 2072 696e 670a 0a57 6179 erator ring..Way │ │ │ │ -00023f80: 7320 746f 2075 7365 2045 7874 4d6f 6475 s to use ExtModu │ │ │ │ -00023f90: 6c65 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d le:.============ │ │ │ │ -00023fa0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -00023fb0: 2245 7874 4d6f 6475 6c65 284d 6f64 756c "ExtModule(Modul │ │ │ │ -00023fc0: 6529 220a 0a46 6f72 2074 6865 2070 726f e)"..For the pro │ │ │ │ -00023fd0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ -00023fe0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -00023ff0: 6f62 6a65 6374 202a 6e6f 7465 2045 7874 object *note Ext │ │ │ │ -00024000: 4d6f 6475 6c65 3a20 4578 744d 6f64 756c Module: ExtModul │ │ │ │ -00024010: 652c 2069 7320 6120 2a6e 6f74 6520 6d65 e, is a *note me │ │ │ │ -00024020: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ -00024030: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ -00024040: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +00023e50: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 ------+..See als │ │ │ │ +00023e60: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +00023e70: 2a6e 6f74 6520 6576 656e 4578 744d 6f64 *note evenExtMod │ │ │ │ +00023e80: 756c 653a 2065 7665 6e45 7874 4d6f 6475 ule: evenExtModu │ │ │ │ +00023e90: 6c65 2c20 2d2d 2065 7665 6e20 7061 7274 le, -- even part │ │ │ │ +00023ea0: 206f 6620 4578 745e 2a28 4d2c 6b29 206f of Ext^*(M,k) o │ │ │ │ +00023eb0: 7665 7220 610a 2020 2020 636f 6d70 6c65 ver a. comple │ │ │ │ +00023ec0: 7465 2069 6e74 6572 7365 6374 696f 6e20 te intersection │ │ │ │ +00023ed0: 6173 206d 6f64 756c 6520 6f76 6572 2043 as module over C │ │ │ │ +00023ee0: 4920 6f70 6572 6174 6f72 2072 696e 670a I operator ring. │ │ │ │ +00023ef0: 2020 2a20 2a6e 6f74 6520 6f64 6445 7874 * *note oddExt │ │ │ │ +00023f00: 4d6f 6475 6c65 3a20 6f64 6445 7874 4d6f Module: oddExtMo │ │ │ │ +00023f10: 6475 6c65 2c20 2d2d 206f 6464 2070 6172 dule, -- odd par │ │ │ │ +00023f20: 7420 6f66 2045 7874 5e2a 284d 2c6b 2920 t of Ext^*(M,k) │ │ │ │ +00023f30: 6f76 6572 2061 2063 6f6d 706c 6574 650a over a complete. │ │ │ │ +00023f40: 2020 2020 696e 7465 7273 6563 7469 6f6e intersection │ │ │ │ +00023f50: 2061 7320 6d6f 6475 6c65 206f 7665 7220 as module over │ │ │ │ +00023f60: 4349 206f 7065 7261 746f 7220 7269 6e67 CI operator ring │ │ │ │ +00023f70: 0a0a 5761 7973 2074 6f20 7573 6520 4578 ..Ways to use Ex │ │ │ │ +00023f80: 744d 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d tModule:.======= │ │ │ │ +00023f90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +00023fa0: 0a20 202a 2022 4578 744d 6f64 756c 6528 . * "ExtModule( │ │ │ │ +00023fb0: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468 Module)"..For th │ │ │ │ +00023fc0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +00023fd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +00023fe0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +00023ff0: 6520 4578 744d 6f64 756c 653a 2045 7874 e ExtModule: Ext │ │ │ │ +00024000: 4d6f 6475 6c65 2c20 6973 2061 202a 6e6f Module, is a *no │ │ │ │ +00024010: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ +00024020: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f on:.(Macaulay2Do │ │ │ │ +00024030: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ +00024040: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ 00024050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -000240a0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -000240b0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -000240c0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -000240d0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -000240e0: 6c61 7932 2d31 2e32 362e 3035 2b64 732f lay2-1.26.05+ds/ │ │ │ │ -000240f0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -00024100: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ -00024110: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -00024120: 7574 696f 6e73 2e6d 323a 3335 3936 3a30 utions.m2:3596:0 │ │ │ │ -00024130: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ -00024140: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ -00024150: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ -00024160: 4e6f 6465 3a20 4578 744d 6f64 756c 6544 Node: ExtModuleD │ │ │ │ -00024170: 6174 612c 204e 6578 743a 2065 7874 5673 ata, Next: extVs │ │ │ │ -00024180: 436f 686f 6d6f 6c6f 6779 2c20 5072 6576 Cohomology, Prev │ │ │ │ -00024190: 3a20 4578 744d 6f64 756c 652c 2055 703a : ExtModule, Up: │ │ │ │ -000241a0: 2054 6f70 0a0a 4578 744d 6f64 756c 6544 Top..ExtModuleD │ │ │ │ -000241b0: 6174 6120 2d2d 2045 7665 6e20 616e 6420 ata -- Even and │ │ │ │ -000241c0: 6f64 6420 4578 7420 6d6f 6475 6c65 7320 odd Ext modules │ │ │ │ -000241d0: 616e 6420 7468 6569 7220 7265 6775 6c61 and their regula │ │ │ │ -000241e0: 7269 7479 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a rity.*********** │ │ │ │ +00024090: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +000240a0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +000240b0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +000240c0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +000240d0: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ +000240e0: 352b 6473 2f4d 322f 4d61 6361 756c 6179 5+ds/M2/Macaulay │ │ │ │ +000240f0: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ +00024100: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +00024110: 5265 736f 6c75 7469 6f6e 732e 6d32 3a33 Resolutions.m2:3 │ │ │ │ +00024120: 3539 363a 302e 0a1f 0a46 696c 653a 2043 596:0....File: C │ │ │ │ +00024130: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ +00024140: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ +00024150: 6e66 6f2c 204e 6f64 653a 2045 7874 4d6f nfo, Node: ExtMo │ │ │ │ +00024160: 6475 6c65 4461 7461 2c20 4e65 7874 3a20 duleData, Next: │ │ │ │ +00024170: 6578 7456 7343 6f68 6f6d 6f6c 6f67 792c extVsCohomology, │ │ │ │ +00024180: 2050 7265 763a 2045 7874 4d6f 6475 6c65 Prev: ExtModule │ │ │ │ +00024190: 2c20 5570 3a20 546f 700a 0a45 7874 4d6f , Up: Top..ExtMo │ │ │ │ +000241a0: 6475 6c65 4461 7461 202d 2d20 4576 656e duleData -- Even │ │ │ │ +000241b0: 2061 6e64 206f 6464 2045 7874 206d 6f64 and odd Ext mod │ │ │ │ +000241c0: 756c 6573 2061 6e64 2074 6865 6972 2072 ules and their r │ │ │ │ +000241d0: 6567 756c 6172 6974 790a 2a2a 2a2a 2a2a egularity.****** │ │ │ │ +000241e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000241f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00024200: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024220: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -00024230: 0a20 2020 2020 2020 204c 203d 2045 7874 . L = Ext │ │ │ │ -00024240: 4d6f 6475 6c65 4461 7461 204d 0a20 202a ModuleData M. * │ │ │ │ -00024250: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00024260: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -00024270: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -00024280: 6329 4d6f 6475 6c65 2c2c 204d 6f64 756c c)Module,, Modul │ │ │ │ -00024290: 6520 6f76 6572 2061 2063 6f6d 706c 6574 e over a complet │ │ │ │ -000242a0: 650a 2020 2020 2020 2020 696e 7465 7273 e. inters │ │ │ │ -000242b0: 6563 7469 6f6e 2053 0a20 202a 204f 7574 ection S. * Out │ │ │ │ -000242c0: 7075 7473 3a0a 2020 2020 2020 2a20 4c2c puts:. * L, │ │ │ │ -000242d0: 2061 202a 6e6f 7465 206c 6973 743a 2028 a *note list: ( │ │ │ │ -000242e0: 4d61 6361 756c 6179 3244 6f63 294c 6973 Macaulay2Doc)Lis │ │ │ │ -000242f0: 742c 2c20 4c20 3d20 5c7b 6576 656e 4578 t,, L = \{evenEx │ │ │ │ -00024300: 744d 6f64 756c 652c 0a20 2020 2020 2020 tModule,. │ │ │ │ -00024310: 206f 6464 4578 744d 6f64 756c 652c 2072 oddExtModule, r │ │ │ │ -00024320: 6567 302c 2072 6567 315c 7d0a 0a44 6573 eg0, reg1\}..Des │ │ │ │ -00024330: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -00024340: 3d3d 3d3d 0a0a 5375 7070 6f73 6520 7468 ====..Suppose th │ │ │ │ -00024350: 6174 204d 2069 7320 6120 6d6f 6475 6c65 at M is a module │ │ │ │ -00024360: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ -00024370: 2069 6e74 6572 7365 6374 696f 6e20 5220 intersection R │ │ │ │ -00024380: 736f 2074 6861 740a 0a45 203a 3d20 4578 so that..E := Ex │ │ │ │ -00024390: 744d 6f64 756c 6520 4d0a 0a69 7320 6120 tModule M..is a │ │ │ │ -000243a0: 6d6f 6475 6c65 2067 656e 6572 6174 6564 module generated │ │ │ │ -000243b0: 2069 6e20 6465 6772 6565 7320 3e3d 3020 in degrees >=0 │ │ │ │ -000243c0: 6f76 6572 2061 2070 6f6c 796e 6f6d 6961 over a polynomia │ │ │ │ -000243d0: 6c20 7269 6e67 2054 2720 6765 6e65 7261 l ring T' genera │ │ │ │ -000243e0: 7465 6420 696e 0a64 6567 7265 6520 322c ted in.degree 2, │ │ │ │ -000243f0: 2061 6e64 0a0a 4530 203a 3d20 6576 656e and..E0 := even │ │ │ │ -00024400: 4578 744d 6f64 756c 6520 4d20 616e 6420 ExtModule M and │ │ │ │ -00024410: 4531 203a 3d20 6f64 6445 7874 4d6f 6475 E1 := oddExtModu │ │ │ │ -00024420: 6c65 204d 0a0a 6172 6520 6d6f 6475 6c65 le M..are module │ │ │ │ -00024430: 7320 6765 6e65 7261 7465 6420 696e 2064 s generated in d │ │ │ │ -00024440: 6567 7265 6520 3e3d 2030 206f 7665 7220 egree >= 0 over │ │ │ │ -00024450: 6120 706f 6c79 6e6f 6d69 616c 2072 696e a polynomial rin │ │ │ │ -00024460: 6720 5420 7769 7468 2067 656e 6572 6174 g T with generat │ │ │ │ -00024470: 6f72 730a 696e 2064 6567 7265 6520 312e ors.in degree 1. │ │ │ │ -00024480: 0a0a 5468 6520 7363 7269 7074 2072 6574 ..The script ret │ │ │ │ -00024490: 7572 6e73 0a0a 4c20 3d20 5c7b 4530 2c45 urns..L = \{E0,E │ │ │ │ -000244a0: 312c 2072 6567 756c 6172 6974 7920 4530 1, regularity E0 │ │ │ │ -000244b0: 2c20 7265 6775 6c61 7269 7479 2045 315c , regularity E1\ │ │ │ │ -000244c0: 7d0a 0a61 6e64 2070 7269 6e74 7320 6120 }..and prints a │ │ │ │ -000244d0: 6d65 7373 6167 6520 6966 207c 7265 6730 message if |reg0 │ │ │ │ -000244e0: 2d72 6567 317c 3e31 2e0a 0a49 6620 7765 -reg1|>1...If we │ │ │ │ -000244f0: 2073 6574 2072 203d 206d 6178 2832 2a72 set r = max(2*r │ │ │ │ -00024500: 6567 302c 2031 2b32 2a72 6567 3129 2c20 eg0, 1+2*reg1), │ │ │ │ -00024510: 616e 6420 4620 6973 2061 2072 6573 6f6c and F is a resol │ │ │ │ -00024520: 7574 696f 6e20 6f66 204d 2c20 7468 656e ution of M, then │ │ │ │ -00024530: 2063 6f6b 6572 0a46 2e64 645f 7b28 722b coker.F.dd_{(r+ │ │ │ │ -00024540: 3129 7d20 6973 2074 6865 2066 6972 7374 1)} is the first │ │ │ │ -00024550: 2073 7a79 6779 206d 6f64 756c 6520 6f66 szygy module of │ │ │ │ -00024560: 204d 2073 7563 6820 7468 6174 2072 6567 M such that reg │ │ │ │ -00024570: 756c 6172 6974 7920 6576 656e 4578 744d ularity evenExtM │ │ │ │ -00024580: 6f64 756c 650a 4d20 3d30 2041 4e44 2072 odule.M =0 AND r │ │ │ │ -00024590: 6567 756c 6172 6974 7920 6f64 6445 7874 egularity oddExt │ │ │ │ -000245a0: 4d6f 6475 6c65 204d 203d 300a 0a57 6520 Module M =0..We │ │ │ │ -000245b0: 6861 7665 2062 6565 6e20 7573 696e 6720 have been using │ │ │ │ -000245c0: 7265 6775 6c61 7269 7479 2045 7874 4d6f regularity ExtMo │ │ │ │ -000245d0: 6475 6c65 204d 2061 7320 6120 7375 6273 dule M as a subs │ │ │ │ -000245e0: 7469 7475 7465 2066 6f72 2072 2c20 6275 titute for r, bu │ │ │ │ -000245f0: 7420 7468 6174 2773 206e 6f74 0a61 6c77 t that's not.alw │ │ │ │ -00024600: 6179 7320 7468 6520 7361 6d65 2e0a 0a54 ays the same...T │ │ │ │ -00024610: 6865 2072 6567 756c 6172 6974 6965 7320 he regularities │ │ │ │ -00024620: 6f66 2074 6865 2065 7665 6e20 616e 6420 of the even and │ │ │ │ -00024630: 6f64 6420 4578 7420 6d6f 6475 6c65 7320 odd Ext modules │ │ │ │ -00024640: 2a63 616e 2a20 6469 6666 6572 2062 7920 *can* differ by │ │ │ │ -00024650: 6d6f 7265 2074 6861 6e20 312e 0a41 6e20 more than 1..An │ │ │ │ -00024660: 6578 616d 706c 6520 6361 6e20 6265 2070 example can be p │ │ │ │ -00024670: 726f 6475 6365 6420 7769 7468 2073 6574 roduced with set │ │ │ │ -00024680: 5261 6e64 6f6d 5365 6564 2030 2053 203d RandomSeed 0 S = │ │ │ │ -00024690: 205a 5a2f 3130 315b 612c 622c 632c 645d ZZ/101[a,b,c,d] │ │ │ │ -000246a0: 2066 660a 3d6d 6174 7269 7822 6134 2c62 ff.=matrix"a4,b │ │ │ │ -000246b0: 342c 6334 2c64 3422 2052 203d 2053 2f69 4,c4,d4" R = S/i │ │ │ │ -000246c0: 6465 616c 2066 6620 4e20 3d20 636f 6b65 deal ff N = coke │ │ │ │ -000246d0: 7220 7261 6e64 6f6d 2852 5e7b 302c 317d r random(R^{0,1} │ │ │ │ -000246e0: 2c20 525e 7b20 2d31 2c2d 322c 2d33 2c2d , R^{ -1,-2,-3,- │ │ │ │ -000246f0: 347d 290a 2d2d 6769 7665 7320 7265 6720 4}).--gives reg │ │ │ │ -00024700: 4578 745e 6576 656e 203d 2034 2c20 7265 Ext^even = 4, re │ │ │ │ -00024710: 6720 4578 745e 6f64 6420 3d20 3320 4c20 g Ext^odd = 3 L │ │ │ │ -00024720: 3d20 4578 744d 6f64 756c 6544 6174 6120 = ExtModuleData │ │ │ │ -00024730: 4e3b 2062 7574 2074 616b 6573 2073 6f6d N; but takes som │ │ │ │ -00024740: 650a 7469 6d65 2074 6f20 636f 6d70 7574 e.time to comput │ │ │ │ -00024750: 652e 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d e.....+--------- │ │ │ │ +00024210: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ +00024220: 6167 653a 200a 2020 2020 2020 2020 4c20 age: . L │ │ │ │ +00024230: 3d20 4578 744d 6f64 756c 6544 6174 6120 = ExtModuleData │ │ │ │ +00024240: 4d0a 2020 2a20 496e 7075 7473 3a0a 2020 M. * Inputs:. │ │ │ │ +00024250: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +00024260: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +00024270: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +00024280: 4d6f 6475 6c65 206f 7665 7220 6120 636f Module over a co │ │ │ │ +00024290: 6d70 6c65 7465 0a20 2020 2020 2020 2069 mplete. i │ │ │ │ +000242a0: 6e74 6572 7365 6374 696f 6e20 530a 2020 ntersection S. │ │ │ │ +000242b0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +000242c0: 202a 204c 2c20 6120 2a6e 6f74 6520 6c69 * L, a *note li │ │ │ │ +000242d0: 7374 3a20 284d 6163 6175 6c61 7932 446f st: (Macaulay2Do │ │ │ │ +000242e0: 6329 4c69 7374 2c2c 204c 203d 205c 7b65 c)List,, L = \{e │ │ │ │ +000242f0: 7665 6e45 7874 4d6f 6475 6c65 2c0a 2020 venExtModule,. │ │ │ │ +00024300: 2020 2020 2020 6f64 6445 7874 4d6f 6475 oddExtModu │ │ │ │ +00024310: 6c65 2c20 7265 6730 2c20 7265 6731 5c7d le, reg0, reg1\} │ │ │ │ +00024320: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +00024330: 3d3d 3d3d 3d3d 3d3d 3d0a 0a53 7570 706f =========..Suppo │ │ │ │ +00024340: 7365 2074 6861 7420 4d20 6973 2061 206d se that M is a m │ │ │ │ +00024350: 6f64 756c 6520 6f76 6572 2061 2063 6f6d odule over a com │ │ │ │ +00024360: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ +00024370: 6f6e 2052 2073 6f20 7468 6174 0a0a 4520 on R so that..E │ │ │ │ +00024380: 3a3d 2045 7874 4d6f 6475 6c65 204d 0a0a := ExtModule M.. │ │ │ │ +00024390: 6973 2061 206d 6f64 756c 6520 6765 6e65 is a module gene │ │ │ │ +000243a0: 7261 7465 6420 696e 2064 6567 7265 6573 rated in degrees │ │ │ │ +000243b0: 203e 3d30 206f 7665 7220 6120 706f 6c79 >=0 over a poly │ │ │ │ +000243c0: 6e6f 6d69 616c 2072 696e 6720 5427 2067 nomial ring T' g │ │ │ │ +000243d0: 656e 6572 6174 6564 2069 6e0a 6465 6772 enerated in.degr │ │ │ │ +000243e0: 6565 2032 2c20 616e 640a 0a45 3020 3a3d ee 2, and..E0 := │ │ │ │ +000243f0: 2065 7665 6e45 7874 4d6f 6475 6c65 204d evenExtModule M │ │ │ │ +00024400: 2061 6e64 2045 3120 3a3d 206f 6464 4578 and E1 := oddEx │ │ │ │ +00024410: 744d 6f64 756c 6520 4d0a 0a61 7265 206d tModule M..are m │ │ │ │ +00024420: 6f64 756c 6573 2067 656e 6572 6174 6564 odules generated │ │ │ │ +00024430: 2069 6e20 6465 6772 6565 203e 3d20 3020 in degree >= 0 │ │ │ │ +00024440: 6f76 6572 2061 2070 6f6c 796e 6f6d 6961 over a polynomia │ │ │ │ +00024450: 6c20 7269 6e67 2054 2077 6974 6820 6765 l ring T with ge │ │ │ │ +00024460: 6e65 7261 746f 7273 0a69 6e20 6465 6772 nerators.in degr │ │ │ │ +00024470: 6565 2031 2e0a 0a54 6865 2073 6372 6970 ee 1...The scrip │ │ │ │ +00024480: 7420 7265 7475 726e 730a 0a4c 203d 205c t returns..L = \ │ │ │ │ +00024490: 7b45 302c 4531 2c20 7265 6775 6c61 7269 {E0,E1, regulari │ │ │ │ +000244a0: 7479 2045 302c 2072 6567 756c 6172 6974 ty E0, regularit │ │ │ │ +000244b0: 7920 4531 5c7d 0a0a 616e 6420 7072 696e y E1\}..and prin │ │ │ │ +000244c0: 7473 2061 206d 6573 7361 6765 2069 6620 ts a message if │ │ │ │ +000244d0: 7c72 6567 302d 7265 6731 7c3e 312e 0a0a |reg0-reg1|>1... │ │ │ │ +000244e0: 4966 2077 6520 7365 7420 7220 3d20 6d61 If we set r = ma │ │ │ │ +000244f0: 7828 322a 7265 6730 2c20 312b 322a 7265 x(2*reg0, 1+2*re │ │ │ │ +00024500: 6731 292c 2061 6e64 2046 2069 7320 6120 g1), and F is a │ │ │ │ +00024510: 7265 736f 6c75 7469 6f6e 206f 6620 4d2c resolution of M, │ │ │ │ +00024520: 2074 6865 6e20 636f 6b65 720a 462e 6464 then coker.F.dd │ │ │ │ +00024530: 5f7b 2872 2b31 297d 2069 7320 7468 6520 _{(r+1)} is the │ │ │ │ +00024540: 6669 7273 7420 737a 7967 7920 6d6f 6475 first szygy modu │ │ │ │ +00024550: 6c65 206f 6620 4d20 7375 6368 2074 6861 le of M such tha │ │ │ │ +00024560: 7420 7265 6775 6c61 7269 7479 2065 7665 t regularity eve │ │ │ │ +00024570: 6e45 7874 4d6f 6475 6c65 0a4d 203d 3020 nExtModule.M =0 │ │ │ │ +00024580: 414e 4420 7265 6775 6c61 7269 7479 206f AND regularity o │ │ │ │ +00024590: 6464 4578 744d 6f64 756c 6520 4d20 3d30 ddExtModule M =0 │ │ │ │ +000245a0: 0a0a 5765 2068 6176 6520 6265 656e 2075 ..We have been u │ │ │ │ +000245b0: 7369 6e67 2072 6567 756c 6172 6974 7920 sing regularity │ │ │ │ +000245c0: 4578 744d 6f64 756c 6520 4d20 6173 2061 ExtModule M as a │ │ │ │ +000245d0: 2073 7562 7374 6974 7574 6520 666f 7220 substitute for │ │ │ │ +000245e0: 722c 2062 7574 2074 6861 7427 7320 6e6f r, but that's no │ │ │ │ +000245f0: 740a 616c 7761 7973 2074 6865 2073 616d t.always the sam │ │ │ │ +00024600: 652e 0a0a 5468 6520 7265 6775 6c61 7269 e...The regulari │ │ │ │ +00024610: 7469 6573 206f 6620 7468 6520 6576 656e ties of the even │ │ │ │ +00024620: 2061 6e64 206f 6464 2045 7874 206d 6f64 and odd Ext mod │ │ │ │ +00024630: 756c 6573 202a 6361 6e2a 2064 6966 6665 ules *can* diffe │ │ │ │ +00024640: 7220 6279 206d 6f72 6520 7468 616e 2031 r by more than 1 │ │ │ │ +00024650: 2e0a 416e 2065 7861 6d70 6c65 2063 616e ..An example can │ │ │ │ +00024660: 2062 6520 7072 6f64 7563 6564 2077 6974 be produced wit │ │ │ │ +00024670: 6820 7365 7452 616e 646f 6d53 6565 6420 h setRandomSeed │ │ │ │ +00024680: 3020 5320 3d20 5a5a 2f31 3031 5b61 2c62 0 S = ZZ/101[a,b │ │ │ │ +00024690: 2c63 2c64 5d20 6666 0a3d 6d61 7472 6978 ,c,d] ff.=matrix │ │ │ │ +000246a0: 2261 342c 6234 2c63 342c 6434 2220 5220 "a4,b4,c4,d4" R │ │ │ │ +000246b0: 3d20 532f 6964 6561 6c20 6666 204e 203d = S/ideal ff N = │ │ │ │ +000246c0: 2063 6f6b 6572 2072 616e 646f 6d28 525e coker random(R^ │ │ │ │ +000246d0: 7b30 2c31 7d2c 2052 5e7b 202d 312c 2d32 {0,1}, R^{ -1,-2 │ │ │ │ +000246e0: 2c2d 332c 2d34 7d29 0a2d 2d67 6976 6573 ,-3,-4}).--gives │ │ │ │ +000246f0: 2072 6567 2045 7874 5e65 7665 6e20 3d20 reg Ext^even = │ │ │ │ +00024700: 342c 2072 6567 2045 7874 5e6f 6464 203d 4, reg Ext^odd = │ │ │ │ +00024710: 2033 204c 203d 2045 7874 4d6f 6475 6c65 3 L = ExtModule │ │ │ │ +00024720: 4461 7461 204e 3b20 6275 7420 7461 6b65 Data N; but take │ │ │ │ +00024730: 7320 736f 6d65 0a74 696d 6520 746f 2063 s some.time to c │ │ │ │ +00024740: 6f6d 7075 7465 2e0a 0a0a 0a2b 2d2d 2d2d ompute.....+---- │ │ │ │ +00024750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00024790: 203a 2073 6574 5261 6e64 6f6d 5365 6564 : setRandomSeed │ │ │ │ -000247a0: 2031 3030 2020 2020 2020 2020 2020 2020 100 │ │ │ │ -000247b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000247c0: 2020 7c0a 7c20 2d2d 2073 6574 7469 6e67 |.| -- setting │ │ │ │ -000247d0: 2072 616e 646f 6d20 7365 6564 2074 6f20 random seed to │ │ │ │ -000247e0: 3130 3020 2020 2020 2020 2020 2020 2020 100 │ │ │ │ -000247f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00024780: 2b0a 7c69 3120 3a20 7365 7452 616e 646f +.|i1 : setRando │ │ │ │ +00024790: 6d53 6565 6420 3130 3020 2020 2020 2020 mSeed 100 │ │ │ │ +000247a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000247b0: 2020 2020 2020 207c 0a7c 202d 2d20 7365 |.| -- se │ │ │ │ +000247c0: 7474 696e 6720 7261 6e64 6f6d 2073 6565 tting random see │ │ │ │ +000247d0: 6420 746f 2031 3030 2020 2020 2020 2020 d to 100 │ │ │ │ +000247e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000247f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00024800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024830: 7c0a 7c6f 3120 3d20 3130 3020 2020 2020 |.|o1 = 100 │ │ │ │ +00024820: 2020 2020 207c 0a7c 6f31 203d 2031 3030 |.|o1 = 100 │ │ │ │ +00024830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024860: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00024850: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00024860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -000248a0: 7c69 3220 3a20 5320 3d20 5a5a 2f31 3031 |i2 : S = ZZ/101 │ │ │ │ -000248b0: 5b61 2c62 2c63 2c64 5d3b 2020 2020 2020 [a,b,c,d]; │ │ │ │ -000248c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000248d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00024890: 2d2d 2d2b 0a7c 6932 203a 2053 203d 205a ---+.|i2 : S = Z │ │ │ │ +000248a0: 5a2f 3130 315b 612c 622c 632c 645d 3b20 Z/101[a,b,c,d]; │ │ │ │ +000248b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000248c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +000248d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000248e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000248f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00024910: 3320 3a20 6620 3d20 6d61 7028 535e 312c 3 : f = map(S^1, │ │ │ │ -00024920: 2053 5e34 2c20 2869 2c6a 2920 2d3e 2053 S^4, (i,j) -> S │ │ │ │ -00024930: 5f6a 5e33 2920 2020 2020 2020 2020 2020 _j^3) │ │ │ │ -00024940: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00024900: 2d2b 0a7c 6933 203a 2066 203d 206d 6170 -+.|i3 : f = map │ │ │ │ +00024910: 2853 5e31 2c20 535e 342c 2028 692c 6a29 (S^1, S^4, (i,j) │ │ │ │ +00024920: 202d 3e20 535f 6a5e 3329 2020 2020 2020 -> S_j^3) │ │ │ │ +00024930: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00024940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024970: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ -00024980: 3d20 7c20 6133 2062 3320 6333 2064 3320 = | a3 b3 c3 d3 │ │ │ │ -00024990: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000249a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000249b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00024960: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00024970: 0a7c 6f33 203d 207c 2061 3320 6233 2063 .|o3 = | a3 b3 c │ │ │ │ +00024980: 3320 6433 207c 2020 2020 2020 2020 2020 3 d3 | │ │ │ │ +00024990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000249a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000249b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000249c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000249d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000249e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000249f0: 2020 2020 2020 2020 3120 2020 2020 2034 1 4 │ │ │ │ +000249d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000249e0: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ +000249f0: 2020 2020 3420 2020 2020 2020 2020 2020 4 │ │ │ │ 00024a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024a10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00024a20: 0a7c 6f33 203a 204d 6174 7269 7820 5320 .|o3 : Matrix S │ │ │ │ -00024a30: 203c 2d2d 2053 2020 2020 2020 2020 2020 <-- S │ │ │ │ -00024a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024a50: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00024a10: 2020 2020 7c0a 7c6f 3320 3a20 4d61 7472 |.|o3 : Matr │ │ │ │ +00024a20: 6978 2053 2020 3c2d 2d20 5320 2020 2020 ix S <-- S │ │ │ │ +00024a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024a40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00024a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00024a90: 6934 203a 2052 203d 2053 2f69 6465 616c i4 : R = S/ideal │ │ │ │ -00024aa0: 2066 3b20 2020 2020 2020 2020 2020 2020 f; │ │ │ │ -00024ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ac0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00024a80: 2d2d 2b0a 7c69 3420 3a20 5220 3d20 532f --+.|i4 : R = S/ │ │ │ │ +00024a90: 6964 6561 6c20 663b 2020 2020 2020 2020 ideal f; │ │ │ │ +00024aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024ab0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00024ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ -00024b00: 203a 204d 203d 2052 5e31 2f69 6465 616c : M = R^1/ideal │ │ │ │ -00024b10: 2261 6232 2b63 6432 223b 2020 2020 2020 "ab2+cd2"; │ │ │ │ -00024b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024b30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00024af0: 2b0a 7c69 3520 3a20 4d20 3d20 525e 312f +.|i5 : M = R^1/ │ │ │ │ +00024b00: 6964 6561 6c22 6162 322b 6364 3222 3b20 ideal"ab2+cd2"; │ │ │ │ +00024b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024b20: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00024b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024b60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ -00024b70: 2062 6574 7469 2028 4620 3d20 6672 6565 betti (F = free │ │ │ │ -00024b80: 5265 736f 6c75 7469 6f6e 284d 2c20 4c65 Resolution(M, Le │ │ │ │ -00024b90: 6e67 7468 4c69 6d69 7420 3d3e 2035 2929 ngthLimit => 5)) │ │ │ │ -00024ba0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00024b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00024b60: 7c69 3620 3a20 6265 7474 6920 2846 203d |i6 : betti (F = │ │ │ │ +00024b70: 2066 7265 6552 6573 6f6c 7574 696f 6e28 freeResolution( │ │ │ │ +00024b80: 4d2c 204c 656e 6774 684c 696d 6974 203d M, LengthLimit = │ │ │ │ +00024b90: 3e20 3529 297c 0a7c 2020 2020 2020 2020 > 5))|.| │ │ │ │ +00024ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024bd0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00024be0: 2020 2020 2020 3020 3120 3220 2033 2020 0 1 2 3 │ │ │ │ -00024bf0: 3420 2035 2020 2020 2020 2020 2020 2020 4 5 │ │ │ │ -00024c00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00024c10: 7c6f 3620 3d20 746f 7461 6c3a 2031 2031 |o6 = total: 1 1 │ │ │ │ -00024c20: 2035 2031 3620 3335 2036 3420 2020 2020 5 16 35 64 │ │ │ │ -00024c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024c40: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00024c50: 2030 3a20 3120 2e20 2e20 202e 2020 2e20 0: 1 . . . . │ │ │ │ -00024c60: 202e 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -00024c70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00024c80: 2020 2020 2020 2020 313a 202e 202e 202e 1: . . . │ │ │ │ -00024c90: 2020 2e20 202e 2020 2e20 2020 2020 2020 . . . │ │ │ │ -00024ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024cb0: 2020 207c 0a7c 2020 2020 2020 2020 2032 |.| 2 │ │ │ │ -00024cc0: 3a20 2e20 3120 2e20 202e 2020 2e20 202e : . 1 . . . . │ │ │ │ -00024cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ce0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00024cf0: 2020 2020 2020 333a 202e 202e 2031 2020 3: . . 1 │ │ │ │ -00024d00: 2e20 202e 2020 2e20 2020 2020 2020 2020 . . . │ │ │ │ -00024d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024d20: 207c 0a7c 2020 2020 2020 2020 2034 3a20 |.| 4: │ │ │ │ -00024d30: 2e20 2e20 3320 2038 2020 3520 202e 2020 . . 3 8 5 . │ │ │ │ -00024d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024d50: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00024d60: 2020 2020 353a 202e 202e 2031 2020 3820 5: . . 1 8 │ │ │ │ -00024d70: 3235 2033 3220 2020 2020 2020 2020 2020 25 32 │ │ │ │ -00024d80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00024d90: 0a7c 2020 2020 2020 2020 2036 3a20 2e20 .| 6: . │ │ │ │ -00024da0: 2e20 2e20 202e 2020 3520 3332 2020 2020 . . . 5 32 │ │ │ │ -00024db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024dc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00024bc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00024bd0: 2020 2020 2020 2020 2020 2030 2031 2032 0 1 2 │ │ │ │ +00024be0: 2020 3320 2034 2020 3520 2020 2020 2020 3 4 5 │ │ │ │ +00024bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024c00: 2020 207c 0a7c 6f36 203d 2074 6f74 616c |.|o6 = total │ │ │ │ +00024c10: 3a20 3120 3120 3520 3136 2033 3520 3634 : 1 1 5 16 35 64 │ │ │ │ +00024c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024c30: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00024c40: 2020 2020 2020 303a 2031 202e 202e 2020 0: 1 . . │ │ │ │ +00024c50: 2e20 202e 2020 2e20 2020 2020 2020 2020 . . . │ │ │ │ +00024c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024c70: 207c 0a7c 2020 2020 2020 2020 2031 3a20 |.| 1: │ │ │ │ +00024c80: 2e20 2e20 2e20 202e 2020 2e20 202e 2020 . . . . . . │ │ │ │ +00024c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024ca0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00024cb0: 2020 2020 323a 202e 2031 202e 2020 2e20 2: . 1 . . │ │ │ │ +00024cc0: 202e 2020 2e20 2020 2020 2020 2020 2020 . . │ │ │ │ +00024cd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00024ce0: 0a7c 2020 2020 2020 2020 2033 3a20 2e20 .| 3: . │ │ │ │ +00024cf0: 2e20 3120 202e 2020 2e20 202e 2020 2020 . 1 . . . │ │ │ │ +00024d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024d10: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00024d20: 2020 343a 202e 202e 2033 2020 3820 2035 4: . . 3 8 5 │ │ │ │ +00024d30: 2020 2e20 2020 2020 2020 2020 2020 2020 . │ │ │ │ +00024d40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00024d50: 2020 2020 2020 2020 2035 3a20 2e20 2e20 5: . . │ │ │ │ +00024d60: 3120 2038 2032 3520 3332 2020 2020 2020 1 8 25 32 │ │ │ │ +00024d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024d80: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00024d90: 363a 202e 202e 202e 2020 2e20 2035 2033 6: . . . . 5 3 │ │ │ │ +00024da0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00024db0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00024dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024df0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024e00: 6f36 203a 2042 6574 7469 5461 6c6c 7920 o6 : BettiTally │ │ │ │ +00024df0: 2020 7c0a 7c6f 3620 3a20 4265 7474 6954 |.|o6 : BettiT │ │ │ │ +00024e00: 616c 6c79 2020 2020 2020 2020 2020 2020 ally │ │ │ │ 00024e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024e30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00024e20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00024e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ -00024e70: 203a 2045 203d 2045 7874 4d6f 6475 6c65 : E = ExtModule │ │ │ │ -00024e80: 4461 7461 204d 3b20 2020 2020 2020 2020 Data M; │ │ │ │ -00024e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ea0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00024e60: 2b0a 7c69 3720 3a20 4520 3d20 4578 744d +.|i7 : E = ExtM │ │ │ │ +00024e70: 6f64 756c 6544 6174 6120 4d3b 2020 2020 oduleData M; │ │ │ │ +00024e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024e90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00024ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024ed0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a ---------+.|i8 : │ │ │ │ -00024ee0: 2045 5f32 2020 2020 2020 2020 2020 2020 E_2 │ │ │ │ +00024ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00024ed0: 7c69 3820 3a20 455f 3220 2020 2020 2020 |i8 : E_2 │ │ │ │ +00024ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f10: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00024f00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00024f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f40: 2020 2020 2020 207c 0a7c 6f38 203d 2032 |.|o8 = 2 │ │ │ │ +00024f30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00024f40: 3820 3d20 3220 2020 2020 2020 2020 2020 8 = 2 │ │ │ │ 00024f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00024f80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00024f70: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00024f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024fb0: 2d2d 2d2d 2d2b 0a7c 6939 203a 2045 5f33 -----+.|i9 : E_3 │ │ │ │ +00024fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 ----------+.|i9 │ │ │ │ +00024fb0: 3a20 455f 3320 2020 2020 2020 2020 2020 : E_3 │ │ │ │ 00024fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024fe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00024fe0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00024ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025020: 2020 207c 0a7c 6f39 203d 2031 2020 2020 |.|o9 = 1 │ │ │ │ +00025010: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20 |.|o9 = │ │ │ │ +00025020: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 00025030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025050: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00025040: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00025050: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00025060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025090: 2d2b 0a7c 6931 3020 3a20 7220 3d20 6d61 -+.|i10 : r = ma │ │ │ │ -000250a0: 7828 322a 455f 322c 322a 455f 332b 3129 x(2*E_2,2*E_3+1) │ │ │ │ -000250b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000250c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00025080: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2072 ------+.|i10 : r │ │ │ │ +00025090: 203d 206d 6178 2832 2a45 5f32 2c32 2a45 = max(2*E_2,2*E │ │ │ │ +000250a0: 5f33 2b31 2920 2020 2020 2020 2020 2020 _3+1) │ │ │ │ +000250b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000250c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000250d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000250e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000250f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00025100: 0a7c 6f31 3020 3d20 3420 2020 2020 2020 .|o10 = 4 │ │ │ │ +000250f0: 2020 2020 7c0a 7c6f 3130 203d 2034 2020 |.|o10 = 4 │ │ │ │ +00025100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025130: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00025120: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00025130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00025170: 6931 3120 3a20 4572 203d 2045 7874 4d6f i11 : Er = ExtMo │ │ │ │ -00025180: 6475 6c65 4461 7461 2063 6f6b 6572 2046 duleData coker F │ │ │ │ -00025190: 2e64 645f 723b 2020 2020 2020 2020 2020 .dd_r; │ │ │ │ -000251a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00025160: 2d2d 2b0a 7c69 3131 203a 2045 7220 3d20 --+.|i11 : Er = │ │ │ │ +00025170: 4578 744d 6f64 756c 6544 6174 6120 636f ExtModuleData co │ │ │ │ +00025180: 6b65 7220 462e 6464 5f72 3b20 2020 2020 ker F.dd_r; │ │ │ │ +00025190: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000251a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000251b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000251c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000251d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -000251e0: 3220 3a20 7265 6775 6c61 7269 7479 2045 2 : regularity E │ │ │ │ -000251f0: 725f 3020 2020 2020 2020 2020 2020 2020 r_0 │ │ │ │ -00025200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025210: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000251d0: 2b0a 7c69 3132 203a 2072 6567 756c 6172 +.|i12 : regular │ │ │ │ +000251e0: 6974 7920 4572 5f30 2020 2020 2020 2020 ity Er_0 │ │ │ │ +000251f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025200: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00025210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025240: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ -00025250: 3d20 3020 2020 2020 2020 2020 2020 2020 = 0 │ │ │ │ +00025230: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00025240: 7c6f 3132 203d 2030 2020 2020 2020 2020 |o12 = 0 │ │ │ │ +00025250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025280: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00025270: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00025280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000252a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000252b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 -------+.|i13 : │ │ │ │ -000252c0: 7265 6775 6c61 7269 7479 2045 725f 3120 regularity Er_1 │ │ │ │ +000252a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000252b0: 3133 203a 2072 6567 756c 6172 6974 7920 13 : regularity │ │ │ │ +000252c0: 4572 5f31 2020 2020 2020 2020 2020 2020 Er_1 │ │ │ │ 000252d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000252e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000252f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000252e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000252f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025320: 2020 2020 207c 0a7c 6f31 3320 3d20 3020 |.|o13 = 0 │ │ │ │ +00025310: 2020 2020 2020 2020 2020 7c0a 7c6f 3133 |.|o13 │ │ │ │ +00025320: 203d 2030 2020 2020 2020 2020 2020 2020 = 0 │ │ │ │ 00025330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025350: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00025350: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00025360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025390: 2d2d 2d2b 0a7c 6931 3420 3a20 7265 6775 ---+.|i14 : regu │ │ │ │ -000253a0: 6c61 7269 7479 2065 7665 6e45 7874 4d6f larity evenExtMo │ │ │ │ -000253b0: 6475 6c65 2863 6f6b 6572 2046 2e64 645f dule(coker F.dd_ │ │ │ │ -000253c0: 2872 2d31 2929 2020 2020 7c0a 7c20 2020 (r-1)) |.| │ │ │ │ +00025380: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a --------+.|i14 : │ │ │ │ +00025390: 2072 6567 756c 6172 6974 7920 6576 656e regularity even │ │ │ │ +000253a0: 4578 744d 6f64 756c 6528 636f 6b65 7220 ExtModule(coker │ │ │ │ +000253b0: 462e 6464 5f28 722d 3129 2920 2020 207c F.dd_(r-1)) | │ │ │ │ +000253c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000253d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000253e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000253f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025400: 207c 0a7c 6f31 3420 3d20 3120 2020 2020 |.|o14 = 1 │ │ │ │ +000253f0: 2020 2020 2020 7c0a 7c6f 3134 203d 2031 |.|o14 = 1 │ │ │ │ +00025400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025430: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00025420: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00025430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00025470: 0a7c 6931 3520 3a20 6666 203d 2066 2a72 .|i15 : ff = f*r │ │ │ │ -00025480: 616e 646f 6d28 736f 7572 6365 2066 2c20 andom(source f, │ │ │ │ -00025490: 736f 7572 6365 2066 293b 2020 2020 2020 source f); │ │ │ │ -000254a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025460: 2d2d 2d2d 2b0a 7c69 3135 203a 2066 6620 ----+.|i15 : ff │ │ │ │ +00025470: 3d20 662a 7261 6e64 6f6d 2873 6f75 7263 = f*random(sourc │ │ │ │ +00025480: 6520 662c 2073 6f75 7263 6520 6629 3b20 e f, source f); │ │ │ │ +00025490: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000254a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000254b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000254c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000254d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000254e0: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ -000254f0: 2020 2020 2034 2020 2020 2020 2020 2020 4 │ │ │ │ -00025500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025510: 2020 2020 7c0a 7c6f 3135 203a 204d 6174 |.|o15 : Mat │ │ │ │ -00025520: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ +000254d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000254e0: 2020 2031 2020 2020 2020 3420 2020 2020 1 4 │ │ │ │ +000254f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025500: 2020 2020 2020 2020 207c 0a7c 6f31 3520 |.|o15 │ │ │ │ +00025510: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20 : Matrix S <-- │ │ │ │ +00025520: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 00025530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025540: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00025540: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00025550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025580: 2d2d 2b0a 7c69 3136 203a 206d 6174 7269 --+.|i16 : matri │ │ │ │ -00025590: 7846 6163 746f 7269 7a61 7469 6f6e 2866 xFactorization(f │ │ │ │ -000255a0: 662c 2063 6f6b 6572 2046 2e64 645f 2872 f, coker F.dd_(r │ │ │ │ -000255b0: 2b31 2929 3b20 2020 207c 0a2b 2d2d 2d2d +1)); |.+---- │ │ │ │ +00025570: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 -------+.|i16 : │ │ │ │ +00025580: 6d61 7472 6978 4661 6374 6f72 697a 6174 matrixFactorizat │ │ │ │ +00025590: 696f 6e28 6666 2c20 636f 6b65 7220 462e ion(ff, coker F. │ │ │ │ +000255a0: 6464 5f28 722b 3129 293b 2020 2020 7c0a dd_(r+1)); |. │ │ │ │ +000255b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000255c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000255d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000255e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000255f0: 2b0a 0a54 6869 7320 7375 6363 6565 6473 +..This succeeds │ │ │ │ -00025600: 2c20 6275 7420 7765 2063 6f75 6c64 2067 , but we could g │ │ │ │ -00025610: 6574 2061 6e20 6572 726f 7220 6672 6f6d et an error from │ │ │ │ -00025620: 0a0a 6d61 7472 6978 4661 6374 6f72 697a ..matrixFactoriz │ │ │ │ -00025630: 6174 696f 6e28 6666 2c20 636f 6b65 7220 ation(ff, coker │ │ │ │ -00025640: 462e 6464 5f72 290a 0a69 6620 6f6e 6520 F.dd_r)..if one │ │ │ │ -00025650: 6f66 2074 6865 2043 4920 6f70 6572 6174 of the CI operat │ │ │ │ -00025660: 6f72 7320 7765 7265 206e 6f74 2073 7572 ors were not sur │ │ │ │ -00025670: 6a65 6374 6976 652e 0a0a 4361 7665 6174 jective...Caveat │ │ │ │ -00025680: 0a3d 3d3d 3d3d 3d0a 0a45 7874 4d6f 6475 .======..ExtModu │ │ │ │ -00025690: 6c65 2063 7265 6174 6573 2061 2072 696e le creates a rin │ │ │ │ -000256a0: 6720 696e 7369 6465 2074 6865 2073 6372 g inside the scr │ │ │ │ -000256b0: 6970 742c 2073 6f20 6966 2069 7427 7320 ipt, so if it's │ │ │ │ -000256c0: 7275 6e20 7477 6963 6520 796f 7520 6765 run twice you ge │ │ │ │ -000256d0: 740a 6d6f 6475 6c65 7320 6f76 6572 2064 t.modules over d │ │ │ │ -000256e0: 6966 6665 7265 6e74 2072 696e 6773 2e20 ifferent rings. │ │ │ │ -000256f0: 5468 6973 2073 686f 756c 6420 6265 2063 This should be c │ │ │ │ -00025700: 6861 6e67 6564 2e0a 0a53 6565 2061 6c73 hanged...See als │ │ │ │ -00025710: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ -00025720: 2a6e 6f74 6520 4578 744d 6f64 756c 653a *note ExtModule: │ │ │ │ -00025730: 2045 7874 4d6f 6475 6c65 2c20 2d2d 2045 ExtModule, -- E │ │ │ │ -00025740: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061 xt^*(M,k) over a │ │ │ │ -00025750: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -00025760: 6563 7469 6f6e 2061 730a 2020 2020 6d6f ection as. mo │ │ │ │ -00025770: 6475 6c65 206f 7665 7220 4349 206f 7065 dule over CI ope │ │ │ │ -00025780: 7261 746f 7220 7269 6e67 0a20 202a 202a rator ring. * * │ │ │ │ -00025790: 6e6f 7465 2065 7665 6e45 7874 4d6f 6475 note evenExtModu │ │ │ │ -000257a0: 6c65 3a20 6576 656e 4578 744d 6f64 756c le: evenExtModul │ │ │ │ -000257b0: 652c 202d 2d20 6576 656e 2070 6172 7420 e, -- even part │ │ │ │ -000257c0: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76 of Ext^*(M,k) ov │ │ │ │ -000257d0: 6572 2061 0a20 2020 2063 6f6d 706c 6574 er a. complet │ │ │ │ -000257e0: 6520 696e 7465 7273 6563 7469 6f6e 2061 e intersection a │ │ │ │ -000257f0: 7320 6d6f 6475 6c65 206f 7665 7220 4349 s module over CI │ │ │ │ -00025800: 206f 7065 7261 746f 7220 7269 6e67 0a20 operator ring. │ │ │ │ -00025810: 202a 202a 6e6f 7465 206f 6464 4578 744d * *note oddExtM │ │ │ │ -00025820: 6f64 756c 653a 206f 6464 4578 744d 6f64 odule: oddExtMod │ │ │ │ -00025830: 756c 652c 202d 2d20 6f64 6420 7061 7274 ule, -- odd part │ │ │ │ -00025840: 206f 6620 4578 745e 2a28 4d2c 6b29 206f of Ext^*(M,k) o │ │ │ │ -00025850: 7665 7220 6120 636f 6d70 6c65 7465 0a20 ver a complete. │ │ │ │ -00025860: 2020 2069 6e74 6572 7365 6374 696f 6e20 intersection │ │ │ │ -00025870: 6173 206d 6f64 756c 6520 6f76 6572 2043 as module over C │ │ │ │ -00025880: 4920 6f70 6572 6174 6f72 2072 696e 670a I operator ring. │ │ │ │ -00025890: 0a57 6179 7320 746f 2075 7365 2045 7874 .Ways to use Ext │ │ │ │ -000258a0: 4d6f 6475 6c65 4461 7461 3a0a 3d3d 3d3d ModuleData:.==== │ │ │ │ -000258b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000258c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245 7874 ======.. * "Ext │ │ │ │ -000258d0: 4d6f 6475 6c65 4461 7461 284d 6f64 756c ModuleData(Modul │ │ │ │ -000258e0: 6529 220a 0a46 6f72 2074 6865 2070 726f e)"..For the pro │ │ │ │ -000258f0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ -00025900: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -00025910: 6f62 6a65 6374 202a 6e6f 7465 2045 7874 object *note Ext │ │ │ │ -00025920: 4d6f 6475 6c65 4461 7461 3a20 4578 744d ModuleData: ExtM │ │ │ │ -00025930: 6f64 756c 6544 6174 612c 2069 7320 6120 oduleData, is a │ │ │ │ -00025940: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e *note method fun │ │ │ │ -00025950: 6374 696f 6e3a 0a28 4d61 6361 756c 6179 ction:.(Macaulay │ │ │ │ -00025960: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ -00025970: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ion,...--------- │ │ │ │ +000255e0: 2d2d 2d2d 2d2b 0a0a 5468 6973 2073 7563 -----+..This suc │ │ │ │ +000255f0: 6365 6564 732c 2062 7574 2077 6520 636f ceeds, but we co │ │ │ │ +00025600: 756c 6420 6765 7420 616e 2065 7272 6f72 uld get an error │ │ │ │ +00025610: 2066 726f 6d0a 0a6d 6174 7269 7846 6163 from..matrixFac │ │ │ │ +00025620: 746f 7269 7a61 7469 6f6e 2866 662c 2063 torization(ff, c │ │ │ │ +00025630: 6f6b 6572 2046 2e64 645f 7229 0a0a 6966 oker F.dd_r)..if │ │ │ │ +00025640: 206f 6e65 206f 6620 7468 6520 4349 206f one of the CI o │ │ │ │ +00025650: 7065 7261 746f 7273 2077 6572 6520 6e6f perators were no │ │ │ │ +00025660: 7420 7375 726a 6563 7469 7665 2e0a 0a43 t surjective...C │ │ │ │ +00025670: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 4578 aveat.======..Ex │ │ │ │ +00025680: 744d 6f64 756c 6520 6372 6561 7465 7320 tModule creates │ │ │ │ +00025690: 6120 7269 6e67 2069 6e73 6964 6520 7468 a ring inside th │ │ │ │ +000256a0: 6520 7363 7269 7074 2c20 736f 2069 6620 e script, so if │ │ │ │ +000256b0: 6974 2773 2072 756e 2074 7769 6365 2079 it's run twice y │ │ │ │ +000256c0: 6f75 2067 6574 0a6d 6f64 756c 6573 206f ou get.modules o │ │ │ │ +000256d0: 7665 7220 6469 6666 6572 656e 7420 7269 ver different ri │ │ │ │ +000256e0: 6e67 732e 2054 6869 7320 7368 6f75 6c64 ngs. This should │ │ │ │ +000256f0: 2062 6520 6368 616e 6765 642e 0a0a 5365 be changed...Se │ │ │ │ +00025700: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ +00025710: 0a20 202a 202a 6e6f 7465 2045 7874 4d6f . * *note ExtMo │ │ │ │ +00025720: 6475 6c65 3a20 4578 744d 6f64 756c 652c dule: ExtModule, │ │ │ │ +00025730: 202d 2d20 4578 745e 2a28 4d2c 6b29 206f -- Ext^*(M,k) o │ │ │ │ +00025740: 7665 7220 6120 636f 6d70 6c65 7465 2069 ver a complete i │ │ │ │ +00025750: 6e74 6572 7365 6374 696f 6e20 6173 0a20 ntersection as. │ │ │ │ +00025760: 2020 206d 6f64 756c 6520 6f76 6572 2043 module over C │ │ │ │ +00025770: 4920 6f70 6572 6174 6f72 2072 696e 670a I operator ring. │ │ │ │ +00025780: 2020 2a20 2a6e 6f74 6520 6576 656e 4578 * *note evenEx │ │ │ │ +00025790: 744d 6f64 756c 653a 2065 7665 6e45 7874 tModule: evenExt │ │ │ │ +000257a0: 4d6f 6475 6c65 2c20 2d2d 2065 7665 6e20 Module, -- even │ │ │ │ +000257b0: 7061 7274 206f 6620 4578 745e 2a28 4d2c part of Ext^*(M, │ │ │ │ +000257c0: 6b29 206f 7665 7220 610a 2020 2020 636f k) over a. co │ │ │ │ +000257d0: 6d70 6c65 7465 2069 6e74 6572 7365 6374 mplete intersect │ │ │ │ +000257e0: 696f 6e20 6173 206d 6f64 756c 6520 6f76 ion as module ov │ │ │ │ +000257f0: 6572 2043 4920 6f70 6572 6174 6f72 2072 er CI operator r │ │ │ │ +00025800: 696e 670a 2020 2a20 2a6e 6f74 6520 6f64 ing. * *note od │ │ │ │ +00025810: 6445 7874 4d6f 6475 6c65 3a20 6f64 6445 dExtModule: oddE │ │ │ │ +00025820: 7874 4d6f 6475 6c65 2c20 2d2d 206f 6464 xtModule, -- odd │ │ │ │ +00025830: 2070 6172 7420 6f66 2045 7874 5e2a 284d part of Ext^*(M │ │ │ │ +00025840: 2c6b 2920 6f76 6572 2061 2063 6f6d 706c ,k) over a compl │ │ │ │ +00025850: 6574 650a 2020 2020 696e 7465 7273 6563 ete. intersec │ │ │ │ +00025860: 7469 6f6e 2061 7320 6d6f 6475 6c65 206f tion as module o │ │ │ │ +00025870: 7665 7220 4349 206f 7065 7261 746f 7220 ver CI operator │ │ │ │ +00025880: 7269 6e67 0a0a 5761 7973 2074 6f20 7573 ring..Ways to us │ │ │ │ +00025890: 6520 4578 744d 6f64 756c 6544 6174 613a e ExtModuleData: │ │ │ │ +000258a0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +000258b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +000258c0: 2022 4578 744d 6f64 756c 6544 6174 6128 "ExtModuleData( │ │ │ │ +000258d0: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468 Module)"..For th │ │ │ │ +000258e0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +000258f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +00025900: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +00025910: 6520 4578 744d 6f64 756c 6544 6174 613a e ExtModuleData: │ │ │ │ +00025920: 2045 7874 4d6f 6475 6c65 4461 7461 2c20 ExtModuleData, │ │ │ │ +00025930: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ +00025940: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ +00025950: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ +00025960: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +00025970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000259a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000259b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000259c0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -000259d0: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -000259e0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -000259f0: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00025a00: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00025a10: 362e 3035 2b64 732f 4d32 2f4d 6163 6175 6.05+ds/M2/Macau │ │ │ │ -00025a20: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ -00025a30: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -00025a40: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d ionResolutions.m │ │ │ │ -00025a50: 323a 3334 3433 3a30 2e0a 1f0a 4669 6c65 2:3443:0....File │ │ │ │ -00025a60: 3a20 436f 6d70 6c65 7465 496e 7465 7273 : CompleteInters │ │ │ │ -00025a70: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -00025a80: 732e 696e 666f 2c20 4e6f 6465 3a20 6578 s.info, Node: ex │ │ │ │ -00025a90: 7456 7343 6f68 6f6d 6f6c 6f67 792c 204e tVsCohomology, N │ │ │ │ -00025aa0: 6578 743a 2066 696e 6974 6542 6574 7469 ext: finiteBetti │ │ │ │ -00025ab0: 4e75 6d62 6572 732c 2050 7265 763a 2045 Numbers, Prev: E │ │ │ │ -00025ac0: 7874 4d6f 6475 6c65 4461 7461 2c20 5570 xtModuleData, Up │ │ │ │ -00025ad0: 3a20 546f 700a 0a65 7874 5673 436f 686f : Top..extVsCoho │ │ │ │ -00025ae0: 6d6f 6c6f 6779 202d 2d20 636f 6d70 6172 mology -- compar │ │ │ │ -00025af0: 6573 2045 7874 5f53 284d 2c6b 2920 6173 es Ext_S(M,k) as │ │ │ │ -00025b00: 2065 7874 6572 696f 7220 6d6f 6475 6c65 exterior module │ │ │ │ -00025b10: 2077 6974 6820 636f 6820 7461 626c 6520 with coh table │ │ │ │ -00025b20: 6f66 2073 6865 6166 2045 7874 5f52 284d of sheaf Ext_R(M │ │ │ │ -00025b30: 2c6b 290a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ,k).************ │ │ │ │ +000259b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +000259c0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +000259d0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +000259e0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +000259f0: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00025a00: 322d 312e 3236 2e30 352b 6473 2f4d 322f 2-1.26.05+ds/M2/ │ │ │ │ +00025a10: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +00025a20: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ +00025a30: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +00025a40: 6f6e 732e 6d32 3a33 3434 333a 302e 0a1f ons.m2:3443:0... │ │ │ │ +00025a50: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ +00025a60: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +00025a70: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ +00025a80: 653a 2065 7874 5673 436f 686f 6d6f 6c6f e: extVsCohomolo │ │ │ │ +00025a90: 6779 2c20 4e65 7874 3a20 6669 6e69 7465 gy, Next: finite │ │ │ │ +00025aa0: 4265 7474 694e 756d 6265 7273 2c20 5072 BettiNumbers, Pr │ │ │ │ +00025ab0: 6576 3a20 4578 744d 6f64 756c 6544 6174 ev: ExtModuleDat │ │ │ │ +00025ac0: 612c 2055 703a 2054 6f70 0a0a 6578 7456 a, Up: Top..extV │ │ │ │ +00025ad0: 7343 6f68 6f6d 6f6c 6f67 7920 2d2d 2063 sCohomology -- c │ │ │ │ +00025ae0: 6f6d 7061 7265 7320 4578 745f 5328 4d2c ompares Ext_S(M, │ │ │ │ +00025af0: 6b29 2061 7320 6578 7465 7269 6f72 206d k) as exterior m │ │ │ │ +00025b00: 6f64 756c 6520 7769 7468 2063 6f68 2074 odule with coh t │ │ │ │ +00025b10: 6162 6c65 206f 6620 7368 6561 6620 4578 able of sheaf Ex │ │ │ │ +00025b20: 745f 5228 4d2c 6b29 0a2a 2a2a 2a2a 2a2a t_R(M,k).******* │ │ │ │ +00025b30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00025b40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00025b50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00025b60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00025b70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00025b80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00025b90: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ -00025ba0: 2020 2020 2020 2845 2c54 2920 3d20 6578 (E,T) = ex │ │ │ │ -00025bb0: 7456 7343 6f68 6f6d 6f6c 6f67 7928 6666 tVsCohomology(ff │ │ │ │ -00025bc0: 2c4e 290a 2020 2a20 496e 7075 7473 3a0a ,N). * Inputs:. │ │ │ │ -00025bd0: 2020 2020 2020 2a20 6666 2c20 6120 2a6e * ff, a *n │ │ │ │ -00025be0: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ -00025bf0: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ -00025c00: 2c2c 2072 6567 756c 6172 2073 6571 7565 ,, regular seque │ │ │ │ -00025c10: 6e63 6520 696e 2061 0a20 2020 2020 2020 nce in a. │ │ │ │ -00025c20: 2072 6567 756c 6172 2072 696e 6720 530a regular ring S. │ │ │ │ -00025c30: 2020 2020 2020 2a20 4e2c 2061 202a 6e6f * N, a *no │ │ │ │ -00025c40: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ -00025c50: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ -00025c60: 2c20 6772 6164 6564 206d 6f64 756c 6520 , graded module │ │ │ │ -00025c70: 6f76 6572 2052 203d 0a20 2020 2020 2020 over R =. │ │ │ │ -00025c80: 2053 2f69 6465 616c 2866 6629 2028 7573 S/ideal(ff) (us │ │ │ │ -00025c90: 7561 6c6c 7920 6120 6869 6768 2073 797a ually a high syz │ │ │ │ -00025ca0: 7967 7929 0a20 202a 204f 7574 7075 7473 ygy). * Outputs │ │ │ │ -00025cb0: 3a0a 2020 2020 2020 2a20 452c 2061 202a :. * E, a * │ │ │ │ -00025cc0: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -00025cd0: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -00025ce0: 652c 2c20 0a20 2020 2020 202a 2054 2c20 e,, . * T, │ │ │ │ -00025cf0: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20 a *note module: │ │ │ │ -00025d00: 284d 6163 6175 6c61 7932 446f 6329 4d6f (Macaulay2Doc)Mo │ │ │ │ -00025d10: 6475 6c65 2c2c 2045 7874 2061 6e64 2054 dule,, Ext and T │ │ │ │ -00025d20: 6f72 2061 7320 6578 7465 7269 6f72 0a20 or as exterior. │ │ │ │ -00025d30: 2020 2020 2020 206d 6f64 756c 6573 0a0a modules.. │ │ │ │ -00025d40: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -00025d50: 3d3d 3d3d 3d3d 3d0a 0a47 6976 656e 2061 =======..Given a │ │ │ │ -00025d60: 206d 6174 7269 7820 6666 2063 6f6e 7461 matrix ff conta │ │ │ │ -00025d70: 696e 696e 6720 6120 7265 6775 6c61 7220 ining a regular │ │ │ │ -00025d80: 7365 7175 656e 6365 2069 6e20 6120 706f sequence in a po │ │ │ │ -00025d90: 6c79 6e6f 6d69 616c 2072 696e 6720 5320 lynomial ring S │ │ │ │ -00025da0: 6f76 6572 206b 2c0a 7365 7420 5220 3d20 over k,.set R = │ │ │ │ -00025db0: 532f 2869 6465 616c 2066 6629 2e20 4966 S/(ideal ff). If │ │ │ │ -00025dc0: 204e 2069 7320 6120 6772 6164 6564 2052 N is a graded R │ │ │ │ -00025dd0: 2d6d 6f64 756c 652c 2061 6e64 204d 2069 -module, and M i │ │ │ │ -00025de0: 7320 7468 6520 6d6f 6475 6c65 204e 2072 s the module N r │ │ │ │ -00025df0: 6567 6172 6465 640a 6173 2061 6e20 532d egarded.as an S- │ │ │ │ -00025e00: 6d6f 6475 6c65 2c20 7468 6520 7363 7269 module, the scri │ │ │ │ -00025e10: 7074 2072 6574 7572 6e73 2045 203d 2045 pt returns E = E │ │ │ │ -00025e20: 7874 5f53 284d 2c6b 2920 616e 6420 5420 xt_S(M,k) and T │ │ │ │ -00025e30: 3d20 546f 725e 5328 4d2c 6b29 2061 7320 = Tor^S(M,k) as │ │ │ │ -00025e40: 6d6f 6475 6c65 730a 6f76 6572 2061 6e20 modules.over an │ │ │ │ -00025e50: 6578 7465 7269 6f72 2061 6c67 6562 7261 exterior algebra │ │ │ │ -00025e60: 2e0a 0a54 6865 2073 6372 6970 7420 7072 ...The script pr │ │ │ │ -00025e70: 696e 7473 2074 6865 2054 6174 6520 7265 ints the Tate re │ │ │ │ -00025e80: 736f 6c75 7469 6f6e 206f 6620 453b 2061 solution of E; a │ │ │ │ -00025e90: 6e64 2074 6865 2063 6f68 6f6d 6f6c 6f67 nd the cohomolog │ │ │ │ -00025ea0: 7920 7461 626c 6520 6f66 2074 6865 0a73 y table of the.s │ │ │ │ -00025eb0: 6865 6166 2061 7373 6f63 6961 7465 6420 heaf associated │ │ │ │ -00025ec0: 746f 2045 7874 5f52 284e 2c6b 2920 6f76 to Ext_R(N,k) ov │ │ │ │ -00025ed0: 6572 2074 6865 2072 696e 6720 6f66 2043 er the ring of C │ │ │ │ -00025ee0: 4920 6f70 6572 6174 6f72 732c 2077 6869 I operators, whi │ │ │ │ -00025ef0: 6368 2069 7320 610a 706f 6c79 6e6f 6d69 ch is a.polynomi │ │ │ │ -00025f00: 616c 2072 696e 6720 6f76 6572 206b 206f al ring over k o │ │ │ │ -00025f10: 6e20 6320 7661 7269 6162 6c65 732e 0a0a n c variables... │ │ │ │ -00025f20: 5468 6520 6f75 7470 7574 2063 616e 2062 The output can b │ │ │ │ -00025f30: 6520 7573 6564 2074 6f20 2873 6f6d 6574 e used to (somet │ │ │ │ -00025f40: 696d 6573 2920 6368 6563 6b20 7768 6574 imes) check whet │ │ │ │ -00025f50: 6865 7220 7468 6520 7375 626d 6f64 756c her the submodul │ │ │ │ -00025f60: 6520 6f66 2045 7874 5f53 284d 2c6b 290a e of Ext_S(M,k). │ │ │ │ -00025f70: 6765 6e65 7261 7465 6420 696e 2064 6567 generated in deg │ │ │ │ -00025f80: 7265 6520 3020 7370 6c69 7473 2028 6173 ree 0 splits (as │ │ │ │ -00025f90: 2061 6e20 6578 7465 7269 6f72 206d 6f64 an exterior mod │ │ │ │ -00025fa0: 756c 650a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ule..+---------- │ │ │ │ +00025b80: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ +00025b90: 3a20 0a20 2020 2020 2020 2028 452c 5429 : . (E,T) │ │ │ │ +00025ba0: 203d 2065 7874 5673 436f 686f 6d6f 6c6f = extVsCohomolo │ │ │ │ +00025bb0: 6779 2866 662c 4e29 0a20 202a 2049 6e70 gy(ff,N). * Inp │ │ │ │ +00025bc0: 7574 733a 0a20 2020 2020 202a 2066 662c uts:. * ff, │ │ │ │ +00025bd0: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ +00025be0: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +00025bf0: 6174 7269 782c 2c20 7265 6775 6c61 7220 atrix,, regular │ │ │ │ +00025c00: 7365 7175 656e 6365 2069 6e20 610a 2020 sequence in a. │ │ │ │ +00025c10: 2020 2020 2020 7265 6775 6c61 7220 7269 regular ri │ │ │ │ +00025c20: 6e67 2053 0a20 2020 2020 202a 204e 2c20 ng S. * N, │ │ │ │ +00025c30: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20 a *note module: │ │ │ │ +00025c40: 284d 6163 6175 6c61 7932 446f 6329 4d6f (Macaulay2Doc)Mo │ │ │ │ +00025c50: 6475 6c65 2c2c 2067 7261 6465 6420 6d6f dule,, graded mo │ │ │ │ +00025c60: 6475 6c65 206f 7665 7220 5220 3d0a 2020 dule over R =. │ │ │ │ +00025c70: 2020 2020 2020 532f 6964 6561 6c28 6666 S/ideal(ff │ │ │ │ +00025c80: 2920 2875 7375 616c 6c79 2061 2068 6967 ) (usually a hig │ │ │ │ +00025c90: 6820 7379 7a79 6779 290a 2020 2a20 4f75 h syzygy). * Ou │ │ │ │ +00025ca0: 7470 7574 733a 0a20 2020 2020 202a 2045 tputs:. * E │ │ │ │ +00025cb0: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +00025cc0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00025cd0: 4d6f 6475 6c65 2c2c 200a 2020 2020 2020 Module,, . │ │ │ │ +00025ce0: 2a20 542c 2061 202a 6e6f 7465 206d 6f64 * T, a *note mod │ │ │ │ +00025cf0: 756c 653a 2028 4d61 6361 756c 6179 3244 ule: (Macaulay2D │ │ │ │ +00025d00: 6f63 294d 6f64 756c 652c 2c20 4578 7420 oc)Module,, Ext │ │ │ │ +00025d10: 616e 6420 546f 7220 6173 2065 7874 6572 and Tor as exter │ │ │ │ +00025d20: 696f 720a 2020 2020 2020 2020 6d6f 6475 ior. modu │ │ │ │ +00025d30: 6c65 730a 0a44 6573 6372 6970 7469 6f6e les..Description │ │ │ │ +00025d40: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4769 .===========..Gi │ │ │ │ +00025d50: 7665 6e20 6120 6d61 7472 6978 2066 6620 ven a matrix ff │ │ │ │ +00025d60: 636f 6e74 6169 6e69 6e67 2061 2072 6567 containing a reg │ │ │ │ +00025d70: 756c 6172 2073 6571 7565 6e63 6520 696e ular sequence in │ │ │ │ +00025d80: 2061 2070 6f6c 796e 6f6d 6961 6c20 7269 a polynomial ri │ │ │ │ +00025d90: 6e67 2053 206f 7665 7220 6b2c 0a73 6574 ng S over k,.set │ │ │ │ +00025da0: 2052 203d 2053 2f28 6964 6561 6c20 6666 R = S/(ideal ff │ │ │ │ +00025db0: 292e 2049 6620 4e20 6973 2061 2067 7261 ). If N is a gra │ │ │ │ +00025dc0: 6465 6420 522d 6d6f 6475 6c65 2c20 616e ded R-module, an │ │ │ │ +00025dd0: 6420 4d20 6973 2074 6865 206d 6f64 756c d M is the modul │ │ │ │ +00025de0: 6520 4e20 7265 6761 7264 6564 0a61 7320 e N regarded.as │ │ │ │ +00025df0: 616e 2053 2d6d 6f64 756c 652c 2074 6865 an S-module, the │ │ │ │ +00025e00: 2073 6372 6970 7420 7265 7475 726e 7320 script returns │ │ │ │ +00025e10: 4520 3d20 4578 745f 5328 4d2c 6b29 2061 E = Ext_S(M,k) a │ │ │ │ +00025e20: 6e64 2054 203d 2054 6f72 5e53 284d 2c6b nd T = Tor^S(M,k │ │ │ │ +00025e30: 2920 6173 206d 6f64 756c 6573 0a6f 7665 ) as modules.ove │ │ │ │ +00025e40: 7220 616e 2065 7874 6572 696f 7220 616c r an exterior al │ │ │ │ +00025e50: 6765 6272 612e 0a0a 5468 6520 7363 7269 gebra...The scri │ │ │ │ +00025e60: 7074 2070 7269 6e74 7320 7468 6520 5461 pt prints the Ta │ │ │ │ +00025e70: 7465 2072 6573 6f6c 7574 696f 6e20 6f66 te resolution of │ │ │ │ +00025e80: 2045 3b20 616e 6420 7468 6520 636f 686f E; and the coho │ │ │ │ +00025e90: 6d6f 6c6f 6779 2074 6162 6c65 206f 6620 mology table of │ │ │ │ +00025ea0: 7468 650a 7368 6561 6620 6173 736f 6369 the.sheaf associ │ │ │ │ +00025eb0: 6174 6564 2074 6f20 4578 745f 5228 4e2c ated to Ext_R(N, │ │ │ │ +00025ec0: 6b29 206f 7665 7220 7468 6520 7269 6e67 k) over the ring │ │ │ │ +00025ed0: 206f 6620 4349 206f 7065 7261 746f 7273 of CI operators │ │ │ │ +00025ee0: 2c20 7768 6963 6820 6973 2061 0a70 6f6c , which is a.pol │ │ │ │ +00025ef0: 796e 6f6d 6961 6c20 7269 6e67 206f 7665 ynomial ring ove │ │ │ │ +00025f00: 7220 6b20 6f6e 2063 2076 6172 6961 626c r k on c variabl │ │ │ │ +00025f10: 6573 2e0a 0a54 6865 206f 7574 7075 7420 es...The output │ │ │ │ +00025f20: 6361 6e20 6265 2075 7365 6420 746f 2028 can be used to ( │ │ │ │ +00025f30: 736f 6d65 7469 6d65 7329 2063 6865 636b sometimes) check │ │ │ │ +00025f40: 2077 6865 7468 6572 2074 6865 2073 7562 whether the sub │ │ │ │ +00025f50: 6d6f 6475 6c65 206f 6620 4578 745f 5328 module of Ext_S( │ │ │ │ +00025f60: 4d2c 6b29 0a67 656e 6572 6174 6564 2069 M,k).generated i │ │ │ │ +00025f70: 6e20 6465 6772 6565 2030 2073 706c 6974 n degree 0 split │ │ │ │ +00025f80: 7320 2861 7320 616e 2065 7874 6572 696f s (as an exterio │ │ │ │ +00025f90: 7220 6d6f 6475 6c65 0a0a 2b2d 2d2d 2d2d r module..+----- │ │ │ │ +00025fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025fe0: 2d2d 2d2b 0a7c 6931 203a 2053 203d 205a ---+.|i1 : S = Z │ │ │ │ -00025ff0: 5a2f 3130 315b 612c 622c 635d 2020 2020 Z/101[a,b,c] │ │ │ │ +00025fd0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +00025fe0: 5320 3d20 5a5a 2f31 3031 5b61 2c62 2c63 S = ZZ/101[a,b,c │ │ │ │ +00025ff0: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ 00026000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026020: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026010: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00026020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026060: 2020 207c 0a7c 6f31 203d 2053 2020 2020 |.|o1 = S │ │ │ │ +00026050: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ +00026060: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 00026070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000260a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026090: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000260a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000260b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000260c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000260d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000260e0: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e |.|o1 : Polyn │ │ │ │ -000260f0: 6f6d 6961 6c52 696e 6720 2020 2020 2020 omialRing │ │ │ │ +000260d0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ +000260e0: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ +000260f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026120: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00026110: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00026120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026160: 2d2d 2d2b 0a7c 6932 203a 2066 6620 3d20 ---+.|i2 : ff = │ │ │ │ -00026170: 6d61 7472 6978 2022 6132 2c62 322c 6332 matrix "a2,b2,c2 │ │ │ │ -00026180: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ -00026190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000261a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026150: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ +00026160: 6666 203d 206d 6174 7269 7820 2261 322c ff = matrix "a2, │ │ │ │ +00026170: 6232 2c63 3222 2020 2020 2020 2020 2020 b2,c2" │ │ │ │ +00026180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026190: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000261a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000261b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000261c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000261d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000261e0: 2020 207c 0a7c 6f32 203d 207c 2061 3220 |.|o2 = | a2 │ │ │ │ -000261f0: 6232 2063 3220 7c20 2020 2020 2020 2020 b2 c2 | │ │ │ │ +000261d0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ +000261e0: 7c20 6132 2062 3220 6332 207c 2020 2020 | a2 b2 c2 | │ │ │ │ +000261f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026220: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026210: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00026220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026260: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00026270: 2020 2031 2020 2020 2020 3320 2020 2020 1 3 │ │ │ │ +00026250: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00026260: 2020 2020 2020 2020 3120 2020 2020 2033 1 3 │ │ │ │ +00026270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000262a0: 2020 207c 0a7c 6f32 203a 204d 6174 7269 |.|o2 : Matri │ │ │ │ -000262b0: 7820 5320 203c 2d2d 2053 2020 2020 2020 x S <-- S │ │ │ │ +00026290: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ +000262a0: 4d61 7472 6978 2053 2020 3c2d 2d20 5320 Matrix S <-- S │ │ │ │ +000262b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000262c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000262d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000262e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +000262d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000262e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000262f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026320: 2d2d 2d2b 0a7c 6933 203a 2052 203d 2053 ---+.|i3 : R = S │ │ │ │ -00026330: 2f28 6964 6561 6c20 6666 2920 2020 2020 /(ideal ff) │ │ │ │ +00026310: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ +00026320: 5220 3d20 532f 2869 6465 616c 2066 6629 R = S/(ideal ff) │ │ │ │ +00026330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026360: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026350: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00026360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000263a0: 2020 207c 0a7c 6f33 203d 2052 2020 2020 |.|o3 = R │ │ │ │ +00026390: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ +000263a0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 000263b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000263c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000263d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000263e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000263d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000263e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000263f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026420: 2020 207c 0a7c 6f33 203a 2051 756f 7469 |.|o3 : Quoti │ │ │ │ -00026430: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ +00026410: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ +00026420: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +00026430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026460: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00026450: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00026460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000264a0: 2d2d 2d2b 0a7c 6934 203a 204e 203d 2068 ---+.|i4 : N = h │ │ │ │ -000264b0: 6967 6853 797a 7967 7928 525e 312f 6964 ighSyzygy(R^1/id │ │ │ │ -000264c0: 6561 6c28 612a 622c 6329 2920 2020 2020 eal(a*b,c)) │ │ │ │ -000264d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000264e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026490: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ +000264a0: 4e20 3d20 6869 6768 5379 7a79 6779 2852 N = highSyzygy(R │ │ │ │ +000264b0: 5e31 2f69 6465 616c 2861 2a62 2c63 2929 ^1/ideal(a*b,c)) │ │ │ │ +000264c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000264d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000264e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000264f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026520: 2020 207c 0a7c 6f34 203d 2063 6f6b 6572 |.|o4 = coker │ │ │ │ -00026530: 6e65 6c20 7b34 7d20 7c20 6320 2d61 6220 nel {4} | c -ab │ │ │ │ -00026540: 3020 3020 3020 2030 2020 3020 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -00026550: 2030 2030 2020 3020 3020 2030 2020 3020 0 0 0 0 0 0 │ │ │ │ -00026560: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -00026570: 2020 2020 7b35 7d20 7c20 3020 6320 2020 {5} | 0 c │ │ │ │ -00026580: 6220 6120 3020 2030 2020 3020 2030 2030 b a 0 0 0 0 0 │ │ │ │ -00026590: 2030 2030 2020 3020 3020 2030 2020 3020 0 0 0 0 0 0 │ │ │ │ -000265a0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -000265b0: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -000265c0: 6320 3020 2d62 2061 2020 3020 2030 2030 c 0 -b a 0 0 0 │ │ │ │ -000265d0: 2030 2030 2020 3020 3020 2030 2020 3020 0 0 0 0 0 0 │ │ │ │ -000265e0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -000265f0: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -00026600: 3020 6320 3020 202d 6220 2d61 2030 2030 0 c 0 -b -a 0 0 │ │ │ │ -00026610: 2030 2030 2020 3020 3020 2030 2020 3020 0 0 0 0 0 0 │ │ │ │ -00026620: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -00026630: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -00026640: 3020 3020 6320 2030 2020 3020 2062 2061 0 0 c 0 0 b a │ │ │ │ -00026650: 2030 2030 2020 3020 3020 2030 2020 3020 0 0 0 0 0 0 │ │ │ │ -00026660: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -00026670: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -00026680: 3020 3020 3020 2063 2020 3020 2030 2062 0 0 0 c 0 0 b │ │ │ │ -00026690: 2030 2030 2020 3020 2d61 2030 2020 3020 0 0 0 -a 0 0 │ │ │ │ -000266a0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -000266b0: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -000266c0: 3020 3020 3020 2030 2020 6320 2030 2030 0 0 0 0 c 0 0 │ │ │ │ -000266d0: 2030 2030 2020 3020 6220 2030 2020 6120 0 0 0 b 0 a │ │ │ │ -000266e0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -000266f0: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -00026700: 3020 3020 3020 2030 2020 3020 2063 2030 0 0 0 0 0 c 0 │ │ │ │ -00026710: 2062 202d 6120 3020 3020 2030 2020 3020 b -a 0 0 0 0 │ │ │ │ -00026720: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -00026730: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -00026740: 3020 3020 3020 2030 2020 3020 2030 2063 0 0 0 0 0 0 c │ │ │ │ -00026750: 2030 2062 2020 6120 3020 2030 2020 3020 0 b a 0 0 0 │ │ │ │ -00026760: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -00026770: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -00026780: 3020 3020 3020 2030 2020 3020 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -00026790: 2030 2030 2020 6220 6320 202d 6120 3020 0 0 b c -a 0 │ │ │ │ -000267a0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -000267b0: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -000267c0: 3020 3020 3020 2030 2020 3020 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -000267d0: 2030 2030 2020 3020 3020 2062 2020 6320 0 0 0 0 b c │ │ │ │ -000267e0: 6120 7c7c 0a7c 2020 2020 2020 2020 2020 a ||.| │ │ │ │ +00026510: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ +00026520: 636f 6b65 726e 656c 207b 347d 207c 2063 cokernel {4} | c │ │ │ │ +00026530: 202d 6162 2030 2030 2030 2020 3020 2030 -ab 0 0 0 0 0 │ │ │ │ +00026540: 2020 3020 3020 3020 3020 2030 2030 2020 0 0 0 0 0 0 │ │ │ │ +00026550: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +00026560: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +00026570: 2063 2020 2062 2061 2030 2020 3020 2030 c b a 0 0 0 │ │ │ │ +00026580: 2020 3020 3020 3020 3020 2030 2030 2020 0 0 0 0 0 0 │ │ │ │ +00026590: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +000265a0: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +000265b0: 2030 2020 2063 2030 202d 6220 6120 2030 0 c 0 -b a 0 │ │ │ │ +000265c0: 2020 3020 3020 3020 3020 2030 2030 2020 0 0 0 0 0 0 │ │ │ │ +000265d0: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +000265e0: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +000265f0: 2030 2020 2030 2063 2030 2020 2d62 202d 0 0 c 0 -b - │ │ │ │ +00026600: 6120 3020 3020 3020 3020 2030 2030 2020 a 0 0 0 0 0 0 │ │ │ │ +00026610: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +00026620: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +00026630: 2030 2020 2030 2030 2063 2020 3020 2030 0 0 0 c 0 0 │ │ │ │ +00026640: 2020 6220 6120 3020 3020 2030 2030 2020 b a 0 0 0 0 │ │ │ │ +00026650: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +00026660: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +00026670: 2030 2020 2030 2030 2030 2020 6320 2030 0 0 0 0 c 0 │ │ │ │ +00026680: 2020 3020 6220 3020 3020 2030 202d 6120 0 b 0 0 0 -a │ │ │ │ +00026690: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +000266a0: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +000266b0: 2030 2020 2030 2030 2030 2020 3020 2063 0 0 0 0 0 c │ │ │ │ +000266c0: 2020 3020 3020 3020 3020 2030 2062 2020 0 0 0 0 0 b │ │ │ │ +000266d0: 3020 2061 2030 207c 7c0a 7c20 2020 2020 0 a 0 ||.| │ │ │ │ +000266e0: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +000266f0: 2030 2020 2030 2030 2030 2020 3020 2030 0 0 0 0 0 0 │ │ │ │ +00026700: 2020 6320 3020 6220 2d61 2030 2030 2020 c 0 b -a 0 0 │ │ │ │ +00026710: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +00026720: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +00026730: 2030 2020 2030 2030 2030 2020 3020 2030 0 0 0 0 0 0 │ │ │ │ +00026740: 2020 3020 6320 3020 6220 2061 2030 2020 0 c 0 b a 0 │ │ │ │ +00026750: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +00026760: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +00026770: 2030 2020 2030 2030 2030 2020 3020 2030 0 0 0 0 0 0 │ │ │ │ +00026780: 2020 3020 3020 3020 3020 2062 2063 2020 0 0 0 0 b c │ │ │ │ +00026790: 2d61 2030 2030 207c 7c0a 7c20 2020 2020 -a 0 0 ||.| │ │ │ │ +000267a0: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +000267b0: 2030 2020 2030 2030 2030 2020 3020 2030 0 0 0 0 0 0 │ │ │ │ +000267c0: 2020 3020 3020 3020 3020 2030 2030 2020 0 0 0 0 0 0 │ │ │ │ +000267d0: 6220 2063 2061 207c 7c0a 7c20 2020 2020 b c a ||.| │ │ │ │ +000267e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000267f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026820: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00026830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026840: 2020 3131 2020 2020 2020 2020 2020 2020 11 │ │ │ │ -00026850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026860: 2020 207c 0a7c 6f34 203a 2052 2d6d 6f64 |.|o4 : R-mod │ │ │ │ -00026870: 756c 652c 2071 756f 7469 656e 7420 6f66 ule, quotient of │ │ │ │ -00026880: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ -00026890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000268a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00026810: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00026820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026830: 2020 2020 2020 2031 3120 2020 2020 2020 11 │ │ │ │ +00026840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026850: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ +00026860: 522d 6d6f 6475 6c65 2c20 7175 6f74 6965 R-module, quotie │ │ │ │ +00026870: 6e74 206f 6620 5220 2020 2020 2020 2020 nt of R │ │ │ │ +00026880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026890: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000268a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000268b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000268c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000268d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000268e0: 2d2d 2d2b 0a7c 6935 203a 2045 203d 2065 ---+.|i5 : E = e │ │ │ │ -000268f0: 7874 5673 436f 686f 6d6f 6c6f 6779 2866 xtVsCohomology(f │ │ │ │ -00026900: 662c 6869 6768 5379 7a79 6779 204e 293b f,highSyzygy N); │ │ │ │ -00026910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026920: 2020 207c 0a7c 5461 7465 2052 6573 6f6c |.|Tate Resol │ │ │ │ -00026930: 7574 696f 6e20 6f66 2045 7874 5f53 284d ution of Ext_S(M │ │ │ │ -00026940: 2c6b 2920 6173 2065 7874 6572 696f 7220 ,k) as exterior │ │ │ │ -00026950: 6d6f 6475 6c65 3a20 2020 2020 2020 2020 module: │ │ │ │ -00026960: 2020 207c 0a7c 4e6f 7465 2074 6861 7420 |.|Note that │ │ │ │ -00026970: 6d61 7073 2067 6f20 6c65 6674 2074 6f20 maps go left to │ │ │ │ -00026980: 7269 6768 7420 2020 2020 2020 2020 2020 right │ │ │ │ -00026990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000269a0: 2020 207c 0a7c 2020 2020 2020 202d 3131 |.| -11 │ │ │ │ -000269b0: 202d 3130 2020 2d39 202d 3820 2d37 202d -10 -9 -8 -7 - │ │ │ │ -000269c0: 3620 2d35 202d 3420 2d33 202d 3220 202d 6 -5 -4 -3 -2 - │ │ │ │ -000269d0: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -000269e0: 2020 207c 0a7c 746f 7461 6c3a 2031 3938 |.|total: 198 │ │ │ │ -000269f0: 2031 3436 2031 3032 2036 3620 3338 2031 146 102 66 38 1 │ │ │ │ -00026a00: 3820 2039 2031 3620 3336 2036 3420 3130 8 9 16 36 64 10 │ │ │ │ -00026a10: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ -00026a20: 2020 207c 0a7c 2020 2020 383a 2031 3036 |.| 8: 106 │ │ │ │ -00026a30: 2020 3739 2020 3536 2033 3720 3232 2031 79 56 37 22 1 │ │ │ │ -00026a40: 3120 2034 2020 3120 2031 2020 3120 2020 1 4 1 1 1 │ │ │ │ -00026a50: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -00026a60: 2020 207c 0a7c 2020 2020 393a 2020 3932 |.| 9: 92 │ │ │ │ -00026a70: 2020 3637 2020 3436 2032 3920 3136 2020 67 46 29 16 │ │ │ │ -00026a80: 3720 2032 2020 2e20 202e 2020 2e20 2020 7 2 . . . │ │ │ │ -00026a90: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -00026aa0: 2020 207c 0a7c 2020 2031 303a 2020 202e |.| 10: . │ │ │ │ -00026ab0: 2020 202e 2020 202e 2020 2e20 202e 2020 . . . . │ │ │ │ -00026ac0: 2e20 202e 2020 3520 3134 2032 3720 2034 . . 5 14 27 4 │ │ │ │ -00026ad0: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -00026ae0: 2020 207c 0a7c 2020 2031 313a 2020 202e |.| 11: . │ │ │ │ -00026af0: 2020 202e 2020 202e 2020 2e20 202e 2020 . . . . │ │ │ │ -00026b00: 2e20 2033 2031 3020 3231 2033 3620 2035 . 3 10 21 36 5 │ │ │ │ -00026b10: 3520 2020 2020 2020 2020 2020 2020 2020 5 │ │ │ │ -00026b20: 2020 207c 0a7c 2d2d 2d20 2020 2020 2020 |.|--- │ │ │ │ +000268d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ +000268e0: 4520 3d20 6578 7456 7343 6f68 6f6d 6f6c E = extVsCohomol │ │ │ │ +000268f0: 6f67 7928 6666 2c68 6967 6853 797a 7967 ogy(ff,highSyzyg │ │ │ │ +00026900: 7920 4e29 3b20 2020 2020 2020 2020 2020 y N); │ │ │ │ +00026910: 2020 2020 2020 2020 7c0a 7c54 6174 6520 |.|Tate │ │ │ │ +00026920: 5265 736f 6c75 7469 6f6e 206f 6620 4578 Resolution of Ex │ │ │ │ +00026930: 745f 5328 4d2c 6b29 2061 7320 6578 7465 t_S(M,k) as exte │ │ │ │ +00026940: 7269 6f72 206d 6f64 756c 653a 2020 2020 rior module: │ │ │ │ +00026950: 2020 2020 2020 2020 7c0a 7c4e 6f74 6520 |.|Note │ │ │ │ +00026960: 7468 6174 206d 6170 7320 676f 206c 6566 that maps go lef │ │ │ │ +00026970: 7420 746f 2072 6967 6874 2020 2020 2020 t to right │ │ │ │ +00026980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026990: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000269a0: 2020 2d31 3120 2d31 3020 202d 3920 2d38 -11 -10 -9 -8 │ │ │ │ +000269b0: 202d 3720 2d36 202d 3520 2d34 202d 3320 -7 -6 -5 -4 -3 │ │ │ │ +000269c0: 2d32 2020 2d31 2020 2020 2020 2020 2020 -2 -1 │ │ │ │ +000269d0: 2020 2020 2020 2020 7c0a 7c74 6f74 616c |.|total │ │ │ │ +000269e0: 3a20 3139 3820 3134 3620 3130 3220 3636 : 198 146 102 66 │ │ │ │ +000269f0: 2033 3820 3138 2020 3920 3136 2033 3620 38 18 9 16 36 │ │ │ │ +00026a00: 3634 2031 3030 2020 2020 2020 2020 2020 64 100 │ │ │ │ +00026a10: 2020 2020 2020 2020 7c0a 7c20 2020 2038 |.| 8 │ │ │ │ +00026a20: 3a20 3130 3620 2037 3920 2035 3620 3337 : 106 79 56 37 │ │ │ │ +00026a30: 2032 3220 3131 2020 3420 2031 2020 3120 22 11 4 1 1 │ │ │ │ +00026a40: 2031 2020 2031 2020 2020 2020 2020 2020 1 1 │ │ │ │ +00026a50: 2020 2020 2020 2020 7c0a 7c20 2020 2039 |.| 9 │ │ │ │ +00026a60: 3a20 2039 3220 2036 3720 2034 3620 3239 : 92 67 46 29 │ │ │ │ +00026a70: 2031 3620 2037 2020 3220 202e 2020 2e20 16 7 2 . . │ │ │ │ +00026a80: 202e 2020 202e 2020 2020 2020 2020 2020 . . │ │ │ │ +00026a90: 2020 2020 2020 2020 7c0a 7c20 2020 3130 |.| 10 │ │ │ │ +00026aa0: 3a20 2020 2e20 2020 2e20 2020 2e20 202e : . . . . │ │ │ │ +00026ab0: 2020 2e20 202e 2020 2e20 2035 2031 3420 . . . 5 14 │ │ │ │ +00026ac0: 3237 2020 3434 2020 2020 2020 2020 2020 27 44 │ │ │ │ +00026ad0: 2020 2020 2020 2020 7c0a 7c20 2020 3131 |.| 11 │ │ │ │ +00026ae0: 3a20 2020 2e20 2020 2e20 2020 2e20 202e : . . . . │ │ │ │ +00026af0: 2020 2e20 202e 2020 3320 3130 2032 3120 . . 3 10 21 │ │ │ │ +00026b00: 3336 2020 3535 2020 2020 2020 2020 2020 36 55 │ │ │ │ +00026b10: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2020 |.|--- │ │ │ │ +00026b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026b60: 2020 207c 0a7c 436f 686f 6d6f 6c6f 6779 |.|Cohomology │ │ │ │ -00026b70: 2074 6162 6c65 206f 6620 6576 656e 4578 table of evenEx │ │ │ │ -00026b80: 744d 6f64 756c 6520 4d3a 2020 2020 2020 tModule M: │ │ │ │ -00026b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ba0: 2020 207c 0a7c 2020 202d 3520 2d34 202d |.| -5 -4 - │ │ │ │ -00026bb0: 3320 2d32 202d 3120 2030 2020 3120 2032 3 -2 -1 0 1 2 │ │ │ │ -00026bc0: 2020 3320 2034 2020 2035 2020 2020 2020 3 4 5 │ │ │ │ -00026bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026be0: 2020 207c 0a7c 323a 2033 3620 3231 2031 |.|2: 36 21 1 │ │ │ │ -00026bf0: 3020 2033 2020 2e20 202e 2020 2e20 202e 0 3 . . . . │ │ │ │ -00026c00: 2020 2e20 202e 2020 202e 2020 2020 2020 . . . │ │ │ │ -00026c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c20: 2020 207c 0a7c 313a 2020 2e20 202e 2020 |.|1: . . │ │ │ │ -00026c30: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -00026c40: 2020 2e20 202e 2020 202e 2020 2020 2020 . . . │ │ │ │ -00026c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c60: 2020 207c 0a7c 303a 2020 3120 2031 2020 |.|0: 1 1 │ │ │ │ -00026c70: 3120 2032 2020 3720 3136 2032 3920 3436 1 2 7 16 29 46 │ │ │ │ -00026c80: 2036 3720 3932 2031 3231 2020 2020 2020 67 92 121 │ │ │ │ -00026c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ca0: 2020 207c 0a7c 2d2d 2d20 2020 2020 2020 |.|--- │ │ │ │ +00026b50: 2020 2020 2020 2020 7c0a 7c43 6f68 6f6d |.|Cohom │ │ │ │ +00026b60: 6f6c 6f67 7920 7461 626c 6520 6f66 2065 ology table of e │ │ │ │ +00026b70: 7665 6e45 7874 4d6f 6475 6c65 204d 3a20 venExtModule M: │ │ │ │ +00026b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026b90: 2020 2020 2020 2020 7c0a 7c20 2020 2d35 |.| -5 │ │ │ │ +00026ba0: 202d 3420 2d33 202d 3220 2d31 2020 3020 -4 -3 -2 -1 0 │ │ │ │ +00026bb0: 2031 2020 3220 2033 2020 3420 2020 3520 1 2 3 4 5 │ │ │ │ +00026bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026bd0: 2020 2020 2020 2020 7c0a 7c32 3a20 3336 |.|2: 36 │ │ │ │ +00026be0: 2032 3120 3130 2020 3320 202e 2020 2e20 21 10 3 . . │ │ │ │ +00026bf0: 202e 2020 2e20 202e 2020 2e20 2020 2e20 . . . . . │ │ │ │ +00026c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026c10: 2020 2020 2020 2020 7c0a 7c31 3a20 202e |.|1: . │ │ │ │ +00026c20: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +00026c30: 202e 2020 2e20 202e 2020 2e20 2020 2e20 . . . . . │ │ │ │ +00026c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026c50: 2020 2020 2020 2020 7c0a 7c30 3a20 2031 |.|0: 1 │ │ │ │ +00026c60: 2020 3120 2031 2020 3220 2037 2031 3620 1 1 2 7 16 │ │ │ │ +00026c70: 3239 2034 3620 3637 2039 3220 3132 3120 29 46 67 92 121 │ │ │ │ +00026c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026c90: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2020 |.|--- │ │ │ │ +00026ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ce0: 2020 207c 0a7c 436f 686f 6d6f 6c6f 6779 |.|Cohomology │ │ │ │ -00026cf0: 2074 6162 6c65 206f 6620 6f64 6445 7874 table of oddExt │ │ │ │ -00026d00: 4d6f 6475 6c65 204d 3a20 2020 2020 2020 Module M: │ │ │ │ -00026d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026d20: 2020 207c 0a7c 2020 202d 3520 2d34 202d |.| -5 -4 - │ │ │ │ -00026d30: 3320 2d32 202d 3120 2030 2020 3120 2032 3 -2 -1 0 1 2 │ │ │ │ -00026d40: 2020 3320 2020 3420 2020 3520 2020 2020 3 4 5 │ │ │ │ -00026d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026d60: 2020 207c 0a7c 323a 2032 3820 3135 2020 |.|2: 28 15 │ │ │ │ -00026d70: 3620 2031 2020 2e20 202e 2020 2e20 202e 6 1 . . . . │ │ │ │ -00026d80: 2020 2e20 2020 2e20 2020 2e20 2020 2020 . . . │ │ │ │ -00026d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026da0: 2020 207c 0a7c 313a 2020 2e20 202e 2020 |.|1: . . │ │ │ │ -00026db0: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -00026dc0: 2020 2e20 2020 2e20 2020 2e20 2020 2020 . . . │ │ │ │ -00026dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026de0: 2020 207c 0a7c 303a 2020 3120 2031 2020 |.|0: 1 1 │ │ │ │ -00026df0: 3120 2034 2031 3120 3232 2033 3720 3536 1 4 11 22 37 56 │ │ │ │ -00026e00: 2037 3920 3130 3620 3133 3720 2020 2020 79 106 137 │ │ │ │ -00026e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026e20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00026cd0: 2020 2020 2020 2020 7c0a 7c43 6f68 6f6d |.|Cohom │ │ │ │ +00026ce0: 6f6c 6f67 7920 7461 626c 6520 6f66 206f ology table of o │ │ │ │ +00026cf0: 6464 4578 744d 6f64 756c 6520 4d3a 2020 ddExtModule M: │ │ │ │ +00026d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026d10: 2020 2020 2020 2020 7c0a 7c20 2020 2d35 |.| -5 │ │ │ │ +00026d20: 202d 3420 2d33 202d 3220 2d31 2020 3020 -4 -3 -2 -1 0 │ │ │ │ +00026d30: 2031 2020 3220 2033 2020 2034 2020 2035 1 2 3 4 5 │ │ │ │ +00026d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026d50: 2020 2020 2020 2020 7c0a 7c32 3a20 3238 |.|2: 28 │ │ │ │ +00026d60: 2031 3520 2036 2020 3120 202e 2020 2e20 15 6 1 . . │ │ │ │ +00026d70: 202e 2020 2e20 202e 2020 202e 2020 202e . . . . . │ │ │ │ +00026d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026d90: 2020 2020 2020 2020 7c0a 7c31 3a20 202e |.|1: . │ │ │ │ +00026da0: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +00026db0: 202e 2020 2e20 202e 2020 202e 2020 202e . . . . . │ │ │ │ +00026dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026dd0: 2020 2020 2020 2020 7c0a 7c30 3a20 2031 |.|0: 1 │ │ │ │ +00026de0: 2020 3120 2031 2020 3420 3131 2032 3220 1 1 4 11 22 │ │ │ │ +00026df0: 3337 2035 3620 3739 2031 3036 2031 3337 37 56 79 106 137 │ │ │ │ +00026e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026e10: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00026e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026e60: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ -00026e70: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ -00026e80: 7465 2068 6967 6853 797a 7967 793a 2068 te highSyzygy: h │ │ │ │ -00026e90: 6967 6853 797a 7967 792c 202d 2d20 5265 ighSyzygy, -- Re │ │ │ │ -00026ea0: 7475 726e 7320 6120 7379 7a79 6779 206d turns a syzygy m │ │ │ │ -00026eb0: 6f64 756c 6520 6f6e 6520 6265 796f 6e64 odule one beyond │ │ │ │ -00026ec0: 2074 6865 0a20 2020 2072 6567 756c 6172 the. regular │ │ │ │ -00026ed0: 6974 7920 6f66 2045 7874 284d 2c6b 290a ity of Ext(M,k). │ │ │ │ -00026ee0: 2020 2a20 2a6e 6f74 6520 6578 7465 7269 * *note exteri │ │ │ │ -00026ef0: 6f72 4578 744d 6f64 756c 653a 2065 7874 orExtModule: ext │ │ │ │ -00026f00: 6572 696f 7245 7874 4d6f 6475 6c65 2c20 eriorExtModule, │ │ │ │ -00026f10: 2d2d 2045 7874 284d 2c6b 2920 6f72 2045 -- Ext(M,k) or E │ │ │ │ -00026f20: 7874 284d 2c4e 2920 6173 2061 0a20 2020 xt(M,N) as a. │ │ │ │ -00026f30: 206d 6f64 756c 6520 6f76 6572 2061 6e20 module over an │ │ │ │ -00026f40: 6578 7465 7269 6f72 2061 6c67 6562 7261 exterior algebra │ │ │ │ -00026f50: 0a0a 5761 7973 2074 6f20 7573 6520 6578 ..Ways to use ex │ │ │ │ -00026f60: 7456 7343 6f68 6f6d 6f6c 6f67 793a 0a3d tVsCohomology:.= │ │ │ │ +00026e50: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ +00026e60: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ +00026e70: 2a20 2a6e 6f74 6520 6869 6768 5379 7a79 * *note highSyzy │ │ │ │ +00026e80: 6779 3a20 6869 6768 5379 7a79 6779 2c20 gy: highSyzygy, │ │ │ │ +00026e90: 2d2d 2052 6574 7572 6e73 2061 2073 797a -- Returns a syz │ │ │ │ +00026ea0: 7967 7920 6d6f 6475 6c65 206f 6e65 2062 ygy module one b │ │ │ │ +00026eb0: 6579 6f6e 6420 7468 650a 2020 2020 7265 eyond the. re │ │ │ │ +00026ec0: 6775 6c61 7269 7479 206f 6620 4578 7428 gularity of Ext( │ │ │ │ +00026ed0: 4d2c 6b29 0a20 202a 202a 6e6f 7465 2065 M,k). * *note e │ │ │ │ +00026ee0: 7874 6572 696f 7245 7874 4d6f 6475 6c65 xteriorExtModule │ │ │ │ +00026ef0: 3a20 6578 7465 7269 6f72 4578 744d 6f64 : exteriorExtMod │ │ │ │ +00026f00: 756c 652c 202d 2d20 4578 7428 4d2c 6b29 ule, -- Ext(M,k) │ │ │ │ +00026f10: 206f 7220 4578 7428 4d2c 4e29 2061 7320 or Ext(M,N) as │ │ │ │ +00026f20: 610a 2020 2020 6d6f 6475 6c65 206f 7665 a. module ove │ │ │ │ +00026f30: 7220 616e 2065 7874 6572 696f 7220 616c r an exterior al │ │ │ │ +00026f40: 6765 6272 610a 0a57 6179 7320 746f 2075 gebra..Ways to u │ │ │ │ +00026f50: 7365 2065 7874 5673 436f 686f 6d6f 6c6f se extVsCohomolo │ │ │ │ +00026f60: 6779 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d gy:.============ │ │ │ │ 00026f70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00026f80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00026f90: 2022 6578 7456 7343 6f68 6f6d 6f6c 6f67 "extVsCohomolog │ │ │ │ -00026fa0: 7928 4d61 7472 6978 2c4d 6f64 756c 6529 y(Matrix,Module) │ │ │ │ -00026fb0: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ -00026fc0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ -00026fd0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ -00026fe0: 6a65 6374 202a 6e6f 7465 2065 7874 5673 ject *note extVs │ │ │ │ -00026ff0: 436f 686f 6d6f 6c6f 6779 3a20 6578 7456 Cohomology: extV │ │ │ │ -00027000: 7343 6f68 6f6d 6f6c 6f67 792c 2069 7320 sCohomology, is │ │ │ │ -00027010: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ -00027020: 756e 6374 696f 6e3a 0a28 4d61 6361 756c unction:.(Macaul │ │ │ │ -00027030: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ -00027040: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +00026f80: 0a0a 2020 2a20 2265 7874 5673 436f 686f .. * "extVsCoho │ │ │ │ +00026f90: 6d6f 6c6f 6779 284d 6174 7269 782c 4d6f mology(Matrix,Mo │ │ │ │ +00026fa0: 6475 6c65 2922 0a0a 466f 7220 7468 6520 dule)"..For the │ │ │ │ +00026fb0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ +00026fc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +00026fd0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +00026fe0: 6578 7456 7343 6f68 6f6d 6f6c 6f67 793a extVsCohomology: │ │ │ │ +00026ff0: 2065 7874 5673 436f 686f 6d6f 6c6f 6779 extVsCohomology │ │ │ │ +00027000: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ +00027010: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod function:.(M │ │ │ │ +00027020: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ +00027030: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d odFunction,...-- │ │ │ │ +00027040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027090: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -000270a0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -000270b0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -000270c0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -000270d0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -000270e0: 2e32 362e 3035 2b64 732f 4d32 2f4d 6163 .26.05+ds/M2/Mac │ │ │ │ -000270f0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -00027100: 0a43 6f6d 706c 6574 6549 6e74 6572 7365 .CompleteInterse │ │ │ │ -00027110: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ -00027120: 2e6d 323a 3238 3236 3a30 2e0a 1f0a 4669 .m2:2826:0....Fi │ │ │ │ -00027130: 6c65 3a20 436f 6d70 6c65 7465 496e 7465 le: CompleteInte │ │ │ │ -00027140: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -00027150: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20 ons.info, Node: │ │ │ │ -00027160: 6669 6e69 7465 4265 7474 694e 756d 6265 finiteBettiNumbe │ │ │ │ -00027170: 7273 2c20 4e65 7874 3a20 6672 6565 4578 rs, Next: freeEx │ │ │ │ -00027180: 7465 7269 6f72 5375 6d6d 616e 642c 2050 teriorSummand, P │ │ │ │ -00027190: 7265 763a 2065 7874 5673 436f 686f 6d6f rev: extVsCohomo │ │ │ │ -000271a0: 6c6f 6779 2c20 5570 3a20 546f 700a 0a66 logy, Up: Top..f │ │ │ │ -000271b0: 696e 6974 6542 6574 7469 4e75 6d62 6572 initeBettiNumber │ │ │ │ -000271c0: 7320 2d2d 2062 6574 7469 206e 756d 6265 s -- betti numbe │ │ │ │ -000271d0: 7273 206f 6620 6669 6e69 7465 2072 6573 rs of finite res │ │ │ │ -000271e0: 6f6c 7574 696f 6e20 636f 6d70 7574 6564 olution computed │ │ │ │ -000271f0: 2066 726f 6d20 6120 6d61 7472 6978 2066 from a matrix f │ │ │ │ -00027200: 6163 746f 7269 7a61 7469 6f6e 0a2a 2a2a actorization.*** │ │ │ │ +00027080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ +00027090: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ +000270a0: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ +000270b0: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ +000270c0: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ +000270d0: 6179 322d 312e 3236 2e30 352b 6473 2f4d ay2-1.26.05+ds/M │ │ │ │ +000270e0: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ +000270f0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ +00027100: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ +00027110: 7469 6f6e 732e 6d32 3a32 3832 363a 302e tions.m2:2826:0. │ │ │ │ +00027120: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ +00027130: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +00027140: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ +00027150: 6f64 653a 2066 696e 6974 6542 6574 7469 ode: finiteBetti │ │ │ │ +00027160: 4e75 6d62 6572 732c 204e 6578 743a 2066 Numbers, Next: f │ │ │ │ +00027170: 7265 6545 7874 6572 696f 7253 756d 6d61 reeExteriorSumma │ │ │ │ +00027180: 6e64 2c20 5072 6576 3a20 6578 7456 7343 nd, Prev: extVsC │ │ │ │ +00027190: 6f68 6f6d 6f6c 6f67 792c 2055 703a 2054 ohomology, Up: T │ │ │ │ +000271a0: 6f70 0a0a 6669 6e69 7465 4265 7474 694e op..finiteBettiN │ │ │ │ +000271b0: 756d 6265 7273 202d 2d20 6265 7474 6920 umbers -- betti │ │ │ │ +000271c0: 6e75 6d62 6572 7320 6f66 2066 696e 6974 numbers of finit │ │ │ │ +000271d0: 6520 7265 736f 6c75 7469 6f6e 2063 6f6d e resolution com │ │ │ │ +000271e0: 7075 7465 6420 6672 6f6d 2061 206d 6174 puted from a mat │ │ │ │ +000271f0: 7269 7820 6661 6374 6f72 697a 6174 696f rix factorizatio │ │ │ │ +00027200: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a n.************** │ │ │ │ 00027210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00027220: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00027230: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00027240: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00027250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00027260: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -00027270: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -00027280: 4c20 3d20 6669 6e69 7465 4265 7474 694e L = finiteBettiN │ │ │ │ -00027290: 756d 6265 7273 204d 460a 2020 2a20 496e umbers MF. * In │ │ │ │ -000272a0: 7075 7473 3a0a 2020 2020 2020 2a20 4d46 puts:. * MF │ │ │ │ -000272b0: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ -000272c0: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ -000272d0: 7374 2c2c 204c 6973 7420 6f66 2048 6173 st,, List of Has │ │ │ │ -000272e0: 6854 6162 6c65 7320 6173 2063 6f6d 7075 hTables as compu │ │ │ │ -000272f0: 7465 640a 2020 2020 2020 2020 6279 2022 ted. by " │ │ │ │ -00027300: 6d61 7472 6978 4661 6374 6f72 697a 6174 matrixFactorizat │ │ │ │ -00027310: 696f 6e22 0a20 202a 204f 7574 7075 7473 ion". * Outputs │ │ │ │ -00027320: 3a0a 2020 2020 2020 2a20 4c2c 2061 202a :. * L, a * │ │ │ │ -00027330: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ -00027340: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ -00027350: 4c69 7374 206f 6620 6265 7474 6920 6e75 List of betti nu │ │ │ │ -00027360: 6d62 6572 730a 0a44 6573 6372 6970 7469 mbers..Descripti │ │ │ │ -00027370: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00027380: 5573 6573 2074 6865 2072 616e 6b73 206f Uses the ranks o │ │ │ │ -00027390: 6620 7468 6520 4220 6d61 7472 6963 6573 f the B matrices │ │ │ │ -000273a0: 2069 6e20 6120 6d61 7472 6978 2066 6163 in a matrix fac │ │ │ │ -000273b0: 746f 7269 7a61 7469 6f6e 2066 6f72 2061 torization for a │ │ │ │ -000273c0: 206d 6f64 756c 6520 4d20 6f76 6572 0a53 module M over.S │ │ │ │ -000273d0: 2f28 665f 312c 2e2e 2c66 5f63 2920 746f /(f_1,..,f_c) to │ │ │ │ -000273e0: 2063 6f6d 7075 7465 2074 6865 2062 6574 compute the bet │ │ │ │ -000273f0: 7469 206e 756d 6265 7273 206f 6620 7468 ti numbers of th │ │ │ │ -00027400: 6520 6d69 6e69 6d61 6c20 7265 736f 6c75 e minimal resolu │ │ │ │ -00027410: 7469 6f6e 206f 6620 4d20 6f76 6572 0a53 tion of M over.S │ │ │ │ -00027420: 2c20 7768 6963 6820 6973 2074 6865 2073 , which is the s │ │ │ │ -00027430: 756d 206f 6620 7468 6520 4b6f 737a 756c um of the Koszul │ │ │ │ -00027440: 2063 6f6d 706c 6578 6573 204b 2866 5f31 complexes K(f_1 │ │ │ │ -00027450: 2e2e 665f 7b6a 2d31 7d29 2074 656e 736f ..f_{j-1}) tenso │ │ │ │ -00027460: 7265 6420 7769 7468 2042 286a 290a 0a2b red with B(j)..+ │ │ │ │ +00027250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +00027260: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +00027270: 2020 2020 204c 203d 2066 696e 6974 6542 L = finiteB │ │ │ │ +00027280: 6574 7469 4e75 6d62 6572 7320 4d46 0a20 ettiNumbers MF. │ │ │ │ +00027290: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +000272a0: 202a 204d 462c 2061 202a 6e6f 7465 206c * MF, a *note l │ │ │ │ +000272b0: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ +000272c0: 6f63 294c 6973 742c 2c20 4c69 7374 206f oc)List,, List o │ │ │ │ +000272d0: 6620 4861 7368 5461 626c 6573 2061 7320 f HashTables as │ │ │ │ +000272e0: 636f 6d70 7574 6564 0a20 2020 2020 2020 computed. │ │ │ │ +000272f0: 2062 7920 226d 6174 7269 7846 6163 746f by "matrixFacto │ │ │ │ +00027300: 7269 7a61 7469 6f6e 220a 2020 2a20 4f75 rization". * Ou │ │ │ │ +00027310: 7470 7574 733a 0a20 2020 2020 202a 204c tputs:. * L │ │ │ │ +00027320: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ +00027330: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ +00027340: 7374 2c2c 204c 6973 7420 6f66 2062 6574 st,, List of bet │ │ │ │ +00027350: 7469 206e 756d 6265 7273 0a0a 4465 7363 ti numbers..Desc │ │ │ │ +00027360: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +00027370: 3d3d 3d0a 0a55 7365 7320 7468 6520 7261 ===..Uses the ra │ │ │ │ +00027380: 6e6b 7320 6f66 2074 6865 2042 206d 6174 nks of the B mat │ │ │ │ +00027390: 7269 6365 7320 696e 2061 206d 6174 7269 rices in a matri │ │ │ │ +000273a0: 7820 6661 6374 6f72 697a 6174 696f 6e20 x factorization │ │ │ │ +000273b0: 666f 7220 6120 6d6f 6475 6c65 204d 206f for a module M o │ │ │ │ +000273c0: 7665 720a 532f 2866 5f31 2c2e 2e2c 665f ver.S/(f_1,..,f_ │ │ │ │ +000273d0: 6329 2074 6f20 636f 6d70 7574 6520 7468 c) to compute th │ │ │ │ +000273e0: 6520 6265 7474 6920 6e75 6d62 6572 7320 e betti numbers │ │ │ │ +000273f0: 6f66 2074 6865 206d 696e 696d 616c 2072 of the minimal r │ │ │ │ +00027400: 6573 6f6c 7574 696f 6e20 6f66 204d 206f esolution of M o │ │ │ │ +00027410: 7665 720a 532c 2077 6869 6368 2069 7320 ver.S, which is │ │ │ │ +00027420: 7468 6520 7375 6d20 6f66 2074 6865 204b the sum of the K │ │ │ │ +00027430: 6f73 7a75 6c20 636f 6d70 6c65 7865 7320 oszul complexes │ │ │ │ +00027440: 4b28 665f 312e 2e66 5f7b 6a2d 317d 2920 K(f_1..f_{j-1}) │ │ │ │ +00027450: 7465 6e73 6f72 6564 2077 6974 6820 4228 tensored with B( │ │ │ │ +00027460: 6a29 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d j)..+----------- │ │ │ │ 00027470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000274a0: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e --+.|i1 : setRan │ │ │ │ -000274b0: 646f 6d53 6565 6420 3020 2020 2020 2020 domSeed 0 │ │ │ │ -000274c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000274d0: 2020 2020 2020 207c 0a7c 202d 2d20 7365 |.| -- se │ │ │ │ -000274e0: 7474 696e 6720 7261 6e64 6f6d 2073 6565 tting random see │ │ │ │ -000274f0: 6420 746f 2030 2020 2020 2020 2020 2020 d to 0 │ │ │ │ -00027500: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00027490: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 -------+.|i1 : s │ │ │ │ +000274a0: 6574 5261 6e64 6f6d 5365 6564 2030 2020 etRandomSeed 0 │ │ │ │ +000274b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000274c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000274d0: 2d2d 2073 6574 7469 6e67 2072 616e 646f -- setting rando │ │ │ │ +000274e0: 6d20 7365 6564 2074 6f20 3020 2020 2020 m seed to 0 │ │ │ │ +000274f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027500: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00027510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027540: 207c 0a7c 6f31 203d 2030 2020 2020 2020 |.|o1 = 0 │ │ │ │ +00027530: 2020 2020 2020 7c0a 7c6f 3120 3d20 3020 |.|o1 = 0 │ │ │ │ +00027540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027570: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00027560: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00027570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000275a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -000275b0: 203a 206b 6b20 3d20 5a5a 2f31 3031 2020 : kk = ZZ/101 │ │ │ │ +000275a0: 2b0a 7c69 3220 3a20 6b6b 203d 205a 5a2f +.|i2 : kk = ZZ/ │ │ │ │ +000275b0: 3130 3120 2020 2020 2020 2020 2020 2020 101 │ │ │ │ 000275c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000275d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000275e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000275d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000275e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000275f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027610: 2020 2020 207c 0a7c 6f32 203d 206b 6b20 |.|o2 = kk │ │ │ │ +00027600: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +00027610: 3d20 6b6b 2020 2020 2020 2020 2020 2020 = kk │ │ │ │ 00027620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027640: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00027630: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00027640: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00027650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027670: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00027680: 0a7c 6f32 203a 2051 756f 7469 656e 7452 .|o2 : QuotientR │ │ │ │ -00027690: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -000276a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000276b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00027670: 2020 2020 7c0a 7c6f 3220 3a20 5175 6f74 |.|o2 : Quot │ │ │ │ +00027680: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +00027690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000276a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000276b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000276c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000276d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000276e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -000276f0: 2053 203d 206b 6b5b 612c 622c 752c 765d S = kk[a,b,u,v] │ │ │ │ +000276d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000276e0: 7c69 3320 3a20 5320 3d20 6b6b 5b61 2c62 |i3 : S = kk[a,b │ │ │ │ +000276f0: 2c75 2c76 5d20 2020 2020 2020 2020 2020 ,u,v] │ │ │ │ 00027700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027710: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027720: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00027710: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00027720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027750: 2020 207c 0a7c 6f33 203d 2053 2020 2020 |.|o3 = S │ │ │ │ +00027740: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ +00027750: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 00027760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027780: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00027770: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00027780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000277a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000277c0: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52 o3 : PolynomialR │ │ │ │ -000277d0: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -000277e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000277b0: 2020 7c0a 7c6f 3320 3a20 506f 6c79 6e6f |.|o3 : Polyno │ │ │ │ +000277c0: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +000277d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000277e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000277f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027820: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2066 -------+.|i4 : f │ │ │ │ -00027830: 6620 3d20 6d61 7472 6978 2261 752c 6276 f = matrix"au,bv │ │ │ │ -00027840: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ -00027850: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00027810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00027820: 3420 3a20 6666 203d 206d 6174 7269 7822 4 : ff = matrix" │ │ │ │ +00027830: 6175 2c62 7622 2020 2020 2020 2020 2020 au,bv" │ │ │ │ +00027840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027850: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00027860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027890: 207c 0a7c 6f34 203d 207c 2061 7520 6276 |.|o4 = | au bv │ │ │ │ -000278a0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000278b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000278c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00027880: 2020 2020 2020 7c0a 7c6f 3420 3d20 7c20 |.|o4 = | │ │ │ │ +00027890: 6175 2062 7620 7c20 2020 2020 2020 2020 au bv | │ │ │ │ +000278a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000278b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000278c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000278d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000278e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000278f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00027900: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ -00027910: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00027920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027930: 7c0a 7c6f 3420 3a20 4d61 7472 6978 2053 |.|o4 : Matrix S │ │ │ │ -00027940: 2020 3c2d 2d20 5320 2020 2020 2020 2020 <-- S │ │ │ │ -00027950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027960: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000278f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00027900: 3120 2020 2020 2032 2020 2020 2020 2020 1 2 │ │ │ │ +00027910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027920: 2020 2020 207c 0a7c 6f34 203a 204d 6174 |.|o4 : Mat │ │ │ │ +00027930: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ +00027940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027950: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027990: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ -000279a0: 3a20 5220 3d20 532f 6964 6561 6c20 6666 : R = S/ideal ff │ │ │ │ +00027980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00027990: 0a7c 6935 203a 2052 203d 2053 2f69 6465 .|i5 : R = S/ide │ │ │ │ +000279a0: 616c 2066 6620 2020 2020 2020 2020 2020 al ff │ │ │ │ 000279b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000279c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000279d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000279c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000279d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000279e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000279f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a00: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020 |.|o5 = R │ │ │ │ +000279f0: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ +00027a00: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 00027a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00027a20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00027a30: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00027a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027a70: 7c6f 3520 3a20 5175 6f74 6965 6e74 5269 |o5 : QuotientRi │ │ │ │ -00027a80: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ -00027a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027aa0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00027a60: 2020 207c 0a7c 6f35 203a 2051 756f 7469 |.|o5 : Quoti │ │ │ │ +00027a70: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ +00027a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027a90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ad0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ -00027ae0: 4d30 203d 2052 5e31 2f69 6465 616c 2261 M0 = R^1/ideal"a │ │ │ │ -00027af0: 2c62 2220 2020 2020 2020 2020 2020 2020 ,b" │ │ │ │ -00027b00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00027ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00027ad0: 6936 203a 204d 3020 3d20 525e 312f 6964 i6 : M0 = R^1/id │ │ │ │ +00027ae0: 6561 6c22 612c 6222 2020 2020 2020 2020 eal"a,b" │ │ │ │ +00027af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027b00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00027b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027b40: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e |.|o6 = cokern │ │ │ │ -00027b50: 656c 207c 2061 2062 207c 2020 2020 2020 el | a b | │ │ │ │ -00027b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027b70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00027b30: 2020 2020 2020 207c 0a7c 6f36 203d 2063 |.|o6 = c │ │ │ │ +00027b40: 6f6b 6572 6e65 6c20 7c20 6120 6220 7c20 okernel | a b | │ │ │ │ +00027b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027b60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00027b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027ba0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00027ba0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00027bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027bc0: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ -00027bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027be0: 207c 0a7c 6f36 203a 2052 2d6d 6f64 756c |.|o6 : R-modul │ │ │ │ -00027bf0: 652c 2071 756f 7469 656e 7420 6f66 2052 e, quotient of R │ │ │ │ -00027c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027c10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00027bc0: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00027bd0: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d |.|o6 : R- │ │ │ │ +00027be0: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74 module, quotient │ │ │ │ +00027bf0: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ +00027c00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00027c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ -00027c50: 203a 2046 203d 2066 7265 6552 6573 6f6c : F = freeResol │ │ │ │ -00027c60: 7574 696f 6e28 4d30 2c20 4c65 6e67 7468 ution(M0, Length │ │ │ │ -00027c70: 4c69 6d69 7420 3d3e 3329 2020 2020 2020 Limit =>3) │ │ │ │ -00027c80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00027c40: 2b0a 7c69 3720 3a20 4620 3d20 6672 6565 +.|i7 : F = free │ │ │ │ +00027c50: 5265 736f 6c75 7469 6f6e 284d 302c 204c Resolution(M0, L │ │ │ │ +00027c60: 656e 6774 684c 696d 6974 203d 3e33 2920 engthLimit =>3) │ │ │ │ +00027c70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00027c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027cb0: 2020 2020 207c 0a7c 2020 2020 2020 3120 |.| 1 │ │ │ │ -00027cc0: 2020 2020 2032 2020 2020 2020 3320 2020 2 3 │ │ │ │ -00027cd0: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -00027ce0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ -00027cf0: 3d20 5220 203c 2d2d 2052 2020 3c2d 2d20 = R <-- R <-- │ │ │ │ -00027d00: 5220 203c 2d2d 2052 2020 2020 2020 2020 R <-- R │ │ │ │ -00027d10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00027d20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00027ca0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00027cb0: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ +00027cc0: 2033 2020 2020 2020 3420 2020 2020 2020 3 4 │ │ │ │ +00027cd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00027ce0: 0a7c 6f37 203d 2052 2020 3c2d 2d20 5220 .|o7 = R <-- R │ │ │ │ +00027cf0: 203c 2d2d 2052 2020 3c2d 2d20 5220 2020 <-- R <-- R │ │ │ │ +00027d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027d10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00027d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027d50: 2020 2020 7c0a 7c20 2020 2020 3020 2020 |.| 0 │ │ │ │ -00027d60: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ -00027d70: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -00027d80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00027d40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00027d50: 2030 2020 2020 2020 3120 2020 2020 2032 0 1 2 │ │ │ │ +00027d60: 2020 2020 2020 3320 2020 2020 2020 2020 3 │ │ │ │ +00027d70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00027d80: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00027d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027db0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027dc0: 7c6f 3720 3a20 436f 6d70 6c65 7820 2020 |o7 : Complex │ │ │ │ +00027db0: 2020 207c 0a7c 6f37 203a 2043 6f6d 706c |.|o7 : Compl │ │ │ │ +00027dc0: 6578 2020 2020 2020 2020 2020 2020 2020 ex │ │ │ │ 00027dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027df0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00027de0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00027df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027e20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 --------+.|i8 : │ │ │ │ -00027e30: 4d20 3d20 636f 6b65 7220 462e 6464 5f33 M = coker F.dd_3 │ │ │ │ -00027e40: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ -00027e50: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00027e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00027e20: 6938 203a 204d 203d 2063 6f6b 6572 2046 i8 : M = coker F │ │ │ │ +00027e30: 2e64 645f 333b 2020 2020 2020 2020 2020 .dd_3; │ │ │ │ +00027e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027e50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00027e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027e90: 2d2d 2b0a 7c69 3920 3a20 4d46 203d 206d --+.|i9 : MF = m │ │ │ │ -00027ea0: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ -00027eb0: 6f6e 2866 662c 4d29 3b20 2020 2020 2020 on(ff,M); │ │ │ │ -00027ec0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00027e80: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 204d -------+.|i9 : M │ │ │ │ +00027e90: 4620 3d20 6d61 7472 6978 4661 6374 6f72 F = matrixFactor │ │ │ │ +00027ea0: 697a 6174 696f 6e28 6666 2c4d 293b 2020 ization(ff,M); │ │ │ │ +00027eb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00027ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00027f00: 3130 203a 2062 6574 7469 2066 7265 6552 10 : betti freeR │ │ │ │ -00027f10: 6573 6f6c 7574 696f 6e20 7075 7368 466f esolution pushFo │ │ │ │ -00027f20: 7277 6172 6428 6d61 7028 522c 5329 2c4d rward(map(R,S),M │ │ │ │ -00027f30: 297c 0a7c 2020 2020 2020 2020 2020 2020 )|.| │ │ │ │ +00027ef0: 2d2b 0a7c 6931 3020 3a20 6265 7474 6920 -+.|i10 : betti │ │ │ │ +00027f00: 6672 6565 5265 736f 6c75 7469 6f6e 2070 freeResolution p │ │ │ │ +00027f10: 7573 6846 6f72 7761 7264 286d 6170 2852 ushForward(map(R │ │ │ │ +00027f20: 2c53 292c 4d29 7c0a 7c20 2020 2020 2020 ,S),M)|.| │ │ │ │ +00027f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027f60: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00027f70: 2020 2020 2020 3020 3120 3220 2020 2020 0 1 2 │ │ │ │ +00027f50: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00027f60: 2020 2020 2020 2020 2020 2030 2031 2032 0 1 2 │ │ │ │ +00027f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027f90: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -00027fa0: 3020 3d20 746f 7461 6c3a 2033 2035 2032 0 = total: 3 5 2 │ │ │ │ +00027f90: 7c0a 7c6f 3130 203d 2074 6f74 616c 3a20 |.|o10 = total: │ │ │ │ +00027fa0: 3320 3520 3220 2020 2020 2020 2020 2020 3 5 2 │ │ │ │ 00027fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027fd0: 7c0a 7c20 2020 2020 2020 2020 2032 3a20 |.| 2: │ │ │ │ -00027fe0: 3320 3420 2e20 2020 2020 2020 2020 2020 3 4 . │ │ │ │ -00027ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028000: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00028010: 2020 333a 202e 2031 2032 2020 2020 2020 3: . 1 2 │ │ │ │ -00028020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028030: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00027fc0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00027fd0: 2020 323a 2033 2034 202e 2020 2020 2020 2: 3 4 . │ │ │ │ +00027fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027ff0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00028000: 2020 2020 2020 2033 3a20 2e20 3120 3220 3: . 1 2 │ │ │ │ +00028010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028020: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028030: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00028040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028060: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028070: 0a7c 6f31 3020 3a20 4265 7474 6954 616c .|o10 : BettiTal │ │ │ │ -00028080: 6c79 2020 2020 2020 2020 2020 2020 2020 ly │ │ │ │ -00028090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000280a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00028060: 2020 2020 7c0a 7c6f 3130 203a 2042 6574 |.|o10 : Bet │ │ │ │ +00028070: 7469 5461 6c6c 7920 2020 2020 2020 2020 tiTally │ │ │ │ +00028080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028090: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000280a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000280b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000280c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000280d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 ---------+.|i11 │ │ │ │ -000280e0: 3a20 6669 6e69 7465 4265 7474 694e 756d : finiteBettiNum │ │ │ │ -000280f0: 6265 7273 204d 4620 2020 2020 2020 2020 bers MF │ │ │ │ -00028100: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00028110: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000280c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000280d0: 7c69 3131 203a 2066 696e 6974 6542 6574 |i11 : finiteBet │ │ │ │ +000280e0: 7469 4e75 6d62 6572 7320 4d46 2020 2020 tiNumbers MF │ │ │ │ +000280f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028100: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028140: 2020 207c 0a7c 6f31 3120 3d20 7b33 2c20 |.|o11 = {3, │ │ │ │ -00028150: 352c 2032 7d20 2020 2020 2020 2020 2020 5, 2} │ │ │ │ -00028160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028170: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00028130: 2020 2020 2020 2020 7c0a 7c6f 3131 203d |.|o11 = │ │ │ │ +00028140: 207b 332c 2035 2c20 327d 2020 2020 2020 {3, 5, 2} │ │ │ │ +00028150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028160: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00028170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000281a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000281b0: 6f31 3120 3a20 4c69 7374 2020 2020 2020 o11 : List │ │ │ │ +000281a0: 2020 7c0a 7c6f 3131 203a 204c 6973 7420 |.|o11 : List │ │ │ │ +000281b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000281c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000281d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000281e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000281d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000281e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000281f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028210: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 -------+.|i12 : │ │ │ │ -00028220: 696e 6669 6e69 7465 4265 7474 694e 756d infiniteBettiNum │ │ │ │ -00028230: 6265 7273 284d 462c 3529 2020 2020 2020 bers(MF,5) │ │ │ │ -00028240: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00028200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00028210: 3132 203a 2069 6e66 696e 6974 6542 6574 12 : infiniteBet │ │ │ │ +00028220: 7469 4e75 6d62 6572 7328 4d46 2c35 2920 tiNumbers(MF,5) │ │ │ │ +00028230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028240: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00028250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028280: 207c 0a7c 6f31 3220 3d20 7b33 2c20 342c |.|o12 = {3, 4, │ │ │ │ -00028290: 2035 2c20 362c 2037 2c20 387d 2020 2020 5, 6, 7, 8} │ │ │ │ -000282a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000282b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00028270: 2020 2020 2020 7c0a 7c6f 3132 203d 207b |.|o12 = { │ │ │ │ +00028280: 332c 2034 2c20 352c 2036 2c20 372c 2038 3, 4, 5, 6, 7, 8 │ │ │ │ +00028290: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +000282a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000282b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000282c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000282d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000282e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -000282f0: 3220 3a20 4c69 7374 2020 2020 2020 2020 2 : List │ │ │ │ +000282e0: 7c0a 7c6f 3132 203a 204c 6973 7420 2020 |.|o12 : List │ │ │ │ +000282f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028320: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00028310: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00028320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028350: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 6265 -----+.|i13 : be │ │ │ │ -00028360: 7474 6920 6672 6565 5265 736f 6c75 7469 tti freeResoluti │ │ │ │ -00028370: 6f6e 2028 4d2c 204c 656e 6774 684c 696d on (M, LengthLim │ │ │ │ -00028380: 6974 203d 3e20 3529 2020 7c0a 7c20 2020 it => 5) |.| │ │ │ │ +00028340: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 ----------+.|i13 │ │ │ │ +00028350: 203a 2062 6574 7469 2066 7265 6552 6573 : betti freeRes │ │ │ │ +00028360: 6f6c 7574 696f 6e20 284d 2c20 4c65 6e67 olution (M, Leng │ │ │ │ +00028370: 7468 4c69 6d69 7420 3d3e 2035 2920 207c thLimit => 5) | │ │ │ │ +00028380: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00028390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000283a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000283b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000283c0: 0a7c 2020 2020 2020 2020 2020 2020 2030 .| 0 │ │ │ │ -000283d0: 2031 2032 2033 2034 2035 2020 2020 2020 1 2 3 4 5 │ │ │ │ -000283e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000283f0: 2020 2020 7c0a 7c6f 3133 203d 2074 6f74 |.|o13 = tot │ │ │ │ -00028400: 616c 3a20 3320 3420 3520 3620 3720 3820 al: 3 4 5 6 7 8 │ │ │ │ -00028410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028420: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00028430: 2020 2020 2020 323a 2033 2034 2035 2036 2: 3 4 5 6 │ │ │ │ -00028440: 2037 2038 2020 2020 2020 2020 2020 2020 7 8 │ │ │ │ -00028450: 2020 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-00028490: 2020 207c 0a7c 6f31 3320 3a20 4265 7474 |.|o13 : Bett │ │ │ │ -000284a0: 6954 616c 6c79 2020 2020 2020 2020 2020 iTally │ │ │ │ -000284b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000284c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00028480: 2020 2020 2020 2020 7c0a 7c6f 3133 203a |.|o13 : │ │ │ │ +00028490: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +000284a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000284b0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000284c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000284d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000284e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000284f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -00028500: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ -00028510: 3d0a 0a20 202a 202a 6e6f 7465 206d 6174 =.. * *note mat │ │ │ │ -00028520: 7269 7846 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7269 7a61 matrix factoriza │ │ │ │ -00028600: 7469 6f6e 0a0a 5761 7973 2074 6f20 7573 tion..Ways to us │ │ │ │ -00028610: 6520 6669 6e69 7465 4265 7474 694e 756d e finiteBettiNum │ │ │ │ -00028620: 6265 7273 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d bers:.========== │ │ │ │ -00028630: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00028640: 3d3d 3d3d 3d0a 0a20 202a 2022 6669 6e69 =====.. * "fini │ │ │ │ -00028650: 7465 4265 7474 694e 756d 6265 7273 284c teBettiNumbers(L │ │ │ │ -00028660: 6973 7429 220a 0a46 6f72 2074 6865 2070 ist)"..For the p │ │ │ │ -00028670: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -00028680: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -00028690: 6520 6f62 6a65 6374 202a 6e6f 7465 2066 e object *note f │ │ │ │ -000286a0: 696e 6974 6542 6574 7469 4e75 6d62 6572 initeBettiNumber │ │ │ │ -000286b0: 733a 2066 696e 6974 6542 6574 7469 4e75 s: finiteBettiNu │ │ │ │ -000286c0: 6d62 6572 732c 2069 7320 6120 2a6e 6f74 mbers, is a *not │ │ │ │ -000286d0: 6520 6d65 7468 6f64 0a66 756e 6374 696f e method.functio │ │ │ │ -000286e0: 6e3a 2028 4d61 6361 756c 6179 3244 6f63 n: (Macaulay2Doc │ │ │ │ -000286f0: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ -00028700: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +000284f0: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d --+..See also.== │ │ │ │ +00028500: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ +00028510: 6520 6d61 7472 6978 4661 6374 6f72 697a e matrixFactoriz │ │ │ │ +00028520: 6174 696f 6e3a 206d 6174 7269 7846 6163 ation: matrixFac │ │ │ │ +00028530: 746f 7269 7a61 7469 6f6e 2c20 2d2d 204d torization, -- M │ │ │ │ +00028540: 6170 7320 696e 2061 2068 6967 6865 720a aps in a higher. │ │ │ │ +00028550: 2020 2020 636f 6469 6d65 6e73 696f 6e20 codimension │ │ │ │ +00028560: 6d61 7472 6978 2066 6163 746f 7269 7a61 matrix factoriza │ │ │ │ +00028570: 7469 6f6e 0a20 202a 202a 6e6f 7465 2069 tion. * *note i │ │ │ │ +00028580: 6e66 696e 6974 6542 6574 7469 4e75 6d62 nfiniteBettiNumb │ │ │ │ +00028590: 6572 733a 2069 6e66 696e 6974 6542 6574 ers: infiniteBet │ │ │ │ +000285a0: 7469 4e75 6d62 6572 732c 202d 2d20 6265 tiNumbers, -- be │ │ │ │ +000285b0: 7474 6920 6e75 6d62 6572 7320 6f66 0a20 tti numbers of. │ │ │ │ +000285c0: 2020 2066 696e 6974 6520 7265 736f 6c75 finite resolu │ │ │ │ +000285d0: 7469 6f6e 2063 6f6d 7075 7465 6420 6672 tion computed fr │ │ │ │ +000285e0: 6f6d 2061 206d 6174 7269 7820 6661 6374 om a matrix fact │ │ │ │ +000285f0: 6f72 697a 6174 696f 6e0a 0a57 6179 7320 orization..Ways │ │ │ │ +00028600: 746f 2075 7365 2066 696e 6974 6542 6574 to use finiteBet │ │ │ │ +00028610: 7469 4e75 6d62 6572 733a 0a3d 3d3d 3d3d tiNumbers:.===== │ │ │ │ +00028620: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00028630: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00028640: 2266 696e 6974 6542 6574 7469 4e75 6d62 "finiteBettiNumb │ │ │ │ +00028650: 6572 7328 4c69 7374 2922 0a0a 466f 7220 ers(List)"..For │ │ │ │ +00028660: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +00028670: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00028680: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +00028690: 6f74 6520 6669 6e69 7465 4265 7474 694e ote finiteBettiN │ │ │ │ +000286a0: 756d 6265 7273 3a20 6669 6e69 7465 4265 umbers: finiteBe │ │ │ │ +000286b0: 7474 694e 756d 6265 7273 2c20 6973 2061 ttiNumbers, is a │ │ │ │ +000286c0: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ +000286d0: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61 nction: (Macaula │ │ │ │ +000286e0: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ +000286f0: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ +00028700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028750: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -00028760: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -00028770: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -00028780: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -00028790: 6163 6175 6c61 7932 2d31 2e32 362e 3035 acaulay2-1.26.05 │ │ │ │ -000287a0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -000287b0: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ -000287c0: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -000287d0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3430 esolutions.m2:40 │ │ │ │ -000287e0: 3732 3a30 2e0a 1f0a 4669 6c65 3a20 436f 72:0....File: Co │ │ │ │ -000287f0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -00028800: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ -00028810: 666f 2c20 4e6f 6465 3a20 6672 6565 4578 fo, Node: freeEx │ │ │ │ -00028820: 7465 7269 6f72 5375 6d6d 616e 642c 204e teriorSummand, N │ │ │ │ -00028830: 6578 743a 2047 7261 6469 6e67 2c20 5072 ext: Grading, Pr │ │ │ │ -00028840: 6576 3a20 6669 6e69 7465 4265 7474 694e ev: finiteBettiN │ │ │ │ -00028850: 756d 6265 7273 2c20 5570 3a20 546f 700a umbers, Up: Top. │ │ │ │ -00028860: 0a66 7265 6545 7874 6572 696f 7253 756d .freeExteriorSum │ │ │ │ -00028870: 6d61 6e64 202d 2d20 6669 6e64 2074 6865 mand -- find the │ │ │ │ -00028880: 2066 7265 6520 7375 6d6d 616e 6473 206f free summands o │ │ │ │ -00028890: 6620 6120 6d6f 6475 6c65 206f 7665 7220 f a module over │ │ │ │ -000288a0: 616e 2065 7874 6572 696f 7220 616c 6765 an exterior alge │ │ │ │ -000288b0: 6272 610a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a bra.************ │ │ │ │ +00028740: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +00028750: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +00028760: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +00028770: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +00028780: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00028790: 3236 2e30 352b 6473 2f4d 322f 4d61 6361 26.05+ds/M2/Maca │ │ │ │ +000287a0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +000287b0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +000287c0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +000287d0: 6d32 3a34 3037 323a 302e 0a1f 0a46 696c m2:4072:0....Fil │ │ │ │ +000287e0: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ +000287f0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +00028800: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2066 ns.info, Node: f │ │ │ │ +00028810: 7265 6545 7874 6572 696f 7253 756d 6d61 reeExteriorSumma │ │ │ │ +00028820: 6e64 2c20 4e65 7874 3a20 4772 6164 696e nd, Next: Gradin │ │ │ │ +00028830: 672c 2050 7265 763a 2066 696e 6974 6542 g, Prev: finiteB │ │ │ │ +00028840: 6574 7469 4e75 6d62 6572 732c 2055 703a ettiNumbers, Up: │ │ │ │ +00028850: 2054 6f70 0a0a 6672 6565 4578 7465 7269 Top..freeExteri │ │ │ │ +00028860: 6f72 5375 6d6d 616e 6420 2d2d 2066 696e orSummand -- fin │ │ │ │ +00028870: 6420 7468 6520 6672 6565 2073 756d 6d61 d the free summa │ │ │ │ +00028880: 6e64 7320 6f66 2061 206d 6f64 756c 6520 nds of a module │ │ │ │ +00028890: 6f76 6572 2061 6e20 6578 7465 7269 6f72 over an exterior │ │ │ │ +000288a0: 2061 6c67 6562 7261 0a2a 2a2a 2a2a 2a2a algebra.******* │ │ │ │ +000288b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000288c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000288d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000288e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000288f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00028900: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -00028910: 653a 200a 2020 2020 2020 2020 4620 3d20 e: . F = │ │ │ │ -00028920: 6672 6565 4578 7465 7269 6f72 5375 6d6d freeExteriorSumm │ │ │ │ -00028930: 616e 6420 4d0a 2020 2a20 496e 7075 7473 and M. * Inputs │ │ │ │ -00028940: 3a0a 2020 2020 2020 2a20 4d2c 2061 202a :. * M, a * │ │ │ │ -00028950: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -00028960: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -00028970: 652c 2c20 6f76 6572 2061 6e20 6578 7465 e,, over an exte │ │ │ │ -00028980: 7269 6f72 2061 6c67 6562 7261 0a20 202a rior algebra. * │ │ │ │ -00028990: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -000289a0: 2a20 462c 2061 202a 6e6f 7465 206d 6174 * F, a *note mat │ │ │ │ -000289b0: 7269 783a 2028 4d61 6361 756c 6179 3244 rix: (Macaulay2D │ │ │ │ -000289c0: 6f63 294d 6174 7269 782c 2c20 4d61 7020 oc)Matrix,, Map │ │ │ │ -000289d0: 6672 6f6d 2061 2066 7265 6520 6d6f 6475 from a free modu │ │ │ │ -000289e0: 6c65 2074 6f20 4d2e 0a20 2020 2020 2020 le to M.. │ │ │ │ -000289f0: 2049 6d61 6765 2069 7320 7468 6520 6c61 Image is the la │ │ │ │ -00028a00: 7267 6573 7420 6672 6565 2073 756d 6d61 rgest free summa │ │ │ │ -00028a10: 6e64 0a0a 4465 7363 7269 7074 696f 6e0a nd..Description. │ │ │ │ -00028a20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d ===========..+-- │ │ │ │ +000288f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00028900: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +00028910: 2046 203d 2066 7265 6545 7874 6572 696f F = freeExterio │ │ │ │ +00028920: 7253 756d 6d61 6e64 204d 0a20 202a 2049 rSummand M. * I │ │ │ │ +00028930: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ +00028940: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +00028950: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00028960: 4d6f 6475 6c65 2c2c 206f 7665 7220 616e Module,, over an │ │ │ │ +00028970: 2065 7874 6572 696f 7220 616c 6765 6272 exterior algebr │ │ │ │ +00028980: 610a 2020 2a20 4f75 7470 7574 733a 0a20 a. * Outputs:. │ │ │ │ +00028990: 2020 2020 202a 2046 2c20 6120 2a6e 6f74 * F, a *not │ │ │ │ +000289a0: 6520 6d61 7472 6978 3a20 284d 6163 6175 e matrix: (Macau │ │ │ │ +000289b0: 6c61 7932 446f 6329 4d61 7472 6978 2c2c lay2Doc)Matrix,, │ │ │ │ +000289c0: 204d 6170 2066 726f 6d20 6120 6672 6565 Map from a free │ │ │ │ +000289d0: 206d 6f64 756c 6520 746f 204d 2e0a 2020 module to M.. │ │ │ │ +000289e0: 2020 2020 2020 496d 6167 6520 6973 2074 Image is t │ │ │ │ +000289f0: 6865 206c 6172 6765 7374 2066 7265 6520 he largest free │ │ │ │ +00028a00: 7375 6d6d 616e 640a 0a44 6573 6372 6970 summand..Descrip │ │ │ │ +00028a10: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +00028a20: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ 00028a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028a60: 2d2b 0a7c 6931 203a 206b 6b3d 205a 5a2f -+.|i1 : kk= ZZ/ │ │ │ │ -00028a70: 3130 3120 2020 2020 2020 2020 2020 2020 101 │ │ │ │ -00028a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00028a50: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b ------+.|i1 : kk │ │ │ │ +00028a60: 3d20 5a5a 2f31 3031 2020 2020 2020 2020 = ZZ/101 │ │ │ │ +00028a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028a80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00028a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00028ad0: 6f31 203d 206b 6b20 2020 2020 2020 2020 o1 = kk │ │ │ │ +00028ac0: 2020 7c0a 7c6f 3120 3d20 6b6b 2020 2020 |.|o1 = kk │ │ │ │ +00028ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b00: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028af0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00028b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b30: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ -00028b40: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +00028b20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00028b30: 7c6f 3120 3a20 5175 6f74 6965 6e74 5269 |o1 : QuotientRi │ │ │ │ +00028b40: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ 00028b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028b70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00028b60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00028b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028ba0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2045 203d -----+.|i2 : E = │ │ │ │ -00028bb0: 206b 6b5b 652c 662c 672c 2053 6b65 7743 kk[e,f,g, SkewC │ │ │ │ -00028bc0: 6f6d 6d75 7461 7469 7665 203d 3e20 7472 ommutative => tr │ │ │ │ -00028bd0: 7565 5d20 2020 2020 2020 207c 0a7c 2020 ue] |.| │ │ │ │ +00028b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ +00028ba0: 3a20 4520 3d20 6b6b 5b65 2c66 2c67 2c20 : E = kk[e,f,g, │ │ │ │ +00028bb0: 536b 6577 436f 6d6d 7574 6174 6976 6520 SkewCommutative │ │ │ │ +00028bc0: 3d3e 2074 7275 655d 2020 2020 2020 2020 => true] │ │ │ │ +00028bd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00028be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c10: 207c 0a7c 6f32 203d 2045 2020 2020 2020 |.|o2 = E │ │ │ │ +00028c00: 2020 2020 2020 7c0a 7c6f 3220 3d20 4520 |.|o2 = E │ │ │ │ +00028c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c40: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00028c30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00028c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00028c80: 6f32 203a 2050 6f6c 796e 6f6d 6961 6c52 o2 : PolynomialR │ │ │ │ -00028c90: 696e 672c 2033 2073 6b65 7720 636f 6d6d ing, 3 skew comm │ │ │ │ -00028ca0: 7574 6174 6976 6520 7661 7269 6162 6c65 utative variable │ │ │ │ -00028cb0: 2873 297c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 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6e67 203d 3e20 3229 0a20 202a 2049 6e70 ng => 2). * Inp │ │ │ │ -00029420: 7574 733a 0a20 2020 2020 202a 2043 6865 uts:. * Che │ │ │ │ -00029430: 636b 2c20 616e 202a 6e6f 7465 2069 6e74 ck, an *note int │ │ │ │ -00029440: 6567 6572 3a20 284d 6163 6175 6c61 7932 eger: (Macaulay2 │ │ │ │ -00029450: 446f 6329 5a5a 2c2c 200a 0a44 6573 6372 Doc)ZZ,, ..Descr │ │ │ │ -00029460: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -00029470: 3d3d 0a0a 6966 2047 7261 6469 6e67 203d ==..if Grading = │ │ │ │ -00029480: 3e31 2c20 7468 656e 2074 6865 206f 7574 >1, then the out │ │ │ │ -00029490: 7075 7420 6973 2063 6f6e 7665 7274 6564 put is converted │ │ │ │ -000294a0: 2074 6f20 7369 6e67 6c65 2d67 7261 6469 to single-gradi │ │ │ │ -000294b0: 6e67 2c20 7573 6566 756c 2069 6e20 7468 ng, useful in th │ │ │ │ -000294c0: 650a 7061 636b 6167 6520 436c 6966 666f e.package Cliffo │ │ │ │ -000294d0: 7264 0a0a 5365 6520 616c 736f 0a3d 3d3d rd..See also.=== │ │ │ │ -000294e0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -000294f0: 2045 6973 656e 6275 6453 6861 6d61 7368 EisenbudShamash │ │ │ │ -00029500: 546f 7461 6c3a 2045 6973 656e 6275 6453 Total: EisenbudS │ │ │ │ -00029510: 6861 6d61 7368 546f 7461 6c2c 202d 2d20 hamashTotal, -- │ │ │ │ -00029520: 5072 6563 7572 736f 7220 636f 6d70 6c65 Precursor comple │ │ │ │ -00029530: 7820 6f66 0a20 2020 2074 6f74 616c 2045 x of. total E │ │ │ │ -00029540: 7874 0a20 202a 202a 6e6f 7465 206e 6577 xt. * *note new │ │ │ │ -00029550: 4578 743a 206e 6577 4578 742c 202d 2d20 Ext: newExt, -- │ │ │ │ -00029560: 476c 6f62 616c 2045 7874 2066 6f72 206d Global Ext for m │ │ │ │ -00029570: 6f64 756c 6573 206f 7665 7220 6120 636f odules over a co │ │ │ │ -00029580: 6d70 6c65 7465 0a20 2020 2049 6e74 6572 mplete. Inter │ │ │ │ -00029590: 7365 6374 696f 6e0a 0a46 756e 6374 696f section..Functio │ │ │ │ -000295a0: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ -000295b0: 2061 7267 756d 656e 7420 6e61 6d65 6420 argument named │ │ │ │ -000295c0: 4772 6164 696e 673a 0a3d 3d3d 3d3d 3d3d Grading:.======= │ │ │ │ +000293d0: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +000293e0: 2020 2020 2020 4569 7365 6e62 7564 5368 EisenbudSh │ │ │ │ +000293f0: 616d 6173 6854 6f74 616c 284d 6261 722c amashTotal(Mbar, │ │ │ │ +00029400: 4772 6164 696e 6720 3d3e 2032 290a 2020 Grading => 2). │ │ │ │ +00029410: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ +00029420: 2a20 4368 6563 6b2c 2061 6e20 2a6e 6f74 * Check, an *not │ │ │ │ +00029430: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ +00029440: 756c 6179 3244 6f63 295a 5a2c 2c20 0a0a ulay2Doc)ZZ,, .. │ │ │ │ +00029450: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00029460: 3d3d 3d3d 3d3d 3d0a 0a69 6620 4772 6164 =======..if Grad │ │ │ │ +00029470: 696e 6720 3d3e 312c 2074 6865 6e20 7468 ing =>1, then th │ │ │ │ +00029480: 6520 6f75 7470 7574 2069 7320 636f 6e76 e output is conv │ │ │ │ +00029490: 6572 7465 6420 746f 2073 696e 676c 652d erted to single- │ │ │ │ +000294a0: 6772 6164 696e 672c 2075 7365 6675 6c20 grading, useful │ │ │ │ +000294b0: 696e 2074 6865 0a70 6163 6b61 6765 2043 in the.package C │ │ │ │ +000294c0: 6c69 6666 6f72 640a 0a53 6565 2061 6c73 lifford..See als │ │ │ │ +000294d0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +000294e0: 2a6e 6f74 6520 4569 7365 6e62 7564 5368 *note EisenbudSh │ │ │ │ +000294f0: 616d 6173 6854 6f74 616c 3a20 4569 7365 amashTotal: Eise │ │ │ │ +00029500: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ +00029510: 2c20 2d2d 2050 7265 6375 7273 6f72 2063 , -- Precursor c │ │ │ │ +00029520: 6f6d 706c 6578 206f 660a 2020 2020 746f omplex of. to │ │ │ │ +00029530: 7461 6c20 4578 740a 2020 2a20 2a6e 6f74 tal Ext. * *not │ │ │ │ +00029540: 6520 6e65 7745 7874 3a20 6e65 7745 7874 e newExt: newExt │ │ │ │ +00029550: 2c20 2d2d 2047 6c6f 6261 6c20 4578 7420 , -- Global Ext │ │ │ │ +00029560: 666f 7220 6d6f 6475 6c65 7320 6f76 6572 for modules over │ │ │ │ +00029570: 2061 2063 6f6d 706c 6574 650a 2020 2020 a complete. │ │ │ │ +00029580: 496e 7465 7273 6563 7469 6f6e 0a0a 4675 Intersection..Fu │ │ │ │ +00029590: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ +000295a0: 696f 6e61 6c20 6172 6775 6d65 6e74 206e ional argument n │ │ │ │ +000295b0: 616d 6564 2047 7261 6469 6e67 3a0a 3d3d amed Grading:.== │ │ │ │ +000295c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 000295d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000295e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000295f0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245 ========.. * "E │ │ │ │ -00029600: 6973 656e 6275 6453 6861 6d61 7368 546f isenbudShamashTo │ │ │ │ -00029610: 7461 6c28 2e2e 2e2c 4772 6164 696e 673d tal(...,Grading= │ │ │ │ -00029620: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e >...)" -- see *n │ │ │ │ -00029630: 6f74 6520 4569 7365 6e62 7564 5368 616d ote EisenbudSham │ │ │ │ -00029640: 6173 6854 6f74 616c 3a0a 2020 2020 4569 ashTotal:. Ei │ │ │ │ -00029650: 7365 6e62 7564 5368 616d 6173 6854 6f74 senbudShamashTot │ │ │ │ -00029660: 616c 2c20 2d2d 2050 7265 6375 7273 6f72 al, -- Precursor │ │ │ │ -00029670: 2063 6f6d 706c 6578 206f 6620 746f 7461 complex of tota │ │ │ │ -00029680: 6c20 4578 740a 2020 2a20 226e 6577 4578 l Ext. * "newEx │ │ │ │ -00029690: 7428 2e2e 2e2c 4772 6164 696e 673d 3e2e t(...,Grading=>. │ │ │ │ -000296a0: 2e2e 2922 202d 2d20 7365 6520 2a6e 6f74 ..)" -- see *not │ │ │ │ -000296b0: 6520 6e65 7745 7874 3a20 6e65 7745 7874 e newExt: newExt │ │ │ │ -000296c0: 2c20 2d2d 2047 6c6f 6261 6c20 4578 7420 , -- Global Ext │ │ │ │ -000296d0: 666f 720a 2020 2020 6d6f 6475 6c65 7320 for. modules │ │ │ │ -000296e0: 6f76 6572 2061 2063 6f6d 706c 6574 6520 over a complete │ │ │ │ -000296f0: 496e 7465 7273 6563 7469 6f6e 0a0a 466f Intersection..Fo │ │ │ │ -00029700: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -00029710: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -00029720: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -00029730: 2a6e 6f74 6520 4772 6164 696e 673a 2047 *note Grading: G │ │ │ │ -00029740: 7261 6469 6e67 2c20 6973 2061 202a 6e6f rading, is a *no │ │ │ │ -00029750: 7465 2073 796d 626f 6c3a 2028 4d61 6361 te symbol: (Maca │ │ │ │ -00029760: 756c 6179 3244 6f63 2953 796d 626f 6c2c ulay2Doc)Symbol, │ │ │ │ -00029770: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +000295e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +000295f0: 202a 2022 4569 7365 6e62 7564 5368 616d * "EisenbudSham │ │ │ │ +00029600: 6173 6854 6f74 616c 282e 2e2e 2c47 7261 ashTotal(...,Gra │ │ │ │ +00029610: 6469 6e67 3d3e 2e2e 2e29 2220 2d2d 2073 ding=>...)" -- s │ │ │ │ +00029620: 6565 202a 6e6f 7465 2045 6973 656e 6275 ee *note Eisenbu │ │ │ │ +00029630: 6453 6861 6d61 7368 546f 7461 6c3a 0a20 dShamashTotal:. │ │ │ │ +00029640: 2020 2045 6973 656e 6275 6453 6861 6d61 EisenbudShama │ │ │ │ +00029650: 7368 546f 7461 6c2c 202d 2d20 5072 6563 shTotal, -- Prec │ │ │ │ +00029660: 7572 736f 7220 636f 6d70 6c65 7820 6f66 ursor complex of │ │ │ │ +00029670: 2074 6f74 616c 2045 7874 0a20 202a 2022 total Ext. * " │ │ │ │ +00029680: 6e65 7745 7874 282e 2e2e 2c47 7261 6469 newExt(...,Gradi │ │ │ │ +00029690: 6e67 3d3e 2e2e 2e29 2220 2d2d 2073 6565 ng=>...)" -- see │ │ │ │ +000296a0: 202a 6e6f 7465 206e 6577 4578 743a 206e *note newExt: n │ │ │ │ +000296b0: 6577 4578 742c 202d 2d20 476c 6f62 616c ewExt, -- Global │ │ │ │ +000296c0: 2045 7874 2066 6f72 0a20 2020 206d 6f64 Ext for. mod │ │ │ │ +000296d0: 756c 6573 206f 7665 7220 6120 636f 6d70 ules over a comp │ │ │ │ +000296e0: 6c65 7465 2049 6e74 6572 7365 6374 696f lete Intersectio │ │ │ │ +000296f0: 6e0a 0a46 6f72 2074 6865 2070 726f 6772 n..For the progr │ │ │ │ +00029700: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +00029710: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +00029720: 6a65 6374 202a 6e6f 7465 2047 7261 6469 ject *note Gradi │ │ │ │ +00029730: 6e67 3a20 4772 6164 696e 672c 2069 7320 ng: Grading, is │ │ │ │ +00029740: 6120 2a6e 6f74 6520 7379 6d62 6f6c 3a20 a *note symbol: │ │ │ │ +00029750: 284d 6163 6175 6c61 7932 446f 6329 5379 (Macaulay2Doc)Sy │ │ │ │ +00029760: 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d mbol,...-------- │ │ │ │ +00029770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000297a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000297b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000297c0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -000297d0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -000297e0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -000297f0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -00029800: 6163 6175 6c61 7932 2d31 2e32 362e 3035 acaulay2-1.26.05 │ │ │ │ -00029810: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -00029820: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ -00029830: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -00029840: 6573 6f6c 7574 696f 6e73 2e6d 323a 3332 esolutions.m2:32 │ │ │ │ -00029850: 3136 3a30 2e0a 1f0a 4669 6c65 3a20 436f 16:0....File: Co │ │ │ │ -00029860: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -00029870: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ -00029880: 666f 2c20 4e6f 6465 3a20 6866 2c20 4e65 fo, Node: hf, Ne │ │ │ │ -00029890: 7874 3a20 6866 4d6f 6475 6c65 4173 4578 xt: hfModuleAsEx │ │ │ │ -000298a0: 742c 2050 7265 763a 2047 7261 6469 6e67 t, Prev: Grading │ │ │ │ -000298b0: 2c20 5570 3a20 546f 700a 0a68 6620 2d2d , Up: Top..hf -- │ │ │ │ -000298c0: 2043 6f6d 7075 7465 7320 7468 6520 6869 Computes the hi │ │ │ │ -000298d0: 6c62 6572 7420 6675 6e63 7469 6f6e 2069 lbert function i │ │ │ │ -000298e0: 6e20 6120 7261 6e67 6520 6f66 2064 6567 n a range of deg │ │ │ │ -000298f0: 7265 6573 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a rees.*********** │ │ │ │ +000297b0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +000297c0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +000297d0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +000297e0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +000297f0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00029800: 3236 2e30 352b 6473 2f4d 322f 4d61 6361 26.05+ds/M2/Maca │ │ │ │ +00029810: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +00029820: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +00029830: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +00029840: 6d32 3a33 3231 363a 302e 0a1f 0a46 696c m2:3216:0....Fil │ │ │ │ +00029850: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ +00029860: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +00029870: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2068 ns.info, Node: h │ │ │ │ +00029880: 662c 204e 6578 743a 2068 664d 6f64 756c f, Next: hfModul │ │ │ │ +00029890: 6541 7345 7874 2c20 5072 6576 3a20 4772 eAsExt, Prev: Gr │ │ │ │ +000298a0: 6164 696e 672c 2055 703a 2054 6f70 0a0a ading, Up: Top.. │ │ │ │ +000298b0: 6866 202d 2d20 436f 6d70 7574 6573 2074 hf -- Computes t │ │ │ │ +000298c0: 6865 2068 696c 6265 7274 2066 756e 6374 he hilbert funct │ │ │ │ +000298d0: 696f 6e20 696e 2061 2072 616e 6765 206f ion in a range o │ │ │ │ +000298e0: 6620 6465 6772 6565 730a 2a2a 2a2a 2a2a f degrees.****** │ │ │ │ +000298f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00029900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00029910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00029920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -00029930: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -00029940: 2020 2020 4820 3d20 6866 2873 2c50 290a H = hf(s,P). │ │ │ │ -00029950: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -00029960: 2020 2a20 732c 2061 202a 6e6f 7465 2073 * s, a *note s │ │ │ │ -00029970: 6571 7565 6e63 653a 2028 4d61 6361 756c equence: (Macaul │ │ │ │ -00029980: 6179 3244 6f63 2953 6571 7565 6e63 652c ay2Doc)Sequence, │ │ │ │ -00029990: 2c20 6f72 204c 6973 740a 2020 2020 2020 , or List. │ │ │ │ -000299a0: 2a20 502c 2061 202a 6e6f 7465 206d 6f64 * P, a *note mod │ │ │ │ -000299b0: 756c 653a 2028 4d61 6361 756c 6179 3244 ule: (Macaulay2D │ │ │ │ -000299c0: 6f63 294d 6f64 756c 652c 2c20 6772 6164 oc)Module,, grad │ │ │ │ -000299d0: 6564 206d 6f64 756c 650a 2020 2a20 4f75 ed module. * Ou │ │ │ │ -000299e0: 7470 7574 733a 0a20 2020 2020 202a 2048 tputs:. * H │ │ │ │ -000299f0: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ -00029a00: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ -00029a10: 7374 2c2c 200a 0a57 6179 7320 746f 2075 st,, ..Ways to u │ │ │ │ -00029a20: 7365 2068 663a 0a3d 3d3d 3d3d 3d3d 3d3d se hf:.========= │ │ │ │ -00029a30: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2268 6628 ======.. * "hf( │ │ │ │ -00029a40: 4c69 7374 2c4d 6f64 756c 6529 220a 2020 List,Module)". │ │ │ │ -00029a50: 2a20 2268 6628 5365 7175 656e 6365 2c4d * "hf(Sequence,M │ │ │ │ -00029a60: 6f64 756c 6529 220a 0a46 6f72 2074 6865 odule)"..For the │ │ │ │ -00029a70: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00029a80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00029a90: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00029aa0: 2068 663a 2068 662c 2069 7320 6120 2a6e hf: hf, is a *n │ │ │ │ -00029ab0: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ -00029ac0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ -00029ad0: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00029ae0: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +00029920: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ +00029930: 0a20 2020 2020 2020 2048 203d 2068 6628 . H = hf( │ │ │ │ +00029940: 732c 5029 0a20 202a 2049 6e70 7574 733a s,P). * Inputs: │ │ │ │ +00029950: 0a20 2020 2020 202a 2073 2c20 6120 2a6e . * s, a *n │ │ │ │ +00029960: 6f74 6520 7365 7175 656e 6365 3a20 284d ote sequence: (M │ │ │ │ +00029970: 6163 6175 6c61 7932 446f 6329 5365 7175 acaulay2Doc)Sequ │ │ │ │ +00029980: 656e 6365 2c2c 206f 7220 4c69 7374 0a20 ence,, or List. │ │ │ │ +00029990: 2020 2020 202a 2050 2c20 6120 2a6e 6f74 * P, a *not │ │ │ │ +000299a0: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ +000299b0: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ +000299c0: 2067 7261 6465 6420 6d6f 6475 6c65 0a20 graded module. │ │ │ │ +000299d0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +000299e0: 2020 2a20 482c 2061 202a 6e6f 7465 206c * H, a *note l │ │ │ │ +000299f0: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ +00029a00: 6f63 294c 6973 742c 2c20 0a0a 5761 7973 oc)List,, ..Ways │ │ │ │ +00029a10: 2074 6f20 7573 6520 6866 3a0a 3d3d 3d3d to use hf:.==== │ │ │ │ +00029a20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +00029a30: 2022 6866 284c 6973 742c 4d6f 6475 6c65 "hf(List,Module │ │ │ │ +00029a40: 2922 0a20 202a 2022 6866 2853 6571 7565 )". * "hf(Seque │ │ │ │ +00029a50: 6e63 652c 4d6f 6475 6c65 2922 0a0a 466f nce,Module)"..Fo │ │ │ │ +00029a60: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00029a70: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00029a80: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00029a90: 2a6e 6f74 6520 6866 3a20 6866 2c20 6973 *note hf: hf, is │ │ │ │ +00029aa0: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ +00029ab0: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ +00029ac0: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ +00029ad0: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +00029ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b30: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00029b40: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00029b50: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00029b60: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00029b70: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -00029b80: 3035 2b64 732f 4d32 2f4d 6163 6175 6c61 05+ds/M2/Macaula │ │ │ │ -00029b90: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d y2/packages/.Com │ │ │ │ -00029ba0: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ -00029bb0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a nResolutions.m2: │ │ │ │ -00029bc0: 3435 3932 3a30 2e0a 1f0a 4669 6c65 3a20 4592:0....File: │ │ │ │ -00029bd0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -00029be0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -00029bf0: 696e 666f 2c20 4e6f 6465 3a20 6866 4d6f info, Node: hfMo │ │ │ │ -00029c00: 6475 6c65 4173 4578 742c 204e 6578 743a duleAsExt, Next: │ │ │ │ -00029c10: 2068 6967 6853 797a 7967 792c 2050 7265 highSyzygy, Pre │ │ │ │ -00029c20: 763a 2068 662c 2055 703a 2054 6f70 0a0a v: hf, Up: Top.. │ │ │ │ -00029c30: 6866 4d6f 6475 6c65 4173 4578 7420 2d2d hfModuleAsExt -- │ │ │ │ -00029c40: 2070 7265 6469 6374 2062 6574 7469 206e predict betti n │ │ │ │ -00029c50: 756d 6265 7273 206f 6620 6d6f 6475 6c65 umbers of module │ │ │ │ -00029c60: 4173 4578 7428 4d2c 5229 0a2a 2a2a 2a2a AsExt(M,R).***** │ │ │ │ +00029b20: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +00029b30: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +00029b40: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +00029b50: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +00029b60: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +00029b70: 312e 3236 2e30 352b 6473 2f4d 322f 4d61 1.26.05+ds/M2/Ma │ │ │ │ +00029b80: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +00029b90: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ +00029ba0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ +00029bb0: 732e 6d32 3a34 3539 323a 302e 0a1f 0a46 s.m2:4592:0....F │ │ │ │ +00029bc0: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ +00029bd0: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00029be0: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ +00029bf0: 2068 664d 6f64 756c 6541 7345 7874 2c20 hfModuleAsExt, │ │ │ │ +00029c00: 4e65 7874 3a20 6869 6768 5379 7a79 6779 Next: highSyzygy │ │ │ │ +00029c10: 2c20 5072 6576 3a20 6866 2c20 5570 3a20 , Prev: hf, Up: │ │ │ │ +00029c20: 546f 700a 0a68 664d 6f64 756c 6541 7345 Top..hfModuleAsE │ │ │ │ +00029c30: 7874 202d 2d20 7072 6564 6963 7420 6265 xt -- predict be │ │ │ │ +00029c40: 7474 6920 6e75 6d62 6572 7320 6f66 206d tti numbers of m │ │ │ │ +00029c50: 6f64 756c 6541 7345 7874 284d 2c52 290a oduleAsExt(M,R). │ │ │ │ +00029c60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00029c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00029c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00029c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00029ca0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ -00029cb0: 3a20 0a20 2020 2020 2020 2073 6571 203d : . seq = │ │ │ │ -00029cc0: 2068 664d 6f64 756c 6541 7345 7874 286e hfModuleAsExt(n │ │ │ │ -00029cd0: 756d 5661 6c75 6573 2c4d 2c6e 756d 6765 umValues,M,numge │ │ │ │ -00029ce0: 6e73 5229 0a20 202a 2049 6e70 7574 733a nsR). * Inputs: │ │ │ │ -00029cf0: 0a20 2020 2020 202a 206e 756d 5661 6c75 . * numValu │ │ │ │ -00029d00: 6573 2c20 616e 202a 6e6f 7465 2069 6e74 es, an *note int │ │ │ │ -00029d10: 6567 6572 3a20 284d 6163 6175 6c61 7932 eger: (Macaulay2 │ │ │ │ -00029d20: 446f 6329 5a5a 2c2c 206e 756d 6265 7220 Doc)ZZ,, number │ │ │ │ -00029d30: 6f66 2076 616c 7565 7320 746f 0a20 2020 of values to. │ │ │ │ -00029d40: 2020 2020 2063 6f6d 7075 7465 0a20 2020 compute. │ │ │ │ -00029d50: 2020 202a 204d 2c20 6120 2a6e 6f74 6520 * M, a *note │ │ │ │ -00029d60: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61 module: (Macaula │ │ │ │ -00029d70: 7932 446f 6329 4d6f 6475 6c65 2c2c 206d y2Doc)Module,, m │ │ │ │ -00029d80: 6f64 756c 6520 6f76 6572 2074 6865 2072 odule over the r │ │ │ │ -00029d90: 696e 6720 6f66 0a20 2020 2020 2020 206f ing of. o │ │ │ │ -00029da0: 7065 7261 746f 7273 0a20 2020 2020 202a perators. * │ │ │ │ -00029db0: 206e 756d 6765 6e73 522c 2061 6e20 2a6e numgensR, an *n │ │ │ │ -00029dc0: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ -00029dd0: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ -00029de0: 6e75 6d62 6572 206f 6620 6765 6e65 7261 number of genera │ │ │ │ -00029df0: 746f 7273 206f 660a 2020 2020 2020 2020 tors of. │ │ │ │ -00029e00: 7468 6520 7461 7267 6574 2072 696e 670a the target ring. │ │ │ │ -00029e10: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -00029e20: 2020 202a 2073 6571 2c20 6120 2a6e 6f74 * seq, a *not │ │ │ │ -00029e30: 6520 7365 7175 656e 6365 3a20 284d 6163 e sequence: (Mac │ │ │ │ -00029e40: 6175 6c61 7932 446f 6329 5365 7175 656e aulay2Doc)Sequen │ │ │ │ -00029e50: 6365 2c2c 2073 6571 7565 6e63 6520 6f66 ce,, sequence of │ │ │ │ -00029e60: 206e 756d 5661 6c75 6573 0a20 2020 2020 numValues. │ │ │ │ -00029e70: 2020 2069 6e74 6567 6572 732c 2074 6865 integers, the │ │ │ │ -00029e80: 2065 7870 6563 7465 6420 746f 7461 6c20 expected total │ │ │ │ -00029e90: 4265 7474 6920 6e75 6d62 6572 730a 0a44 Betti numbers..D │ │ │ │ -00029ea0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -00029eb0: 3d3d 3d3d 3d3d 0a0a 4769 7665 6e20 6120 ======..Given a │ │ │ │ -00029ec0: 6d6f 6475 6c65 204d 206f 7665 7220 7468 module M over th │ │ │ │ -00029ed0: 6520 7269 6e67 206f 6620 6f70 6572 6174 e ring of operat │ │ │ │ -00029ee0: 6f72 7320 246b 5b78 5f31 2e2e 785f 635d ors $k[x_1..x_c] │ │ │ │ -00029ef0: 242c 2074 6865 2063 616c 6c20 244e 203d $, the call $N = │ │ │ │ -00029f00: 0a6d 6f64 756c 6541 7345 7874 284d 2c52 .moduleAsExt(M,R │ │ │ │ -00029f10: 2924 2070 726f 6475 6365 7320 6120 6d6f )$ produces a mo │ │ │ │ -00029f20: 6475 6c65 204e 206f 7665 7220 7468 6520 dule N over the │ │ │ │ -00029f30: 7269 6e67 2052 2077 686f 7365 2065 7874 ring R whose ext │ │ │ │ -00029f40: 206d 6f64 756c 6520 6973 2074 6865 0a65 module is the.e │ │ │ │ -00029f50: 7874 6572 696f 7220 616c 6765 6272 6120 xterior algebra │ │ │ │ -00029f60: 6f6e 206e 3d6e 756d 6765 6e73 5220 6765 on n=numgensR ge │ │ │ │ -00029f70: 6e65 7261 746f 7273 2074 656e 736f 7265 nerators tensore │ │ │ │ -00029f80: 6420 7769 7468 204d 2e20 5468 6973 2073 d with M. This s │ │ │ │ -00029f90: 6372 6970 7420 636f 6d70 7574 6573 0a6e cript computes.n │ │ │ │ -00029fa0: 756d 5661 6c75 6573 2076 616c 7565 7320 umValues values │ │ │ │ -00029fb0: 6f66 2074 6865 2048 696c 6265 7274 2066 of the Hilbert f │ │ │ │ -00029fc0: 756e 6374 696f 6e20 6f66 2024 2420 4d20 unction of $$ M │ │ │ │ -00029fd0: 5c6f 7469 6d65 7320 5c77 6564 6765 206b \otimes \wedge k │ │ │ │ -00029fe0: 5e6e 2c20 2424 2077 6869 6368 0a73 686f ^n, $$ which.sho │ │ │ │ -00029ff0: 756c 6420 6265 2065 7175 616c 2074 6f20 uld be equal to │ │ │ │ -0002a000: 7468 6520 746f 7461 6c20 6265 7474 6920 the total betti │ │ │ │ -0002a010: 6e75 6d62 6572 7320 6f66 204e 2e0a 0a2b numbers of N...+ │ │ │ │ +00029c90: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +00029ca0: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ +00029cb0: 7365 7120 3d20 6866 4d6f 6475 6c65 4173 seq = hfModuleAs │ │ │ │ +00029cc0: 4578 7428 6e75 6d56 616c 7565 732c 4d2c Ext(numValues,M, │ │ │ │ +00029cd0: 6e75 6d67 656e 7352 290a 2020 2a20 496e numgensR). * In │ │ │ │ +00029ce0: 7075 7473 3a0a 2020 2020 2020 2a20 6e75 puts:. * nu │ │ │ │ +00029cf0: 6d56 616c 7565 732c 2061 6e20 2a6e 6f74 mValues, an *not │ │ │ │ +00029d00: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ +00029d10: 756c 6179 3244 6f63 295a 5a2c 2c20 6e75 ulay2Doc)ZZ,, nu │ │ │ │ +00029d20: 6d62 6572 206f 6620 7661 6c75 6573 2074 mber of values t │ │ │ │ +00029d30: 6f0a 2020 2020 2020 2020 636f 6d70 7574 o. comput │ │ │ │ +00029d40: 650a 2020 2020 2020 2a20 4d2c 2061 202a e. * M, a * │ │ │ │ +00029d50: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ +00029d60: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ +00029d70: 652c 2c20 6d6f 6475 6c65 206f 7665 7220 e,, module over │ │ │ │ +00029d80: 7468 6520 7269 6e67 206f 660a 2020 2020 the ring of. │ │ │ │ +00029d90: 2020 2020 6f70 6572 6174 6f72 730a 2020 operators. │ │ │ │ +00029da0: 2020 2020 2a20 6e75 6d67 656e 7352 2c20 * numgensR, │ │ │ │ +00029db0: 616e 202a 6e6f 7465 2069 6e74 6567 6572 an *note integer │ │ │ │ +00029dc0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00029dd0: 5a5a 2c2c 206e 756d 6265 7220 6f66 2067 ZZ,, number of g │ │ │ │ +00029de0: 656e 6572 6174 6f72 7320 6f66 0a20 2020 enerators of. │ │ │ │ +00029df0: 2020 2020 2074 6865 2074 6172 6765 7420 the target │ │ │ │ +00029e00: 7269 6e67 0a20 202a 204f 7574 7075 7473 ring. * Outputs │ │ │ │ +00029e10: 3a0a 2020 2020 2020 2a20 7365 712c 2061 :. * seq, a │ │ │ │ +00029e20: 202a 6e6f 7465 2073 6571 7565 6e63 653a *note sequence: │ │ │ │ +00029e30: 2028 4d61 6361 756c 6179 3244 6f63 2953 (Macaulay2Doc)S │ │ │ │ +00029e40: 6571 7565 6e63 652c 2c20 7365 7175 656e equence,, sequen │ │ │ │ +00029e50: 6365 206f 6620 6e75 6d56 616c 7565 730a ce of numValues. │ │ │ │ +00029e60: 2020 2020 2020 2020 696e 7465 6765 7273 integers │ │ │ │ +00029e70: 2c20 7468 6520 6578 7065 6374 6564 2074 , the expected t │ │ │ │ +00029e80: 6f74 616c 2042 6574 7469 206e 756d 6265 otal Betti numbe │ │ │ │ +00029e90: 7273 0a0a 4465 7363 7269 7074 696f 6e0a rs..Description. │ │ │ │ +00029ea0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a47 6976 ===========..Giv │ │ │ │ +00029eb0: 656e 2061 206d 6f64 756c 6520 4d20 6f76 en a module M ov │ │ │ │ +00029ec0: 6572 2074 6865 2072 696e 6720 6f66 206f er the ring of o │ │ │ │ +00029ed0: 7065 7261 746f 7273 2024 6b5b 785f 312e perators $k[x_1. │ │ │ │ +00029ee0: 2e78 5f63 5d24 2c20 7468 6520 6361 6c6c .x_c]$, the call │ │ │ │ +00029ef0: 2024 4e20 3d0a 6d6f 6475 6c65 4173 4578 $N =.moduleAsEx │ │ │ │ +00029f00: 7428 4d2c 5229 2420 7072 6f64 7563 6573 t(M,R)$ produces │ │ │ │ +00029f10: 2061 206d 6f64 756c 6520 4e20 6f76 6572 a module N over │ │ │ │ +00029f20: 2074 6865 2072 696e 6720 5220 7768 6f73 the ring R whos │ │ │ │ +00029f30: 6520 6578 7420 6d6f 6475 6c65 2069 7320 e ext module is │ │ │ │ +00029f40: 7468 650a 6578 7465 7269 6f72 2061 6c67 the.exterior alg │ │ │ │ +00029f50: 6562 7261 206f 6e20 6e3d 6e75 6d67 656e ebra on n=numgen │ │ │ │ +00029f60: 7352 2067 656e 6572 6174 6f72 7320 7465 sR generators te │ │ │ │ +00029f70: 6e73 6f72 6564 2077 6974 6820 4d2e 2054 nsored with M. T │ │ │ │ +00029f80: 6869 7320 7363 7269 7074 2063 6f6d 7075 his script compu │ │ │ │ +00029f90: 7465 730a 6e75 6d56 616c 7565 7320 7661 tes.numValues va │ │ │ │ +00029fa0: 6c75 6573 206f 6620 7468 6520 4869 6c62 lues of the Hilb │ │ │ │ +00029fb0: 6572 7420 6675 6e63 7469 6f6e 206f 6620 ert function of │ │ │ │ +00029fc0: 2424 204d 205c 6f74 696d 6573 205c 7765 $$ M \otimes \we │ │ │ │ +00029fd0: 6467 6520 6b5e 6e2c 2024 2420 7768 6963 dge k^n, $$ whic │ │ │ │ +00029fe0: 680a 7368 6f75 6c64 2062 6520 6571 7561 h.should be equa │ │ │ │ +00029ff0: 6c20 746f 2074 6865 2074 6f74 616c 2062 l to the total b │ │ │ │ +0002a000: 6574 7469 206e 756d 6265 7273 206f 6620 etti numbers of │ │ │ │ +0002a010: 4e2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d N...+----------- │ │ │ │ 0002a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a050: 2b0a 7c69 3120 3a20 6b6b 203d 205a 5a2f +.|i1 : kk = ZZ/ │ │ │ │ -0002a060: 3130 313b 2020 2020 2020 2020 2020 2020 101; │ │ │ │ -0002a070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a080: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002a040: 2d2d 2d2d 2d2b 0a7c 6931 203a 206b 6b20 -----+.|i1 : kk │ │ │ │ +0002a050: 3d20 5a5a 2f31 3031 3b20 2020 2020 2020 = ZZ/101; │ │ │ │ +0002a060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a070: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0002a080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a0b0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 ------+.|i2 : S │ │ │ │ -0002a0c0: 3d20 6b6b 5b61 2c62 2c63 5d3b 2020 2020 = kk[a,b,c]; │ │ │ │ -0002a0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a0e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0002a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +0002a0b0: 203a 2053 203d 206b 6b5b 612c 622c 635d : S = kk[a,b,c] │ │ │ │ +0002a0c0: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +0002a0d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002a0e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0002a0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002a120: 3320 3a20 6666 203d 206d 6174 7269 787b 3 : ff = matrix{ │ │ │ │ -0002a130: 7b61 5e34 2c20 625e 342c 635e 347d 7d3b {a^4, b^4,c^4}}; │ │ │ │ -0002a140: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002a150: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002a110: 2d2b 0a7c 6933 203a 2066 6620 3d20 6d61 -+.|i3 : ff = ma │ │ │ │ +0002a120: 7472 6978 7b7b 615e 342c 2062 5e34 2c63 trix{{a^4, b^4,c │ │ │ │ +0002a130: 5e34 7d7d 3b20 2020 2020 2020 2020 2020 ^4}}; │ │ │ │ +0002a140: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a180: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002a190: 2020 3120 2020 2020 2033 2020 2020 2020 1 3 │ │ │ │ -0002a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a1b0: 2020 2020 207c 0a7c 6f33 203a 204d 6174 |.|o3 : Mat │ │ │ │ -0002a1c0: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ -0002a1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a1e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0002a170: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002a180: 2020 2020 2020 2031 2020 2020 2020 3320 1 3 │ │ │ │ +0002a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a1a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +0002a1b0: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20 : Matrix S <-- │ │ │ │ +0002a1c0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +0002a1d0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0002a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ -0002a220: 203a 2052 203d 2053 2f69 6465 616c 2066 : R = S/ideal f │ │ │ │ -0002a230: 663b 2020 2020 2020 2020 2020 2020 2020 f; │ │ │ │ -0002a240: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a250: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002a210: 2b0a 7c69 3420 3a20 5220 3d20 532f 6964 +.|i4 : R = S/id │ │ │ │ +0002a220: 6561 6c20 6666 3b20 2020 2020 2020 2020 eal ff; │ │ │ │ +0002a230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a240: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002a250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a280: 2d2b 0a7c 6935 203a 204f 7073 203d 206b -+.|i5 : Ops = k │ │ │ │ -0002a290: 6b5b 785f 312c 785f 322c 785f 335d 3b20 k[x_1,x_2,x_3]; │ │ │ │ -0002a2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a2b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002a270: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 4f70 ------+.|i5 : Op │ │ │ │ +0002a280: 7320 3d20 6b6b 5b78 5f31 2c78 5f32 2c78 s = kk[x_1,x_2,x │ │ │ │ +0002a290: 5f33 5d3b 2020 2020 2020 2020 2020 2020 _3]; │ │ │ │ +0002a2a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0002a2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a2e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204d -------+.|i6 : M │ │ │ │ -0002a2f0: 4d20 3d20 4f70 735e 312f 2878 5f31 2a69 M = Ops^1/(x_1*i │ │ │ │ -0002a300: 6465 616c 2878 5f32 5e32 2c78 5f33 2929 deal(x_2^2,x_3)) │ │ │ │ -0002a310: 3b20 2020 2020 2020 2020 7c0a 2b2d 2d2d ; |.+--- │ │ │ │ +0002a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0002a2e0: 3620 3a20 4d4d 203d 204f 7073 5e31 2f28 6 : MM = Ops^1/( │ │ │ │ +0002a2f0: 785f 312a 6964 6561 6c28 785f 325e 322c x_1*ideal(x_2^2, │ │ │ │ +0002a300: 785f 3329 293b 2020 2020 2020 2020 207c x_3)); | │ │ │ │ +0002a310: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0002a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0002a350: 6937 203a 204e 203d 206d 6f64 756c 6541 i7 : N = moduleA │ │ │ │ -0002a360: 7345 7874 284d 4d2c 5229 3b20 2020 2020 sExt(MM,R); │ │ │ │ -0002a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a380: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0002a340: 2d2d 2b0a 7c69 3720 3a20 4e20 3d20 6d6f --+.|i7 : N = mo │ │ │ │ +0002a350: 6475 6c65 4173 4578 7428 4d4d 2c52 293b duleAsExt(MM,R); │ │ │ │ +0002a360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a370: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002a380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a3b0: 2d2d 2d2b 0a7c 6938 203a 2062 6574 7469 ---+.|i8 : betti │ │ │ │ -0002a3c0: 2066 7265 6552 6573 6f6c 7574 696f 6e28 freeResolution( │ │ │ │ -0002a3d0: 204e 2c20 4c65 6e67 7468 4c69 6d69 7420 N, LengthLimit │ │ │ │ -0002a3e0: 3d3e 2031 3029 7c0a 7c20 2020 2020 2020 => 10)|.| │ │ │ │ +0002a3a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 --------+.|i8 : │ │ │ │ +0002a3b0: 6265 7474 6920 6672 6565 5265 736f 6c75 betti freeResolu │ │ │ │ +0002a3c0: 7469 6f6e 2820 4e2c 204c 656e 6774 684c tion( N, LengthL │ │ │ │ +0002a3d0: 696d 6974 203d 3e20 3130 297c 0a7c 2020 imit => 10)|.| │ │ │ │ +0002a3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a410: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0002a420: 2020 2020 2020 2020 2030 2020 3120 2032 0 1 2 │ │ │ │ -0002a430: 2020 3320 2034 2020 3520 2036 2020 3720 3 4 5 6 7 │ │ │ │ -0002a440: 2038 2020 3920 3130 2020 2020 7c0a 7c6f 8 9 10 |.|o │ │ │ │ -0002a450: 3820 3d20 746f 7461 6c3a 2033 3620 3237 8 = total: 36 27 │ │ │ │ -0002a460: 2032 3920 3331 2033 3320 3335 2033 3720 29 31 33 35 37 │ │ │ │ -0002a470: 3339 2034 3120 3433 2034 3520 2020 207c 39 41 43 45 | │ │ │ │ -0002a480: 0a7c 2020 2020 2020 2020 2d36 3a20 3138 .| -6: 18 │ │ │ │ -0002a490: 2020 3620 202e 2020 2e20 202e 2020 2e20 6 . . . . │ │ │ │ -0002a4a0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0002a4b0: 2020 7c0a 7c20 2020 2020 2020 202d 353a |.| -5: │ │ │ │ -0002a4c0: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ -0002a4d0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0002a4e0: 2e20 2020 207c 0a7c 2020 2020 2020 2020 . |.| │ │ │ │ -0002a4f0: 2d34 3a20 3138 2032 3120 3231 2020 3720 -4: 18 21 21 7 │ │ │ │ -0002a500: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0002a510: 2e20 202e 2020 2020 7c0a 7c20 2020 2020 . . |.| │ │ │ │ -0002a520: 2020 202d 333a 2020 2e20 202e 2020 2e20 -3: . . . │ │ │ │ -0002a530: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0002a540: 2e20 202e 2020 2e20 2020 207c 0a7c 2020 . . . |.| │ │ │ │ -0002a550: 2020 2020 2020 2d32 3a20 202e 2020 2e20 -2: . . │ │ │ │ -0002a560: 2038 2032 3420 3234 2020 3820 202e 2020 8 24 24 8 . │ │ │ │ -0002a570: 2e20 202e 2020 2e20 202e 2020 2020 7c0a . . . . |. │ │ │ │ -0002a580: 7c20 2020 2020 2020 202d 313a 2020 2e20 | -1: . │ │ │ │ -0002a590: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0002a5a0: 2e20 202e 2020 2e20 202e 2020 2e20 2020 . . . . . │ │ │ │ -0002a5b0: 207c 0a7c 2020 2020 2020 2020 2030 3a20 |.| 0: │ │ │ │ -0002a5c0: 202e 2020 2e20 202e 2020 2e20 2039 2032 . . . . 9 2 │ │ │ │ -0002a5d0: 3720 3237 2020 3920 202e 2020 2e20 202e 7 27 9 . . . │ │ │ │ -0002a5e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002a5f0: 313a 2020 2e20 202e 2020 2e20 202e 2020 1: . . . . │ │ │ │ -0002a600: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0002a610: 2020 2e20 2020 207c 0a7c 2020 2020 2020 . |.| │ │ │ │ -0002a620: 2020 2032 3a20 202e 2020 2e20 202e 2020 2: . . . │ │ │ │ -0002a630: 2e20 202e 2020 2e20 3130 2033 3020 3330 . . . 10 30 30 │ │ │ │ -0002a640: 2031 3020 202e 2020 2020 7c0a 7c20 2020 10 . |.| │ │ │ │ -0002a650: 2020 2020 2020 333a 2020 2e20 202e 2020 3: . . │ │ │ │ -0002a660: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0002a670: 2020 2e20 202e 2020 2e20 2020 207c 0a7c . . . |.| │ │ │ │ -0002a680: 2020 2020 2020 2020 2034 3a20 202e 2020 4: . │ │ │ │ -0002a690: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0002a6a0: 2020 2e20 3131 2033 3320 3333 2020 2020 . 11 33 33 │ │ │ │ -0002a6b0: 7c0a 7c20 2020 2020 2020 2020 353a 2020 |.| 5: │ │ │ │ -0002a6c0: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0002a6d0: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ -0002a6e0: 2020 207c 0a7c 2020 2020 2020 2020 2036 |.| 6 │ │ │ │ -0002a6f0: 3a20 202e 2020 2e20 202e 2020 2e20 202e : . . . . . │ │ │ │ -0002a700: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ -0002a710: 3132 2020 2020 7c0a 7c20 2020 2020 2020 12 |.| │ │ │ │ +0002a400: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002a410: 7c20 2020 2020 2020 2020 2020 2020 3020 | 0 │ │ │ │ +0002a420: 2031 2020 3220 2033 2020 3420 2035 2020 1 2 3 4 5 │ │ │ │ +0002a430: 3620 2037 2020 3820 2039 2031 3020 2020 6 7 8 9 10 │ │ │ │ +0002a440: 207c 0a7c 6f38 203d 2074 6f74 616c 3a20 |.|o8 = total: │ │ │ │ +0002a450: 3336 2032 3720 3239 2033 3120 3333 2033 36 27 29 31 33 3 │ │ │ │ +0002a460: 3520 3337 2033 3920 3431 2034 3320 3435 5 37 39 41 43 45 │ │ │ │ +0002a470: 2020 2020 7c0a 7c20 2020 2020 2020 202d |.| - │ │ │ │ +0002a480: 363a 2031 3820 2036 2020 2e20 202e 2020 6: 18 6 . . │ │ │ │ +0002a490: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ +0002a4a0: 2020 2e20 2020 207c 0a7c 2020 2020 2020 . |.| │ │ │ │ +0002a4b0: 2020 2d35 3a20 202e 2020 2e20 202e 2020 -5: . . . │ │ │ │ +0002a4c0: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ +0002a4d0: 2020 2e20 202e 2020 2020 7c0a 7c20 2020 . . |.| │ │ │ │ +0002a4e0: 2020 2020 202d 343a 2031 3820 3231 2032 -4: 18 21 2 │ │ │ │ +0002a4f0: 3120 2037 2020 2e20 202e 2020 2e20 202e 1 7 . . . . │ │ │ │ +0002a500: 2020 2e20 202e 2020 2e20 2020 207c 0a7c . . . |.| │ │ │ │ +0002a510: 2020 2020 2020 2020 2d33 3a20 202e 2020 -3: . │ │ │ │ +0002a520: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ +0002a530: 2020 2e20 202e 2020 2e20 202e 2020 2020 . . . . │ │ │ │ +0002a540: 7c0a 7c20 2020 2020 2020 202d 323a 2020 |.| -2: │ │ │ │ +0002a550: 2e20 202e 2020 3820 3234 2032 3420 2038 . . 8 24 24 8 │ │ │ │ +0002a560: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0002a570: 2020 207c 0a7c 2020 2020 2020 2020 2d31 |.| -1 │ │ │ │ +0002a580: 3a20 202e 2020 2e20 202e 2020 2e20 202e : . . . . . │ │ │ │ +0002a590: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0002a5a0: 202e 2020 2020 7c0a 7c20 2020 2020 2020 . |.| │ │ │ │ +0002a5b0: 2020 303a 2020 2e20 202e 2020 2e20 202e 0: . . . . │ │ │ │ +0002a5c0: 2020 3920 3237 2032 3720 2039 2020 2e20 9 27 27 9 . │ │ │ │ +0002a5d0: 202e 2020 2e20 2020 207c 0a7c 2020 2020 . . |.| │ │ │ │ +0002a5e0: 2020 2020 2031 3a20 202e 2020 2e20 202e 1: . . . │ │ │ │ +0002a5f0: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0002a600: 202e 2020 2e20 202e 2020 2020 7c0a 7c20 . . . |.| │ │ │ │ +0002a610: 2020 2020 2020 2020 323a 2020 2e20 202e 2: . . │ │ │ │ +0002a620: 2020 2e20 202e 2020 2e20 202e 2031 3020 . . . . 10 │ │ │ │ +0002a630: 3330 2033 3020 3130 2020 2e20 2020 207c 30 30 10 . | │ │ │ │ +0002a640: 0a7c 2020 2020 2020 2020 2033 3a20 202e .| 3: . │ │ │ │ +0002a650: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0002a660: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ +0002a670: 2020 7c0a 7c20 2020 2020 2020 2020 343a |.| 4: │ │ │ │ +0002a680: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0002a690: 202e 2020 2e20 202e 2031 3120 3333 2033 . . . 11 33 3 │ │ │ │ +0002a6a0: 3320 2020 207c 0a7c 2020 2020 2020 2020 3 |.| │ │ │ │ +0002a6b0: 2035 3a20 202e 2020 2e20 202e 2020 2e20 5: . . . . │ │ │ │ +0002a6c0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ +0002a6d0: 2e20 202e 2020 2020 7c0a 7c20 2020 2020 . . |.| │ │ │ │ +0002a6e0: 2020 2020 363a 2020 2e20 202e 2020 2e20 6: . . . │ │ │ │ +0002a6f0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ +0002a700: 2e20 202e 2031 3220 2020 207c 0a7c 2020 . . 12 |.| │ │ │ │ +0002a710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a740: 2020 2020 2020 2020 207c 0a7c 6f38 203a |.|o8 : │ │ │ │ -0002a750: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +0002a730: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002a740: 7c6f 3820 3a20 4265 7474 6954 616c 6c79 |o8 : BettiTally │ │ │ │ +0002a750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a770: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0002a770: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0002a780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002a7b0: 0a7c 6939 203a 2068 664d 6f64 756c 6541 .|i9 : hfModuleA │ │ │ │ -0002a7c0: 7345 7874 2831 322c 4d4d 2c33 2920 2020 sExt(12,MM,3) │ │ │ │ -0002a7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a7e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002a7a0: 2d2d 2d2d 2b0a 7c69 3920 3a20 6866 4d6f ----+.|i9 : hfMo │ │ │ │ +0002a7b0: 6475 6c65 4173 4578 7428 3132 2c4d 4d2c duleAsExt(12,MM, │ │ │ │ +0002a7c0: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ +0002a7d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002a7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a810: 2020 2020 207c 0a7c 6f39 203d 2028 3233 |.|o9 = (23 │ │ │ │ -0002a820: 2c20 3235 2c20 3237 2c20 3239 2c20 3331 , 25, 27, 29, 31 │ │ │ │ -0002a830: 2c20 3333 2c20 3335 2c20 3337 2c20 3339 , 33, 35, 37, 39 │ │ │ │ -0002a840: 2c20 3431 2920 2020 7c0a 7c20 2020 2020 , 41) |.| │ │ │ │ +0002a800: 2020 2020 2020 2020 2020 7c0a 7c6f 3920 |.|o9 │ │ │ │ +0002a810: 3d20 2832 332c 2032 352c 2032 372c 2032 = (23, 25, 27, 2 │ │ │ │ +0002a820: 392c 2033 312c 2033 332c 2033 352c 2033 9, 31, 33, 35, 3 │ │ │ │ +0002a830: 372c 2033 392c 2034 3129 2020 207c 0a7c 7, 39, 41) |.| │ │ │ │ +0002a840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a870: 2020 2020 2020 2020 2020 207c 0a7c 6f39 |.|o9 │ │ │ │ -0002a880: 203a 2053 6571 7565 6e63 6520 2020 2020 : Sequence │ │ │ │ +0002a870: 7c0a 7c6f 3920 3a20 5365 7175 656e 6365 |.|o9 : Sequence │ │ │ │ +0002a880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a8a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a8b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002a8a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002a8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a8e0: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d -+..See also.=== │ │ │ │ -0002a8f0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -0002a900: 206d 6f64 756c 6541 7345 7874 3a20 6d6f moduleAsExt: mo │ │ │ │ -0002a910: 6475 6c65 4173 4578 742c 202d 2d20 4669 duleAsExt, -- Fi │ │ │ │ -0002a920: 6e64 2061 206d 6f64 756c 6520 7769 7468 nd a module with │ │ │ │ -0002a930: 2067 6976 656e 2061 7379 6d70 746f 7469 given asymptoti │ │ │ │ -0002a940: 630a 2020 2020 7265 736f 6c75 7469 6f6e c. resolution │ │ │ │ -0002a950: 0a0a 5761 7973 2074 6f20 7573 6520 6866 ..Ways to use hf │ │ │ │ -0002a960: 4d6f 6475 6c65 4173 4578 743a 0a3d 3d3d ModuleAsExt:.=== │ │ │ │ -0002a970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002a980: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6866 =======.. * "hf │ │ │ │ -0002a990: 4d6f 6475 6c65 4173 4578 7428 5a5a 2c4d ModuleAsExt(ZZ,M │ │ │ │ -0002a9a0: 6f64 756c 652c 5a5a 2922 0a0a 466f 7220 odule,ZZ)"..For │ │ │ │ -0002a9b0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -0002a9c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002a9d0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -0002a9e0: 6f74 6520 6866 4d6f 6475 6c65 4173 4578 ote hfModuleAsEx │ │ │ │ -0002a9f0: 743a 2068 664d 6f64 756c 6541 7345 7874 t: hfModuleAsExt │ │ │ │ -0002aa00: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -0002aa10: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod function:.(M │ │ │ │ -0002aa20: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -0002aa30: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d odFunction,...-- │ │ │ │ +0002a8d0: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 ------+..See als │ │ │ │ +0002a8e0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +0002a8f0: 2a6e 6f74 6520 6d6f 6475 6c65 4173 4578 *note moduleAsEx │ │ │ │ +0002a900: 743a 206d 6f64 756c 6541 7345 7874 2c20 t: moduleAsExt, │ │ │ │ +0002a910: 2d2d 2046 696e 6420 6120 6d6f 6475 6c65 -- Find a module │ │ │ │ +0002a920: 2077 6974 6820 6769 7665 6e20 6173 796d with given asym │ │ │ │ +0002a930: 7074 6f74 6963 0a20 2020 2072 6573 6f6c ptotic. resol │ │ │ │ +0002a940: 7574 696f 6e0a 0a57 6179 7320 746f 2075 ution..Ways to u │ │ │ │ +0002a950: 7365 2068 664d 6f64 756c 6541 7345 7874 se hfModuleAsExt │ │ │ │ +0002a960: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +0002a970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +0002a980: 2a20 2268 664d 6f64 756c 6541 7345 7874 * "hfModuleAsExt │ │ │ │ +0002a990: 285a 5a2c 4d6f 6475 6c65 2c5a 5a29 220a (ZZ,Module,ZZ)". │ │ │ │ +0002a9a0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0002a9b0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0002a9c0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0002a9d0: 6374 202a 6e6f 7465 2068 664d 6f64 756c ct *note hfModul │ │ │ │ +0002a9e0: 6541 7345 7874 3a20 6866 4d6f 6475 6c65 eAsExt: hfModule │ │ │ │ +0002a9f0: 4173 4578 742c 2069 7320 6120 2a6e 6f74 AsExt, is a *not │ │ │ │ +0002aa00: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +0002aa10: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63 n:.(Macaulay2Doc │ │ │ │ +0002aa20: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ +0002aa30: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 0002aa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002aa50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002aa60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002aa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002aa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -0002aa90: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -0002aaa0: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -0002aab0: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -0002aac0: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -0002aad0: 6179 322d 312e 3236 2e30 352b 6473 2f4d ay2-1.26.05+ds/M │ │ │ │ -0002aae0: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -0002aaf0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ -0002ab00: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -0002ab10: 7469 6f6e 732e 6d32 3a33 3134 313a 302e tions.m2:3141:0. │ │ │ │ -0002ab20: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ -0002ab30: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -0002ab40: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ -0002ab50: 6f64 653a 2068 6967 6853 797a 7967 792c ode: highSyzygy, │ │ │ │ -0002ab60: 204e 6578 743a 2068 4d61 7073 2c20 5072 Next: hMaps, Pr │ │ │ │ -0002ab70: 6576 3a20 6866 4d6f 6475 6c65 4173 4578 ev: hfModuleAsEx │ │ │ │ -0002ab80: 742c 2055 703a 2054 6f70 0a0a 6869 6768 t, Up: Top..high │ │ │ │ -0002ab90: 5379 7a79 6779 202d 2d20 5265 7475 726e Syzygy -- Return │ │ │ │ -0002aba0: 7320 6120 7379 7a79 6779 206d 6f64 756c s a syzygy modul │ │ │ │ -0002abb0: 6520 6f6e 6520 6265 796f 6e64 2074 6865 e one beyond the │ │ │ │ -0002abc0: 2072 6567 756c 6172 6974 7920 6f66 2045 regularity of E │ │ │ │ -0002abd0: 7874 284d 2c6b 290a 2a2a 2a2a 2a2a 2a2a xt(M,k).******** │ │ │ │ +0002aa80: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0002aa90: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0002aaa0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0002aab0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0002aac0: 6163 6175 6c61 7932 2d31 2e32 362e 3035 acaulay2-1.26.05 │ │ │ │ +0002aad0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +0002aae0: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ +0002aaf0: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +0002ab00: 6573 6f6c 7574 696f 6e73 2e6d 323a 3331 esolutions.m2:31 │ │ │ │ +0002ab10: 3431 3a30 2e0a 1f0a 4669 6c65 3a20 436f 41:0....File: Co │ │ │ │ +0002ab20: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +0002ab30: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ +0002ab40: 666f 2c20 4e6f 6465 3a20 6869 6768 5379 fo, Node: highSy │ │ │ │ +0002ab50: 7a79 6779 2c20 4e65 7874 3a20 684d 6170 zygy, Next: hMap │ │ │ │ +0002ab60: 732c 2050 7265 763a 2068 664d 6f64 756c s, Prev: hfModul │ │ │ │ +0002ab70: 6541 7345 7874 2c20 5570 3a20 546f 700a eAsExt, Up: Top. │ │ │ │ +0002ab80: 0a68 6967 6853 797a 7967 7920 2d2d 2052 .highSyzygy -- R │ │ │ │ +0002ab90: 6574 7572 6e73 2061 2073 797a 7967 7920 eturns a syzygy │ │ │ │ +0002aba0: 6d6f 6475 6c65 206f 6e65 2062 6579 6f6e module one beyon │ │ │ │ +0002abb0: 6420 7468 6520 7265 6775 6c61 7269 7479 d the regularity │ │ │ │ +0002abc0: 206f 6620 4578 7428 4d2c 6b29 0a2a 2a2a of Ext(M,k).*** │ │ │ │ +0002abd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002abe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002abf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002ac00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002ac10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002ac20: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -0002ac30: 0a20 2020 2020 2020 204d 203d 2068 6967 . M = hig │ │ │ │ -0002ac40: 6853 797a 7967 7920 4d30 0a20 202a 2049 hSyzygy M0. * I │ │ │ │ -0002ac50: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ -0002ac60: 302c 2061 202a 6e6f 7465 206d 6f64 756c 0, a *note modul │ │ │ │ -0002ac70: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -0002ac80: 294d 6f64 756c 652c 2c20 6f76 6572 2061 )Module,, over a │ │ │ │ -0002ac90: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -0002aca0: 6563 7469 6f6e 0a20 2020 2020 2020 2072 ection. r │ │ │ │ -0002acb0: 696e 670a 2020 2a20 2a6e 6f74 6520 4f70 ing. * *note Op │ │ │ │ -0002acc0: 7469 6f6e 616c 2069 6e70 7574 733a 2028 tional inputs: ( │ │ │ │ -0002acd0: 4d61 6361 756c 6179 3244 6f63 2975 7369 Macaulay2Doc)usi │ │ │ │ -0002ace0: 6e67 2066 756e 6374 696f 6e73 2077 6974 ng functions wit │ │ │ │ -0002acf0: 6820 6f70 7469 6f6e 616c 2069 6e70 7574 h optional input │ │ │ │ -0002ad00: 732c 3a0a 2020 2020 2020 2a20 4f70 7469 s,:. * Opti │ │ │ │ -0002ad10: 6d69 736d 203d 3e20 2e2e 2e2c 2064 6566 mism => ..., def │ │ │ │ -0002ad20: 6175 6c74 2076 616c 7565 2030 0a20 202a ault value 0. * │ │ │ │ -0002ad30: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -0002ad40: 2a20 4d2c 2061 202a 6e6f 7465 206d 6f64 * M, a *note mod │ │ │ │ -0002ad50: 756c 653a 2028 4d61 6361 756c 6179 3244 ule: (Macaulay2D │ │ │ │ -0002ad60: 6f63 294d 6f64 756c 652c 2c20 6120 7379 oc)Module,, a sy │ │ │ │ -0002ad70: 7a79 6779 206d 6f64 756c 6520 6f66 204d zygy module of M │ │ │ │ -0002ad80: 300a 0a44 6573 6372 6970 7469 6f6e 0a3d 0..Description.= │ │ │ │ -0002ad90: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4120 2268 ==========..A "h │ │ │ │ -0002ada0: 6967 6820 7379 7a79 6779 2220 6f76 6572 igh syzygy" over │ │ │ │ -0002adb0: 2061 2063 6f6d 706c 6574 6520 696e 7465 a complete inte │ │ │ │ -0002adc0: 7273 6563 7469 6f6e 2069 7320 6f6e 6520 rsection is one │ │ │ │ -0002add0: 7375 6368 2074 6861 7420 6765 6e65 7261 such that genera │ │ │ │ -0002ade0: 6c0a 6369 2d6f 7065 7261 746f 7273 2068 l.ci-operators h │ │ │ │ -0002adf0: 6176 6520 7370 6c69 7420 6b65 726e 656c ave split kernel │ │ │ │ -0002ae00: 7320 7768 656e 2061 7070 6c69 6564 2072 s when applied r │ │ │ │ -0002ae10: 6563 7572 7369 7665 6c79 206f 6e20 636f ecursively on co │ │ │ │ -0002ae20: 7379 7a79 6779 2063 6861 696e 7320 6f66 syzygy chains of │ │ │ │ -0002ae30: 0a70 7265 7669 6f75 7320 6b65 726e 656c .previous kernel │ │ │ │ -0002ae40: 732e 0a0a 4966 2070 203d 206d 6642 6f75 s...If p = mfBou │ │ │ │ -0002ae50: 6e64 204d 302c 2074 6865 6e20 6869 6768 nd M0, then high │ │ │ │ -0002ae60: 5379 7a79 6779 204d 3020 7265 7475 726e Syzygy M0 return │ │ │ │ -0002ae70: 7320 7468 6520 702d 7468 2073 797a 7967 s the p-th syzyg │ │ │ │ -0002ae80: 7920 6f66 204d 302e 2028 6966 2046 2069 y of M0. (if F i │ │ │ │ -0002ae90: 7320 610a 7265 736f 6c75 7469 6f6e 206f s a.resolution o │ │ │ │ -0002aea0: 6620 4d20 7468 6973 2069 7320 7468 6520 f M this is the │ │ │ │ -0002aeb0: 636f 6b65 726e 656c 206f 6620 462e 6464 cokernel of F.dd │ │ │ │ -0002aec0: 5f7b 702b 317d 292e 204f 7074 696d 6973 _{p+1}). Optimis │ │ │ │ -0002aed0: 6d20 3d3e 2072 2061 7320 6f70 7469 6f6e m => r as option │ │ │ │ -0002aee0: 616c 0a61 7267 756d 656e 742c 2068 6967 al.argument, hig │ │ │ │ -0002aef0: 6853 797a 7967 7928 4d30 2c4f 7074 696d hSyzygy(M0,Optim │ │ │ │ -0002af00: 6973 6d3d 3e72 2920 7265 7475 726e 7320 ism=>r) returns │ │ │ │ -0002af10: 7468 6520 2870 2d72 292d 7468 2073 797a the (p-r)-th syz │ │ │ │ -0002af20: 7967 792e 2054 6865 2073 6372 6970 7420 ygy. The script │ │ │ │ -0002af30: 6973 0a75 7365 6675 6c20 7769 7468 206d is.useful with m │ │ │ │ -0002af40: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ -0002af50: 6f6e 2866 662c 2068 6967 6853 797a 7967 on(ff, highSyzyg │ │ │ │ -0002af60: 7920 4d30 292e 0a0a 2b2d 2d2d 2d2d 2d2d y M0)...+------- │ │ │ │ +0002ac10: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ +0002ac20: 6167 653a 200a 2020 2020 2020 2020 4d20 age: . M │ │ │ │ +0002ac30: 3d20 6869 6768 5379 7a79 6779 204d 300a = highSyzygy M0. │ │ │ │ +0002ac40: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0002ac50: 2020 2a20 4d30 2c20 6120 2a6e 6f74 6520 * M0, a *note │ │ │ │ +0002ac60: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61 module: (Macaula │ │ │ │ +0002ac70: 7932 446f 6329 4d6f 6475 6c65 2c2c 206f y2Doc)Module,, o │ │ │ │ +0002ac80: 7665 7220 6120 636f 6d70 6c65 7465 2069 ver a complete i │ │ │ │ +0002ac90: 6e74 6572 7365 6374 696f 6e0a 2020 2020 ntersection. │ │ │ │ +0002aca0: 2020 2020 7269 6e67 0a20 202a 202a 6e6f ring. * *no │ │ │ │ +0002acb0: 7465 204f 7074 696f 6e61 6c20 696e 7075 te Optional inpu │ │ │ │ +0002acc0: 7473 3a20 284d 6163 6175 6c61 7932 446f ts: (Macaulay2Do │ │ │ │ +0002acd0: 6329 7573 696e 6720 6675 6e63 7469 6f6e c)using function │ │ │ │ +0002ace0: 7320 7769 7468 206f 7074 696f 6e61 6c20 s with optional │ │ │ │ +0002acf0: 696e 7075 7473 2c3a 0a20 2020 2020 202a inputs,:. * │ │ │ │ +0002ad00: 204f 7074 696d 6973 6d20 3d3e 202e 2e2e Optimism => ... │ │ │ │ +0002ad10: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +0002ad20: 300a 2020 2a20 4f75 7470 7574 733a 0a20 0. * Outputs:. │ │ │ │ +0002ad30: 2020 2020 202a 204d 2c20 6120 2a6e 6f74 * M, a *not │ │ │ │ +0002ad40: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ +0002ad50: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ +0002ad60: 2061 2073 797a 7967 7920 6d6f 6475 6c65 a syzygy module │ │ │ │ +0002ad70: 206f 6620 4d30 0a0a 4465 7363 7269 7074 of M0..Descript │ │ │ │ +0002ad80: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +0002ad90: 0a41 2022 6869 6768 2073 797a 7967 7922 .A "high syzygy" │ │ │ │ +0002ada0: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ +0002adb0: 2069 6e74 6572 7365 6374 696f 6e20 6973 intersection is │ │ │ │ +0002adc0: 206f 6e65 2073 7563 6820 7468 6174 2067 one such that g │ │ │ │ +0002add0: 656e 6572 616c 0a63 692d 6f70 6572 6174 eneral.ci-operat │ │ │ │ +0002ade0: 6f72 7320 6861 7665 2073 706c 6974 206b ors have split k │ │ │ │ +0002adf0: 6572 6e65 6c73 2077 6865 6e20 6170 706c ernels when appl │ │ │ │ +0002ae00: 6965 6420 7265 6375 7273 6976 656c 7920 ied recursively │ │ │ │ +0002ae10: 6f6e 2063 6f73 797a 7967 7920 6368 6169 on cosyzygy chai │ │ │ │ +0002ae20: 6e73 206f 660a 7072 6576 696f 7573 206b ns of.previous k │ │ │ │ +0002ae30: 6572 6e65 6c73 2e0a 0a49 6620 7020 3d20 ernels...If p = │ │ │ │ +0002ae40: 6d66 426f 756e 6420 4d30 2c20 7468 656e mfBound M0, then │ │ │ │ +0002ae50: 2068 6967 6853 797a 7967 7920 4d30 2072 highSyzygy M0 r │ │ │ │ +0002ae60: 6574 7572 6e73 2074 6865 2070 2d74 6820 eturns the p-th │ │ │ │ +0002ae70: 7379 7a79 6779 206f 6620 4d30 2e20 2869 syzygy of M0. (i │ │ │ │ +0002ae80: 6620 4620 6973 2061 0a72 6573 6f6c 7574 f F is a.resolut │ │ │ │ +0002ae90: 696f 6e20 6f66 204d 2074 6869 7320 6973 ion of M this is │ │ │ │ +0002aea0: 2074 6865 2063 6f6b 6572 6e65 6c20 6f66 the cokernel of │ │ │ │ +0002aeb0: 2046 2e64 645f 7b70 2b31 7d29 2e20 4f70 F.dd_{p+1}). Op │ │ │ │ +0002aec0: 7469 6d69 736d 203d 3e20 7220 6173 206f timism => r as o │ │ │ │ +0002aed0: 7074 696f 6e61 6c0a 6172 6775 6d65 6e74 ptional.argument │ │ │ │ +0002aee0: 2c20 6869 6768 5379 7a79 6779 284d 302c , highSyzygy(M0, │ │ │ │ +0002aef0: 4f70 7469 6d69 736d 3d3e 7229 2072 6574 Optimism=>r) ret │ │ │ │ +0002af00: 7572 6e73 2074 6865 2028 702d 7229 2d74 urns the (p-r)-t │ │ │ │ +0002af10: 6820 7379 7a79 6779 2e20 5468 6520 7363 h syzygy. The sc │ │ │ │ +0002af20: 7269 7074 2069 730a 7573 6566 756c 2077 ript is.useful w │ │ │ │ +0002af30: 6974 6820 6d61 7472 6978 4661 6374 6f72 ith matrixFactor │ │ │ │ +0002af40: 697a 6174 696f 6e28 6666 2c20 6869 6768 ization(ff, high │ │ │ │ +0002af50: 5379 7a79 6779 204d 3029 2e0a 0a2b 2d2d Syzygy M0)...+-- │ │ │ │ +0002af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002af80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002afa0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 -------+.|i1 : s │ │ │ │ -0002afb0: 6574 5261 6e64 6f6d 5365 6564 2031 3030 etRandomSeed 100 │ │ │ │ +0002af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0002afa0: 3120 3a20 7365 7452 616e 646f 6d53 6565 1 : setRandomSee │ │ │ │ +0002afb0: 6420 3130 3020 2020 2020 2020 2020 2020 d 100 │ │ │ │ 0002afc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002afd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002afe0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2073 |.| -- s │ │ │ │ -0002aff0: 6574 7469 6e67 2072 616e 646f 6d20 7365 etting random se │ │ │ │ -0002b000: 6564 2074 6f20 3130 3020 2020 2020 2020 ed to 100 │ │ │ │ -0002b010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b020: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002afd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002afe0: 202d 2d20 7365 7474 696e 6720 7261 6e64 -- setting rand │ │ │ │ +0002aff0: 6f6d 2073 6565 6420 746f 2031 3030 2020 om seed to 100 │ │ │ │ +0002b000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b010: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002b020: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b060: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ -0002b070: 3d20 3130 3020 2020 2020 2020 2020 2020 = 100 │ │ │ │ +0002b050: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b060: 0a7c 6f31 203d 2031 3030 2020 2020 2020 .|o1 = 100 │ │ │ │ +0002b070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b0a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002b0a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0002b0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002b0f0: 3220 3a20 5320 3d20 5a5a 2f31 3031 5b78 2 : S = ZZ/101[x │ │ │ │ -0002b100: 2c79 2c7a 5d20 2020 2020 2020 2020 2020 ,y,z] │ │ │ │ +0002b0e0: 2d2b 0a7c 6932 203a 2053 203d 205a 5a2f -+.|i2 : S = ZZ/ │ │ │ │ +0002b0f0: 3130 315b 782c 792c 7a5d 2020 2020 2020 101[x,y,z] │ │ │ │ +0002b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b120: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002b120: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002b130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b160: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b170: 7c6f 3220 3d20 5320 2020 2020 2020 2020 |o2 = S │ │ │ │ +0002b160: 2020 207c 0a7c 6f32 203d 2053 2020 2020 |.|o2 = S │ │ │ │ +0002b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b1a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b1b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002b1a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b1f0: 7c0a 7c6f 3220 3a20 506f 6c79 6e6f 6d69 |.|o2 : Polynomi │ │ │ │ -0002b200: 616c 5269 6e67 2020 2020 2020 2020 2020 alRing │ │ │ │ +0002b1e0: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c |.|o2 : Pol │ │ │ │ +0002b1f0: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ +0002b200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b230: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002b220: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002b230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b270: 2d2d 2b0a 7c69 3320 3a20 6620 3d20 6d61 --+.|i3 : f = ma │ │ │ │ -0002b280: 7472 6978 2278 332c 7933 2b78 332c 7a33 trix"x3,y3+x3,z3 │ │ │ │ -0002b290: 2b78 332b 7933 2220 2020 2020 2020 2020 +x3+y3" │ │ │ │ -0002b2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b2b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002b260: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2066 -------+.|i3 : f │ │ │ │ +0002b270: 203d 206d 6174 7269 7822 7833 2c79 332b = matrix"x3,y3+ │ │ │ │ +0002b280: 7833 2c7a 332b 7833 2b79 3322 2020 2020 x3,z3+x3+y3" │ │ │ │ +0002b290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b2a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002b2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b2f0: 2020 2020 7c0a 7c6f 3320 3d20 7c20 7833 |.|o3 = | x3 │ │ │ │ -0002b300: 2078 332b 7933 2078 332b 7933 2b7a 3320 x3+y3 x3+y3+z3 │ │ │ │ -0002b310: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b330: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002b2e0: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ +0002b2f0: 207c 2078 3320 7833 2b79 3320 7833 2b79 | x3 x3+y3 x3+y │ │ │ │ +0002b300: 332b 7a33 207c 2020 2020 2020 2020 2020 3+z3 | │ │ │ │ +0002b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b320: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002b330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b370: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0002b380: 2020 2020 2020 3120 2020 2020 2033 2020 1 3 │ │ │ │ +0002b360: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002b370: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +0002b380: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0002b390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b3b0: 2020 2020 2020 207c 0a7c 6f33 203a 204d |.|o3 : M │ │ │ │ -0002b3c0: 6174 7269 7820 5320 203c 2d2d 2053 2020 atrix S <-- S │ │ │ │ +0002b3a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0002b3b0: 3320 3a20 4d61 7472 6978 2053 2020 3c2d 3 : Matrix S <- │ │ │ │ +0002b3c0: 2d20 5320 2020 2020 2020 2020 2020 2020 - S │ │ │ │ 0002b3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b3f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0002b3e0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0002b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b430: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ -0002b440: 2066 6620 3d20 662a 7261 6e64 6f6d 2873 ff = f*random(s │ │ │ │ -0002b450: 6f75 7263 6520 662c 2073 6f75 7263 6520 ource f, source │ │ │ │ -0002b460: 6629 2020 2020 2020 2020 2020 2020 2020 f) │ │ │ │ -0002b470: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002b420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002b430: 7c69 3420 3a20 6666 203d 2066 2a72 616e |i4 : ff = f*ran │ │ │ │ +0002b440: 646f 6d28 736f 7572 6365 2066 2c20 736f dom(source f, so │ │ │ │ +0002b450: 7572 6365 2066 2920 2020 2020 2020 2020 urce f) │ │ │ │ +0002b460: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b470: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002b480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b4b0: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ -0002b4c0: 203d 207c 2031 3078 332d 3232 7933 2d34 = | 10x3-22y3-4 │ │ │ │ -0002b4d0: 7a33 202d 3230 7833 2d32 3079 332d 367a z3 -20x3-20y3-6z │ │ │ │ -0002b4e0: 3320 2d32 3778 332d 3431 7933 2b7a 3320 3 -27x3-41y3+z3 │ │ │ │ -0002b4f0: 7c20 2020 2020 2020 2020 2020 7c0a 7c20 | |.| │ │ │ │ +0002b4b0: 7c0a 7c6f 3420 3d20 7c20 3130 7833 2d32 |.|o4 = | 10x3-2 │ │ │ │ +0002b4c0: 3279 332d 347a 3320 2d32 3078 332d 3230 2y3-4z3 -20x3-20 │ │ │ │ +0002b4d0: 7933 2d36 7a33 202d 3237 7833 2d34 3179 y3-6z3 -27x3-41y │ │ │ │ +0002b4e0: 332b 7a33 207c 2020 2020 2020 2020 2020 3+z3 | │ │ │ │ +0002b4f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b530: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002b540: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ -0002b550: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ +0002b530: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002b540: 2020 3120 2020 2020 2033 2020 2020 2020 1 3 │ │ │ │ +0002b550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b570: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b580: 7c6f 3420 3a20 4d61 7472 6978 2053 2020 |o4 : Matrix S │ │ │ │ -0002b590: 3c2d 2d20 5320 2020 2020 2020 2020 2020 <-- S │ │ │ │ +0002b570: 2020 207c 0a7c 6f34 203a 204d 6174 7269 |.|o4 : Matri │ │ │ │ +0002b580: 7820 5320 203c 2d2d 2053 2020 2020 2020 x S <-- S │ │ │ │ +0002b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b5b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b5c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0002b5b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002b5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b600: 2b0a 7c69 3520 3a20 5220 3d20 532f 6964 +.|i5 : R = S/id │ │ │ │ -0002b610: 6561 6c20 6620 2020 2020 2020 2020 2020 eal f │ │ │ │ +0002b5f0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2052 203d -----+.|i5 : R = │ │ │ │ +0002b600: 2053 2f69 6465 616c 2066 2020 2020 2020 S/ideal f │ │ │ │ +0002b610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b640: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002b630: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002b640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b680: 2020 7c0a 7c6f 3520 3d20 5220 2020 2020 |.|o5 = R │ │ │ │ +0002b670: 2020 2020 2020 207c 0a7c 6f35 203d 2052 |.|o5 = R │ │ │ │ +0002b680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b6c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002b6b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002b6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b700: 2020 2020 7c0a 7c6f 3520 3a20 5175 6f74 |.|o5 : Quot │ │ │ │ -0002b710: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +0002b6f0: 2020 2020 2020 2020 207c 0a7c 6f35 203a |.|o5 : │ │ │ │ +0002b700: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +0002b710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b740: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002b730: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002b740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b780: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 4d30 ------+.|i6 : M0 │ │ │ │ -0002b790: 203d 2052 5e31 2f69 6465 616c 2278 327a = R^1/ideal"x2z │ │ │ │ -0002b7a0: 322c 7879 7a22 2020 2020 2020 2020 2020 2,xyz" │ │ │ │ -0002b7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b7c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 -----------+.|i6 │ │ │ │ +0002b780: 203a 204d 3020 3d20 525e 312f 6964 6561 : M0 = R^1/idea │ │ │ │ +0002b790: 6c22 7832 7a32 2c78 797a 2220 2020 2020 l"x2z2,xyz" │ │ │ │ +0002b7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b7b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002b7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b800: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ -0002b810: 636f 6b65 726e 656c 207c 2078 327a 3220 cokernel | x2z2 │ │ │ │ -0002b820: 7879 7a20 7c20 2020 2020 2020 2020 2020 xyz | │ │ │ │ -0002b830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b840: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002b7f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002b800: 6f36 203d 2063 6f6b 6572 6e65 6c20 7c20 o6 = cokernel | │ │ │ │ +0002b810: 7832 7a32 2078 797a 207c 2020 2020 2020 x2z2 xyz | │ │ │ │ +0002b820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b830: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002b840: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002b850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b880: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0002b890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b8a0: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ +0002b870: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b880: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002b890: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ +0002b8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b8c0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -0002b8d0: 203a 2052 2d6d 6f64 756c 652c 2071 756f : R-module, quo │ │ │ │ -0002b8e0: 7469 656e 7420 6f66 2052 2020 2020 2020 tient of R │ │ │ │ +0002b8c0: 7c0a 7c6f 3620 3a20 522d 6d6f 6475 6c65 |.|o6 : R-module │ │ │ │ +0002b8d0: 2c20 7175 6f74 6965 6e74 206f 6620 5220 , quotient of R │ │ │ │ +0002b8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b900: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0002b900: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0002b910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0002b950: 6937 203a 2062 6574 7469 2066 7265 6552 i7 : betti freeR │ │ │ │ -0002b960: 6573 6f6c 7574 696f 6e20 284d 302c 204c esolution (M0, L │ │ │ │ -0002b970: 656e 6774 684c 696d 6974 203d 3e20 3729 engthLimit => 7) │ │ │ │ -0002b980: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b990: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002b940: 2d2d 2b0a 7c69 3720 3a20 6265 7474 6920 --+.|i7 : betti │ │ │ │ +0002b950: 6672 6565 5265 736f 6c75 7469 6f6e 2028 freeResolution ( │ │ │ │ +0002b960: 4d30 2c20 4c65 6e67 7468 4c69 6d69 7420 M0, LengthLimit │ │ │ │ +0002b970: 3d3e 2037 2920 2020 2020 2020 2020 2020 => 7) │ │ │ │ +0002b980: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002b990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b9c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b9d0: 0a7c 2020 2020 2020 2020 2020 2020 3020 .| 0 │ │ │ │ -0002b9e0: 3120 3220 2033 2020 3420 2035 2020 3620 1 2 3 4 5 6 │ │ │ │ -0002b9f0: 2037 2020 2020 2020 2020 2020 2020 2020 7 │ │ │ │ -0002ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba10: 7c0a 7c6f 3720 3d20 746f 7461 6c3a 2031 |.|o7 = total: 1 │ │ │ │ -0002ba20: 2032 2036 2031 3120 3138 2032 3620 3336 2 6 11 18 26 36 │ │ │ │ -0002ba30: 2034 3720 2020 2020 2020 2020 2020 2020 47 │ │ │ │ -0002ba40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba50: 207c 0a7c 2020 2020 2020 2020 2030 3a20 |.| 0: │ │ │ │ -0002ba60: 3120 2e20 2e20 202e 2020 2e20 202e 2020 1 . . . . . │ │ │ │ -0002ba70: 2e20 202e 2020 2020 2020 2020 2020 2020 . . │ │ │ │ -0002ba80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba90: 2020 7c0a 7c20 2020 2020 2020 2020 313a |.| 1: │ │ │ │ -0002baa0: 202e 202e 202e 2020 2e20 202e 2020 2e20 . . . . . . │ │ │ │ -0002bab0: 202e 2020 2e20 2020 2020 2020 2020 2020 . . │ │ │ │ -0002bac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bad0: 2020 207c 0a7c 2020 2020 2020 2020 2032 |.| 2 │ │ │ │ -0002bae0: 3a20 2e20 3120 2e20 202e 2020 2e20 202e : . 1 . . . . │ │ │ │ -0002baf0: 2020 2e20 202e 2020 2020 2020 2020 2020 . . │ │ │ │ -0002bb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002bb20: 333a 202e 2031 2036 2020 3620 202e 2020 3: . 1 6 6 . │ │ │ │ -0002bb30: 2e20 202e 2020 2e20 2020 2020 2020 2020 . . . │ │ │ │ -0002bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb50: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0002bb60: 2034 3a20 2e20 2e20 2e20 2035 2031 3820 4: . . . 5 18 │ │ │ │ -0002bb70: 3134 2020 2e20 202e 2020 2020 2020 2020 14 . . │ │ │ │ -0002bb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb90: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0002bba0: 2020 353a 202e 202e 202e 2020 2e20 202e 5: . . . . . │ │ │ │ -0002bbb0: 2031 3220 3336 2032 3520 2020 2020 2020 12 36 25 │ │ │ │ -0002bbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bbd0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002bbe0: 2020 2036 3a20 2e20 2e20 2e20 202e 2020 6: . . . . │ │ │ │ -0002bbf0: 2e20 202e 2020 2e20 3232 2020 2020 2020 . . . 22 │ │ │ │ -0002bc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc10: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002b9c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b9d0: 2020 2030 2031 2032 2020 3320 2034 2020 0 1 2 3 4 │ │ │ │ +0002b9e0: 3520 2036 2020 3720 2020 2020 2020 2020 5 6 7 │ │ │ │ +0002b9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ba00: 2020 2020 207c 0a7c 6f37 203d 2074 6f74 |.|o7 = tot │ │ │ │ +0002ba10: 616c 3a20 3120 3220 3620 3131 2031 3820 al: 1 2 6 11 18 │ │ │ │ +0002ba20: 3236 2033 3620 3437 2020 2020 2020 2020 26 36 47 │ │ │ │ +0002ba30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ba40: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002ba50: 2020 303a 2031 202e 202e 2020 2e20 202e 0: 1 . . . . │ │ │ │ +0002ba60: 2020 2e20 202e 2020 2e20 2020 2020 2020 . . . │ │ │ │ +0002ba70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ba80: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002ba90: 2020 2031 3a20 2e20 2e20 2e20 202e 2020 1: . . . . │ │ │ │ +0002baa0: 2e20 202e 2020 2e20 202e 2020 2020 2020 . . . . │ │ │ │ +0002bab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bac0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002bad0: 2020 2020 323a 202e 2031 202e 2020 2e20 2: . 1 . . │ │ │ │ +0002bae0: 202e 2020 2e20 202e 2020 2e20 2020 2020 . . . . │ │ │ │ +0002baf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002bb10: 2020 2020 2033 3a20 2e20 3120 3620 2036 3: . 1 6 6 │ │ │ │ +0002bb20: 2020 2e20 202e 2020 2e20 202e 2020 2020 . . . . │ │ │ │ +0002bb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002bb50: 2020 2020 2020 343a 202e 202e 202e 2020 4: . . . │ │ │ │ +0002bb60: 3520 3138 2031 3420 202e 2020 2e20 2020 5 18 14 . . │ │ │ │ +0002bb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb80: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002bb90: 2020 2020 2020 2035 3a20 2e20 2e20 2e20 5: . . . │ │ │ │ +0002bba0: 202e 2020 2e20 3132 2033 3620 3235 2020 . . 12 36 25 │ │ │ │ +0002bbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bbc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002bbd0: 2020 2020 2020 2020 363a 202e 202e 202e 6: . . . │ │ │ │ +0002bbe0: 2020 2e20 202e 2020 2e20 202e 2032 3220 . . . . 22 │ │ │ │ +0002bbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bc00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002bc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc50: 2020 2020 2020 2020 207c 0a7c 6f37 203a |.|o7 : │ │ │ │ -0002bc60: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +0002bc40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002bc50: 7c6f 3720 3a20 4265 7474 6954 616c 6c79 |o7 : BettiTally │ │ │ │ +0002bc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc90: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002bc80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002bc90: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0002bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ -0002bce0: 203a 206d 6642 6f75 6e64 204d 3020 2020 : mfBound M0 │ │ │ │ +0002bcd0: 2b0a 7c69 3820 3a20 6d66 426f 756e 6420 +.|i8 : mfBound │ │ │ │ +0002bce0: 4d30 2020 2020 2020 2020 2020 2020 2020 M0 │ │ │ │ 0002bcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bd10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002bd10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002bd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bd50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002bd60: 6f38 203d 2033 2020 2020 2020 2020 2020 o8 = 3 │ │ │ │ +0002bd50: 2020 7c0a 7c6f 3820 3d20 3320 2020 2020 |.|o8 = 3 │ │ │ │ +0002bd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bd90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002bda0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002bd90: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002bda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002bdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002bdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002bde0: 0a7c 6939 203a 204d 203d 2062 6574 7469 .|i9 : M = betti │ │ │ │ -0002bdf0: 2066 7265 6552 6573 6f6c 7574 696f 6e28 freeResolution( │ │ │ │ -0002be00: 6869 6768 5379 7a79 6779 204d 302c 204c highSyzygy M0, L │ │ │ │ -0002be10: 656e 6774 684c 696d 6974 203d 3e20 3729 engthLimit => 7) │ │ │ │ -0002be20: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002bdd0: 2d2d 2d2d 2b0a 7c69 3920 3a20 4d20 3d20 ----+.|i9 : M = │ │ │ │ +0002bde0: 6265 7474 6920 6672 6565 5265 736f 6c75 betti freeResolu │ │ │ │ +0002bdf0: 7469 6f6e 2868 6967 6853 797a 7967 7920 tion(highSyzygy │ │ │ │ +0002be00: 4d30 2c20 4c65 6e67 7468 4c69 6d69 7420 M0, LengthLimit │ │ │ │ +0002be10: 3d3e 2037 297c 0a7c 2020 2020 2020 2020 => 7)|.| │ │ │ │ +0002be20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002be30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002be40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002be50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002be60: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002be70: 2030 2020 3120 2032 2020 3320 2034 2020 0 1 2 3 4 │ │ │ │ -0002be80: 3520 2036 2020 3720 2020 2020 2020 2020 5 6 7 │ │ │ │ -0002be90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bea0: 2020 7c0a 7c6f 3920 3d20 746f 7461 6c3a |.|o9 = total: │ │ │ │ -0002beb0: 2031 3120 3138 2032 3620 3336 2034 3720 11 18 26 36 47 │ │ │ │ -0002bec0: 3630 2037 3420 3930 2020 2020 2020 2020 60 74 90 │ │ │ │ -0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bee0: 2020 207c 0a7c 2020 2020 2020 2020 2036 |.| 6 │ │ │ │ -0002bef0: 3a20 2036 2020 2e20 202e 2020 2e20 202e : 6 . . . . │ │ │ │ -0002bf00: 2020 2e20 202e 2020 2e20 2020 2020 2020 . . . │ │ │ │ -0002bf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002bf30: 373a 2020 3520 3138 2031 3420 202e 2020 7: 5 18 14 . │ │ │ │ -0002bf40: 2e20 202e 2020 2e20 202e 2020 2020 2020 . . . . │ │ │ │ -0002bf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf60: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0002bf70: 2038 3a20 202e 2020 2e20 3132 2033 3620 8: . . 12 36 │ │ │ │ -0002bf80: 3235 2020 2e20 202e 2020 2e20 2020 2020 25 . . . │ │ │ │ -0002bf90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0002bfb0: 2020 393a 2020 2e20 202e 2020 2e20 202e 9: . . . . │ │ │ │ -0002bfc0: 2032 3220 3630 2033 3920 202e 2020 2020 22 60 39 . │ │ │ │ -0002bfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfe0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002bff0: 2020 3130 3a20 202e 2020 2e20 202e 2020 10: . . . │ │ │ │ -0002c000: 2e20 202e 2020 2e20 3335 2039 3020 2020 . . . 35 90 │ │ │ │ -0002c010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c020: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002be50: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002be60: 2020 2020 2020 3020 2031 2020 3220 2033 0 1 2 3 │ │ │ │ +0002be70: 2020 3420 2035 2020 3620 2037 2020 2020 4 5 6 7 │ │ │ │ +0002be80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be90: 2020 2020 2020 207c 0a7c 6f39 203d 2074 |.|o9 = t │ │ │ │ +0002bea0: 6f74 616c 3a20 3131 2031 3820 3236 2033 otal: 11 18 26 3 │ │ │ │ +0002beb0: 3620 3437 2036 3020 3734 2039 3020 2020 6 47 60 74 90 │ │ │ │ +0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bed0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002bee0: 2020 2020 363a 2020 3620 202e 2020 2e20 6: 6 . . │ │ │ │ +0002bef0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ +0002bf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002bf20: 2020 2020 2037 3a20 2035 2031 3820 3134 7: 5 18 14 │ │ │ │ +0002bf30: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0002bf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002bf60: 2020 2020 2020 383a 2020 2e20 202e 2031 8: . . 1 │ │ │ │ +0002bf70: 3220 3336 2032 3520 202e 2020 2e20 202e 2 36 25 . . . │ │ │ │ +0002bf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf90: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002bfa0: 2020 2020 2020 2039 3a20 202e 2020 2e20 9: . . │ │ │ │ +0002bfb0: 202e 2020 2e20 3232 2036 3020 3339 2020 . . 22 60 39 │ │ │ │ +0002bfc0: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ +0002bfd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002bfe0: 2020 2020 2020 2031 303a 2020 2e20 202e 10: . . │ │ │ │ +0002bff0: 2020 2e20 202e 2020 2e20 202e 2033 3520 . . . . 35 │ │ │ │ +0002c000: 3930 2020 2020 2020 2020 2020 2020 2020 90 │ │ │ │ +0002c010: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c060: 2020 2020 2020 2020 207c 0a7c 6f39 203a |.|o9 : │ │ │ │ -0002c070: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +0002c050: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c060: 7c6f 3920 3a20 4265 7474 6954 616c 6c79 |o9 : BettiTally │ │ │ │ +0002c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c0a0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002c090: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002c0a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0002c0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -0002c0f0: 3020 3a20 6e65 744c 6973 7420 4252 616e 0 : netList BRan │ │ │ │ -0002c100: 6b73 206d 6174 7269 7846 6163 746f 7269 ks matrixFactori │ │ │ │ -0002c110: 7a61 7469 6f6e 2866 662c 2068 6967 6853 zation(ff, highS │ │ │ │ -0002c120: 797a 7967 7920 4d30 2920 2020 7c0a 7c20 yzygy M0) |.| │ │ │ │ +0002c0e0: 2b0a 7c69 3130 203a 206e 6574 4c69 7374 +.|i10 : netList │ │ │ │ +0002c0f0: 2042 5261 6e6b 7320 6d61 7472 6978 4661 BRanks matrixFa │ │ │ │ +0002c100: 6374 6f72 697a 6174 696f 6e28 6666 2c20 ctorization(ff, │ │ │ │ +0002c110: 6869 6768 5379 7a79 6779 204d 3029 2020 highSyzygy M0) │ │ │ │ +0002c120: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002c130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c160: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002c170: 2020 2020 2020 2b2d 2b2d 2b20 2020 2020 +-+-+ │ │ │ │ +0002c160: 2020 7c0a 7c20 2020 2020 202b 2d2b 2d2b |.| +-+-+ │ │ │ │ +0002c170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c1a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002c1b0: 7c6f 3130 203d 207c 367c 367c 2020 2020 |o10 = |6|6| │ │ │ │ +0002c1a0: 2020 207c 0a7c 6f31 3020 3d20 7c36 7c36 |.|o10 = |6|6 │ │ │ │ +0002c1b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002c1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c1e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002c1f0: 0a7c 2020 2020 2020 2b2d 2b2d 2b20 2020 .| +-+-+ │ │ │ │ +0002c1e0: 2020 2020 7c0a 7c20 2020 2020 202b 2d2b |.| +-+ │ │ │ │ +0002c1f0: 2d2b 2020 2020 2020 2020 2020 2020 2020 -+ │ │ │ │ 0002c200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c230: 7c0a 7c20 2020 2020 207c 337c 367c 2020 |.| |3|6| │ │ │ │ +0002c220: 2020 2020 207c 0a7c 2020 2020 2020 7c33 |.| |3 │ │ │ │ +0002c230: 7c36 7c20 2020 2020 2020 2020 2020 2020 |6| │ │ │ │ 0002c240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c270: 207c 0a7c 2020 2020 2020 2b2d 2b2d 2b20 |.| +-+-+ │ │ │ │ +0002c260: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ +0002c270: 2d2b 2d2b 2020 2020 2020 2020 2020 2020 -+-+ │ │ │ │ 0002c280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2b0: 2020 7c0a 7c20 2020 2020 207c 327c 367c |.| |2|6| │ │ │ │ +0002c2a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002c2b0: 7c32 7c36 7c20 2020 2020 2020 2020 2020 |2|6| │ │ │ │ 0002c2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2f0: 2020 207c 0a7c 2020 2020 2020 2b2d 2b2d |.| +-+- │ │ │ │ -0002c300: 2b20 2020 2020 2020 2020 2020 2020 2020 + │ │ │ │ +0002c2e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002c2f0: 202b 2d2b 2d2b 2020 2020 2020 2020 2020 +-+-+ │ │ │ │ +0002c300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c330: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002c320: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0002c330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c370: 2d2d 2d2d 2d2b 0a0a 496e 2074 6869 7320 -----+..In this │ │ │ │ -0002c380: 6361 7365 2061 7320 696e 2061 6c6c 206f case as in all o │ │ │ │ -0002c390: 7468 6572 7320 7765 2068 6176 6520 6578 thers we have ex │ │ │ │ -0002c3a0: 616d 696e 6564 2c20 6772 6561 7465 7220 amined, greater │ │ │ │ -0002c3b0: 224f 7074 696d 6973 6d22 2069 7320 6e6f "Optimism" is no │ │ │ │ -0002c3c0: 740a 6a75 7374 6966 6965 642c 2061 6e64 t.justified, and │ │ │ │ -0002c3d0: 2074 6875 7320 6d61 7472 6978 4661 6374 thus matrixFact │ │ │ │ -0002c3e0: 6f72 697a 6174 696f 6e28 6666 2c20 6869 orization(ff, hi │ │ │ │ -0002c3f0: 6768 5379 7a79 6779 284d 302c 204f 7074 ghSyzygy(M0, Opt │ │ │ │ -0002c400: 696d 6973 6d3d 3e31 2929 3b20 776f 756c imism=>1)); woul │ │ │ │ -0002c410: 640a 7072 6f64 7563 6520 616e 2065 7272 d.produce an err │ │ │ │ -0002c420: 6f72 2e0a 0a43 6176 6561 740a 3d3d 3d3d or...Caveat.==== │ │ │ │ -0002c430: 3d3d 0a0a 4120 6275 6720 696e 2074 6865 ==..A bug in the │ │ │ │ -0002c440: 2074 6f74 616c 2045 7874 2073 6372 6970 total Ext scrip │ │ │ │ -0002c450: 7420 6d65 616e 7320 7468 6174 2074 6865 t means that the │ │ │ │ -0002c460: 206f 6464 4578 744d 6f64 756c 6520 6973 oddExtModule is │ │ │ │ -0002c470: 2073 6f6d 6574 696d 6573 207a 6572 6f2c sometimes zero, │ │ │ │ -0002c480: 0a61 6e64 2074 6869 7320 6361 6e20 6361 .and this can ca │ │ │ │ -0002c490: 7573 6520 6120 7772 6f6e 6720 7661 6c75 use a wrong valu │ │ │ │ -0002c4a0: 6520 746f 2062 6520 7265 7475 726e 6564 e to be returned │ │ │ │ -0002c4b0: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d ...See also.==== │ │ │ │ -0002c4c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -0002c4d0: 6576 656e 4578 744d 6f64 756c 653a 2065 evenExtModule: e │ │ │ │ -0002c4e0: 7665 6e45 7874 4d6f 6475 6c65 2c20 2d2d venExtModule, -- │ │ │ │ -0002c4f0: 2065 7665 6e20 7061 7274 206f 6620 4578 even part of Ex │ │ │ │ -0002c500: 745e 2a28 4d2c 6b29 206f 7665 7220 610a t^*(M,k) over a. │ │ │ │ -0002c510: 2020 2020 636f 6d70 6c65 7465 2069 6e74 complete int │ │ │ │ -0002c520: 6572 7365 6374 696f 6e20 6173 206d 6f64 ersection as mod │ │ │ │ -0002c530: 756c 6520 6f76 6572 2043 4920 6f70 6572 ule over CI oper │ │ │ │ -0002c540: 6174 6f72 2072 696e 670a 2020 2a20 2a6e ator ring. * *n │ │ │ │ -0002c550: 6f74 6520 6f64 6445 7874 4d6f 6475 6c65 ote oddExtModule │ │ │ │ -0002c560: 3a20 6f64 6445 7874 4d6f 6475 6c65 2c20 : oddExtModule, │ │ │ │ -0002c570: 2d2d 206f 6464 2070 6172 7420 6f66 2045 -- odd part of E │ │ │ │ -0002c580: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061 xt^*(M,k) over a │ │ │ │ -0002c590: 2063 6f6d 706c 6574 650a 2020 2020 696e complete. in │ │ │ │ -0002c5a0: 7465 7273 6563 7469 6f6e 2061 7320 6d6f tersection as mo │ │ │ │ -0002c5b0: 6475 6c65 206f 7665 7220 4349 206f 7065 dule over CI ope │ │ │ │ -0002c5c0: 7261 746f 7220 7269 6e67 0a20 202a 202a rator ring. * * │ │ │ │ -0002c5d0: 6e6f 7465 206d 6642 6f75 6e64 3a20 6d66 note mfBound: mf │ │ │ │ -0002c5e0: 426f 756e 642c 202d 2d20 6465 7465 726d Bound, -- determ │ │ │ │ -0002c5f0: 696e 6573 2068 6f77 2068 6967 6820 6120 ines how high a │ │ │ │ -0002c600: 7379 7a79 6779 2074 6f20 7461 6b65 2066 syzygy to take f │ │ │ │ -0002c610: 6f72 0a20 2020 2022 6d61 7472 6978 4661 or. "matrixFa │ │ │ │ -0002c620: 6374 6f72 697a 6174 696f 6e22 0a20 202a ctorization". * │ │ │ │ -0002c630: 202a 6e6f 7465 206d 6174 7269 7846 6163 *note matrixFac │ │ │ │ -0002c640: 746f 7269 7a61 7469 6f6e 3a20 6d61 7472 torization: matr │ │ │ │ -0002c650: 6978 4661 6374 6f72 697a 6174 696f 6e2c ixFactorization, │ │ │ │ -0002c660: 202d 2d20 4d61 7073 2069 6e20 6120 6869 -- Maps in a hi │ │ │ │ -0002c670: 6768 6572 0a20 2020 2063 6f64 696d 656e gher. codimen │ │ │ │ -0002c680: 7369 6f6e 206d 6174 7269 7820 6661 6374 sion matrix fact │ │ │ │ -0002c690: 6f72 697a 6174 696f 6e0a 0a57 6179 7320 orization..Ways │ │ │ │ -0002c6a0: 746f 2075 7365 2068 6967 6853 797a 7967 to use highSyzyg │ │ │ │ -0002c6b0: 793a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d y:.============= │ │ │ │ -0002c6c0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -0002c6d0: 2268 6967 6853 797a 7967 7928 4d6f 6475 "highSyzygy(Modu │ │ │ │ -0002c6e0: 6c65 2922 0a0a 466f 7220 7468 6520 7072 le)"..For the pr │ │ │ │ -0002c6f0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ -0002c700: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -0002c710: 206f 626a 6563 7420 2a6e 6f74 6520 6869 object *note hi │ │ │ │ -0002c720: 6768 5379 7a79 6779 3a20 6869 6768 5379 ghSyzygy: highSy │ │ │ │ -0002c730: 7a79 6779 2c20 6973 2061 202a 6e6f 7465 zygy, is a *note │ │ │ │ -0002c740: 206d 6574 686f 6420 6675 6e63 7469 6f6e method function │ │ │ │ -0002c750: 2077 6974 680a 6f70 7469 6f6e 733a 2028 with.options: ( │ │ │ │ -0002c760: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ -0002c770: 686f 6446 756e 6374 696f 6e57 6974 684f hodFunctionWithO │ │ │ │ -0002c780: 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d ptions,...------ │ │ │ │ +0002c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6e20 ----------+..In │ │ │ │ +0002c370: 7468 6973 2063 6173 6520 6173 2069 6e20 this case as in │ │ │ │ +0002c380: 616c 6c20 6f74 6865 7273 2077 6520 6861 all others we ha │ │ │ │ +0002c390: 7665 2065 7861 6d69 6e65 642c 2067 7265 ve examined, gre │ │ │ │ +0002c3a0: 6174 6572 2022 4f70 7469 6d69 736d 2220 ater "Optimism" │ │ │ │ +0002c3b0: 6973 206e 6f74 0a6a 7573 7469 6669 6564 is not.justified │ │ │ │ +0002c3c0: 2c20 616e 6420 7468 7573 206d 6174 7269 , and thus matri │ │ │ │ +0002c3d0: 7846 6163 746f 7269 7a61 7469 6f6e 2866 xFactorization(f │ │ │ │ +0002c3e0: 662c 2068 6967 6853 797a 7967 7928 4d30 f, highSyzygy(M0 │ │ │ │ +0002c3f0: 2c20 4f70 7469 6d69 736d 3d3e 3129 293b , Optimism=>1)); │ │ │ │ +0002c400: 2077 6f75 6c64 0a70 726f 6475 6365 2061 would.produce a │ │ │ │ +0002c410: 6e20 6572 726f 722e 0a0a 4361 7665 6174 n error...Caveat │ │ │ │ +0002c420: 0a3d 3d3d 3d3d 3d0a 0a41 2062 7567 2069 .======..A bug i │ │ │ │ +0002c430: 6e20 7468 6520 746f 7461 6c20 4578 7420 n the total Ext │ │ │ │ +0002c440: 7363 7269 7074 206d 6561 6e73 2074 6861 script means tha │ │ │ │ +0002c450: 7420 7468 6520 6f64 6445 7874 4d6f 6475 t the oddExtModu │ │ │ │ +0002c460: 6c65 2069 7320 736f 6d65 7469 6d65 7320 le is sometimes │ │ │ │ +0002c470: 7a65 726f 2c0a 616e 6420 7468 6973 2063 zero,.and this c │ │ │ │ +0002c480: 616e 2063 6175 7365 2061 2077 726f 6e67 an cause a wrong │ │ │ │ +0002c490: 2076 616c 7565 2074 6f20 6265 2072 6574 value to be ret │ │ │ │ +0002c4a0: 7572 6e65 642e 0a0a 5365 6520 616c 736f urned...See also │ │ │ │ +0002c4b0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +0002c4c0: 6e6f 7465 2065 7665 6e45 7874 4d6f 6475 note evenExtModu │ │ │ │ +0002c4d0: 6c65 3a20 6576 656e 4578 744d 6f64 756c le: evenExtModul │ │ │ │ +0002c4e0: 652c 202d 2d20 6576 656e 2070 6172 7420 e, -- even part │ │ │ │ +0002c4f0: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76 of Ext^*(M,k) ov │ │ │ │ +0002c500: 6572 2061 0a20 2020 2063 6f6d 706c 6574 er a. complet │ │ │ │ +0002c510: 6520 696e 7465 7273 6563 7469 6f6e 2061 e intersection a │ │ │ │ +0002c520: 7320 6d6f 6475 6c65 206f 7665 7220 4349 s module over CI │ │ │ │ +0002c530: 206f 7065 7261 746f 7220 7269 6e67 0a20 operator ring. │ │ │ │ +0002c540: 202a 202a 6e6f 7465 206f 6464 4578 744d * *note oddExtM │ │ │ │ +0002c550: 6f64 756c 653a 206f 6464 4578 744d 6f64 odule: oddExtMod │ │ │ │ +0002c560: 756c 652c 202d 2d20 6f64 6420 7061 7274 ule, -- odd part │ │ │ │ +0002c570: 206f 6620 4578 745e 2a28 4d2c 6b29 206f of Ext^*(M,k) o │ │ │ │ +0002c580: 7665 7220 6120 636f 6d70 6c65 7465 0a20 ver a complete. │ │ │ │ +0002c590: 2020 2069 6e74 6572 7365 6374 696f 6e20 intersection │ │ │ │ +0002c5a0: 6173 206d 6f64 756c 6520 6f76 6572 2043 as module over C │ │ │ │ +0002c5b0: 4920 6f70 6572 6174 6f72 2072 696e 670a I operator ring. │ │ │ │ +0002c5c0: 2020 2a20 2a6e 6f74 6520 6d66 426f 756e * *note mfBoun │ │ │ │ +0002c5d0: 643a 206d 6642 6f75 6e64 2c20 2d2d 2064 d: mfBound, -- d │ │ │ │ +0002c5e0: 6574 6572 6d69 6e65 7320 686f 7720 6869 etermines how hi │ │ │ │ +0002c5f0: 6768 2061 2073 797a 7967 7920 746f 2074 gh a syzygy to t │ │ │ │ +0002c600: 616b 6520 666f 720a 2020 2020 226d 6174 ake for. "mat │ │ │ │ +0002c610: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ +0002c620: 220a 2020 2a20 2a6e 6f74 6520 6d61 7472 ". * *note matr │ │ │ │ +0002c630: 6978 4661 6374 6f72 697a 6174 696f 6e3a ixFactorization: │ │ │ │ +0002c640: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ +0002c650: 7469 6f6e 2c20 2d2d 204d 6170 7320 696e tion, -- Maps in │ │ │ │ +0002c660: 2061 2068 6967 6865 720a 2020 2020 636f a higher. co │ │ │ │ +0002c670: 6469 6d65 6e73 696f 6e20 6d61 7472 6978 dimension matrix │ │ │ │ +0002c680: 2066 6163 746f 7269 7a61 7469 6f6e 0a0a factorization.. │ │ │ │ +0002c690: 5761 7973 2074 6f20 7573 6520 6869 6768 Ways to use high │ │ │ │ +0002c6a0: 5379 7a79 6779 3a0a 3d3d 3d3d 3d3d 3d3d Syzygy:.======== │ │ │ │ +0002c6b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0002c6c0: 0a20 202a 2022 6869 6768 5379 7a79 6779 . * "highSyzygy │ │ │ │ +0002c6d0: 284d 6f64 756c 6529 220a 0a46 6f72 2074 (Module)"..For t │ │ │ │ +0002c6e0: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ +0002c6f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002c700: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ +0002c710: 7465 2068 6967 6853 797a 7967 793a 2068 te highSyzygy: h │ │ │ │ +0002c720: 6967 6853 797a 7967 792c 2069 7320 6120 ighSyzygy, is a │ │ │ │ +0002c730: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e *note method fun │ │ │ │ +0002c740: 6374 696f 6e20 7769 7468 0a6f 7074 696f ction with.optio │ │ │ │ +0002c750: 6e73 3a20 284d 6163 6175 6c61 7932 446f ns: (Macaulay2Do │ │ │ │ +0002c760: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ +0002c770: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d WithOptions,...- │ │ │ │ +0002c780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c7d0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -0002c7e0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -0002c7f0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -0002c800: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -0002c810: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -0002c820: 312e 3236 2e30 352b 6473 2f4d 322f 4d61 1.26.05+ds/M2/Ma │ │ │ │ -0002c830: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -0002c840: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ -0002c850: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -0002c860: 732e 6d32 3a33 3330 393a 302e 0a1f 0a46 s.m2:3309:0....F │ │ │ │ -0002c870: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ -0002c880: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ -0002c890: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ -0002c8a0: 2068 4d61 7073 2c20 4e65 7874 3a20 486f hMaps, Next: Ho │ │ │ │ -0002c8b0: 6d57 6974 6843 6f6d 706f 6e65 6e74 732c mWithComponents, │ │ │ │ -0002c8c0: 2050 7265 763a 2068 6967 6853 797a 7967 Prev: highSyzyg │ │ │ │ -0002c8d0: 792c 2055 703a 2054 6f70 0a0a 684d 6170 y, Up: Top..hMap │ │ │ │ -0002c8e0: 7320 2d2d 206c 6973 7420 7468 6520 6d61 s -- list the ma │ │ │ │ -0002c8f0: 7073 2020 6828 7029 3a20 415f 3028 7029 ps h(p): A_0(p) │ │ │ │ -0002c900: 2d2d 3e20 415f 3128 7029 2069 6e20 6120 --> A_1(p) in a │ │ │ │ -0002c910: 6d61 7472 6978 4661 6374 6f72 697a 6174 matrixFactorizat │ │ │ │ -0002c920: 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ion.************ │ │ │ │ +0002c7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +0002c7d0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +0002c7e0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +0002c7f0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +0002c800: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +0002c810: 6c61 7932 2d31 2e32 362e 3035 2b64 732f lay2-1.26.05+ds/ │ │ │ │ +0002c820: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +0002c830: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ +0002c840: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +0002c850: 7574 696f 6e73 2e6d 323a 3333 3039 3a30 utions.m2:3309:0 │ │ │ │ +0002c860: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ +0002c870: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ +0002c880: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ +0002c890: 4e6f 6465 3a20 684d 6170 732c 204e 6578 Node: hMaps, Nex │ │ │ │ +0002c8a0: 743a 2048 6f6d 5769 7468 436f 6d70 6f6e t: HomWithCompon │ │ │ │ +0002c8b0: 656e 7473 2c20 5072 6576 3a20 6869 6768 ents, Prev: high │ │ │ │ +0002c8c0: 5379 7a79 6779 2c20 5570 3a20 546f 700a Syzygy, Up: Top. │ │ │ │ +0002c8d0: 0a68 4d61 7073 202d 2d20 6c69 7374 2074 .hMaps -- list t │ │ │ │ +0002c8e0: 6865 206d 6170 7320 2068 2870 293a 2041 he maps h(p): A │ │ │ │ +0002c8f0: 5f30 2870 292d 2d3e 2041 5f31 2870 2920 _0(p)--> A_1(p) │ │ │ │ +0002c900: 696e 2061 206d 6174 7269 7846 6163 746f in a matrixFacto │ │ │ │ +0002c910: 7269 7a61 7469 6f6e 0a2a 2a2a 2a2a 2a2a rization.******* │ │ │ │ +0002c920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002c930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002c940: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002c950: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002c960: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -0002c970: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -0002c980: 2068 4d61 7073 203d 2068 4d61 7073 206d hMaps = hMaps m │ │ │ │ -0002c990: 660a 2020 2a20 496e 7075 7473 3a0a 2020 f. * Inputs:. │ │ │ │ -0002c9a0: 2020 2020 2a20 6d66 2c20 6120 2a6e 6f74 * mf, a *not │ │ │ │ -0002c9b0: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ -0002c9c0: 7932 446f 6329 4c69 7374 2c2c 206f 7574 y2Doc)List,, out │ │ │ │ -0002c9d0: 7075 7420 6f66 2061 206d 6174 7269 7846 put of a matrixF │ │ │ │ -0002c9e0: 6163 746f 7269 7a61 7469 6f6e 0a20 2020 actorization. │ │ │ │ -0002c9f0: 2020 2020 2063 6f6d 7075 7461 7469 6f6e computation │ │ │ │ -0002ca00: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -0002ca10: 2020 2020 2a20 684d 6170 732c 2061 202a * hMaps, a * │ │ │ │ -0002ca20: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ -0002ca30: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ -0002ca40: 6c69 7374 206d 6174 7269 6365 7320 2468 list matrices $h │ │ │ │ -0002ca50: 5f70 3a20 415f 3028 7029 5c74 6f0a 2020 _p: A_0(p)\to. │ │ │ │ -0002ca60: 2020 2020 2020 415f 3128 7029 242e 2054 A_1(p)$. T │ │ │ │ -0002ca70: 6865 2073 6f75 7263 6573 2061 6e64 2074 he sources and t │ │ │ │ -0002ca80: 6172 6765 7473 206f 6620 7468 6573 6520 argets of these │ │ │ │ -0002ca90: 6d61 7073 2068 6176 6520 7468 6520 636f maps have the co │ │ │ │ -0002caa0: 6d70 6f6e 656e 7473 0a20 2020 2020 2020 mponents. │ │ │ │ -0002cab0: 2042 5f73 2870 292e 0a0a 4465 7363 7269 B_s(p)...Descri │ │ │ │ -0002cac0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -0002cad0: 3d0a 0a53 6565 2074 6865 2064 6f63 756d =..See the docum │ │ │ │ -0002cae0: 656e 7461 7469 6f6e 2066 6f72 206d 6174 entation for mat │ │ │ │ -0002caf0: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -0002cb00: 2066 6f72 2061 6e20 6578 616d 706c 652e for an example. │ │ │ │ -0002cb10: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -0002cb20: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d ===.. * *note m │ │ │ │ -0002cb30: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ -0002cb40: 6f6e 3a20 6d61 7472 6978 4661 6374 6f72 on: matrixFactor │ │ │ │ -0002cb50: 697a 6174 696f 6e2c 202d 2d20 4d61 7073 ization, -- Maps │ │ │ │ -0002cb60: 2069 6e20 6120 6869 6768 6572 0a20 2020 in a higher. │ │ │ │ -0002cb70: 2063 6f64 696d 656e 7369 6f6e 206d 6174 codimension mat │ │ │ │ -0002cb80: 7269 7820 6661 6374 6f72 697a 6174 696f rix factorizatio │ │ │ │ -0002cb90: 6e0a 2020 2a20 2a6e 6f74 6520 644d 6170 n. * *note dMap │ │ │ │ -0002cba0: 733a 2064 4d61 7073 2c20 2d2d 206c 6973 s: dMaps, -- lis │ │ │ │ -0002cbb0: 7420 7468 6520 6d61 7073 2020 6428 7029 t the maps d(p) │ │ │ │ -0002cbc0: 3a41 5f31 2870 292d 2d3e 2041 5f30 2870 :A_1(p)--> A_0(p │ │ │ │ -0002cbd0: 2920 696e 2061 0a20 2020 206d 6174 7269 ) in a. matri │ │ │ │ -0002cbe0: 7846 6163 746f 7269 7a61 7469 6f6e 0a20 xFactorization. │ │ │ │ -0002cbf0: 202a 202a 6e6f 7465 2042 5261 6e6b 733a * *note BRanks: │ │ │ │ -0002cc00: 2042 5261 6e6b 732c 202d 2d20 7261 6e6b BRanks, -- rank │ │ │ │ -0002cc10: 7320 6f66 2074 6865 206d 6f64 756c 6573 s of the modules │ │ │ │ -0002cc20: 2042 5f69 2864 2920 696e 2061 0a20 2020 B_i(d) in a. │ │ │ │ -0002cc30: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ -0002cc40: 7469 6f6e 0a20 202a 202a 6e6f 7465 2062 tion. * *note b │ │ │ │ -0002cc50: 4d61 7073 3a20 624d 6170 732c 202d 2d20 Maps: bMaps, -- │ │ │ │ -0002cc60: 6c69 7374 2074 6865 206d 6170 7320 2064 list the maps d │ │ │ │ -0002cc70: 5f70 3a42 5f31 2870 292d 2d3e 425f 3028 _p:B_1(p)-->B_0( │ │ │ │ -0002cc80: 7029 2069 6e20 610a 2020 2020 6d61 7472 p) in a. matr │ │ │ │ -0002cc90: 6978 4661 6374 6f72 697a 6174 696f 6e0a ixFactorization. │ │ │ │ -0002cca0: 2020 2a20 2a6e 6f74 6520 7073 694d 6170 * *note psiMap │ │ │ │ -0002ccb0: 733a 2070 7369 4d61 7073 2c20 2d2d 206c s: psiMaps, -- l │ │ │ │ -0002ccc0: 6973 7420 7468 6520 6d61 7073 2020 7073 ist the maps ps │ │ │ │ -0002ccd0: 6928 7029 3a20 425f 3128 7029 202d 2d3e i(p): B_1(p) --> │ │ │ │ -0002cce0: 2041 5f30 2870 2d31 2920 696e 2061 0a20 A_0(p-1) in a. │ │ │ │ -0002ccf0: 2020 206d 6174 7269 7846 6163 746f 7269 matrixFactori │ │ │ │ -0002cd00: 7a61 7469 6f6e 0a0a 5761 7973 2074 6f20 zation..Ways to │ │ │ │ -0002cd10: 7573 6520 684d 6170 733a 0a3d 3d3d 3d3d use hMaps:.===== │ │ │ │ -0002cd20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -0002cd30: 202a 2022 684d 6170 7328 4c69 7374 2922 * "hMaps(List)" │ │ │ │ -0002cd40: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -0002cd50: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -0002cd60: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -0002cd70: 6563 7420 2a6e 6f74 6520 684d 6170 733a ect *note hMaps: │ │ │ │ -0002cd80: 2068 4d61 7073 2c20 6973 2061 202a 6e6f hMaps, is a *no │ │ │ │ -0002cd90: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ -0002cda0: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f on:.(Macaulay2Do │ │ │ │ -0002cdb0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ -0002cdc0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ +0002c960: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +0002c970: 2020 2020 2020 684d 6170 7320 3d20 684d hMaps = hM │ │ │ │ +0002c980: 6170 7320 6d66 0a20 202a 2049 6e70 7574 aps mf. * Input │ │ │ │ +0002c990: 733a 0a20 2020 2020 202a 206d 662c 2061 s:. * mf, a │ │ │ │ +0002c9a0: 202a 6e6f 7465 206c 6973 743a 2028 4d61 *note list: (Ma │ │ │ │ +0002c9b0: 6361 756c 6179 3244 6f63 294c 6973 742c caulay2Doc)List, │ │ │ │ +0002c9c0: 2c20 6f75 7470 7574 206f 6620 6120 6d61 , output of a ma │ │ │ │ +0002c9d0: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ +0002c9e0: 6e0a 2020 2020 2020 2020 636f 6d70 7574 n. comput │ │ │ │ +0002c9f0: 6174 696f 6e0a 2020 2a20 4f75 7470 7574 ation. * Output │ │ │ │ +0002ca00: 733a 0a20 2020 2020 202a 2068 4d61 7073 s:. * hMaps │ │ │ │ +0002ca10: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ +0002ca20: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ +0002ca30: 7374 2c2c 206c 6973 7420 6d61 7472 6963 st,, list matric │ │ │ │ +0002ca40: 6573 2024 685f 703a 2041 5f30 2870 295c es $h_p: A_0(p)\ │ │ │ │ +0002ca50: 746f 0a20 2020 2020 2020 2041 5f31 2870 to. A_1(p │ │ │ │ +0002ca60: 2924 2e20 5468 6520 736f 7572 6365 7320 )$. The sources │ │ │ │ +0002ca70: 616e 6420 7461 7267 6574 7320 6f66 2074 and targets of t │ │ │ │ +0002ca80: 6865 7365 206d 6170 7320 6861 7665 2074 hese maps have t │ │ │ │ +0002ca90: 6865 2063 6f6d 706f 6e65 6e74 730a 2020 he components. │ │ │ │ +0002caa0: 2020 2020 2020 425f 7328 7029 2e0a 0a44 B_s(p)...D │ │ │ │ +0002cab0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +0002cac0: 3d3d 3d3d 3d3d 0a0a 5365 6520 7468 6520 ======..See the │ │ │ │ +0002cad0: 646f 6375 6d65 6e74 6174 696f 6e20 666f documentation fo │ │ │ │ +0002cae0: 7220 6d61 7472 6978 4661 6374 6f72 697a r matrixFactoriz │ │ │ │ +0002caf0: 6174 696f 6e20 666f 7220 616e 2065 7861 ation for an exa │ │ │ │ +0002cb00: 6d70 6c65 2e0a 0a53 6565 2061 6c73 6f0a mple...See also. │ │ │ │ +0002cb10: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +0002cb20: 6f74 6520 6d61 7472 6978 4661 6374 6f72 ote matrixFactor │ │ │ │ +0002cb30: 697a 6174 696f 6e3a 206d 6174 7269 7846 ization: matrixF │ │ │ │ +0002cb40: 6163 746f 7269 7a61 7469 6f6e 2c20 2d2d actorization, -- │ │ │ │ +0002cb50: 204d 6170 7320 696e 2061 2068 6967 6865 Maps in a highe │ │ │ │ +0002cb60: 720a 2020 2020 636f 6469 6d65 6e73 696f r. codimensio │ │ │ │ +0002cb70: 6e20 6d61 7472 6978 2066 6163 746f 7269 n matrix factori │ │ │ │ +0002cb80: 7a61 7469 6f6e 0a20 202a 202a 6e6f 7465 zation. * *note │ │ │ │ +0002cb90: 2064 4d61 7073 3a20 644d 6170 732c 202d dMaps: dMaps, - │ │ │ │ +0002cba0: 2d20 6c69 7374 2074 6865 206d 6170 7320 - list the maps │ │ │ │ +0002cbb0: 2064 2870 293a 415f 3128 7029 2d2d 3e20 d(p):A_1(p)--> │ │ │ │ +0002cbc0: 415f 3028 7029 2069 6e20 610a 2020 2020 A_0(p) in a. │ │ │ │ +0002cbd0: 6d61 7472 6978 4661 6374 6f72 697a 6174 matrixFactorizat │ │ │ │ +0002cbe0: 696f 6e0a 2020 2a20 2a6e 6f74 6520 4252 ion. * *note BR │ │ │ │ +0002cbf0: 616e 6b73 3a20 4252 616e 6b73 2c20 2d2d anks: BRanks, -- │ │ │ │ +0002cc00: 2072 616e 6b73 206f 6620 7468 6520 6d6f ranks of the mo │ │ │ │ +0002cc10: 6475 6c65 7320 425f 6928 6429 2069 6e20 dules B_i(d) in │ │ │ │ +0002cc20: 610a 2020 2020 6d61 7472 6978 4661 6374 a. matrixFact │ │ │ │ +0002cc30: 6f72 697a 6174 696f 6e0a 2020 2a20 2a6e orization. * *n │ │ │ │ +0002cc40: 6f74 6520 624d 6170 733a 2062 4d61 7073 ote bMaps: bMaps │ │ │ │ +0002cc50: 2c20 2d2d 206c 6973 7420 7468 6520 6d61 , -- list the ma │ │ │ │ +0002cc60: 7073 2020 645f 703a 425f 3128 7029 2d2d ps d_p:B_1(p)-- │ │ │ │ +0002cc70: 3e42 5f30 2870 2920 696e 2061 0a20 2020 >B_0(p) in a. │ │ │ │ +0002cc80: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ +0002cc90: 7469 6f6e 0a20 202a 202a 6e6f 7465 2070 tion. * *note p │ │ │ │ +0002cca0: 7369 4d61 7073 3a20 7073 694d 6170 732c siMaps: psiMaps, │ │ │ │ +0002ccb0: 202d 2d20 6c69 7374 2074 6865 206d 6170 -- list the map │ │ │ │ +0002ccc0: 7320 2070 7369 2870 293a 2042 5f31 2870 s psi(p): B_1(p │ │ │ │ +0002ccd0: 2920 2d2d 3e20 415f 3028 702d 3129 2069 ) --> A_0(p-1) i │ │ │ │ +0002cce0: 6e20 610a 2020 2020 6d61 7472 6978 4661 n a. matrixFa │ │ │ │ +0002ccf0: 6374 6f72 697a 6174 696f 6e0a 0a57 6179 ctorization..Way │ │ │ │ +0002cd00: 7320 746f 2075 7365 2068 4d61 7073 3a0a s to use hMaps:. │ │ │ │ +0002cd10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002cd20: 3d3d 0a0a 2020 2a20 2268 4d61 7073 284c ==.. * "hMaps(L │ │ │ │ +0002cd30: 6973 7429 220a 0a46 6f72 2074 6865 2070 ist)"..For the p │ │ │ │ +0002cd40: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +0002cd50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +0002cd60: 6520 6f62 6a65 6374 202a 6e6f 7465 2068 e object *note h │ │ │ │ +0002cd70: 4d61 7073 3a20 684d 6170 732c 2069 7320 Maps: hMaps, is │ │ │ │ +0002cd80: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ +0002cd90: 756e 6374 696f 6e3a 0a28 4d61 6361 756c unction:.(Macaul │ │ │ │ +0002cda0: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +0002cdb0: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +0002cdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ce10: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -0002ce20: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -0002ce30: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -0002ce40: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -0002ce50: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -0002ce60: 352b 6473 2f4d 322f 4d61 6361 756c 6179 5+ds/M2/Macaulay │ │ │ │ -0002ce70: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ -0002ce80: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -0002ce90: 5265 736f 6c75 7469 6f6e 732e 6d32 3a34 Resolutions.m2:4 │ │ │ │ -0002cea0: 3435 373a 302e 0a1f 0a46 696c 653a 2043 457:0....File: C │ │ │ │ -0002ceb0: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -0002cec0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ -0002ced0: 6e66 6f2c 204e 6f64 653a 2048 6f6d 5769 nfo, Node: HomWi │ │ │ │ -0002cee0: 7468 436f 6d70 6f6e 656e 7473 2c20 4e65 thComponents, Ne │ │ │ │ -0002cef0: 7874 3a20 696e 6669 6e69 7465 4265 7474 xt: infiniteBett │ │ │ │ -0002cf00: 694e 756d 6265 7273 2c20 5072 6576 3a20 iNumbers, Prev: │ │ │ │ -0002cf10: 684d 6170 732c 2055 703a 2054 6f70 0a0a hMaps, Up: Top.. │ │ │ │ -0002cf20: 486f 6d57 6974 6843 6f6d 706f 6e65 6e74 HomWithComponent │ │ │ │ -0002cf30: 7320 2d2d 2063 6f6d 7075 7465 7320 486f s -- computes Ho │ │ │ │ -0002cf40: 6d2c 2070 7265 7365 7276 696e 6720 6469 m, preserving di │ │ │ │ -0002cf50: 7265 6374 2073 756d 2069 6e66 6f72 6d61 rect sum informa │ │ │ │ -0002cf60: 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a tion.*********** │ │ │ │ +0002ce00: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +0002ce10: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0002ce20: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0002ce30: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0002ce40: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0002ce50: 2e32 362e 3035 2b64 732f 4d32 2f4d 6163 .26.05+ds/M2/Mac │ │ │ │ +0002ce60: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +0002ce70: 0a43 6f6d 706c 6574 6549 6e74 6572 7365 .CompleteInterse │ │ │ │ +0002ce80: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ +0002ce90: 2e6d 323a 3434 3537 3a30 2e0a 1f0a 4669 .m2:4457:0....Fi │ │ │ │ +0002cea0: 6c65 3a20 436f 6d70 6c65 7465 496e 7465 le: CompleteInte │ │ │ │ +0002ceb0: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +0002cec0: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20 ons.info, Node: │ │ │ │ +0002ced0: 486f 6d57 6974 6843 6f6d 706f 6e65 6e74 HomWithComponent │ │ │ │ +0002cee0: 732c 204e 6578 743a 2069 6e66 696e 6974 s, Next: infinit │ │ │ │ +0002cef0: 6542 6574 7469 4e75 6d62 6572 732c 2050 eBettiNumbers, P │ │ │ │ +0002cf00: 7265 763a 2068 4d61 7073 2c20 5570 3a20 rev: hMaps, Up: │ │ │ │ +0002cf10: 546f 700a 0a48 6f6d 5769 7468 436f 6d70 Top..HomWithComp │ │ │ │ +0002cf20: 6f6e 656e 7473 202d 2d20 636f 6d70 7574 onents -- comput │ │ │ │ +0002cf30: 6573 2048 6f6d 2c20 7072 6573 6572 7669 es Hom, preservi │ │ │ │ +0002cf40: 6e67 2064 6972 6563 7420 7375 6d20 696e ng direct sum in │ │ │ │ +0002cf50: 666f 726d 6174 696f 6e0a 2a2a 2a2a 2a2a formation.****** │ │ │ │ +0002cf60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002cf70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002cf80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002cf90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002cfa0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -0002cfb0: 7361 6765 3a20 0a20 2020 2020 2020 2048 sage: . H │ │ │ │ -0002cfc0: 203d 2048 6f6d 284d 2c4e 290a 2020 2a20 = Hom(M,N). * │ │ │ │ -0002cfd0: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -0002cfe0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c M, a *note modul │ │ │ │ -0002cff0: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -0002d000: 294d 6f64 756c 652c 2c20 0a20 2020 2020 )Module,, . │ │ │ │ -0002d010: 202a 204e 2c20 6120 2a6e 6f74 6520 6d6f * N, a *note mo │ │ │ │ -0002d020: 6475 6c65 3a20 284d 6163 6175 6c61 7932 dule: (Macaulay2 │ │ │ │ -0002d030: 446f 6329 4d6f 6475 6c65 2c2c 200a 2020 Doc)Module,, . │ │ │ │ -0002d040: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ -0002d050: 202a 2048 2c20 6120 2a6e 6f74 6520 6d6f * H, a *note mo │ │ │ │ -0002d060: 6475 6c65 3a20 284d 6163 6175 6c61 7932 dule: (Macaulay2 │ │ │ │ -0002d070: 446f 6329 4d6f 6475 6c65 2c2c 200a 0a44 Doc)Module,, ..D │ │ │ │ -0002d080: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -0002d090: 3d3d 3d3d 3d3d 0a0a 4966 204d 2061 6e64 ======..If M and │ │ │ │ -0002d0a0: 2f6f 7220 4e20 6172 6520 6469 7265 6374 /or N are direct │ │ │ │ -0002d0b0: 2073 756d 206d 6f64 756c 6573 2028 6973 sum modules (is │ │ │ │ -0002d0c0: 4469 7265 6374 5375 6d20 4d20 3d3d 2074 DirectSum M == t │ │ │ │ -0002d0d0: 7275 6529 2074 6865 6e20 4820 6973 2074 rue) then H is t │ │ │ │ -0002d0e0: 6865 0a64 6972 6563 7420 7375 6d20 6f66 he.direct sum of │ │ │ │ -0002d0f0: 2074 6865 2048 6f6d 7320 6265 7477 6565 the Homs betwee │ │ │ │ -0002d100: 6e20 7468 6520 636f 6d70 6f6e 656e 7473 n the components │ │ │ │ -0002d110: 2e20 5468 6973 2053 484f 554c 4420 6265 . This SHOULD be │ │ │ │ -0002d120: 2062 7569 6c74 2069 6e74 6f0a 486f 6d28 built into.Hom( │ │ │ │ -0002d130: 4d2c 4e29 2c20 6275 7420 6973 6e27 7420 M,N), but isn't │ │ │ │ -0002d140: 6173 206f 6620 4d32 2c20 762e 2031 2e37 as of M2, v. 1.7 │ │ │ │ -0002d150: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -0002d160: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2074 ===.. * *note t │ │ │ │ -0002d170: 656e 736f 7257 6974 6843 6f6d 706f 6e65 ensorWithCompone │ │ │ │ -0002d180: 6e74 733a 2074 656e 736f 7257 6974 6843 nts: tensorWithC │ │ │ │ -0002d190: 6f6d 706f 6e65 6e74 732c 202d 2d20 666f omponents, -- fo │ │ │ │ -0002d1a0: 726d 7320 7468 6520 7465 6e73 6f72 0a20 rms the tensor. │ │ │ │ -0002d1b0: 2020 2070 726f 6475 6374 2c20 7072 6573 product, pres │ │ │ │ -0002d1c0: 6572 7669 6e67 2064 6972 6563 7420 7375 erving direct su │ │ │ │ -0002d1d0: 6d20 696e 666f 726d 6174 696f 6e0a 2020 m information. │ │ │ │ -0002d1e0: 2a20 2a6e 6f74 6520 6475 616c 5769 7468 * *note dualWith │ │ │ │ -0002d1f0: 436f 6d70 6f6e 656e 7473 3a20 6475 616c Components: dual │ │ │ │ -0002d200: 5769 7468 436f 6d70 6f6e 656e 7473 2c20 WithComponents, │ │ │ │ -0002d210: 2d2d 2064 7561 6c20 6d6f 6475 6c65 2070 -- dual module p │ │ │ │ -0002d220: 7265 7365 7276 696e 670a 2020 2020 6469 reserving. di │ │ │ │ -0002d230: 7265 6374 2073 756d 2069 6e66 6f72 6d61 rect sum informa │ │ │ │ -0002d240: 7469 6f6e 0a0a 5761 7973 2074 6f20 7573 tion..Ways to us │ │ │ │ -0002d250: 6520 486f 6d57 6974 6843 6f6d 706f 6e65 e HomWithCompone │ │ │ │ -0002d260: 6e74 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d nts:.=========== │ │ │ │ -0002d270: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002d280: 3d3d 3d0a 0a20 202a 2022 486f 6d57 6974 ===.. * "HomWit │ │ │ │ -0002d290: 6843 6f6d 706f 6e65 6e74 7328 4d6f 6475 hComponents(Modu │ │ │ │ -0002d2a0: 6c65 2c4d 6f64 756c 6529 220a 0a46 6f72 le,Module)"..For │ │ │ │ -0002d2b0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -0002d2c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002d2d0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -0002d2e0: 6e6f 7465 2048 6f6d 5769 7468 436f 6d70 note HomWithComp │ │ │ │ -0002d2f0: 6f6e 656e 7473 3a20 486f 6d57 6974 6843 onents: HomWithC │ │ │ │ -0002d300: 6f6d 706f 6e65 6e74 732c 2069 7320 6120 omponents, is a │ │ │ │ -0002d310: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e *note method.fun │ │ │ │ -0002d320: 6374 696f 6e3a 2028 4d61 6361 756c 6179 ction: (Macaulay │ │ │ │ -0002d330: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ -0002d340: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ion,...--------- │ │ │ │ +0002cf90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +0002cfa0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +0002cfb0: 2020 2020 4820 3d20 486f 6d28 4d2c 4e29 H = Hom(M,N) │ │ │ │ +0002cfc0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ +0002cfd0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520 * M, a *note │ │ │ │ +0002cfe0: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61 module: (Macaula │ │ │ │ +0002cff0: 7932 446f 6329 4d6f 6475 6c65 2c2c 200a y2Doc)Module,, . │ │ │ │ +0002d000: 2020 2020 2020 2a20 4e2c 2061 202a 6e6f * N, a *no │ │ │ │ +0002d010: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ +0002d020: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ +0002d030: 2c20 0a20 202a 204f 7574 7075 7473 3a0a , . * Outputs:. │ │ │ │ +0002d040: 2020 2020 2020 2a20 482c 2061 202a 6e6f * H, a *no │ │ │ │ +0002d050: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ +0002d060: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ +0002d070: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a , ..Description. │ │ │ │ +0002d080: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620 ===========..If │ │ │ │ +0002d090: 4d20 616e 642f 6f72 204e 2061 7265 2064 M and/or N are d │ │ │ │ +0002d0a0: 6972 6563 7420 7375 6d20 6d6f 6475 6c65 irect sum module │ │ │ │ +0002d0b0: 7320 2869 7344 6972 6563 7453 756d 204d s (isDirectSum M │ │ │ │ +0002d0c0: 203d 3d20 7472 7565 2920 7468 656e 2048 == true) then H │ │ │ │ +0002d0d0: 2069 7320 7468 650a 6469 7265 6374 2073 is the.direct s │ │ │ │ +0002d0e0: 756d 206f 6620 7468 6520 486f 6d73 2062 um of the Homs b │ │ │ │ +0002d0f0: 6574 7765 656e 2074 6865 2063 6f6d 706f etween the compo │ │ │ │ +0002d100: 6e65 6e74 732e 2054 6869 7320 5348 4f55 nents. This SHOU │ │ │ │ +0002d110: 4c44 2062 6520 6275 696c 7420 696e 746f LD be built into │ │ │ │ +0002d120: 0a48 6f6d 284d 2c4e 292c 2062 7574 2069 .Hom(M,N), but i │ │ │ │ +0002d130: 736e 2774 2061 7320 6f66 204d 322c 2076 sn't as of M2, v │ │ │ │ +0002d140: 2e20 312e 370a 0a53 6565 2061 6c73 6f0a . 1.7..See also. │ │ │ │ +0002d150: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +0002d160: 6f74 6520 7465 6e73 6f72 5769 7468 436f ote tensorWithCo │ │ │ │ +0002d170: 6d70 6f6e 656e 7473 3a20 7465 6e73 6f72 mponents: tensor │ │ │ │ +0002d180: 5769 7468 436f 6d70 6f6e 656e 7473 2c20 WithComponents, │ │ │ │ +0002d190: 2d2d 2066 6f72 6d73 2074 6865 2074 656e -- forms the ten │ │ │ │ +0002d1a0: 736f 720a 2020 2020 7072 6f64 7563 742c sor. product, │ │ │ │ +0002d1b0: 2070 7265 7365 7276 696e 6720 6469 7265 preserving dire │ │ │ │ +0002d1c0: 6374 2073 756d 2069 6e66 6f72 6d61 7469 ct sum informati │ │ │ │ +0002d1d0: 6f6e 0a20 202a 202a 6e6f 7465 2064 7561 on. * *note dua │ │ │ │ +0002d1e0: 6c57 6974 6843 6f6d 706f 6e65 6e74 733a lWithComponents: │ │ │ │ +0002d1f0: 2064 7561 6c57 6974 6843 6f6d 706f 6e65 dualWithCompone │ │ │ │ +0002d200: 6e74 732c 202d 2d20 6475 616c 206d 6f64 nts, -- dual mod │ │ │ │ +0002d210: 756c 6520 7072 6573 6572 7669 6e67 0a20 ule preserving. │ │ │ │ +0002d220: 2020 2064 6972 6563 7420 7375 6d20 696e direct sum in │ │ │ │ +0002d230: 666f 726d 6174 696f 6e0a 0a57 6179 7320 formation..Ways │ │ │ │ +0002d240: 746f 2075 7365 2048 6f6d 5769 7468 436f to use HomWithCo │ │ │ │ +0002d250: 6d70 6f6e 656e 7473 3a0a 3d3d 3d3d 3d3d mponents:.====== │ │ │ │ +0002d260: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002d270: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2248 ========.. * "H │ │ │ │ +0002d280: 6f6d 5769 7468 436f 6d70 6f6e 656e 7473 omWithComponents │ │ │ │ +0002d290: 284d 6f64 756c 652c 4d6f 6475 6c65 2922 (Module,Module)" │ │ │ │ +0002d2a0: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ +0002d2b0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ +0002d2c0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ +0002d2d0: 6563 7420 2a6e 6f74 6520 486f 6d57 6974 ect *note HomWit │ │ │ │ +0002d2e0: 6843 6f6d 706f 6e65 6e74 733a 2048 6f6d hComponents: Hom │ │ │ │ +0002d2f0: 5769 7468 436f 6d70 6f6e 656e 7473 2c20 WithComponents, │ │ │ │ +0002d300: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ +0002d310: 640a 6675 6e63 7469 6f6e 3a20 284d 6163 d.function: (Mac │ │ │ │ +0002d320: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ +0002d330: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +0002d340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d390: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -0002d3a0: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -0002d3b0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -0002d3c0: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -0002d3d0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -0002d3e0: 362e 3035 2b64 732f 4d32 2f4d 6163 6175 6.05+ds/M2/Macau │ │ │ │ -0002d3f0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ -0002d400: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -0002d410: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d ionResolutions.m │ │ │ │ -0002d420: 323a 3236 3435 3a30 2e0a 1f0a 4669 6c65 2:2645:0....File │ │ │ │ -0002d430: 3a20 436f 6d70 6c65 7465 496e 7465 7273 : CompleteInters │ │ │ │ -0002d440: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -0002d450: 732e 696e 666f 2c20 4e6f 6465 3a20 696e s.info, Node: in │ │ │ │ -0002d460: 6669 6e69 7465 4265 7474 694e 756d 6265 finiteBettiNumbe │ │ │ │ -0002d470: 7273 2c20 4e65 7874 3a20 6973 4c69 6e65 rs, Next: isLine │ │ │ │ -0002d480: 6172 2c20 5072 6576 3a20 486f 6d57 6974 ar, Prev: HomWit │ │ │ │ -0002d490: 6843 6f6d 706f 6e65 6e74 732c 2055 703a hComponents, Up: │ │ │ │ -0002d4a0: 2054 6f70 0a0a 696e 6669 6e69 7465 4265 Top..infiniteBe │ │ │ │ -0002d4b0: 7474 694e 756d 6265 7273 202d 2d20 6265 ttiNumbers -- be │ │ │ │ -0002d4c0: 7474 6920 6e75 6d62 6572 7320 6f66 2066 tti numbers of f │ │ │ │ -0002d4d0: 696e 6974 6520 7265 736f 6c75 7469 6f6e inite resolution │ │ │ │ -0002d4e0: 2063 6f6d 7075 7465 6420 6672 6f6d 2061 computed from a │ │ │ │ -0002d4f0: 206d 6174 7269 7820 6661 6374 6f72 697a matrix factoriz │ │ │ │ -0002d500: 6174 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a ation.********** │ │ │ │ +0002d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +0002d390: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +0002d3a0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +0002d3b0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +0002d3c0: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +0002d3d0: 322d 312e 3236 2e30 352b 6473 2f4d 322f 2-1.26.05+ds/M2/ │ │ │ │ +0002d3e0: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +0002d3f0: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ +0002d400: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +0002d410: 6f6e 732e 6d32 3a32 3634 353a 302e 0a1f ons.m2:2645:0... │ │ │ │ +0002d420: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ +0002d430: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +0002d440: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ +0002d450: 653a 2069 6e66 696e 6974 6542 6574 7469 e: infiniteBetti │ │ │ │ +0002d460: 4e75 6d62 6572 732c 204e 6578 743a 2069 Numbers, Next: i │ │ │ │ +0002d470: 734c 696e 6561 722c 2050 7265 763a 2048 sLinear, Prev: H │ │ │ │ +0002d480: 6f6d 5769 7468 436f 6d70 6f6e 656e 7473 omWithComponents │ │ │ │ +0002d490: 2c20 5570 3a20 546f 700a 0a69 6e66 696e , Up: Top..infin │ │ │ │ +0002d4a0: 6974 6542 6574 7469 4e75 6d62 6572 7320 iteBettiNumbers │ │ │ │ +0002d4b0: 2d2d 2062 6574 7469 206e 756d 6265 7273 -- betti numbers │ │ │ │ +0002d4c0: 206f 6620 6669 6e69 7465 2072 6573 6f6c of finite resol │ │ │ │ +0002d4d0: 7574 696f 6e20 636f 6d70 7574 6564 2066 ution computed f │ │ │ │ +0002d4e0: 726f 6d20 6120 6d61 7472 6978 2066 6163 rom a matrix fac │ │ │ │ +0002d4f0: 746f 7269 7a61 7469 6f6e 0a2a 2a2a 2a2a torization.***** │ │ │ │ +0002d500: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002d510: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002d520: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002d530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002d540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002d550: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002d560: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ -0002d570: 3a20 0a20 2020 2020 2020 204c 203d 2066 : . L = f │ │ │ │ -0002d580: 696e 6974 6542 6574 7469 4e75 6d62 6572 initeBettiNumber │ │ │ │ -0002d590: 7320 284d 462c 6c65 6e29 0a20 202a 2049 s (MF,len). * I │ │ │ │ -0002d5a0: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ -0002d5b0: 462c 2061 202a 6e6f 7465 206c 6973 743a F, a *note list: │ │ │ │ -0002d5c0: 2028 4d61 6361 756c 6179 3244 6f63 294c (Macaulay2Doc)L │ │ │ │ -0002d5d0: 6973 742c 2c20 4c69 7374 206f 6620 4861 ist,, List of Ha │ │ │ │ -0002d5e0: 7368 5461 626c 6573 2061 7320 636f 6d70 shTables as comp │ │ │ │ -0002d5f0: 7574 6564 0a20 2020 2020 2020 2062 7920 uted. by │ │ │ │ -0002d600: 226d 6174 7269 7846 6163 746f 7269 7a61 "matrixFactoriza │ │ │ │ -0002d610: 7469 6f6e 220a 2020 2020 2020 2a20 6c65 tion". * le │ │ │ │ -0002d620: 6e2c 2061 6e20 2a6e 6f74 6520 696e 7465 n, an *note inte │ │ │ │ -0002d630: 6765 723a 2028 4d61 6361 756c 6179 3244 ger: (Macaulay2D │ │ │ │ -0002d640: 6f63 295a 5a2c 2c20 6c65 6e67 7468 206f oc)ZZ,, length o │ │ │ │ -0002d650: 6620 6265 7474 6920 6e75 6d62 6572 0a20 f betti number. │ │ │ │ -0002d660: 2020 2020 2020 2073 6571 7565 6e63 6520 sequence │ │ │ │ -0002d670: 746f 2070 726f 6475 6365 0a20 202a 204f to produce. * O │ │ │ │ -0002d680: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -0002d690: 4c2c 2061 202a 6e6f 7465 206c 6973 743a L, a *note list: │ │ │ │ -0002d6a0: 2028 4d61 6361 756c 6179 3244 6f63 294c (Macaulay2Doc)L │ │ │ │ -0002d6b0: 6973 742c 2c20 4c69 7374 206f 6620 6265 ist,, List of be │ │ │ │ -0002d6c0: 7474 6920 6e75 6d62 6572 730a 0a44 6573 tti numbers..Des │ │ │ │ -0002d6d0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -0002d6e0: 3d3d 3d3d 0a0a 5573 6573 2074 6865 2072 ====..Uses the r │ │ │ │ -0002d6f0: 616e 6b73 206f 6620 7468 6520 4220 6d61 anks of the B ma │ │ │ │ -0002d700: 7472 6963 6573 2069 6e20 6120 6d61 7472 trices in a matr │ │ │ │ -0002d710: 6978 2066 6163 746f 7269 7a61 7469 6f6e ix factorization │ │ │ │ -0002d720: 2066 6f72 2061 206d 6f64 756c 6520 4d20 for a module M │ │ │ │ -0002d730: 6f76 6572 0a53 2f28 665f 312c 2e2e 2c66 over.S/(f_1,..,f │ │ │ │ -0002d740: 5f63 2920 746f 2063 6f6d 7075 7465 2074 _c) to compute t │ │ │ │ -0002d750: 6865 2062 6574 7469 206e 756d 6265 7273 he betti numbers │ │ │ │ -0002d760: 206f 6620 7468 6520 6d69 6e69 6d61 6c20 of the minimal │ │ │ │ -0002d770: 7265 736f 6c75 7469 6f6e 206f 6620 4d20 resolution of M │ │ │ │ -0002d780: 6f76 6572 0a52 2c20 7768 6963 6820 6973 over.R, which is │ │ │ │ -0002d790: 2074 6865 2073 756d 206f 6620 7468 6520 the sum of the │ │ │ │ -0002d7a0: 6469 7669 6465 6420 706f 7765 7220 616c divided power al │ │ │ │ -0002d7b0: 6765 6272 6173 206f 6e20 632d 6a2b 3120 gebras on c-j+1 │ │ │ │ -0002d7c0: 7661 7269 6162 6c65 7320 7465 6e73 6f72 variables tensor │ │ │ │ -0002d7d0: 6564 0a77 6974 6820 4228 6a29 2e0a 0a2b ed.with B(j)...+ │ │ │ │ +0002d550: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +0002d560: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ +0002d570: 4c20 3d20 6669 6e69 7465 4265 7474 694e L = finiteBettiN │ │ │ │ +0002d580: 756d 6265 7273 2028 4d46 2c6c 656e 290a umbers (MF,len). │ │ │ │ +0002d590: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0002d5a0: 2020 2a20 4d46 2c20 6120 2a6e 6f74 6520 * MF, a *note │ │ │ │ +0002d5b0: 6c69 7374 3a20 284d 6163 6175 6c61 7932 list: (Macaulay2 │ │ │ │ +0002d5c0: 446f 6329 4c69 7374 2c2c 204c 6973 7420 Doc)List,, List │ │ │ │ +0002d5d0: 6f66 2048 6173 6854 6162 6c65 7320 6173 of HashTables as │ │ │ │ +0002d5e0: 2063 6f6d 7075 7465 640a 2020 2020 2020 computed. │ │ │ │ +0002d5f0: 2020 6279 2022 6d61 7472 6978 4661 6374 by "matrixFact │ │ │ │ +0002d600: 6f72 697a 6174 696f 6e22 0a20 2020 2020 orization". │ │ │ │ +0002d610: 202a 206c 656e 2c20 616e 202a 6e6f 7465 * len, an *note │ │ │ │ +0002d620: 2069 6e74 6567 6572 3a20 284d 6163 6175 integer: (Macau │ │ │ │ +0002d630: 6c61 7932 446f 6329 5a5a 2c2c 206c 656e lay2Doc)ZZ,, len │ │ │ │ +0002d640: 6774 6820 6f66 2062 6574 7469 206e 756d gth of betti num │ │ │ │ +0002d650: 6265 720a 2020 2020 2020 2020 7365 7175 ber. sequ │ │ │ │ +0002d660: 656e 6365 2074 6f20 7072 6f64 7563 650a ence to produce. │ │ │ │ +0002d670: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ +0002d680: 2020 202a 204c 2c20 6120 2a6e 6f74 6520 * L, a *note │ │ │ │ +0002d690: 6c69 7374 3a20 284d 6163 6175 6c61 7932 list: (Macaulay2 │ │ │ │ +0002d6a0: 446f 6329 4c69 7374 2c2c 204c 6973 7420 Doc)List,, List │ │ │ │ +0002d6b0: 6f66 2062 6574 7469 206e 756d 6265 7273 of betti numbers │ │ │ │ +0002d6c0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +0002d6d0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a55 7365 7320 =========..Uses │ │ │ │ +0002d6e0: 7468 6520 7261 6e6b 7320 6f66 2074 6865 the ranks of the │ │ │ │ +0002d6f0: 2042 206d 6174 7269 6365 7320 696e 2061 B matrices in a │ │ │ │ +0002d700: 206d 6174 7269 7820 6661 6374 6f72 697a matrix factoriz │ │ │ │ +0002d710: 6174 696f 6e20 666f 7220 6120 6d6f 6475 ation for a modu │ │ │ │ +0002d720: 6c65 204d 206f 7665 720a 532f 2866 5f31 le M over.S/(f_1 │ │ │ │ +0002d730: 2c2e 2e2c 665f 6329 2074 6f20 636f 6d70 ,..,f_c) to comp │ │ │ │ +0002d740: 7574 6520 7468 6520 6265 7474 6920 6e75 ute the betti nu │ │ │ │ +0002d750: 6d62 6572 7320 6f66 2074 6865 206d 696e mbers of the min │ │ │ │ +0002d760: 696d 616c 2072 6573 6f6c 7574 696f 6e20 imal resolution │ │ │ │ +0002d770: 6f66 204d 206f 7665 720a 522c 2077 6869 of M over.R, whi │ │ │ │ +0002d780: 6368 2069 7320 7468 6520 7375 6d20 6f66 ch is the sum of │ │ │ │ +0002d790: 2074 6865 2064 6976 6964 6564 2070 6f77 the divided pow │ │ │ │ +0002d7a0: 6572 2061 6c67 6562 7261 7320 6f6e 2063 er algebras on c │ │ │ │ +0002d7b0: 2d6a 2b31 2076 6172 6961 626c 6573 2074 -j+1 variables t │ │ │ │ +0002d7c0: 656e 736f 7265 640a 7769 7468 2042 286a ensored.with B(j │ │ │ │ +0002d7d0: 292e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d )...+----------- │ │ │ │ 0002d7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d810: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e --+.|i1 : setRan │ │ │ │ -0002d820: 646f 6d53 6565 6420 3020 2020 2020 2020 domSeed 0 │ │ │ │ -0002d830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d840: 2020 2020 2020 207c 0a7c 202d 2d20 7365 |.| -- se │ │ │ │ -0002d850: 7474 696e 6720 7261 6e64 6f6d 2073 6565 tting random see │ │ │ │ -0002d860: 6420 746f 2030 2020 2020 2020 2020 2020 d to 0 │ │ │ │ -0002d870: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002d800: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 -------+.|i1 : s │ │ │ │ +0002d810: 6574 5261 6e64 6f6d 5365 6564 2030 2020 etRandomSeed 0 │ │ │ │ +0002d820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d830: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002d840: 2d2d 2073 6574 7469 6e67 2072 616e 646f -- setting rando │ │ │ │ +0002d850: 6d20 7365 6564 2074 6f20 3020 2020 2020 m seed to 0 │ │ │ │ +0002d860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d870: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002d880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d8b0: 207c 0a7c 6f31 203d 2030 2020 2020 2020 |.|o1 = 0 │ │ │ │ +0002d8a0: 2020 2020 2020 7c0a 7c6f 3120 3d20 3020 |.|o1 = 0 │ │ │ │ +0002d8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d8e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002d8d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002d8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -0002d920: 203a 206b 6b20 3d20 5a5a 2f31 3031 2020 : kk = ZZ/101 │ │ │ │ +0002d910: 2b0a 7c69 3220 3a20 6b6b 203d 205a 5a2f +.|i2 : kk = ZZ/ │ │ │ │ +0002d920: 3130 3120 2020 2020 2020 2020 2020 2020 101 │ │ │ │ 0002d930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d950: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002d940: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002d950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d980: 2020 2020 207c 0a7c 6f32 203d 206b 6b20 |.|o2 = kk │ │ │ │ +0002d970: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0002d980: 3d20 6b6b 2020 2020 2020 2020 2020 2020 = kk │ │ │ │ 0002d990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d9b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002d9a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002d9b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002d9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d9e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002d9f0: 0a7c 6f32 203a 2051 756f 7469 656e 7452 .|o2 : QuotientR │ │ │ │ -0002da00: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -0002da10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002da20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002d9e0: 2020 2020 7c0a 7c6f 3220 3a20 5175 6f74 |.|o2 : Quot │ │ │ │ +0002d9f0: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +0002da00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002da10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0002da20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002da50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -0002da60: 2053 203d 206b 6b5b 612c 622c 752c 765d S = kk[a,b,u,v] │ │ │ │ +0002da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002da50: 7c69 3320 3a20 5320 3d20 6b6b 5b61 2c62 |i3 : S = kk[a,b │ │ │ │ +0002da60: 2c75 2c76 5d20 2020 2020 2020 2020 2020 ,u,v] │ │ │ │ 0002da70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002da80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002da90: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002da80: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002da90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002daa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dac0: 2020 207c 0a7c 6f33 203d 2053 2020 2020 |.|o3 = S │ │ │ │ +0002dab0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ +0002dac0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 0002dad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002daf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002dae0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002daf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002db00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002db10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002db30: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52 o3 : PolynomialR │ │ │ │ -0002db40: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -0002db50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db60: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0002db20: 2020 7c0a 7c6f 3320 3a20 506f 6c79 6e6f |.|o3 : Polyno │ │ │ │ +0002db30: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +0002db40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002db50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0002db60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002db70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002db90: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2066 -------+.|i4 : f │ │ │ │ -0002dba0: 6620 3d20 6d61 7472 6978 2261 752c 6276 f = matrix"au,bv │ │ │ │ -0002dbb0: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ -0002dbc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0002db90: 3420 3a20 6666 203d 206d 6174 7269 7822 4 : ff = matrix" │ │ │ │ +0002dba0: 6175 2c62 7622 2020 2020 2020 2020 2020 au,bv" │ │ │ │ +0002dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dbc0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc00: 207c 0a7c 6f34 203d 207c 2061 7520 6276 |.|o4 = | au bv │ │ │ │ -0002dc10: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc30: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002dbf0: 2020 2020 2020 7c0a 7c6f 3420 3d20 7c20 |.|o4 = | │ │ │ │ +0002dc00: 6175 2062 7620 7c20 2020 2020 2020 2020 au bv | │ │ │ │ +0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc20: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002dc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc60: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002dc70: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ -0002dc80: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dca0: 7c0a 7c6f 3420 3a20 4d61 7472 6978 2053 |.|o4 : Matrix S │ │ │ │ -0002dcb0: 2020 3c2d 2d20 5320 2020 2020 2020 2020 <-- S │ │ │ │ -0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dcd0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002dc60: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002dc70: 3120 2020 2020 2032 2020 2020 2020 2020 1 2 │ │ │ │ +0002dc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc90: 2020 2020 207c 0a7c 6f34 203a 204d 6174 |.|o4 : Mat │ │ │ │ +0002dca0: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ +0002dcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dcc0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002dcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002dce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ -0002dd10: 3a20 5220 3d20 532f 6964 6561 6c20 6666 : R = S/ideal ff │ │ │ │ +0002dcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002dd00: 0a7c 6935 203a 2052 203d 2053 2f69 6465 .|i5 : R = S/ide │ │ │ │ +0002dd10: 616c 2066 6620 2020 2020 2020 2020 2020 al ff │ │ │ │ 0002dd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002dd40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002dd30: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002dd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd70: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020 |.|o5 = R │ │ │ │ +0002dd60: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ +0002dd70: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 0002dd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dda0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002dd90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002dda0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002ddb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ddc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ddd0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002dde0: 7c6f 3520 3a20 5175 6f74 6965 6e74 5269 |o5 : QuotientRi │ │ │ │ -0002ddf0: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ -0002de00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002de10: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002ddd0: 2020 207c 0a7c 6f35 203a 2051 756f 7469 |.|o5 : Quoti │ │ │ │ +0002dde0: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ +0002ddf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002de00: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0002de10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de40: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ -0002de50: 4d30 203d 2052 5e31 2f69 6465 616c 2261 M0 = R^1/ideal"a │ │ │ │ -0002de60: 2c62 2220 2020 2020 2020 2020 2020 2020 ,b" │ │ │ │ -0002de70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0002de40: 6936 203a 204d 3020 3d20 525e 312f 6964 i6 : M0 = R^1/id │ │ │ │ +0002de50: 6561 6c22 612c 6222 2020 2020 2020 2020 eal"a,b" │ │ │ │ +0002de60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002de70: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002de80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002de90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002deb0: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e |.|o6 = cokern │ │ │ │ -0002dec0: 656c 207c 2061 2062 207c 2020 2020 2020 el | a b | │ │ │ │ -0002ded0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dee0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002dea0: 2020 2020 2020 207c 0a7c 6f36 203d 2063 |.|o6 = c │ │ │ │ +0002deb0: 6f6b 6572 6e65 6c20 7c20 6120 6220 7c20 okernel | a b | │ │ │ │ +0002dec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ded0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002dee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002def0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002df00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002df10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df30: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ -0002df40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df50: 207c 0a7c 6f36 203a 2052 2d6d 6f64 756c |.|o6 : R-modul │ │ │ │ -0002df60: 652c 2071 756f 7469 656e 7420 6f66 2052 e, quotient of R │ │ │ │ -0002df70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002df30: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0002df40: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d |.|o6 : R- │ │ │ │ +0002df50: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74 module, quotient │ │ │ │ +0002df60: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ +0002df70: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ -0002dfc0: 203a 2046 203d 2066 7265 6552 6573 6f6c : F = freeResol │ │ │ │ -0002dfd0: 7574 696f 6e28 4d30 2c20 4c65 6e67 7468 ution(M0, Length │ │ │ │ -0002dfe0: 4c69 6d69 7420 3d3e 3329 2020 2020 2020 Limit =>3) │ │ │ │ -0002dff0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002dfb0: 2b0a 7c69 3720 3a20 4620 3d20 6672 6565 +.|i7 : F = free │ │ │ │ +0002dfc0: 5265 736f 6c75 7469 6f6e 284d 302c 204c Resolution(M0, L │ │ │ │ +0002dfd0: 656e 6774 684c 696d 6974 203d 3e33 2920 engthLimit =>3) │ │ │ │ +0002dfe0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002dff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e020: 2020 2020 207c 0a7c 2020 2020 2020 3120 |.| 1 │ │ │ │ -0002e030: 2020 2020 2032 2020 2020 2020 3320 2020 2 3 │ │ │ │ -0002e040: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -0002e050: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ -0002e060: 3d20 5220 203c 2d2d 2052 2020 3c2d 2d20 = R <-- R <-- │ │ │ │ -0002e070: 5220 203c 2d2d 2052 2020 2020 2020 2020 R <-- R │ │ │ │ -0002e080: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002e090: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002e010: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002e020: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ +0002e030: 2033 2020 2020 2020 3420 2020 2020 2020 3 4 │ │ │ │ +0002e040: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002e050: 0a7c 6f37 203d 2052 2020 3c2d 2d20 5220 .|o7 = R <-- R │ │ │ │ +0002e060: 203c 2d2d 2052 2020 3c2d 2d20 5220 2020 <-- R <-- R │ │ │ │ +0002e070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e080: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002e090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e0c0: 2020 2020 7c0a 7c20 2020 2020 3020 2020 |.| 0 │ │ │ │ -0002e0d0: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ -0002e0e0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -0002e0f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002e0b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002e0c0: 2030 2020 2020 2020 3120 2020 2020 2032 0 1 2 │ │ │ │ +0002e0d0: 2020 2020 2020 3320 2020 2020 2020 2020 3 │ │ │ │ +0002e0e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002e0f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002e100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e120: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002e130: 7c6f 3720 3a20 436f 6d70 6c65 7820 2020 |o7 : Complex │ │ │ │ +0002e120: 2020 207c 0a7c 6f37 203a 2043 6f6d 706c |.|o7 : Compl │ │ │ │ +0002e130: 6578 2020 2020 2020 2020 2020 2020 2020 ex │ │ │ │ 0002e140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e160: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002e150: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0002e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e190: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 --------+.|i8 : │ │ │ │ -0002e1a0: 4d20 3d20 636f 6b65 7220 462e 6464 5f33 M = coker F.dd_3 │ │ │ │ -0002e1b0: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ -0002e1c0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0002e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0002e190: 6938 203a 204d 203d 2063 6f6b 6572 2046 i8 : M = coker F │ │ │ │ +0002e1a0: 2e64 645f 333b 2020 2020 2020 2020 2020 .dd_3; │ │ │ │ +0002e1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e1c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0002e1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e200: 2d2d 2b0a 7c69 3920 3a20 4d46 203d 206d --+.|i9 : MF = m │ │ │ │ -0002e210: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ -0002e220: 6f6e 2866 662c 4d29 3b20 2020 2020 2020 on(ff,M); │ │ │ │ -0002e230: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0002e1f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 204d -------+.|i9 : M │ │ │ │ +0002e200: 4620 3d20 6d61 7472 6978 4661 6374 6f72 F = matrixFactor │ │ │ │ +0002e210: 697a 6174 696f 6e28 6666 2c4d 293b 2020 ization(ff,M); │ │ │ │ +0002e220: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0002e230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002e270: 3130 203a 2062 6574 7469 2066 7265 6552 10 : betti freeR │ │ │ │ -0002e280: 6573 6f6c 7574 696f 6e20 7075 7368 466f esolution pushFo │ │ │ │ -0002e290: 7277 6172 6428 6d61 7028 522c 5329 2c4d rward(map(R,S),M │ │ │ │ -0002e2a0: 297c 0a7c 2020 2020 2020 2020 2020 2020 )|.| │ │ │ │ +0002e260: 2d2b 0a7c 6931 3020 3a20 6265 7474 6920 -+.|i10 : betti │ │ │ │ +0002e270: 6672 6565 5265 736f 6c75 7469 6f6e 2070 freeResolution p │ │ │ │ +0002e280: 7573 6846 6f72 7761 7264 286d 6170 2852 ushForward(map(R │ │ │ │ +0002e290: 2c53 292c 4d29 7c0a 7c20 2020 2020 2020 ,S),M)|.| │ │ │ │ +0002e2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e2d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0002e2e0: 2020 2020 2020 3020 3120 3220 2020 2020 0 1 2 │ │ │ │ +0002e2c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002e2d0: 2020 2020 2020 2020 2020 2030 2031 2032 0 1 2 │ │ │ │ +0002e2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e300: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0002e310: 3020 3d20 746f 7461 6c3a 2033 2035 2032 0 = total: 3 5 2 │ │ │ │ +0002e300: 7c0a 7c6f 3130 203d 2074 6f74 616c 3a20 |.|o10 = total: │ │ │ │ +0002e310: 3320 3520 3220 2020 2020 2020 2020 2020 3 5 2 │ │ │ │ 0002e320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e340: 7c0a 7c20 2020 2020 2020 2020 2032 3a20 |.| 2: │ │ │ │ -0002e350: 3320 3420 2e20 2020 2020 2020 2020 2020 3 4 . │ │ │ │ -0002e360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e370: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0002e380: 2020 333a 202e 2031 2032 2020 2020 2020 3: . 1 2 │ │ │ │ -0002e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e3a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002e330: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002e340: 2020 323a 2033 2034 202e 2020 2020 2020 2: 3 4 . │ │ │ │ +0002e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e360: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002e370: 2020 2020 2020 2033 3a20 2e20 3120 3220 3: . 1 2 │ │ │ │ +0002e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e390: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002e3a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002e3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e3d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002e3e0: 0a7c 6f31 3020 3a20 4265 7474 6954 616c .|o10 : BettiTal │ │ │ │ -0002e3f0: 6c79 2020 2020 2020 2020 2020 2020 2020 ly │ │ │ │ -0002e400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e410: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002e3d0: 2020 2020 7c0a 7c6f 3130 203a 2042 6574 |.|o10 : Bet │ │ │ │ +0002e3e0: 7469 5461 6c6c 7920 2020 2020 2020 2020 tiTally │ │ │ │ +0002e3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e400: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0002e410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e440: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 ---------+.|i11 │ │ │ │ -0002e450: 3a20 6669 6e69 7465 4265 7474 694e 756d : finiteBettiNum │ │ │ │ -0002e460: 6265 7273 204d 4620 2020 2020 2020 2020 bers MF │ │ │ │ -0002e470: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002e480: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002e440: 7c69 3131 203a 2066 696e 6974 6542 6574 |i11 : finiteBet │ │ │ │ +0002e450: 7469 4e75 6d62 6572 7320 4d46 2020 2020 tiNumbers MF │ │ │ │ +0002e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e470: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4b0: 2020 207c 0a7c 6f31 3120 3d20 7b33 2c20 |.|o11 = {3, │ │ │ │ -0002e4c0: 352c 2032 7d20 2020 2020 2020 2020 2020 5, 2} │ │ │ │ -0002e4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002e4a0: 2020 2020 2020 2020 7c0a 7c6f 3131 203d |.|o11 = │ │ │ │ +0002e4b0: 207b 332c 2035 2c20 327d 2020 2020 2020 {3, 5, 2} │ │ │ │ +0002e4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e4d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002e4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e510: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002e520: 6f31 3120 3a20 4c69 7374 2020 2020 2020 o11 : List │ │ │ │ +0002e510: 2020 7c0a 7c6f 3131 203a 204c 6973 7420 |.|o11 : List │ │ │ │ +0002e520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e550: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0002e540: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0002e550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e580: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 -------+.|i12 : │ │ │ │ -0002e590: 696e 6669 6e69 7465 4265 7474 694e 756d infiniteBettiNum │ │ │ │ -0002e5a0: 6265 7273 284d 462c 3529 2020 2020 2020 bers(MF,5) │ │ │ │ -0002e5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0002e580: 3132 203a 2069 6e66 696e 6974 6542 6574 12 : infiniteBet │ │ │ │ +0002e590: 7469 4e75 6d62 6572 7328 4d46 2c35 2920 tiNumbers(MF,5) │ │ │ │ +0002e5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e5b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002e5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e5f0: 207c 0a7c 6f31 3220 3d20 7b33 2c20 342c |.|o12 = {3, 4, │ │ │ │ -0002e600: 2035 2c20 362c 2037 2c20 387d 2020 2020 5, 6, 7, 8} │ │ │ │ -0002e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e620: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002e5e0: 2020 2020 2020 7c0a 7c6f 3132 203d 207b |.|o12 = { │ │ │ │ +0002e5f0: 332c 2034 2c20 352c 2036 2c20 372c 2038 3, 4, 5, 6, 7, 8 │ │ │ │ +0002e600: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0002e610: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002e620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e650: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0002e660: 3220 3a20 4c69 7374 2020 2020 2020 2020 2 : List │ │ │ │ +0002e650: 7c0a 7c6f 3132 203a 204c 6973 7420 2020 |.|o12 : List │ │ │ │ +0002e660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e690: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0002e680: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002e690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e6c0: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 6265 -----+.|i13 : be │ │ │ │ -0002e6d0: 7474 6920 6672 6565 5265 736f 6c75 7469 tti freeResoluti │ │ │ │ -0002e6e0: 6f6e 2028 4d2c 204c 656e 6774 684c 696d on (M, LengthLim │ │ │ │ -0002e6f0: 6974 203d 3e20 3529 2020 7c0a 7c20 2020 it => 5) |.| │ │ │ │ +0002e6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 ----------+.|i13 │ │ │ │ +0002e6c0: 203a 2062 6574 7469 2066 7265 6552 6573 : betti freeRes │ │ │ │ +0002e6d0: 6f6c 7574 696f 6e20 284d 2c20 4c65 6e67 olution (M, Leng │ │ │ │ +0002e6e0: 7468 4c69 6d69 7420 3d3e 2035 2920 207c thLimit => 5) | │ │ │ │ +0002e6f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e720: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002e730: 0a7c 2020 2020 2020 2020 2020 2020 2030 .| 0 │ │ │ │ -0002e740: 2031 2032 2033 2034 2035 2020 2020 2020 1 2 3 4 5 │ │ │ │ -0002e750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e760: 2020 2020 7c0a 7c6f 3133 203d 2074 6f74 |.|o13 = tot │ │ │ │ -0002e770: 616c 3a20 3320 3420 3520 3620 3720 3820 al: 3 4 5 6 7 8 │ │ │ │ -0002e780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e790: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0002e7a0: 2020 2020 2020 323a 2033 2034 2035 2036 2: 3 4 5 6 │ │ │ │ -0002e7b0: 2037 2038 2020 2020 2020 2020 2020 2020 7 8 │ │ │ │ -0002e7c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002e7d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002e720: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002e730: 2020 2020 3020 3120 3220 3320 3420 3520 0 1 2 3 4 5 │ │ │ │ +0002e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e750: 2020 2020 2020 2020 207c 0a7c 6f31 3320 |.|o13 │ │ │ │ +0002e760: 3d20 746f 7461 6c3a 2033 2034 2035 2036 = total: 3 4 5 6 │ │ │ │ +0002e770: 2037 2038 2020 2020 2020 2020 2020 2020 7 8 │ │ │ │ +0002e780: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002e790: 7c20 2020 2020 2020 2020 2032 3a20 3320 | 2: 3 │ │ │ │ +0002e7a0: 3420 3520 3620 3720 3820 2020 2020 2020 4 5 6 7 8 │ │ │ │ +0002e7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e7c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002e7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e800: 2020 207c 0a7c 6f31 3320 3a20 4265 7474 |.|o13 : Bett │ │ │ │ -0002e810: 6954 616c 6c79 2020 2020 2020 2020 2020 iTally │ │ │ │ -0002e820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e830: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0002e7f0: 2020 2020 2020 2020 7c0a 7c6f 3133 203a |.|o13 : │ │ │ │ +0002e800: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +0002e810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e820: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0002e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -0002e870: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ -0002e880: 3d0a 0a20 202a 202a 6e6f 7465 206d 6174 =.. * *note mat │ │ │ │ -0002e890: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -0002e8a0: 3a20 6d61 7472 6978 4661 6374 6f72 697a : matrixFactoriz │ │ │ │ -0002e8b0: 6174 696f 6e2c 202d 2d20 4d61 7073 2069 ation, -- Maps i │ │ │ │ -0002e8c0: 6e20 6120 6869 6768 6572 0a20 2020 2063 n a higher. c │ │ │ │ -0002e8d0: 6f64 696d 656e 7369 6f6e 206d 6174 7269 odimension matri │ │ │ │ -0002e8e0: 7820 6661 6374 6f72 697a 6174 696f 6e0a x factorization. │ │ │ │ -0002e8f0: 2020 2a20 2a6e 6f74 6520 6669 6e69 7465 * *note finite │ │ │ │ -0002e900: 4265 7474 694e 756d 6265 7273 3a20 6669 BettiNumbers: fi │ │ │ │ -0002e910: 6e69 7465 4265 7474 694e 756d 6265 7273 niteBettiNumbers │ │ │ │ -0002e920: 2c20 2d2d 2062 6574 7469 206e 756d 6265 , -- betti numbe │ │ │ │ -0002e930: 7273 206f 6620 6669 6e69 7465 0a20 2020 rs of finite. │ │ │ │ -0002e940: 2072 6573 6f6c 7574 696f 6e20 636f 6d70 resolution comp │ │ │ │ -0002e950: 7574 6564 2066 726f 6d20 6120 6d61 7472 uted from a matr │ │ │ │ -0002e960: 6978 2066 6163 746f 7269 7a61 7469 6f6e ix factorization │ │ │ │ -0002e970: 0a0a 5761 7973 2074 6f20 7573 6520 696e ..Ways to use in │ │ │ │ -0002e980: 6669 6e69 7465 4265 7474 694e 756d 6265 finiteBettiNumbe │ │ │ │ -0002e990: 7273 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d rs:.============ │ │ │ │ -0002e9a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002e9b0: 3d3d 3d3d 3d0a 0a20 202a 2022 696e 6669 =====.. * "infi │ │ │ │ -0002e9c0: 6e69 7465 4265 7474 694e 756d 6265 7273 niteBettiNumbers │ │ │ │ -0002e9d0: 284c 6973 742c 5a5a 2922 0a0a 466f 7220 (List,ZZ)"..For │ │ │ │ -0002e9e0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -0002e9f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002ea00: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -0002ea10: 6f74 6520 696e 6669 6e69 7465 4265 7474 ote infiniteBett │ │ │ │ -0002ea20: 694e 756d 6265 7273 3a20 696e 6669 6e69 iNumbers: infini │ │ │ │ -0002ea30: 7465 4265 7474 694e 756d 6265 7273 2c20 teBettiNumbers, │ │ │ │ -0002ea40: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -0002ea50: 640a 6675 6e63 7469 6f6e 3a20 284d 6163 d.function: (Mac │ │ │ │ -0002ea60: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -0002ea70: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +0002e860: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d --+..See also.== │ │ │ │ +0002e870: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ +0002e880: 6520 6d61 7472 6978 4661 6374 6f72 697a e matrixFactoriz │ │ │ │ +0002e890: 6174 696f 6e3a 206d 6174 7269 7846 6163 ation: matrixFac │ │ │ │ +0002e8a0: 746f 7269 7a61 7469 6f6e 2c20 2d2d 204d torization, -- M │ │ │ │ +0002e8b0: 6170 7320 696e 2061 2068 6967 6865 720a aps in a higher. │ │ │ │ +0002e8c0: 2020 2020 636f 6469 6d65 6e73 696f 6e20 codimension │ │ │ │ +0002e8d0: 6d61 7472 6978 2066 6163 746f 7269 7a61 matrix factoriza │ │ │ │ +0002e8e0: 7469 6f6e 0a20 202a 202a 6e6f 7465 2066 tion. * *note f │ │ │ │ +0002e8f0: 696e 6974 6542 6574 7469 4e75 6d62 6572 initeBettiNumber │ │ │ │ +0002e900: 733a 2066 696e 6974 6542 6574 7469 4e75 s: finiteBettiNu │ │ │ │ +0002e910: 6d62 6572 732c 202d 2d20 6265 7474 6920 mbers, -- betti │ │ │ │ +0002e920: 6e75 6d62 6572 7320 6f66 2066 696e 6974 numbers of finit │ │ │ │ +0002e930: 650a 2020 2020 7265 736f 6c75 7469 6f6e e. resolution │ │ │ │ +0002e940: 2063 6f6d 7075 7465 6420 6672 6f6d 2061 computed from a │ │ │ │ +0002e950: 206d 6174 7269 7820 6661 6374 6f72 697a matrix factoriz │ │ │ │ +0002e960: 6174 696f 6e0a 0a57 6179 7320 746f 2075 ation..Ways to u │ │ │ │ +0002e970: 7365 2069 6e66 696e 6974 6542 6574 7469 se infiniteBetti │ │ │ │ +0002e980: 4e75 6d62 6572 733a 0a3d 3d3d 3d3d 3d3d Numbers:.======= │ │ │ │ +0002e990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002e9a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +0002e9b0: 2269 6e66 696e 6974 6542 6574 7469 4e75 "infiniteBettiNu │ │ │ │ +0002e9c0: 6d62 6572 7328 4c69 7374 2c5a 5a29 220a mbers(List,ZZ)". │ │ │ │ +0002e9d0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0002e9e0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0002e9f0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0002ea00: 6374 202a 6e6f 7465 2069 6e66 696e 6974 ct *note infinit │ │ │ │ +0002ea10: 6542 6574 7469 4e75 6d62 6572 733a 2069 eBettiNumbers: i │ │ │ │ +0002ea20: 6e66 696e 6974 6542 6574 7469 4e75 6d62 nfiniteBettiNumb │ │ │ │ +0002ea30: 6572 732c 2069 7320 6120 2a6e 6f74 6520 ers, is a *note │ │ │ │ +0002ea40: 6d65 7468 6f64 0a66 756e 6374 696f 6e3a method.function: │ │ │ │ +0002ea50: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +0002ea60: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +0002ea70: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 0002ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ea90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002eaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -0002ead0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -0002eae0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -0002eaf0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -0002eb00: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -0002eb10: 322d 312e 3236 2e30 352b 6473 2f4d 322f 2-1.26.05+ds/M2/ │ │ │ │ -0002eb20: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -0002eb30: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ -0002eb40: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -0002eb50: 6f6e 732e 6d32 3a34 3131 333a 302e 0a1f ons.m2:4113:0... │ │ │ │ -0002eb60: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ -0002eb70: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -0002eb80: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ -0002eb90: 653a 2069 734c 696e 6561 722c 204e 6578 e: isLinear, Nex │ │ │ │ -0002eba0: 743a 2069 7351 7561 7369 5265 6775 6c61 t: isQuasiRegula │ │ │ │ -0002ebb0: 722c 2050 7265 763a 2069 6e66 696e 6974 r, Prev: infinit │ │ │ │ -0002ebc0: 6542 6574 7469 4e75 6d62 6572 732c 2055 eBettiNumbers, U │ │ │ │ -0002ebd0: 703a 2054 6f70 0a0a 6973 4c69 6e65 6172 p: Top..isLinear │ │ │ │ -0002ebe0: 202d 2d20 6368 6563 6b20 7768 6574 6865 -- check whethe │ │ │ │ -0002ebf0: 7220 6d61 7472 6978 2065 6e74 7269 6573 r matrix entries │ │ │ │ -0002ec00: 2068 6176 6520 6465 6772 6565 2031 0a2a have degree 1.* │ │ │ │ +0002eac0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +0002ead0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +0002eae0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +0002eaf0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +0002eb00: 6175 6c61 7932 2d31 2e32 362e 3035 2b64 aulay2-1.26.05+d │ │ │ │ +0002eb10: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +0002eb20: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ +0002eb30: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +0002eb40: 6f6c 7574 696f 6e73 2e6d 323a 3431 3133 olutions.m2:4113 │ │ │ │ +0002eb50: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ +0002eb60: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +0002eb70: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ +0002eb80: 2c20 4e6f 6465 3a20 6973 4c69 6e65 6172 , Node: isLinear │ │ │ │ +0002eb90: 2c20 4e65 7874 3a20 6973 5175 6173 6952 , Next: isQuasiR │ │ │ │ +0002eba0: 6567 756c 6172 2c20 5072 6576 3a20 696e egular, Prev: in │ │ │ │ +0002ebb0: 6669 6e69 7465 4265 7474 694e 756d 6265 finiteBettiNumbe │ │ │ │ +0002ebc0: 7273 2c20 5570 3a20 546f 700a 0a69 734c rs, Up: Top..isL │ │ │ │ +0002ebd0: 696e 6561 7220 2d2d 2063 6865 636b 2077 inear -- check w │ │ │ │ +0002ebe0: 6865 7468 6572 206d 6174 7269 7820 656e hether matrix en │ │ │ │ +0002ebf0: 7472 6965 7320 6861 7665 2064 6567 7265 tries have degre │ │ │ │ +0002ec00: 6520 310a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a e 1.************ │ │ │ │ 0002ec10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002ec20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002ec30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002ec40: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ -0002ec50: 3a20 0a20 2020 2020 2020 2062 203d 2069 : . b = i │ │ │ │ -0002ec60: 734c 696e 6561 7220 4d0a 2020 2a20 496e sLinear M. * In │ │ │ │ -0002ec70: 7075 7473 3a0a 2020 2020 2020 2a20 4d2c puts:. * M, │ │ │ │ -0002ec80: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ -0002ec90: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ -0002eca0: 6174 7269 782c 2c20 0a20 202a 204f 7574 atrix,, . * Out │ │ │ │ -0002ecb0: 7075 7473 3a0a 2020 2020 2020 2a20 622c puts:. * b, │ │ │ │ -0002ecc0: 2061 202a 6e6f 7465 2042 6f6f 6c65 616e a *note Boolean │ │ │ │ -0002ecd0: 2076 616c 7565 3a20 284d 6163 6175 6c61 value: (Macaula │ │ │ │ -0002ece0: 7932 446f 6329 426f 6f6c 6561 6e2c 2c20 y2Doc)Boolean,, │ │ │ │ -0002ecf0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -0002ed00: 3d3d 3d3d 3d3d 3d3d 3d0a 0a4e 6f74 6520 =========..Note │ │ │ │ -0002ed10: 7468 6174 2061 206c 696e 6561 7220 6d61 that a linear ma │ │ │ │ -0002ed20: 7472 6978 2c20 696e 2074 6869 7320 7365 trix, in this se │ │ │ │ -0002ed30: 6e73 652c 2063 616e 2073 7469 6c6c 2068 nse, can still h │ │ │ │ -0002ed40: 6176 6520 6469 6666 6572 656e 7420 7461 ave different ta │ │ │ │ -0002ed50: 7267 6574 0a64 6567 7265 6573 2028 696e rget.degrees (in │ │ │ │ -0002ed60: 2077 6869 6368 2063 6173 6520 7468 6520 which case the │ │ │ │ -0002ed70: 636f 6b65 726e 656c 2064 6563 6f6d 706f cokernel decompo │ │ │ │ -0002ed80: 7365 7320 696e 746f 2061 2064 6972 6563 ses into a direc │ │ │ │ -0002ed90: 7420 7375 6d20 6279 2067 656e 6572 6174 t sum by generat │ │ │ │ -0002eda0: 6f72 0a64 6567 7265 652e 290a 0a57 6179 or.degree.)..Way │ │ │ │ -0002edb0: 7320 746f 2075 7365 2069 734c 696e 6561 s to use isLinea │ │ │ │ -0002edc0: 723a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r:.============= │ │ │ │ -0002edd0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269 ========.. * "i │ │ │ │ -0002ede0: 734c 696e 6561 7228 4d61 7472 6978 2922 sLinear(Matrix)" │ │ │ │ -0002edf0: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -0002ee00: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -0002ee10: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -0002ee20: 6563 7420 2a6e 6f74 6520 6973 4c69 6e65 ect *note isLine │ │ │ │ -0002ee30: 6172 3a20 6973 4c69 6e65 6172 2c20 6973 ar: isLinear, is │ │ │ │ -0002ee40: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ -0002ee50: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ -0002ee60: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ -0002ee70: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +0002ec30: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +0002ec40: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ +0002ec50: 6220 3d20 6973 4c69 6e65 6172 204d 0a20 b = isLinear M. │ │ │ │ +0002ec60: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +0002ec70: 202a 204d 2c20 6120 2a6e 6f74 6520 6d61 * M, a *note ma │ │ │ │ +0002ec80: 7472 6978 3a20 284d 6163 6175 6c61 7932 trix: (Macaulay2 │ │ │ │ +0002ec90: 446f 6329 4d61 7472 6978 2c2c 200a 2020 Doc)Matrix,, . │ │ │ │ +0002eca0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +0002ecb0: 202a 2062 2c20 6120 2a6e 6f74 6520 426f * b, a *note Bo │ │ │ │ +0002ecc0: 6f6c 6561 6e20 7661 6c75 653a 2028 4d61 olean value: (Ma │ │ │ │ +0002ecd0: 6361 756c 6179 3244 6f63 2942 6f6f 6c65 caulay2Doc)Boole │ │ │ │ +0002ece0: 616e 2c2c 200a 0a44 6573 6372 6970 7469 an,, ..Descripti │ │ │ │ +0002ecf0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +0002ed00: 4e6f 7465 2074 6861 7420 6120 6c69 6e65 Note that a line │ │ │ │ +0002ed10: 6172 206d 6174 7269 782c 2069 6e20 7468 ar matrix, in th │ │ │ │ +0002ed20: 6973 2073 656e 7365 2c20 6361 6e20 7374 is sense, can st │ │ │ │ +0002ed30: 696c 6c20 6861 7665 2064 6966 6665 7265 ill have differe │ │ │ │ +0002ed40: 6e74 2074 6172 6765 740a 6465 6772 6565 nt target.degree │ │ │ │ +0002ed50: 7320 2869 6e20 7768 6963 6820 6361 7365 s (in which case │ │ │ │ +0002ed60: 2074 6865 2063 6f6b 6572 6e65 6c20 6465 the cokernel de │ │ │ │ +0002ed70: 636f 6d70 6f73 6573 2069 6e74 6f20 6120 composes into a │ │ │ │ +0002ed80: 6469 7265 6374 2073 756d 2062 7920 6765 direct sum by ge │ │ │ │ +0002ed90: 6e65 7261 746f 720a 6465 6772 6565 2e29 nerator.degree.) │ │ │ │ +0002eda0: 0a0a 5761 7973 2074 6f20 7573 6520 6973 ..Ways to use is │ │ │ │ +0002edb0: 4c69 6e65 6172 3a0a 3d3d 3d3d 3d3d 3d3d Linear:.======== │ │ │ │ +0002edc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +0002edd0: 202a 2022 6973 4c69 6e65 6172 284d 6174 * "isLinear(Mat │ │ │ │ +0002ede0: 7269 7829 220a 0a46 6f72 2074 6865 2070 rix)"..For the p │ │ │ │ +0002edf0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +0002ee00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +0002ee10: 6520 6f62 6a65 6374 202a 6e6f 7465 2069 e object *note i │ │ │ │ +0002ee20: 734c 696e 6561 723a 2069 734c 696e 6561 sLinear: isLinea │ │ │ │ +0002ee30: 722c 2069 7320 6120 2a6e 6f74 6520 6d65 r, is a *note me │ │ │ │ +0002ee40: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ +0002ee50: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ +0002ee60: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +0002ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ee90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eec0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -0002eed0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -0002eee0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -0002eef0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -0002ef00: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -0002ef10: 312e 3236 2e30 352b 6473 2f4d 322f 4d61 1.26.05+ds/M2/Ma │ │ │ │ -0002ef20: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -0002ef30: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ -0002ef40: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -0002ef50: 732e 6d32 3a33 3436 333a 302e 0a1f 0a46 s.m2:3463:0....F │ │ │ │ -0002ef60: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ -0002ef70: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ -0002ef80: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ -0002ef90: 2069 7351 7561 7369 5265 6775 6c61 722c isQuasiRegular, │ │ │ │ -0002efa0: 204e 6578 743a 2069 7353 7461 626c 7954 Next: isStablyT │ │ │ │ -0002efb0: 7269 7669 616c 2c20 5072 6576 3a20 6973 rivial, Prev: is │ │ │ │ -0002efc0: 4c69 6e65 6172 2c20 5570 3a20 546f 700a Linear, Up: Top. │ │ │ │ -0002efd0: 0a69 7351 7561 7369 5265 6775 6c61 7220 .isQuasiRegular │ │ │ │ -0002efe0: 2d2d 2074 6573 7473 2061 206d 6174 7269 -- tests a matri │ │ │ │ -0002eff0: 7820 6f72 2073 6571 7565 6e63 6520 6f72 x or sequence or │ │ │ │ -0002f000: 206c 6973 7420 666f 7220 7175 6173 692d list for quasi- │ │ │ │ -0002f010: 7265 6775 6c61 7269 7479 206f 6e20 6120 regularity on a │ │ │ │ -0002f020: 6d6f 6475 6c65 0a2a 2a2a 2a2a 2a2a 2a2a module.********* │ │ │ │ +0002eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +0002eec0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +0002eed0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +0002eee0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +0002eef0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +0002ef00: 6c61 7932 2d31 2e32 362e 3035 2b64 732f lay2-1.26.05+ds/ │ │ │ │ +0002ef10: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +0002ef20: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ +0002ef30: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +0002ef40: 7574 696f 6e73 2e6d 323a 3334 3633 3a30 utions.m2:3463:0 │ │ │ │ +0002ef50: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ +0002ef60: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ +0002ef70: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ +0002ef80: 4e6f 6465 3a20 6973 5175 6173 6952 6567 Node: isQuasiReg │ │ │ │ +0002ef90: 756c 6172 2c20 4e65 7874 3a20 6973 5374 ular, Next: isSt │ │ │ │ +0002efa0: 6162 6c79 5472 6976 6961 6c2c 2050 7265 ablyTrivial, Pre │ │ │ │ +0002efb0: 763a 2069 734c 696e 6561 722c 2055 703a v: isLinear, Up: │ │ │ │ +0002efc0: 2054 6f70 0a0a 6973 5175 6173 6952 6567 Top..isQuasiReg │ │ │ │ +0002efd0: 756c 6172 202d 2d20 7465 7374 7320 6120 ular -- tests a │ │ │ │ +0002efe0: 6d61 7472 6978 206f 7220 7365 7175 656e matrix or sequen │ │ │ │ +0002eff0: 6365 206f 7220 6c69 7374 2066 6f72 2071 ce or list for q │ │ │ │ +0002f000: 7561 7369 2d72 6567 756c 6172 6974 7920 uasi-regularity │ │ │ │ +0002f010: 6f6e 2061 206d 6f64 756c 650a 2a2a 2a2a on a module.**** │ │ │ │ +0002f020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002f030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002f040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002f050: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002f060: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002f070: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ -0002f080: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -0002f090: 2020 7420 3d20 6973 5175 6173 6952 6567 t = isQuasiReg │ │ │ │ -0002f0a0: 756c 6172 2866 662c 4d29 0a20 202a 2049 ular(ff,M). * I │ │ │ │ -0002f0b0: 6e70 7574 733a 0a20 2020 2020 202a 2066 nputs:. * f │ │ │ │ -0002f0c0: 662c 2061 202a 6e6f 7465 206d 6174 7269 f, a *note matri │ │ │ │ -0002f0d0: 783a 2028 4d61 6361 756c 6179 3244 6f63 x: (Macaulay2Doc │ │ │ │ -0002f0e0: 294d 6174 7269 782c 2c20 0a20 2020 2020 )Matrix,, . │ │ │ │ -0002f0f0: 202a 2066 662c 2061 202a 6e6f 7465 206c * ff, a *note l │ │ │ │ -0002f100: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ -0002f110: 6f63 294c 6973 742c 2c20 0a20 2020 2020 oc)List,, . │ │ │ │ -0002f120: 202a 2066 662c 2061 202a 6e6f 7465 2073 * ff, a *note s │ │ │ │ -0002f130: 6571 7565 6e63 653a 2028 4d61 6361 756c equence: (Macaul │ │ │ │ -0002f140: 6179 3244 6f63 2953 6571 7565 6e63 652c ay2Doc)Sequence, │ │ │ │ -0002f150: 2c20 0a20 2020 2020 202a 204d 2c20 6120 , . * M, a │ │ │ │ -0002f160: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ -0002f170: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ -0002f180: 6c65 2c2c 200a 2020 2a20 4f75 7470 7574 le,, . * Output │ │ │ │ -0002f190: 733a 0a20 2020 2020 202a 2074 2c20 6120 s:. * t, a │ │ │ │ -0002f1a0: 2a6e 6f74 6520 426f 6f6c 6561 6e20 7661 *note Boolean va │ │ │ │ -0002f1b0: 6c75 653a 2028 4d61 6361 756c 6179 3244 lue: (Macaulay2D │ │ │ │ -0002f1c0: 6f63 2942 6f6f 6c65 616e 2c2c 200a 0a44 oc)Boolean,, ..D │ │ │ │ -0002f1d0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -0002f1e0: 3d3d 3d3d 3d3d 0a0a 6666 2069 7320 7175 ======..ff is qu │ │ │ │ -0002f1f0: 6173 692d 7265 6775 6c61 7220 6966 2074 asi-regular if t │ │ │ │ -0002f200: 6865 206c 656e 6774 6820 6f66 2066 6620 he length of ff │ │ │ │ -0002f210: 6973 203c 3d20 6469 6d20 4d20 616e 6420 is <= dim M and │ │ │ │ -0002f220: 7468 6520 616e 6e69 6869 6c61 746f 7220 the annihilator │ │ │ │ -0002f230: 6f66 2066 665f 690a 6f6e 204d 2f28 6666 of ff_i.on M/(ff │ │ │ │ -0002f240: 5f30 2e2e 6666 5f7b 2869 2d31 2929 7d4d _0..ff_{(i-1))}M │ │ │ │ -0002f250: 2068 6173 2066 696e 6974 6520 6c65 6e67 has finite leng │ │ │ │ -0002f260: 7468 2066 6f72 2061 6c6c 2069 3d30 2e2e th for all i=0.. │ │ │ │ -0002f270: 286c 656e 6774 6820 6666 292d 312e 0a0a (length ff)-1... │ │ │ │ -0002f280: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002f290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ -0002f2b0: 3a20 6b6b 3d5a 5a2f 3130 313b 2020 2020 : kk=ZZ/101; │ │ │ │ -0002f2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f2d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f070: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +0002f080: 2020 2020 2020 2074 203d 2069 7351 7561 t = isQua │ │ │ │ +0002f090: 7369 5265 6775 6c61 7228 6666 2c4d 290a siRegular(ff,M). │ │ │ │ +0002f0a0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0002f0b0: 2020 2a20 6666 2c20 6120 2a6e 6f74 6520 * ff, a *note │ │ │ │ +0002f0c0: 6d61 7472 6978 3a20 284d 6163 6175 6c61 matrix: (Macaula │ │ │ │ +0002f0d0: 7932 446f 6329 4d61 7472 6978 2c2c 200a y2Doc)Matrix,, . │ │ │ │ +0002f0e0: 2020 2020 2020 2a20 6666 2c20 6120 2a6e * ff, a *n │ │ │ │ +0002f0f0: 6f74 6520 6c69 7374 3a20 284d 6163 6175 ote list: (Macau │ │ │ │ +0002f100: 6c61 7932 446f 6329 4c69 7374 2c2c 200a lay2Doc)List,, . │ │ │ │ +0002f110: 2020 2020 2020 2a20 6666 2c20 6120 2a6e * ff, a *n │ │ │ │ +0002f120: 6f74 6520 7365 7175 656e 6365 3a20 284d ote sequence: (M │ │ │ │ +0002f130: 6163 6175 6c61 7932 446f 6329 5365 7175 acaulay2Doc)Sequ │ │ │ │ +0002f140: 656e 6365 2c2c 200a 2020 2020 2020 2a20 ence,, . * │ │ │ │ +0002f150: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c M, a *note modul │ │ │ │ +0002f160: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ +0002f170: 294d 6f64 756c 652c 2c20 0a20 202a 204f )Module,, . * O │ │ │ │ +0002f180: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ +0002f190: 742c 2061 202a 6e6f 7465 2042 6f6f 6c65 t, a *note Boole │ │ │ │ +0002f1a0: 616e 2076 616c 7565 3a20 284d 6163 6175 an value: (Macau │ │ │ │ +0002f1b0: 6c61 7932 446f 6329 426f 6f6c 6561 6e2c lay2Doc)Boolean, │ │ │ │ +0002f1c0: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a , ..Description. │ │ │ │ +0002f1d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a66 6620 ===========..ff │ │ │ │ +0002f1e0: 6973 2071 7561 7369 2d72 6567 756c 6172 is quasi-regular │ │ │ │ +0002f1f0: 2069 6620 7468 6520 6c65 6e67 7468 206f if the length o │ │ │ │ +0002f200: 6620 6666 2069 7320 3c3d 2064 696d 204d f ff is <= dim M │ │ │ │ +0002f210: 2061 6e64 2074 6865 2061 6e6e 6968 696c and the annihil │ │ │ │ +0002f220: 6174 6f72 206f 6620 6666 5f69 0a6f 6e20 ator of ff_i.on │ │ │ │ +0002f230: 4d2f 2866 665f 302e 2e66 665f 7b28 692d M/(ff_0..ff_{(i- │ │ │ │ +0002f240: 3129 297d 4d20 6861 7320 6669 6e69 7465 1))}M has finite │ │ │ │ +0002f250: 206c 656e 6774 6820 666f 7220 616c 6c20 length for all │ │ │ │ +0002f260: 693d 302e 2e28 6c65 6e67 7468 2066 6629 i=0..(length ff) │ │ │ │ +0002f270: 2d31 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d -1...+---------- │ │ │ │ +0002f280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002f2a0: 0a7c 6931 203a 206b 6b3d 5a5a 2f31 3031 .|i1 : kk=ZZ/101 │ │ │ │ +0002f2b0: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +0002f2c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f300: 2d2d 2b0a 7c69 3220 3a20 5320 3d20 6b6b --+.|i2 : S = kk │ │ │ │ -0002f310: 5b61 2c62 2c63 5d3b 2020 2020 2020 2020 [a,b,c]; │ │ │ │ -0002f320: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f330: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002f340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f350: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ -0002f360: 3a20 4520 3d20 535e 312f 6964 6561 6c22 : E = S^1/ideal" │ │ │ │ -0002f370: 6162 222b 2b53 5e31 2f69 6465 616c 2076 ab"++S^1/ideal v │ │ │ │ -0002f380: 6172 7320 533b 7c0a 2b2d 2d2d 2d2d 2d2d ars S;|.+------- │ │ │ │ +0002f2f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 -------+.|i2 : S │ │ │ │ +0002f300: 203d 206b 6b5b 612c 622c 635d 3b20 2020 = kk[a,b,c]; │ │ │ │ +0002f310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f320: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002f330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002f350: 0a7c 6933 203a 2045 203d 2053 5e31 2f69 .|i3 : E = S^1/i │ │ │ │ +0002f360: 6465 616c 2261 6222 2b2b 535e 312f 6964 deal"ab"++S^1/id │ │ │ │ +0002f370: 6561 6c20 7661 7273 2053 3b7c 0a2b 2d2d eal vars S;|.+-- │ │ │ │ +0002f380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f3b0: 2d2d 2b0a 7c69 3420 3a20 6631 203d 6d61 --+.|i4 : f1 =ma │ │ │ │ -0002f3c0: 7472 6978 2261 223b 2020 2020 2020 2020 trix"a"; │ │ │ │ -0002f3d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f3e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f400: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0002f410: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ -0002f420: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -0002f430: 2020 2020 2020 7c0a 7c6f 3420 3a20 4d61 |.|o4 : Ma │ │ │ │ -0002f440: 7472 6978 2053 2020 3c2d 2d20 5320 2020 trix S <-- S │ │ │ │ -0002f450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f460: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0002f3a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2066 -------+.|i4 : f │ │ │ │ +0002f3b0: 3120 3d6d 6174 7269 7822 6122 3b20 2020 1 =matrix"a"; │ │ │ │ +0002f3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f3d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f3f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f400: 0a7c 2020 2020 2020 2020 2020 2020 2031 .| 1 │ │ │ │ +0002f410: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +0002f420: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ +0002f430: 203a 204d 6174 7269 7820 5320 203c 2d2d : Matrix S <-- │ │ │ │ +0002f440: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +0002f450: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0002f460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002f490: 7c69 3520 3a20 6632 203d 6d61 7472 6978 |i5 : f2 =matrix │ │ │ │ -0002f4a0: 2261 2b62 2c63 223b 2020 2020 2020 2020 "a+b,c"; │ │ │ │ -0002f4b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002f480: 2d2d 2d2b 0a7c 6935 203a 2066 3220 3d6d ---+.|i5 : f2 =m │ │ │ │ +0002f490: 6174 7269 7822 612b 622c 6322 3b20 2020 atrix"a+b,c"; │ │ │ │ +0002f4a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f4b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002f4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f4e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0002f4f0: 2020 2020 2020 3120 2020 2020 2032 2020 1 2 │ │ │ │ -0002f500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f510: 2020 7c0a 7c6f 3520 3a20 4d61 7472 6978 |.|o5 : Matrix │ │ │ │ -0002f520: 2053 2020 3c2d 2d20 5320 2020 2020 2020 S <-- S │ │ │ │ -0002f530: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f540: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002f550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f560: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ -0002f570: 3a20 6633 203d 206d 6174 7269 7822 612b : f3 = matrix"a+ │ │ │ │ -0002f580: 6222 3b20 2020 2020 2020 2020 2020 2020 b"; │ │ │ │ -0002f590: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002f4d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002f4e0: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +0002f4f0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002f500: 2020 2020 2020 207c 0a7c 6f35 203a 204d |.|o5 : M │ │ │ │ +0002f510: 6174 7269 7820 5320 203c 2d2d 2053 2020 atrix S <-- S │ │ │ │ +0002f520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f530: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002f540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002f560: 0a7c 6936 203a 2066 3320 3d20 6d61 7472 .|i6 : f3 = matr │ │ │ │ +0002f570: 6978 2261 2b62 223b 2020 2020 2020 2020 ix"a+b"; │ │ │ │ +0002f580: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002f590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f5c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002f5d0: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ -0002f5e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f5f0: 7c6f 3620 3a20 4d61 7472 6978 2053 2020 |o6 : Matrix S │ │ │ │ -0002f600: 3c2d 2d20 5320 2020 2020 2020 2020 2020 <-- S │ │ │ │ -0002f610: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002f5b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002f5c0: 2020 2020 2020 2031 2020 2020 2020 3120 1 1 │ │ │ │ +0002f5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f5e0: 2020 207c 0a7c 6f36 203a 204d 6174 7269 |.|o6 : Matri │ │ │ │ +0002f5f0: 7820 5320 203c 2d2d 2053 2020 2020 2020 x S <-- S │ │ │ │ +0002f600: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f610: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0002f620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f640: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6634 ------+.|i7 : f4 │ │ │ │ -0002f650: 203d 206d 6174 7269 7822 612b 622c 2061 = matrix"a+b, a │ │ │ │ -0002f660: 322b 6222 3b20 2020 2020 2020 2020 2020 2+b"; │ │ │ │ -0002f670: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ +0002f640: 203a 2066 3420 3d20 6d61 7472 6978 2261 : f4 = matrix"a │ │ │ │ +0002f650: 2b62 2c20 6132 2b62 223b 2020 2020 2020 +b, a2+b"; │ │ │ │ +0002f660: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002f670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f690: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f6a0: 7c20 2020 2020 2020 2020 2020 2020 3120 | 1 │ │ │ │ -0002f6b0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -0002f6c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ -0002f6d0: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20 : Matrix S <-- │ │ │ │ -0002f6e0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ -0002f6f0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f690: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f6a0: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ +0002f6b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f6c0: 0a7c 6f37 203a 204d 6174 7269 7820 5320 .|o7 : Matrix S │ │ │ │ +0002f6d0: 203c 2d2d 2053 2020 2020 2020 2020 2020 <-- S │ │ │ │ +0002f6e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f720: 2d2d 2b0a 7c69 3820 3a20 6973 5175 6173 --+.|i8 : isQuas │ │ │ │ -0002f730: 6952 6567 756c 6172 2866 312c 4529 2020 iRegular(f1,E) │ │ │ │ -0002f740: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f750: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f770: 2020 2020 2020 2020 2020 7c0a 7c6f 3820 |.|o8 │ │ │ │ -0002f780: 3d20 6661 6c73 6520 2020 2020 2020 2020 = false │ │ │ │ -0002f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f7a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f710: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2069 -------+.|i8 : i │ │ │ │ +0002f720: 7351 7561 7369 5265 6775 6c61 7228 6631 sQuasiRegular(f1 │ │ │ │ +0002f730: 2c45 2920 2020 2020 2020 2020 2020 2020 ,E) │ │ │ │ +0002f740: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f760: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f770: 0a7c 6f38 203d 2066 616c 7365 2020 2020 .|o8 = false │ │ │ │ +0002f780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f790: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f7d0: 2d2d 2b0a 7c69 3920 3a20 6973 5175 6173 --+.|i9 : isQuas │ │ │ │ -0002f7e0: 6952 6567 756c 6172 2866 322c 4529 2020 iRegular(f2,E) │ │ │ │ -0002f7f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f800: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f820: 2020 2020 2020 2020 2020 7c0a 7c6f 3920 |.|o9 │ │ │ │ -0002f830: 3d20 7472 7565 2020 2020 2020 2020 2020 = true │ │ │ │ -0002f840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f850: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f7c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2069 -------+.|i9 : i │ │ │ │ +0002f7d0: 7351 7561 7369 5265 6775 6c61 7228 6632 sQuasiRegular(f2 │ │ │ │ +0002f7e0: 2c45 2920 2020 2020 2020 2020 2020 2020 ,E) │ │ │ │ +0002f7f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f810: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f820: 0a7c 6f39 203d 2074 7275 6520 2020 2020 .|o9 = true │ │ │ │ +0002f830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f840: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f880: 2d2d 2b0a 7c69 3130 203a 2069 7351 7561 --+.|i10 : isQua │ │ │ │ -0002f890: 7369 5265 6775 6c61 7228 6633 2c45 2920 siRegular(f3,E) │ │ │ │ -0002f8a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f8b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f8d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3130 |.|o10 │ │ │ │ -0002f8e0: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ -0002f8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f900: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f870: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 -------+.|i10 : │ │ │ │ +0002f880: 6973 5175 6173 6952 6567 756c 6172 2866 isQuasiRegular(f │ │ │ │ +0002f890: 332c 4529 2020 2020 2020 2020 2020 2020 3,E) │ │ │ │ +0002f8a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f8c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f8d0: 0a7c 6f31 3020 3d20 7472 7565 2020 2020 .|o10 = true │ │ │ │ +0002f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f8f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f930: 2d2d 2b0a 7c69 3131 203a 2069 7351 7561 --+.|i11 : isQua │ │ │ │ -0002f940: 7369 5265 6775 6c61 7228 6634 2c45 2920 siRegular(f4,E) │ │ │ │ -0002f950: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f960: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f980: 2020 2020 2020 2020 2020 7c0a 7c6f 3131 |.|o11 │ │ │ │ -0002f990: 203d 2066 616c 7365 2020 2020 2020 2020 = false │ │ │ │ -0002f9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f9b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f920: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ +0002f930: 6973 5175 6173 6952 6567 756c 6172 2866 isQuasiRegular(f │ │ │ │ +0002f940: 342c 4529 2020 2020 2020 2020 2020 2020 4,E) │ │ │ │ +0002f950: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f970: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f980: 0a7c 6f31 3120 3d20 6661 6c73 6520 2020 .|o11 = false │ │ │ │ +0002f990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f9a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f9e0: 2d2d 2b0a 0a57 6179 7320 746f 2075 7365 --+..Ways to use │ │ │ │ -0002f9f0: 2069 7351 7561 7369 5265 6775 6c61 723a isQuasiRegular: │ │ │ │ -0002fa00: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0002fa10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ -0002fa20: 2a20 2269 7351 7561 7369 5265 6775 6c61 * "isQuasiRegula │ │ │ │ -0002fa30: 7228 4c69 7374 2c4d 6f64 756c 6529 220a r(List,Module)". │ │ │ │ -0002fa40: 2020 2a20 2269 7351 7561 7369 5265 6775 * "isQuasiRegu │ │ │ │ -0002fa50: 6c61 7228 4d61 7472 6978 2c4d 6f64 756c lar(Matrix,Modul │ │ │ │ -0002fa60: 6529 220a 2020 2a20 2269 7351 7561 7369 e)". * "isQuasi │ │ │ │ -0002fa70: 5265 6775 6c61 7228 5365 7175 656e 6365 Regular(Sequence │ │ │ │ -0002fa80: 2c4d 6f64 756c 6529 220a 0a46 6f72 2074 ,Module)"..For t │ │ │ │ -0002fa90: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -0002faa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002fab0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -0002fac0: 7465 2069 7351 7561 7369 5265 6775 6c61 te isQuasiRegula │ │ │ │ -0002fad0: 723a 2069 7351 7561 7369 5265 6775 6c61 r: isQuasiRegula │ │ │ │ -0002fae0: 722c 2069 7320 6120 2a6e 6f74 6520 6d65 r, is a *note me │ │ │ │ -0002faf0: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ -0002fb00: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ -0002fb10: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +0002f9d0: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 -------+..Ways t │ │ │ │ +0002f9e0: 6f20 7573 6520 6973 5175 6173 6952 6567 o use isQuasiReg │ │ │ │ +0002f9f0: 756c 6172 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d ular:.========== │ │ │ │ +0002fa00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002fa10: 3d0a 0a20 202a 2022 6973 5175 6173 6952 =.. * "isQuasiR │ │ │ │ +0002fa20: 6567 756c 6172 284c 6973 742c 4d6f 6475 egular(List,Modu │ │ │ │ +0002fa30: 6c65 2922 0a20 202a 2022 6973 5175 6173 le)". * "isQuas │ │ │ │ +0002fa40: 6952 6567 756c 6172 284d 6174 7269 782c iRegular(Matrix, │ │ │ │ +0002fa50: 4d6f 6475 6c65 2922 0a20 202a 2022 6973 Module)". * "is │ │ │ │ +0002fa60: 5175 6173 6952 6567 756c 6172 2853 6571 QuasiRegular(Seq │ │ │ │ +0002fa70: 7565 6e63 652c 4d6f 6475 6c65 2922 0a0a uence,Module)".. │ │ │ │ +0002fa80: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ +0002fa90: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ +0002faa0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ +0002fab0: 7420 2a6e 6f74 6520 6973 5175 6173 6952 t *note isQuasiR │ │ │ │ +0002fac0: 6567 756c 6172 3a20 6973 5175 6173 6952 egular: isQuasiR │ │ │ │ +0002fad0: 6567 756c 6172 2c20 6973 2061 202a 6e6f egular, is a *no │ │ │ │ +0002fae0: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ +0002faf0: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f on:.(Macaulay2Do │ │ │ │ +0002fb00: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ +0002fb10: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ 0002fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -0002fb70: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -0002fb80: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -0002fb90: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -0002fba0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -0002fbb0: 6c61 7932 2d31 2e32 362e 3035 2b64 732f lay2-1.26.05+ds/ │ │ │ │ -0002fbc0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -0002fbd0: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ -0002fbe0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -0002fbf0: 7574 696f 6e73 2e6d 323a 3436 3237 3a30 utions.m2:4627:0 │ │ │ │ -0002fc00: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ -0002fc10: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ -0002fc20: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ -0002fc30: 4e6f 6465 3a20 6973 5374 6162 6c79 5472 Node: isStablyTr │ │ │ │ -0002fc40: 6976 6961 6c2c 204e 6578 743a 206b 6f73 ivial, Next: kos │ │ │ │ -0002fc50: 7a75 6c45 7874 656e 7369 6f6e 2c20 5072 zulExtension, Pr │ │ │ │ -0002fc60: 6576 3a20 6973 5175 6173 6952 6567 756c ev: isQuasiRegul │ │ │ │ -0002fc70: 6172 2c20 5570 3a20 546f 700a 0a69 7353 ar, Up: Top..isS │ │ │ │ -0002fc80: 7461 626c 7954 7269 7669 616c 202d 2d20 tablyTrivial -- │ │ │ │ -0002fc90: 7265 7475 726e 7320 7472 7565 2069 6620 returns true if │ │ │ │ -0002fca0: 7468 6520 6d61 7020 676f 6573 2074 6f20 the map goes to │ │ │ │ -0002fcb0: 3020 756e 6465 7220 7374 6162 6c65 486f 0 under stableHo │ │ │ │ -0002fcc0: 6d0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a m.************** │ │ │ │ +0002fb60: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +0002fb70: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +0002fb80: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +0002fb90: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +0002fba0: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ +0002fbb0: 352b 6473 2f4d 322f 4d61 6361 756c 6179 5+ds/M2/Macaulay │ │ │ │ +0002fbc0: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ +0002fbd0: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +0002fbe0: 5265 736f 6c75 7469 6f6e 732e 6d32 3a34 Resolutions.m2:4 │ │ │ │ +0002fbf0: 3632 373a 302e 0a1f 0a46 696c 653a 2043 627:0....File: C │ │ │ │ +0002fc00: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ +0002fc10: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ +0002fc20: 6e66 6f2c 204e 6f64 653a 2069 7353 7461 nfo, Node: isSta │ │ │ │ +0002fc30: 626c 7954 7269 7669 616c 2c20 4e65 7874 blyTrivial, Next │ │ │ │ +0002fc40: 3a20 6b6f 737a 756c 4578 7465 6e73 696f : koszulExtensio │ │ │ │ +0002fc50: 6e2c 2050 7265 763a 2069 7351 7561 7369 n, Prev: isQuasi │ │ │ │ +0002fc60: 5265 6775 6c61 722c 2055 703a 2054 6f70 Regular, Up: Top │ │ │ │ +0002fc70: 0a0a 6973 5374 6162 6c79 5472 6976 6961 ..isStablyTrivia │ │ │ │ +0002fc80: 6c20 2d2d 2072 6574 7572 6e73 2074 7275 l -- returns tru │ │ │ │ +0002fc90: 6520 6966 2074 6865 206d 6170 2067 6f65 e if the map goe │ │ │ │ +0002fca0: 7320 746f 2030 2075 6e64 6572 2073 7461 s to 0 under sta │ │ │ │ +0002fcb0: 626c 6548 6f6d 0a2a 2a2a 2a2a 2a2a 2a2a bleHom.********* │ │ │ │ +0002fcc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002fcd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002fce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002fcf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002fd00: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -0002fd10: 653a 200a 2020 2020 2020 2020 6220 3d20 e: . b = │ │ │ │ -0002fd20: 6973 5374 6162 6c79 5472 6976 6961 6c20 isStablyTrivial │ │ │ │ -0002fd30: 660a 2020 2a20 496e 7075 7473 3a0a 2020 f. * Inputs:. │ │ │ │ -0002fd40: 2020 2020 2a20 662c 2061 202a 6e6f 7465 * f, a *note │ │ │ │ -0002fd50: 206d 6174 7269 783a 2028 4d61 6361 756c matrix: (Macaul │ │ │ │ -0002fd60: 6179 3244 6f63 294d 6174 7269 782c 2c20 ay2Doc)Matrix,, │ │ │ │ -0002fd70: 6d61 7020 4d20 746f 204e 0a20 202a 204f map M to N. * O │ │ │ │ -0002fd80: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -0002fd90: 622c 2061 202a 6e6f 7465 2042 6f6f 6c65 b, a *note Boole │ │ │ │ -0002fda0: 616e 2076 616c 7565 3a20 284d 6163 6175 an value: (Macau │ │ │ │ -0002fdb0: 6c61 7932 446f 6329 426f 6f6c 6561 6e2c lay2Doc)Boolean, │ │ │ │ -0002fdc0: 2c20 7472 7565 2069 6666 2066 2066 6163 , true iff f fac │ │ │ │ -0002fdd0: 746f 7273 0a20 2020 2020 2020 2074 6872 tors. thr │ │ │ │ -0002fde0: 6f75 6768 2061 2070 726f 6a65 6374 6976 ough a projectiv │ │ │ │ -0002fdf0: 650a 0a44 6573 6372 6970 7469 6f6e 0a3d e..Description.= │ │ │ │ -0002fe00: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4120 706f ==========..A po │ │ │ │ -0002fe10: 7373 6962 6c65 206f 6273 7472 7563 7469 ssible obstructi │ │ │ │ -0002fe20: 6f6e 2074 6f20 7468 6520 636f 6d6d 7574 on to the commut │ │ │ │ -0002fe30: 6174 6976 6974 7920 6f66 2074 6865 2043 ativity of the C │ │ │ │ -0002fe40: 4920 6f70 6572 6174 6f72 7320 696e 2063 I operators in c │ │ │ │ -0002fe50: 6f64 696d 2063 2c0a 6576 656e 2061 7379 odim c,.even asy │ │ │ │ -0002fe60: 6d70 746f 7469 6361 6c6c 792c 2077 6f75 mptotically, wou │ │ │ │ -0002fe70: 6c64 2062 6520 7468 6520 6e6f 6e2d 7472 ld be the non-tr │ │ │ │ -0002fe80: 6976 6961 6c69 7479 206f 6620 7468 6520 iviality of the │ │ │ │ -0002fe90: 6d61 7020 4d5f 7b28 6b2b 3429 7d20 2d2d map M_{(k+4)} -- │ │ │ │ -0002fea0: 3e20 4d5f 6b0a 5c6f 7469 6d65 7320 5c77 > M_k.\otimes \w │ │ │ │ -0002feb0: 6564 6765 5e32 2853 5e63 2920 696e 2074 edge^2(S^c) in t │ │ │ │ -0002fec0: 6865 2073 7461 626c 6520 6361 7465 676f he stable catego │ │ │ │ -0002fed0: 7279 206f 6620 6d61 7869 6d61 6c20 436f ry of maximal Co │ │ │ │ -0002fee0: 6865 6e2d 4d61 6361 756c 6179 206d 6f64 hen-Macaulay mod │ │ │ │ -0002fef0: 756c 6573 2e0a 0a49 6e20 7468 6520 666f ules...In the fo │ │ │ │ -0002ff00: 6c6c 6f77 696e 6720 6578 616d 706c 652c llowing example, │ │ │ │ -0002ff10: 2073 7475 6469 6564 2069 6e20 7468 6520 studied in the │ │ │ │ -0002ff20: 7061 7065 7220 2254 6f72 2061 7320 6120 paper "Tor as a │ │ │ │ -0002ff30: 6d6f 6475 6c65 206f 7665 7220 616e 0a65 module over an.e │ │ │ │ -0002ff40: 7874 6572 696f 7220 616c 6765 6272 6122 xterior algebra" │ │ │ │ -0002ff50: 206f 6620 4569 7365 6e62 7564 2c20 5065 of Eisenbud, Pe │ │ │ │ -0002ff60: 6576 6120 616e 6420 5363 6872 6579 6572 eva and Schreyer │ │ │ │ -0002ff70: 2c20 7468 6520 6d61 7020 6973 206e 6f6e , the map is non │ │ │ │ -0002ff80: 2d74 7269 7669 616c 2e2e 2e62 7574 0a69 -trivial...but.i │ │ │ │ -0002ff90: 7420 6973 2073 7461 626c 7920 7472 6976 t is stably triv │ │ │ │ -0002ffa0: 6961 6c2e 2054 6865 2073 616d 6520 676f ial. The same go │ │ │ │ -0002ffb0: 6573 2066 6f72 2068 6967 6865 7220 7661 es for higher va │ │ │ │ -0002ffc0: 6c75 6573 206f 6620 6b20 2877 6869 6368 lues of k (which │ │ │ │ -0002ffd0: 2074 616b 6520 6c6f 6e67 6572 0a74 6f20 take longer.to │ │ │ │ -0002ffe0: 636f 6d70 7574 6529 2e20 286e 6f74 6520 compute). (note │ │ │ │ -0002fff0: 7468 6174 2069 6e20 7468 6973 2063 6173 that in this cas │ │ │ │ -00030000: 652c 2077 6974 6820 6320 3d20 332c 2074 e, with c = 3, t │ │ │ │ -00030010: 776f 206f 6620 7468 6520 7468 7265 6520 wo of the three │ │ │ │ -00030020: 616c 7465 726e 6174 696e 670a 7072 6f64 alternating.prod │ │ │ │ -00030030: 7563 7473 2061 7265 2061 6374 7561 6c6c ucts are actuall │ │ │ │ -00030040: 7920 6571 7561 6c20 746f 2030 2c20 736f y equal to 0, so │ │ │ │ -00030050: 2077 6520 7465 7374 206f 6e6c 7920 7468 we test only th │ │ │ │ -00030060: 6520 7468 6972 642e 290a 0a4e 6f74 6520 e third.)..Note │ │ │ │ -00030070: 7468 6174 2054 2069 7320 7765 6c6c 2d64 that T is well-d │ │ │ │ -00030080: 6566 696e 6564 2075 7020 746f 2068 6f6d efined up to hom │ │ │ │ -00030090: 6f74 6f70 793b 2073 6f20 545e 3220 6973 otopy; so T^2 is │ │ │ │ -000300a0: 2077 656c 6c2d 6465 6669 6e65 6420 6d6f well-defined mo │ │ │ │ -000300b0: 6420 6d6d 5e32 2e0a 0a2b 2d2d 2d2d 2d2d d mm^2...+------ │ │ │ │ +0002fcf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +0002fd00: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +0002fd10: 2062 203d 2069 7353 7461 626c 7954 7269 b = isStablyTri │ │ │ │ +0002fd20: 7669 616c 2066 0a20 202a 2049 6e70 7574 vial f. * Input │ │ │ │ +0002fd30: 733a 0a20 2020 2020 202a 2066 2c20 6120 s:. * f, a │ │ │ │ +0002fd40: 2a6e 6f74 6520 6d61 7472 6978 3a20 284d *note matrix: (M │ │ │ │ +0002fd50: 6163 6175 6c61 7932 446f 6329 4d61 7472 acaulay2Doc)Matr │ │ │ │ +0002fd60: 6978 2c2c 206d 6170 204d 2074 6f20 4e0a ix,, map M to N. │ │ │ │ +0002fd70: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ +0002fd80: 2020 202a 2062 2c20 6120 2a6e 6f74 6520 * b, a *note │ │ │ │ +0002fd90: 426f 6f6c 6561 6e20 7661 6c75 653a 2028 Boolean value: ( │ │ │ │ +0002fda0: 4d61 6361 756c 6179 3244 6f63 2942 6f6f Macaulay2Doc)Boo │ │ │ │ +0002fdb0: 6c65 616e 2c2c 2074 7275 6520 6966 6620 lean,, true iff │ │ │ │ +0002fdc0: 6620 6661 6374 6f72 730a 2020 2020 2020 f factors. │ │ │ │ +0002fdd0: 2020 7468 726f 7567 6820 6120 7072 6f6a through a proj │ │ │ │ +0002fde0: 6563 7469 7665 0a0a 4465 7363 7269 7074 ective..Descript │ │ │ │ +0002fdf0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +0002fe00: 0a41 2070 6f73 7369 626c 6520 6f62 7374 .A possible obst │ │ │ │ +0002fe10: 7275 6374 696f 6e20 746f 2074 6865 2063 ruction to the c │ │ │ │ +0002fe20: 6f6d 6d75 7461 7469 7669 7479 206f 6620 ommutativity of │ │ │ │ +0002fe30: 7468 6520 4349 206f 7065 7261 746f 7273 the CI operators │ │ │ │ +0002fe40: 2069 6e20 636f 6469 6d20 632c 0a65 7665 in codim c,.eve │ │ │ │ +0002fe50: 6e20 6173 796d 7074 6f74 6963 616c 6c79 n asymptotically │ │ │ │ +0002fe60: 2c20 776f 756c 6420 6265 2074 6865 206e , would be the n │ │ │ │ +0002fe70: 6f6e 2d74 7269 7669 616c 6974 7920 6f66 on-triviality of │ │ │ │ +0002fe80: 2074 6865 206d 6170 204d 5f7b 286b 2b34 the map M_{(k+4 │ │ │ │ +0002fe90: 297d 202d 2d3e 204d 5f6b 0a5c 6f74 696d )} --> M_k.\otim │ │ │ │ +0002fea0: 6573 205c 7765 6467 655e 3228 535e 6329 es \wedge^2(S^c) │ │ │ │ +0002feb0: 2069 6e20 7468 6520 7374 6162 6c65 2063 in the stable c │ │ │ │ +0002fec0: 6174 6567 6f72 7920 6f66 206d 6178 696d ategory of maxim │ │ │ │ +0002fed0: 616c 2043 6f68 656e 2d4d 6163 6175 6c61 al Cohen-Macaula │ │ │ │ +0002fee0: 7920 6d6f 6475 6c65 732e 0a0a 496e 2074 y modules...In t │ │ │ │ +0002fef0: 6865 2066 6f6c 6c6f 7769 6e67 2065 7861 he following exa │ │ │ │ +0002ff00: 6d70 6c65 2c20 7374 7564 6965 6420 696e mple, studied in │ │ │ │ +0002ff10: 2074 6865 2070 6170 6572 2022 546f 7220 the paper "Tor │ │ │ │ +0002ff20: 6173 2061 206d 6f64 756c 6520 6f76 6572 as a module over │ │ │ │ +0002ff30: 2061 6e0a 6578 7465 7269 6f72 2061 6c67 an.exterior alg │ │ │ │ +0002ff40: 6562 7261 2220 6f66 2045 6973 656e 6275 ebra" of Eisenbu │ │ │ │ +0002ff50: 642c 2050 6565 7661 2061 6e64 2053 6368 d, Peeva and Sch │ │ │ │ +0002ff60: 7265 7965 722c 2074 6865 206d 6170 2069 reyer, the map i │ │ │ │ +0002ff70: 7320 6e6f 6e2d 7472 6976 6961 6c2e 2e2e s non-trivial... │ │ │ │ +0002ff80: 6275 740a 6974 2069 7320 7374 6162 6c79 but.it is stably │ │ │ │ +0002ff90: 2074 7269 7669 616c 2e20 5468 6520 7361 trivial. The sa │ │ │ │ +0002ffa0: 6d65 2067 6f65 7320 666f 7220 6869 6768 me goes for high │ │ │ │ +0002ffb0: 6572 2076 616c 7565 7320 6f66 206b 2028 er values of k ( │ │ │ │ +0002ffc0: 7768 6963 6820 7461 6b65 206c 6f6e 6765 which take longe │ │ │ │ +0002ffd0: 720a 746f 2063 6f6d 7075 7465 292e 2028 r.to compute). ( │ │ │ │ +0002ffe0: 6e6f 7465 2074 6861 7420 696e 2074 6869 note that in thi │ │ │ │ +0002fff0: 7320 6361 7365 2c20 7769 7468 2063 203d s case, with c = │ │ │ │ +00030000: 2033 2c20 7477 6f20 6f66 2074 6865 2074 3, two of the t │ │ │ │ +00030010: 6872 6565 2061 6c74 6572 6e61 7469 6e67 hree alternating │ │ │ │ +00030020: 0a70 726f 6475 6374 7320 6172 6520 6163 .products are ac │ │ │ │ +00030030: 7475 616c 6c79 2065 7175 616c 2074 6f20 tually equal to │ │ │ │ +00030040: 302c 2073 6f20 7765 2074 6573 7420 6f6e 0, so we test on │ │ │ │ +00030050: 6c79 2074 6865 2074 6869 7264 2e29 0a0a ly the third.).. │ │ │ │ +00030060: 4e6f 7465 2074 6861 7420 5420 6973 2077 Note that T is w │ │ │ │ +00030070: 656c 6c2d 6465 6669 6e65 6420 7570 2074 ell-defined up t │ │ │ │ +00030080: 6f20 686f 6d6f 746f 7079 3b20 736f 2054 o homotopy; so T │ │ │ │ +00030090: 5e32 2069 7320 7765 6c6c 2d64 6566 696e ^2 is well-defin │ │ │ │ +000300a0: 6564 206d 6f64 206d 6d5e 322e 0a0a 2b2d ed mod mm^2...+- │ │ │ │ +000300b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000300c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000300d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000300e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000300f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030100: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b -------+.|i1 : k │ │ │ │ -00030110: 6b20 3d20 5a5a 2f31 3031 2020 2020 2020 k = ZZ/101 │ │ │ │ +000300f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00030100: 3120 3a20 6b6b 203d 205a 5a2f 3130 3120 1 : kk = ZZ/101 │ │ │ │ +00030110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030150: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030140: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000301a0: 2020 2020 2020 207c 0a7c 6f31 203d 206b |.|o1 = k │ │ │ │ -000301b0: 6b20 2020 2020 2020 2020 2020 2020 2020 k │ │ │ │ +00030190: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000301a0: 3120 3d20 6b6b 2020 2020 2020 2020 2020 1 = kk │ │ │ │ +000301b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000301c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000301d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000301e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000301f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000301e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000301f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030240: 2020 2020 2020 207c 0a7c 6f31 203a 2051 |.|o1 : Q │ │ │ │ -00030250: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +00030230: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030240: 3120 3a20 5175 6f74 6965 6e74 5269 6e67 1 : QuotientRing │ │ │ │ +00030250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030290: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030280: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00030290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000302a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000302b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000302c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000302d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000302e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 -------+.|i2 : S │ │ │ │ -000302f0: 203d 206b 6b5b 612c 622c 635d 2020 2020 = kk[a,b,c] │ │ │ │ +000302d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000302e0: 3220 3a20 5320 3d20 6b6b 5b61 2c62 2c63 2 : S = kk[a,b,c │ │ │ │ +000302f0: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ 00030300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030330: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030320: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030380: 2020 2020 2020 207c 0a7c 6f32 203d 2053 |.|o2 = S │ │ │ │ +00030370: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030380: 3220 3d20 5320 2020 2020 2020 2020 2020 2 = S │ │ │ │ 00030390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000303a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000303b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000303c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000303d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000303c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000303d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000303e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000303f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030420: 2020 2020 2020 207c 0a7c 6f32 203a 2050 |.|o2 : P │ │ │ │ -00030430: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ +00030410: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030420: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269 2 : PolynomialRi │ │ │ │ +00030430: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ 00030440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030470: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030460: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00030470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000304a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000304b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000304c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2066 -------+.|i3 : f │ │ │ │ -000304d0: 6620 3d20 6d61 7472 6978 2261 322c 6232 f = matrix"a2,b2 │ │ │ │ -000304e0: 2c63 3222 2020 2020 2020 2020 2020 2020 ,c2" │ │ │ │ +000304b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000304c0: 3320 3a20 6666 203d 206d 6174 7269 7822 3 : ff = matrix" │ │ │ │ +000304d0: 6132 2c62 322c 6332 2220 2020 2020 2020 a2,b2,c2" │ │ │ │ +000304e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000304f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030510: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030500: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030560: 2020 2020 2020 207c 0a7c 6f33 203d 207c |.|o3 = | │ │ │ │ -00030570: 2061 3220 6232 2063 3220 7c20 2020 2020 a2 b2 c2 | │ │ │ │ +00030550: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030560: 3320 3d20 7c20 6132 2062 3220 6332 207c 3 = | a2 b2 c2 | │ │ │ │ +00030570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000305a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000305b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000305a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000305b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000305c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000305d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000305e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000305f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030600: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00030610: 2020 2020 2020 2031 2020 2020 2020 3320 1 3 │ │ │ │ +000305f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030600: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ +00030610: 2020 2033 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00030620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030650: 2020 2020 2020 207c 0a7c 6f33 203a 204d |.|o3 : M │ │ │ │ -00030660: 6174 7269 7820 5320 203c 2d2d 2053 2020 atrix S <-- S │ │ │ │ +00030640: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030650: 3320 3a20 4d61 7472 6978 2053 2020 3c2d 3 : Matrix S <- │ │ │ │ +00030660: 2d20 5320 2020 2020 2020 2020 2020 2020 - S │ │ │ │ 00030670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000306a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030690: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000306a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000306b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000306c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000306d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000306e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000306f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 -------+.|i4 : R │ │ │ │ -00030700: 203d 2053 2f69 6465 616c 2066 6620 2020 = S/ideal ff │ │ │ │ +000306e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000306f0: 3420 3a20 5220 3d20 532f 6964 6561 6c20 4 : R = S/ideal │ │ │ │ +00030700: 6666 2020 2020 2020 2020 2020 2020 2020 ff │ │ │ │ 00030710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030740: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030730: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030790: 2020 2020 2020 207c 0a7c 6f34 203d 2052 |.|o4 = R │ │ │ │ +00030780: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030790: 3420 3d20 5220 2020 2020 2020 2020 2020 4 = R │ │ │ │ 000307a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000307b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000307c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000307d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000307e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000307d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000307e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000307f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030830: 2020 2020 2020 207c 0a7c 6f34 203a 2051 |.|o4 : Q │ │ │ │ -00030840: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +00030820: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030830: 3420 3a20 5175 6f74 6965 6e74 5269 6e67 4 : QuotientRing │ │ │ │ +00030840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030880: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030870: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00030880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000308a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000308b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000308c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000308d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 204d -------+.|i5 : M │ │ │ │ -000308e0: 203d 2052 5e31 2f69 6465 616c 2261 2c62 = R^1/ideal"a,b │ │ │ │ -000308f0: 6322 2020 2020 2020 2020 2020 2020 2020 c" │ │ │ │ +000308c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000308d0: 3520 3a20 4d20 3d20 525e 312f 6964 6561 5 : M = R^1/idea │ │ │ │ +000308e0: 6c22 612c 6263 2220 2020 2020 2020 2020 l"a,bc" │ │ │ │ +000308f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030920: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030910: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030970: 2020 2020 2020 207c 0a7c 6f35 203d 2063 |.|o5 = c │ │ │ │ -00030980: 6f6b 6572 6e65 6c20 7c20 6120 6263 207c okernel | a bc | │ │ │ │ +00030960: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030970: 3520 3d20 636f 6b65 726e 656c 207c 2061 5 = cokernel | a │ │ │ │ +00030980: 2062 6320 7c20 2020 2020 2020 2020 2020 bc | │ │ │ │ 00030990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000309a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000309b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000309c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000309b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000309c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000309d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000309e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000309f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a10: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00030a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a30: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +00030a00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030a20: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +00030a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a60: 2020 2020 2020 207c 0a7c 6f35 203a 2052 |.|o5 : R │ │ │ │ -00030a70: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ -00030a80: 7420 6f66 2052 2020 2020 2020 2020 2020 t of R │ │ │ │ +00030a50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030a60: 3520 3a20 522d 6d6f 6475 6c65 2c20 7175 5 : R-module, qu │ │ │ │ +00030a70: 6f74 6965 6e74 206f 6620 5220 2020 2020 otient of R │ │ │ │ +00030a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ab0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030aa0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00030ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030b00: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 206b -------+.|i6 : k │ │ │ │ -00030b10: 203d 2031 2020 2020 2020 2020 2020 2020 = 1 │ │ │ │ +00030af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00030b00: 3620 3a20 6b20 3d20 3120 2020 2020 2020 6 : k = 1 │ │ │ │ +00030b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b50: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030b40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ba0: 2020 2020 2020 207c 0a7c 6f36 203d 2031 |.|o6 = 1 │ │ │ │ +00030b90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030ba0: 3620 3d20 3120 2020 2020 2020 2020 2020 6 = 1 │ │ │ │ 00030bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030bf0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030be0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00030bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030c40: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 206d -------+.|i7 : m │ │ │ │ -00030c50: 203d 206b 2b35 2020 2020 2020 2020 2020 = k+5 │ │ │ │ +00030c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00030c40: 3720 3a20 6d20 3d20 6b2b 3520 2020 2020 7 : m = k+5 │ │ │ │ +00030c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030c90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030c80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ce0: 2020 2020 2020 207c 0a7c 6f37 203d 2036 |.|o7 = 6 │ │ │ │ +00030cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030ce0: 3720 3d20 3620 2020 2020 2020 2020 2020 7 = 6 │ │ │ │ 00030cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030d30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030d20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00030d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030d80: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2046 -------+.|i8 : F │ │ │ │ -00030d90: 203d 2066 7265 6552 6573 6f6c 7574 696f = freeResolutio │ │ │ │ -00030da0: 6e28 4d2c 204c 656e 6774 684c 696d 6974 n(M, LengthLimit │ │ │ │ -00030db0: 203d 3e20 6d29 2020 2020 2020 2020 2020 => m) │ │ │ │ -00030dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030dd0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00030d80: 3820 3a20 4620 3d20 6672 6565 5265 736f 8 : F = freeReso │ │ │ │ +00030d90: 6c75 7469 6f6e 284d 2c20 4c65 6e67 7468 lution(M, Length │ │ │ │ +00030da0: 4c69 6d69 7420 3d3e 206d 2920 2020 2020 Limit => m) │ │ │ │ +00030db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030dc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e20: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00030e30: 3120 2020 2020 2032 2020 2020 2020 3420 1 2 4 │ │ │ │ -00030e40: 2020 2020 2037 2020 2020 2020 3131 2020 7 11 │ │ │ │ -00030e50: 2020 2020 3136 2020 2020 2020 3232 2020 16 22 │ │ │ │ -00030e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e70: 2020 2020 2020 207c 0a7c 6f38 203d 2052 |.|o8 = R │ │ │ │ -00030e80: 2020 3c2d 2d20 5220 203c 2d2d 2052 2020 <-- R <-- R │ │ │ │ -00030e90: 3c2d 2d20 5220 203c 2d2d 2052 2020 203c <-- R <-- R < │ │ │ │ -00030ea0: 2d2d 2052 2020 203c 2d2d 2052 2020 2020 -- R <-- R │ │ │ │ -00030eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030e10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030e20: 2020 2020 2031 2020 2020 2020 3220 2020 1 2 │ │ │ │ +00030e30: 2020 2034 2020 2020 2020 3720 2020 2020 4 7 │ │ │ │ +00030e40: 2031 3120 2020 2020 2031 3620 2020 2020 11 16 │ │ │ │ +00030e50: 2032 3220 2020 2020 2020 2020 2020 2020 22 │ │ │ │ +00030e60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030e70: 3820 3d20 5220 203c 2d2d 2052 2020 3c2d 8 = R <-- R <- │ │ │ │ +00030e80: 2d20 5220 203c 2d2d 2052 2020 3c2d 2d20 - R <-- R <-- │ │ │ │ +00030e90: 5220 2020 3c2d 2d20 5220 2020 3c2d 2d20 R <-- R <-- │ │ │ │ +00030ea0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +00030eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030f10: 2020 2020 2020 207c 0a7c 2020 2020 2030 |.| 0 │ │ │ │ -00030f20: 2020 2020 2020 3120 2020 2020 2032 2020 1 2 │ │ │ │ -00030f30: 2020 2020 3320 2020 2020 2034 2020 2020 3 4 │ │ │ │ -00030f40: 2020 2035 2020 2020 2020 2036 2020 2020 5 6 │ │ │ │ -00030f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030f60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030f00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030f10: 2020 2020 3020 2020 2020 2031 2020 2020 0 1 │ │ │ │ +00030f20: 2020 3220 2020 2020 2033 2020 2020 2020 2 3 │ │ │ │ +00030f30: 3420 2020 2020 2020 3520 2020 2020 2020 4 5 │ │ │ │ +00030f40: 3620 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ +00030f50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030fb0: 2020 2020 2020 207c 0a7c 6f38 203a 2043 |.|o8 : C │ │ │ │ -00030fc0: 6f6d 706c 6578 2020 2020 2020 2020 2020 omplex │ │ │ │ +00030fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030fb0: 3820 3a20 436f 6d70 6c65 7820 2020 2020 8 : Complex │ │ │ │ +00030fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031000: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030ff0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00031000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031050: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2073 -------+.|i9 : s │ │ │ │ -00031060: 797a 7967 6965 7320 3d20 6170 706c 7928 yzygies = apply( │ │ │ │ -00031070: 312e 2e6d 2c20 692d 3e63 6f6b 6572 2046 1..m, i->coker F │ │ │ │ -00031080: 2e64 645f 6929 3b20 2020 2020 2020 2020 .dd_i); │ │ │ │ -00031090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000310a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00031040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00031050: 3920 3a20 7379 7a79 6769 6573 203d 2061 9 : syzygies = a │ │ │ │ +00031060: 7070 6c79 2831 2e2e 6d2c 2069 2d3e 636f pply(1..m, i->co │ │ │ │ +00031070: 6b65 7220 462e 6464 5f69 293b 2020 2020 ker F.dd_i); │ │ │ │ +00031080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031090: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000310a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000310b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000310c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000310d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000310e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000310f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 -------+.|i10 : │ │ │ │ -00031100: 7431 203d 206d 616b 6554 2866 662c 462c t1 = makeT(ff,F, │ │ │ │ -00031110: 6b2b 3429 3b20 2020 2020 2020 2020 2020 k+4); │ │ │ │ +000310e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000310f0: 3130 203a 2074 3120 3d20 6d61 6b65 5428 10 : t1 = makeT( │ │ │ │ +00031100: 6666 2c46 2c6b 2b34 293b 2020 2020 2020 ff,F,k+4); │ │ │ │ +00031110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031140: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00031130: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00031140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031190: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ -000311a0: 7432 203d 206d 616b 6554 2866 662c 462c t2 = makeT(ff,F, │ │ │ │ -000311b0: 6b2b 3229 3b20 2020 2020 2020 2020 2020 k+2); │ │ │ │ +00031180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00031190: 3131 203a 2074 3220 3d20 6d61 6b65 5428 11 : t2 = makeT( │ │ │ │ +000311a0: 6666 2c46 2c6b 2b32 293b 2020 2020 2020 ff,F,k+2); │ │ │ │ +000311b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000311c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000311d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000311e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000311d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000311e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000311f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031230: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 -------+.|i12 : │ │ │ │ -00031240: 5432 436f 6d70 6f6e 656e 7473 203d 2066 T2Components = f │ │ │ │ -00031250: 6c61 7474 656e 2066 6f72 2069 2066 726f latten for i fro │ │ │ │ -00031260: 6d20 3020 746f 2031 206c 6973 7428 666f m 0 to 1 list(fo │ │ │ │ -00031270: 7220 6a20 6672 6f6d 2069 2b31 2074 6f20 r j from i+1 to │ │ │ │ -00031280: 3220 6c69 7374 207c 0a7c 2d2d 2d2d 2d2d 2 list |.|------ │ │ │ │ +00031220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00031230: 3132 203a 2054 3243 6f6d 706f 6e65 6e74 12 : T2Component │ │ │ │ +00031240: 7320 3d20 666c 6174 7465 6e20 666f 7220 s = flatten for │ │ │ │ +00031250: 6920 6672 6f6d 2030 2074 6f20 3120 6c69 i from 0 to 1 li │ │ │ │ +00031260: 7374 2866 6f72 206a 2066 726f 6d20 692b st(for j from i+ │ │ │ │ +00031270: 3120 746f 2032 206c 6973 7420 7c0a 7c2d 1 to 2 list |.|- │ │ │ │ +00031280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000312a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000312b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000312c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000312d0: 2d2d 2d2d 2d2d 2d7c 0a7c 6d61 7028 465f -------|.|map(F_ │ │ │ │ -000312e0: 6b2c 2046 5f28 6b2b 3429 2c20 7432 5f69 k, F_(k+4), t2_i │ │ │ │ -000312f0: 2a74 315f 6a2d 7432 5f6a 2a74 315f 6929 *t1_j-t2_j*t1_i) │ │ │ │ -00031300: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ -00031310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031320: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000312c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c6d ------------|.|m │ │ │ │ +000312d0: 6170 2846 5f6b 2c20 465f 286b 2b34 292c ap(F_k, F_(k+4), │ │ │ │ +000312e0: 2074 325f 692a 7431 5f6a 2d74 325f 6a2a t2_i*t1_j-t2_j* │ │ │ │ +000312f0: 7431 5f69 2929 3b20 2020 2020 2020 2020 t1_i)); │ │ │ │ +00031300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031310: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00031320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031370: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 -------+.|i13 : │ │ │ │ -00031380: 6720 3d20 6d61 7028 7379 7a79 6769 6573 g = map(syzygies │ │ │ │ -00031390: 5f6b 2c20 7379 7a79 6769 6573 5f28 6b2b _k, syzygies_(k+ │ │ │ │ -000313a0: 3429 2c20 5432 436f 6d70 6f6e 656e 7473 4), T2Components │ │ │ │ -000313b0: 5f32 2920 2020 2020 2020 2020 2020 2020 _2) │ │ │ │ -000313c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00031360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00031370: 3133 203a 2067 203d 206d 6170 2873 797a 13 : g = map(syz │ │ │ │ +00031380: 7967 6965 735f 6b2c 2073 797a 7967 6965 ygies_k, syzygie │ │ │ │ +00031390: 735f 286b 2b34 292c 2054 3243 6f6d 706f s_(k+4), T2Compo │ │ │ │ +000313a0: 6e65 6e74 735f 3229 2020 2020 2020 2020 nents_2) │ │ │ │ +000313b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000313c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000313d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000313e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000313f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031410: 2020 2020 2020 207c 0a7c 6f31 3320 3d20 |.|o13 = │ │ │ │ -00031420: 7b31 7d20 7c20 3020 3020 3020 3020 3020 {1} | 0 0 0 0 0 │ │ │ │ -00031430: 2d63 2030 2030 2062 2030 2030 2030 2030 -c 0 0 b 0 0 0 0 │ │ │ │ -00031440: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ -00031450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031460: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00031470: 7b32 7d20 7c20 3020 3020 3020 3020 3020 {2} | 0 0 0 0 0 │ │ │ │ -00031480: 3020 2030 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 0 │ │ │ │ -00031490: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ -000314a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000314b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00031400: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00031410: 3133 203d 207b 317d 207c 2030 2030 2030 13 = {1} | 0 0 0 │ │ │ │ +00031420: 2030 2030 202d 6320 3020 3020 6220 3020 0 0 -c 0 0 b 0 │ │ │ │ +00031430: 3020 3020 3020 3020 3020 3020 7c20 2020 0 0 0 0 0 0 | │ │ │ │ +00031440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031450: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00031460: 2020 2020 207b 327d 207c 2030 2030 2030 {2} | 0 0 0 │ │ │ │ +00031470: 2030 2030 2030 2020 3020 3020 3020 3020 0 0 0 0 0 0 0 │ │ │ │ +00031480: 3020 3020 3020 3020 3020 3020 7c20 2020 0 0 0 0 0 0 | │ │ │ │ +00031490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000314a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000314b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000314c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000314d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000314e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000314f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031500: 2020 2020 2020 207c 0a7c 6f31 3320 3a20 |.|o13 : │ │ │ │ -00031510: 4d61 7472 6978 2020 2020 2020 2020 2020 Matrix │ │ │ │ +000314f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00031500: 3133 203a 204d 6174 7269 7820 2020 2020 13 : Matrix │ │ │ │ +00031510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031550: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00031540: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00031550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000315a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 -------+.|i14 : │ │ │ │ -000315b0: 6973 5374 6162 6c79 5472 6976 6961 6c20 isStablyTrivial │ │ │ │ -000315c0: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ +00031590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000315a0: 3134 203a 2069 7353 7461 626c 7954 7269 14 : isStablyTri │ │ │ │ +000315b0: 7669 616c 2067 2020 2020 2020 2020 2020 vial g │ │ │ │ +000315c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000315d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000315e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000315f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000315e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000315f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031640: 2020 2020 2020 207c 0a7c 6f31 3420 3d20 |.|o14 = │ │ │ │ -00031650: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ +00031630: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00031640: 3134 203d 2074 7275 6520 2020 2020 2020 14 = true │ │ │ │ +00031650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031690: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00031680: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00031690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000316a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000316b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000316c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000316d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000316e0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ -000316f0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -00031700: 202a 6e6f 7465 2073 7461 626c 6548 6f6d *note stableHom │ │ │ │ -00031710: 3a20 7374 6162 6c65 486f 6d2c 202d 2d20 : stableHom, -- │ │ │ │ -00031720: 6d61 7020 6672 6f6d 2048 6f6d 284d 2c4e map from Hom(M,N │ │ │ │ -00031730: 2920 746f 2074 6865 2073 7461 626c 6520 ) to the stable │ │ │ │ -00031740: 486f 6d20 6d6f 6475 6c65 0a0a 5761 7973 Hom module..Ways │ │ │ │ -00031750: 2074 6f20 7573 6520 6973 5374 6162 6c79 to use isStably │ │ │ │ -00031760: 5472 6976 6961 6c3a 0a3d 3d3d 3d3d 3d3d Trivial:.======= │ │ │ │ -00031770: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00031780: 3d3d 3d3d 3d0a 0a20 202a 2022 6973 5374 =====.. * "isSt │ │ │ │ -00031790: 6162 6c79 5472 6976 6961 6c28 4d61 7472 ablyTrivial(Matr │ │ │ │ -000317a0: 6978 2922 0a0a 466f 7220 7468 6520 7072 ix)"..For the pr │ │ │ │ -000317b0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ -000317c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -000317d0: 206f 626a 6563 7420 2a6e 6f74 6520 6973 object *note is │ │ │ │ -000317e0: 5374 6162 6c79 5472 6976 6961 6c3a 2069 StablyTrivial: i │ │ │ │ -000317f0: 7353 7461 626c 7954 7269 7669 616c 2c20 sStablyTrivial, │ │ │ │ -00031800: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -00031810: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ -00031820: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -00031830: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +000316d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 ------------+..S │ │ │ │ +000316e0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +000316f0: 0a0a 2020 2a20 2a6e 6f74 6520 7374 6162 .. * *note stab │ │ │ │ +00031700: 6c65 486f 6d3a 2073 7461 626c 6548 6f6d leHom: stableHom │ │ │ │ +00031710: 2c20 2d2d 206d 6170 2066 726f 6d20 486f , -- map from Ho │ │ │ │ +00031720: 6d28 4d2c 4e29 2074 6f20 7468 6520 7374 m(M,N) to the st │ │ │ │ +00031730: 6162 6c65 2048 6f6d 206d 6f64 756c 650a able Hom module. │ │ │ │ +00031740: 0a57 6179 7320 746f 2075 7365 2069 7353 .Ways to use isS │ │ │ │ +00031750: 7461 626c 7954 7269 7669 616c 3a0a 3d3d tablyTrivial:.== │ │ │ │ +00031760: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00031770: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00031780: 2269 7353 7461 626c 7954 7269 7669 616c "isStablyTrivial │ │ │ │ +00031790: 284d 6174 7269 7829 220a 0a46 6f72 2074 (Matrix)"..For t │ │ │ │ +000317a0: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ +000317b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000317c0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ +000317d0: 7465 2069 7353 7461 626c 7954 7269 7669 te isStablyTrivi │ │ │ │ +000317e0: 616c 3a20 6973 5374 6162 6c79 5472 6976 al: isStablyTriv │ │ │ │ +000317f0: 6961 6c2c 2069 7320 6120 2a6e 6f74 6520 ial, is a *note │ │ │ │ +00031800: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ +00031810: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ +00031820: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +00031830: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 00031840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -00031890: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -000318a0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -000318b0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -000318c0: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -000318d0: 322d 312e 3236 2e30 352b 6473 2f4d 322f 2-1.26.05+ds/M2/ │ │ │ │ -000318e0: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -000318f0: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ -00031900: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -00031910: 6f6e 732e 6d32 3a34 3639 393a 302e 0a1f ons.m2:4699:0... │ │ │ │ -00031920: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ -00031930: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -00031940: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ -00031950: 653a 206b 6f73 7a75 6c45 7874 656e 7369 e: koszulExtensi │ │ │ │ -00031960: 6f6e 2c20 4e65 7874 3a20 4c61 7965 7265 on, Next: Layere │ │ │ │ -00031970: 642c 2050 7265 763a 2069 7353 7461 626c d, Prev: isStabl │ │ │ │ -00031980: 7954 7269 7669 616c 2c20 5570 3a20 546f yTrivial, Up: To │ │ │ │ -00031990: 700a 0a6b 6f73 7a75 6c45 7874 656e 7369 p..koszulExtensi │ │ │ │ -000319a0: 6f6e 202d 2d20 6372 6561 7465 7320 7468 on -- creates th │ │ │ │ -000319b0: 6520 4b6f 737a 756c 2065 7874 656e 7369 e Koszul extensi │ │ │ │ -000319c0: 6f6e 2063 6f6d 706c 6578 206f 6620 6120 on complex of a │ │ │ │ -000319d0: 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a map.************ │ │ │ │ +00031880: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +00031890: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +000318a0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +000318b0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +000318c0: 6175 6c61 7932 2d31 2e32 362e 3035 2b64 aulay2-1.26.05+d │ │ │ │ +000318d0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +000318e0: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ +000318f0: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +00031900: 6f6c 7574 696f 6e73 2e6d 323a 3436 3939 olutions.m2:4699 │ │ │ │ +00031910: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ +00031920: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +00031930: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ +00031940: 2c20 4e6f 6465 3a20 6b6f 737a 756c 4578 , Node: koszulEx │ │ │ │ +00031950: 7465 6e73 696f 6e2c 204e 6578 743a 204c tension, Next: L │ │ │ │ +00031960: 6179 6572 6564 2c20 5072 6576 3a20 6973 ayered, Prev: is │ │ │ │ +00031970: 5374 6162 6c79 5472 6976 6961 6c2c 2055 StablyTrivial, U │ │ │ │ +00031980: 703a 2054 6f70 0a0a 6b6f 737a 756c 4578 p: Top..koszulEx │ │ │ │ +00031990: 7465 6e73 696f 6e20 2d2d 2063 7265 6174 tension -- creat │ │ │ │ +000319a0: 6573 2074 6865 204b 6f73 7a75 6c20 6578 es the Koszul ex │ │ │ │ +000319b0: 7465 6e73 696f 6e20 636f 6d70 6c65 7820 tension complex │ │ │ │ +000319c0: 6f66 2061 206d 6170 0a2a 2a2a 2a2a 2a2a of a map.******* │ │ │ │ +000319d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000319e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000319f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00031a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00031a10: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -00031a20: 200a 2020 2020 2020 2020 4d4d 203d 206b . MM = k │ │ │ │ -00031a30: 6f73 7a75 6c45 7874 656e 7369 6f6e 2846 oszulExtension(F │ │ │ │ -00031a40: 462c 4242 2c70 7369 312c 6666 290a 2020 F,BB,psi1,ff). │ │ │ │ -00031a50: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ -00031a60: 2a20 4646 2c20 6120 2a6e 6f74 6520 636f * FF, a *note co │ │ │ │ -00031a70: 6d70 6c65 783a 2028 436f 6d70 6c65 7865 mplex: (Complexe │ │ │ │ -00031a80: 7329 436f 6d70 6c65 782c 2c20 7265 736f s)Complex,, reso │ │ │ │ -00031a90: 6c75 7469 6f6e 206f 7665 7220 530a 2020 lution over S. │ │ │ │ -00031aa0: 2020 2020 2a20 4242 2c20 6120 2a6e 6f74 * BB, a *not │ │ │ │ -00031ab0: 6520 636f 6d70 6c65 783a 2028 436f 6d70 e complex: (Comp │ │ │ │ -00031ac0: 6c65 7865 7329 436f 6d70 6c65 782c 2c20 lexes)Complex,, │ │ │ │ -00031ad0: 7477 6f2d 7465 726d 2063 6f6d 706c 6578 two-term complex │ │ │ │ -00031ae0: 2042 425f 312d 2d3e 4242 5f30 0a20 2020 BB_1-->BB_0. │ │ │ │ -00031af0: 2020 202a 2070 7369 312c 2061 202a 6e6f * psi1, a *no │ │ │ │ -00031b00: 7465 206d 6174 7269 783a 2028 4d61 6361 te matrix: (Maca │ │ │ │ -00031b10: 756c 6179 3244 6f63 294d 6174 7269 782c ulay2Doc)Matrix, │ │ │ │ -00031b20: 2c20 6672 6f6d 2042 425f 3120 746f 2046 , from BB_1 to F │ │ │ │ -00031b30: 465f 300a 2020 2020 2020 2a20 6666 2c20 F_0. * ff, │ │ │ │ -00031b40: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ -00031b50: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ -00031b60: 7472 6978 2c2c 2072 6567 756c 6172 2073 trix,, regular s │ │ │ │ -00031b70: 6571 7565 6e63 650a 2020 2020 2020 2020 equence. │ │ │ │ -00031b80: 616e 6e69 6869 6c61 7469 6e67 2074 6865 annihilating the │ │ │ │ -00031b90: 206d 6f64 756c 6520 7265 736f 6c76 6564 module resolved │ │ │ │ -00031ba0: 2062 7920 4646 0a20 202a 204f 7574 7075 by FF. * Outpu │ │ │ │ -00031bb0: 7473 3a0a 2020 2020 2020 2a20 4d4d 2c20 ts:. * MM, │ │ │ │ -00031bc0: 6120 2a6e 6f74 6520 636f 6d70 6c65 783a a *note complex: │ │ │ │ -00031bd0: 2028 436f 6d70 6c65 7865 7329 436f 6d70 (Complexes)Comp │ │ │ │ -00031be0: 6c65 782c 2c20 7468 6520 6d61 7070 696e lex,, the mappin │ │ │ │ -00031bf0: 6720 636f 6e65 206f 6620 7468 650a 2020 g cone of the. │ │ │ │ -00031c00: 2020 2020 2020 696e 6475 6365 6420 6d61 induced ma │ │ │ │ -00031c10: 7020 425b 2d31 5d5c 6f74 696d 6573 204b p B[-1]\otimes K │ │ │ │ -00031c20: 4b28 6666 2920 746f 2057 2065 7874 656e K(ff) to W exten │ │ │ │ -00031c30: 6469 6e67 2070 7369 0a0a 4465 7363 7269 ding psi..Descri │ │ │ │ -00031c40: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -00031c50: 3d0a 0a49 6d70 6c65 6d65 6e74 7320 7468 =..Implements th │ │ │ │ -00031c60: 6520 636f 6e73 7472 7563 7469 6f6e 2069 e construction i │ │ │ │ -00031c70: 6e20 7468 6520 7061 7065 7220 224d 6174 n the paper "Mat │ │ │ │ -00031c80: 7269 7820 4661 6374 6f72 697a 6174 696f rix Factorizatio │ │ │ │ -00031c90: 6e73 2069 6e20 4869 6768 6572 0a43 6f64 ns in Higher.Cod │ │ │ │ -00031ca0: 696d 656e 7369 6f6e 2220 6279 2045 6973 imension" by Eis │ │ │ │ -00031cb0: 656e 6275 6420 616e 6420 5065 6576 612e enbud and Peeva. │ │ │ │ -00031cc0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -00031cd0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d ===.. * *note m │ │ │ │ -00031ce0: 616b 6546 696e 6974 6552 6573 6f6c 7574 akeFiniteResolut │ │ │ │ -00031cf0: 696f 6e3a 206d 616b 6546 696e 6974 6552 ion: makeFiniteR │ │ │ │ -00031d00: 6573 6f6c 7574 696f 6e2c 202d 2d20 6669 esolution, -- fi │ │ │ │ -00031d10: 6e69 7465 2072 6573 6f6c 7574 696f 6e20 nite resolution │ │ │ │ -00031d20: 6f66 2061 0a20 2020 206d 6174 7269 7820 of a. matrix │ │ │ │ -00031d30: 6661 6374 6f72 697a 6174 696f 6e20 6d6f factorization mo │ │ │ │ -00031d40: 6475 6c65 204d 0a0a 5761 7973 2074 6f20 dule M..Ways to │ │ │ │ -00031d50: 7573 6520 6b6f 737a 756c 4578 7465 6e73 use koszulExtens │ │ │ │ -00031d60: 696f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ion:.=========== │ │ │ │ -00031d70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00031d80: 3d0a 0a20 202a 2022 6b6f 737a 756c 4578 =.. * "koszulEx │ │ │ │ -00031d90: 7465 6e73 696f 6e28 436f 6d70 6c65 782c tension(Complex, │ │ │ │ -00031da0: 436f 6d70 6c65 782c 4d61 7472 6978 2c4d Complex,Matrix,M │ │ │ │ -00031db0: 6174 7269 7829 220a 0a46 6f72 2074 6865 atrix)"..For the │ │ │ │ -00031dc0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00031dd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00031de0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00031df0: 206b 6f73 7a75 6c45 7874 656e 7369 6f6e koszulExtension │ │ │ │ -00031e00: 3a20 6b6f 737a 756c 4578 7465 6e73 696f : koszulExtensio │ │ │ │ -00031e10: 6e2c 2069 7320 6120 2a6e 6f74 6520 6d65 n, is a *note me │ │ │ │ -00031e20: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ -00031e30: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ -00031e40: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +00031a00: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +00031a10: 7361 6765 3a20 0a20 2020 2020 2020 204d sage: . M │ │ │ │ +00031a20: 4d20 3d20 6b6f 737a 756c 4578 7465 6e73 M = koszulExtens │ │ │ │ +00031a30: 696f 6e28 4646 2c42 422c 7073 6931 2c66 ion(FF,BB,psi1,f │ │ │ │ +00031a40: 6629 0a20 202a 2049 6e70 7574 733a 0a20 f). * Inputs:. │ │ │ │ +00031a50: 2020 2020 202a 2046 462c 2061 202a 6e6f * FF, a *no │ │ │ │ +00031a60: 7465 2063 6f6d 706c 6578 3a20 2843 6f6d te complex: (Com │ │ │ │ +00031a70: 706c 6578 6573 2943 6f6d 706c 6578 2c2c plexes)Complex,, │ │ │ │ +00031a80: 2072 6573 6f6c 7574 696f 6e20 6f76 6572 resolution over │ │ │ │ +00031a90: 2053 0a20 2020 2020 202a 2042 422c 2061 S. * BB, a │ │ │ │ +00031aa0: 202a 6e6f 7465 2063 6f6d 706c 6578 3a20 *note complex: │ │ │ │ +00031ab0: 2843 6f6d 706c 6578 6573 2943 6f6d 706c (Complexes)Compl │ │ │ │ +00031ac0: 6578 2c2c 2074 776f 2d74 6572 6d20 636f ex,, two-term co │ │ │ │ +00031ad0: 6d70 6c65 7820 4242 5f31 2d2d 3e42 425f mplex BB_1-->BB_ │ │ │ │ +00031ae0: 300a 2020 2020 2020 2a20 7073 6931 2c20 0. * psi1, │ │ │ │ +00031af0: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ +00031b00: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ +00031b10: 7472 6978 2c2c 2066 726f 6d20 4242 5f31 trix,, from BB_1 │ │ │ │ +00031b20: 2074 6f20 4646 5f30 0a20 2020 2020 202a to FF_0. * │ │ │ │ +00031b30: 2066 662c 2061 202a 6e6f 7465 206d 6174 ff, a *note mat │ │ │ │ +00031b40: 7269 783a 2028 4d61 6361 756c 6179 3244 rix: (Macaulay2D │ │ │ │ +00031b50: 6f63 294d 6174 7269 782c 2c20 7265 6775 oc)Matrix,, regu │ │ │ │ +00031b60: 6c61 7220 7365 7175 656e 6365 0a20 2020 lar sequence. │ │ │ │ +00031b70: 2020 2020 2061 6e6e 6968 696c 6174 696e annihilatin │ │ │ │ +00031b80: 6720 7468 6520 6d6f 6475 6c65 2072 6573 g the module res │ │ │ │ +00031b90: 6f6c 7665 6420 6279 2046 460a 2020 2a20 olved by FF. * │ │ │ │ +00031ba0: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ +00031bb0: 204d 4d2c 2061 202a 6e6f 7465 2063 6f6d MM, a *note com │ │ │ │ +00031bc0: 706c 6578 3a20 2843 6f6d 706c 6578 6573 plex: (Complexes │ │ │ │ +00031bd0: 2943 6f6d 706c 6578 2c2c 2074 6865 206d )Complex,, the m │ │ │ │ +00031be0: 6170 7069 6e67 2063 6f6e 6520 6f66 2074 apping cone of t │ │ │ │ +00031bf0: 6865 0a20 2020 2020 2020 2069 6e64 7563 he. induc │ │ │ │ +00031c00: 6564 206d 6170 2042 5b2d 315d 5c6f 7469 ed map B[-1]\oti │ │ │ │ +00031c10: 6d65 7320 4b4b 2866 6629 2074 6f20 5720 mes KK(ff) to W │ │ │ │ +00031c20: 6578 7465 6e64 696e 6720 7073 690a 0a44 extending psi..D │ │ │ │ +00031c30: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +00031c40: 3d3d 3d3d 3d3d 0a0a 496d 706c 656d 656e ======..Implemen │ │ │ │ +00031c50: 7473 2074 6865 2063 6f6e 7374 7275 6374 ts the construct │ │ │ │ +00031c60: 696f 6e20 696e 2074 6865 2070 6170 6572 ion in the paper │ │ │ │ +00031c70: 2022 4d61 7472 6978 2046 6163 746f 7269 "Matrix Factori │ │ │ │ +00031c80: 7a61 7469 6f6e 7320 696e 2048 6967 6865 zations in Highe │ │ │ │ +00031c90: 720a 436f 6469 6d65 6e73 696f 6e22 2062 r.Codimension" b │ │ │ │ +00031ca0: 7920 4569 7365 6e62 7564 2061 6e64 2050 y Eisenbud and P │ │ │ │ +00031cb0: 6565 7661 2e0a 0a53 6565 2061 6c73 6f0a eeva...See also. │ │ │ │ +00031cc0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +00031cd0: 6f74 6520 6d61 6b65 4669 6e69 7465 5265 ote makeFiniteRe │ │ │ │ +00031ce0: 736f 6c75 7469 6f6e 3a20 6d61 6b65 4669 solution: makeFi │ │ │ │ +00031cf0: 6e69 7465 5265 736f 6c75 7469 6f6e 2c20 niteResolution, │ │ │ │ +00031d00: 2d2d 2066 696e 6974 6520 7265 736f 6c75 -- finite resolu │ │ │ │ +00031d10: 7469 6f6e 206f 6620 610a 2020 2020 6d61 tion of a. ma │ │ │ │ +00031d20: 7472 6978 2066 6163 746f 7269 7a61 7469 trix factorizati │ │ │ │ +00031d30: 6f6e 206d 6f64 756c 6520 4d0a 0a57 6179 on module M..Way │ │ │ │ +00031d40: 7320 746f 2075 7365 206b 6f73 7a75 6c45 s to use koszulE │ │ │ │ +00031d50: 7874 656e 7369 6f6e 3a0a 3d3d 3d3d 3d3d xtension:.====== │ │ │ │ +00031d60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00031d70: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226b 6f73 ======.. * "kos │ │ │ │ +00031d80: 7a75 6c45 7874 656e 7369 6f6e 2843 6f6d zulExtension(Com │ │ │ │ +00031d90: 706c 6578 2c43 6f6d 706c 6578 2c4d 6174 plex,Complex,Mat │ │ │ │ +00031da0: 7269 782c 4d61 7472 6978 2922 0a0a 466f rix,Matrix)"..Fo │ │ │ │ +00031db0: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00031dc0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00031dd0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00031de0: 2a6e 6f74 6520 6b6f 737a 756c 4578 7465 *note koszulExte │ │ │ │ +00031df0: 6e73 696f 6e3a 206b 6f73 7a75 6c45 7874 nsion: koszulExt │ │ │ │ +00031e00: 656e 7369 6f6e 2c20 6973 2061 202a 6e6f ension, is a *no │ │ │ │ +00031e10: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ +00031e20: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f on:.(Macaulay2Do │ │ │ │ +00031e30: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ +00031e40: 2c2e 0a0a 2d2d 2d2d 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6f6c ntersectionResol │ │ │ │ -00031f20: 7574 696f 6e73 2e6d 323a 3330 3038 3a30 utions.m2:3008:0 │ │ │ │ -00031f30: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ -00031f40: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ -00031f50: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ -00031f60: 4e6f 6465 3a20 4c61 7965 7265 642c 204e Node: Layered, N │ │ │ │ -00031f70: 6578 743a 206c 6179 6572 6564 5265 736f ext: layeredReso │ │ │ │ -00031f80: 6c75 7469 6f6e 2c20 5072 6576 3a20 6b6f lution, Prev: ko │ │ │ │ -00031f90: 737a 756c 4578 7465 6e73 696f 6e2c 2055 szulExtension, U │ │ │ │ -00031fa0: 703a 2054 6f70 0a0a 4c61 7965 7265 6420 p: Top..Layered │ │ │ │ -00031fb0: 2d2d 204f 7074 696f 6e20 666f 7220 6d61 -- Option for ma │ │ │ │ -00031fc0: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ -00031fd0: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a n.************** │ │ │ │ +00031e90: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +00031ea0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +00031eb0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +00031ec0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +00031ed0: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ +00031ee0: 352b 6473 2f4d 322f 4d61 6361 756c 6179 5+ds/M2/Macaulay │ │ │ │ +00031ef0: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ +00031f00: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +00031f10: 5265 736f 6c75 7469 6f6e 732e 6d32 3a33 Resolutions.m2:3 │ │ │ │ +00031f20: 3030 383a 302e 0a1f 0a46 696c 653a 2043 008:0....File: C │ │ │ │ +00031f30: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ +00031f40: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ +00031f50: 6e66 6f2c 204e 6f64 653a 204c 6179 6572 nfo, Node: Layer │ │ │ │ +00031f60: 6564 2c20 4e65 7874 3a20 6c61 7965 7265 ed, Next: layere │ │ │ │ +00031f70: 6452 6573 6f6c 7574 696f 6e2c 2050 7265 dResolution, Pre │ │ │ │ +00031f80: 763a 206b 6f73 7a75 6c45 7874 656e 7369 v: koszulExtensi │ │ │ │ +00031f90: 6f6e 2c20 5570 3a20 546f 700a 0a4c 6179 on, Up: Top..Lay │ │ │ │ +00031fa0: 6572 6564 202d 2d20 4f70 7469 6f6e 2066 ered -- Option f │ │ │ │ +00031fb0: 6f72 206d 6174 7269 7846 6163 746f 7269 or matrixFactori │ │ │ │ +00031fc0: 7a61 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a zation.********* │ │ │ │ +00031fd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00031fe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00031ff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -00032000: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -00032010: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ -00032020: 7469 6f6e 2866 662c 6d2c 4c61 7965 7265 tion(ff,m,Layere │ │ │ │ -00032030: 6420 3d3e 2074 7275 6529 0a20 202a 2049 d => true). * I │ │ │ │ -00032040: 6e70 7574 733a 0a20 2020 2020 202a 2043 nputs:. * C │ │ │ │ -00032050: 6865 636b 2c20 6120 2a6e 6f74 6520 426f heck, a *note Bo │ │ │ │ -00032060: 6f6c 6561 6e20 7661 6c75 653a 2028 4d61 olean value: (Ma │ │ │ │ -00032070: 6361 756c 6179 3244 6f63 2942 6f6f 6c65 caulay2Doc)Boole │ │ │ │ -00032080: 616e 2c2c 200a 0a44 6573 6372 6970 7469 an,, ..Descripti │ │ │ │ -00032090: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -000320a0: 4d61 6b65 7320 6d61 7472 6978 4661 6374 Makes matrixFact │ │ │ │ -000320b0: 6f72 697a 6174 696f 6e20 7573 6520 7468 orization use th │ │ │ │ -000320c0: 6520 226c 6179 6572 6564 2220 616c 676f e "layered" algo │ │ │ │ -000320d0: 7269 7468 6d2c 2077 6869 6368 2077 6f72 rithm, which wor │ │ │ │ -000320e0: 6b73 2066 6f72 2061 6e79 204d 434d 0a6d ks for any MCM.m │ │ │ │ -000320f0: 6f64 756c 652c 2062 7574 2072 6574 7572 odule, but retur │ │ │ │ -00032100: 6e73 2073 6f6d 6574 6869 6e67 206e 6f6e ns something non │ │ │ │ -00032110: 2d6d 696e 696d 616c 2069 6620 7468 6520 -minimal if the │ │ │ │ -00032120: 6d6f 6475 6c65 2069 7320 6e6f 7420 6120 module is not a │ │ │ │ -00032130: 2268 6967 6820 7379 7a79 6779 220a 696e "high syzygy".in │ │ │ │ -00032140: 2061 2073 7569 7461 626c 6520 7365 6e73 a suitable sens │ │ │ │ -00032150: 652e 2044 6566 6175 6c74 2069 7320 2274 e. Default is "t │ │ │ │ -00032160: 7275 6522 2e20 4e6f 7465 2074 6861 7420 rue". Note that │ │ │ │ -00032170: 7768 656e 2074 6865 206d 6f64 756c 6520 when the module │ │ │ │ -00032180: 6973 2061 2068 6967 680a 7379 7a79 6779 is a high.syzygy │ │ │ │ -00032190: 2c20 4c61 7965 7265 643d 3e20 6661 6c73 , Layered=> fals │ │ │ │ -000321a0: 6520 6973 206d 7563 6820 6661 7374 6572 e is much faster │ │ │ │ -000321b0: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d ...See also.==== │ │ │ │ -000321c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -000321d0: 6d61 7472 6978 4661 6374 6f72 697a 6174 matrixFactorizat │ │ │ │ -000321e0: 696f 6e3a 206d 6174 7269 7846 6163 746f ion: matrixFacto │ │ │ │ -000321f0: 7269 7a61 7469 6f6e 2c20 2d2d 204d 6170 rization, -- Map │ │ │ │ -00032200: 7320 696e 2061 2068 6967 6865 720a 2020 s in a higher. │ │ │ │ -00032210: 2020 636f 6469 6d65 6e73 696f 6e20 6d61 codimension ma │ │ │ │ -00032220: 7472 6978 2066 6163 746f 7269 7a61 7469 trix factorizati │ │ │ │ -00032230: 6f6e 0a0a 4675 6e63 7469 6f6e 7320 7769 on..Functions wi │ │ │ │ -00032240: 7468 206f 7074 696f 6e61 6c20 6172 6775 th optional argu │ │ │ │ -00032250: 6d65 6e74 206e 616d 6564 204c 6179 6572 ment named Layer │ │ │ │ -00032260: 6564 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ed:.============ │ │ │ │ +00031ff0: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +00032000: 2020 2020 2020 6d61 7472 6978 4661 6374 matrixFact │ │ │ │ +00032010: 6f72 697a 6174 696f 6e28 6666 2c6d 2c4c orization(ff,m,L │ │ │ │ +00032020: 6179 6572 6564 203d 3e20 7472 7565 290a ayered => true). │ │ │ │ +00032030: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +00032040: 2020 2a20 4368 6563 6b2c 2061 202a 6e6f * Check, a *no │ │ │ │ +00032050: 7465 2042 6f6f 6c65 616e 2076 616c 7565 te Boolean value │ │ │ │ +00032060: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00032070: 426f 6f6c 6561 6e2c 2c20 0a0a 4465 7363 Boolean,, ..Desc │ │ │ │ +00032080: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +00032090: 3d3d 3d0a 0a4d 616b 6573 206d 6174 7269 ===..Makes matri │ │ │ │ +000320a0: 7846 6163 746f 7269 7a61 7469 6f6e 2075 xFactorization u │ │ │ │ +000320b0: 7365 2074 6865 2022 6c61 7965 7265 6422 se the "layered" │ │ │ │ +000320c0: 2061 6c67 6f72 6974 686d 2c20 7768 6963 algorithm, whic │ │ │ │ +000320d0: 6820 776f 726b 7320 666f 7220 616e 7920 h works for any │ │ │ │ +000320e0: 4d43 4d0a 6d6f 6475 6c65 2c20 6275 7420 MCM.module, but │ │ │ │ +000320f0: 7265 7475 726e 7320 736f 6d65 7468 696e returns somethin │ │ │ │ +00032100: 6720 6e6f 6e2d 6d69 6e69 6d61 6c20 6966 g non-minimal if │ │ │ │ +00032110: 2074 6865 206d 6f64 756c 6520 6973 206e the module is n │ │ │ │ +00032120: 6f74 2061 2022 6869 6768 2073 797a 7967 ot a "high syzyg │ │ │ │ +00032130: 7922 0a69 6e20 6120 7375 6974 6162 6c65 y".in a suitable │ │ │ │ +00032140: 2073 656e 7365 2e20 4465 6661 756c 7420 sense. Default │ │ │ │ +00032150: 6973 2022 7472 7565 222e 204e 6f74 6520 is "true". Note │ │ │ │ +00032160: 7468 6174 2077 6865 6e20 7468 6520 6d6f that when the mo │ │ │ │ +00032170: 6475 6c65 2069 7320 6120 6869 6768 0a73 dule is a high.s │ │ │ │ +00032180: 797a 7967 792c 204c 6179 6572 6564 3d3e yzygy, Layered=> │ │ │ │ +00032190: 2066 616c 7365 2069 7320 6d75 6368 2066 false is much f │ │ │ │ +000321a0: 6173 7465 722e 0a0a 5365 6520 616c 736f aster...See also │ │ │ │ +000321b0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +000321c0: 6e6f 7465 206d 6174 7269 7846 6163 746f note matrixFacto │ │ │ │ +000321d0: 7269 7a61 7469 6f6e 3a20 6d61 7472 6978 rization: matrix │ │ │ │ +000321e0: 4661 6374 6f72 697a 6174 696f 6e2c 202d Factorization, - │ │ │ │ +000321f0: 2d20 4d61 7073 2069 6e20 6120 6869 6768 - Maps in a high │ │ │ │ +00032200: 6572 0a20 2020 2063 6f64 696d 656e 7369 er. codimensi │ │ │ │ +00032210: 6f6e 206d 6174 7269 7820 6661 6374 6f72 on matrix factor │ │ │ │ +00032220: 697a 6174 696f 6e0a 0a46 756e 6374 696f ization..Functio │ │ │ │ +00032230: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ +00032240: 2061 7267 756d 656e 7420 6e61 6d65 6420 argument named │ │ │ │ +00032250: 4c61 7965 7265 643a 0a3d 3d3d 3d3d 3d3d Layered:.======= │ │ │ │ +00032260: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00032270: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00032280: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00032290: 3d3d 3d0a 0a20 202a 2022 6d61 7472 6978 ===.. * "matrix │ │ │ │ -000322a0: 4661 6374 6f72 697a 6174 696f 6e28 2e2e Factorization(.. │ │ │ │ -000322b0: 2e2c 4c61 7965 7265 643d 3e2e 2e2e 2922 .,Layered=>...)" │ │ │ │ -000322c0: 202d 2d20 7365 6520 2a6e 6f74 6520 6d61 -- see *note ma │ │ │ │ -000322d0: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ -000322e0: 6e3a 0a20 2020 206d 6174 7269 7846 6163 n:. matrixFac │ │ │ │ -000322f0: 746f 7269 7a61 7469 6f6e 2c20 2d2d 204d torization, -- M │ │ │ │ -00032300: 6170 7320 696e 2061 2068 6967 6865 7220 aps in a higher │ │ │ │ -00032310: 636f 6469 6d65 6e73 696f 6e20 6d61 7472 codimension matr │ │ │ │ -00032320: 6978 2066 6163 746f 7269 7a61 7469 6f6e ix factorization │ │ │ │ -00032330: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -00032340: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -00032350: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -00032360: 6563 7420 2a6e 6f74 6520 4c61 7965 7265 ect *note Layere │ │ │ │ -00032370: 643a 204c 6179 6572 6564 2c20 6973 2061 d: Layered, is a │ │ │ │ -00032380: 202a 6e6f 7465 2073 796d 626f 6c3a 2028 *note symbol: ( │ │ │ │ -00032390: 4d61 6361 756c 6179 3244 6f63 2953 796d Macaulay2Doc)Sym │ │ │ │ -000323a0: 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d bol,...--------- │ │ │ │ +00032280: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d ========.. * "m │ │ │ │ +00032290: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ +000322a0: 6f6e 282e 2e2e 2c4c 6179 6572 6564 3d3e on(...,Layered=> │ │ │ │ +000322b0: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f ...)" -- see *no │ │ │ │ +000322c0: 7465 206d 6174 7269 7846 6163 746f 7269 te matrixFactori │ │ │ │ +000322d0: 7a61 7469 6f6e 3a0a 2020 2020 6d61 7472 zation:. matr │ │ │ │ +000322e0: 6978 4661 6374 6f72 697a 6174 696f 6e2c ixFactorization, │ │ │ │ +000322f0: 202d 2d20 4d61 7073 2069 6e20 6120 6869 -- Maps in a hi │ │ │ │ +00032300: 6768 6572 2063 6f64 696d 656e 7369 6f6e gher codimension │ │ │ │ +00032310: 206d 6174 7269 7820 6661 6374 6f72 697a matrix factoriz │ │ │ │ +00032320: 6174 696f 6e0a 0a46 6f72 2074 6865 2070 ation..For the p │ │ │ │ +00032330: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +00032340: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +00032350: 6520 6f62 6a65 6374 202a 6e6f 7465 204c e object *note L │ │ │ │ +00032360: 6179 6572 6564 3a20 4c61 7965 7265 642c ayered: Layered, │ │ │ │ +00032370: 2069 7320 6120 2a6e 6f74 6520 7379 6d62 is a *note symb │ │ │ │ +00032380: 6f6c 3a20 284d 6163 6175 6c61 7932 446f ol: (Macaulay2Do │ │ │ │ +00032390: 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d c)Symbol,...---- │ │ │ │ +000323a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000323b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000323c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000323d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000323e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000323f0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -00032400: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -00032410: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00032420: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00032430: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00032440: 362e 3035 2b64 732f 4d32 2f4d 6163 6175 6.05+ds/M2/Macau │ │ │ │ -00032450: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ -00032460: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -00032470: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d ionResolutions.m │ │ │ │ -00032480: 323a 3332 3531 3a30 2e0a 1f0a 4669 6c65 2:3251:0....File │ │ │ │ -00032490: 3a20 436f 6d70 6c65 7465 496e 7465 7273 : CompleteInters │ │ │ │ -000324a0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -000324b0: 732e 696e 666f 2c20 4e6f 6465 3a20 6c61 s.info, Node: la │ │ │ │ -000324c0: 7965 7265 6452 6573 6f6c 7574 696f 6e2c yeredResolution, │ │ │ │ -000324d0: 204e 6578 743a 204c 6966 742c 2050 7265 Next: Lift, Pre │ │ │ │ -000324e0: 763a 204c 6179 6572 6564 2c20 5570 3a20 v: Layered, Up: │ │ │ │ -000324f0: 546f 700a 0a6c 6179 6572 6564 5265 736f Top..layeredReso │ │ │ │ -00032500: 6c75 7469 6f6e 202d 2d20 6c61 7965 7265 lution -- layere │ │ │ │ -00032510: 6420 6669 6e69 7465 2061 6e64 2069 6e66 d finite and inf │ │ │ │ -00032520: 696e 6974 6520 6c61 7965 7265 6420 7265 inite layered re │ │ │ │ -00032530: 736f 6c75 7469 6f6e 7320 6f66 2043 4d20 solutions of CM │ │ │ │ -00032540: 6d6f 6475 6c65 730a 2a2a 2a2a 2a2a 2a2a modules.******** │ │ │ │ +000323e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +000323f0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +00032400: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +00032410: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +00032420: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00032430: 322d 312e 3236 2e30 352b 6473 2f4d 322f 2-1.26.05+ds/M2/ │ │ │ │ +00032440: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +00032450: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ +00032460: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +00032470: 6f6e 732e 6d32 3a33 3235 313a 302e 0a1f ons.m2:3251:0... │ │ │ │ +00032480: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ +00032490: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +000324a0: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ +000324b0: 653a 206c 6179 6572 6564 5265 736f 6c75 e: layeredResolu │ │ │ │ +000324c0: 7469 6f6e 2c20 4e65 7874 3a20 4c69 6674 tion, Next: Lift │ │ │ │ +000324d0: 2c20 5072 6576 3a20 4c61 7965 7265 642c , Prev: Layered, │ │ │ │ +000324e0: 2055 703a 2054 6f70 0a0a 6c61 7965 7265 Up: Top..layere │ │ │ │ +000324f0: 6452 6573 6f6c 7574 696f 6e20 2d2d 206c dResolution -- l │ │ │ │ +00032500: 6179 6572 6564 2066 696e 6974 6520 616e ayered finite an │ │ │ │ +00032510: 6420 696e 6669 6e69 7465 206c 6179 6572 d infinite layer │ │ │ │ +00032520: 6564 2072 6573 6f6c 7574 696f 6e73 206f ed resolutions o │ │ │ │ +00032530: 6620 434d 206d 6f64 756c 6573 0a2a 2a2a f CM modules.*** │ │ │ │ +00032540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00032550: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00032560: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00032570: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00032580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00032590: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -000325a0: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -000325b0: 2846 462c 2061 7567 2920 3d20 6c61 7965 (FF, aug) = laye │ │ │ │ -000325c0: 7265 6452 6573 6f6c 7574 696f 6e28 6666 redResolution(ff │ │ │ │ -000325d0: 2c4d 290a 2020 2020 2020 2020 2846 462c ,M). (FF, │ │ │ │ -000325e0: 2061 7567 2920 3d20 6c61 7965 7265 6452 aug) = layeredR │ │ │ │ -000325f0: 6573 6f6c 7574 696f 6e28 6666 2c4d 2c6c esolution(ff,M,l │ │ │ │ -00032600: 656e 290a 2020 2a20 496e 7075 7473 3a0a en). * Inputs:. │ │ │ │ -00032610: 2020 2020 2020 2a20 6666 2c20 6120 2a6e * ff, a *n │ │ │ │ -00032620: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ -00032630: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ -00032640: 2c2c 2031 2078 2063 206d 6174 7269 7820 ,, 1 x c matrix │ │ │ │ -00032650: 7768 6f73 6520 656e 7472 6965 730a 2020 whose entries. │ │ │ │ -00032660: 2020 2020 2020 6172 6520 6120 7265 6775 are a regu │ │ │ │ -00032670: 6c61 7220 7365 7175 656e 6365 2069 6e20 lar sequence in │ │ │ │ -00032680: 7468 6520 476f 7265 6e73 7465 696e 2072 the Gorenstein r │ │ │ │ -00032690: 696e 6720 530a 2020 2020 2020 2a20 4d2c ing S. * M, │ │ │ │ -000326a0: 2061 202a 6e6f 7465 206d 6f64 756c 653a a *note module: │ │ │ │ -000326b0: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ -000326c0: 6f64 756c 652c 2c20 4d43 4d20 6d6f 6475 odule,, MCM modu │ │ │ │ -000326d0: 6c65 206f 7665 7220 522c 0a20 2020 2020 le over R,. │ │ │ │ -000326e0: 2020 2072 6570 7265 7365 6e74 6564 2061 represented a │ │ │ │ -000326f0: 7320 616e 2053 2d6d 6f64 756c 6520 696e s an S-module in │ │ │ │ -00032700: 2074 6865 2066 6972 7374 2063 6173 6520 the first case │ │ │ │ -00032710: 616e 6420 6173 2061 6e20 522d 6d6f 6475 and as an R-modu │ │ │ │ -00032720: 6c65 2069 6e20 7468 650a 2020 2020 2020 le in the. │ │ │ │ -00032730: 2020 7365 636f 6e64 0a20 2020 2020 202a second. * │ │ │ │ -00032740: 206c 656e 2c20 616e 202a 6e6f 7465 2069 len, an *note i │ │ │ │ -00032750: 6e74 6567 6572 3a20 284d 6163 6175 6c61 nteger: (Macaula │ │ │ │ -00032760: 7932 446f 6329 5a5a 2c2c 206c 656e 6774 y2Doc)ZZ,, lengt │ │ │ │ -00032770: 6820 6f66 2074 6865 2073 6567 6d65 6e74 h of the segment │ │ │ │ -00032780: 206f 6620 7468 650a 2020 2020 2020 2020 of the. │ │ │ │ -00032790: 7265 736f 6c75 7469 6f6e 2074 6f20 6265 resolution to be │ │ │ │ -000327a0: 2063 6f6d 7075 7465 6420 6f76 6572 2052 computed over R │ │ │ │ -000327b0: 2c20 696e 2074 6865 2073 6563 6f6e 6420 , in the second │ │ │ │ -000327c0: 666f 726d 2e0a 2020 2a20 2a6e 6f74 6520 form.. * *note │ │ │ │ -000327d0: 4f70 7469 6f6e 616c 2069 6e70 7574 733a Optional inputs: │ │ │ │ -000327e0: 2028 4d61 6361 756c 6179 3244 6f63 2975 (Macaulay2Doc)u │ │ │ │ -000327f0: 7369 6e67 2066 756e 6374 696f 6e73 2077 sing functions w │ │ │ │ -00032800: 6974 6820 6f70 7469 6f6e 616c 2069 6e70 ith optional inp │ │ │ │ -00032810: 7574 732c 3a0a 2020 2020 2020 2a20 4368 uts,:. * Ch │ │ │ │ -00032820: 6563 6b20 3d3e 202e 2e2e 2c20 6465 6661 eck => ..., defa │ │ │ │ -00032830: 756c 7420 7661 6c75 6520 6661 6c73 650a ult value false. │ │ │ │ -00032840: 2020 2020 2020 2a20 5665 7262 6f73 6520 * Verbose │ │ │ │ -00032850: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ -00032860: 7661 6c75 6520 6661 6c73 650a 2020 2a20 value false. * │ │ │ │ -00032870: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -00032880: 2046 462c 2061 202a 6e6f 7465 2063 6f6d FF, a *note com │ │ │ │ -00032890: 706c 6578 3a20 2843 6f6d 706c 6578 6573 plex: (Complexes │ │ │ │ -000328a0: 2943 6f6d 706c 6578 2c2c 2072 6573 6f6c )Complex,, resol │ │ │ │ -000328b0: 7574 696f 6e20 6f66 204d 206f 7665 7220 ution of M over │ │ │ │ -000328c0: 5320 696e 2074 6865 0a20 2020 2020 2020 S in the. │ │ │ │ -000328d0: 2066 6972 7374 2063 6173 653b 206c 656e first case; len │ │ │ │ -000328e0: 6774 6820 6c65 6e20 7365 676d 656e 7420 gth len segment │ │ │ │ -000328f0: 6f66 2074 6865 2072 6573 6f6c 7574 696f of the resolutio │ │ │ │ -00032900: 6e20 6f76 6572 2052 2069 6e20 7468 6520 n over R in the │ │ │ │ -00032910: 7365 636f 6e64 2e0a 0a44 6573 6372 6970 second...Descrip │ │ │ │ -00032920: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -00032930: 0a0a 5468 6520 7265 736f 6c75 7469 6f6e ..The resolution │ │ │ │ -00032940: 7320 636f 6d70 7574 6564 2061 7265 2074 s computed are t │ │ │ │ -00032950: 686f 7365 2064 6573 6372 6962 6564 2069 hose described i │ │ │ │ -00032960: 6e20 7468 6520 7061 7065 7220 224c 6179 n the paper "Lay │ │ │ │ -00032970: 6572 6564 2052 6573 6f6c 7574 696f 6e73 ered Resolutions │ │ │ │ -00032980: 0a6f 6620 436f 6865 6e2d 4d61 6361 756c .of Cohen-Macaul │ │ │ │ -00032990: 6179 206d 6f64 756c 6573 2220 6279 2045 ay modules" by E │ │ │ │ -000329a0: 6973 656e 6275 6420 616e 6420 5065 6576 isenbud and Peev │ │ │ │ -000329b0: 612e 2054 6865 7920 6172 6520 626f 7468 a. They are both │ │ │ │ -000329c0: 206d 696e 696d 616c 2077 6865 6e20 4d0a minimal when M. │ │ │ │ -000329d0: 6973 2061 2073 7566 6669 6369 656e 746c is a sufficientl │ │ │ │ -000329e0: 7920 6869 6768 2073 797a 7967 7920 6f66 y high syzygy of │ │ │ │ -000329f0: 2061 206d 6f64 756c 6520 4e2e 2049 6620 a module N. If │ │ │ │ -00032a00: 7468 6520 6f70 7469 6f6e 2056 6572 626f the option Verbo │ │ │ │ -00032a10: 7365 3d3e 7472 7565 2069 730a 7365 742c se=>true is.set, │ │ │ │ -00032a20: 2074 6865 6e20 2869 6e20 7468 6520 6361 then (in the ca │ │ │ │ -00032a30: 7365 206f 6620 7468 6520 7265 736f 6c75 se of the resolu │ │ │ │ -00032a40: 7469 6f6e 206f 7665 7220 5329 2074 6865 tion over S) the │ │ │ │ -00032a50: 2072 616e 6b73 206f 6620 7468 6520 6d6f ranks of the mo │ │ │ │ -00032a60: 6475 6c65 7320 425f 730a 696e 2074 6865 dules B_s.in the │ │ │ │ -00032a70: 2072 6573 6f6c 7574 696f 6e20 6172 6520 resolution are │ │ │ │ -00032a80: 6f75 7470 7574 2e0a 0a48 6572 6520 6973 output...Here is │ │ │ │ -00032a90: 2061 6e20 6578 616d 706c 6520 636f 6d70 an example comp │ │ │ │ -00032aa0: 7574 696e 6720 3520 7465 726d 7320 6f66 uting 5 terms of │ │ │ │ -00032ab0: 2061 6e20 696e 6669 6e69 7465 2072 6573 an infinite res │ │ │ │ -00032ac0: 6f6c 7574 696f 6e3a 0a0a 2b2d 2d2d 2d2d olution:..+----- │ │ │ │ +00032580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +00032590: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +000325a0: 2020 2020 2028 4646 2c20 6175 6729 203d (FF, aug) = │ │ │ │ +000325b0: 206c 6179 6572 6564 5265 736f 6c75 7469 layeredResoluti │ │ │ │ +000325c0: 6f6e 2866 662c 4d29 0a20 2020 2020 2020 on(ff,M). │ │ │ │ +000325d0: 2028 4646 2c20 6175 6729 203d 206c 6179 (FF, aug) = lay │ │ │ │ +000325e0: 6572 6564 5265 736f 6c75 7469 6f6e 2866 eredResolution(f │ │ │ │ +000325f0: 662c 4d2c 6c65 6e29 0a20 202a 2049 6e70 f,M,len). * Inp │ │ │ │ +00032600: 7574 733a 0a20 2020 2020 202a 2066 662c uts:. * ff, │ │ │ │ +00032610: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ +00032620: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +00032630: 6174 7269 782c 2c20 3120 7820 6320 6d61 atrix,, 1 x c ma │ │ │ │ +00032640: 7472 6978 2077 686f 7365 2065 6e74 7269 trix whose entri │ │ │ │ +00032650: 6573 0a20 2020 2020 2020 2061 7265 2061 es. are a │ │ │ │ +00032660: 2072 6567 756c 6172 2073 6571 7565 6e63 regular sequenc │ │ │ │ +00032670: 6520 696e 2074 6865 2047 6f72 656e 7374 e in the Gorenst │ │ │ │ +00032680: 6569 6e20 7269 6e67 2053 0a20 2020 2020 ein ring S. │ │ │ │ +00032690: 202a 204d 2c20 6120 2a6e 6f74 6520 6d6f * M, a *note mo │ │ │ │ +000326a0: 6475 6c65 3a20 284d 6163 6175 6c61 7932 dule: (Macaulay2 │ │ │ │ +000326b0: 446f 6329 4d6f 6475 6c65 2c2c 204d 434d Doc)Module,, MCM │ │ │ │ +000326c0: 206d 6f64 756c 6520 6f76 6572 2052 2c0a module over R,. │ │ │ │ +000326d0: 2020 2020 2020 2020 7265 7072 6573 656e represen │ │ │ │ +000326e0: 7465 6420 6173 2061 6e20 532d 6d6f 6475 ted as an S-modu │ │ │ │ +000326f0: 6c65 2069 6e20 7468 6520 6669 7273 7420 le in the first │ │ │ │ +00032700: 6361 7365 2061 6e64 2061 7320 616e 2052 case and as an R │ │ │ │ +00032710: 2d6d 6f64 756c 6520 696e 2074 6865 0a20 -module in the. │ │ │ │ +00032720: 2020 2020 2020 2073 6563 6f6e 640a 2020 second. │ │ │ │ +00032730: 2020 2020 2a20 6c65 6e2c 2061 6e20 2a6e * len, an *n │ │ │ │ +00032740: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ +00032750: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ +00032760: 6c65 6e67 7468 206f 6620 7468 6520 7365 length of the se │ │ │ │ +00032770: 676d 656e 7420 6f66 2074 6865 0a20 2020 gment of the. │ │ │ │ +00032780: 2020 2020 2072 6573 6f6c 7574 696f 6e20 resolution │ │ │ │ +00032790: 746f 2062 6520 636f 6d70 7574 6564 206f to be computed o │ │ │ │ +000327a0: 7665 7220 522c 2069 6e20 7468 6520 7365 ver R, in the se │ │ │ │ +000327b0: 636f 6e64 2066 6f72 6d2e 0a20 202a 202a cond form.. * * │ │ │ │ +000327c0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e note Optional in │ │ │ │ +000327d0: 7075 7473 3a20 284d 6163 6175 6c61 7932 puts: (Macaulay2 │ │ │ │ +000327e0: 446f 6329 7573 696e 6720 6675 6e63 7469 Doc)using functi │ │ │ │ +000327f0: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ +00032800: 6c20 696e 7075 7473 2c3a 0a20 2020 2020 l inputs,:. │ │ │ │ +00032810: 202a 2043 6865 636b 203d 3e20 2e2e 2e2c * Check => ..., │ │ │ │ +00032820: 2064 6566 6175 6c74 2076 616c 7565 2066 default value f │ │ │ │ +00032830: 616c 7365 0a20 2020 2020 202a 2056 6572 alse. * Ver │ │ │ │ +00032840: 626f 7365 203d 3e20 2e2e 2e2c 2064 6566 bose => ..., def │ │ │ │ +00032850: 6175 6c74 2076 616c 7565 2066 616c 7365 ault value false │ │ │ │ +00032860: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +00032870: 2020 2020 2a20 4646 2c20 6120 2a6e 6f74 * FF, a *not │ │ │ │ +00032880: 6520 636f 6d70 6c65 783a 2028 436f 6d70 e complex: (Comp │ │ │ │ +00032890: 6c65 7865 7329 436f 6d70 6c65 782c 2c20 lexes)Complex,, │ │ │ │ +000328a0: 7265 736f 6c75 7469 6f6e 206f 6620 4d20 resolution of M │ │ │ │ +000328b0: 6f76 6572 2053 2069 6e20 7468 650a 2020 over S in the. │ │ │ │ +000328c0: 2020 2020 2020 6669 7273 7420 6361 7365 first case │ │ │ │ +000328d0: 3b20 6c65 6e67 7468 206c 656e 2073 6567 ; length len seg │ │ │ │ +000328e0: 6d65 6e74 206f 6620 7468 6520 7265 736f ment of the reso │ │ │ │ +000328f0: 6c75 7469 6f6e 206f 7665 7220 5220 696e lution over R in │ │ │ │ +00032900: 2074 6865 2073 6563 6f6e 642e 0a0a 4465 the second...De │ │ │ │ +00032910: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +00032920: 3d3d 3d3d 3d0a 0a54 6865 2072 6573 6f6c =====..The resol │ │ │ │ +00032930: 7574 696f 6e73 2063 6f6d 7075 7465 6420 utions computed │ │ │ │ +00032940: 6172 6520 7468 6f73 6520 6465 7363 7269 are those descri │ │ │ │ +00032950: 6265 6420 696e 2074 6865 2070 6170 6572 bed in the paper │ │ │ │ +00032960: 2022 4c61 7965 7265 6420 5265 736f 6c75 "Layered Resolu │ │ │ │ +00032970: 7469 6f6e 730a 6f66 2043 6f68 656e 2d4d tions.of Cohen-M │ │ │ │ +00032980: 6163 6175 6c61 7920 6d6f 6475 6c65 7322 acaulay modules" │ │ │ │ +00032990: 2062 7920 4569 7365 6e62 7564 2061 6e64 by Eisenbud and │ │ │ │ +000329a0: 2050 6565 7661 2e20 5468 6579 2061 7265 Peeva. They are │ │ │ │ +000329b0: 2062 6f74 6820 6d69 6e69 6d61 6c20 7768 both minimal wh │ │ │ │ +000329c0: 656e 204d 0a69 7320 6120 7375 6666 6963 en M.is a suffic │ │ │ │ +000329d0: 6965 6e74 6c79 2068 6967 6820 7379 7a79 iently high syzy │ │ │ │ +000329e0: 6779 206f 6620 6120 6d6f 6475 6c65 204e gy of a module N │ │ │ │ +000329f0: 2e20 4966 2074 6865 206f 7074 696f 6e20 . If the option │ │ │ │ +00032a00: 5665 7262 6f73 653d 3e74 7275 6520 6973 Verbose=>true is │ │ │ │ +00032a10: 0a73 6574 2c20 7468 656e 2028 696e 2074 .set, then (in t │ │ │ │ +00032a20: 6865 2063 6173 6520 6f66 2074 6865 2072 he case of the r │ │ │ │ +00032a30: 6573 6f6c 7574 696f 6e20 6f76 6572 2053 esolution over S │ │ │ │ +00032a40: 2920 7468 6520 7261 6e6b 7320 6f66 2074 ) the ranks of t │ │ │ │ +00032a50: 6865 206d 6f64 756c 6573 2042 5f73 0a69 he modules B_s.i │ │ │ │ +00032a60: 6e20 7468 6520 7265 736f 6c75 7469 6f6e n the resolution │ │ │ │ +00032a70: 2061 7265 206f 7574 7075 742e 0a0a 4865 are output...He │ │ │ │ +00032a80: 7265 2069 7320 616e 2065 7861 6d70 6c65 re is an example │ │ │ │ +00032a90: 2063 6f6d 7075 7469 6e67 2035 2074 6572 computing 5 ter │ │ │ │ +00032aa0: 6d73 206f 6620 616e 2069 6e66 696e 6974 ms of an infinit │ │ │ │ +00032ab0: 6520 7265 736f 6c75 7469 6f6e 3a0a 0a2b e resolution:..+ │ │ │ │ +00032ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00032b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00032b10: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -00032b20: 5320 3d20 5a5a 2f31 3031 5b61 2c62 2c63 S = ZZ/101[a,b,c │ │ │ │ -00032b30: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +00032b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00032b10: 6931 203a 2053 203d 205a 5a2f 3130 315b i1 : S = ZZ/101[ │ │ │ │ +00032b20: 612c 622c 635d 2020 2020 2020 2020 2020 a,b,c] │ │ │ │ +00032b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032b60: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00032b50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032bb0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -00032bc0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +00032ba0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032bb0: 6f31 203d 2053 2020 2020 2020 2020 2020 o1 = S │ │ │ │ +00032bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032c00: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00032bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032c50: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ -00032c60: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ +00032c40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032c50: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52 o1 : PolynomialR │ │ │ │ +00032c60: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ 00032c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032ca0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00032c90: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00032ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00032ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00032cf0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -00032d00: 6666 203d 206d 6174 7269 7822 6133 2c20 ff = matrix"a3, │ │ │ │ -00032d10: 6233 2c20 6333 2220 2020 2020 2020 2020 b3, c3" │ │ │ │ +00032ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00032cf0: 6932 203a 2066 6620 3d20 6d61 7472 6978 i2 : ff = matrix │ │ │ │ +00032d00: 2261 332c 2062 332c 2063 3322 2020 2020 "a3, b3, c3" │ │ │ │ +00032d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032d40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00032d30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032d90: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ -00032da0: 7c20 6133 2062 3320 6333 207c 2020 2020 | a3 b3 c3 | │ │ │ │ +00032d80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032d90: 6f32 203d 207c 2061 3320 6233 2063 3320 o2 = | a3 b3 c3 │ │ │ │ +00032da0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00032db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032de0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00032dd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00032e40: 2020 2020 2020 2020 3120 2020 2020 2033 1 3 │ │ │ │ +00032e20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032e30: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ +00032e40: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ 00032e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032e80: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ -00032e90: 4d61 7472 6978 2053 2020 3c2d 2d20 5320 Matrix S <-- S │ │ │ │ +00032e70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032e80: 6f32 203a 204d 6174 7269 7820 5320 203c o2 : Matrix S < │ │ │ │ +00032e90: 2d2d 2053 2020 2020 2020 2020 2020 2020 -- S │ │ │ │ 00032ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032ed0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00032ec0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00032ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00032f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00032f20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ -00032f30: 5220 3d20 532f 6964 6561 6c20 6666 2020 R = S/ideal ff │ │ │ │ +00032f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00032f20: 6933 203a 2052 203d 2053 2f69 6465 616c i3 : R = S/ideal │ │ │ │ +00032f30: 2066 6620 2020 2020 2020 2020 2020 2020 ff │ │ │ │ 00032f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032f70: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00032f60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032fc0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -00032fd0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +00032fb0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032fc0: 6f33 203d 2052 2020 2020 2020 2020 2020 o3 = R │ │ │ │ +00032fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033010: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00033000: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00033010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033060: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ -00033070: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +00033050: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00033060: 6f33 203a 2051 756f 7469 656e 7452 696e o3 : QuotientRin │ │ │ │ +00033070: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ 00033080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000330a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000330b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000330a0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000330b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000330c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000330d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000330e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000330f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00033100: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ -00033110: 4d20 3d20 7379 7a79 6779 4d6f 6475 6c65 M = syzygyModule │ │ │ │ -00033120: 2832 2c63 6f6b 6572 2076 6172 7320 5229 (2,coker vars R) │ │ │ │ +000330f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00033100: 6934 203a 204d 203d 2073 797a 7967 794d i4 : M = syzygyM │ │ │ │ +00033110: 6f64 756c 6528 322c 636f 6b65 7220 7661 odule(2,coker va │ │ │ │ +00033120: 7273 2052 2920 2020 2020 2020 2020 2020 rs R) │ │ │ │ 00033130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033150: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00033140: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00033150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000331a0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ -000331b0: 636f 6b65 726e 656c 207b 327d 207c 2061 cokernel {2} | a │ │ │ │ -000331c0: 2020 3020 2d63 3220 3020 2020 6232 2030 0 -c2 0 b2 0 │ │ │ │ -000331d0: 2030 2020 2030 2020 3020 2030 207c 2020 0 0 0 0 | │ │ │ │ -000331e0: 2020 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7c0a 7c74 6f74 616c |.|total │ │ │ │ -00033570: 3a20 3420 3420 3420 3420 3420 3420 2020 : 4 4 4 4 4 4 │ │ │ │ +00033550: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00033560: 746f 7461 6c3a 2034 2034 2034 2034 2034 total: 4 4 4 4 4 │ │ │ │ +00033570: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ 00033580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000335a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000335b0: 2020 2020 2020 2020 7c0a 7c20 2020 2032 |.| 2 │ │ │ │ -000335c0: 3a20 3320 3120 2e20 2e20 2e20 2e20 2020 : 3 1 . . . . │ │ │ │ +000335a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000335b0: 2020 2020 323a 2033 2031 202e 202e 202e 2: 3 1 . . . │ │ │ │ +000335c0: 202e 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ 000335d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000335e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000335f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 │ │ │ │ -000336a0: 2020 2020 2020 2020 7c0a 7c20 2020 2035 |.| 5 │ │ │ │ -000336b0: 3a20 2e20 2e20 2e20 2e20 3120 3320 2020 : . . . . 1 3 │ │ │ │ +00033690: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000336a0: 2020 2020 353a 202e 202e 202e 202e 2031 5: . . . . 1 │ │ │ │ +000336b0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 000336c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000336d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000336e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000336f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00033700: 2020 3020 3120 3220 2033 2020 3420 2035 0 1 2 3 4 5 │ │ │ │ +000336e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000336f0: 2020 2020 2020 2030 2031 2032 2020 3320 0 1 2 3 │ │ │ │ +00033700: 2034 2020 3520 2020 2020 2020 2020 2020 4 5 │ │ │ │ 00033710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033740: 2020 2020 2020 2020 7c0a 7c74 6f74 616c |.|total │ │ │ │ -00033750: 3a20 3520 3720 3920 3131 2031 3320 3135 : 5 7 9 11 13 15 │ │ │ │ +00033730: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00033740: 746f 7461 6c3a 2035 2037 2039 2031 3120 total: 5 7 9 11 │ │ │ │ +00033750: 3133 2031 3520 2020 2020 2020 2020 2020 13 15 │ │ │ │ 00033760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033790: 2020 2020 2020 2020 7c0a 7c20 2020 2032 |.| 2 │ │ │ │ -000337a0: 3a20 3320 3120 2e20 202e 2020 2e20 202e : 3 1 . . . . │ │ │ │ +00033780: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00033790: 2020 2020 323a 2033 2031 202e 2020 2e20 2: 3 1 . . │ │ │ │ +000337a0: 202e 2020 2e20 2020 2020 2020 2020 2020 . . │ │ │ │ 000337b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000337c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 2020 │ │ │ │ 00036410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036440: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00036430: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00036440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036490: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a --------+.|i10 : │ │ │ │ -000364a0: 2061 7070 6c79 2835 2c20 6a2d 3e20 7072 apply(5, j-> pr │ │ │ │ -000364b0: 756e 6520 4848 5f6a 2043 203d 3d20 3029 une HH_j C == 0) │ │ │ │ +00036480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00036490: 6931 3020 3a20 6170 706c 7928 352c 206a i10 : apply(5, j │ │ │ │ +000364a0: 2d3e 2070 7275 6e65 2048 485f 6a20 4320 -> prune HH_j C │ │ │ │ +000364b0: 3d3d 2030 2920 2020 2020 2020 2020 2020 == 0) │ │ │ │ 000364c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000364d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000364e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000364d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000364e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000364f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036530: 2020 2020 2020 2020 7c0a 7c6f 3130 203d |.|o10 = │ │ │ │ -00036540: 207b 7472 7565 2c20 6661 6c73 652c 2066 {true, false, f │ │ │ │ -00036550: 616c 7365 2c20 6661 6c73 652c 2066 616c alse, false, fal │ │ │ │ -00036560: 7365 7d20 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+000365c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000365d0: 6f31 3020 3a20 4c69 7374 2020 2020 2020 o10 : List │ │ │ │ +000365e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000365f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036620: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00036610: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00036620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036670: 2d2d 2d2d 2d2d 2d2d 2b0a 0a41 6e64 206f --------+..And o │ │ │ │ -00036680: 6e65 2063 6f6d 7075 7469 6e67 2074 6865 ne computing the │ │ │ │ -00036690: 2077 686f 6c65 2066 696e 6974 6520 7265 whole finite re │ │ │ │ -000366a0: 736f 6c75 7469 6f6e 3a0a 0a2b 2d2d 2d2d solution:..+---- │ │ │ │ +00036660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +00036670: 416e 6420 6f6e 6520 636f 6d70 7574 696e And one computin │ │ │ │ +00036680: 6720 7468 6520 7768 6f6c 6520 6669 6e69 g the whole fini │ │ │ │ +00036690: 7465 2072 6573 6f6c 7574 696f 6e3a 0a0a te resolution:.. │ │ │ │ +000366a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000366b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000366c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000366d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000366e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000366f0: 2d2d 2d2b 0a7c 6931 3120 3a20 4d53 203d ---+.|i11 : MS = │ │ │ │ -00036700: 2070 7573 6846 6f72 7761 7264 286d 6170 pushForward(map │ │ │ │ -00036710: 2852 2c53 292c 204d 293b 2020 2020 2020 (R,S), M); │ │ │ │ +000366e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a --------+.|i11 : │ │ │ │ +000366f0: 204d 5320 3d20 7075 7368 466f 7277 6172 MS = pushForwar │ │ │ │ +00036700: 6428 6d61 7028 522c 5329 2c20 4d29 3b20 d(map(R,S), M); │ │ │ │ +00036710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036730: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00036730: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00036740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036780: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 -------+.|i12 : │ │ │ │ -00036790: 2847 472c 2061 7567 2920 3d20 6c61 7965 (GG, aug) = laye │ │ │ │ -000367a0: 7265 6452 6573 6f6c 7574 696f 6e28 6666 redResolution(ff │ │ │ │ -000367b0: 2c4d 5329 2020 2020 2020 2020 2020 2020 ,MS) │ │ │ │ -000367c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000367d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00036770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00036780: 3132 203a 2028 4747 2c20 6175 6729 203d 12 : (GG, aug) = │ │ │ │ +00036790: 206c 6179 6572 6564 5265 736f 6c75 7469 layeredResoluti │ │ │ │ +000367a0: 6f6e 2866 662c 4d53 2920 2020 2020 2020 on(ff,MS) │ │ │ │ +000367b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000367c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000367d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000367e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000367f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036810: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00036820: 2020 2020 2020 3620 2020 2020 2031 3320 6 13 │ │ │ │ -00036830: 2020 2020 2031 3020 2020 2020 2033 2020 10 3 │ │ │ │ +00036810: 7c0a 7c20 2020 2020 2020 2036 2020 2020 |.| 6 │ │ │ │ +00036820: 2020 3133 2020 2020 2020 3130 2020 2020 13 10 │ │ │ │ +00036830: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00036840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036860: 2020 2020 207c 0a7c 6f31 3220 3d20 2853 |.|o12 = (S │ │ │ │ -00036870: 2020 3c2d 2d20 5320 2020 3c2d 2d20 5320 <-- S <-- S │ │ │ │ -00036880: 2020 3c2d 2d20 5320 2c20 7b32 7d20 7c20 <-- S , {2} | │ │ │ │ -00036890: 3020 3020 3020 3020 2030 2020 3120 7c29 0 0 0 0 0 1 |) │ │ │ │ -000368a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000368b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000368c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000368d0: 2020 2020 7b32 7d20 7c20 3020 3020 3020 {2} | 0 0 0 │ │ │ │ -000368e0: 2d31 2030 2020 3020 7c20 2020 2020 2020 -1 0 0 | │ │ │ │ -000368f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00036900: 2020 2030 2020 2020 2020 3120 2020 2020 0 1 │ │ │ │ -00036910: 2020 3220 2020 2020 2020 3320 2020 7b32 2 3 {2 │ │ │ │ -00036920: 7d20 7c20 3020 3020 3020 3020 202d 3120 } | 0 0 0 0 -1 │ │ │ │ -00036930: 3020 7c20 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ -00036940: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00036950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036960: 2020 2020 2020 2020 7b33 7d20 7c20 3120 {3} | 1 │ │ │ │ -00036970: 3020 3020 3020 2030 2020 3020 7c20 2020 0 0 0 0 0 | │ │ │ │ -00036980: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00036850: 2020 2020 2020 2020 2020 7c0a 7c6f 3132 |.|o12 │ │ │ │ +00036860: 203d 2028 5320 203c 2d2d 2053 2020 203c = (S <-- S < │ │ │ │ +00036870: 2d2d 2053 2020 203c 2d2d 2053 202c 207b -- S <-- S , { │ │ │ │ +00036880: 327d 207c 2030 2030 2030 2030 2020 3020 2} | 0 0 0 0 0 │ │ │ │ +00036890: 2031 207c 2920 2020 2020 2020 2020 2020 1 |) │ │ │ │ +000368a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000368b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000368c0: 2020 2020 2020 2020 207b 327d 207c 2030 {2} | 0 │ │ │ │ +000368d0: 2030 2030 202d 3120 3020 2030 207c 2020 0 0 -1 0 0 | │ │ │ │ +000368e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000368f0: 7c20 2020 2020 2020 3020 2020 2020 2031 | 0 1 │ │ │ │ +00036900: 2020 2020 2020 2032 2020 2020 2020 2033 2 3 │ │ │ │ +00036910: 2020 207b 327d 207c 2030 2030 2030 2030 {2} | 0 0 0 0 │ │ │ │ +00036920: 2020 2d31 2030 207c 2020 2020 2020 2020 -1 0 | │ │ │ │ +00036930: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00036940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036950: 2020 2020 2020 2020 2020 2020 207b 337d {3} │ │ │ │ +00036960: 207c 2031 2030 2030 2030 2020 3020 2030 | 1 0 0 0 0 0 │ │ │ │ +00036970: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00036980: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00036990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000369a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000369b0: 2020 7b33 7d20 7c20 3020 3120 3020 3020 {3} | 0 1 0 0 │ │ │ │ -000369c0: 2030 2020 3020 7c20 2020 2020 2020 2020 0 0 | │ │ │ │ -000369d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000369a0: 2020 2020 2020 207b 337d 207c 2030 2031 {3} | 0 1 │ │ │ │ +000369b0: 2030 2030 2020 3020 2030 207c 2020 2020 0 0 0 0 | │ │ │ │ +000369c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000369d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000369e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000369f0: 2020 2020 2020 2020 2020 2020 7b33 7d20 {3} │ │ │ │ -00036a00: 7c20 3020 3020 3120 3020 2030 2020 3020 | 0 0 1 0 0 0 │ │ │ │ -00036a10: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00036a20: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000369f0: 207b 337d 207c 2030 2030 2031 2030 2020 {3} | 0 0 1 0 │ │ │ │ +00036a00: 3020 2030 207c 2020 2020 2020 2020 2020 0 0 | │ │ │ │ +00036a10: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00036a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036a60: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -00036a70: 3220 3a20 5365 7175 656e 6365 2020 2020 2 : Sequence │ │ │ │ +00036a60: 7c0a 7c6f 3132 203a 2053 6571 7565 6e63 |.|o12 : Sequenc │ │ │ │ +00036a70: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ 00036a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036ab0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00036aa0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00036ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00036b00: 0a7c 6931 3320 3a20 2847 472c 2061 7567 .|i13 : (GG, aug │ │ │ │ -00036b10: 2920 3d20 6c61 7965 7265 6452 6573 6f6c ) = layeredResol │ │ │ │ -00036b20: 7574 696f 6e28 6666 2c4d 532c 2056 6572 ution(ff,MS, Ver │ │ │ │ -00036b30: 626f 7365 203d 3e74 7275 6529 2020 2020 bose =>true) │ │ │ │ -00036b40: 2020 2020 2020 2020 207c 0a7c 7b33 2c20 |.|{3, │ │ │ │ -00036b50: 317d 2069 6e20 636f 6469 6d65 6e73 696f 1} in codimensio │ │ │ │ -00036b60: 6e20 3320 2020 2020 2020 2020 2020 2020 n 3 │ │ │ │ +00036af0: 2d2d 2d2d 2b0a 7c69 3133 203a 2028 4747 ----+.|i13 : (GG │ │ │ │ +00036b00: 2c20 6175 6729 203d 206c 6179 6572 6564 , aug) = layered │ │ │ │ +00036b10: 5265 736f 6c75 7469 6f6e 2866 662c 4d53 Resolution(ff,MS │ │ │ │ +00036b20: 2c20 5665 7262 6f73 6520 3d3e 7472 7565 , Verbose =>true │ │ │ │ +00036b30: 2920 2020 2020 2020 2020 2020 2020 7c0a ) |. │ │ │ │ +00036b40: 7c7b 332c 2031 7d20 696e 2063 6f64 696d |{3, 1} in codim │ │ │ │ +00036b50: 656e 7369 6f6e 2033 2020 2020 2020 2020 ension 3 │ │ │ │ +00036b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036b90: 2020 207c 0a7c 7b33 2c20 317d 2069 6e20 |.|{3, 1} in │ │ │ │ -00036ba0: 636f 6469 6d65 6e73 696f 6e20 3220 2020 codimension 2 │ │ │ │ +00036b80: 2020 2020 2020 2020 7c0a 7c7b 332c 2031 |.|{3, 1 │ │ │ │ +00036b90: 7d20 696e 2063 6f64 696d 656e 7369 6f6e } in codimension │ │ │ │ +00036ba0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00036bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036bd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00036bd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00036be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036c20: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00036c30: 2020 3620 2020 2020 2031 3320 2020 2020 6 13 │ │ │ │ -00036c40: 2031 3020 2020 2020 2033 2020 2020 2020 10 3 │ │ │ │ +00036c10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00036c20: 2020 2020 2020 2036 2020 2020 2020 3133 6 13 │ │ │ │ +00036c30: 2020 2020 2020 3130 2020 2020 2020 3320 10 3 │ │ │ │ +00036c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036c70: 207c 0a7c 6f31 3320 3d20 2853 2020 3c2d |.|o13 = (S <- │ │ │ │ -00036c80: 2d20 5320 2020 3c2d 2d20 5320 2020 3c2d - S <-- S <- │ │ │ │ -00036c90: 2d20 5320 2c20 7b32 7d20 7c20 3020 3020 - S , {2} | 0 0 │ │ │ │ -00036ca0: 3020 3020 2030 2020 3120 7c29 2020 2020 0 0 0 1 |) │ │ │ │ -00036cb0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00036c60: 2020 2020 2020 7c0a 7c6f 3133 203d 2028 |.|o13 = ( │ │ │ │ +00036c70: 5320 203c 2d2d 2053 2020 203c 2d2d 2053 S <-- S <-- S │ │ │ │ +00036c80: 2020 203c 2d2d 2053 202c 207b 327d 207c <-- S , {2} | │ │ │ │ +00036c90: 2030 2030 2030 2030 2020 3020 2031 207c 0 0 0 0 0 1 | │ │ │ │ +00036ca0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00036cb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00036cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036ce0: 7b32 7d20 7c20 3020 3020 3020 2d31 2030 {2} | 0 0 0 -1 0 │ │ │ │ -00036cf0: 2020 3020 7c20 2020 2020 2020 2020 2020 0 | │ │ │ │ -00036d00: 2020 2020 207c 0a7c 2020 2020 2020 2030 |.| 0 │ │ │ │ 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2020 S │ │ │ │ 00037790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000377a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000377b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000377a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000377b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000377c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000377d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000377e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000377f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037800: 207c 0a7c 2020 2020 2020 3020 2020 2020 |.| 0 │ │ │ │ -00037810: 2031 2020 2020 2020 3220 2020 2020 2020 1 2 │ │ │ │ -00037820: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +000377f0: 2020 2020 2020 7c0a 7c20 2020 2020 2030 |.| 0 │ │ │ │ +00037800: 2020 2020 2020 3120 2020 2020 2032 2020 1 2 │ │ │ │ +00037810: 2020 2020 2033 2020 2020 2020 2020 2020 3 │ │ │ │ +00037820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037830: 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j-> │ │ │ │ +00037940: 7072 756e 6520 4848 5f6a 2043 203d 3d20 prune HH_j C == │ │ │ │ +00037950: 3029 2020 2020 2020 2020 2020 2020 2020 0) │ │ │ │ +00037960: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00037970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000379a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000379b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000379c0: 6f31 3720 3d20 7b74 7275 652c 2074 7275 o17 = {true, tru │ │ │ │ -000379d0: 652c 2074 7275 652c 2066 616c 7365 7d20 e, true, false} │ │ │ │ +000379b0: 2020 7c0a 7c6f 3137 203d 207b 7472 7565 |.|o17 = {true │ │ │ │ +000379c0: 2c20 7472 7565 2c20 7472 7565 2c20 6661 , true, true, fa │ │ │ │ +000379d0: 6c73 657d 2020 2020 2020 2020 2020 2020 lse} │ │ │ │ 000379e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000379f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037a00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000379f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00037a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037a50: 207c 0a7c 6f31 3720 3a20 4c69 7374 2020 |.|o17 : List │ │ │ │ +00037a40: 2020 2020 2020 7c0a 7c6f 3137 203a 204c |.|o17 : L │ │ │ │ +00037a50: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ 00037a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037a90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00037a90: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00037aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00037ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00037ae0: 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 6f20 -----+..Ways to │ │ │ │ -00037af0: 7573 6520 6c61 7965 7265 6452 6573 6f6c use layeredResol │ │ │ │ -00037b00: 7574 696f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d ution:.========= │ │ │ │ -00037b10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00037b20: 3d3d 3d3d 3d0a 0a20 202a 2022 6c61 7965 =====.. * "laye │ │ │ │ -00037b30: 7265 6452 6573 6f6c 7574 696f 6e28 4d61 redResolution(Ma │ │ │ │ -00037b40: 7472 6978 2c4d 6f64 756c 6529 220a 2020 trix,Module)". │ │ │ │ -00037b50: 2a20 226c 6179 6572 6564 5265 736f 6c75 * "layeredResolu │ │ │ │ -00037b60: 7469 6f6e 284d 6174 7269 782c 4d6f 6475 tion(Matrix,Modu │ │ │ │ -00037b70: 6c65 2c5a 5a29 220a 0a46 6f72 2074 6865 le,ZZ)"..For the │ │ │ │ -00037b80: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00037b90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00037ba0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00037bb0: 206c 6179 6572 6564 5265 736f 6c75 7469 layeredResoluti │ │ │ │ -00037bc0: 6f6e 3a20 6c61 7965 7265 6452 6573 6f6c on: layeredResol │ │ │ │ -00037bd0: 7574 696f 6e2c 2069 7320 6120 2a6e 6f74 ution, is a *not │ │ │ │ -00037be0: 6520 6d65 7468 6f64 0a66 756e 6374 696f e method.functio │ │ │ │ -00037bf0: 6e20 7769 7468 206f 7074 696f 6e73 3a20 n with options: │ │ │ │ -00037c00: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -00037c10: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ -00037c20: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ +00037ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6179 ----------+..Way │ │ │ │ +00037ae0: 7320 746f 2075 7365 206c 6179 6572 6564 s to use layered │ │ │ │ +00037af0: 5265 736f 6c75 7469 6f6e 3a0a 3d3d 3d3d Resolution:.==== │ │ │ │ +00037b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00037b10: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00037b20: 226c 6179 6572 6564 5265 736f 6c75 7469 "layeredResoluti │ │ │ │ +00037b30: 6f6e 284d 6174 7269 782c 4d6f 6475 6c65 on(Matrix,Module │ │ │ │ +00037b40: 2922 0a20 202a 2022 6c61 7965 7265 6452 )". * "layeredR │ │ │ │ +00037b50: 6573 6f6c 7574 696f 6e28 4d61 7472 6978 esolution(Matrix │ │ │ │ +00037b60: 2c4d 6f64 756c 652c 5a5a 2922 0a0a 466f ,Module,ZZ)"..Fo │ │ │ │ +00037b70: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00037b80: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00037b90: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00037ba0: 2a6e 6f74 6520 6c61 7965 7265 6452 6573 *note layeredRes │ │ │ │ +00037bb0: 6f6c 7574 696f 6e3a 206c 6179 6572 6564 olution: layered │ │ │ │ +00037bc0: 5265 736f 6c75 7469 6f6e 2c20 6973 2061 Resolution, is a │ │ │ │ +00037bd0: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ +00037be0: 6e63 7469 6f6e 2077 6974 6820 6f70 7469 nction with opti │ │ │ │ +00037bf0: 6f6e 733a 2028 4d61 6361 756c 6179 3244 ons: (Macaulay2D │ │ │ │ +00037c00: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ +00037c10: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a nWithOptions,... │ │ │ │ +00037c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00037c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00037c70: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -00037c80: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -00037c90: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -00037ca0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -00037cb0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -00037cc0: 2d31 2e32 362e 3035 2b64 732f 4d32 2f4d -1.26.05+ds/M2/M │ │ │ │ -00037cd0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00037ce0: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ -00037cf0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -00037d00: 6e73 2e6d 323a 3438 3935 3a30 2e0a 1f0a ns.m2:4895:0.... │ │ │ │ -00037d10: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ -00037d20: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -00037d30: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ -00037d40: 3a20 4c69 6674 2c20 4e65 7874 3a20 6d61 : Lift, Next: ma │ │ │ │ -00037d50: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ -00037d60: 6f6e 2c20 5072 6576 3a20 6c61 7965 7265 on, Prev: layere │ │ │ │ -00037d70: 6452 6573 6f6c 7574 696f 6e2c 2055 703a dResolution, Up: │ │ │ │ -00037d80: 2054 6f70 0a0a 4c69 6674 202d 2d20 4f70 Top..Lift -- Op │ │ │ │ -00037d90: 7469 6f6e 2066 6f72 206e 6577 4578 740a tion for newExt. │ │ │ │ -00037da0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00037db0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -00037dc0: 7361 6765 3a20 0a20 2020 2020 2020 206e sage: . n │ │ │ │ -00037dd0: 6577 4578 7428 4d2c 4e2c 4368 6563 6b20 ewExt(M,N,Check │ │ │ │ -00037de0: 3d3e 7472 7565 290a 2020 2a20 496e 7075 =>true). * Inpu │ │ │ │ -00037df0: 7473 3a0a 2020 2020 2020 2a20 4368 6563 ts:. * Chec │ │ │ │ -00037e00: 6b2c 2061 202a 6e6f 7465 2042 6f6f 6c65 k, a *note Boole │ │ │ │ -00037e10: 616e 2076 616c 7565 3a20 284d 6163 6175 an value: (Macau │ │ │ │ -00037e20: 6c61 7932 446f 6329 426f 6f6c 6561 6e2c lay2Doc)Boolean, │ │ │ │ -00037e30: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a , ..Description. │ │ │ │ -00037e40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a4d 616b ===========..Mak │ │ │ │ -00037e50: 6573 206e 6577 4578 7420 7065 7266 6f72 es newExt perfor │ │ │ │ -00037e60: 6d20 7661 7269 6f75 7320 6368 6563 6b73 m various checks │ │ │ │ -00037e70: 2061 7320 6974 2063 6f6d 7075 7465 732e as it computes. │ │ │ │ -00037e80: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -00037e90: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206e ===.. * *note n │ │ │ │ -00037ea0: 6577 4578 743a 206e 6577 4578 742c 202d ewExt: newExt, - │ │ │ │ -00037eb0: 2d20 476c 6f62 616c 2045 7874 2066 6f72 - Global Ext for │ │ │ │ -00037ec0: 206d 6f64 756c 6573 206f 7665 7220 6120 modules over a │ │ │ │ -00037ed0: 636f 6d70 6c65 7465 0a20 2020 2049 6e74 complete. Int │ │ │ │ -00037ee0: 6572 7365 6374 696f 6e0a 0a46 756e 6374 ersection..Funct │ │ │ │ -00037ef0: 696f 6e73 2077 6974 6820 6f70 7469 6f6e ions with option │ │ │ │ -00037f00: 616c 2061 7267 756d 656e 7420 6e61 6d65 al argument name │ │ │ │ -00037f10: 6420 4c69 6674 3a0a 3d3d 3d3d 3d3d 3d3d d Lift:.======== │ │ │ │ +00037c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +00037c70: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +00037c80: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +00037c90: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +00037ca0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +00037cb0: 756c 6179 322d 312e 3236 2e30 352b 6473 ulay2-1.26.05+ds │ │ │ │ +00037cc0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +00037cd0: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ +00037ce0: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +00037cf0: 6c75 7469 6f6e 732e 6d32 3a34 3839 353a lutions.m2:4895: │ │ │ │ +00037d00: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ +00037d10: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +00037d20: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ +00037d30: 204e 6f64 653a 204c 6966 742c 204e 6578 Node: Lift, Nex │ │ │ │ +00037d40: 743a 206d 616b 6546 696e 6974 6552 6573 t: makeFiniteRes │ │ │ │ +00037d50: 6f6c 7574 696f 6e2c 2050 7265 763a 206c olution, Prev: l │ │ │ │ +00037d60: 6179 6572 6564 5265 736f 6c75 7469 6f6e ayeredResolution │ │ │ │ +00037d70: 2c20 5570 3a20 546f 700a 0a4c 6966 7420 , Up: Top..Lift │ │ │ │ +00037d80: 2d2d 204f 7074 696f 6e20 666f 7220 6e65 -- Option for ne │ │ │ │ +00037d90: 7745 7874 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a wExt.*********** │ │ │ │ +00037da0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +00037db0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +00037dc0: 2020 2020 6e65 7745 7874 284d 2c4e 2c43 newExt(M,N,C │ │ │ │ +00037dd0: 6865 636b 203d 3e74 7275 6529 0a20 202a heck =>true). * │ │ │ │ +00037de0: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +00037df0: 2043 6865 636b 2c20 6120 2a6e 6f74 6520 Check, a *note │ │ │ │ +00037e00: 426f 6f6c 6561 6e20 7661 6c75 653a 2028 Boolean value: ( │ │ │ │ +00037e10: 4d61 6361 756c 6179 3244 6f63 2942 6f6f Macaulay2Doc)Boo │ │ │ │ +00037e20: 6c65 616e 2c2c 200a 0a44 6573 6372 6970 lean,, ..Descrip │ │ │ │ +00037e30: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +00037e40: 0a0a 4d61 6b65 7320 6e65 7745 7874 2070 ..Makes newExt p │ │ │ │ +00037e50: 6572 666f 726d 2076 6172 696f 7573 2063 erform various c │ │ │ │ +00037e60: 6865 636b 7320 6173 2069 7420 636f 6d70 hecks as it comp │ │ │ │ +00037e70: 7574 6573 2e0a 0a53 6565 2061 6c73 6f0a utes...See also. │ │ │ │ +00037e80: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +00037e90: 6f74 6520 6e65 7745 7874 3a20 6e65 7745 ote newExt: newE │ │ │ │ +00037ea0: 7874 2c20 2d2d 2047 6c6f 6261 6c20 4578 xt, -- Global Ex │ │ │ │ +00037eb0: 7420 666f 7220 6d6f 6475 6c65 7320 6f76 t for modules ov │ │ │ │ +00037ec0: 6572 2061 2063 6f6d 706c 6574 650a 2020 er a complete. │ │ │ │ +00037ed0: 2020 496e 7465 7273 6563 7469 6f6e 0a0a Intersection.. │ │ │ │ +00037ee0: 4675 6e63 7469 6f6e 7320 7769 7468 206f Functions with o │ │ │ │ +00037ef0: 7074 696f 6e61 6c20 6172 6775 6d65 6e74 ptional argument │ │ │ │ +00037f00: 206e 616d 6564 204c 6966 743a 0a3d 3d3d named Lift:.=== │ │ │ │ +00037f10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00037f20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00037f30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00037f40: 3d3d 3d3d 0a0a 2020 2a20 226e 6577 4578 ====.. * "newEx │ │ │ │ -00037f50: 7428 2e2e 2e2c 4c69 6674 3d3e 2e2e 2e29 t(...,Lift=>...) │ │ │ │ -00037f60: 2220 2d2d 2073 6565 202a 6e6f 7465 206e " -- see *note n │ │ │ │ -00037f70: 6577 4578 743a 206e 6577 4578 742c 202d ewExt: newExt, - │ │ │ │ -00037f80: 2d20 476c 6f62 616c 2045 7874 2066 6f72 - Global Ext for │ │ │ │ -00037f90: 0a20 2020 206d 6f64 756c 6573 206f 7665 . modules ove │ │ │ │ -00037fa0: 7220 6120 636f 6d70 6c65 7465 2049 6e74 r a complete Int │ │ │ │ -00037fb0: 6572 7365 6374 696f 6e0a 0a46 6f72 2074 ersection..For t │ │ │ │ -00037fc0: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -00037fd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00037fe0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -00037ff0: 7465 204c 6966 743a 204c 6966 742c 2069 te Lift: Lift, i │ │ │ │ -00038000: 7320 6120 2a6e 6f74 6520 7379 6d62 6f6c s a *note symbol │ │ │ │ -00038010: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00038020: 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d Symbol,...------ │ │ │ │ +00037f30: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ +00037f40: 6e65 7745 7874 282e 2e2e 2c4c 6966 743d newExt(...,Lift= │ │ │ │ +00037f50: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e >...)" -- see *n │ │ │ │ +00037f60: 6f74 6520 6e65 7745 7874 3a20 6e65 7745 ote newExt: newE │ │ │ │ +00037f70: 7874 2c20 2d2d 2047 6c6f 6261 6c20 4578 xt, -- Global Ex │ │ │ │ +00037f80: 7420 666f 720a 2020 2020 6d6f 6475 6c65 t for. module │ │ │ │ +00037f90: 7320 6f76 6572 2061 2063 6f6d 706c 6574 s over a complet │ │ │ │ +00037fa0: 6520 496e 7465 7273 6563 7469 6f6e 0a0a e Intersection.. │ │ │ │ +00037fb0: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ +00037fc0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ +00037fd0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ +00037fe0: 7420 2a6e 6f74 6520 4c69 6674 3a20 4c69 t *note Lift: Li │ │ │ │ +00037ff0: 6674 2c20 6973 2061 202a 6e6f 7465 2073 ft, is a *note s │ │ │ │ +00038000: 796d 626f 6c3a 2028 4d61 6361 756c 6179 ymbol: (Macaulay │ │ │ │ +00038010: 3244 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2Doc)Symbol,...- │ │ │ │ +00038020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038070: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -00038080: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -00038090: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -000380a0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -000380b0: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -000380c0: 312e 3236 2e30 352b 6473 2f4d 322f 4d61 1.26.05+ds/M2/Ma │ │ │ │ -000380d0: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -000380e0: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ -000380f0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -00038100: 732e 6d32 3a33 3139 383a 302e 0a1f 0a46 s.m2:3198:0....F │ │ │ │ -00038110: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ -00038120: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ -00038130: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ -00038140: 206d 616b 6546 696e 6974 6552 6573 6f6c makeFiniteResol │ │ │ │ -00038150: 7574 696f 6e2c 204e 6578 743a 206d 616b ution, Next: mak │ │ │ │ -00038160: 6546 696e 6974 6552 6573 6f6c 7574 696f eFiniteResolutio │ │ │ │ -00038170: 6e43 6f64 696d 322c 2050 7265 763a 204c nCodim2, Prev: L │ │ │ │ -00038180: 6966 742c 2055 703a 2054 6f70 0a0a 6d61 ift, Up: Top..ma │ │ │ │ -00038190: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ -000381a0: 6f6e 202d 2d20 6669 6e69 7465 2072 6573 on -- finite res │ │ │ │ -000381b0: 6f6c 7574 696f 6e20 6f66 2061 206d 6174 olution of a mat │ │ │ │ -000381c0: 7269 7820 6661 6374 6f72 697a 6174 696f rix factorizatio │ │ │ │ -000381d0: 6e20 6d6f 6475 6c65 204d 0a2a 2a2a 2a2a n module M.***** │ │ │ │ +00038060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +00038070: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +00038080: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +00038090: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +000380a0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +000380b0: 6c61 7932 2d31 2e32 362e 3035 2b64 732f lay2-1.26.05+ds/ │ │ │ │ +000380c0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +000380d0: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ +000380e0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +000380f0: 7574 696f 6e73 2e6d 323a 3331 3938 3a30 utions.m2:3198:0 │ │ │ │ +00038100: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ +00038110: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ +00038120: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ +00038130: 4e6f 6465 3a20 6d61 6b65 4669 6e69 7465 Node: makeFinite │ │ │ │ +00038140: 5265 736f 6c75 7469 6f6e 2c20 4e65 7874 Resolution, Next │ │ │ │ +00038150: 3a20 6d61 6b65 4669 6e69 7465 5265 736f : makeFiniteReso │ │ │ │ +00038160: 6c75 7469 6f6e 436f 6469 6d32 2c20 5072 lutionCodim2, Pr │ │ │ │ +00038170: 6576 3a20 4c69 6674 2c20 5570 3a20 546f ev: Lift, Up: To │ │ │ │ +00038180: 700a 0a6d 616b 6546 696e 6974 6552 6573 p..makeFiniteRes │ │ │ │ +00038190: 6f6c 7574 696f 6e20 2d2d 2066 696e 6974 olution -- finit │ │ │ │ +000381a0: 6520 7265 736f 6c75 7469 6f6e 206f 6620 e resolution of │ │ │ │ +000381b0: 6120 6d61 7472 6978 2066 6163 746f 7269 a matrix factori │ │ │ │ +000381c0: 7a61 7469 6f6e 206d 6f64 756c 6520 4d0a zation module M. │ │ │ │ +000381d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000381e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000381f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00038200: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00038210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00038220: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ -00038230: 6765 3a20 0a20 2020 2020 2020 2041 203d ge: . A = │ │ │ │ -00038240: 206d 616b 6546 696e 6974 6552 6573 6f6c makeFiniteResol │ │ │ │ -00038250: 7574 696f 6e28 6666 2c6d 6629 0a20 202a ution(ff,mf). * │ │ │ │ -00038260: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00038270: 206d 662c 2061 202a 6e6f 7465 206c 6973 mf, a *note lis │ │ │ │ -00038280: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ -00038290: 294c 6973 742c 2c20 6f75 7470 7574 206f )List,, output o │ │ │ │ -000382a0: 6620 6d61 7472 6978 4661 6374 6f72 697a f matrixFactoriz │ │ │ │ -000382b0: 6174 696f 6e0a 2020 2020 2020 2a20 6666 ation. * ff │ │ │ │ -000382c0: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978 , a *note matrix │ │ │ │ -000382d0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -000382e0: 4d61 7472 6978 2c2c 2074 6865 2072 6567 Matrix,, the reg │ │ │ │ -000382f0: 756c 6172 2073 6571 7565 6e63 6520 7573 ular sequence us │ │ │ │ -00038300: 6564 0a20 2020 2020 2020 2066 6f72 2074 ed. for t │ │ │ │ -00038310: 6865 206d 6174 7269 7846 6163 746f 7269 he matrixFactori │ │ │ │ -00038320: 7a61 7469 6f6e 2063 6f6d 7075 7461 7469 zation computati │ │ │ │ -00038330: 6f6e 0a20 202a 204f 7574 7075 7473 3a0a on. * Outputs:. │ │ │ │ -00038340: 2020 2020 2020 2a20 412c 2061 202a 6e6f * A, a *no │ │ │ │ -00038350: 7465 2063 6f6d 706c 6578 3a20 2843 6f6d te complex: (Com │ │ │ │ -00038360: 706c 6578 6573 2943 6f6d 706c 6578 2c2c plexes)Complex,, │ │ │ │ -00038370: 2041 2069 7320 7468 6520 6d69 6e69 6d61 A is the minima │ │ │ │ -00038380: 6c20 6669 6e69 7465 0a20 2020 2020 2020 l finite. │ │ │ │ -00038390: 2072 6573 6f6c 7574 696f 6e20 6f66 204d resolution of M │ │ │ │ -000383a0: 206f 7665 7220 522e 0a0a 4465 7363 7269 over R...Descri │ │ │ │ -000383b0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -000383c0: 3d0a 0a53 7570 706f 7365 2074 6861 7420 =..Suppose that │ │ │ │ -000383d0: 665f 312e 2e66 5f63 2069 7320 6120 686f f_1..f_c is a ho │ │ │ │ -000383e0: 6d6f 6765 6e65 6f75 7320 7265 6775 6c61 mogeneous regula │ │ │ │ -000383f0: 7220 7365 7175 656e 6365 206f 6620 666f r sequence of fo │ │ │ │ -00038400: 726d 7320 6f66 2074 6865 2073 616d 650a rms of the same. │ │ │ │ -00038410: 6465 6772 6565 2069 6e20 6120 706f 6c79 degree in a poly │ │ │ │ -00038420: 6e6f 6d69 616c 2072 696e 6720 5320 616e nomial ring S an │ │ │ │ -00038430: 6420 4d20 6973 2061 2068 6967 6820 7379 d M is a high sy │ │ │ │ -00038440: 7a79 6779 206d 6f64 756c 6520 6f76 6572 zygy module over │ │ │ │ -00038450: 2053 2f28 665f 312c 2e2e 2c66 5f63 290a S/(f_1,..,f_c). │ │ │ │ -00038460: 3d20 5228 6329 2c20 616e 6420 6d66 203d = R(c), and mf = │ │ │ │ -00038470: 2028 642c 6829 2069 7320 7468 6520 6f75 (d,h) is the ou │ │ │ │ -00038480: 7470 7574 206f 6620 6d61 7472 6978 4661 tput of matrixFa │ │ │ │ -00038490: 6374 6f72 697a 6174 696f 6e28 4d2c 6666 ctorization(M,ff │ │ │ │ -000384a0: 292e 2049 6620 7468 650a 636f 6d70 6c65 ). If the.comple │ │ │ │ -000384b0: 7869 7479 206f 6620 4d20 6973 2063 272c xity of M is c', │ │ │ │ -000384c0: 2074 6865 6e20 4d20 6861 7320 6120 6669 then M has a fi │ │ │ │ -000384d0: 6e69 7465 2066 7265 6520 7265 736f 6c75 nite free resolu │ │ │ │ -000384e0: 7469 6f6e 206f 7665 7220 5220 3d0a 532f tion over R =.S/ │ │ │ │ -000384f0: 2866 5f31 2c2e 2e2c 665f 7b28 632d 6327 (f_1,..,f_{(c-c' │ │ │ │ -00038500: 297d 2920 2861 6e64 2c20 6d6f 7265 2067 )}) (and, more g │ │ │ │ -00038510: 656e 6572 616c 6c79 2c20 6861 7320 636f enerally, has co │ │ │ │ -00038520: 6d70 6c65 7869 7479 2063 2d64 206f 7665 mplexity c-d ove │ │ │ │ -00038530: 720a 532f 2866 5f31 2c2e 2e2c 665f 7b28 r.S/(f_1,..,f_{( │ │ │ │ -00038540: 632d 6429 7d29 2066 6f72 2064 3e3d 6327 c-d)}) for d>=c' │ │ │ │ -00038550: 292e 0a0a 5468 6520 636f 6d70 6c65 7820 )...The complex │ │ │ │ -00038560: 4120 6973 2074 6865 206d 696e 696d 616c A is the minimal │ │ │ │ -00038570: 2066 696e 6974 6520 6672 6565 2072 6573 finite free res │ │ │ │ -00038580: 6f6c 7574 696f 6e20 6f66 204d 206f 7665 olution of M ove │ │ │ │ -00038590: 7220 412c 2063 6f6e 7374 7275 6374 6564 r A, constructed │ │ │ │ -000385a0: 2061 730a 616e 2069 7465 7261 7465 6420 as.an iterated │ │ │ │ -000385b0: 4b6f 737a 756c 2065 7874 656e 7369 6f6e Koszul extension │ │ │ │ -000385c0: 2c20 6d61 6465 2066 726f 6d20 7468 6520 , made from the │ │ │ │ -000385d0: 6d61 7073 2069 6e20 624d 6170 7320 6d66 maps in bMaps mf │ │ │ │ -000385e0: 2061 6e64 2070 7369 4d61 7073 206d 662c and psiMaps mf, │ │ │ │ -000385f0: 2061 730a 6465 7363 7269 6265 6420 696e as.described in │ │ │ │ -00038600: 2045 6973 656e 6275 642d 5065 6576 612e Eisenbud-Peeva. │ │ │ │ -00038610: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +00038210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ +00038220: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ +00038230: 2020 4120 3d20 6d61 6b65 4669 6e69 7465 A = makeFinite │ │ │ │ +00038240: 5265 736f 6c75 7469 6f6e 2866 662c 6d66 Resolution(ff,mf │ │ │ │ +00038250: 290a 2020 2a20 496e 7075 7473 3a0a 2020 ). * Inputs:. │ │ │ │ +00038260: 2020 2020 2a20 6d66 2c20 6120 2a6e 6f74 * mf, a *not │ │ │ │ +00038270: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ +00038280: 7932 446f 6329 4c69 7374 2c2c 206f 7574 y2Doc)List,, out │ │ │ │ +00038290: 7075 7420 6f66 206d 6174 7269 7846 6163 put of matrixFac │ │ │ │ +000382a0: 746f 7269 7a61 7469 6f6e 0a20 2020 2020 torization. │ │ │ │ +000382b0: 202a 2066 662c 2061 202a 6e6f 7465 206d * ff, a *note m │ │ │ │ +000382c0: 6174 7269 783a 2028 4d61 6361 756c 6179 atrix: (Macaulay │ │ │ │ +000382d0: 3244 6f63 294d 6174 7269 782c 2c20 7468 2Doc)Matrix,, th │ │ │ │ +000382e0: 6520 7265 6775 6c61 7220 7365 7175 656e e regular sequen │ │ │ │ +000382f0: 6365 2075 7365 640a 2020 2020 2020 2020 ce used. │ │ │ │ +00038300: 666f 7220 7468 6520 6d61 7472 6978 4661 for the matrixFa │ │ │ │ +00038310: 6374 6f72 697a 6174 696f 6e20 636f 6d70 ctorization comp │ │ │ │ +00038320: 7574 6174 696f 6e0a 2020 2a20 4f75 7470 utation. * Outp │ │ │ │ +00038330: 7574 733a 0a20 2020 2020 202a 2041 2c20 uts:. * A, │ │ │ │ +00038340: 6120 2a6e 6f74 6520 636f 6d70 6c65 783a a *note complex: │ │ │ │ +00038350: 2028 436f 6d70 6c65 7865 7329 436f 6d70 (Complexes)Comp │ │ │ │ +00038360: 6c65 782c 2c20 4120 6973 2074 6865 206d lex,, A is the m │ │ │ │ +00038370: 696e 696d 616c 2066 696e 6974 650a 2020 inimal finite. │ │ │ │ +00038380: 2020 2020 2020 7265 736f 6c75 7469 6f6e resolution │ │ │ │ +00038390: 206f 6620 4d20 6f76 6572 2052 2e0a 0a44 of M over R...D │ │ │ │ +000383a0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +000383b0: 3d3d 3d3d 3d3d 0a0a 5375 7070 6f73 6520 ======..Suppose │ │ │ │ +000383c0: 7468 6174 2066 5f31 2e2e 665f 6320 6973 that f_1..f_c is │ │ │ │ +000383d0: 2061 2068 6f6d 6f67 656e 656f 7573 2072 a homogeneous r │ │ │ │ +000383e0: 6567 756c 6172 2073 6571 7565 6e63 6520 egular sequence │ │ │ │ +000383f0: 6f66 2066 6f72 6d73 206f 6620 7468 6520 of forms of the │ │ │ │ +00038400: 7361 6d65 0a64 6567 7265 6520 696e 2061 same.degree in a │ │ │ │ +00038410: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ +00038420: 2053 2061 6e64 204d 2069 7320 6120 6869 S and M is a hi │ │ │ │ +00038430: 6768 2073 797a 7967 7920 6d6f 6475 6c65 gh syzygy module │ │ │ │ +00038440: 206f 7665 7220 532f 2866 5f31 2c2e 2e2c over S/(f_1,.., │ │ │ │ +00038450: 665f 6329 0a3d 2052 2863 292c 2061 6e64 f_c).= R(c), and │ │ │ │ +00038460: 206d 6620 3d20 2864 2c68 2920 6973 2074 mf = (d,h) is t │ │ │ │ +00038470: 6865 206f 7574 7075 7420 6f66 206d 6174 he output of mat │ │ │ │ +00038480: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ +00038490: 284d 2c66 6629 2e20 4966 2074 6865 0a63 (M,ff). If the.c │ │ │ │ +000384a0: 6f6d 706c 6578 6974 7920 6f66 204d 2069 omplexity of M i │ │ │ │ +000384b0: 7320 6327 2c20 7468 656e 204d 2068 6173 s c', then M has │ │ │ │ +000384c0: 2061 2066 696e 6974 6520 6672 6565 2072 a finite free r │ │ │ │ +000384d0: 6573 6f6c 7574 696f 6e20 6f76 6572 2052 esolution over R │ │ │ │ +000384e0: 203d 0a53 2f28 665f 312c 2e2e 2c66 5f7b =.S/(f_1,..,f_{ │ │ │ │ +000384f0: 2863 2d63 2729 7d29 2028 616e 642c 206d (c-c')}) (and, m │ │ │ │ +00038500: 6f72 6520 6765 6e65 7261 6c6c 792c 2068 ore generally, h │ │ │ │ +00038510: 6173 2063 6f6d 706c 6578 6974 7920 632d as complexity c- │ │ │ │ +00038520: 6420 6f76 6572 0a53 2f28 665f 312c 2e2e d over.S/(f_1,.. │ │ │ │ +00038530: 2c66 5f7b 2863 2d64 297d 2920 666f 7220 ,f_{(c-d)}) for │ │ │ │ +00038540: 643e 3d63 2729 2e0a 0a54 6865 2063 6f6d d>=c')...The com │ │ │ │ +00038550: 706c 6578 2041 2069 7320 7468 6520 6d69 plex A is the mi │ │ │ │ +00038560: 6e69 6d61 6c20 6669 6e69 7465 2066 7265 nimal finite fre │ │ │ │ +00038570: 6520 7265 736f 6c75 7469 6f6e 206f 6620 e resolution of │ │ │ │ +00038580: 4d20 6f76 6572 2041 2c20 636f 6e73 7472 M over A, constr │ │ │ │ +00038590: 7563 7465 6420 6173 0a61 6e20 6974 6572 ucted as.an iter │ │ │ │ +000385a0: 6174 6564 204b 6f73 7a75 6c20 6578 7465 ated Koszul exte │ │ │ │ +000385b0: 6e73 696f 6e2c 206d 6164 6520 6672 6f6d nsion, made from │ │ │ │ +000385c0: 2074 6865 206d 6170 7320 696e 2062 4d61 the maps in bMa │ │ │ │ +000385d0: 7073 206d 6620 616e 6420 7073 694d 6170 ps mf and psiMap │ │ │ │ +000385e0: 7320 6d66 2c20 6173 0a64 6573 6372 6962 s mf, as.describ │ │ │ │ +000385f0: 6564 2069 6e20 4569 7365 6e62 7564 2d50 ed in Eisenbud-P │ │ │ │ +00038600: 6565 7661 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d eeva...+-------- │ │ │ │ +00038610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038660: 2b0a 7c69 3120 3a20 7365 7452 616e 646f +.|i1 : setRando │ │ │ │ -00038670: 6d53 6565 6420 3020 2020 2020 2020 2020 mSeed 0 │ │ │ │ +00038650: 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 6574 -----+.|i1 : set │ │ │ │ +00038660: 5261 6e64 6f6d 5365 6564 2030 2020 2020 RandomSeed 0 │ │ │ │ +00038670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000386a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000386b0: 7c0a 7c20 2d2d 2073 6574 7469 6e67 2072 |.| -- setting r │ │ │ │ -000386c0: 616e 646f 6d20 7365 6564 2074 6f20 3020 andom seed to 0 │ │ │ │ +000386a0: 2020 2020 207c 0a7c 202d 2d20 7365 7474 |.| -- sett │ │ │ │ +000386b0: 696e 6720 7261 6e64 6f6d 2073 6565 6420 ing random seed │ │ │ │ +000386c0: 746f 2030 2020 2020 2020 2020 2020 2020 to 0 │ │ │ │ 000386d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000386e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000386f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038700: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000386f0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038750: 7c0a 7c6f 3120 3d20 3020 2020 2020 2020 |.|o1 = 0 │ │ │ │ +00038740: 2020 2020 207c 0a7c 6f31 203d 2030 2020 |.|o1 = 0 │ │ │ │ +00038750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000387a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00038790: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000387a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000387b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000387c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000387d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000387e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000387f0: 2b0a 7c69 3220 3a20 5320 3d20 5a5a 2f31 +.|i2 : S = ZZ/1 │ │ │ │ -00038800: 3031 5b61 2c62 2c63 5d3b 2020 2020 2020 01[a,b,c]; │ │ │ │ +000387e0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 203d -----+.|i2 : S = │ │ │ │ +000387f0: 205a 5a2f 3130 315b 612c 622c 635d 3b20 ZZ/101[a,b,c]; │ │ │ │ +00038800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038840: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00038830: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00038840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038890: 2b0a 7c69 3320 3a20 6666 203d 206d 6174 +.|i3 : ff = mat │ │ │ │ -000388a0: 7269 7822 6133 2c62 3322 3b20 2020 2020 rix"a3,b3"; │ │ │ │ +00038880: 2d2d 2d2d 2d2b 0a7c 6933 203a 2066 6620 -----+.|i3 : ff │ │ │ │ +00038890: 3d20 6d61 7472 6978 2261 332c 6233 223b = matrix"a3,b3"; │ │ │ │ +000388a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000388b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000388c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000388d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000388e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000388d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000388e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000388f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038930: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00038940: 3120 2020 2020 2032 2020 2020 2020 2020 1 2 │ │ │ │ +00038920: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038930: 2020 2020 2031 2020 2020 2020 3220 2020 1 2 │ │ │ │ +00038940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038980: 7c0a 7c6f 3320 3a20 4d61 7472 6978 2053 |.|o3 : Matrix S │ │ │ │ -00038990: 2020 3c2d 2d20 5320 2020 2020 2020 2020 <-- S │ │ │ │ +00038970: 2020 2020 207c 0a7c 6f33 203a 204d 6174 |.|o3 : Mat │ │ │ │ +00038980: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ +00038990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000389a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000389b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000389c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000389d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +000389c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000389d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000389e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000389f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038a20: 2b0a 7c69 3420 3a20 5220 3d20 532f 6964 +.|i4 : R = S/id │ │ │ │ -00038a30: 6561 6c20 6666 3b20 2020 2020 2020 2020 eal ff; │ │ │ │ +00038a10: 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 203d -----+.|i4 : R = │ │ │ │ +00038a20: 2053 2f69 6465 616c 2066 663b 2020 2020 S/ideal ff; │ │ │ │ +00038a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038a70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00038a60: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00038a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038ac0: 2b0a 7c69 3520 3a20 4d20 3d20 6869 6768 +.|i5 : M = high │ │ │ │ -00038ad0: 5379 7a79 6779 2028 525e 312f 6964 6561 Syzygy (R^1/idea │ │ │ │ -00038ae0: 6c20 7661 7273 2052 293b 2020 2020 2020 l vars R); │ │ │ │ +00038ab0: 2d2d 2d2d 2d2b 0a7c 6935 203a 204d 203d -----+.|i5 : M = │ │ │ │ +00038ac0: 2068 6967 6853 797a 7967 7920 2852 5e31 highSyzygy (R^1 │ │ │ │ +00038ad0: 2f69 6465 616c 2076 6172 7320 5229 3b20 /ideal vars R); │ │ │ │ +00038ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038b10: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00038b00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00038b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038b60: 2b0a 7c69 3620 3a20 6d66 203d 206d 6174 +.|i6 : mf = mat │ │ │ │ -00038b70: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -00038b80: 2028 6666 2c20 4d29 2020 2020 2020 2020 (ff, M) │ │ │ │ +00038b50: 2d2d 2d2d 2d2b 0a7c 6936 203a 206d 6620 -----+.|i6 : mf │ │ │ │ +00038b60: 3d20 6d61 7472 6978 4661 6374 6f72 697a = matrixFactoriz │ │ │ │ +00038b70: 6174 696f 6e20 2866 662c 204d 2920 2020 ation (ff, M) │ │ │ │ +00038b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038bb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00038ba0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038c00: 7c0a 7c6f 3620 3d20 7b7b 347d 207c 202d |.|o6 = {{4} | - │ │ │ │ -00038c10: 6320 6220 3020 2061 3220 3020 2030 2020 c b 0 a2 0 0 │ │ │ │ -00038c20: 3020 2030 2020 3020 2020 7c2c 207b 357d 0 0 0 |, {5} │ │ │ │ -00038c30: 207c 2030 2061 3220 3020 202d 6220 3020 | 0 a2 0 -b 0 │ │ │ │ -00038c40: 2030 2020 2030 2020 2030 2030 2020 3020 0 0 0 0 0 │ │ │ │ -00038c50: 7c0a 7c20 2020 2020 207b 347d 207c 2061 |.| {4} | a │ │ │ │ -00038c60: 2020 3020 6220 2030 2020 3020 2030 2020 0 b 0 0 0 │ │ │ │ -00038c70: 3020 2030 2020 3020 2020 7c20 207b 357d 0 0 0 | {5} │ │ │ │ -00038c80: 207c 2030 2030 2020 6132 202d 6320 6232 | 0 0 a2 -c b2 │ │ │ │ -00038c90: 2030 2020 2030 2020 2030 2030 2020 3020 0 0 0 0 0 │ │ │ │ -00038ca0: 7c0a 7c20 2020 2020 207b 347d 207c 2030 |.| {4} | 0 │ │ │ │ -00038cb0: 2020 6120 6320 2030 2020 3020 2030 2020 a c 0 0 0 │ │ │ │ -00038cc0: 3020 2030 2020 2d62 3220 7c20 207b 357d 0 0 -b2 | {5} │ │ │ │ -00038cd0: 207c 2030 2030 2020 3020 2061 2020 3020 | 0 0 0 a 0 │ │ │ │ -00038ce0: 2062 3220 2030 2020 2030 2030 2020 3020 b2 0 0 0 0 │ │ │ │ -00038cf0: 7c0a 7c20 2020 2020 207b 337d 207c 2030 |.| {3} | 0 │ │ │ │ -00038d00: 2020 3020 6132 2030 2020 3020 2062 3220 0 a2 0 0 b2 │ │ │ │ -00038d10: 3020 2030 2020 3020 2020 7c20 207b 367d 0 0 0 | {6} │ │ │ │ -00038d20: 207c 2061 2063 2020 2d62 2030 2020 3020 | a c -b 0 0 │ │ │ │ -00038d30: 2030 2020 2030 2020 2030 2030 2020 3020 0 0 0 0 0 │ │ │ │ -00038d40: 7c0a 7c20 2020 2020 207b 347d 207c 2030 |.| {4} | 0 │ │ │ │ -00038d50: 2020 3020 3020 2030 2020 6220 202d 6120 0 0 0 b -a │ │ │ │ -00038d60: 3020 2030 2020 3020 2020 7c20 207b 357d 0 0 0 | {5} │ │ │ │ -00038d70: 207c 2030 2030 2020 3020 2030 2020 3020 | 0 0 0 0 0 │ │ │ │ -00038d80: 2030 2020 2030 2020 2061 2062 3220 3020 0 0 a b2 0 │ │ │ │ -00038d90: 7c0a 7c20 2020 2020 207b 347d 207c 2030 |.| {4} | 0 │ │ │ │ -00038da0: 2020 3020 3020 2030 2020 2d63 2030 2020 0 0 0 -c 0 │ │ │ │ -00038db0: 6120 2062 3220 3020 2020 7c20 207b 357d a b2 0 | {5} │ │ │ │ -00038dc0: 207c 2030 2030 2020 3020 2030 2020 3020 | 0 0 0 0 0 │ │ │ │ -00038dd0: 202d 6132 2030 2020 2062 2030 2020 3020 -a2 0 b 0 0 │ │ │ │ -00038de0: 7c0a 7c20 2020 2020 207b 347d 207c 2030 |.| {4} | 0 │ │ │ │ -00038df0: 2020 3020 3020 2030 2020 3020 2063 2020 0 0 0 0 c │ │ │ │ -00038e00: 2d62 2030 2020 6132 2020 7c20 207b 357d -b 0 a2 | {5} │ │ │ │ -00038e10: 207c 2030 2030 2020 3020 2030 2020 3020 | 0 0 0 0 0 │ │ │ │ -00038e20: 2030 2020 202d 6132 2063 2030 2020 3020 0 -a2 c 0 0 │ │ │ │ -00038e30: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00038bf0: 2020 2020 207c 0a7c 6f36 203d 207b 7b34 |.|o6 = {{4 │ │ │ │ +00038c00: 7d20 7c20 2d63 2062 2030 2020 6132 2030 } | -c b 0 a2 0 │ │ │ │ +00038c10: 2020 3020 2030 2020 3020 2030 2020 207c 0 0 0 0 | │ │ │ │ +00038c20: 2c20 7b35 7d20 7c20 3020 6132 2030 2020 , {5} | 0 a2 0 │ │ │ │ +00038c30: 2d62 2030 2020 3020 2020 3020 2020 3020 -b 0 0 0 0 │ │ │ │ +00038c40: 3020 2030 207c 0a7c 2020 2020 2020 7b34 0 0 |.| {4 │ │ │ │ +00038c50: 7d20 7c20 6120 2030 2062 2020 3020 2030 } | a 0 b 0 0 │ │ │ │ +00038c60: 2020 3020 2030 2020 3020 2030 2020 207c 0 0 0 0 | │ │ │ │ +00038c70: 2020 7b35 7d20 7c20 3020 3020 2061 3220 {5} | 0 0 a2 │ │ │ │ +00038c80: 2d63 2062 3220 3020 2020 3020 2020 3020 -c b2 0 0 0 │ │ │ │ +00038c90: 3020 2030 207c 0a7c 2020 2020 2020 7b34 0 0 |.| {4 │ │ │ │ +00038ca0: 7d20 7c20 3020 2061 2063 2020 3020 2030 } | 0 a c 0 0 │ │ │ │ +00038cb0: 2020 3020 2030 2020 3020 202d 6232 207c 0 0 0 -b2 | │ │ │ │ +00038cc0: 2020 7b35 7d20 7c20 3020 3020 2030 2020 {5} | 0 0 0 │ │ │ │ +00038cd0: 6120 2030 2020 6232 2020 3020 2020 3020 a 0 b2 0 0 │ │ │ │ +00038ce0: 3020 2030 207c 0a7c 2020 2020 2020 7b33 0 0 |.| {3 │ │ │ │ +00038cf0: 7d20 7c20 3020 2030 2061 3220 3020 2030 } | 0 0 a2 0 0 │ │ │ │ +00038d00: 2020 6232 2030 2020 3020 2030 2020 207c b2 0 0 0 | │ │ │ │ +00038d10: 2020 7b36 7d20 7c20 6120 6320 202d 6220 {6} | a c -b │ │ │ │ +00038d20: 3020 2030 2020 3020 2020 3020 2020 3020 0 0 0 0 0 │ │ │ │ +00038d30: 3020 2030 207c 0a7c 2020 2020 2020 7b34 0 0 |.| {4 │ │ │ │ +00038d40: 7d20 7c20 3020 2030 2030 2020 3020 2062 } | 0 0 0 0 b │ │ │ │ +00038d50: 2020 2d61 2030 2020 3020 2030 2020 207c -a 0 0 0 | │ │ │ │ +00038d60: 2020 7b35 7d20 7c20 3020 3020 2030 2020 {5} | 0 0 0 │ │ │ │ +00038d70: 3020 2030 2020 3020 2020 3020 2020 6120 0 0 0 0 a │ │ │ │ +00038d80: 6232 2030 207c 0a7c 2020 2020 2020 7b34 b2 0 |.| {4 │ │ │ │ +00038d90: 7d20 7c20 3020 2030 2030 2020 3020 202d } | 0 0 0 0 - │ │ │ │ +00038da0: 6320 3020 2061 2020 6232 2030 2020 207c c 0 a b2 0 | │ │ │ │ +00038db0: 2020 7b35 7d20 7c20 3020 3020 2030 2020 {5} | 0 0 0 │ │ │ │ +00038dc0: 3020 2030 2020 2d61 3220 3020 2020 6220 0 0 -a2 0 b │ │ │ │ +00038dd0: 3020 2030 207c 0a7c 2020 2020 2020 7b34 0 0 |.| {4 │ │ │ │ +00038de0: 7d20 7c20 3020 2030 2030 2020 3020 2030 } | 0 0 0 0 0 │ │ │ │ +00038df0: 2020 6320 202d 6220 3020 2061 3220 207c c -b 0 a2 | │ │ │ │ +00038e00: 2020 7b35 7d20 7c20 3020 3020 2030 2020 {5} | 0 0 0 │ │ │ │ +00038e10: 3020 2030 2020 3020 2020 2d61 3220 6320 0 0 0 -a2 c │ │ │ │ +00038e20: 3020 2030 207c 0a7c 2020 2020 2020 2020 0 0 |.| │ │ │ │ +00038e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038e50: 2020 2020 2020 2020 2020 2020 207b 367d {6} │ │ │ │ -00038e60: 207c 2030 2030 2020 3020 2030 2020 3020 | 0 0 0 0 0 │ │ │ │ -00038e70: 2030 2020 2030 2020 2030 2063 2020 6220 0 0 0 c b │ │ │ │ -00038e80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00038e50: 2020 7b36 7d20 7c20 3020 3020 2030 2020 {6} | 0 0 0 │ │ │ │ +00038e60: 3020 2030 2020 3020 2020 3020 2020 3020 0 0 0 0 0 │ │ │ │ +00038e70: 6320 2062 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7468 656e 2074 maximal, then t │ │ │ │ -0003ac10: 6865 2066 696e 6974 6520 7265 736f 6c75 he finite resolu │ │ │ │ -0003ac20: 7469 6f6e 2074 616b 6573 2070 6c61 6365 tion takes place │ │ │ │ -0003ac30: 0a6f 7665 7220 616e 2069 6e74 6572 6d65 .over an interme │ │ │ │ -0003ac40: 6469 6174 6520 636f 6d70 6c65 7465 2069 diate complete i │ │ │ │ -0003ac50: 6e74 6572 7365 6374 696f 6e3a 0a0a 2b2d ntersection:..+- │ │ │ │ +0003abd0: 2d2d 2d2d 2d2b 0a0a 4966 2074 6865 2063 -----+..If the c │ │ │ │ +0003abe0: 6f6d 706c 6578 6974 7920 6f66 204d 2069 omplexity of M i │ │ │ │ +0003abf0: 7320 6e6f 7420 6d61 7869 6d61 6c2c 2074 s not maximal, t │ │ │ │ +0003ac00: 6865 6e20 7468 6520 6669 6e69 7465 2072 hen the finite r │ │ │ │ +0003ac10: 6573 6f6c 7574 696f 6e20 7461 6b65 7320 esolution takes │ │ │ │ +0003ac20: 706c 6163 650a 6f76 6572 2061 6e20 696e place.over an in │ │ │ │ +0003ac30: 7465 726d 6564 6961 7465 2063 6f6d 706c termediate compl │ │ │ │ +0003ac40: 6574 6520 696e 7465 7273 6563 7469 6f6e ete intersection │ │ │ │ +0003ac50: 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d :..+------------ │ │ │ │ 0003ac60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003ac70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003ac80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003ac90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003aca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0003acb0: 3133 203a 2053 203d 205a 5a2f 3130 315b 13 : S = ZZ/101[ │ │ │ │ -0003acc0: 612c 622c 632c 645d 2020 2020 2020 2020 a,b,c,d] │ │ │ │ +0003aca0: 2d2b 0a7c 6931 3320 3a20 5320 3d20 5a5a -+.|i13 : S = ZZ │ │ │ │ +0003acb0: 2f31 3031 5b61 2c62 2c63 2c64 5d20 2020 /101[a,b,c,d] │ │ │ │ +0003acc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003acd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ace0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003acf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003acf0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ad00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ad10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ad20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ad30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ad40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003ad50: 3133 203d 2053 2020 2020 2020 2020 2020 13 = S │ │ │ │ +0003ad40: 207c 0a7c 6f31 3320 3d20 5320 2020 2020 |.|o13 = S │ │ │ │ +0003ad50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ad60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ad70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ad80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ad90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003ad90: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ada0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003adb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003adc0: 2020 2020 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2020 2020 2020 2020 2020 4 │ │ │ │ +0003afc0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003afd0: 2020 3120 2020 2020 2034 2020 2020 2020 1 4 │ │ │ │ +0003afe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b010: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b020: 3134 203a 204d 6174 7269 7820 5320 203c 14 : Matrix S < │ │ │ │ -0003b030: 2d2d 2053 2020 2020 2020 2020 2020 2020 -- S │ │ │ │ +0003b010: 207c 0a7c 6f31 3420 3a20 4d61 7472 6978 |.|o14 : Matrix │ │ │ │ +0003b020: 2053 2020 3c2d 2d20 5320 2020 2020 2020 S <-- S │ │ │ │ +0003b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b060: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0003b060: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 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│ │ +0003b100: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b150: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b160: 3135 203d 207c 2032 3461 332d 3336 6233 15 = | 24a3-36b3 │ │ │ │ -0003b170: 2d33 3063 332d 3239 6433 2031 3961 332b -30c3-29d3 19a3+ │ │ │ │ -0003b180: 3139 6233 2d31 3063 332d 3239 6433 202d 19b3-10c3-29d3 - │ │ │ │ -0003b190: 3861 332d 3232 6233 2d32 3963 332d 3234 8a3-22b3-29c3-24 │ │ │ │ -0003b1a0: 6433 2020 2020 2020 2020 2020 7c0a 7c20 d3 |.| │ │ │ │ -0003b1b0: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d ----------- │ │ │ │ +0003b150: 207c 0a7c 6f31 3520 3d20 7c20 3234 6133 |.|o15 = | 24a3 │ │ │ │ +0003b160: 2d33 3662 332d 3330 6333 2d32 3964 3320 -36b3-30c3-29d3 │ │ │ │ +0003b170: 3139 6133 2b31 3962 332d 3130 6333 2d32 19a3+19b3-10c3-2 │ │ │ │ +0003b180: 3964 3320 2d38 6133 2d32 3262 332d 3239 9d3 -8a3-22b3-29 │ │ │ │ +0003b190: 6333 2d32 3464 3320 2020 2020 2020 2020 c3-24d3 │ │ │ │ +0003b1a0: 207c 0a7c 2020 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +0003b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ -0003b200: 2020 2020 202d 3338 6133 2d31 3662 332b -38a3-16b3+ │ │ │ │ -0003b210: 3339 6333 2b32 3164 3320 7c20 2020 2020 39c3+21d3 | │ │ │ │ +0003b1f0: 2d7c 0a7c 2020 2020 2020 2d33 3861 332d -|.| -38a3- │ │ │ │ +0003b200: 3136 6233 2b33 3963 332b 3231 6433 207c 16b3+39c3+21d3 | │ │ │ │ +0003b210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b240: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b240: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b290: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0003b2a0: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ -0003b2b0: 2020 2020 3420 2020 2020 2020 2020 2020 4 │ │ │ │ +0003b290: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003b2a0: 2020 3120 2020 2020 2034 2020 2020 2020 1 4 │ │ │ │ +0003b2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b2e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b2f0: 3135 203a 204d 6174 7269 7820 5320 203c 15 : Matrix S < │ │ │ │ -0003b300: 2d2d 2053 2020 2020 2020 2020 2020 2020 -- S │ │ │ │ +0003b2e0: 207c 0a7c 6f31 3520 3a20 4d61 7472 6978 |.|o15 : Matrix │ │ │ │ +0003b2f0: 2053 2020 3c2d 2d20 5320 2020 2020 2020 S <-- S │ │ │ │ +0003b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b330: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0003b330: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0003b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0003b390: 3136 203a 2052 203d 2053 2f69 6465 616c 16 : R = S/ideal │ │ │ │ -0003b3a0: 2066 6620 2020 2020 2020 2020 2020 2020 ff │ │ │ │ +0003b380: 2d2b 0a7c 6931 3620 3a20 5220 3d20 532f -+.|i16 : R = S/ │ │ │ │ +0003b390: 6964 6561 6c20 6666 2020 2020 2020 2020 ideal ff │ │ │ │ +0003b3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b3d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b3d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b420: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b430: 3136 203d 2052 2020 2020 2020 2020 2020 16 = R │ │ │ │ +0003b420: 207c 0a7c 6f31 3620 3d20 5220 2020 2020 |.|o16 = R │ │ │ │ +0003b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b470: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b470: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b4c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b4d0: 3136 203a 2051 756f 7469 656e 7452 696e 16 : QuotientRin │ │ │ │ -0003b4e0: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ +0003b4c0: 207c 0a7c 6f31 3620 3a20 5175 6f74 6965 |.|o16 : Quotie │ │ │ │ +0003b4d0: 6e74 5269 6e67 2020 2020 2020 2020 2020 ntRing │ │ │ │ +0003b4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b510: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0003b510: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0003b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0003b570: 3137 203a 204d 203d 2068 6967 6853 797a 17 : M = highSyz │ │ │ │ -0003b580: 7967 7920 2852 5e31 2f69 6465 616c 2261 ygy (R^1/ideal"a │ │ │ │ -0003b590: 3262 3222 2920 2020 2020 2020 2020 2020 2b2") │ │ │ │ +0003b560: 2d2b 0a7c 6931 3720 3a20 4d20 3d20 6869 -+.|i17 : M = hi │ │ │ │ +0003b570: 6768 5379 7a79 6779 2028 525e 312f 6964 ghSyzygy (R^1/id │ │ │ │ +0003b580: 6561 6c22 6132 6232 2229 2020 2020 2020 eal"a2b2") │ │ │ │ +0003b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b5b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b600: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b610: 3137 203d 2063 6f6b 6572 6e65 6c20 7b36 17 = cokernel {6 │ │ │ │ -0003b620: 7d20 7c20 6232 2030 202d 6132 2030 207c } | b2 0 -a2 0 | │ │ │ │ +0003b600: 207c 0a7c 6f31 3720 3d20 636f 6b65 726e |.|o17 = cokern │ │ │ │ +0003b610: 656c 207b 367d 207c 2062 3220 3020 2d61 el {6} | b2 0 -a │ │ │ │ +0003b620: 3220 3020 7c20 2020 2020 2020 2020 2020 2 0 | │ │ │ │ 0003b630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b650: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0003b660: 2020 2020 2020 2020 2020 2020 2020 7b37 {7 │ │ │ │ -0003b670: 7d20 7c20 6120 2062 2030 2020 2030 207c } | a b 0 0 | │ │ │ │ +0003b650: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003b660: 2020 207b 377d 207c 2061 2020 6220 3020 {7} | a b 0 │ │ │ │ +0003b670: 2020 3020 7c20 2020 2020 2020 2020 2020 0 | │ │ │ │ 0003b680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b6a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0003b6b0: 2020 2020 2020 2020 2020 2020 2020 7b37 {7 │ │ │ │ -0003b6c0: 7d20 7c20 3020 2030 2062 2020 2061 207c } | 0 0 b a | │ │ │ │ +0003b6a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003b6b0: 2020 207b 377d 207c 2030 2020 3020 6220 {7} | 0 0 b │ │ │ │ +0003b6c0: 2020 6120 7c20 2020 2020 2020 2020 2020 a | │ │ │ │ 0003b6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b6f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b6f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b740: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b750: 3137 203a 2052 2d6d 6f64 756c 652c 2071 17 : R-module, q │ │ │ │ -0003b760: 756f 7469 6520 2020 2020 2020 3320 2020 uotie 3 │ │ │ │ +0003b740: 207c 0a7c 6f31 3720 3a20 522d 6d6f 6475 |.|o17 : R-modu │ │ │ │ +0003b750: 6c65 2c20 7175 6f74 6965 2020 2020 2020 le, quotie │ │ │ │ +0003b760: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0003b770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b790: 2020 2020 2020 2020 2020 2020 7c0a 7c6e |.|n │ │ │ │ -0003b7a0: 7420 6f66 2052 2020 2020 2020 2020 2020 t of R │ │ │ │ +0003b790: 207c 0a7c 6e74 206f 6620 5220 2020 2020 |.|nt of R │ │ │ │ +0003b7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b7e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0003b7e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0003b7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0003b840: 3138 203a 2063 6f6d 706c 6578 6974 7920 18 : complexity │ │ │ │ -0003b850: 4d20 2020 2020 2020 2020 2020 2020 2020 M │ │ │ │ +0003b830: 2d2b 0a7c 6931 3820 3a20 636f 6d70 6c65 -+.|i18 : comple │ │ │ │ +0003b840: 7869 7479 204d 2020 2020 2020 2020 2020 xity M │ │ │ │ +0003b850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b880: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b880: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b8d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b8e0: 3138 203d 2032 2020 2020 2020 2020 2020 18 = 2 │ │ │ │ +0003b8d0: 207c 0a7c 6f31 3820 3d20 3220 2020 2020 |.|o18 = 2 │ │ │ │ +0003b8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b920: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0003b920: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0003b930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0003b980: 3139 203a 206d 6620 3d20 6d61 7472 6978 19 : mf = matrix │ │ │ │ -0003b990: 4661 6374 6f72 697a 6174 696f 6e20 2866 Factorization (f │ │ │ │ -0003b9a0: 662c 204d 2920 2020 2020 2020 2020 2020 f, M) │ │ │ │ +0003b970: 2d2b 0a7c 6931 3920 3a20 6d66 203d 206d -+.|i19 : mf = m │ │ │ │ +0003b980: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ +0003b990: 6f6e 2028 6666 2c20 4d29 2020 2020 2020 on (ff, M) │ │ │ │ +0003b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b9c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b9c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ba10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003ba20: 3139 203d 207b 7b37 7d20 7c20 2d61 202d 19 = {{7} | -a - │ │ │ │ -0003ba30: 3336 6220 3020 6120 7c2c 207b 387d 207c 36b 0 a |, {8} | │ │ │ │ -0003ba40: 2033 3561 3220 2034 3862 2020 3020 2020 35a2 48b 0 │ │ │ │ -0003ba50: 2020 2d33 3362 2030 2020 2020 207c 2c20 -33b 0 |, │ │ │ │ -0003ba60: 7b36 7d20 7c20 3020 2020 3336 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+0003cfa0: 6d61 7472 6978 4661 6374 6f72 697a 6174 matrixFactorizat │ │ │ │ +0003cfb0: 696f 6e0a 2020 2a20 2a6e 6f74 6520 636f ion. * *note co │ │ │ │ +0003cfc0: 6d70 6c65 7869 7479 3a20 636f 6d70 6c65 mplexity: comple │ │ │ │ +0003cfd0: 7869 7479 2c20 2d2d 2063 6f6d 706c 6578 xity, -- complex │ │ │ │ +0003cfe0: 6974 7920 6f66 2061 206d 6f64 756c 6520 ity of a module │ │ │ │ +0003cff0: 6f76 6572 2061 2063 6f6d 706c 6574 650a over a complete. │ │ │ │ +0003d000: 2020 2020 696e 7465 7273 6563 7469 6f6e intersection │ │ │ │ +0003d010: 0a0a 5761 7973 2074 6f20 7573 6520 6d61 ..Ways to use ma │ │ │ │ +0003d020: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ +0003d030: 6f6e 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d on:.============ │ │ │ │ 0003d040: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0003d050: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0003d060: 0a0a 2020 2a20 226d 616b 6546 696e 6974 .. * "makeFinit │ │ │ │ -0003d070: 6552 6573 6f6c 7574 696f 6e28 4d61 7472 eResolution(Matr │ │ │ │ -0003d080: 6978 2c4c 6973 7429 220a 0a46 6f72 2074 ix,List)"..For t │ │ │ │ -0003d090: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -0003d0a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0003d0b0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -0003d0c0: 7465 206d 616b 6546 696e 6974 6552 6573 te makeFiniteRes │ │ │ │ -0003d0d0: 6f6c 7574 696f 6e3a 206d 616b 6546 696e olution: makeFin │ │ │ │ -0003d0e0: 6974 6552 6573 6f6c 7574 696f 6e2c 2069 iteResolution, i │ │ │ │ -0003d0f0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ -0003d100: 0a66 756e 6374 696f 6e3a 2028 4d61 6361 .function: (Maca │ │ │ │ -0003d110: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ -0003d120: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ +0003d050: 3d3d 3d3d 3d0a 0a20 202a 2022 6d61 6b65 =====.. * "make │ │ │ │ +0003d060: 4669 6e69 7465 5265 736f 6c75 7469 6f6e FiniteResolution │ │ │ │ +0003d070: 284d 6174 7269 782c 4c69 7374 2922 0a0a (Matrix,List)".. │ │ │ │ +0003d080: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ +0003d090: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ +0003d0a0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ +0003d0b0: 7420 2a6e 6f74 6520 6d61 6b65 4669 6e69 t *note makeFini │ │ │ │ +0003d0c0: 7465 5265 736f 6c75 7469 6f6e 3a20 6d61 teResolution: ma │ │ │ │ +0003d0d0: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ +0003d0e0: 6f6e 2c20 6973 2061 202a 6e6f 7465 206d on, is a *note m │ │ │ │ +0003d0f0: 6574 686f 640a 6675 6e63 7469 6f6e 3a20 ethod.function: │ │ │ │ +0003d100: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +0003d110: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +0003d120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d170: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -0003d180: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -0003d190: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -0003d1a0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -0003d1b0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -0003d1c0: 2d31 2e32 362e 3035 2b64 732f 4d32 2f4d -1.26.05+ds/M2/M │ │ │ │ -0003d1d0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -0003d1e0: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ -0003d1f0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -0003d200: 6e73 2e6d 323a 3238 3939 3a30 2e0a 1f0a ns.m2:2899:0.... │ │ │ │ -0003d210: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ -0003d220: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -0003d230: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ -0003d240: 3a20 6d61 6b65 4669 6e69 7465 5265 736f : makeFiniteReso │ │ │ │ -0003d250: 6c75 7469 6f6e 436f 6469 6d32 2c20 4e65 lutionCodim2, Ne │ │ │ │ -0003d260: 7874 3a20 6d61 6b65 486f 6d6f 746f 7069 xt: makeHomotopi │ │ │ │ -0003d270: 6573 2c20 5072 6576 3a20 6d61 6b65 4669 es, Prev: makeFi │ │ │ │ -0003d280: 6e69 7465 5265 736f 6c75 7469 6f6e 2c20 niteResolution, │ │ │ │ -0003d290: 5570 3a20 546f 700a 0a6d 616b 6546 696e Up: Top..makeFin │ │ │ │ -0003d2a0: 6974 6552 6573 6f6c 7574 696f 6e43 6f64 iteResolutionCod │ │ │ │ -0003d2b0: 696d 3220 2d2d 204d 6170 7320 6173 736f im2 -- Maps asso │ │ │ │ -0003d2c0: 6369 6174 6564 2074 6f20 7468 6520 6669 ciated to the fi │ │ │ │ -0003d2d0: 6e69 7465 2072 6573 6f6c 7574 696f 6e20 nite resolution │ │ │ │ -0003d2e0: 6f66 2061 2068 6967 6820 7379 7a79 6779 of a high syzygy │ │ │ │ -0003d2f0: 206d 6f64 756c 6520 696e 2063 6f64 696d module in codim │ │ │ │ -0003d300: 2032 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2.************* │ │ │ │ +0003d160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +0003d170: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +0003d180: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +0003d190: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +0003d1a0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +0003d1b0: 756c 6179 322d 312e 3236 2e30 352b 6473 ulay2-1.26.05+ds │ │ │ │ +0003d1c0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +0003d1d0: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ +0003d1e0: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +0003d1f0: 6c75 7469 6f6e 732e 6d32 3a32 3839 393a lutions.m2:2899: │ │ │ │ +0003d200: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ +0003d210: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +0003d220: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ +0003d230: 204e 6f64 653a 206d 616b 6546 696e 6974 Node: makeFinit │ │ │ │ +0003d240: 6552 6573 6f6c 7574 696f 6e43 6f64 696d eResolutionCodim │ │ │ │ +0003d250: 322c 204e 6578 743a 206d 616b 6548 6f6d 2, Next: makeHom │ │ │ │ +0003d260: 6f74 6f70 6965 732c 2050 7265 763a 206d otopies, Prev: m │ │ │ │ +0003d270: 616b 6546 696e 6974 6552 6573 6f6c 7574 akeFiniteResolut │ │ │ │ +0003d280: 696f 6e2c 2055 703a 2054 6f70 0a0a 6d61 ion, Up: Top..ma │ │ │ │ +0003d290: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ +0003d2a0: 6f6e 436f 6469 6d32 202d 2d20 4d61 7073 onCodim2 -- Maps │ │ │ │ +0003d2b0: 2061 7373 6f63 6961 7465 6420 746f 2074 associated to t │ │ │ │ +0003d2c0: 6865 2066 696e 6974 6520 7265 736f 6c75 he finite resolu │ │ │ │ +0003d2d0: 7469 6f6e 206f 6620 6120 6869 6768 2073 tion of a high s │ │ │ │ +0003d2e0: 797a 7967 7920 6d6f 6475 6c65 2069 6e20 yzygy module in │ │ │ │ +0003d2f0: 636f 6469 6d20 320a 2a2a 2a2a 2a2a 2a2a codim 2.******** │ │ │ │ +0003d300: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003d310: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003d320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003d330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003d340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003d350: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0003d360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ -0003d370: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -0003d380: 2020 6d61 7073 203d 206d 616b 6546 696e maps = makeFin │ │ │ │ -0003d390: 6974 6552 6573 6f6c 7574 696f 6e43 6f64 iteResolutionCod │ │ │ │ -0003d3a0: 696d 3228 6666 2c6d 6629 0a20 202a 2049 im2(ff,mf). * I │ │ │ │ -0003d3b0: 6e70 7574 733a 0a20 2020 2020 202a 206d nputs:. * m │ │ │ │ -0003d3c0: 662c 2061 202a 6e6f 7465 206c 6973 743a f, a *note list: │ │ │ │ -0003d3d0: 2028 4d61 6361 756c 6179 3244 6f63 294c (Macaulay2Doc)L │ │ │ │ -0003d3e0: 6973 742c 2c20 6d61 7472 6978 2066 6163 ist,, matrix fac │ │ │ │ -0003d3f0: 746f 7269 7a61 7469 6f6e 0a20 2020 2020 torization. │ │ │ │ -0003d400: 202a 2066 662c 2061 202a 6e6f 7465 206d * ff, a *note m │ │ │ │ -0003d410: 6174 7269 783a 2028 4d61 6361 756c 6179 atrix: (Macaulay │ │ │ │ -0003d420: 3244 6f63 294d 6174 7269 782c 2c20 7265 2Doc)Matrix,, re │ │ │ │ -0003d430: 6775 6c61 7220 7365 7175 656e 6365 0a20 gular sequence. │ │ │ │ -0003d440: 202a 202a 6e6f 7465 204f 7074 696f 6e61 * *note Optiona │ │ │ │ -0003d450: 6c20 696e 7075 7473 3a20 284d 6163 6175 l inputs: (Macau │ │ │ │ -0003d460: 6c61 7932 446f 6329 7573 696e 6720 6675 lay2Doc)using fu │ │ │ │ -0003d470: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ -0003d480: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20 ional inputs,:. │ │ │ │ -0003d490: 2020 2020 202a 2043 6865 636b 203d 3e20 * Check => │ │ │ │ -0003d4a0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -0003d4b0: 7565 2066 616c 7365 0a20 202a 204f 7574 ue false. * Out │ │ │ │ -0003d4c0: 7075 7473 3a0a 2020 2020 2020 2a20 6d61 puts:. * ma │ │ │ │ -0003d4d0: 7073 2c20 6120 2a6e 6f74 6520 6861 7368 ps, a *note hash │ │ │ │ -0003d4e0: 2074 6162 6c65 3a20 284d 6163 6175 6c61 table: (Macaula │ │ │ │ -0003d4f0: 7932 446f 6329 4861 7368 5461 626c 652c y2Doc)HashTable, │ │ │ │ -0003d500: 2c20 6d61 6e79 206d 6170 730a 0a44 6573 , many maps..Des │ │ │ │ -0003d510: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -0003d520: 3d3d 3d3d 0a0a 4769 7665 6e20 6120 636f ====..Given a co │ │ │ │ -0003d530: 6469 6d20 3220 6d61 7472 6978 2066 6163 dim 2 matrix fac │ │ │ │ -0003d540: 746f 7269 7a61 7469 6f6e 2c20 6d61 6b65 torization, make │ │ │ │ -0003d550: 7320 616c 6c20 7468 6520 636f 6d70 6f6e s all the compon │ │ │ │ -0003d560: 656e 7473 206f 6620 7468 650a 6469 6666 ents of the.diff │ │ │ │ -0003d570: 6572 656e 7469 616c 2061 6e64 206f 6620 erential and of │ │ │ │ -0003d580: 7468 6520 686f 6d6f 746f 7069 6573 2074 the homotopies t │ │ │ │ -0003d590: 6861 7420 6172 6520 7265 6c65 7661 6e74 hat are relevant │ │ │ │ -0003d5a0: 2074 6f20 7468 6520 6669 6e69 7465 2072 to the finite r │ │ │ │ -0003d5b0: 6573 6f6c 7574 696f 6e2c 0a61 7320 696e esolution,.as in │ │ │ │ -0003d5c0: 2034 2e32 2e33 206f 6620 4569 7365 6e62 4.2.3 of Eisenb │ │ │ │ -0003d5d0: 7564 2d50 6565 7661 2022 4d69 6e69 6d61 ud-Peeva "Minima │ │ │ │ -0003d5e0: 6c20 4672 6565 2052 6573 6f6c 7574 696f l Free Resolutio │ │ │ │ -0003d5f0: 6e73 2061 6e64 2048 6967 6865 7220 4d61 ns and Higher Ma │ │ │ │ -0003d600: 7472 6978 0a46 6163 746f 7269 7a61 7469 trix.Factorizati │ │ │ │ -0003d610: 6f6e 7322 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ons"..+--------- │ │ │ │ +0003d360: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +0003d370: 2020 2020 2020 206d 6170 7320 3d20 6d61 maps = ma │ │ │ │ +0003d380: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ +0003d390: 6f6e 436f 6469 6d32 2866 662c 6d66 290a onCodim2(ff,mf). │ │ │ │ +0003d3a0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0003d3b0: 2020 2a20 6d66 2c20 6120 2a6e 6f74 6520 * mf, a *note │ │ │ │ +0003d3c0: 6c69 7374 3a20 284d 6163 6175 6c61 7932 list: (Macaulay2 │ │ │ │ +0003d3d0: 446f 6329 4c69 7374 2c2c 206d 6174 7269 Doc)List,, matri │ │ │ │ +0003d3e0: 7820 6661 6374 6f72 697a 6174 696f 6e0a x factorization. │ │ │ │ +0003d3f0: 2020 2020 2020 2a20 6666 2c20 6120 2a6e * ff, a *n │ │ │ │ +0003d400: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ +0003d410: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ +0003d420: 2c2c 2072 6567 756c 6172 2073 6571 7565 ,, regular seque │ │ │ │ +0003d430: 6e63 650a 2020 2a20 2a6e 6f74 6520 4f70 nce. * *note Op │ │ │ │ +0003d440: 7469 6f6e 616c 2069 6e70 7574 733a 2028 tional inputs: ( │ │ │ │ +0003d450: 4d61 6361 756c 6179 3244 6f63 2975 7369 Macaulay2Doc)usi │ │ │ │ +0003d460: 6e67 2066 756e 6374 696f 6e73 2077 6974 ng functions wit │ │ │ │ +0003d470: 6820 6f70 7469 6f6e 616c 2069 6e70 7574 h optional input │ │ │ │ +0003d480: 732c 3a0a 2020 2020 2020 2a20 4368 6563 s,:. * Chec │ │ │ │ +0003d490: 6b20 3d3e 202e 2e2e 2c20 6465 6661 756c k => ..., defaul │ │ │ │ +0003d4a0: 7420 7661 6c75 6520 6661 6c73 650a 2020 t value false. │ │ │ │ +0003d4b0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +0003d4c0: 202a 206d 6170 732c 2061 202a 6e6f 7465 * maps, a *note │ │ │ │ +0003d4d0: 2068 6173 6820 7461 626c 653a 2028 4d61 hash table: (Ma │ │ │ │ +0003d4e0: 6361 756c 6179 3244 6f63 2948 6173 6854 caulay2Doc)HashT │ │ │ │ +0003d4f0: 6162 6c65 2c2c 206d 616e 7920 6d61 7073 able,, many maps │ │ │ │ +0003d500: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +0003d510: 3d3d 3d3d 3d3d 3d3d 3d0a 0a47 6976 656e =========..Given │ │ │ │ +0003d520: 2061 2063 6f64 696d 2032 206d 6174 7269 a codim 2 matri │ │ │ │ +0003d530: 7820 6661 6374 6f72 697a 6174 696f 6e2c x factorization, │ │ │ │ +0003d540: 206d 616b 6573 2061 6c6c 2074 6865 2063 makes all the c │ │ │ │ +0003d550: 6f6d 706f 6e65 6e74 7320 6f66 2074 6865 omponents of the │ │ │ │ +0003d560: 0a64 6966 6665 7265 6e74 6961 6c20 616e .differential an │ │ │ │ +0003d570: 6420 6f66 2074 6865 2068 6f6d 6f74 6f70 d of the homotop │ │ │ │ +0003d580: 6965 7320 7468 6174 2061 7265 2072 656c ies that are rel │ │ │ │ +0003d590: 6576 616e 7420 746f 2074 6865 2066 696e evant to the fin │ │ │ │ +0003d5a0: 6974 6520 7265 736f 6c75 7469 6f6e 2c0a ite resolution,. │ │ │ │ +0003d5b0: 6173 2069 6e20 342e 322e 3320 6f66 2045 as in 4.2.3 of E │ │ │ │ +0003d5c0: 6973 656e 6275 642d 5065 6576 6120 224d isenbud-Peeva "M │ │ │ │ +0003d5d0: 696e 696d 616c 2046 7265 6520 5265 736f inimal Free Reso │ │ │ │ +0003d5e0: 6c75 7469 6f6e 7320 616e 6420 4869 6768 lutions and High │ │ │ │ +0003d5f0: 6572 204d 6174 7269 780a 4661 6374 6f72 er Matrix.Factor │ │ │ │ +0003d600: 697a 6174 696f 6e73 220a 0a2b 2d2d 2d2d izations"..+---- │ │ │ │ +0003d610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d650: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b -------+.|i1 : k │ │ │ │ -0003d660: 6b3d 5a5a 2f31 3031 2020 2020 2020 2020 k=ZZ/101 │ │ │ │ +0003d640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0003d650: 3120 3a20 6b6b 3d5a 5a2f 3130 3120 2020 1 : kk=ZZ/101 │ │ │ │ +0003d660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d690: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003d680: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003d690: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0003d6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d6d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0003d6e0: 6f31 203d 206b 6b20 2020 2020 2020 2020 o1 = kk │ │ │ │ +0003d6d0: 2020 7c0a 7c6f 3120 3d20 6b6b 2020 2020 |.|o1 = kk │ │ │ │ +0003d6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d720: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003d710: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003d720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d760: 2020 207c 0a7c 6f31 203a 2051 756f 7469 |.|o1 : Quoti │ │ │ │ -0003d770: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ +0003d750: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ +0003d760: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +0003d770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d7a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0003d790: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0003d7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d7e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -0003d7f0: 2053 203d 206b 6b5b 612c 625d 2020 2020 S = kk[a,b] │ │ │ │ +0003d7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0003d7e0: 7c69 3220 3a20 5320 3d20 6b6b 5b61 2c62 |i2 : S = kk[a,b │ │ │ │ +0003d7f0: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ 0003d800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d820: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003d820: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d860: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003d870: 0a7c 6f32 203d 2053 2020 2020 2020 2020 .|o2 = S │ │ │ │ +0003d860: 2020 2020 7c0a 7c6f 3220 3d20 5320 2020 |.|o2 = S │ │ │ │ +0003d870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d8b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003d8a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0003d8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d8f0: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c |.|o2 : Pol │ │ │ │ -0003d900: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ +0003d8e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0003d8f0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +0003d900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d930: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0003d920: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0003d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ -0003d980: 203a 2066 6620 3d20 6d61 7472 6978 2261 : ff = matrix"a │ │ │ │ -0003d990: 342c 6234 2220 2020 2020 2020 2020 2020 4,b4" │ │ │ │ +0003d970: 2b0a 7c69 3320 3a20 6666 203d 206d 6174 +.|i3 : ff = mat │ │ │ │ +0003d980: 7269 7822 6134 2c62 3422 2020 2020 2020 rix"a4,b4" │ │ │ │ +0003d990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d9b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0003d9c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0003d9b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003d9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003da00: 207c 0a7c 6f33 203d 207c 2061 3420 6234 |.|o3 = | a4 b4 │ │ │ │ -0003da10: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0003d9f0: 2020 2020 2020 7c0a 7c6f 3320 3d20 7c20 |.|o3 = | │ │ │ │ +0003da00: 6134 2062 3420 7c20 2020 2020 2020 2020 a4 b4 | │ │ │ │ +0003da10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003da20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003da30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003da40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0003da30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003da40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003da50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003da60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003da70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003da80: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0003da90: 2020 2020 2020 2031 2020 2020 2020 3220 1 2 │ │ │ │ +0003da70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003da80: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ +0003da90: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0003daa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dac0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ -0003dad0: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20 : Matrix S <-- │ │ │ │ -0003dae0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +0003dab0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003dac0: 0a7c 6f33 203a 204d 6174 7269 7820 5320 .|o3 : Matrix S │ │ │ │ +0003dad0: 203c 2d2d 2053 2020 2020 2020 2020 2020 <-- S │ │ │ │ +0003dae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003daf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003db00: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0003db00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0003db10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003db20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003db30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003db40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003db50: 2b0a 7c69 3420 3a20 5220 3d20 532f 6964 +.|i4 : R = S/id │ │ │ │ -0003db60: 6561 6c20 6666 2020 2020 2020 2020 2020 eal ff │ │ │ │ +0003db40: 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 203d -----+.|i4 : R = │ │ │ │ +0003db50: 2053 2f69 6465 616c 2066 6620 2020 2020 S/ideal ff │ │ │ │ +0003db60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003db70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003db80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003db90: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003db80: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0003db90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dbd0: 2020 2020 2020 7c0a 7c6f 3420 3d20 5220 |.|o4 = R │ │ │ │ +0003dbc0: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ +0003dbd0: 203d 2052 2020 2020 2020 2020 2020 2020 = R │ │ │ │ 0003dbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dc10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003dc00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0003dc10: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0003dc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dc50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003dc60: 3420 3a20 5175 6f74 6965 6e74 5269 6e67 4 : QuotientRing │ │ │ │ +0003dc50: 207c 0a7c 6f34 203a 2051 756f 7469 656e |.|o4 : Quotien │ │ │ │ +0003dc60: 7452 696e 6720 2020 2020 2020 2020 2020 tRing │ │ │ │ 0003dc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dc90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003dca0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0003dc90: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0003dca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003dcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003dcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003dcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003dce0: 2d2d 2b0a 7c69 3520 3a20 4e20 3d20 525e --+.|i5 : N = R^ │ │ │ │ -0003dcf0: 312f 6964 6561 6c22 6132 2c20 6162 2c20 1/ideal"a2, ab, │ │ │ │ -0003dd00: 6233 2220 2020 2020 2020 2020 2020 2020 b3" │ │ │ │ -0003dd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dd20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003dcd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 204e -------+.|i5 : N │ │ │ │ +0003dce0: 203d 2052 5e31 2f69 6465 616c 2261 322c = R^1/ideal"a2, │ │ │ │ +0003dcf0: 2061 622c 2062 3322 2020 2020 2020 2020 ab, b3" │ │ │ │ +0003dd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dd10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003dd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dd60: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ -0003dd70: 636f 6b65 726e 656c 207c 2061 3220 6162 cokernel | a2 ab │ │ │ │ -0003dd80: 2062 3320 7c20 2020 2020 2020 2020 2020 b3 | │ │ │ │ +0003dd50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0003dd60: 6f35 203d 2063 6f6b 6572 6e65 6c20 7c20 o5 = cokernel | │ │ │ │ +0003dd70: 6132 2061 6220 6233 207c 2020 2020 2020 a2 ab b3 | │ │ │ │ +0003dd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dda0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0003dda0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ddb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ddc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ddd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dde0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0003ddf0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0003de00: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ +0003dde0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003ddf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003de00: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 0003de10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003de20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003de30: 207c 0a7c 6f35 203a 2052 2d6d 6f64 756c |.|o5 : R-modul │ │ │ │ -0003de40: 652c 2071 756f 7469 656e 7420 6f66 2052 e, quotient of R │ │ │ │ +0003de20: 2020 2020 2020 7c0a 7c6f 3520 3a20 522d |.|o5 : R- │ │ │ │ +0003de30: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74 module, quotient │ │ │ │ +0003de40: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ 0003de50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003de60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003de70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0003de60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0003de70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003de80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003de90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003dea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003deb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204e -------+.|i6 : N │ │ │ │ -0003dec0: 203d 2063 6f6b 6572 2076 6172 7320 5220 = coker vars R │ │ │ │ +0003dea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0003deb0: 3620 3a20 4e20 3d20 636f 6b65 7220 7661 6 : N = coker va │ │ │ │ +0003dec0: 7273 2052 2020 2020 2020 2020 2020 2020 rs R │ │ │ │ 0003ded0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003def0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003dee0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003def0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0003df00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003df10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003df30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0003df40: 6f36 203d 2063 6f6b 6572 6e65 6c20 7c20 o6 = cokernel | │ │ │ │ -0003df50: 6120 6220 7c20 2020 2020 2020 2020 2020 a b | │ │ │ │ +0003df30: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e |.|o6 = cokern │ │ │ │ +0003df40: 656c 207c 2061 2062 207c 2020 2020 2020 el | a b | │ │ │ │ +0003df50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003df60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003df70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003df80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003df70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003df80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003df90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dfc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003dfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dfe0: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -0003dff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e000: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d |.|o6 : R- │ │ │ │ -0003e010: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74 module, quotient │ │ │ │ -0003e020: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ -0003e030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e040: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0003dfb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0003dfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dfd0: 2020 2020 2020 2031 2020 2020 2020 2020 1 │ │ │ │ +0003dfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dff0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ +0003e000: 203a 2052 2d6d 6f64 756c 652c 2071 756f : R-module, quo │ │ │ │ +0003e010: 7469 656e 7420 6f66 2052 2020 2020 2020 tient of R │ │ │ │ +0003e020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e030: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0003e040: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0003e050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0003e090: 3720 3a20 4d20 3d20 6869 6768 5379 7a79 7 : M = highSyzy │ │ │ │ -0003e0a0: 6779 204e 2020 2020 2020 2020 2020 2020 gy N │ │ │ │ +0003e080: 2d2b 0a7c 6937 203a 204d 203d 2068 6967 -+.|i7 : M = hig │ │ │ │ +0003e090: 6853 797a 7967 7920 4e20 2020 2020 2020 hSyzygy N │ │ │ │ +0003e0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e0c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e0d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0003e0c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0003e0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e110: 2020 7c0a 7c6f 3720 3d20 636f 6b65 726e |.|o7 = cokern │ │ │ │ -0003e120: 656c 207b 327d 207c 2030 202d 6233 2061 el {2} | 0 -b3 a │ │ │ │ -0003e130: 3320 3020 7c20 2020 2020 2020 2020 2020 3 0 | │ │ │ │ -0003e140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e150: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0003e160: 2020 2020 2020 7b34 7d20 7c20 6220 6120 {4} | b a │ │ │ │ -0003e170: 2020 3020 2030 207c 2020 2020 2020 2020 0 0 | │ │ │ │ -0003e180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e190: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0003e1a0: 2020 2020 2020 2020 207b 347d 207c 2030 {4} | 0 │ │ │ │ -0003e1b0: 2030 2020 2062 2020 6120 7c20 2020 2020 0 b a | │ │ │ │ +0003e100: 2020 2020 2020 207c 0a7c 6f37 203d 2063 |.|o7 = c │ │ │ │ +0003e110: 6f6b 6572 6e65 6c20 7b32 7d20 7c20 3020 okernel {2} | 0 │ │ │ │ +0003e120: 2d62 3320 6133 2030 207c 2020 2020 2020 -b3 a3 0 | │ │ │ │ +0003e130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e140: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e150: 2020 2020 2020 2020 2020 207b 347d 207c {4} | │ │ │ │ +0003e160: 2062 2061 2020 2030 2020 3020 7c20 2020 b a 0 0 | │ │ │ │ +0003e170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e180: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0003e190: 2020 2020 2020 2020 2020 2020 2020 7b34 {4 │ │ │ │ +0003e1a0: 7d20 7c20 3020 3020 2020 6220 2061 207c } | 0 0 b a | │ │ │ │ +0003e1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e1d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0003e1d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e210: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0003e220: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0003e230: 2020 2020 2020 2020 2020 2020 2033 2020 3 │ │ │ │ +0003e210: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003e220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e230: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0003e240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e260: 207c 0a7c 6f37 203a 2052 2d6d 6f64 756c |.|o7 : R-modul │ │ │ │ -0003e270: 652c 2071 756f 7469 656e 7420 6f66 2052 e, quotient of R │ │ │ │ +0003e250: 2020 2020 2020 7c0a 7c6f 3720 3a20 522d |.|o7 : R- │ │ │ │ +0003e260: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74 module, quotient │ │ │ │ +0003e270: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ 0003e280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e2a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0003e290: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0003e2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e2e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 204d -------+.|i8 : M │ │ │ │ -0003e2f0: 5320 3d20 7075 7368 466f 7277 6172 6428 S = pushForward( │ │ │ │ -0003e300: 6d61 7028 522c 5329 2c4d 2920 2020 2020 map(R,S),M) │ │ │ │ -0003e310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e320: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0003e2e0: 3820 3a20 4d53 203d 2070 7573 6846 6f72 8 : MS = pushFor │ │ │ │ +0003e2f0: 7761 7264 286d 6170 2852 2c53 292c 4d29 ward(map(R,S),M) │ │ │ │ +0003e300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e310: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003e320: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0003e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e360: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0003e370: 6f38 203d 2063 6f6b 6572 6e65 6c20 7b32 o8 = cokernel {2 │ │ │ │ -0003e380: 7d20 7c20 3020 6233 2061 3320 3020 3020 } | 0 b3 a3 0 0 │ │ │ │ -0003e390: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0003e3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e3b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003e3c0: 207b 347d 207c 2062 202d 6120 3020 2030 {4} | b -a 0 0 │ │ │ │ -0003e3d0: 2030 2020 7c20 2020 2020 2020 2020 2020 0 | │ │ │ │ -0003e3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e3f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003e400: 2020 2020 7b34 7d20 7c20 3020 3020 2062 {4} | 0 0 b │ │ │ │ -0003e410: 2020 6120 6234 207c 2020 2020 2020 2020 a b4 | │ │ │ │ -0003e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e430: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0003e360: 2020 7c0a 7c6f 3820 3d20 636f 6b65 726e |.|o8 = cokern │ │ │ │ +0003e370: 656c 207b 327d 207c 2030 2062 3320 6133 el {2} | 0 b3 a3 │ │ │ │ +0003e380: 2030 2030 2020 7c20 2020 2020 2020 2020 0 0 | │ │ │ │ +0003e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e3a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003e3b0: 2020 2020 2020 7b34 7d20 7c20 6220 2d61 {4} | b -a │ │ │ │ +0003e3c0: 2030 2020 3020 3020 207c 2020 2020 2020 0 0 0 | │ │ │ │ +0003e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e3e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0003e3f0: 2020 2020 2020 2020 207b 347d 207c 2030 {4} | 0 │ │ │ │ +0003e400: 2030 2020 6220 2061 2062 3420 7c20 2020 0 b a b4 | │ │ │ │ +0003e410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e420: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0003e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e470: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0003e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e490: 2020 2020 2020 2020 3320 2020 2020 2020 3 │ │ │ │ +0003e460: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0003e470: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0003e480: 2020 2020 2020 2020 2020 2020 2033 2020 3 │ │ │ │ +0003e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e4b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003e4c0: 3820 3a20 532d 6d6f 6475 6c65 2c20 7175 8 : S-module, qu │ │ │ │ -0003e4d0: 6f74 6965 6e74 206f 6620 5320 2020 2020 otient of S │ │ │ │ +0003e4b0: 207c 0a7c 6f38 203a 2053 2d6d 6f64 756c |.|o8 : S-modul │ │ │ │ +0003e4c0: 652c 2071 756f 7469 656e 7420 6f66 2053 e, quotient of S │ │ │ │ +0003e4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e4f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e500: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0003e4f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0003e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e540: 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2020 3020 3020 2020 2d61 5} | 0 0 0 -a │ │ │ │ +0003e670: 2062 3320 7c20 207b 347d 207c 2031 2030 b3 | {4} | 1 0 │ │ │ │ +0003e680: 2020 3020 7c20 7c0a 7c20 2020 2020 2020 0 | |.| │ │ │ │ +0003e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e6a0: 2020 7b35 7d20 7c20 3020 2030 2061 3320 {5} | 0 0 a3 │ │ │ │ +0003e6b0: 2062 2020 3020 207c 2020 2020 2020 2020 b 0 | │ │ │ │ +0003e6c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003e6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e710: 2020 2020 2020 207c 0a7c 6f39 203a 204c |.|o9 : L │ │ │ │ -0003e720: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ +0003e700: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0003e710: 3920 3a20 4c69 7374 2020 2020 2020 2020 9 : List │ │ │ │ +0003e720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e750: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0003e740: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003e750: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0003e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0003e7a0: 6931 3020 3a20 4720 3d20 6d61 6b65 4669 i10 : G = makeFi │ │ │ │ -0003e7b0: 6e69 7465 5265 736f 6c75 7469 6f6e 436f niteResolutionCo │ │ │ │ -0003e7c0: 6469 6d32 2866 662c 6d66 2920 2020 2020 dim2(ff,mf) │ │ │ │ -0003e7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e7e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003e790: 2d2d 2b0a 7c69 3130 203a 2047 203d 206d --+.|i10 : G = m │ │ │ │ +0003e7a0: 616b 6546 696e 6974 6552 6573 6f6c 7574 akeFiniteResolut │ │ │ │ +0003e7b0: 696f 6e43 6f64 696d 3228 6666 2c6d 6629 ionCodim2(ff,mf) │ │ │ │ +0003e7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e7d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003e7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e820: 2020 207c 0a7c 6f31 3020 3d20 4861 7368 |.|o10 = Hash │ │ │ │ -0003e830: 5461 626c 657b 2261 6c70 6861 2220 3d3e Table{"alpha" => │ │ │ │ -0003e840: 207b 357d 207c 2030 2020 2030 207c 2020 {5} | 0 0 | │ │ │ │ -0003e850: 2020 2020 2020 2020 2020 2020 7d20 2020 } │ │ │ │ -0003e860: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0003e870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e880: 2020 2020 7b35 7d20 7c20 2d62 3320 3020 {5} | -b3 0 │ │ │ │ -0003e890: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0003e8a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0003e8b0: 2020 2020 2020 2020 2020 2020 2262 2220 "b" │ │ │ │ -0003e8c0: 3d3e 207b 347d 207c 2062 2061 207c 2020 => {4} | b a | │ │ │ │ +0003e810: 2020 2020 2020 2020 7c0a 7c6f 3130 203d |.|o10 = │ │ │ │ +0003e820: 2048 6173 6854 6162 6c65 7b22 616c 7068 HashTable{"alph │ │ │ │ +0003e830: 6122 203d 3e20 7b35 7d20 7c20 3020 2020 a" => {5} | 0 │ │ │ │ +0003e840: 3020 7c20 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ +0003e850: 207d 2020 2020 2020 2020 207c 0a7c 2020 } |.| │ │ │ │ +0003e860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e870: 2020 2020 2020 2020 207b 357d 207c 202d {5} | - │ │ │ │ +0003e880: 6233 2030 207c 2020 2020 2020 2020 2020 b3 0 | │ │ │ │ +0003e890: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0003e8a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0003e8b0: 2022 6222 203d 3e20 7b34 7d20 7c20 6220 "b" => {4} | b │ │ │ │ +0003e8c0: 6120 7c20 2020 2020 2020 2020 2020 2020 a | │ │ │ │ 0003e8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e8e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0003e8f0: 2020 2020 2020 2020 2020 2020 2020 2022 " │ │ │ │ -0003e900: 6831 2722 203d 3e20 7b35 7d20 7c20 3020 h1'" => {5} | 0 │ │ │ │ -0003e910: 2020 3020 2030 2020 7c20 2020 2020 2020 0 0 | │ │ │ │ -0003e920: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e930: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003e940: 2020 2020 2020 2020 2020 207b 357d 207c {5} | │ │ │ │ -0003e950: 202d 6233 2030 2020 3020 207c 2020 2020 -b3 0 0 | │ │ │ │ -0003e960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e970: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003e980: 2020 2020 2020 2020 2020 2020 2020 7b35 {5 │ │ │ │ -0003e990: 7d20 7c20 3020 2020 2d61 2062 3320 7c20 } | 0 -a b3 | │ │ │ │ -0003e9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e9b0: 2020 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3e20 7b34 7d20 7c20 rtial" => {4} | │ │ │ │ -0003ebb0: 6120 2d62 207c 2020 2020 2020 2020 2020 a -b | │ │ │ │ -0003ebc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0003e8e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003e8f0: 2020 2020 2268 3127 2220 3d3e 207b 357d "h1'" => {5} │ │ │ │ +0003e900: 207c 2030 2020 2030 2020 3020 207c 2020 | 0 0 0 | │ │ │ │ +0003e910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e920: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0003e930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e940: 7b35 7d20 7c20 2d62 3320 3020 2030 2020 {5} | -b3 0 0 │ │ │ │ +0003e950: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0003e960: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0003e970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e980: 2020 207b 357d 207c 2030 2020 202d 6120 {5} | 0 -a │ │ │ │ +0003e990: 6233 207c 2020 2020 2020 2020 2020 2020 b3 | │ │ │ │ +0003e9a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 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6f6d 706c 0....File: Compl │ │ │ │ +0003f590: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +0003f5a0: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ +0003f5b0: 204e 6f64 653a 206d 616b 6548 6f6d 6f74 Node: makeHomot │ │ │ │ +0003f5c0: 6f70 6965 732c 204e 6578 743a 206d 616b opies, Next: mak │ │ │ │ +0003f5d0: 6548 6f6d 6f74 6f70 6965 7331 2c20 5072 eHomotopies1, Pr │ │ │ │ +0003f5e0: 6576 3a20 6d61 6b65 4669 6e69 7465 5265 ev: makeFiniteRe │ │ │ │ +0003f5f0: 736f 6c75 7469 6f6e 436f 6469 6d32 2c20 solutionCodim2, │ │ │ │ +0003f600: 5570 3a20 546f 700a 0a6d 616b 6548 6f6d Up: Top..makeHom │ │ │ │ +0003f610: 6f74 6f70 6965 7320 2d2d 2072 6574 7572 otopies -- retur │ │ │ │ +0003f620: 6e73 2061 2073 7973 7465 6d20 6f66 2068 ns a system of h │ │ │ │ +0003f630: 6967 6865 7220 686f 6d6f 746f 7069 6573 igher homotopies │ │ │ │ +0003f640: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ 0003f650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 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-0003f920: 7920 6f66 2068 6967 6865 7220 686f 6d6f y of higher homo │ │ │ │ -0003f930: 746f 7069 6573 206f 6e20 4620 666f 7220 topies on F for │ │ │ │ -0003f940: 7468 6520 656c 656d 656e 7473 206f 6620 the elements of │ │ │ │ -0003f950: 662c 2069 6e20 7468 6520 7365 6e73 650a f, in the sense. │ │ │ │ -0003f960: 6465 7363 7269 6265 642c 2066 6f72 2065 described, for e │ │ │ │ -0003f970: 7861 6d70 6c65 2c20 696e 2045 6973 656e xample, in Eisen │ │ │ │ -0003f980: 6275 6420 2245 6e72 6963 6865 6420 4672 bud "Enriched Fr │ │ │ │ -0003f990: 6565 2052 6573 6f6c 7574 696f 6e73 2061 ee Resolutions a │ │ │ │ -0003f9a0: 6e64 2043 6861 6e67 6520 6f66 0a52 696e nd Change of.Rin │ │ │ │ -0003f9b0: 6773 222e 0a0a 5468 6520 6f75 7470 7574 gs"...The output │ │ │ │ -0003f9c0: 2069 7320 6120 6861 7368 2074 6162 6c65 is a hash table │ │ │ │ -0003f9d0: 2077 6974 6820 656e 7472 6965 7320 6f66 with entries of │ │ │ │ -0003f9e0: 2074 6865 2066 6f72 6d20 245c 7b4a 2c69 the form $\{J,i │ │ │ │ -0003f9f0: 5c7d 3d3e 7324 2c20 7768 6572 6520 4a20 \}=>s$, where J │ │ │ │ -0003fa00: 6973 2061 0a6c 6973 7420 6f66 206e 6f6e is a.list of non │ │ │ │ -0003fa10: 2d6e 6567 6174 6976 6520 696e 7465 6765 -negative intege │ │ │ │ -0003fa20: 7273 2c20 6f66 206c 656e 6774 6820 6e20 rs, of length n │ │ │ │ -0003fa30: 616e 6420 2448 5c23 5c7b 4a2c 695c 7d3a and $H\#\{J,i\}: │ │ │ │ -0003fa40: 2046 5f69 2d3e 465f 7b69 2b32 7c4a 7c2d F_i->F_{i+2|J|- │ │ │ │ -0003fa50: 317d 240a 6172 6520 6d61 7073 2073 6174 1}$.are maps sat │ │ │ │ -0003fa60: 6973 6679 696e 6720 7468 6520 636f 6e64 isfying the cond │ │ │ │ -0003fa70: 6974 696f 6e73 2024 2420 485c 235c 7b65 itions $$ H\#\{e │ │ │ │ -0003fa80: 302c 695c 7d20 3d20 643b 2024 2420 616e 0,i\} = d; $$ an │ │ │ │ -0003fa90: 6420 2424 0a48 235c 7b65 302c 692b 315c d $$.H#\{e0,i+1\ │ │ │ │ -0003faa0: 7d2a 4823 5c7b 652c 695c 7d2b 4823 5c7b }*H#\{e,i\}+H#\{ │ │ │ │ -0003fab0: 652c 692d 315c 7d48 235c 7b65 302c 695c e,i-1\}H#\{e0,i\ │ │ │ │ -0003fac0: 7d20 3d20 665f 692c 2024 2420 7768 6572 } = f_i, $$ wher │ │ │ │ -0003fad0: 6520 2465 3020 3d0a 5c7b 302c 5c64 6f74 e $e0 =.\{0,\dot │ │ │ │ -0003fae0: 732c 305c 7d24 2061 6e64 2024 6524 2069 s,0\}$ and $e$ i │ │ │ │ -0003faf0: 7320 7468 6520 696e 6465 7820 6f66 2064 s the index of d │ │ │ │ -0003fb00: 6567 7265 6520 3120 7769 7468 2061 2031 egree 1 with a 1 │ │ │ │ -0003fb10: 2069 6e20 7468 6520 2469 242d 7468 2070 in the $i$-th p │ │ │ │ -0003fb20: 6c61 6365 3b0a 616e 642c 2066 6f72 2065 lace;.and, for e │ │ │ │ -0003fb30: 6163 6820 696e 6465 7820 6c69 7374 2049 ach index list I │ │ │ │ -0003fb40: 2077 6974 6820 7c49 7c3c 3d64 2c20 2424 with |I|<=d, $$ │ │ │ │ -0003fb50: 2073 756d 5f7b 4a3c 497d 2048 235c 7b49 sum_{Js$, whe │ │ │ │ +0003f9f0: 7265 204a 2069 7320 610a 6c69 7374 206f re J is a.list o │ │ │ │ +0003fa00: 6620 6e6f 6e2d 6e65 6761 7469 7665 2069 f non-negative i │ │ │ │ +0003fa10: 6e74 6567 6572 732c 206f 6620 6c65 6e67 ntegers, of leng │ │ │ │ +0003fa20: 7468 206e 2061 6e64 2024 485c 235c 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-00040430: 2030 2020 7c20 7c0a 7c20 2020 2020 2020 0 | |.| │ │ │ │ -00040440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040450: 2020 2020 2020 2020 2020 7b31 7d20 7c20 {1} | │ │ │ │ -00040460: 6120 2030 2020 2d63 2030 2020 2d64 2030 a 0 -c 0 -d 0 │ │ │ │ -00040470: 2020 7c20 7c0a 7c20 2020 2020 2020 2020 | |.| │ │ │ │ -00040480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040490: 2020 2020 2020 2020 7b31 7d20 7c20 3020 {1} | 0 │ │ │ │ -000404a0: 2061 2020 6220 2030 2020 3020 202d 6420 a b 0 0 -d │ │ │ │ -000404b0: 7c20 7c0a 7c20 2020 2020 2020 2020 2020 | |.| │ │ │ │ -000404c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000404d0: 2020 2020 2020 7b31 7d20 7c20 3020 2030 {1} | 0 0 │ │ │ │ -000404e0: 2020 3020 2061 2020 6220 2063 2020 7c20 0 a b c | │ │ │ │ -000404f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00040500: 2020 7b7b 302c 2030 2c20 307d 2c20 337d {{0, 0, 0}, 3} │ │ │ │ -00040510: 203d 3e20 7b32 7d20 7c20 6320 2064 2020 => {2} | c d │ │ │ │ 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│ -00040eb0: 2020 7b7b 312c 2030 2c20 307d 2c20 307d {{1, 0, 0}, 0} │ │ │ │ -00040ec0: 203d 3e20 7b31 7d20 7c20 3120 7c20 2020 => {1} | 1 | │ │ │ │ -00040ed0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00040ee0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00040ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040f00: 2020 7b31 7d20 7c20 3020 7c20 2020 2020 {1} | 0 | │ │ │ │ -00040f10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00040dc0: 207b 337d 207c 2030 2030 2020 3020 3020 {3} | 0 0 0 0 │ │ │ │ +00040dd0: 2030 2031 207c 2020 2020 207c 0a7c 2020 0 1 | |.| │ │ │ │ +00040de0: 2020 2020 2020 2020 2020 2020 207b 7b30 {{0 │ │ │ │ +00040df0: 2c20 312c 2031 7d2c 202d 317d 203d 3e20 , 1, 1}, -1} => │ │ │ │ +00040e00: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +00040e10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00040e20: 2020 2020 2020 2020 2020 207b 7b30 2c20 {{0, │ │ │ │ +00040e30: 322c 2030 7d2c 202d 317d 203d 3e20 3020 2, 0}, -1} => 0 │ │ │ │ +00040e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040e50: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00040e60: 2020 2020 2020 2020 207b 7b31 2c20 302c {{1, 0, │ │ │ │ +00040e70: 2030 7d2c 202d 317d 203d 3e20 3020 2020 0}, -1} => 0 │ │ │ │ +00040e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040e90: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00040ea0: 2020 2020 2020 207b 7b31 2c20 302c 2030 {{1, 0, 0 │ │ │ │ +00040eb0: 7d2c 2030 7d20 3d3e 207b 317d 207c 2031 }, 0} => {1} | 1 │ │ │ │ +00040ec0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00040ed0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00040ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040ef0: 2020 2020 2020 207b 317d 207c 2030 207c {1} | 0 | │ │ │ │ +00040f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040f10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00040f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040f30: 2020 2020 2020 2020 2020 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2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00041010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041020: 2020 2020 2020 2020 2020 2020 207b 327d {2} │ │ │ │ +00041030: 207c 2030 2030 2030 2030 207c 2020 2020 | 0 0 0 0 | │ │ │ │ +00041040: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00041050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041060: 2020 2020 2020 2020 2020 207b 327d 207c {2} | │ │ │ │ +00041070: 2030 2030 2030 2031 207c 2020 2020 2020 0 0 0 1 | │ │ │ │ +00041080: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00041090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000410a0: 2020 2020 2020 2020 207b 327d 207c 2030 {2} | 0 │ │ │ │ +000410b0: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ +000410c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000410d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000410e0: 2020 2020 2020 207b 327d 207c 2030 2030 {2} | 0 0 │ │ │ │ +000410f0: 2030 2030 207c 2020 2020 2020 2020 2020 0 0 | │ │ │ │ +00041100: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00041110: 2020 207b 7b31 2c20 302c 2030 7d2c 2032 {{1, 0, 0}, 2 │ │ │ │ +00041120: 7d20 3d3e 207b 337d 207c 2030 2030 2031 } => {3} | 0 0 1 │ │ │ │ +00041130: 2030 2030 2030 207c 2020 2020 2020 207c 0 0 0 | | │ │ │ │ +00041140: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00041150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041160: 2020 2020 2020 2020 2020 2020 2020 7b33 {3 │ │ │ │ -00041170: 7d20 7c20 3020 3020 3020 3020 3120 3020 } | 0 0 0 0 1 0 │ │ │ │ -00041180: 7c20 2020 2020 2020 7c0a 7c20 2020 2020 | |.| │ │ │ │ +00041160: 2020 207b 337d 207c 2030 2030 2030 2030 {3} | 0 0 0 0 │ │ │ │ +00041170: 2031 2030 207c 2020 2020 2020 207c 0a7c 1 0 | |.| │ │ │ │ +00041180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000411a0: 2020 2020 2020 2020 2020 2020 7b33 7d20 {3} │ │ │ │ -000411b0: 7c20 3020 3020 3020 3020 3020 3120 7c20 | 0 0 0 0 0 1 | │ │ │ │ -000411c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -000411d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000411e0: 2020 2020 2020 2020 2020 7b33 7d20 7c20 {3} | │ │ │ │ -000411f0: 3020 3020 3020 3020 3020 3020 7c20 2020 0 0 0 0 0 0 | │ │ │ │ -00041200: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00041210: 2020 2020 2020 7b7b 312c 2030 2c20 317d {{1, 0, 1} │ │ │ │ -00041220: 2c20 2d31 7d20 3d3e 2030 2020 2020 2020 , -1} => 0 │ │ │ │ -00041230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041240: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00041250: 2020 2020 7b7b 312c 2031 2c20 307d 2c20 {{1, 1, 0}, │ │ │ │ -00041260: 2d31 7d20 3d3e 2030 2020 2020 2020 2020 -1} => 0 │ │ │ │ -00041270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041280: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00041290: 2020 7b7b 322c 2030 2c20 307d 2c20 2d31 {{2, 0, 0}, -1 │ │ │ │ -000412a0: 7d20 3d3e 2030 2020 2020 2020 2020 2020 } => 0 │ │ │ │ -000412b0: 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+00041330: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00041340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041370: 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6e20 7468 --------+..In th │ │ │ │ -00041380: 6973 2063 6173 6520 7468 6520 6869 6768 is case the high │ │ │ │ -00041390: 6572 2068 6f6d 6f74 6f70 6965 7320 6172 er homotopies ar │ │ │ │ -000413a0: 6520 303a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d e 0:..+--------- │ │ │ │ +00041360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +00041370: 496e 2074 6869 7320 6361 7365 2074 6865 In this case the │ │ │ │ +00041380: 2068 6967 6865 7220 686f 6d6f 746f 7069 higher homotopi │ │ │ │ +00041390: 6573 2061 7265 2030 3a0a 0a2b 2d2d 2d2d es are 0:..+---- │ │ │ │ +000413a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000413b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000413c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000413d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000413e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204c -------+.|i6 : L │ │ │ │ -000413f0: 203d 2073 6f72 7420 7365 6c65 6374 286b = sort select(k │ │ │ │ -00041400: 6579 7320 686f 6d6f 742c 206b 2d3e 2868 eys homot, k->(h │ │ │ │ -00041410: 6f6d 6f74 236b 213d 3020 616e 6420 7375 omot#k!=0 and su │ │ │ │ -00041420: 6d28 6b5f 3029 3e31 2929 7c0a 7c20 2020 m(k_0)>1))|.| │ │ │ │ +000413d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000413e0: 3620 3a20 4c20 3d20 736f 7274 2073 656c 6 : L = sort sel │ │ │ │ +000413f0: 6563 7428 6b65 7973 2068 6f6d 6f74 2c20 ect(keys homot, │ │ │ │ +00041400: 6b2d 3e28 686f 6d6f 7423 6b21 3d30 2061 k->(homot#k!=0 a │ │ │ │ +00041410: 6e64 2073 756d 286b 5f30 293e 3129 297c nd sum(k_0)>1))| │ │ │ │ +00041420: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00041430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041460: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041470: 6f36 203d 207b 7d20 2020 2020 2020 2020 o6 = {} │ │ │ │ +00041460: 2020 7c0a 7c6f 3620 3d20 7b7d 2020 2020 |.|o6 = {} │ │ │ │ +00041470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000414a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000414b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000414a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000414b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000414c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000414d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000414e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000414f0: 2020 207c 0a7c 6f36 203a 204c 6973 7420 |.|o6 : List │ │ │ │ +000414e0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20 |.|o6 : │ │ │ │ +000414f0: 4c69 7374 2020 2020 2020 2020 2020 2020 List │ │ │ │ 00041500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041530: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00041520: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00041530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041570: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4f6e 2074 ---------+..On t │ │ │ │ -00041580: 6865 206f 7468 6572 2068 616e 642c 2069 he other hand, i │ │ │ │ -00041590: 6620 7765 2074 616b 6520 6120 636f 6d70 f we take a comp │ │ │ │ -000415a0: 6c65 7465 2069 6e74 6572 7365 6374 696f lete intersectio │ │ │ │ -000415b0: 6e20 616e 6420 736f 6d65 7468 696e 6720 n and something │ │ │ │ -000415c0: 636f 6e74 6169 6e65 640a 696e 2069 7420 contained.in it │ │ │ │ -000415d0: 696e 2061 206d 6f72 6520 636f 6d70 6c69 in a more compli │ │ │ │ -000415e0: 6361 7465 6420 7369 7475 6174 696f 6e2c cated situation, │ │ │ │ -000415f0: 2074 6865 2070 726f 6772 616d 2067 6976 the program giv │ │ │ │ -00041600: 6573 206e 6f6e 7a65 726f 2068 6967 6865 es nonzero highe │ │ │ │ -00041610: 720a 686f 6d6f 746f 7069 6573 3a0a 0a2b r.homotopies:..+ │ │ │ │ +00041560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00041570: 0a4f 6e20 7468 6520 6f74 6865 7220 6861 .On the other ha │ │ │ │ +00041580: 6e64 2c20 6966 2077 6520 7461 6b65 2061 nd, if we take a │ │ │ │ +00041590: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ +000415a0: 6563 7469 6f6e 2061 6e64 2073 6f6d 6574 ection and somet │ │ │ │ +000415b0: 6869 6e67 2063 6f6e 7461 696e 6564 0a69 hing contained.i │ │ │ │ +000415c0: 6e20 6974 2069 6e20 6120 6d6f 7265 2063 n it in a more c │ │ │ │ +000415d0: 6f6d 706c 6963 6174 6564 2073 6974 7561 omplicated situa │ │ │ │ +000415e0: 7469 6f6e 2c20 7468 6520 7072 6f67 7261 tion, the progra │ │ │ │ +000415f0: 6d20 6769 7665 7320 6e6f 6e7a 6572 6f20 m gives nonzero │ │ │ │ +00041600: 6869 6768 6572 0a68 6f6d 6f74 6f70 6965 higher.homotopie │ │ │ │ +00041610: 733a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d s:..+----------- │ │ │ │ 00041620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00041670: 6937 203a 206b 6b3d 205a 5a2f 3332 3030 i7 : kk= ZZ/3200 │ │ │ │ -00041680: 333b 2020 2020 2020 2020 2020 2020 2020 3; │ │ │ │ +00041660: 2d2d 2b0a 7c69 3720 3a20 6b6b 3d20 5a5a --+.|i7 : kk= ZZ │ │ │ │ +00041670: 2f33 3230 3033 3b20 2020 2020 2020 2020 /32003; │ │ │ │ +00041680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000416a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000416b0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000416b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 000416c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000416d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000416e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000416f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00041710: 6938 203a 2053 203d 206b 6b5b 612c 622c i8 : S = kk[a,b, │ │ │ │ -00041720: 632c 645d 3b20 2020 2020 2020 2020 2020 c,d]; │ │ │ │ +00041700: 2d2d 2b0a 7c69 3820 3a20 5320 3d20 6b6b --+.|i8 : S = kk │ │ │ │ +00041710: 5b61 2c62 2c63 2c64 5d3b 2020 2020 2020 [a,b,c,d]; │ │ │ │ +00041720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041750: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00041750: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00041760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000417a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -000417b0: 6939 203a 204d 203d 2053 5e31 2f28 6964 i9 : M = S^1/(id │ │ │ │ -000417c0: 6561 6c22 6132 2c62 322c 6332 2c64 3222 eal"a2,b2,c2,d2" │ │ │ │ -000417d0: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ +000417a0: 2d2d 2b0a 7c69 3920 3a20 4d20 3d20 535e --+.|i9 : M = S^ │ │ │ │ +000417b0: 312f 2869 6465 616c 2261 322c 6232 2c63 1/(ideal"a2,b2,c │ │ │ │ +000417c0: 322c 6432 2229 3b20 2020 2020 2020 2020 2,d2"); │ │ │ │ +000417d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000417e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000417f0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000417f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00041800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00041850: 6931 3020 3a20 4620 3d20 6672 6565 5265 i10 : F = freeRe │ │ │ │ -00041860: 736f 6c75 7469 6f6e 204d 2020 2020 2020 solution M │ │ │ │ +00041840: 2d2d 2b0a 7c69 3130 203a 2046 203d 2066 --+.|i10 : F = f │ │ │ │ +00041850: 7265 6552 6573 6f6c 7574 696f 6e20 4d20 reeResolution M │ │ │ │ +00041860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041890: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00041890: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000418a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000418b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000418c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000418d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000418e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000418f0: 2020 2020 2020 2031 2020 2020 2020 3420 1 4 │ │ │ │ -00041900: 2020 2020 2036 2020 2020 2020 3420 2020 6 4 │ │ │ │ -00041910: 2020 2031 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +000418e0: 2020 7c0a 7c20 2020 2020 2020 3120 2020 |.| 1 │ │ │ │ +000418f0: 2020 2034 2020 2020 2020 3620 2020 2020 4 6 │ │ │ │ +00041900: 2034 2020 2020 2020 3120 2020 2020 2020 4 1 │ │ │ │ +00041910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041930: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041940: 6f31 3020 3d20 5320 203c 2d2d 2053 2020 o10 = S <-- S │ │ │ │ -00041950: 3c2d 2d20 5320 203c 2d2d 2053 2020 3c2d <-- S <-- S <- │ │ │ │ -00041960: 2d20 5320 2020 2020 2020 2020 2020 2020 - S │ │ │ │ +00041930: 2020 7c0a 7c6f 3130 203d 2053 2020 3c2d |.|o10 = S <- │ │ │ │ +00041940: 2d20 5320 203c 2d2d 2053 2020 3c2d 2d20 - S <-- S <-- │ │ │ │ +00041950: 5320 203c 2d2d 2053 2020 2020 2020 2020 S <-- S │ │ │ │ +00041960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041980: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 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2020 2020 2020 2020 2020 2020 │ │ │ │ 00041a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041a70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041a80: 6f31 3020 3a20 436f 6d70 6c65 7820 2020 o10 : Complex │ │ │ │ +00041a70: 2020 7c0a 7c6f 3130 203a 2043 6f6d 706c |.|o10 : Compl │ │ │ │ +00041a80: 6578 2020 2020 2020 2020 2020 2020 2020 ex │ │ │ │ 00041a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041ac0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00041ac0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00041ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00041b20: 6931 3120 3a20 7365 7452 616e 646f 6d53 i11 : setRandomS │ │ │ │ -00041b30: 6565 6420 3020 2020 2020 2020 2020 2020 eed 0 │ │ │ │ +00041b10: 2d2d 2b0a 7c69 3131 203a 2073 6574 5261 --+.|i11 : setRa │ │ │ │ +00041b20: 6e64 6f6d 5365 6564 2030 2020 2020 2020 ndomSeed 0 │ │ │ │ +00041b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041b60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041b70: 202d 2d20 7365 7474 696e 6720 7261 6e64 -- setting rand │ │ │ │ -00041b80: 6f6d 2073 6565 6420 746f 2030 2020 2020 om seed to 0 │ │ │ │ +00041b60: 2020 7c0a 7c20 2d2d 2073 6574 7469 6e67 |.| -- setting │ │ │ │ +00041b70: 2072 616e 646f 6d20 7365 6564 2074 6f20 random seed to │ │ │ │ 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2020 2020 207c 0a2b |.+ │ │ │ │ +00041c50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00041c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00041cb0: 6931 3220 3a20 6620 3d20 7261 6e64 6f6d i12 : f = random │ │ │ │ -00041cc0: 2853 5e31 2c53 5e7b 323a 2d35 7d29 3b20 (S^1,S^{2:-5}); │ │ │ │ +00041ca0: 2d2d 2b0a 7c69 3132 203a 2066 203d 2072 --+.|i12 : f = r │ │ │ │ +00041cb0: 616e 646f 6d28 535e 312c 535e 7b32 3a2d andom(S^1,S^{2:- │ │ │ │ +00041cc0: 357d 293b 2020 2020 2020 2020 2020 2020 5}); │ │ │ │ 00041cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041cf0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00041cf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00041d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041d40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041d50: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ -00041d60: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00041d40: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00041d50: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ +00041d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041d90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041da0: 6f31 3220 3a20 4d61 7472 6978 2053 2020 o12 : Matrix S │ │ │ │ -00041db0: 3c2d 2d20 5320 2020 2020 2020 2020 2020 <-- S │ │ │ │ +00041d90: 2020 7c0a 7c6f 3132 203a 204d 6174 7269 |.|o12 : Matri │ │ │ │ +00041da0: 7820 5320 203c 2d2d 2053 2020 2020 2020 x S <-- S │ │ │ │ +00041db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041de0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00041de0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00041df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00041e40: 6931 3320 3a20 686f 6d6f 7420 3d20 6d61 i13 : homot = ma │ │ │ │ -00041e50: 6b65 486f 6d6f 746f 7069 6573 2866 2c46 keHomotopies(f,F │ │ │ │ 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3133 203d 2048 6173 6854 |.|o13 = HashT │ │ │ │ +00041ee0: 6162 6c65 7b7b 7b30 2c20 307d 2c20 307d able{{{0, 0}, 0} │ │ │ │ +00041ef0: 203d 3e20 3020 2020 2020 2020 2020 2020 => 0 │ │ │ │ 00041f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041f20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041f40: 7b7b 302c 2030 7d2c 2031 7d20 3d3e 207c {{0, 0}, 1} => | │ │ │ │ -00041f50: 2061 3220 2020 2020 2020 2020 2020 2020 a2 │ │ │ │ +00041f20: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00041f30: 2020 2020 207b 7b30 2c20 307d 2c20 317d {{0, 0}, 1} │ │ │ │ +00041f40: 203d 3e20 7c20 6132 2020 2020 2020 2020 => | a2 │ │ │ │ +00041f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041f70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ 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0a7c |.| │ │ │ │ +00042010: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00042020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042030: 2020 2020 2020 2020 2020 2020 2020 207b { │ │ │ │ -00042040: 327d 207c 2020 2020 2020 2020 2020 2020 2} | │ │ │ │ +00042030: 2020 2020 7b32 7d20 7c20 2020 2020 2020 {2} | │ │ │ │ +00042040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00042050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042060: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00042060: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00042070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042080: 2020 2020 2020 2020 2020 2020 2020 207b { │ │ │ │ -00042090: 327d 207c 2020 2020 2020 2020 2020 2020 2} | │ │ │ │ +00042080: 2020 2020 7b32 7d20 7c20 2020 2020 2020 {2} | │ │ │ │ +00042090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000420a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000420b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 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2020 2020 2020 2020 2020 2020 │ │ │ │ 00053530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053540: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00053540: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00053550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053590: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00053590: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000535a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000535b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000535c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000535d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000535e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000535f0: 3633 6364 332b 3633 3431 6434 207c 2020 63cd3+6341d4 | │ │ │ │ +000535e0: 2020 7c0a 7c36 3363 6433 2b36 3334 3164 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2020 2020 2020 │ │ │ │ 000538c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000538d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000538e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000538f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053900: 207c 0a7c 6f31 3420 3a20 4c69 7374 2020 |.|o14 : List │ │ │ │ +000538f0: 2020 2020 2020 7c0a 7c6f 3134 203a 204c |.|o14 : L │ │ │ │ +00053900: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ 00053910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053950: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00053940: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00053950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053970: 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2020 2020 │ │ │ │ -00053a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053a40: 207c 0a7c 6f31 3520 3d20 3620 2020 2020 |.|o15 = 6 │ │ │ │ +00053a30: 2020 2020 2020 7c0a 7c6f 3135 203d 2036 |.|o15 = 6 │ │ │ │ +00053a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053a90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00053a80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00053a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00053ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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│ │ -00053ea0: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -00053eb0: 2d2b 2d2b 2020 2020 2020 2020 2020 2020 -+-+ │ │ │ │ +00053e90: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ +00053ea0: 2d2d 2d2d 2d2d 2b2d 2b20 2020 2020 2020 ------+-+ │ │ │ │ +00053eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053ef0: 207c 0a7c 2020 2020 2020 7c7b 322c 2030 |.| |{2, 0 │ │ │ │ -00053f00: 7d7c 317c 2020 2020 2020 2020 2020 2020 }|1| │ │ │ │ +00053ee0: 2020 2020 2020 7c0a 7c20 2020 2020 207c |.| | │ │ │ │ +00053ef0: 7b32 2c20 307d 7c31 7c20 2020 2020 2020 {2, 0}|1| │ │ │ │ +00053f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053f40: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -00053f50: 2d2b 2d2b 2020 2020 2020 2020 2020 2020 -+-+ │ │ │ │ +00053f30: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ +00053f40: 2d2d 2d2d 2d2d 2b2d 2b20 2020 2020 2020 ------+-+ │ │ │ │ +00053f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053f90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00053f80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00053f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00053fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00053fe0: 2d2b 0a0a 466f 7220 6578 616d 706c 6520 -+..For example │ │ │ │ -00053ff0: 7765 2068 6176 653a 0a0a 2b2d 2d2d 2d2d we have:..+----- │ │ │ │ +00053fd0: 2d2d 2d2d 2d2d 2b0a 0a46 6f72 2065 7861 ------+..For exa │ │ │ │ +00053fe0: 6d70 6c65 2077 6520 6861 7665 3a0a 0a2b mple we have:..+ │ │ │ │ +00053ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00054000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00054010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00054020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00054030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00054040: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a --------+.|i17 : │ │ │ │ -00054050: 2068 6f6d 6f74 2328 4c5f 3029 2020 2020 homot#(L_0) │ │ │ │ +00054030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00054040: 6931 3720 3a20 686f 6d6f 7423 284c 5f30 i17 : homot#(L_0 │ │ │ │ +00054050: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00054060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054090: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00054080: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00054090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000540a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000540b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000540c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000540d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000540e0: 2020 2020 2020 2020 7c0a 7c6f 3137 203d |.|o17 = │ │ │ │ -000540f0: 207b 367d 207c 202d 3133 3739 3561 342b {6} | -13795a4+ │ │ │ │ -00054100: 3230 3139 6133 622b 3133 3736 3961 3262 2019a3b+13769a2b │ │ │ │ -00054110: 322b 3735 3836 6162 332b 3836 3439 6234 2+7586ab3+8649b4 │ │ │ │ -00054120: 2b36 3435 3461 3363 2d31 3031 3837 6132 +6454a3c-10187a2 │ │ │ │ -00054130: 6263 2d31 3738 3361 7c0a 7c20 2020 2020 bc-1783a|.| │ │ │ │ -00054140: 207b 367d 207c 2031 3131 3532 6134 2d31 {6} | 11152a4-1 │ │ │ │ -00054150: 3333 3661 3362 2b31 3138 3436 6132 6232 336a3b+11846a2b2 │ │ │ │ -00054160: 2b31 3032 3634 6162 332b 3631 3862 342d +10264ab3+618b4- │ │ │ │ -00054170: 3131 3035 3161 3363 2b31 3231 3239 6132 11051a3c+12129a2 │ │ │ │ -00054180: 6263 2b35 3932 3761 7c0a 7c20 2020 2020 bc+5927a|.| │ │ │ │ -00054190: 207b 367d 207c 202d 3633 3338 6134 2b31 {6} | -6338a4+1 │ │ │ │ -000541a0: 3030 3235 6133 622b 3134 3938 3761 3363 0025a3b+14987a3c │ │ │ │ -000541b0: 2d39 3935 3961 3262 632d 3131 3639 3161 -9959a2bc-11691a │ │ │ │ -000541c0: 3263 322b 3132 3333 3661 6263 322d 3737 2c2+12336abc2-77 │ │ │ │ -000541d0: 3836 6133 642d 3131 7c0a 7c20 2020 2020 86a3d-11|.| │ │ │ │ -000541e0: 207b 367d 207c 2032 3237 3561 342d 3233 {6} | 2275a4-23 │ │ │ │ -000541f0: 3961 3362 2b31 3435 3934 6132 6232 2d38 9a3b+14594a2b2-8 │ │ │ │ -00054200: 3135 3361 6233 2d31 3139 3435 6234 2d38 153ab3-11945b4-8 │ │ │ │ -00054210: 3431 3661 3363 2b36 3235 3161 3262 632d 416a3c+6251a2bc- │ │ │ │ -00054220: 3330 3233 6162 3263 7c0a 7c20 2020 2020 3023ab2c|.| │ │ │ │ -00054230: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d --------------- │ │ │ │ +000540d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000540e0: 6f31 3720 3d20 7b36 7d20 7c20 2d31 3337 o17 = {6} | -137 │ │ │ │ +000540f0: 3935 6134 2b32 3031 3961 3362 2b31 3337 95a4+2019a3b+137 │ │ │ │ +00054100: 3639 6132 6232 2b37 3538 3661 6233 2b38 69a2b2+7586ab3+8 │ │ │ │ +00054110: 3634 3962 342b 3634 3534 6133 632d 3130 649b4+6454a3c-10 │ │ │ │ +00054120: 3138 3761 3262 632d 3137 3833 617c 0a7c 187a2bc-1783a|.| │ │ │ │ +00054130: 2020 2020 2020 7b36 7d20 7c20 3131 3135 {6} | 1115 │ │ │ │ +00054140: 3261 342d 3133 3336 6133 622b 3131 3834 2a4-1336a3b+1184 │ │ │ │ +00054150: 3661 3262 322b 3130 3236 3461 6233 2b36 6a2b2+10264ab3+6 │ │ │ │ +00054160: 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2d2d 2d2d ---------------- │ │ │ │ 00054240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00054250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00054260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00054270: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 --------|.| │ │ │ │ -00054280: 2062 3263 2b39 3231 3962 3363 2b35 3531 b2c+9219b3c+551 │ │ │ │ -00054290: 3361 3263 322b 3130 3535 3861 6263 322b 3a2c2+10558abc2+ │ │ │ │ -000542a0: 3235 3930 6232 6332 2b31 3136 3234 6133 2590b2c2+11624a3 │ │ │ │ -000542b0: 642d 3536 3033 6132 6264 2b31 3430 3538 d-5603a2bd+14058 │ │ │ │ -000542c0: 6162 3264 2d31 3236 7c0a 7c20 2020 2020 ab2d-126|.| │ │ │ │ -000542d0: 2062 3263 2b34 3839 6233 632d 3135 3338 b2c+489b3c-1538 │ │ │ │ -000542e0: 3361 3263 322b 3530 3761 6263 322d 3133 3a2c2+507abc2-13 │ │ │ │ -000542f0: 3830 3462 3263 322d 3834 3136 6163 332b 804b2c2-8416ac3+ │ │ │ │ -00054300: 3932 6334 2d31 3130 3537 6133 642d 3531 92c4-11057a3d-51 │ │ │ │ -00054310: 3133 6132 6264 2d32 7c0a 7c20 2020 2020 13a2bd-2|.| │ │ │ │ -00054320: 2035 3661 3262 642b 3439 3630 6132 6364 56a2bd+4960a2cd │ │ │ │ -00054330: 2d35 3538 3961 6263 642d 3831 3633 6163 -5589abcd-8163ac │ │ │ │ -00054340: 3264 2d31 3839 3562 6332 642b 3934 3634 2d-1895bc2d+9464 │ │ │ │ -00054350: 6132 6432 2d37 3235 3361 6264 322b 3132 a2d2-7253abd2+12 │ │ │ │ -00054360: 3634 3261 6364 322d 7c0a 7c20 2020 2020 642acd2-|.| │ │ │ │ -00054370: 202b 3539 3333 6233 632b 3932 6132 6332 +5933b3c+92a2c2 │ │ │ │ -00054380: 2b35 3334 3361 6263 322b 3337 3938 6232 +5343abc2+3798b2 │ │ │ │ -00054390: 6332 2d31 3539 3638 6133 642b 3437 3361 c2-15968a3d+473a │ │ │ │ -000543a0: 3262 642b 3133 3239 3361 6232 642d 3337 2bd+13293ab2d-37 │ │ │ │ -000543b0: 3631 6233 642d 3737 7c0a 7c20 2020 2020 61b3d-77|.| │ │ │ │ -000543c0: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d --------------- │ │ │ │ +00054260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +00054270: 2020 2020 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+9464a2d2-7253ab │ │ │ │ +00054350: 6432 2b31 3236 3432 6163 6432 2d7c 0a7c d2+12642acd2-|.| │ │ │ │ +00054360: 2020 2020 2020 2b35 3933 3362 3363 2b39 +5933b3c+9 │ │ │ │ +00054370: 3261 3263 322b 3533 3433 6162 6332 2b33 2a2c2+5343abc2+3 │ │ │ │ +00054380: 3739 3862 3263 322d 3135 3936 3861 3364 798b2c2-15968a3d │ │ │ │ +00054390: 2b34 3733 6132 6264 2b31 3332 3933 6162 +473a2bd+13293ab │ │ │ │ +000543a0: 3264 2d33 3736 3162 3364 2d37 377c 0a7c 2d-3761b3d-77|.| │ │ │ │ +000543b0: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d ---------- │ │ │ │ +000543c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000543d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000543e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000543f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00054400: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 --------|.| │ │ │ │ -00054410: 2031 3562 3364 2b37 3836 3961 3263 642d 15b3d+7869a2cd- │ │ │ │ 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3261 6332 642d 31b2cd+6042ac2d- │ │ │ │ +00054430: 3235 3631 6263 3264 2d38 3730 3961 3264 2561bc2d-8709a2d │ │ │ │ +00054440: 322d 3133 3231 3961 6264 322b 347c 0a7c 2-13219abd2+4|.| │ │ │ │ +00054450: 2020 2020 2020 3736 3261 6232 642b 3134 762ab2d+14 │ │ │ │ +00054460: 3039 3562 3364 2d31 3538 3861 3263 642b 095b3d-1588a2cd+ │ │ │ │ +00054470: 3230 3030 6162 6364 2d32 3038 3062 3263 2000abcd-2080b2c │ │ │ │ +00054480: 642b 3931 3735 6163 3264 2d36 3439 6263 d+9175ac2d-649bc │ │ │ │ +00054490: 3264 2b38 3832 3963 3364 2b32 317c 0a7c 2d+8829c3d+21|.| │ │ │ │ +000544a0: 2020 2020 2020 3139 3538 6263 6432 2020 1958bcd2 │ │ │ │ +000544b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000544c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000544d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000544e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000544f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00054500: 2031 3761 3263 642d 3733 3839 6162 6364 17a2cd-7389abcd │ │ │ │ -00054510: 2b34 3732 3362 3263 642d 3133 3236 3261 +4723b2cd-13262a │ │ │ │ -00054520: 6332 642b 3534 3331 6263 3264 2b31 3132 c2d+5431bc2d+112 │ │ │ │ -00054530: 3734 6132 6432 2d32 3137 6162 6432 2b31 74a2d2-217abd2+1 │ │ │ │ -00054540: 3236 3162 3264 322b 7c0a 7c20 2020 2020 261b2d2+|.| │ │ │ │ -00054550: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d --------------- │ │ │ │ +000544e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000544f0: 2020 2020 2020 3137 6132 6364 2d37 3338 17a2cd-738 │ │ │ │ +00054500: 3961 6263 642b 3437 3233 6232 6364 2d31 9abcd+4723b2cd-1 │ │ │ │ +00054510: 3332 3632 6163 3264 2b35 3433 3162 6332 3262ac2d+5431bc2 │ │ │ │ +00054520: 642b 3131 3237 3461 3264 322d 3231 3761 d+11274a2d2-217a │ │ │ │ +00054530: 6264 322b 3132 3631 6232 6432 2b7c 0a7c bd2+1261b2d2+|.| │ │ │ │ +00054540: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d ---------- │ │ │ │ +00054550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00054560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00054570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00054580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00054590: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 --------|.| │ │ │ │ -000545a0: 2032 3039 6232 6432 2b31 3232 3235 6163 209b2d2+12225ac │ │ │ │ -000545b0: 6432 2d32 3630 3562 6364 322d 3932 6332 d2-2605bcd2-92c2 │ │ │ │ -000545c0: 6432 2b31 3539 3638 6164 332b 3134 3836 d2+15968ad3+1486 │ │ │ │ -000545d0: 3062 6433 2d38 3832 3963 6433 2d31 3132 0bd3-8829cd3-112 │ │ │ │ -000545e0: 3734 6434 207c 2020 7c0a 7c20 2020 2020 74d4 | |.| │ │ │ │ -000545f0: 2036 3461 3264 322b 3836 3335 6162 6432 64a2d2+8635abd2 │ │ │ │ -00054600: 2d37 3136 3162 3264 322b 3939 3761 6364 -7161b2d2+997acd │ │ │ │ -00054610: 322b 3330 3135 6263 6432 2b31 3132 3734 2+3015bcd2+11274 │ │ │ │ -00054620: 6332 6432 2020 2020 2020 2020 2020 2020 c2d2 │ │ │ │ -00054630: 2020 2020 207c 2020 7c0a 7c20 2020 2020 | |.| │ │ │ │ +00054580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +00054590: 2020 2020 2020 3230 3962 3264 322b 3132 209b2d2+12 │ │ │ │ +000545a0: 3232 3561 6364 322d 3236 3035 6263 6432 225acd2-2605bcd2 │ │ │ │ +000545b0: 2d39 3263 3264 322b 3135 3936 3861 6433 -92c2d2+15968ad3 │ │ │ │ +000545c0: 2b31 3438 3630 6264 332d 3838 3239 6364 +14860bd3-8829cd │ │ │ │ +000545d0: 332d 3131 3237 3464 3420 7c20 207c 0a7c 3-11274d4 | |.| │ │ │ │ +000545e0: 2020 2020 2020 3634 6132 6432 2b38 3633 64a2d2+863 │ │ │ │ +000545f0: 3561 6264 322d 3731 3631 6232 6432 2b39 5abd2-7161b2d2+9 │ │ │ │ +00054600: 3937 6163 6432 2b33 3031 3562 6364 322b 97acd2+3015bcd2+ │ │ │ │ +00054610: 3131 3237 3463 3264 3220 2020 2020 2020 11274c2d2 │ │ │ │ +00054620: 2020 2020 2020 2020 2020 7c20 207c 0a7c | |.| │ │ │ │ +00054630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054680: 2020 2020 207c 2020 7c0a 7c20 2020 2020 | |.| │ │ │ │ -00054690: 2038 3230 3161 6364 322d 3134 3038 3062 8201acd2-14080b │ │ │ │ -000546a0: 6364 3220 2020 2020 2020 2020 2020 2020 cd2 │ │ │ │ +00054670: 2020 2020 2020 2020 2020 7c20 207c 0a7c | |.| │ │ │ │ +00054680: 2020 2020 2020 3832 3031 6163 6432 2d31 8201acd2-1 │ │ │ │ +00054690: 3430 3830 6263 6432 2020 2020 2020 2020 4080bcd2 │ │ │ │ +000546a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000546b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000546c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000546d0: 2020 2020 207c 2020 7c0a 7c20 2020 2020 | |.| │ │ │ │ +000546c0: 2020 2020 2020 2020 2020 7c20 207c 0a7c | |.| │ │ │ │ +000546d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000546e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000546f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054700: 2020 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-00054a50: 0a20 2020 2068 6f6d 6f74 6f70 6965 730a . homotopies. │ │ │ │ -00054a60: 0a57 6179 7320 746f 2075 7365 206d 616b .Ways to use mak │ │ │ │ -00054a70: 6548 6f6d 6f74 6f70 6965 733a 0a3d 3d3d eHomotopies:.=== │ │ │ │ -00054a80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00054a90: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d ========.. * "m │ │ │ │ -00054aa0: 616b 6548 6f6d 6f74 6f70 6965 7328 4d61 akeHomotopies(Ma │ │ │ │ -00054ab0: 7472 6978 2c43 6f6d 706c 6578 2922 0a20 trix,Complex)". │ │ │ │ -00054ac0: 202a 2022 6d61 6b65 486f 6d6f 746f 7069 * "makeHomotopi │ │ │ │ -00054ad0: 6573 284d 6174 7269 782c 436f 6d70 6c65 es(Matrix,Comple │ │ │ │ -00054ae0: 782c 5a5a 2922 0a0a 466f 7220 7468 6520 x,ZZ)"..For the │ │ │ │ -00054af0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -00054b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -00054b10: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -00054b20: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20 makeHomotopies: │ │ │ │ -00054b30: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20 makeHomotopies, │ │ │ │ -00054b40: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -00054b50: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ -00054b60: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -00054b70: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +000549e0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ +000549f0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ +00054a00: 202a 6e6f 7465 206d 616b 6548 6f6d 6f74 *note makeHomot │ │ │ │ +00054a10: 6f70 6965 7331 3a20 6d61 6b65 486f 6d6f opies1: makeHomo │ │ │ │ +00054a20: 746f 7069 6573 312c 202d 2d20 7265 7475 topies1, -- retu │ │ │ │ +00054a30: 726e 7320 6120 7379 7374 656d 206f 6620 rns a system of │ │ │ │ +00054a40: 6669 7273 740a 2020 2020 686f 6d6f 746f first. homoto │ │ │ │ +00054a50: 7069 6573 0a0a 5761 7973 2074 6f20 7573 pies..Ways to us │ │ │ │ +00054a60: 6520 6d61 6b65 486f 6d6f 746f 7069 6573 e makeHomotopies │ │ │ │ +00054a70: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +00054a80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +00054a90: 202a 2022 6d61 6b65 486f 6d6f 746f 7069 * "makeHomotopi │ │ │ │ +00054aa0: 6573 284d 6174 7269 782c 436f 6d70 6c65 es(Matrix,Comple │ │ │ │ +00054ab0: 7829 220a 2020 2a20 226d 616b 6548 6f6d x)". * "makeHom │ │ │ │ +00054ac0: 6f74 6f70 6965 7328 4d61 7472 6978 2c43 otopies(Matrix,C │ │ │ │ +00054ad0: 6f6d 706c 6578 2c5a 5a29 220a 0a46 6f72 omplex,ZZ)"..For │ │ │ │ +00054ae0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +00054af0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00054b00: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +00054b10: 6e6f 7465 206d 616b 6548 6f6d 6f74 6f70 note makeHomotop │ │ │ │ +00054b20: 6965 733a 206d 616b 6548 6f6d 6f74 6f70 ies: 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2020 2a20 482c puts:. * H, │ │ │ │ -00054ed0: 2061 202a 6e6f 7465 2068 6173 6820 7461 a *note hash ta │ │ │ │ -00054ee0: 626c 653a 2028 4d61 6361 756c 6179 3244 ble: (Macaulay2D │ │ │ │ -00054ef0: 6f63 2948 6173 6854 6162 6c65 2c2c 2067 oc)HashTable,, g │ │ │ │ -00054f00: 6976 6573 2074 6865 2068 6f6d 6f74 6f70 ives the homotop │ │ │ │ -00054f10: 790a 2020 2020 2020 2020 6672 6f6d 2046 y. from F │ │ │ │ -00054f20: 5f69 2063 6f72 7265 7370 6f6e 6469 6e67 _i corresponding │ │ │ │ -00054f30: 2074 6f20 665f 6a20 6173 2074 6865 2076 to f_j as the v │ │ │ │ -00054f40: 616c 7565 2024 4823 5c7b 6a2c 695c 7d24 alue $H#\{j,i\}$ │ │ │ │ -00054f50: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -00054f60: 3d3d 3d3d 3d3d 3d3d 3d0a 0a53 616d 6520 =========..Same │ │ │ │ -00054f70: 6173 206d 616b 6548 6f6d 6f74 6f70 6965 as makeHomotopie │ │ │ │ -00054f80: 732c 2062 7574 206f 6e6c 7920 636f 6d70 s, but only comp │ │ │ │ -00054f90: 7574 6573 2074 6865 206f 7264 696e 6172 utes the ordinar │ │ │ │ -00054fa0: 7920 686f 6d6f 746f 7069 6573 2c20 6e6f y homotopies, no │ │ │ │ -00054fb0: 7420 7468 650a 6869 6768 6572 206f 6e65 t the.higher one │ │ │ │ -00054fc0: 732e 2055 7365 6420 696e 2065 7874 6572 s. Used in exter │ │ │ │ -00054fd0: 696f 7254 6f72 4d6f 6475 6c65 0a0a 5365 iorTorModule..Se │ │ │ │ -00054fe0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ -00054ff0: 0a20 202a 202a 6e6f 7465 206d 616b 6548 . * *note makeH │ │ │ │ -00055000: 6f6d 6f74 6f70 6965 733a 206d 616b 6548 omotopies: makeH │ │ │ │ -00055010: 6f6d 6f74 6f70 6965 732c 202d 2d20 7265 omotopies, -- re │ │ │ │ -00055020: 7475 726e 7320 6120 7379 7374 656d 206f turns a system o │ │ │ │ -00055030: 6620 6869 6768 6572 0a20 2020 2068 6f6d f higher. hom │ │ │ │ -00055040: 6f74 6f70 6965 730a 2020 2a20 2a6e 6f74 otopies. * *not │ │ │ │ -00055050: 6520 6578 7465 7269 6f72 546f 724d 6f64 e exteriorTorMod │ │ │ │ -00055060: 756c 653a 2065 7874 6572 696f 7254 6f72 ule: exteriorTor │ │ │ │ -00055070: 4d6f 6475 6c65 2c20 2d2d 2054 6f72 2061 Module, -- Tor a │ │ │ │ -00055080: 7320 6120 6d6f 6475 6c65 206f 7665 7220 s a module over │ │ │ │ -00055090: 616e 0a20 2020 2065 7874 6572 696f 7220 an. exterior │ │ │ │ -000550a0: 616c 6765 6272 6120 6f72 2062 6967 7261 algebra or bigra │ │ │ │ -000550b0: 6465 6420 616c 6765 6272 610a 0a57 6179 ded algebra..Way │ │ │ │ -000550c0: 7320 746f 2075 7365 206d 616b 6548 6f6d s to use makeHom │ │ │ │ -000550d0: 6f74 6f70 6965 7331 3a0a 3d3d 3d3d 3d3d otopies1:.====== │ │ │ │ -000550e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000550f0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d 616b ======.. * "mak │ │ │ │ -00055100: 6548 6f6d 6f74 6f70 6965 7331 284d 6174 eHomotopies1(Mat │ │ │ │ -00055110: 7269 782c 436f 6d70 6c65 7829 220a 2020 rix,Complex)". │ │ │ │ -00055120: 2a20 226d 616b 6548 6f6d 6f74 6f70 6965 * "makeHomotopie │ │ │ │ -00055130: 7331 284d 6174 7269 782c 436f 6d70 6c65 s1(Matrix,Comple │ │ │ │ -00055140: 782c 5a5a 2922 0a0a 466f 7220 7468 6520 x,ZZ)"..For the │ │ │ │ -00055150: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -00055160: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -00055170: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -00055180: 6d61 6b65 486f 6d6f 746f 7069 6573 313a makeHomotopies1: │ │ │ │ -00055190: 206d 616b 6548 6f6d 6f74 6f70 6965 7331 makeHomotopies1 │ │ │ │ -000551a0: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -000551b0: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod function:.(M │ │ │ │ -000551c0: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -000551d0: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d odFunction,...-- │ │ │ │ +00054d40: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ +00054d50: 6765 3a20 0a20 2020 2020 2020 2048 203d ge: . H = │ │ │ │ +00054d60: 206d 616b 6548 6f6d 6f74 6f70 6965 7331 makeHomotopies1 │ │ │ │ +00054d70: 2866 2c46 2c64 290a 2020 2a20 496e 7075 (f,F,d). * Inpu │ │ │ │ +00054d80: 7473 3a0a 2020 2020 2020 2a20 662c 2061 ts:. * f, a │ │ │ │ +00054d90: 202a 6e6f 7465 206d 6174 7269 783a 2028 *note matrix: ( │ │ │ │ +00054da0: 4d61 6361 756c 6179 3244 6f63 294d 6174 Macaulay2Doc)Mat │ │ │ │ +00054db0: 7269 782c 2c20 3178 6e20 6d61 7472 6978 rix,, 1xn matrix │ │ │ │ +00054dc0: 206f 6620 656c 656d 656e 7473 206f 6620 of elements of │ │ │ │ +00054dd0: 530a 2020 2020 2020 2a20 462c 2061 202a S. * F, a * │ │ │ │ +00054de0: 6e6f 7465 2063 6f6d 706c 6578 3a20 2843 note complex: (C │ │ │ │ +00054df0: 6f6d 706c 6578 6573 2943 6f6d 706c 6578 omplexes)Complex │ │ │ │ +00054e00: 2c2c 2061 646d 6974 7469 6e67 2068 6f6d ,, admitting hom │ │ │ │ +00054e10: 6f74 6f70 6965 7320 666f 7220 7468 650a otopies for the. │ │ │ │ +00054e20: 2020 2020 2020 2020 656e 7472 6965 7320 entries │ │ │ │ +00054e30: 6f66 2066 0a20 2020 2020 202a 2064 2c20 of f. * d, │ │ │ │ +00054e40: 616e 202a 6e6f 7465 2069 6e74 6567 6572 an *note integer │ │ │ │ +00054e50: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00054e60: 5a5a 2c2c 2068 6f77 2066 6172 2062 6163 ZZ,, how far bac │ │ │ │ +00054e70: 6b20 746f 2063 6f6d 7075 7465 2074 6865 k to compute the │ │ │ │ +00054e80: 0a20 2020 2020 2020 2068 6f6d 6f74 6f70 . homotop │ │ │ │ +00054e90: 6965 7320 2864 6566 6175 6c74 7320 746f ies (defaults to │ │ │ │ +00054ea0: 206c 656e 6774 6820 6f66 2046 290a 2020 length of F). │ │ │ │ +00054eb0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +00054ec0: 202a 2048 2c20 6120 2a6e 6f74 6520 6861 * H, a *note ha │ │ │ │ +00054ed0: 7368 2074 6162 6c65 3a20 284d 6163 6175 sh table: (Macau │ │ │ │ +00054ee0: 6c61 7932 446f 6329 4861 7368 5461 626c lay2Doc)HashTabl │ │ │ │ +00054ef0: 652c 2c20 6769 7665 7320 7468 6520 686f e,, gives the ho │ │ │ │ +00054f00: 6d6f 746f 7079 0a20 2020 2020 2020 2066 motopy. f │ │ │ │ +00054f10: 726f 6d20 465f 6920 636f 7272 6573 706f rom F_i correspo │ │ │ │ +00054f20: 6e64 696e 6720 746f 2066 5f6a 2061 7320 nding to f_j as │ │ │ │ +00054f30: 7468 6520 7661 6c75 6520 2448 235c 7b6a the value $H#\{j │ │ │ │ +00054f40: 2c69 5c7d 240a 0a44 6573 6372 6970 7469 ,i\}$..Descripti │ │ │ │ +00054f50: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +00054f60: 5361 6d65 2061 7320 6d61 6b65 486f 6d6f Same as makeHomo │ │ │ │ +00054f70: 746f 7069 6573 2c20 6275 7420 6f6e 6c79 topies, but only │ │ │ │ +00054f80: 2063 6f6d 7075 7465 7320 7468 6520 6f72 computes the or │ │ │ │ +00054f90: 6469 6e61 7279 2068 6f6d 6f74 6f70 6965 dinary homotopie │ │ │ │ +00054fa0: 732c 206e 6f74 2074 6865 0a68 6967 6865 s, not the.highe │ │ │ │ +00054fb0: 7220 6f6e 6573 2e20 5573 6564 2069 6e20 r ones. Used in │ │ │ │ +00054fc0: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ +00054fd0: 650a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d e..See also.==== │ │ │ │ +00054fe0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ +00054ff0: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20 makeHomotopies: │ │ │ │ +00055000: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20 makeHomotopies, │ │ │ │ +00055010: 2d2d 2072 6574 7572 6e73 2061 2073 7973 -- returns a sys │ │ │ │ +00055020: 7465 6d20 6f66 2068 6967 6865 720a 2020 tem of higher. │ │ │ │ +00055030: 2020 686f 6d6f 746f 7069 6573 0a20 202a homotopies. * │ │ │ │ +00055040: 202a 6e6f 7465 2065 7874 6572 696f 7254 *note exteriorT │ │ │ │ +00055050: 6f72 4d6f 6475 6c65 3a20 6578 7465 7269 orModule: exteri │ │ │ │ +00055060: 6f72 546f 724d 6f64 756c 652c 202d 2d20 orTorModule, -- │ │ │ │ +00055070: 546f 7220 6173 2061 206d 6f64 756c 6520 Tor as a module │ │ │ │ +00055080: 6f76 6572 2061 6e0a 2020 2020 6578 7465 over an. exte │ │ │ │ +00055090: 7269 6f72 2061 6c67 6562 7261 206f 7220 rior algebra or │ │ │ │ +000550a0: 6269 6772 6164 6564 2061 6c67 6562 7261 bigraded algebra │ │ │ │ +000550b0: 0a0a 5761 7973 2074 6f20 7573 6520 6d61 ..Ways to use ma │ │ │ │ +000550c0: 6b65 486f 6d6f 746f 7069 6573 313a 0a3d keHomotopies1:.= │ │ │ │ +000550d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000550e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +000550f0: 2022 6d61 6b65 486f 6d6f 746f 7069 6573 "makeHomotopies │ │ │ │ +00055100: 3128 4d61 7472 6978 2c43 6f6d 706c 6578 1(Matrix,Complex │ │ │ │ +00055110: 2922 0a20 202a 2022 6d61 6b65 486f 6d6f )". * "makeHomo │ │ │ │ +00055120: 746f 7069 6573 3128 4d61 7472 6978 2c43 topies1(Matrix,C │ │ │ │ +00055130: 6f6d 706c 6578 2c5a 5a29 220a 0a46 6f72 omplex,ZZ)"..For │ │ │ │ +00055140: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +00055150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00055160: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +00055170: 6e6f 7465 206d 616b 6548 6f6d 6f74 6f70 note makeHomotop │ │ │ │ +00055180: 6965 7331 3a20 6d61 6b65 486f 6d6f 746f ies1: makeHomoto │ │ │ │ +00055190: 7069 6573 312c 2069 7320 6120 2a6e 6f74 pies1, is a *not │ │ │ │ +000551a0: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +000551b0: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63 n:.(Macaulay2Doc │ │ │ │ +000551c0: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ +000551d0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 000551e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000551f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00055200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00055210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00055220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -00055230: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -00055240: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -00055250: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -00055260: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -00055270: 6179 322d 312e 3236 2e30 352b 6473 2f4d ay2-1.26.05+ds/M │ │ │ │ -00055280: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -00055290: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ -000552a0: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -000552b0: 7469 6f6e 732e 6d32 3a33 3830 313a 302e tions.m2:3801:0. │ │ │ │ -000552c0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ -000552d0: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -000552e0: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ -000552f0: 6f64 653a 206d 616b 6548 6f6d 6f74 6f70 ode: makeHomotop │ │ │ │ -00055300: 6965 734f 6e48 6f6d 6f6c 6f67 792c 204e iesOnHomology, N │ │ │ │ -00055310: 6578 743a 206d 616b 654d 6f64 756c 652c ext: makeModule, │ │ │ │ -00055320: 2050 7265 763a 206d 616b 6548 6f6d 6f74 Prev: makeHomot │ │ │ │ -00055330: 6f70 6965 7331 2c20 5570 3a20 546f 700a opies1, Up: Top. │ │ │ │ -00055340: 0a6d 616b 6548 6f6d 6f74 6f70 6965 734f .makeHomotopiesO │ │ │ │ -00055350: 6e48 6f6d 6f6c 6f67 7920 2d2d 2048 6f6d nHomology -- Hom │ │ │ │ -00055360: 6f6c 6f67 7920 6f66 2061 2063 6f6d 706c ology of a compl │ │ │ │ -00055370: 6578 2061 7320 6578 7465 7269 6f72 206d ex as exterior m │ │ │ │ -00055380: 6f64 756c 650a 2a2a 2a2a 2a2a 2a2a 2a2a odule.********** │ │ │ │ +00055220: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +00055230: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +00055240: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +00055250: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +00055260: 6163 6175 6c61 7932 2d31 2e32 362e 3035 acaulay2-1.26.05 │ │ │ │ +00055270: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +00055280: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ +00055290: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +000552a0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3338 esolutions.m2:38 │ │ │ │ +000552b0: 3031 3a30 2e0a 1f0a 4669 6c65 3a20 436f 01:0....File: Co │ │ │ │ +000552c0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +000552d0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ +000552e0: 666f 2c20 4e6f 6465 3a20 6d61 6b65 486f fo, Node: makeHo │ │ │ │ +000552f0: 6d6f 746f 7069 6573 4f6e 486f 6d6f 6c6f motopiesOnHomolo │ │ │ │ +00055300: 6779 2c20 4e65 7874 3a20 6d61 6b65 4d6f gy, Next: makeMo │ │ │ │ +00055310: 6475 6c65 2c20 5072 6576 3a20 6d61 6b65 dule, Prev: make │ │ │ │ +00055320: 486f 6d6f 746f 7069 6573 312c 2055 703a Homotopies1, Up: │ │ │ │ +00055330: 2054 6f70 0a0a 6d61 6b65 486f 6d6f 746f Top..makeHomoto │ │ │ │ +00055340: 7069 6573 4f6e 486f 6d6f 6c6f 6779 202d piesOnHomology - │ │ │ │ +00055350: 2d20 486f 6d6f 6c6f 6779 206f 6620 6120 - Homology of a │ │ │ │ +00055360: 636f 6d70 6c65 7820 6173 2065 7874 6572 complex as exter │ │ │ │ +00055370: 696f 7220 6d6f 6475 6c65 0a2a 2a2a 2a2a ior module.***** │ │ │ │ +00055380: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00055390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000553a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000553b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000553c0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -000553d0: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -000553e0: 2848 2c68 2920 3d20 6d61 6b65 486f 6d6f (H,h) = makeHomo │ │ │ │ -000553f0: 746f 7069 6573 4f6e 486f 6d6f 6c6f 6779 topiesOnHomology │ │ │ │ -00055400: 2866 662c 2043 290a 2020 2a20 496e 7075 (ff, C). * Inpu │ │ │ │ -00055410: 7473 3a0a 2020 2020 2020 2a20 6666 2c20 ts:. * ff, │ │ │ │ -00055420: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ -00055430: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ -00055440: 7472 6978 2c2c 206d 6174 7269 7820 6f66 trix,, matrix of │ │ │ │ -00055450: 2065 6c65 6d65 6e74 7320 686f 6d6f 746f elements homoto │ │ │ │ -00055460: 7069 630a 2020 2020 2020 2020 746f 2030 pic. to 0 │ │ │ │ -00055470: 206f 6e20 430a 2020 2020 2020 2a20 432c on C. * C, │ │ │ │ -00055480: 2061 202a 6e6f 7465 2063 6f6d 706c 6578 a *note complex │ │ │ │ -00055490: 3a20 2843 6f6d 706c 6578 6573 2943 6f6d : (Complexes)Com │ │ │ │ -000554a0: 706c 6578 2c2c 200a 2020 2a20 4f75 7470 plex,, . * Outp │ │ │ │ -000554b0: 7574 733a 0a20 2020 2020 202a 2048 2c20 uts:. * H, │ │ │ │ -000554c0: 6120 2a6e 6f74 6520 6861 7368 2074 6162 a *note hash tab │ │ │ │ -000554d0: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -000554e0: 6329 4861 7368 5461 626c 652c 2c20 486f c)HashTable,, Ho │ │ │ │ -000554f0: 6d6f 6c6f 6779 206f 6620 432c 2069 6e64 mology of C, ind │ │ │ │ -00055500: 6578 6564 0a20 2020 2020 2020 2062 7920 exed. by │ │ │ │ -00055510: 706c 6163 6573 2069 6e20 7468 6520 430a places in the C. │ │ │ │ -00055520: 2020 2020 2020 2a20 682c 2061 202a 6e6f * h, a *no │ │ │ │ -00055530: 7465 2068 6173 6820 7461 626c 653a 2028 te hash table: ( │ │ │ │ -00055540: 4d61 6361 756c 6179 3244 6f63 2948 6173 Macaulay2Doc)Has │ │ │ │ -00055550: 6854 6162 6c65 2c2c 2068 6f6d 6f74 6f70 hTable,, homotop │ │ │ │ -00055560: 6965 7320 666f 720a 2020 2020 2020 2020 ies for. │ │ │ │ -00055570: 656c 656d 656e 7473 206f 6620 6620 6f6e elements of f on │ │ │ │ -00055580: 2074 6865 2068 6f6d 6f6c 6f67 7920 6f66 the homology of │ │ │ │ -00055590: 2043 0a0a 4465 7363 7269 7074 696f 6e0a C..Description. │ │ │ │ -000555a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -000555b0: 2073 6372 6970 7420 6361 6c6c 7320 6d61 script calls ma │ │ │ │ -000555c0: 6b65 486f 6d6f 746f 7069 6573 3120 746f keHomotopies1 to │ │ │ │ -000555d0: 2070 726f 6475 6365 2068 6f6d 6f74 6f70 produce homotop │ │ │ │ -000555e0: 6965 7320 666f 7220 7468 6520 6666 5f69 ies for the ff_i │ │ │ │ -000555f0: 206f 6e20 432c 2061 6e64 0a74 6865 6e20 on C, and.then │ │ │ │ -00055600: 636f 6d70 7574 6573 2074 6865 6972 2061 computes their a │ │ │ │ -00055610: 6374 696f 6e20 6f6e 2074 6865 2048 6f6d ction on the Hom │ │ │ │ -00055620: 6f6c 6f67 7920 6f66 2043 2e0a 0a53 6565 ology of C...See │ │ │ │ -00055630: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -00055640: 2020 2a20 2a6e 6f74 6520 6578 7465 7269 * *note exteri │ │ │ │ -00055650: 6f72 546f 724d 6f64 756c 653a 2065 7874 orTorModule: ext │ │ │ │ -00055660: 6572 696f 7254 6f72 4d6f 6475 6c65 2c20 eriorTorModule, │ │ │ │ -00055670: 2d2d 2054 6f72 2061 7320 6120 6d6f 6475 -- Tor as a modu │ │ │ │ -00055680: 6c65 206f 7665 7220 616e 0a20 2020 2065 le over an. e │ │ │ │ -00055690: 7874 6572 696f 7220 616c 6765 6272 6120 xterior algebra │ │ │ │ -000556a0: 6f72 2062 6967 7261 6465 6420 616c 6765 or bigraded alge │ │ │ │ -000556b0: 6272 610a 2020 2a20 2a6e 6f74 6520 6578 bra. * *note ex │ │ │ │ -000556c0: 7465 7269 6f72 4578 744d 6f64 756c 653a teriorExtModule: │ │ │ │ -000556d0: 2065 7874 6572 696f 7245 7874 4d6f 6475 exteriorExtModu │ │ │ │ -000556e0: 6c65 2c20 2d2d 2045 7874 284d 2c6b 2920 le, -- Ext(M,k) │ │ │ │ -000556f0: 6f72 2045 7874 284d 2c4e 2920 6173 2061 or Ext(M,N) as a │ │ │ │ -00055700: 0a20 2020 206d 6f64 756c 6520 6f76 6572 . module over │ │ │ │ -00055710: 2061 6e20 6578 7465 7269 6f72 2061 6c67 an exterior alg │ │ │ │ -00055720: 6562 7261 0a0a 5761 7973 2074 6f20 7573 ebra..Ways to us │ │ │ │ -00055730: 6520 6d61 6b65 486f 6d6f 746f 7069 6573 e makeHomotopies │ │ │ │ -00055740: 4f6e 486f 6d6f 6c6f 6779 3a0a 3d3d 3d3d OnHomology:.==== │ │ │ │ +000553b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +000553c0: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +000553d0: 2020 2020 2028 482c 6829 203d 206d 616b (H,h) = mak │ │ │ │ +000553e0: 6548 6f6d 6f74 6f70 6965 734f 6e48 6f6d eHomotopiesOnHom │ │ │ │ +000553f0: 6f6c 6f67 7928 6666 2c20 4329 0a20 202a ology(ff, C). * │ │ │ │ +00055400: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +00055410: 2066 662c 2061 202a 6e6f 7465 206d 6174 ff, a *note mat │ │ │ │ +00055420: 7269 783a 2028 4d61 6361 756c 6179 3244 rix: (Macaulay2D │ │ │ │ +00055430: 6f63 294d 6174 7269 782c 2c20 6d61 7472 oc)Matrix,, matr │ │ │ │ +00055440: 6978 206f 6620 656c 656d 656e 7473 2068 ix of elements h │ │ │ │ +00055450: 6f6d 6f74 6f70 6963 0a20 2020 2020 2020 omotopic. │ │ │ │ +00055460: 2074 6f20 3020 6f6e 2043 0a20 2020 2020 to 0 on C. │ │ │ │ +00055470: 202a 2043 2c20 6120 2a6e 6f74 6520 636f * C, a *note co │ │ │ │ +00055480: 6d70 6c65 783a 2028 436f 6d70 6c65 7865 mplex: (Complexe │ │ │ │ +00055490: 7329 436f 6d70 6c65 782c 2c20 0a20 202a s)Complex,, . * │ │ │ │ +000554a0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ +000554b0: 2a20 482c 2061 202a 6e6f 7465 2068 6173 * H, a *note has │ │ │ │ +000554c0: 6820 7461 626c 653a 2028 4d61 6361 756c h table: (Macaul │ │ │ │ +000554d0: 6179 3244 6f63 2948 6173 6854 6162 6c65 ay2Doc)HashTable │ │ │ │ +000554e0: 2c2c 2048 6f6d 6f6c 6f67 7920 6f66 2043 ,, Homology of C │ │ │ │ +000554f0: 2c20 696e 6465 7865 640a 2020 2020 2020 , indexed. │ │ │ │ +00055500: 2020 6279 2070 6c61 6365 7320 696e 2074 by places in t │ │ │ │ +00055510: 6865 2043 0a20 2020 2020 202a 2068 2c20 he C. * h, │ │ │ │ +00055520: 6120 2a6e 6f74 6520 6861 7368 2074 6162 a *note hash tab │ │ │ │ +00055530: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ +00055540: 6329 4861 7368 5461 626c 652c 2c20 686f c)HashTable,, ho │ │ │ │ +00055550: 6d6f 746f 7069 6573 2066 6f72 0a20 2020 motopies for. │ │ │ │ +00055560: 2020 2020 2065 6c65 6d65 6e74 7320 6f66 elements of │ │ │ │ +00055570: 2066 206f 6e20 7468 6520 686f 6d6f 6c6f f on the homolo │ │ │ │ +00055580: 6779 206f 6620 430a 0a44 6573 6372 6970 gy of C..Descrip │ │ │ │ +00055590: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +000555a0: 0a0a 5468 6520 7363 7269 7074 2063 616c ..The script cal │ │ │ │ +000555b0: 6c73 206d 616b 6548 6f6d 6f74 6f70 6965 ls makeHomotopie │ │ │ │ +000555c0: 7331 2074 6f20 7072 6f64 7563 6520 686f s1 to produce ho │ │ │ │ +000555d0: 6d6f 746f 7069 6573 2066 6f72 2074 6865 motopies for the │ │ │ │ +000555e0: 2066 665f 6920 6f6e 2043 2c20 616e 640a ff_i on C, and. │ │ │ │ +000555f0: 7468 656e 2063 6f6d 7075 7465 7320 7468 then computes th │ │ │ │ +00055600: 6569 7220 6163 7469 6f6e 206f 6e20 7468 eir action on th │ │ │ │ +00055610: 6520 486f 6d6f 6c6f 6779 206f 6620 432e e Homology of C. │ │ │ │ +00055620: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ +00055630: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2065 ===.. * *note e │ │ │ │ +00055640: 7874 6572 696f 7254 6f72 4d6f 6475 6c65 xteriorTorModule │ │ │ │ +00055650: 3a20 6578 7465 7269 6f72 546f 724d 6f64 : exteriorTorMod │ │ │ │ +00055660: 756c 652c 202d 2d20 546f 7220 6173 2061 ule, -- Tor as a │ │ │ │ +00055670: 206d 6f64 756c 6520 6f76 6572 2061 6e0a module over an. │ │ │ │ +00055680: 2020 2020 6578 7465 7269 6f72 2061 6c67 exterior alg │ │ │ │ +00055690: 6562 7261 206f 7220 6269 6772 6164 6564 ebra or bigraded │ │ │ │ +000556a0: 2061 6c67 6562 7261 0a20 202a 202a 6e6f algebra. * *no │ │ │ │ +000556b0: 7465 2065 7874 6572 696f 7245 7874 4d6f te exteriorExtMo │ │ │ │ +000556c0: 6475 6c65 3a20 6578 7465 7269 6f72 4578 dule: exteriorEx │ │ │ │ +000556d0: 744d 6f64 756c 652c 202d 2d20 4578 7428 tModule, -- Ext( │ │ │ │ +000556e0: 4d2c 6b29 206f 7220 4578 7428 4d2c 4e29 M,k) or Ext(M,N) │ │ │ │ +000556f0: 2061 7320 610a 2020 2020 6d6f 6475 6c65 as a. module │ │ │ │ +00055700: 206f 7665 7220 616e 2065 7874 6572 696f over an exterio │ │ │ │ +00055710: 7220 616c 6765 6272 610a 0a57 6179 7320 r algebra..Ways │ │ │ │ +00055720: 746f 2075 7365 206d 616b 6548 6f6d 6f74 to use makeHomot │ │ │ │ +00055730: 6f70 6965 734f 6e48 6f6d 6f6c 6f67 793a opiesOnHomology: │ │ │ │ +00055740: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ 00055750: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00055760: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00055770: 3d0a 0a20 202a 2022 6d61 6b65 486f 6d6f =.. * "makeHomo │ │ │ │ -00055780: 746f 7069 6573 4f6e 486f 6d6f 6c6f 6779 topiesOnHomology │ │ │ │ -00055790: 284d 6174 7269 782c 436f 6d70 6c65 7829 (Matrix,Complex) │ │ │ │ -000557a0: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ -000557b0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ -000557c0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ -000557d0: 6a65 6374 202a 6e6f 7465 206d 616b 6548 ject *note makeH │ │ │ │ -000557e0: 6f6d 6f74 6f70 6965 734f 6e48 6f6d 6f6c omotopiesOnHomol │ │ │ │ -000557f0: 6f67 793a 206d 616b 6548 6f6d 6f74 6f70 ogy: makeHomotop │ │ │ │ -00055800: 6965 734f 6e48 6f6d 6f6c 6f67 792c 2069 iesOnHomology, i │ │ │ │ -00055810: 7320 6120 2a6e 6f74 650a 6d65 7468 6f64 s a *note.method │ │ │ │ -00055820: 2066 756e 6374 696f 6e3a 2028 4d61 6361 function: (Maca │ │ │ │ -00055830: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ -00055840: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ +00055760: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d 616b ======.. * "mak │ │ │ │ +00055770: 6548 6f6d 6f74 6f70 6965 734f 6e48 6f6d eHomotopiesOnHom │ │ │ │ +00055780: 6f6c 6f67 7928 4d61 7472 6978 2c43 6f6d ology(Matrix,Com │ │ │ │ +00055790: 706c 6578 2922 0a0a 466f 7220 7468 6520 plex)"..For the │ │ │ │ +000557a0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ +000557b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +000557c0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +000557d0: 6d61 6b65 486f 6d6f 746f 7069 6573 4f6e makeHomotopiesOn │ │ │ │ +000557e0: 486f 6d6f 6c6f 6779 3a20 6d61 6b65 486f Homology: makeHo │ │ │ │ +000557f0: 6d6f 746f 7069 6573 4f6e 486f 6d6f 6c6f motopiesOnHomolo │ │ │ │ +00055800: 6779 2c20 6973 2061 202a 6e6f 7465 0a6d gy, is a *note.m │ │ │ │ +00055810: 6574 686f 6420 6675 6e63 7469 6f6e 3a20 ethod function: │ │ │ │ +00055820: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +00055830: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +00055840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00055850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00055860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00055870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00055880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00055890: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -000558a0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -000558b0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -000558c0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -000558d0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -000558e0: 2d31 2e32 362e 3035 2b64 732f 4d32 2f4d -1.26.05+ds/M2/M │ │ │ │ -000558f0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00055900: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ -00055910: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -00055920: 6e73 2e6d 323a 3236 3934 3a30 2e0a 1f0a ns.m2:2694:0.... │ │ │ │ -00055930: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ -00055940: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -00055950: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ -00055960: 3a20 6d61 6b65 4d6f 6475 6c65 2c20 4e65 : makeModule, Ne │ │ │ │ -00055970: 7874 3a20 6d61 6b65 542c 2050 7265 763a xt: makeT, Prev: │ │ │ │ -00055980: 206d 616b 6548 6f6d 6f74 6f70 6965 734f makeHomotopiesO │ │ │ │ -00055990: 6e48 6f6d 6f6c 6f67 792c 2055 703a 2054 nHomology, Up: T │ │ │ │ -000559a0: 6f70 0a0a 6d61 6b65 4d6f 6475 6c65 202d op..makeModule - │ │ │ │ -000559b0: 2d20 6d61 6b65 7320 6120 4d6f 6475 6c65 - makes a Module │ │ │ │ -000559c0: 206f 7574 206f 6620 6120 636f 6c6c 6563 out of a collec │ │ │ │ -000559d0: 7469 6f6e 206f 6620 6d6f 6475 6c65 7320 tion of modules │ │ │ │ -000559e0: 616e 6420 6d61 7073 0a2a 2a2a 2a2a 2a2a and maps.******* │ │ │ │ +00055880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +00055890: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +000558a0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +000558b0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +000558c0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +000558d0: 756c 6179 322d 312e 3236 2e30 352b 6473 ulay2-1.26.05+ds │ │ │ │ +000558e0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +000558f0: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ +00055900: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +00055910: 6c75 7469 6f6e 732e 6d32 3a32 3639 343a lutions.m2:2694: │ │ │ │ +00055920: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ +00055930: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +00055940: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ +00055950: 204e 6f64 653a 206d 616b 654d 6f64 756c Node: makeModul │ │ │ │ +00055960: 652c 204e 6578 743a 206d 616b 6554 2c20 e, Next: makeT, │ │ │ │ +00055970: 5072 6576 3a20 6d61 6b65 486f 6d6f 746f Prev: makeHomoto │ │ │ │ +00055980: 7069 6573 4f6e 486f 6d6f 6c6f 6779 2c20 piesOnHomology, │ │ │ │ +00055990: 5570 3a20 546f 700a 0a6d 616b 654d 6f64 Up: Top..makeMod │ │ │ │ +000559a0: 756c 6520 2d2d 206d 616b 6573 2061 204d ule -- makes a M │ │ │ │ +000559b0: 6f64 756c 6520 6f75 7420 6f66 2061 2063 odule out of a c │ │ │ │ +000559c0: 6f6c 6c65 6374 696f 6e20 6f66 206d 6f64 ollection of mod │ │ │ │ +000559d0: 756c 6573 2061 6e64 206d 6170 730a 2a2a ules and maps.** │ │ │ │ +000559e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000559f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00055a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00055a10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00055a20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ -00055a30: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -00055a40: 2020 204d 203d 206d 616b 654d 6f64 756c M = makeModul │ │ │ │ -00055a50: 6528 482c 452c 7068 6929 0a20 202a 2049 e(H,E,phi). * I │ │ │ │ -00055a60: 6e70 7574 733a 0a20 2020 2020 202a 2048 nputs:. * H │ │ │ │ -00055a70: 2c20 6120 2a6e 6f74 6520 6861 7368 2074 , a *note hash t │ │ │ │ -00055a80: 6162 6c65 3a20 284d 6163 6175 6c61 7932 able: (Macaulay2 │ │ │ │ -00055a90: 446f 6329 4861 7368 5461 626c 652c 2c20 Doc)HashTable,, │ │ │ │ -00055aa0: 6772 6164 6564 2063 6f6d 706f 6e65 6e74 graded component │ │ │ │ -00055ab0: 7320 7468 6174 0a20 2020 2020 2020 2061 s that. a │ │ │ │ -00055ac0: 7265 206d 6f64 756c 6573 2c20 746f 206d re modules, to m │ │ │ │ -00055ad0: 616b 6520 696e 746f 2061 7320 7369 6e67 ake into as sing │ │ │ │ -00055ae0: 6c65 206d 6f64 756c 650a 2020 2020 2020 le module. │ │ │ │ -00055af0: 2a20 452c 2061 202a 6e6f 7465 206d 6174 * E, a *note mat │ │ │ │ -00055b00: 7269 783a 2028 4d61 6361 756c 6179 3244 rix: (Macaulay2D │ │ │ │ -00055b10: 6f63 294d 6174 7269 782c 2c20 4d61 7472 oc)Matrix,, Matr │ │ │ │ -00055b20: 6978 206f 6620 7661 7269 6162 6c65 7320 ix of variables │ │ │ │ -00055b30: 7768 6f73 650a 2020 2020 2020 2020 6163 whose. ac │ │ │ │ -00055b40: 7469 6f6e 2077 696c 6c20 6465 6669 6e65 tion will define │ │ │ │ -00055b50: 640a 2020 2020 2020 2a20 7068 692c 2061 d. * phi, a │ │ │ │ -00055b60: 202a 6e6f 7465 2068 6173 6820 7461 626c *note hash tabl │ │ │ │ -00055b70: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -00055b80: 2948 6173 6854 6162 6c65 2c2c 206d 6170 )HashTable,, map │ │ │ │ -00055b90: 7320 6265 7477 6565 6e20 7468 650a 2020 s between the. │ │ │ │ -00055ba0: 2020 2020 2020 6772 6164 6564 2063 6f6d graded com │ │ │ │ -00055bb0: 706f 6e65 6e74 7320 7468 6174 2077 696c ponents that wil │ │ │ │ -00055bc0: 6c20 6265 2074 6865 2061 6374 696f 6e20 l be the action │ │ │ │ -00055bd0: 6f66 2074 6865 2076 6172 6961 626c 6573 of the variables │ │ │ │ -00055be0: 2069 6e20 450a 2020 2a20 4f75 7470 7574 in E. * Output │ │ │ │ -00055bf0: 733a 0a20 2020 2020 202a 204d 2c20 6120 s:. * M, a │ │ │ │ -00055c00: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ -00055c10: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ -00055c20: 6c65 2c2c 2067 7261 6465 6420 6d6f 6475 le,, graded modu │ │ │ │ -00055c30: 6c65 7320 7768 6f73 650a 2020 2020 2020 les whose. │ │ │ │ -00055c40: 2020 636f 6d70 6f6e 656e 7473 2061 7265 components are │ │ │ │ -00055c50: 2067 6976 656e 2062 7920 480a 0a44 6573 given by H..Des │ │ │ │ -00055c60: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -00055c70: 3d3d 3d3d 0a0a 5468 6520 4861 7368 7461 ====..The Hashta │ │ │ │ -00055c80: 626c 6520 4820 7368 6f75 6c64 2068 6176 ble H should hav │ │ │ │ -00055c90: 6520 636f 6e73 6563 7574 6976 6520 696e e consecutive in │ │ │ │ -00055ca0: 7465 6765 7220 6b65 7973 2069 5f30 2e2e teger keys i_0.. │ │ │ │ -00055cb0: 695f 302c 2073 6179 2c20 7769 7468 2076 i_0, say, with v │ │ │ │ -00055cc0: 616c 7565 730a 4823 6920 7468 6174 2061 alues.H#i that a │ │ │ │ -00055cd0: 7265 206d 6f64 756c 6573 206f 7665 7220 re modules over │ │ │ │ -00055ce0: 6120 7269 6e67 2053 4520 7768 6f73 6520 a ring SE whose │ │ │ │ -00055cf0: 7661 7269 6162 6c65 7320 696e 636c 7564 variables includ │ │ │ │ -00055d00: 6520 7468 6520 656c 656d 656e 7473 206f e the elements o │ │ │ │ -00055d10: 6620 452e 0a45 3a20 5c6f 706c 7573 2053 f E..E: \oplus S │ │ │ │ -00055d20: 455e 7b64 5f69 7d20 5c74 6f20 5345 5e31 E^{d_i} \to SE^1 │ │ │ │ -00055d30: 2069 7320 6120 6d61 7472 6978 206f 6620 is a matrix of │ │ │ │ -00055d40: 6320 7661 7269 6162 6c65 7320 6672 6f6d c variables from │ │ │ │ -00055d50: 2053 4520 4820 6973 2061 2068 6173 6854 SE H is a hashT │ │ │ │ -00055d60: 6162 6c65 0a6f 6620 6d20 7061 6972 7320 able.of m pairs │ │ │ │ -00055d70: 7b69 2c20 745f 697d 2c20 7768 6572 6520 {i, t_i}, where │ │ │ │ -00055d80: 7468 6520 745f 6920 6172 6520 5245 2d6d the t_i are RE-m │ │ │ │ -00055d90: 6f64 756c 6573 2c20 616e 6420 7468 6520 odules, and the │ │ │ │ -00055da0: 6920 6172 6520 636f 6e73 6563 7574 6976 i are consecutiv │ │ │ │ -00055db0: 650a 696e 7465 6765 722e 2070 6869 2069 e.integer. phi i │ │ │ │ -00055dc0: 7320 6120 6861 7368 2d74 6162 6c65 206f s a hash-table o │ │ │ │ -00055dd0: 6620 686f 6d6f 6765 6e65 6f75 7320 6d61 f homogeneous ma │ │ │ │ -00055de0: 7073 2070 6869 237b 6a2c 697d 3a20 4823 ps phi#{j,i}: H# │ │ │ │ -00055df0: 692a 2a46 5f6a 5c74 6f20 4823 2869 2b31 i**F_j\to H#(i+1 │ │ │ │ -00055e00: 290a 7768 6572 6520 465f 6a20 3d20 736f ).where F_j = so │ │ │ │ -00055e10: 7572 6365 2028 455f 7b6a 7d20 3d20 6d61 urce (E_{j} = ma │ │ │ │ -00055e20: 7472 6978 207b 7b65 5f6a 7d7d 292e 2054 trix {{e_j}}). T │ │ │ │ -00055e30: 6875 7320 7468 6520 6d61 7073 2070 237b hus the maps p#{ │ │ │ │ -00055e40: 6a2c 697d 203d 2028 455f 6a20 7c7c 0a2d j,i} = (E_j ||.- │ │ │ │ -00055e50: 7068 6923 7b6a 2c69 7d29 3a20 745f 692a phi#{j,i}): t_i* │ │ │ │ -00055e60: 2a46 5f6a 205c 746f 2074 5f69 2b2b 745f *F_j \to t_i++t_ │ │ │ │ -00055e70: 7b28 692b 3129 7d2c 2061 7265 2068 6f6d {(i+1)}, are hom │ │ │ │ -00055e80: 6f67 656e 656f 7573 2e20 5468 6520 7363 ogeneous. The sc │ │ │ │ -00055e90: 7269 7074 2072 6574 7572 6e73 204d 0a3d ript returns M.= │ │ │ │ -00055ea0: 205c 6f70 6c75 735f 6920 545f 2061 7320 \oplus_i T_ as │ │ │ │ -00055eb0: 616e 2053 452d 6d6f 6475 6c65 2c20 636f an SE-module, co │ │ │ │ -00055ec0: 6d70 7574 6564 2061 7320 7468 6520 7175 mputed as the qu │ │ │ │ -00055ed0: 6f74 6965 6e74 206f 6620 5020 3a3d 205c otient of P := \ │ │ │ │ -00055ee0: 6f70 6c75 7320 545f 690a 6f62 7461 696e oplus T_i.obtain │ │ │ │ -00055ef0: 6564 2062 7920 6661 6374 6f72 696e 6720 ed by factoring │ │ │ │ -00055f00: 6f75 7420 7468 6520 7375 6d20 6f66 2074 out the sum of t │ │ │ │ -00055f10: 6865 2069 6d61 6765 7320 6f66 2074 6865 he images of the │ │ │ │ -00055f20: 206d 6170 7320 7023 7b6a 2c69 7d0a 0a54 maps p#{j,i}..T │ │ │ │ -00055f30: 6865 2048 6173 6874 6162 6c65 2070 6869 he Hashtable phi │ │ │ │ -00055f40: 2068 6173 206b 6579 7320 6f66 2074 6865 has keys of the │ │ │ │ -00055f50: 2066 6f72 6d20 7b6a 2c69 7d20 7768 6572 form {j,i} wher │ │ │ │ -00055f60: 6520 6a20 7275 6e73 2066 726f 6d20 3020 e j runs from 0 │ │ │ │ -00055f70: 746f 2063 2d31 2c20 6920 616e 640a 692b to c-1, i and.i+ │ │ │ │ -00055f80: 3120 6172 6520 6b65 7973 206f 6620 482c 1 are keys of H, │ │ │ │ -00055f90: 2061 6e64 2070 6869 237b 6a2c 697d 2069 and phi#{j,i} i │ │ │ │ -00055fa0: 7320 7468 6520 6d61 7020 6672 6f6d 2028 s the map from ( │ │ │ │ -00055fb0: 736f 7572 6365 2045 5f7b 697d 292a 2a48 source E_{i})**H │ │ │ │ -00055fc0: 2369 2074 6f20 4823 2869 2b31 290a 7468 #i to H#(i+1).th │ │ │ │ -00055fd0: 6174 2077 696c 6c20 6265 2069 6465 6e74 at will be ident │ │ │ │ -00055fe0: 6966 6965 6420 7769 7468 2074 6865 2061 ified with the a │ │ │ │ -00055ff0: 6374 696f 6e20 6f66 2045 5f7b 6a7d 2e0a ction of E_{j}.. │ │ │ │ -00056000: 0a54 6865 2073 6372 6970 7420 6973 2075 .The script is u │ │ │ │ -00056010: 7365 6420 696e 2062 6f74 6820 7468 6520 sed in both the │ │ │ │ -00056020: 7369 6e67 6c79 2067 7261 6465 6420 6361 singly graded ca │ │ │ │ -00056030: 7365 2c20 666f 7220 6578 616d 706c 6520 se, for example │ │ │ │ -00056040: 696e 0a65 7874 6572 696f 7254 6f72 4d6f in.exteriorTorMo │ │ │ │ -00056050: 6475 6c65 2866 662c 4d29 2061 6e64 2069 dule(ff,M) and i │ │ │ │ -00056060: 6e20 7468 6520 6269 6772 6164 6564 2063 n the bigraded c │ │ │ │ -00056070: 6173 652c 2066 6f72 2065 7861 6d70 6c65 ase, for example │ │ │ │ -00056080: 2069 6e0a 6578 7465 7269 6f72 546f 724d in.exteriorTorM │ │ │ │ -00056090: 6f64 756c 6528 6666 2c4d 2c4e 292e 0a0a odule(ff,M,N)... │ │ │ │ -000560a0: 496e 2074 6865 2066 6f6c 6c6f 7769 6e67 In the following │ │ │ │ -000560b0: 2077 6520 7573 6520 6d61 6b65 4d6f 6475 we use makeModu │ │ │ │ -000560c0: 6c65 2074 6f20 636f 6e73 7472 7563 7420 le to construct │ │ │ │ -000560d0: 6279 2068 616e 6420 6120 6672 6565 206d by hand a free m │ │ │ │ -000560e0: 6f64 756c 6520 6f66 2072 616e 6b20 310a odule of rank 1. │ │ │ │ -000560f0: 6f76 6572 2074 6865 2065 7874 6572 696f over the exterio │ │ │ │ -00056100: 7220 616c 6765 6272 6120 6f6e 2078 2c79 r algebra on x,y │ │ │ │ -00056110: 2c20 7374 6172 7469 6e67 2077 6974 6820 , starting with │ │ │ │ -00056120: 7468 6520 636f 6e73 7472 7563 7469 6f6e the construction │ │ │ │ -00056130: 206f 6620 6120 6d6f 6475 6c65 0a6f 7665 of a module.ove │ │ │ │ -00056140: 7220 6120 6269 686f 6d6f 6765 6e65 6f75 r a bihomogeneou │ │ │ │ -00056150: 7320 7269 6e67 2e0a 0a2b 2d2d 2d2d 2d2d s ring...+------ │ │ │ │ +00055a20: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ +00055a30: 2020 2020 2020 2020 4d20 3d20 6d61 6b65 M = make │ │ │ │ +00055a40: 4d6f 6475 6c65 2848 2c45 2c70 6869 290a Module(H,E,phi). │ │ │ │ +00055a50: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +00055a60: 2020 2a20 482c 2061 202a 6e6f 7465 2068 * H, a *note h │ │ │ │ +00055a70: 6173 6820 7461 626c 653a 2028 4d61 6361 ash table: (Maca │ │ │ │ +00055a80: 756c 6179 3244 6f63 2948 6173 6854 6162 ulay2Doc)HashTab │ │ │ │ +00055a90: 6c65 2c2c 2067 7261 6465 6420 636f 6d70 le,, graded comp │ │ │ │ +00055aa0: 6f6e 656e 7473 2074 6861 740a 2020 2020 onents that. │ │ │ │ +00055ab0: 2020 2020 6172 6520 6d6f 6475 6c65 732c are modules, │ │ │ │ +00055ac0: 2074 6f20 6d61 6b65 2069 6e74 6f20 6173 to make into as │ │ │ │ +00055ad0: 2073 696e 676c 6520 6d6f 6475 6c65 0a20 single module. │ │ │ │ +00055ae0: 2020 2020 202a 2045 2c20 6120 2a6e 6f74 * E, a *not │ │ │ │ +00055af0: 6520 6d61 7472 6978 3a20 284d 6163 6175 e matrix: (Macau │ │ │ │ +00055b00: 6c61 7932 446f 6329 4d61 7472 6978 2c2c lay2Doc)Matrix,, │ │ │ │ +00055b10: 204d 6174 7269 7820 6f66 2076 6172 6961 Matrix of varia │ │ │ │ +00055b20: 626c 6573 2077 686f 7365 0a20 2020 2020 bles whose. │ │ │ │ +00055b30: 2020 2061 6374 696f 6e20 7769 6c6c 2064 action will d │ │ │ │ +00055b40: 6566 696e 6564 0a20 2020 2020 202a 2070 efined. * p │ │ │ │ +00055b50: 6869 2c20 6120 2a6e 6f74 6520 6861 7368 hi, a *note hash │ │ │ │ +00055b60: 2074 6162 6c65 3a20 284d 6163 6175 6c61 table: (Macaula │ │ │ │ +00055b70: 7932 446f 6329 4861 7368 5461 626c 652c y2Doc)HashTable, │ │ │ │ +00055b80: 2c20 6d61 7073 2062 6574 7765 656e 2074 , maps between t │ │ │ │ +00055b90: 6865 0a20 2020 2020 2020 2067 7261 6465 he. grade │ │ │ │ +00055ba0: 6420 636f 6d70 6f6e 656e 7473 2074 6861 d components tha │ │ │ │ +00055bb0: 7420 7769 6c6c 2062 6520 7468 6520 6163 t will be the ac │ │ │ │ +00055bc0: 7469 6f6e 206f 6620 7468 6520 7661 7269 tion of the vari │ │ │ │ +00055bd0: 6162 6c65 7320 696e 2045 0a20 202a 204f ables in E. * O │ │ │ │ +00055be0: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ +00055bf0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c M, a *note modul │ │ │ │ +00055c00: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ +00055c10: 294d 6f64 756c 652c 2c20 6772 6164 6564 )Module,, graded │ │ │ │ +00055c20: 206d 6f64 756c 6573 2077 686f 7365 0a20 modules whose. │ │ │ │ +00055c30: 2020 2020 2020 2063 6f6d 706f 6e65 6e74 component │ │ │ │ +00055c40: 7320 6172 6520 6769 7665 6e20 6279 2048 s are given by H │ │ │ │ +00055c50: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +00055c60: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 2048 =========..The H │ │ │ │ +00055c70: 6173 6874 6162 6c65 2048 2073 686f 756c ashtable H shoul │ │ │ │ +00055c80: 6420 6861 7665 2063 6f6e 7365 6375 7469 d have consecuti │ │ │ │ +00055c90: 7665 2069 6e74 6567 6572 206b 6579 7320 ve integer keys │ │ │ │ +00055ca0: 695f 302e 2e69 5f30 2c20 7361 792c 2077 i_0..i_0, say, w │ │ │ │ +00055cb0: 6974 6820 7661 6c75 6573 0a48 2369 2074 ith values.H#i t │ │ │ │ +00055cc0: 6861 7420 6172 6520 6d6f 6475 6c65 7320 hat are modules │ │ │ │ +00055cd0: 6f76 6572 2061 2072 696e 6720 5345 2077 over a ring SE w │ │ │ │ +00055ce0: 686f 7365 2076 6172 6961 626c 6573 2069 hose variables i │ │ │ │ +00055cf0: 6e63 6c75 6465 2074 6865 2065 6c65 6d65 nclude the eleme │ │ │ │ +00055d00: 6e74 7320 6f66 2045 2e0a 453a 205c 6f70 nts of E..E: \op │ │ │ │ +00055d10: 6c75 7320 5345 5e7b 645f 697d 205c 746f lus SE^{d_i} \to │ │ │ │ +00055d20: 2053 455e 3120 6973 2061 206d 6174 7269 SE^1 is a matri │ │ │ │ +00055d30: 7820 6f66 2063 2076 6172 6961 626c 6573 x of c variables │ │ │ │ +00055d40: 2066 726f 6d20 5345 2048 2069 7320 6120 from SE H is a │ │ │ │ +00055d50: 6861 7368 5461 626c 650a 6f66 206d 2070 hashTable.of m p │ │ │ │ +00055d60: 6169 7273 207b 692c 2074 5f69 7d2c 2077 airs {i, t_i}, w │ │ │ │ +00055d70: 6865 7265 2074 6865 2074 5f69 2061 7265 here the t_i are │ │ │ │ +00055d80: 2052 452d 6d6f 6475 6c65 732c 2061 6e64 RE-modules, and │ │ │ │ +00055d90: 2074 6865 2069 2061 7265 2063 6f6e 7365 the i are conse │ │ │ │ +00055da0: 6375 7469 7665 0a69 6e74 6567 6572 2e20 cutive.integer. │ │ │ │ +00055db0: 7068 6920 6973 2061 2068 6173 682d 7461 phi is a hash-ta │ │ │ │ +00055dc0: 626c 6520 6f66 2068 6f6d 6f67 656e 656f ble of homogeneo │ │ │ │ +00055dd0: 7573 206d 6170 7320 7068 6923 7b6a 2c69 us maps phi#{j,i │ │ │ │ +00055de0: 7d3a 2048 2369 2a2a 465f 6a5c 746f 2048 }: H#i**F_j\to H │ │ │ │ +00055df0: 2328 692b 3129 0a77 6865 7265 2046 5f6a #(i+1).where F_j │ │ │ │ +00055e00: 203d 2073 6f75 7263 6520 2845 5f7b 6a7d = source (E_{j} │ │ │ │ +00055e10: 203d 206d 6174 7269 7820 7b7b 655f 6a7d = matrix {{e_j} │ │ │ │ +00055e20: 7d29 2e20 5468 7573 2074 6865 206d 6170 }). Thus the map │ │ │ │ +00055e30: 7320 7023 7b6a 2c69 7d20 3d20 2845 5f6a s p#{j,i} = (E_j │ │ │ │ +00055e40: 207c 7c0a 2d70 6869 237b 6a2c 697d 293a ||.-phi#{j,i}): │ │ │ │ +00055e50: 2074 5f69 2a2a 465f 6a20 5c74 6f20 745f t_i**F_j \to t_ │ │ │ │ +00055e60: 692b 2b74 5f7b 2869 2b31 297d 2c20 6172 i++t_{(i+1)}, ar │ │ │ │ +00055e70: 6520 686f 6d6f 6765 6e65 6f75 732e 2054 e homogeneous. T │ │ │ │ +00055e80: 6865 2073 6372 6970 7420 7265 7475 726e he script return │ │ │ │ +00055e90: 7320 4d0a 3d20 5c6f 706c 7573 5f69 2054 s M.= \oplus_i T │ │ │ │ +00055ea0: 5f20 6173 2061 6e20 5345 2d6d 6f64 756c _ as an SE-modul │ │ │ │ +00055eb0: 652c 2063 6f6d 7075 7465 6420 6173 2074 e, computed as t │ │ │ │ +00055ec0: 6865 2071 756f 7469 656e 7420 6f66 2050 he quotient of P │ │ │ │ +00055ed0: 203a 3d20 5c6f 706c 7573 2054 5f69 0a6f := \oplus T_i.o │ │ │ │ +00055ee0: 6274 6169 6e65 6420 6279 2066 6163 746f btained by facto │ │ │ │ +00055ef0: 7269 6e67 206f 7574 2074 6865 2073 756d ring out the sum │ │ │ │ +00055f00: 206f 6620 7468 6520 696d 6167 6573 206f of the images o │ │ │ │ +00055f10: 6620 7468 6520 6d61 7073 2070 237b 6a2c f the maps p#{j, │ │ │ │ +00055f20: 697d 0a0a 5468 6520 4861 7368 7461 626c i}..The Hashtabl │ │ │ │ +00055f30: 6520 7068 6920 6861 7320 6b65 7973 206f e phi has keys o │ │ │ │ +00055f40: 6620 7468 6520 666f 726d 207b 6a2c 697d f the form {j,i} │ │ │ │ +00055f50: 2077 6865 7265 206a 2072 756e 7320 6672 where j runs fr │ │ │ │ +00055f60: 6f6d 2030 2074 6f20 632d 312c 2069 2061 om 0 to c-1, i a │ │ │ │ +00055f70: 6e64 0a69 2b31 2061 7265 206b 6579 7320 nd.i+1 are keys │ │ │ │ +00055f80: 6f66 2048 2c20 616e 6420 7068 6923 7b6a of H, and phi#{j │ │ │ │ +00055f90: 2c69 7d20 6973 2074 6865 206d 6170 2066 ,i} is the map f │ │ │ │ +00055fa0: 726f 6d20 2873 6f75 7263 6520 455f 7b69 rom (source E_{i │ │ │ │ +00055fb0: 7d29 2a2a 4823 6920 746f 2048 2328 692b })**H#i to H#(i+ │ │ │ │ +00055fc0: 3129 0a74 6861 7420 7769 6c6c 2062 6520 1).that will be │ │ │ │ +00055fd0: 6964 656e 7469 6669 6564 2077 6974 6820 identified with │ │ │ │ +00055fe0: 7468 6520 6163 7469 6f6e 206f 6620 455f the action of E_ │ │ │ │ +00055ff0: 7b6a 7d2e 0a0a 5468 6520 7363 7269 7074 {j}...The script │ │ │ │ +00056000: 2069 7320 7573 6564 2069 6e20 626f 7468 is used in both │ │ │ │ +00056010: 2074 6865 2073 696e 676c 7920 6772 6164 the singly grad │ │ │ │ +00056020: 6564 2063 6173 652c 2066 6f72 2065 7861 ed case, for exa │ │ │ │ +00056030: 6d70 6c65 2069 6e0a 6578 7465 7269 6f72 mple in.exterior │ │ │ │ +00056040: 546f 724d 6f64 756c 6528 6666 2c4d 2920 TorModule(ff,M) │ │ │ │ +00056050: 616e 6420 696e 2074 6865 2062 6967 7261 and in the bigra │ │ │ │ +00056060: 6465 6420 6361 7365 2c20 666f 7220 6578 ded case, for ex │ │ │ │ +00056070: 616d 706c 6520 696e 0a65 7874 6572 696f ample in.exterio │ │ │ │ +00056080: 7254 6f72 4d6f 6475 6c65 2866 662c 4d2c rTorModule(ff,M, │ │ │ │ +00056090: 4e29 2e0a 0a49 6e20 7468 6520 666f 6c6c N)...In the foll │ │ │ │ +000560a0: 6f77 696e 6720 7765 2075 7365 206d 616b owing we use mak │ │ │ │ +000560b0: 654d 6f64 756c 6520 746f 2063 6f6e 7374 eModule to const │ │ │ │ +000560c0: 7275 6374 2062 7920 6861 6e64 2061 2066 ruct by hand a f │ │ │ │ +000560d0: 7265 6520 6d6f 6475 6c65 206f 6620 7261 ree module of ra │ │ │ │ +000560e0: 6e6b 2031 0a6f 7665 7220 7468 6520 6578 nk 1.over the ex │ │ │ │ +000560f0: 7465 7269 6f72 2061 6c67 6562 7261 206f terior algebra o │ │ │ │ +00056100: 6e20 782c 792c 2073 7461 7274 696e 6720 n x,y, starting │ │ │ │ +00056110: 7769 7468 2074 6865 2063 6f6e 7374 7275 with the constru │ │ │ │ +00056120: 6374 696f 6e20 6f66 2061 206d 6f64 756c ction of a modul │ │ │ │ +00056130: 650a 6f76 6572 2061 2062 6968 6f6d 6f67 e.over a bihomog │ │ │ │ +00056140: 656e 656f 7573 2072 696e 672e 0a0a 2b2d eneous ring...+- │ │ │ │ +00056150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000561a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053 -------+.|i1 : S │ │ │ │ -000561b0: 4520 3d20 5a5a 2f31 3031 5b61 2c62 2c63 E = ZZ/101[a,b,c │ │ │ │ -000561c0: 2c78 2c79 2c44 6567 7265 6573 3d3e 746f ,x,y,Degrees=>to │ │ │ │ -000561d0: 4c69 7374 2833 3a7b 312c 307d 297c 746f List(3:{1,0})|to │ │ │ │ -000561e0: 4c69 7374 2832 3a7b 312c 317d 292c 2020 List(2:{1,1}), │ │ │ │ -000561f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00056190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000561a0: 3120 3a20 5345 203d 205a 5a2f 3130 315b 1 : SE = ZZ/101[ │ │ │ │ +000561b0: 612c 622c 632c 782c 792c 4465 6772 6565 a,b,c,x,y,Degree │ │ │ │ +000561c0: 733d 3e74 6f4c 6973 7428 333a 7b31 2c30 s=>toList(3:{1,0 │ │ │ │ +000561d0: 7d29 7c74 6f4c 6973 7428 323a 7b31 2c31 })|toList(2:{1,1 │ │ │ │ +000561e0: 7d29 2c20 2020 2020 2020 2020 7c0a 7c20 }), |.| │ │ │ │ +000561f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056240: 2020 2020 2020 207c 0a7c 6f31 203d 2053 |.|o1 = S │ │ │ │ -00056250: 4520 2020 2020 2020 2020 2020 2020 2020 E │ │ │ │ +00056230: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00056240: 3120 3d20 5345 2020 2020 2020 2020 2020 1 = SE │ │ │ │ +00056250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056290: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00056280: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000562a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000562b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000562c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000562d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000562e0: 2020 2020 2020 207c 0a7c 6f31 203a 2050 |.|o1 : P │ │ │ │ -000562f0: 6f6c 796e 6f6d 6961 6c52 696e 672c 2032 olynomialRing, 2 │ │ │ │ -00056300: 2073 6b65 7720 636f 6d6d 7574 6174 6976 skew commutativ │ │ │ │ -00056310: 6520 7661 7269 6162 6c65 2873 2920 2020 e variable(s) │ │ │ │ -00056320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056330: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d |.|------ │ │ │ │ +000562d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000562e0: 3120 3a20 506f 6c79 6e6f 6d69 616c 5269 1 : PolynomialRi │ │ │ │ +000562f0: 6e67 2c20 3220 736b 6577 2063 6f6d 6d75 ng, 2 skew commu │ │ │ │ +00056300: 7461 7469 7665 2076 6172 6961 626c 6528 tative variable( │ │ │ │ +00056310: 7329 2020 2020 2020 2020 2020 2020 2020 s) │ │ │ │ +00056320: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ +00056330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056380: 2d2d 2d2d 2d2d 2d7c 0a7c 536b 6577 436f -------|.|SkewCo │ │ │ │ -00056390: 6d6d 7574 6174 6976 653d 3e7b 782c 797d mmutative=>{x,y} │ │ │ │ -000563a0: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +00056370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c53 ------------|.|S │ │ │ │ +00056380: 6b65 7743 6f6d 6d75 7461 7469 7665 3d3e kewCommutative=> │ │ │ │ +00056390: 7b78 2c79 7d5d 2020 2020 2020 2020 2020 {x,y}] │ │ │ │ +000563a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000563b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000563c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000563d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000563c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000563d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000563e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000563f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056420: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052 -------+.|i2 : R │ │ │ │ -00056430: 4520 3d20 5345 2f69 6465 616c 2261 322c E = SE/ideal"a2, │ │ │ │ -00056440: 6232 2c63 3222 2020 2020 2020 2020 2020 b2,c2" │ │ │ │ +00056410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00056420: 3220 3a20 5245 203d 2053 452f 6964 6561 2 : RE = SE/idea │ │ │ │ +00056430: 6c22 6132 2c62 322c 6332 2220 2020 2020 l"a2,b2,c2" │ │ │ │ +00056440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056470: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00056460: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000564a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000564b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000564c0: 2020 2020 2020 207c 0a7c 6f32 203d 2052 |.|o2 = R │ │ │ │ -000564d0: 4520 2020 2020 2020 2020 2020 2020 2020 E │ │ │ │ +000564b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000564c0: 3220 3d20 5245 2020 2020 2020 2020 2020 2 = RE │ │ │ │ +000564d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000564e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000564f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056510: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00056500: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056560: 2020 2020 2020 207c 0a7c 6f32 203a 2051 |.|o2 : Q │ │ │ │ -00056570: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +00056550: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00056560: 3220 3a20 5175 6f74 6965 6e74 5269 6e67 2 : QuotientRing │ │ │ │ +00056570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000565a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000565b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000565a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000565b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000565c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000565d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000565e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000565f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056600: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2054 -------+.|i3 : T │ │ │ │ -00056610: 203d 2068 6173 6854 6162 6c65 207b 7b30 = hashTable {{0 │ │ │ │ -00056620: 2c52 455e 317d 2c7b 312c 5245 5e7b 323a ,RE^1},{1,RE^{2: │ │ │ │ -00056630: 7b20 2d31 2c2d 317d 7d7d 2c20 7b32 2c52 { -1,-1}}}, {2,R │ │ │ │ -00056640: 455e 7b7b 202d 322c 2d32 7d7d 7d7d 2020 E^{{ -2,-2}}}} │ │ │ │ -00056650: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000565f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00056600: 3320 3a20 5420 3d20 6861 7368 5461 626c 3 : T = hashTabl │ │ │ │ +00056610: 6520 7b7b 302c 5245 5e31 7d2c 7b31 2c52 e {{0,RE^1},{1,R │ │ │ │ +00056620: 455e 7b32 3a7b 202d 312c 2d31 7d7d 7d2c E^{2:{ -1,-1}}}, │ │ │ │ +00056630: 207b 322c 5245 5e7b 7b20 2d32 2c2d 327d {2,RE^{{ -2,-2} │ │ │ │ +00056640: 7d7d 7d20 2020 2020 2020 2020 7c0a 7c20 }}} |.| │ │ │ │ +00056650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000566a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000566b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000566c0: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00056690: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000566a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000566b0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +000566c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000566d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000566e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000566f0: 2020 2020 2020 207c 0a7c 6f33 203d 2048 |.|o3 = H │ │ │ │ -00056700: 6173 6854 6162 6c65 7b30 203d 3e20 5245 ashTable{0 => RE │ │ │ │ -00056710: 207d 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +000566e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000566f0: 3320 3d20 4861 7368 5461 626c 657b 3020 3 = HashTable{0 │ │ │ │ +00056700: 3d3e 2052 4520 7d20 2020 2020 2020 2020 => RE } │ │ │ │ +00056710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056740: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00056750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056760: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00056730: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00056750: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00056760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056790: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000567a0: 2020 2020 2020 2020 2031 203d 3e20 5245 1 => RE │ │ │ │ +00056780: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056790: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ +000567a0: 3d3e 2052 4520 2020 2020 2020 2020 2020 => RE │ │ │ │ 000567b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000567c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000567d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000567e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000567f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056800: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +000567d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000567e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000567f0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +00056800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056830: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00056840: 2020 2020 2020 2020 2032 203d 3e20 5245 2 => RE │ │ │ │ +00056820: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056830: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +00056840: 3d3e 2052 4520 2020 2020 2020 2020 2020 => RE │ │ │ │ 00056850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056880: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00056870: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000568a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000568b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000568c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000568d0: 2020 2020 2020 207c 0a7c 6f33 203a 2048 |.|o3 : H │ │ │ │ -000568e0: 6173 6854 6162 6c65 2020 2020 2020 2020 ashTable │ │ │ │ +000568c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000568d0: 3320 3a20 4861 7368 5461 626c 6520 2020 3 : HashTable │ │ │ │ +000568e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000568f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056920: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00056910: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00056920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056970: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2045 -------+.|i4 : E │ │ │ │ -00056980: 203d 206d 6174 7269 787b 7b78 2c79 7d7d = matrix{{x,y}} │ │ │ │ +00056960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00056970: 3420 3a20 4520 3d20 6d61 7472 6978 7b7b 4 : E = matrix{{ │ │ │ │ +00056980: 782c 797d 7d20 2020 2020 2020 2020 2020 x,y}} │ │ │ │ 00056990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000569a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000569b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000569c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000569b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000569c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000569d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000569e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000569f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056a10: 2020 2020 2020 207c 0a7c 6f34 203d 207c |.|o4 = | │ │ │ │ -00056a20: 2078 2079 207c 2020 2020 2020 2020 2020 x y | │ │ │ │ +00056a00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00056a10: 3420 3d20 7c20 7820 7920 7c20 2020 2020 4 = | x y | │ │ │ │ +00056a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056a60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00056a50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056ab0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00056ac0: 2020 2020 2020 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2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00057270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00057280: 3720 3a20 6170 706c 7928 6b65 7973 2070 7 : apply(keys p │ │ │ │ +00057290: 6869 2c20 6b2d 3e69 7348 6f6d 6f67 656e hi, k->isHomogen │ │ │ │ +000572a0: 656f 7573 2070 6869 236b 2920 2020 2020 eous phi#k) │ │ │ │ +000572b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000572c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000572d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000572e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000572f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00057310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00057320: 2020 2020 2020 207c 0a7c 6f37 203d 207b |.|o7 = { │ │ │ │ -00057330: 7472 7565 2c20 7472 7565 2c20 7472 7565 true, true, true │ │ │ │ -00057340: 2c20 7472 7565 7d20 2020 2020 2020 2020 , true} │ │ │ │ +00057310: 2020 2020 2020 2020 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-00057fb0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -00057fc0: 202a 6e6f 7465 2065 7874 6572 696f 7248 *note exteriorH │ │ │ │ -00057fd0: 6f6d 6f6c 6f67 794d 6f64 756c 653a 2065 omologyModule: e │ │ │ │ -00057fe0: 7874 6572 696f 7248 6f6d 6f6c 6f67 794d xteriorHomologyM │ │ │ │ -00057ff0: 6f64 756c 652c 202d 2d20 4d61 6b65 2074 odule, -- Make t │ │ │ │ -00058000: 6865 2068 6f6d 6f6c 6f67 790a 2020 2020 he homology. │ │ │ │ -00058010: 6f66 2061 2063 6f6d 706c 6578 2069 6e74 of a complex int │ │ │ │ -00058020: 6f20 6120 6d6f 6475 6c65 206f 7665 7220 o a module over │ │ │ │ -00058030: 616e 2065 7874 6572 696f 7220 616c 6765 an exterior alge │ │ │ │ -00058040: 6272 610a 2020 2a20 2a6e 6f74 6520 6578 bra. * *note ex │ │ │ │ -00058050: 7465 7269 6f72 546f 724d 6f64 756c 653a teriorTorModule: │ │ │ │ -00058060: 2065 7874 6572 696f 7254 6f72 4d6f 6475 exteriorTorModu │ │ │ │ -00058070: 6c65 2c20 2d2d 2054 6f72 2061 7320 6120 le, -- Tor as a │ │ │ │ -00058080: 6d6f 6475 6c65 206f 7665 7220 616e 0a20 module over an. │ │ │ │ -00058090: 2020 2065 7874 6572 696f 7220 616c 6765 exterior alge │ │ │ │ -000580a0: 6272 6120 6f72 2062 6967 7261 6465 6420 bra or bigraded │ │ │ │ -000580b0: 616c 6765 6272 610a 2020 2a20 2a6e 6f74 algebra. * *not │ │ │ │ -000580c0: 6520 6578 7465 7269 6f72 4578 744d 6f64 e exteriorExtMod │ │ │ │ -000580d0: 756c 653a 2065 7874 6572 696f 7245 7874 ule: exteriorExt │ │ │ │ -000580e0: 4d6f 6475 6c65 2c20 2d2d 2045 7874 284d Module, -- Ext(M │ │ │ │ -000580f0: 2c6b 2920 6f72 2045 7874 284d 2c4e 2920 ,k) or Ext(M,N) │ │ │ │ -00058100: 6173 2061 0a20 2020 206d 6f64 756c 6520 as a. module │ │ │ │ -00058110: 6f76 6572 2061 6e20 6578 7465 7269 6f72 over an exterior │ │ │ │ -00058120: 2061 6c67 6562 7261 0a0a 5761 7973 2074 algebra..Ways t │ │ │ │ -00058130: 6f20 7573 6520 6d61 6b65 4d6f 6475 6c65 o use makeModule │ │ │ │ -00058140: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -00058150: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ -00058160: 6d61 6b65 4d6f 6475 6c65 2848 6173 6854 makeModule(HashT │ │ │ │ -00058170: 6162 6c65 2c4d 6174 7269 782c 4861 7368 able,Matrix,Hash │ │ │ │ -00058180: 5461 626c 6529 220a 0a46 6f72 2074 6865 Table)"..For the │ │ │ │ -00058190: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -000581a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -000581b0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -000581c0: 206d 616b 654d 6f64 756c 653a 206d 616b makeModule: mak │ │ │ │ -000581d0: 654d 6f64 756c 652c 2069 7320 6120 2a6e eModule, is a *n │ │ │ │ -000581e0: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ -000581f0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ -00058200: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00058210: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +00057f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 ------------+..S │ │ │ │ +00057fa0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +00057fb0: 0a0a 2020 2a20 2a6e 6f74 6520 6578 7465 .. * *note exte │ │ │ │ +00057fc0: 7269 6f72 486f 6d6f 6c6f 6779 4d6f 6475 riorHomologyModu │ │ │ │ +00057fd0: 6c65 3a20 6578 7465 7269 6f72 486f 6d6f le: exteriorHomo │ │ │ │ +00057fe0: 6c6f 6779 4d6f 6475 6c65 2c20 2d2d 204d logyModule, -- M │ │ │ │ +00057ff0: 616b 6520 7468 6520 686f 6d6f 6c6f 6779 ake the homology │ │ │ │ +00058000: 0a20 2020 206f 6620 6120 636f 6d70 6c65 . of a comple │ │ │ │ +00058010: 7820 696e 746f 2061 206d 6f64 756c 6520 x into a module │ │ │ │ +00058020: 6f76 6572 2061 6e20 6578 7465 7269 6f72 over an exterior │ │ │ │ +00058030: 2061 6c67 6562 7261 0a20 202a 202a 6e6f algebra. * *no │ │ │ │ +00058040: 7465 2065 7874 6572 696f 7254 6f72 4d6f te exteriorTorMo │ │ │ │ +00058050: 6475 6c65 3a20 6578 7465 7269 6f72 546f dule: exteriorTo │ │ │ │ +00058060: 724d 6f64 756c 652c 202d 2d20 546f 7220 rModule, -- Tor │ │ 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323a nResolutions.m2: │ │ │ │ -000582f0: 3237 3539 3a30 2e0a 1f0a 4669 6c65 3a20 2759:0....File: │ │ │ │ -00058300: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -00058310: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -00058320: 696e 666f 2c20 4e6f 6465 3a20 6d61 6b65 info, Node: make │ │ │ │ -00058330: 542c 204e 6578 743a 206d 6174 7269 7846 T, Next: matrixF │ │ │ │ -00058340: 6163 746f 7269 7a61 7469 6f6e 2c20 5072 actorization, Pr │ │ │ │ -00058350: 6576 3a20 6d61 6b65 4d6f 6475 6c65 2c20 ev: makeModule, │ │ │ │ -00058360: 5570 3a20 546f 700a 0a6d 616b 6554 202d Up: Top..makeT - │ │ │ │ -00058370: 2d20 6d61 6b65 2074 6865 2043 4920 6f70 - make the CI op │ │ │ │ -00058380: 6572 6174 6f72 7320 6f6e 2061 2063 6f6d erators on a com │ │ │ │ -00058390: 706c 6578 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a plex.*********** │ │ │ │ +00058250: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +00058260: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +00058270: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +00058280: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +00058290: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +000582a0: 312e 3236 2e30 352b 6473 2f4d 322f 4d61 1.26.05+ds/M2/Ma │ │ │ │ +000582b0: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +000582c0: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ +000582d0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ +000582e0: 732e 6d32 3a32 3735 393a 302e 0a1f 0a46 s.m2:2759:0....F │ │ │ │ +000582f0: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ +00058300: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00058310: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ +00058320: 206d 616b 6554 2c20 4e65 7874 3a20 6d61 makeT, Next: ma │ │ │ │ +00058330: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ +00058340: 6e2c 2050 7265 763a 206d 616b 654d 6f64 n, Prev: makeMod │ │ │ │ +00058350: 756c 652c 2055 703a 2054 6f70 0a0a 6d61 ule, Up: Top..ma │ │ │ │ +00058360: 6b65 5420 2d2d 206d 616b 6520 7468 6520 keT -- make the │ │ │ │ +00058370: 4349 206f 7065 7261 746f 7273 206f 6e20 CI operators on │ │ │ │ +00058380: 6120 636f 6d70 6c65 780a 2a2a 2a2a 2a2a a complex.****** │ │ │ │ +00058390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000583a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000583b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000583c0: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ -000583d0: 2020 2020 2020 5420 3d20 6d61 6b65 5428 T = makeT( │ │ │ │ -000583e0: 6666 2c46 2c69 290a 2020 2020 2020 2020 ff,F,i). │ │ │ │ -000583f0: 5420 3d20 6d61 6b65 5428 6666 2c46 2c74 T = makeT(ff,F,t │ │ │ │ -00058400: 302c 6929 0a20 202a 2049 6e70 7574 733a 0,i). * Inputs: │ │ │ │ -00058410: 0a20 2020 2020 202a 2066 662c 2061 202a . * ff, a * │ │ │ │ -00058420: 6e6f 7465 206d 6174 7269 783a 2028 4d61 note matrix: (Ma │ │ │ │ -00058430: 6361 756c 6179 3244 6f63 294d 6174 7269 caulay2Doc)Matri │ │ │ │ -00058440: 782c 2c20 3178 6320 6d61 7472 6978 2077 x,, 1xc matrix w │ │ │ │ -00058450: 686f 7365 2065 6e74 7269 6573 2061 7265 hose entries are │ │ │ │ -00058460: 0a20 2020 2020 2020 2061 2063 6f6d 706c . a compl │ │ │ │ -00058470: 6574 6520 696e 7465 7273 6563 7469 6f6e ete intersection │ │ │ │ -00058480: 2069 6e20 530a 2020 2020 2020 2a20 462c in S. * F, │ │ │ │ -00058490: 2061 202a 6e6f 7465 2063 6f6d 706c 6578 a *note complex │ │ │ │ -000584a0: 3a20 2843 6f6d 706c 6578 6573 2943 6f6d : (Complexes)Com │ │ │ │ -000584b0: 706c 6578 2c2c 206f 7665 7220 532f 6964 plex,, over S/id │ │ │ │ -000584c0: 6561 6c20 6666 0a20 2020 2020 202a 2074 eal ff. * t │ │ │ │ -000584d0: 302c 2061 202a 6e6f 7465 206d 6174 7269 0, a *note matri │ │ │ │ -000584e0: 783a 2028 4d61 6361 756c 6179 3244 6f63 x: (Macaulay2Doc │ │ │ │ -000584f0: 294d 6174 7269 782c 2c20 4349 2d6f 7065 )Matrix,, CI-ope │ │ │ │ -00058500: 7261 746f 7220 6f6e 2046 2066 6f72 2066 rator on F for f │ │ │ │ -00058510: 665f 3020 746f 0a20 2020 2020 2020 2062 f_0 to. b │ │ │ │ -00058520: 6520 7072 6573 6572 7665 640a 2020 2020 e preserved. │ │ │ │ -00058530: 2020 2a20 692c 2061 6e20 2a6e 6f74 6520 * i, an *note │ │ │ │ -00058540: 696e 7465 6765 723a 2028 4d61 6361 756c integer: (Macaul │ │ │ │ -00058550: 6179 3244 6f63 295a 5a2c 2c20 6465 6669 ay2Doc)ZZ,, defi │ │ │ │ -00058560: 6e65 2043 4920 6f70 6572 6174 6f72 7320 ne CI operators │ │ │ │ -00058570: 6672 6f6d 2046 5f69 0a20 2020 2020 2020 from F_i. │ │ │ │ -00058580: 205c 746f 2046 5f7b 692d 327d 0a20 202a \to F_{i-2}. * │ │ │ │ -00058590: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -000585a0: 2a20 4c2c 2061 202a 6e6f 7465 206c 6973 * L, a *note lis │ │ │ │ -000585b0: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ -000585c0: 294c 6973 742c 2c20 6f66 2043 4920 6f70 )List,, of CI op │ │ │ │ -000585d0: 6572 6174 6f72 7320 465f 6920 5c74 6f20 erators F_i \to │ │ │ │ -000585e0: 465f 7b69 2d32 7d0a 2020 2020 2020 2020 F_{i-2}. │ │ │ │ -000585f0: 636f 7272 6573 706f 6e64 696e 6720 746f corresponding to │ │ │ │ -00058600: 2065 6e74 7269 6573 206f 6620 6666 0a0a entries of ff.. │ │ │ │ -00058610: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -00058620: 3d3d 3d3d 3d3d 3d0a 0a73 7562 7374 6974 =======..substit │ │ │ │ -00058630: 7574 6520 6d61 7472 6963 6573 206f 6620 ute matrices of │ │ │ │ -00058640: 7477 6f20 6469 6666 6572 656e 7469 616c two differential │ │ │ │ -00058650: 7320 6f66 2046 2069 6e74 6f20 5320 3d20 s of F into S = │ │ │ │ -00058660: 7269 6e67 2066 662c 2063 6f6d 706f 7365 ring ff, compose │ │ │ │ -00058670: 2074 6865 6d2c 0a61 6e64 2064 6976 6964 them,.and divid │ │ │ │ -00058680: 6520 6279 2065 6e74 7269 6573 206f 6620 e by entries of │ │ │ │ -00058690: 6666 2c20 696e 206f 7264 6572 2e20 4966 ff, in order. If │ │ │ │ -000586a0: 2074 6865 2073 6563 6f6e 6420 4d61 7472 the second Matr │ │ │ │ -000586b0: 6978 2061 7267 756d 656e 7420 7430 2069 ix argument t0 i │ │ │ │ -000586c0: 730a 7072 6573 656e 742c 2075 7365 2069 s.present, use i │ │ │ │ -000586d0: 7420 6173 2074 6865 2066 6972 7374 2043 t as the first C │ │ │ │ -000586e0: 4920 6f70 6572 6174 6f72 2e0a 0a54 6865 I operator...The │ │ │ │ -000586f0: 2064 6567 7265 6573 206f 6620 7468 6520 degrees of the │ │ │ │ -00058700: 7461 7267 6574 7320 6f66 2074 6865 2054 targets of the T │ │ │ │ -00058710: 5f6a 2061 7265 2063 6861 6e67 6564 2062 _j are changed b │ │ │ │ -00058720: 7920 7468 6520 6465 6772 6565 7320 6f66 y the degrees of │ │ │ │ -00058730: 2074 6865 2066 5f6a 2074 6f0a 6d61 6b65 the f_j to.make │ │ │ │ -00058740: 2074 6865 2054 5f6a 2068 6f6d 6f67 656e the T_j homogen │ │ │ │ -00058750: 656f 7573 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d eous...+-------- │ │ │ │ +000583b0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ +000583c0: 3a20 0a20 2020 2020 2020 2054 203d 206d : . T = m │ │ │ │ +000583d0: 616b 6554 2866 662c 462c 6929 0a20 2020 akeT(ff,F,i). │ │ │ │ +000583e0: 2020 2020 2054 203d 206d 616b 6554 2866 T = makeT(f │ │ │ │ +000583f0: 662c 462c 7430 2c69 290a 2020 2a20 496e f,F,t0,i). * In │ │ │ │ +00058400: 7075 7473 3a0a 2020 2020 2020 2a20 6666 puts:. * ff │ │ │ │ +00058410: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978 , a *note matrix │ │ │ │ +00058420: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00058430: 4d61 7472 6978 2c2c 2031 7863 206d 6174 Matrix,, 1xc mat │ │ │ │ +00058440: 7269 7820 7768 6f73 6520 656e 7472 6965 rix whose entrie │ │ │ │ +00058450: 7320 6172 650a 2020 2020 2020 2020 6120 s are. a │ │ │ │ +00058460: 636f 6d70 6c65 7465 2069 6e74 6572 7365 complete interse │ │ │ │ +00058470: 6374 696f 6e20 696e 2053 0a20 2020 2020 ction in S. │ │ │ │ +00058480: 202a 2046 2c20 6120 2a6e 6f74 6520 636f * F, a *note co │ │ │ │ +00058490: 6d70 6c65 783a 2028 436f 6d70 6c65 7865 mplex: (Complexe │ │ │ │ +000584a0: 7329 436f 6d70 6c65 782c 2c20 6f76 6572 s)Complex,, over │ │ │ │ +000584b0: 2053 2f69 6465 616c 2066 660a 2020 2020 S/ideal ff. │ │ │ │ +000584c0: 2020 2a20 7430 2c20 6120 2a6e 6f74 6520 * t0, a *note │ │ │ │ +000584d0: 6d61 7472 6978 3a20 284d 6163 6175 6c61 matrix: (Macaula │ │ │ │ +000584e0: 7932 446f 6329 4d61 7472 6978 2c2c 2043 y2Doc)Matrix,, C │ │ │ │ +000584f0: 492d 6f70 6572 6174 6f72 206f 6e20 4620 I-operator on F │ │ │ │ +00058500: 666f 7220 6666 5f30 2074 6f0a 2020 2020 for ff_0 to. │ │ │ │ +00058510: 2020 2020 6265 2070 7265 7365 7276 6564 be preserved │ │ │ │ +00058520: 0a20 2020 2020 202a 2069 2c20 616e 202a . * i, an * │ │ │ │ +00058530: 6e6f 7465 2069 6e74 6567 6572 3a20 284d note integer: (M │ │ │ │ +00058540: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c acaulay2Doc)ZZ,, │ │ │ │ +00058550: 2064 6566 696e 6520 4349 206f 7065 7261 define CI opera │ │ │ │ +00058560: 746f 7273 2066 726f 6d20 465f 690a 2020 tors from F_i. │ │ │ │ +00058570: 2020 2020 2020 5c74 6f20 465f 7b69 2d32 \to F_{i-2 │ │ │ │ +00058580: 7d0a 2020 2a20 4f75 7470 7574 733a 0a20 }. * Outputs:. │ │ │ │ +00058590: 2020 2020 202a 204c 2c20 6120 2a6e 6f74 * L, a *not │ │ │ │ +000585a0: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ +000585b0: 7932 446f 6329 4c69 7374 2c2c 206f 6620 y2Doc)List,, of │ │ │ │ +000585c0: 4349 206f 7065 7261 746f 7273 2046 5f69 CI operators F_i │ │ │ │ +000585d0: 205c 746f 2046 5f7b 692d 327d 0a20 2020 \to F_{i-2}. │ │ │ │ +000585e0: 2020 2020 2063 6f72 7265 7370 6f6e 6469 correspondi │ │ │ │ +000585f0: 6e67 2074 6f20 656e 7472 6965 7320 6f66 ng to entries of │ │ │ │ +00058600: 2066 660a 0a44 6573 6372 6970 7469 6f6e ff..Description │ │ │ │ +00058610: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 7375 .===========..su │ │ │ │ +00058620: 6273 7469 7475 7465 206d 6174 7269 6365 bstitute matrice │ │ │ │ +00058630: 7320 6f66 2074 776f 2064 6966 6665 7265 s of two differe │ │ │ │ +00058640: 6e74 6961 6c73 206f 6620 4620 696e 746f ntials of F into │ │ │ │ +00058650: 2053 203d 2072 696e 6720 6666 2c20 636f S = ring ff, co │ │ │ │ +00058660: 6d70 6f73 6520 7468 656d 2c0a 616e 6420 mpose them,.and │ │ │ │ +00058670: 6469 7669 6465 2062 7920 656e 7472 6965 divide by entrie │ │ │ │ +00058680: 7320 6f66 2066 662c 2069 6e20 6f72 6465 s of ff, in orde │ │ │ │ +00058690: 722e 2049 6620 7468 6520 7365 636f 6e64 r. If the second │ │ │ │ +000586a0: 204d 6174 7269 7820 6172 6775 6d65 6e74 Matrix argument │ │ │ │ +000586b0: 2074 3020 6973 0a70 7265 7365 6e74 2c20 t0 is.present, │ │ │ │ +000586c0: 7573 6520 6974 2061 7320 7468 6520 6669 use it as the fi │ │ │ │ +000586d0: 7273 7420 4349 206f 7065 7261 746f 722e rst CI operator. │ │ │ │ +000586e0: 0a0a 5468 6520 6465 6772 6565 7320 6f66 ..The degrees of │ │ │ │ +000586f0: 2074 6865 2074 6172 6765 7473 206f 6620 the targets of │ │ │ │ +00058700: 7468 6520 545f 6a20 6172 6520 6368 616e the T_j are chan │ │ │ │ +00058710: 6765 6420 6279 2074 6865 2064 6567 7265 ged by the degre │ │ │ │ +00058720: 6573 206f 6620 7468 6520 665f 6a20 746f es of the f_j to │ │ │ │ +00058730: 0a6d 616b 6520 7468 6520 545f 6a20 686f .make the T_j ho │ │ │ │ +00058740: 6d6f 6765 6e65 6f75 732e 0a0a 2b2d 2d2d mogeneous...+--- │ │ │ │ +00058750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00058760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00058770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00058790: 3120 3a20 5320 3d20 5a5a 2f31 3031 5b78 1 : S = ZZ/101[x │ │ │ │ -000587a0: 2c79 2c7a 5d3b 2020 2020 2020 2020 2020 ,y,z]; │ │ │ │ -000587b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000587c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00058780: 2d2b 0a7c 6931 203a 2053 203d 205a 5a2f -+.|i1 : S = ZZ/ │ │ │ │ +00058790: 3130 315b 782c 792c 7a5d 3b20 2020 2020 101[x,y,z]; │ │ │ │ +000587a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000587b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000587c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000587d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000587e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000587f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ -00058800: 3a20 6666 203d 206d 6174 7269 7822 7833 : ff = matrix"x3 │ │ │ │ -00058810: 2c79 332c 7a33 223b 2020 2020 2020 2020 ,y3,z3"; │ │ │ │ -00058820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058830: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000587e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +000587f0: 0a7c 6932 203a 2066 6620 3d20 6d61 7472 .|i2 : ff = matr │ │ │ │ +00058800: 6978 2278 332c 7933 2c7a 3322 3b20 2020 ix"x3,y3,z3"; │ │ │ │ +00058810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00058820: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00058830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00058840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058860: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00058870: 2020 2020 2020 2020 3120 2020 2020 2033 1 3 │ │ │ │ +00058850: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00058860: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ +00058870: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ 00058880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058890: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000588a0: 0a7c 6f32 203a 204d 6174 7269 7820 5320 .|o2 : Matrix S │ │ │ │ -000588b0: 203c 2d2d 2053 2020 2020 2020 2020 2020 <-- S │ │ │ │ -000588c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000588d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00058890: 2020 2020 7c0a 7c6f 3220 3a20 4d61 7472 |.|o2 : Matr │ │ │ │ +000588a0: 6978 2053 2020 3c2d 2d20 5320 2020 2020 ix S <-- S │ │ │ │ +000588b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000588c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +000588d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000588e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000588f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00058910: 6933 203a 2052 203d 2053 2f69 6465 616c i3 : R = S/ideal │ │ │ │ -00058920: 2066 663b 2020 2020 2020 2020 2020 2020 ff; │ │ │ │ -00058930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058940: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00058900: 2d2d 2b0a 7c69 3320 3a20 5220 3d20 532f --+.|i3 : R = S/ │ │ │ │ +00058910: 6964 6561 6c20 6666 3b20 2020 2020 2020 ideal ff; │ │ │ │ +00058920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00058930: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00058940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00058950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00058960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ -00058980: 203a 204d 203d 2063 6f6b 6572 206d 6174 : M = coker mat │ │ │ │ -00058990: 7269 7822 782c 792c 7a3b 792c 7a2c 7822 rix"x,y,z;y,z,x" │ │ │ │ -000589a0: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ -000589b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00058970: 2b0a 7c69 3420 3a20 4d20 3d20 636f 6b65 +.|i4 : M = coke │ │ │ │ +00058980: 7220 6d61 7472 6978 2278 2c79 2c7a 3b79 r matrix"x,y,z;y │ │ │ │ +00058990: 2c7a 2c78 223b 2020 2020 2020 2020 2020 ,z,x"; │ │ │ │ +000589a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000589b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000589c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000589d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000589e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ -000589f0: 2062 6574 7469 2028 4620 3d20 6672 6565 betti (F = free │ │ │ │ -00058a00: 5265 736f 6c75 7469 6f6e 284d 2c20 4c65 Resolution(M, Le │ │ │ │ -00058a10: 6e67 7468 4c69 6d69 7420 3d3e 2033 2929 ngthLimit => 3)) │ │ │ │ -00058a20: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000589d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000589e0: 7c69 3520 3a20 6265 7474 6920 2846 203d |i5 : betti (F = │ │ │ │ +000589f0: 2066 7265 6552 6573 6f6c 7574 696f 6e28 freeResolution( │ │ │ │ +00058a00: 4d2c 204c 656e 6774 684c 696d 6974 203d M, LengthLimit = │ │ │ │ +00058a10: 3e20 3329 297c 0a7c 2020 2020 2020 2020 > 3))|.| │ │ │ │ +00058a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00058a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058a50: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00058a60: 2020 2020 2020 3020 3120 3220 3320 2020 0 1 2 3 │ │ │ │ +00058a40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00058a50: 2020 2020 2020 2020 2020 2030 2031 2032 0 1 2 │ │ │ │ +00058a60: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00058a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058a80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00058a90: 7c6f 3520 3d20 746f 7461 6c3a 2032 2033 |o5 = total: 2 3 │ │ │ │ -00058aa0: 2035 2036 2020 2020 2020 2020 2020 2020 5 6 │ │ │ │ -00058ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058ac0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00058ad0: 2030 3a20 3220 3320 2e20 2e20 2020 2020 0: 2 3 . . │ │ │ │ +00058a80: 2020 207c 0a7c 6f35 203d 2074 6f74 616c |.|o5 = total │ │ │ │ +00058a90: 3a20 3220 3320 3520 3620 2020 2020 2020 : 2 3 5 6 │ │ │ │ +00058aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00058ab0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00058ac0: 2020 2020 2020 303a 2032 2033 202e 202e 0: 2 3 . . │ │ │ │ +00058ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00058ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058af0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00058b00: 2020 2020 2020 2020 313a 202e 202e 2035 1: . . 5 │ │ │ │ -00058b10: 2036 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ -00058b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058b30: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00058af0: 207c 0a7c 2020 2020 2020 2020 2031 3a20 |.| 1: │ │ │ │ +00058b00: 2e20 2e20 3520 3620 2020 2020 2020 2020 . . 5 6 │ │ │ │ +00058b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00058b20: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00058b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00058b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058b60: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ -00058b70: 3a20 4265 7474 6954 616c 6c79 2020 2020 : BettiTally │ │ │ │ +00058b50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00058b60: 0a7c 6f35 203a 2042 6574 7469 5461 6c6c .|o5 : BettiTall │ │ │ │ +00058b70: 7920 2020 2020 2020 2020 2020 2020 2020 y │ │ │ │ 00058b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058ba0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00058b90: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00058ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00058bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058bd0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ -00058be0: 5420 3d20 6d61 6b65 5428 6666 2c46 2c33 T = makeT(ff,F,3 │ │ │ │ -00058bf0: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ -00058c00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00058c10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00058bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00058bd0: 6936 203a 2054 203d 206d 616b 6554 2866 i6 : T = makeT(f │ │ │ │ +00058be0: 662c 462c 3329 3b20 2020 2020 2020 2020 f,F,3); │ │ │ │ +00058bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00058c00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00058c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00058c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058c40: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6e65 ------+.|i7 : ne │ │ │ │ -00058c50: 744c 6973 7420 5420 2020 2020 2020 2020 tList T │ │ │ │ +00058c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ +00058c40: 203a 206e 6574 4c69 7374 2054 2020 2020 : netList T │ │ │ │ +00058c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00058c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058c70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00058c70: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00058c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00058c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058cb0: 2020 2020 7c0a 7c20 2020 2020 2b2d 2d2d |.| +--- │ │ │ │ -00058cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058cd0: 2d2d 2d2d 2d2b 2020 2020 2020 2020 2020 -----+ │ │ │ │ -00058ce0: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -00058cf0: 203d 207c 7b34 7d20 7c20 3020 3020 3020 = |{4} | 0 0 0 │ │ │ │ -00058d00: 3020 2031 2030 207c 2020 2020 7c20 2020 0 1 0 | | │ │ │ │ -00058d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058d20: 2020 7c0a 7c20 2020 2020 7c7b 347d 207c |.| |{4} | │ │ │ │ -00058d30: 2030 2030 2030 202d 3120 3020 3020 7c20 0 0 0 -1 0 0 | │ │ │ │ -00058d40: 2020 207c 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00058d50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00058d60: 207c 7b34 7d20 7c20 3020 3020 3020 3020 |{4} | 0 0 0 0 │ │ │ │ -00058d70: 2030 2031 207c 2020 2020 7c20 2020 2020 0 1 | | │ │ │ │ -00058d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058d90: 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d 2d2d |.| +------- │ │ │ │ -00058da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058db0: 2d2b 2020 2020 2020 2020 2020 2020 2020 -+ │ │ │ │ -00058dc0: 2020 2020 2020 207c 0a7c 2020 2020 207c |.| | │ │ │ │ -00058dd0: 7b34 7d20 7c20 3020 3120 3020 3020 3020 {4} | 0 1 0 0 0 │ │ │ │ -00058de0: 3020 7c20 2020 2020 7c20 2020 2020 2020 0 | | │ │ │ │ -00058df0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00058e00: 7c20 2020 2020 7c7b 347d 207c 2031 2030 | |{4} | 1 0 │ │ │ │ -00058e10: 2030 2030 2030 2030 207c 2020 2020 207c 0 0 0 0 | | │ │ │ │ -00058e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058e30: 2020 2020 207c 0a7c 2020 2020 207c 7b34 |.| |{4 │ │ │ │ -00058e40: 7d20 7c20 3020 3020 3120 3020 3020 3020 } | 0 0 1 0 0 0 │ │ │ │ -00058e50: 7c20 2020 2020 7c20 2020 2020 2020 2020 | | │ │ │ │ -00058e60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00058e70: 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d +----------- │ │ │ │ -00058e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 2020 -------------+ │ │ │ │ -00058e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058ea0: 2020 207c 0a7c 2020 2020 207c 7b34 7d20 |.| |{4} │ │ │ │ -00058eb0: 7c20 3020 202d 3120 3020 2030 202d 3120 | 0 -1 0 0 -1 │ │ │ │ -00058ec0: 3020 207c 7c20 2020 2020 2020 2020 2020 0 || │ │ │ │ -00058ed0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00058ee0: 2020 7c7b 347d 207c 202d 3120 3020 2030 |{4} | -1 0 0 │ │ │ │ -00058ef0: 2020 3120 3020 2030 2020 7c7c 2020 2020 1 0 0 || │ │ │ │ -00058f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058f10: 207c 0a7c 2020 2020 207c 7b34 7d20 7c20 |.| |{4} | │ │ │ │ -00058f20: 3020 2030 2020 2d31 2030 2030 2020 2d31 0 0 -1 0 0 -1 │ │ │ │ -00058f30: 207c 7c20 2020 2020 2020 2020 2020 2020 || │ │ │ │ -00058f40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00058f50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00058f60: 2d2d 2d2d 2d2d 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| │ │ │ │ +00058e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00058e60: 207c 0a7c 2020 2020 202b 2d2d 2d2d 2d2d |.| +------ │ │ │ │ +00058e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00058e80: 2d2d 2b20 2020 2020 2020 2020 2020 2020 --+ │ │ │ │ +00058e90: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00058ea0: 7c7b 347d 207c 2030 2020 2d31 2030 2020 |{4} | 0 -1 0 │ │ │ │ +00058eb0: 3020 2d31 2030 2020 7c7c 2020 2020 2020 0 -1 0 || │ │ │ │ +00058ec0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00058ed0: 0a7c 2020 2020 207c 7b34 7d20 7c20 2d31 .| |{4} | -1 │ │ │ │ +00058ee0: 2030 2020 3020 2031 2030 2020 3020 207c 0 0 1 0 0 | │ │ │ │ +00058ef0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00058f00: 2020 2020 2020 7c0a 7c20 2020 2020 7c7b |.| |{ │ │ │ │ +00058f10: 347d 207c 2030 2020 3020 202d 3120 3020 4} | 0 0 -1 0 │ │ │ │ +00058f20: 3020 202d 3120 7c7c 2020 2020 2020 2020 0 -1 || │ │ │ │ +00058f30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00058f40: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ +00058f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 --------------+ │ │ │ │ +00058f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00058f70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00058f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00058f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058fb0: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 6973 ------+.|i8 : is │ │ │ │ -00058fc0: 486f 6d6f 6765 6e65 6f75 7320 545f 3220 Homogeneous T_2 │ │ │ │ +00058fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ +00058fb0: 203a 2069 7348 6f6d 6f67 656e 656f 7573 : isHomogeneous │ │ │ │ +00058fc0: 2054 5f32 2020 2020 2020 2020 2020 2020 T_2 │ │ │ │ 00058fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058fe0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00058fe0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00058ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059020: 2020 2020 7c0a 7c6f 3820 3d20 7472 7565 |.|o8 = true │ │ │ │ +00059010: 2020 2020 2020 2020 207c 0a7c 6f38 203d |.|o8 = │ │ │ │ +00059020: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ 00059030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059050: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00059050: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00059060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00059070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00059080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00059090: 2d2d 2b0a 0a43 6176 6561 740a 3d3d 3d3d --+..Caveat.==== │ │ │ │ -000590a0: 3d3d 0a0a 5363 7269 7074 2061 7373 756d ==..Script assum │ │ │ │ -000590b0: 6573 2074 6861 7420 7269 6e67 2046 203d es that ring F = │ │ │ │ -000590c0: 3d20 2872 696e 6720 6666 292f 2869 6465 = (ring ff)/(ide │ │ │ │ -000590d0: 616c 2066 6629 2e20 4974 206d 6967 6874 al ff). It might │ │ │ │ -000590e0: 2062 6520 6d6f 7265 2075 7365 6675 6c20 be more useful │ │ │ │ -000590f0: 746f 0a72 6574 7572 6e20 7468 6520 6f70 to.return the op │ │ │ │ -00059100: 6572 6174 6f72 7320 6173 206d 6174 7269 erators as matri │ │ │ │ -00059110: 6365 7320 6f76 6572 2053 2072 6174 6865 ces over S rathe │ │ │ │ -00059120: 7220 7468 616e 206f 7665 7220 522c 2073 r than over R, s │ │ │ │ -00059130: 696e 6365 2074 6869 7320 6973 2077 6861 ince this is wha │ │ │ │ -00059140: 740a 7765 2764 206e 6565 6420 666f 7220 t.we'd need for │ │ │ │ -00059150: 7468 696e 6773 206c 696b 6520 6d61 7472 things like matr │ │ │ │ -00059160: 6978 4661 6374 6f72 697a 6174 696f 6e20 ixFactorization │ │ │ │ -00059170: 2877 6865 7265 2074 6869 7320 7072 6f63 (where this proc │ │ │ │ -00059180: 6573 7320 6375 7272 656e 746c 790a 646f ess currently.do │ │ │ │ -00059190: 6e65 206f 6e20 7468 6520 666c 792c 206e ne on the fly, n │ │ │ │ -000591a0: 6f74 2063 616c 6c69 6e67 206d 616b 6554 ot calling makeT │ │ │ │ -000591b0: 290a 0a57 6179 7320 746f 2075 7365 206d )..Ways to use m │ │ │ │ -000591c0: 616b 6554 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d akeT:.========== │ │ │ │ -000591d0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d ========.. * "m │ │ │ │ -000591e0: 616b 6554 284d 6174 7269 782c 436f 6d70 akeT(Matrix,Comp │ │ │ │ -000591f0: 6c65 782c 5a5a 2922 0a0a 466f 7220 7468 lex,ZZ)"..For th │ │ │ │ -00059200: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -00059210: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00059220: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -00059230: 6520 6d61 6b65 543a 206d 616b 6554 2c20 e makeT: makeT, │ │ │ │ -00059240: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -00059250: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ -00059260: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -00059270: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +00059080: 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 6174 -------+..Caveat │ │ │ │ +00059090: 0a3d 3d3d 3d3d 3d0a 0a53 6372 6970 7420 .======..Script │ │ │ │ +000590a0: 6173 7375 6d65 7320 7468 6174 2072 696e assumes that rin │ │ │ │ +000590b0: 6720 4620 3d3d 2028 7269 6e67 2066 6629 g F == (ring ff) │ │ │ │ +000590c0: 2f28 6964 6561 6c20 6666 292e 2049 7420 /(ideal ff). It │ │ │ │ +000590d0: 6d69 6768 7420 6265 206d 6f72 6520 7573 might be more us │ │ │ │ +000590e0: 6566 756c 2074 6f0a 7265 7475 726e 2074 eful to.return t │ │ │ │ +000590f0: 6865 206f 7065 7261 746f 7273 2061 7320 he operators as │ │ │ │ +00059100: 6d61 7472 6963 6573 206f 7665 7220 5320 matrices over S │ │ │ │ +00059110: 7261 7468 6572 2074 6861 6e20 6f76 6572 rather than over │ │ │ │ +00059120: 2052 2c20 7369 6e63 6520 7468 6973 2069 R, since this i │ │ │ │ +00059130: 7320 7768 6174 0a77 6527 6420 6e65 6564 s what.we'd need │ │ │ │ +00059140: 2066 6f72 2074 6869 6e67 7320 6c69 6b65 for things like │ │ │ │ +00059150: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ +00059160: 7469 6f6e 2028 7768 6572 6520 7468 6973 tion (where this │ │ │ │ +00059170: 2070 726f 6365 7373 2063 7572 7265 6e74 process current │ │ │ │ +00059180: 6c79 0a64 6f6e 6520 6f6e 2074 6865 2066 ly.done on the f │ │ │ │ +00059190: 6c79 2c20 6e6f 7420 6361 6c6c 696e 6720 ly, not calling │ │ │ │ +000591a0: 6d61 6b65 5429 0a0a 5761 7973 2074 6f20 makeT)..Ways to │ │ │ │ +000591b0: 7573 6520 6d61 6b65 543a 0a3d 3d3d 3d3d use makeT:.===== │ │ │ │ +000591c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +000591d0: 202a 2022 6d61 6b65 5428 4d61 7472 6978 * "makeT(Matrix │ │ │ │ +000591e0: 2c43 6f6d 706c 6578 2c5a 5a29 220a 0a46 ,Complex,ZZ)"..F │ │ │ │ +000591f0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +00059200: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +00059210: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +00059220: 202a 6e6f 7465 206d 616b 6554 3a20 6d61 *note makeT: ma │ │ │ │ +00059230: 6b65 542c 2069 7320 6120 2a6e 6f74 6520 keT, is a *note │ │ │ │ +00059240: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ +00059250: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ +00059260: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +00059270: 0a2d 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5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -00059350: 6f6e 732e 6d32 3a33 3534 383a 302e 0a1f ons.m2:3548:0... │ │ │ │ -00059360: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ -00059370: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -00059380: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ -00059390: 653a 206d 6174 7269 7846 6163 746f 7269 e: matrixFactori │ │ │ │ -000593a0: 7a61 7469 6f6e 2c20 4e65 7874 3a20 6d66 zation, Next: mf │ │ │ │ -000593b0: 426f 756e 642c 2050 7265 763a 206d 616b Bound, Prev: mak │ │ │ │ -000593c0: 6554 2c20 5570 3a20 546f 700a 0a6d 6174 eT, Up: Top..mat │ │ │ │ -000593d0: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -000593e0: 202d 2d20 4d61 7073 2069 6e20 6120 6869 -- Maps in a hi │ │ │ │ -000593f0: 6768 6572 2063 6f64 696d 656e 7369 6f6e gher codimension │ │ │ │ -00059400: 206d 6174 7269 7820 6661 6374 6f72 697a matrix factoriz │ │ │ │ -00059410: 6174 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a ation.********** │ │ │ │ +000592c0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +000592d0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +000592e0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +000592f0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00059300: 6175 6c61 7932 2d31 2e32 362e 3035 2b64 aulay2-1.26.05+d │ │ │ │ +00059310: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00059320: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ +00059330: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +00059340: 6f6c 7574 696f 6e73 2e6d 323a 3335 3438 olutions.m2:3548 │ │ │ │ +00059350: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ +00059360: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +00059370: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ +00059380: 2c20 4e6f 6465 3a20 6d61 7472 6978 4661 , Node: matrixFa │ │ │ │ +00059390: 6374 6f72 697a 6174 696f 6e2c 204e 6578 ctorization, Nex │ │ │ │ +000593a0: 743a 206d 6642 6f75 6e64 2c20 5072 6576 t: mfBound, Prev │ │ │ │ +000593b0: 3a20 6d61 6b65 542c 2055 703a 2054 6f70 : makeT, Up: Top │ │ │ │ +000593c0: 0a0a 6d61 7472 6978 4661 6374 6f72 697a ..matrixFactoriz │ │ │ │ +000593d0: 6174 696f 6e20 2d2d 204d 6170 7320 696e ation -- Maps in │ │ │ │ +000593e0: 2061 2068 6967 6865 7220 636f 6469 6d65 a higher codime │ │ │ │ +000593f0: 6e73 696f 6e20 6d61 7472 6978 2066 6163 nsion matrix fac │ │ │ │ +00059400: 746f 7269 7a61 7469 6f6e 0a2a 2a2a 2a2a torization.***** │ │ │ │ +00059410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00059420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00059430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00059440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00059450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -00059460: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -00059470: 2020 2020 4d46 203d 206d 6174 7269 7846 MF = matrixF │ │ │ │ -00059480: 6163 746f 7269 7a61 7469 6f6e 2866 662c actorization(ff, │ │ │ │ -00059490: 4d29 0a20 202a 2049 6e70 7574 733a 0a20 M). * Inputs:. │ │ │ │ -000594a0: 2020 2020 202a 2066 662c 2061 202a 6e6f * ff, a *no │ │ │ │ -000594b0: 7465 206d 6174 7269 783a 2028 4d61 6361 te matrix: (Maca │ │ │ │ -000594c0: 756c 6179 3244 6f63 294d 6174 7269 782c ulay2Doc)Matrix, │ │ │ │ -000594d0: 2c20 6120 7375 6666 6963 6965 6e74 6c79 , a sufficiently │ │ │ │ -000594e0: 2067 656e 6572 616c 0a20 2020 2020 2020 general. │ │ │ │ -000594f0: 2072 6567 756c 6172 2073 6571 7565 6e63 regular sequenc │ │ │ │ -00059500: 6520 696e 2061 2072 696e 6720 530a 2020 e in a ring S. │ │ │ │ -00059510: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ -00059520: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ -00059530: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ -00059540: 6120 6d61 7869 6d61 6c20 436f 6865 6e2d a maximal Cohen- │ │ │ │ -00059550: 4d61 6361 756c 6179 0a20 2020 2020 2020 Macaulay. │ │ │ │ -00059560: 206d 6f64 756c 6520 6f76 6572 2053 2f69 module over S/i │ │ │ │ -00059570: 6465 616c 2066 660a 2020 2a20 2a6e 6f74 deal ff. * *not │ │ │ │ -00059580: 6520 4f70 7469 6f6e 616c 2069 6e70 7574 e Optional input │ │ │ │ -00059590: 733a 2028 4d61 6361 756c 6179 3244 6f63 s: (Macaulay2Doc │ │ │ │ -000595a0: 2975 7369 6e67 2066 756e 6374 696f 6e73 )using functions │ │ │ │ -000595b0: 2077 6974 6820 6f70 7469 6f6e 616c 2069 with optional i │ │ │ │ -000595c0: 6e70 7574 732c 3a0a 2020 2020 2020 2a20 nputs,:. * │ │ │ │ -000595d0: 4175 676d 656e 7461 7469 6f6e 203d 3e20 Augmentation => │ │ │ │ -000595e0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -000595f0: 7565 2074 7275 650a 2020 2020 2020 2a20 ue true. * │ │ │ │ -00059600: 4368 6563 6b20 3d3e 202e 2e2e 2c20 6465 Check => ..., de │ │ │ │ -00059610: 6661 756c 7420 7661 6c75 6520 6661 6c73 fault value fals │ │ │ │ -00059620: 650a 2020 2020 2020 2a20 4c61 7965 7265 e. * Layere │ │ │ │ -00059630: 6420 3d3e 202e 2e2e 2c20 6465 6661 756c d => ..., defaul │ │ │ │ -00059640: 7420 7661 6c75 6520 7472 7565 0a20 2020 t value true. │ │ │ │ -00059650: 2020 202a 2056 6572 626f 7365 203d 3e20 * Verbose => │ │ │ │ -00059660: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -00059670: 7565 2066 616c 7365 0a20 202a 204f 7574 ue false. * Out │ │ │ │ -00059680: 7075 7473 3a0a 2020 2020 2020 2a20 4d46 puts:. * MF │ │ │ │ -00059690: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ -000596a0: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ -000596b0: 7374 2c2c 205c 7b64 2c68 2c67 616d 6d61 st,, \{d,h,gamma │ │ │ │ -000596c0: 5c7d 2c20 7768 6572 6520 643a 415f 3120 \}, where d:A_1 │ │ │ │ -000596d0: 5c74 6f0a 2020 2020 2020 2020 415f 3020 \to. A_0 │ │ │ │ -000596e0: 616e 6420 683a 205c 6f70 6c75 7320 415f and h: \oplus A_ │ │ │ │ -000596f0: 3028 7029 205c 746f 2041 5f31 2069 7320 0(p) \to A_1 is │ │ │ │ -00059700: 7468 6520 6469 7265 6374 2073 756d 206f the direct sum o │ │ │ │ -00059710: 6620 7061 7274 6961 6c0a 2020 2020 2020 f partial. │ │ │ │ -00059720: 2020 686f 6d6f 746f 7069 6573 2c20 616e homotopies, an │ │ │ │ -00059730: 6420 6761 6d6d 613a 2041 5f30 202d 3e4d d gamma: A_0 ->M │ │ │ │ -00059740: 2069 7320 7468 6520 6175 676d 656e 7461 is the augmenta │ │ │ │ -00059750: 7469 6f6e 2028 7265 7475 726e 6564 206f tion (returned o │ │ │ │ -00059760: 6e6c 7920 6966 0a20 2020 2020 2020 2041 nly if. A │ │ │ │ -00059770: 7567 6d65 6e74 6174 696f 6e20 3d3e 7472 ugmentation =>tr │ │ │ │ -00059780: 7565 290a 0a44 6573 6372 6970 7469 6f6e ue)..Description │ │ │ │ -00059790: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 .===========..Th │ │ │ │ -000597a0: 6520 696e 7075 7420 6d6f 6475 6c65 204d e input module M │ │ │ │ -000597b0: 2073 686f 756c 6420 6265 2061 206d 6178 should be a max │ │ │ │ -000597c0: 696d 616c 2043 6f68 656e 2d4d 6163 6175 imal Cohen-Macau │ │ │ │ -000597d0: 6c61 7920 6d6f 6475 6c65 206f 7665 7220 lay module over │ │ │ │ -000597e0: 5220 3d20 532f 6964 6561 6c0a 6666 2e20 R = S/ideal.ff. │ │ │ │ -000597f0: 2049 6620 4d20 6973 2069 6e20 6661 6374 If M is in fact │ │ │ │ -00059800: 2061 2022 6869 6768 2073 797a 7967 7922 a "high syzygy" │ │ │ │ -00059810: 2c20 7468 656e 2074 6865 2066 756e 6374 , then the funct │ │ │ │ -00059820: 696f 6e0a 6d61 7472 6978 4661 6374 6f72 ion.matrixFactor │ │ │ │ -00059830: 697a 6174 696f 6e28 6666 2c4d 2c4c 6179 ization(ff,M,Lay │ │ │ │ -00059840: 6572 6564 3d3e 6661 6c73 6529 2075 7365 ered=>false) use │ │ │ │ -00059850: 7320 6120 6469 6666 6572 656e 742c 2066 s a different, f │ │ │ │ -00059860: 6173 7465 7220 616c 676f 7269 7468 6d0a aster algorithm. │ │ │ │ -00059870: 7768 6963 6820 6f6e 6c79 2077 6f72 6b73 which only works │ │ │ │ -00059880: 2069 6e20 7468 6520 6869 6768 2073 797a in the high syz │ │ │ │ -00059890: 7967 7920 6361 7365 2e0a 0a49 6e20 616c ygy case...In al │ │ │ │ -000598a0: 6c20 6578 616d 706c 6573 2077 6520 6b6e l examples we kn │ │ │ │ -000598b0: 6f77 2c20 4d20 6361 6e20 6265 2063 6f6e ow, M can be con │ │ │ │ -000598c0: 7369 6465 7265 6420 6120 2268 6967 6820 sidered a "high │ │ │ │ -000598d0: 7379 7a79 6779 2220 6173 206c 6f6e 6720 syzygy" as long │ │ │ │ -000598e0: 6173 0a45 7874 5e7b 6576 656e 7d5f 5228 as.Ext^{even}_R( │ │ │ │ -000598f0: 4d2c 6b29 2061 6e64 2045 7874 5e7b 6f64 M,k) and Ext^{od │ │ │ │ -00059900: 647d 5f52 284d 2c6b 2920 6861 7665 206e d}_R(M,k) have n │ │ │ │ -00059910: 6567 6174 6976 6520 7265 6775 6c61 7269 egative regulari │ │ │ │ -00059920: 7479 206f 7665 7220 7468 6520 7269 6e67 ty over the ring │ │ │ │ -00059930: 0a6f 6620 4349 206f 7065 7261 746f 7273 .of CI operators │ │ │ │ -00059940: 2028 7265 6772 6164 6564 2077 6974 6820 (regraded with │ │ │ │ -00059950: 7661 7269 6162 6c65 7320 6f66 2064 6567 variables of deg │ │ │ │ -00059960: 7265 6520 312e 2048 6f77 6576 6572 2c20 ree 1. However, │ │ │ │ -00059970: 7468 6520 6265 7374 2072 6573 756c 740a the best result. │ │ │ │ -00059980: 7765 2063 616e 2070 726f 7665 2069 7320 we can prove is │ │ │ │ -00059990: 7468 6174 2069 7420 7375 6666 6963 6573 that it suffices │ │ │ │ -000599a0: 2074 6f20 6861 7665 2072 6567 756c 6172 to have regular │ │ │ │ -000599b0: 6974 7920 3c20 2d28 322a 6469 6d20 522b ity < -(2*dim R+ │ │ │ │ -000599c0: 3129 2e0a 0a57 6865 6e20 7468 6520 6f70 1)...When the op │ │ │ │ -000599d0: 7469 6f6e 616c 2069 6e70 7574 2043 6865 tional input Che │ │ │ │ -000599e0: 636b 3d3d 7472 7565 2028 7468 6520 6465 ck==true (the de │ │ │ │ -000599f0: 6661 756c 7420 6973 2043 6865 636b 3d3d fault is Check== │ │ │ │ -00059a00: 6661 6c73 6529 2c20 7468 650a 7072 6f70 false), the.prop │ │ │ │ -00059a10: 6572 7469 6573 2069 6e20 7468 6520 6465 erties in the de │ │ │ │ -00059a20: 6669 6e69 7469 6f6e 206f 6620 4d61 7472 finition of Matr │ │ │ │ -00059a30: 6978 2046 6163 746f 7269 7a61 7469 6f6e ix Factorization │ │ │ │ -00059a40: 2061 7265 2076 6572 6966 6965 640a 0a54 are verified..T │ │ │ │ -00059a50: 6865 206f 7574 7075 7420 6973 2061 206c he output is a l │ │ │ │ -00059a60: 6973 7420 6f66 206d 6170 7320 5c7b 642c ist of maps \{d, │ │ │ │ -00059a70: 685c 7d20 6f72 205c 7b64 2c68 2c67 616d h\} or \{d,h,gam │ │ │ │ -00059a80: 6d61 5c7d 2c20 7768 6572 6520 6761 6d6d ma\}, where gamm │ │ │ │ -00059a90: 6120 6973 2061 6e0a 6175 676d 656e 7461 a is an.augmenta │ │ │ │ -00059aa0: 7469 6f6e 2c20 7468 6174 2069 732c 2061 tion, that is, a │ │ │ │ -00059ab0: 206d 6170 2066 726f 6d20 7461 7267 6574 map from target │ │ │ │ -00059ac0: 2064 2074 6f20 4d2e 0a0a 5468 6520 6d61 d to M...The ma │ │ │ │ -00059ad0: 7020 6420 6973 2061 2073 7065 6369 616c p d is a special │ │ │ │ -00059ae0: 206c 6966 7469 6e67 2074 6f20 5320 6f66 lifting to S of │ │ │ │ -00059af0: 2061 2070 7265 7365 6e74 6174 696f 6e20 a presentation │ │ │ │ -00059b00: 6f66 204d 206f 7665 7220 522e 2054 6f20 of M over R. To │ │ │ │ -00059b10: 6578 706c 6169 6e0a 7468 6520 636f 6e74 explain.the cont │ │ │ │ -00059b20: 656e 7473 2c20 7765 2069 6e74 726f 6475 ents, we introdu │ │ │ │ -00059b30: 6365 2073 6f6d 6520 6e6f 7461 7469 6f6e ce some notation │ │ │ │ -00059b40: 2028 6672 6f6d 2045 6973 656e 6275 6420 (from Eisenbud │ │ │ │ -00059b50: 616e 6420 5065 6576 612c 2022 4d69 6e69 and Peeva, "Mini │ │ │ │ -00059b60: 6d61 6c0a 6672 6565 2072 6573 6f6c 7574 mal.free resolut │ │ │ │ -00059b70: 696f 6e73 206f 7665 7220 636f 6d70 6c65 ions over comple │ │ │ │ -00059b80: 7465 2069 6e74 6572 7365 6374 696f 6e73 te intersections │ │ │ │ -00059b90: 2220 4c65 6374 7572 6520 4e6f 7465 7320 " Lecture Notes │ │ │ │ -00059ba0: 696e 204d 6174 6865 6d61 7469 6373 2c0a in Mathematics,. │ │ │ │ -00059bb0: 3231 3532 2e20 5370 7269 6e67 6572 2c20 2152. Springer, │ │ │ │ -00059bc0: 4368 616d 2c20 3230 3136 2e20 782b 3130 Cham, 2016. x+10 │ │ │ │ -00059bd0: 3720 7070 2e20 4953 424e 3a20 3937 382d 7 pp. ISBN: 978- │ │ │ │ -00059be0: 332d 3331 392d 3236 3433 362d 333b 0a39 3-319-26436-3;.9 │ │ │ │ -00059bf0: 3738 2d33 2d33 3139 2d32 3634 3337 2d30 78-3-319-26437-0 │ │ │ │ -00059c00: 292e 0a0a 5228 6929 203d 2053 2f28 6666 )...R(i) = S/(ff │ │ │ │ -00059c10: 5f30 2c2e 2e2c 6666 5f7b 692d 317d 292e _0,..,ff_{i-1}). │ │ │ │ -00059c20: 2048 6572 6520 303c 3d20 6920 3c3d 2063 Here 0<= i <= c │ │ │ │ -00059c30: 2c20 616e 6420 5220 3d20 5228 6329 2061 , and R = R(c) a │ │ │ │ -00059c40: 6e64 2053 203d 2052 2830 292e 0a0a 4228 nd S = R(0)...B( │ │ │ │ -00059c50: 6929 203d 2074 6865 206d 6174 7269 7820 i) = the matrix │ │ │ │ -00059c60: 286f 7665 7220 5329 2072 6570 7265 7365 (over S) represe │ │ │ │ -00059c70: 6e74 696e 6720 645f 693a 2042 5f31 2869 nting d_i: B_1(i │ │ │ │ -00059c80: 2920 5c74 6f20 425f 3028 6929 0a0a 6428 ) \to B_0(i)..d( │ │ │ │ -00059c90: 6929 3a20 415f 3128 6929 205c 746f 2041 i): A_1(i) \to A │ │ │ │ -00059ca0: 5f30 2869 2920 7468 6520 7265 7374 7269 _0(i) the restri │ │ │ │ -00059cb0: 6374 696f 6e20 6f66 2064 203d 2064 2863 ction of d = d(c │ │ │ │ -00059cc0: 292e 2077 6865 7265 2041 2869 2920 3d0a ). where A(i) =. │ │ │ │ -00059cd0: 5c6f 706c 7573 5f7b 693d 317d 5e70 2042 \oplus_{i=1}^p B │ │ │ │ -00059ce0: 2869 290a 0a0a 0a54 6865 206d 6170 2068 (i)....The map h │ │ │ │ -00059cf0: 2069 7320 6120 6469 7265 6374 2073 756d is a direct sum │ │ │ │ -00059d00: 206f 6620 6d61 7073 2074 6172 6765 7420 of maps target │ │ │ │ -00059d10: 6428 7029 205c 746f 2073 6f75 7263 6520 d(p) \to source │ │ │ │ -00059d20: 6428 7029 2074 6861 7420 6172 650a 686f d(p) that are.ho │ │ │ │ -00059d30: 6d6f 746f 7069 6573 2066 6f72 2066 665f motopies for ff_ │ │ │ │ -00059d40: 7020 6f6e 2074 6865 2072 6573 7472 6963 p on the restric │ │ │ │ -00059d50: 7469 6f6e 2064 2870 293a 206f 7665 7220 tion d(p): over │ │ │ │ -00059d60: 7468 6520 7269 6e67 2052 2328 702d 3129 the ring R#(p-1) │ │ │ │ -00059d70: 203d 0a53 2f28 6666 2331 2e2e 6666 2328 =.S/(ff#1..ff#( │ │ │ │ -00059d80: 702d 3129 2c20 736f 2064 2870 2920 2a20 p-1), so d(p) * │ │ │ │ -00059d90: 6823 7020 3d20 6666 2370 206d 6f64 2028 h#p = ff#p mod ( │ │ │ │ -00059da0: 6666 2331 2e2e 6666 2328 702d 3129 2e0a ff#1..ff#(p-1).. │ │ │ │ -00059db0: 0a49 6e20 6164 6469 7469 6f6e 2c20 6823 .In addition, h# │ │ │ │ -00059dc0: 7020 2a20 6428 7029 2069 6e64 7563 6573 p * d(p) induces │ │ │ │ -00059dd0: 2066 6623 7020 6f6e 2042 3123 7020 6d6f ff#p on B1#p mo │ │ │ │ -00059de0: 6420 2866 6623 312e 2e66 6623 2870 2d31 d (ff#1..ff#(p-1 │ │ │ │ -00059df0: 292e 0a0a 4865 7265 2069 7320 6120 7369 )...Here is a si │ │ │ │ -00059e00: 6d70 6c65 2065 7861 6d70 6c65 3a0a 0a2b mple example:..+ │ │ │ │ +00059450: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ +00059460: 0a20 2020 2020 2020 204d 4620 3d20 6d61 . MF = ma │ │ │ │ +00059470: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ +00059480: 6e28 6666 2c4d 290a 2020 2a20 496e 7075 n(ff,M). * Inpu │ │ │ │ +00059490: 7473 3a0a 2020 2020 2020 2a20 6666 2c20 ts:. * ff, │ │ │ │ +000594a0: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ +000594b0: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ +000594c0: 7472 6978 2c2c 2061 2073 7566 6669 6369 trix,, a suffici │ │ │ │ +000594d0: 656e 746c 7920 6765 6e65 7261 6c0a 2020 ently general. │ │ │ │ +000594e0: 2020 2020 2020 7265 6775 6c61 7220 7365 regular se │ │ │ │ +000594f0: 7175 656e 6365 2069 6e20 6120 7269 6e67 quence in a ring │ │ │ │ +00059500: 2053 0a20 2020 2020 202a 204d 2c20 6120 S. * M, a │ │ │ │ +00059510: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ +00059520: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ +00059530: 6c65 2c2c 2061 206d 6178 696d 616c 2043 le,, a maximal C │ │ │ │ +00059540: 6f68 656e 2d4d 6163 6175 6c61 790a 2020 ohen-Macaulay. │ │ │ │ +00059550: 2020 2020 2020 6d6f 6475 6c65 206f 7665 module ove │ │ │ │ +00059560: 7220 532f 6964 6561 6c20 6666 0a20 202a r S/ideal ff. * │ │ │ │ +00059570: 202a 6e6f 7465 204f 7074 696f 6e61 6c20 *note Optional │ │ │ │ +00059580: 696e 7075 7473 3a20 284d 6163 6175 6c61 inputs: (Macaula │ │ │ │ +00059590: 7932 446f 6329 7573 696e 6720 6675 6e63 y2Doc)using func │ │ │ │ +000595a0: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ +000595b0: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020 nal inputs,:. │ │ │ │ +000595c0: 2020 202a 2041 7567 6d65 6e74 6174 696f * Augmentatio │ │ │ │ +000595d0: 6e20 3d3e 202e 2e2e 2c20 6465 6661 756c n => ..., defaul │ │ │ │ +000595e0: 7420 7661 6c75 6520 7472 7565 0a20 2020 t value true. │ │ │ │ +000595f0: 2020 202a 2043 6865 636b 203d 3e20 2e2e * Check => .. │ │ │ │ +00059600: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ +00059610: 2066 616c 7365 0a20 2020 2020 202a 204c false. * L │ │ │ │ +00059620: 6179 6572 6564 203d 3e20 2e2e 2e2c 2064 ayered => ..., d │ │ │ │ +00059630: 6566 6175 6c74 2076 616c 7565 2074 7275 efault value tru │ │ │ │ +00059640: 650a 2020 2020 2020 2a20 5665 7262 6f73 e. * Verbos │ │ │ │ +00059650: 6520 3d3e 202e 2e2e 2c20 6465 6661 756c e => ..., defaul │ │ │ │ +00059660: 7420 7661 6c75 6520 6661 6c73 650a 2020 t value false. │ │ │ │ +00059670: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +00059680: 202a 204d 462c 2061 202a 6e6f 7465 206c * MF, a *note l │ │ │ │ +00059690: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ +000596a0: 6f63 294c 6973 742c 2c20 5c7b 642c 682c oc)List,, \{d,h, │ │ │ │ +000596b0: 6761 6d6d 615c 7d2c 2077 6865 7265 2064 gamma\}, where d │ │ │ │ +000596c0: 3a41 5f31 205c 746f 0a20 2020 2020 2020 :A_1 \to. │ │ │ │ +000596d0: 2041 5f30 2061 6e64 2068 3a20 5c6f 706c A_0 and h: \opl │ │ │ │ +000596e0: 7573 2041 5f30 2870 2920 5c74 6f20 415f us A_0(p) \to A_ │ │ │ │ +000596f0: 3120 6973 2074 6865 2064 6972 6563 7420 1 is the direct │ │ │ │ +00059700: 7375 6d20 6f66 2070 6172 7469 616c 0a20 sum of partial. │ │ │ │ +00059710: 2020 2020 2020 2068 6f6d 6f74 6f70 6965 homotopie │ │ │ │ +00059720: 732c 2061 6e64 2067 616d 6d61 3a20 415f s, and gamma: A_ │ │ │ │ +00059730: 3020 2d3e 4d20 6973 2074 6865 2061 7567 0 ->M is the aug │ │ │ │ +00059740: 6d65 6e74 6174 696f 6e20 2872 6574 7572 mentation (retur │ │ │ │ +00059750: 6e65 6420 6f6e 6c79 2069 660a 2020 2020 ned only if. │ │ │ │ +00059760: 2020 2020 4175 676d 656e 7461 7469 6f6e Augmentation │ │ │ │ +00059770: 203d 3e74 7275 6529 0a0a 4465 7363 7269 =>true)..Descri │ │ │ │ +00059780: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ +00059790: 3d0a 0a54 6865 2069 6e70 7574 206d 6f64 =..The input mod │ │ │ │ +000597a0: 756c 6520 4d20 7368 6f75 6c64 2062 6520 ule M should be │ │ │ │ +000597b0: 6120 6d61 7869 6d61 6c20 436f 6865 6e2d a maximal Cohen- │ │ │ │ +000597c0: 4d61 6361 756c 6179 206d 6f64 756c 6520 Macaulay module │ │ │ │ +000597d0: 6f76 6572 2052 203d 2053 2f69 6465 616c over R = S/ideal │ │ │ │ +000597e0: 0a66 662e 2020 4966 204d 2069 7320 696e .ff. If M is in │ │ │ │ +000597f0: 2066 6163 7420 6120 2268 6967 6820 7379 fact a "high sy │ │ │ │ +00059800: 7a79 6779 222c 2074 6865 6e20 7468 6520 zygy", then the │ │ │ │ +00059810: 6675 6e63 7469 6f6e 0a6d 6174 7269 7846 function.matrixF │ │ │ │ +00059820: 6163 746f 7269 7a61 7469 6f6e 2866 662c actorization(ff, │ │ │ │ +00059830: 4d2c 4c61 7965 7265 643d 3e66 616c 7365 M,Layered=>false │ │ │ │ +00059840: 2920 7573 6573 2061 2064 6966 6665 7265 ) uses a differe │ │ │ │ +00059850: 6e74 2c20 6661 7374 6572 2061 6c67 6f72 nt, faster algor │ │ │ │ +00059860: 6974 686d 0a77 6869 6368 206f 6e6c 7920 ithm.which only │ │ │ │ +00059870: 776f 726b 7320 696e 2074 6865 2068 6967 works in the hig │ │ │ │ +00059880: 6820 7379 7a79 6779 2063 6173 652e 0a0a h syzygy case... │ │ │ │ +00059890: 496e 2061 6c6c 2065 7861 6d70 6c65 7320 In all examples │ │ │ │ +000598a0: 7765 206b 6e6f 772c 204d 2063 616e 2062 we know, M can b │ │ │ │ +000598b0: 6520 636f 6e73 6964 6572 6564 2061 2022 e considered a " │ │ │ │ +000598c0: 6869 6768 2073 797a 7967 7922 2061 7320 high syzygy" as │ │ │ │ +000598d0: 6c6f 6e67 2061 730a 4578 745e 7b65 7665 long as.Ext^{eve │ │ │ │ +000598e0: 6e7d 5f52 284d 2c6b 2920 616e 6420 4578 n}_R(M,k) and Ex │ │ │ │ +000598f0: 745e 7b6f 6464 7d5f 5228 4d2c 6b29 2068 t^{odd}_R(M,k) h │ │ │ │ +00059900: 6176 6520 6e65 6761 7469 7665 2072 6567 ave negative reg │ │ │ │ +00059910: 756c 6172 6974 7920 6f76 6572 2074 6865 ularity over the │ │ │ │ +00059920: 2072 696e 670a 6f66 2043 4920 6f70 6572 ring.of CI oper │ │ │ │ +00059930: 6174 6f72 7320 2872 6567 7261 6465 6420 ators (regraded │ │ │ │ +00059940: 7769 7468 2076 6172 6961 626c 6573 206f with variables o │ │ │ │ +00059950: 6620 6465 6772 6565 2031 2e20 486f 7765 f degree 1. Howe │ │ │ │ +00059960: 7665 722c 2074 6865 2062 6573 7420 7265 ver, the best re │ │ │ │ +00059970: 7375 6c74 0a77 6520 6361 6e20 7072 6f76 sult.we can prov │ │ │ │ +00059980: 6520 6973 2074 6861 7420 6974 2073 7566 e is that it suf │ │ │ │ +00059990: 6669 6365 7320 746f 2068 6176 6520 7265 fices to have re │ │ │ │ +000599a0: 6775 6c61 7269 7479 203c 202d 2832 2a64 gularity < -(2*d │ │ │ │ +000599b0: 696d 2052 2b31 292e 0a0a 5768 656e 2074 im R+1)...When t │ │ │ │ +000599c0: 6865 206f 7074 696f 6e61 6c20 696e 7075 he optional inpu │ │ │ │ +000599d0: 7420 4368 6563 6b3d 3d74 7275 6520 2874 t Check==true (t │ │ │ │ +000599e0: 6865 2064 6566 6175 6c74 2069 7320 4368 he default is Ch │ │ │ │ +000599f0: 6563 6b3d 3d66 616c 7365 292c 2074 6865 eck==false), the │ │ │ │ +00059a00: 0a70 726f 7065 7274 6965 7320 696e 2074 .properties in t │ │ │ │ +00059a10: 6865 2064 6566 696e 6974 696f 6e20 6f66 he definition of │ │ │ │ +00059a20: 204d 6174 7269 7820 4661 6374 6f72 697a Matrix Factoriz │ │ │ │ +00059a30: 6174 696f 6e20 6172 6520 7665 7269 6669 ation are verifi │ │ │ │ +00059a40: 6564 0a0a 5468 6520 6f75 7470 7574 2069 ed..The output i │ │ │ │ +00059a50: 7320 6120 6c69 7374 206f 6620 6d61 7073 s a list of maps │ │ │ │ +00059a60: 205c 7b64 2c68 5c7d 206f 7220 5c7b 642c \{d,h\} or \{d, │ │ │ │ +00059a70: 682c 6761 6d6d 615c 7d2c 2077 6865 7265 h,gamma\}, where │ │ │ │ +00059a80: 2067 616d 6d61 2069 7320 616e 0a61 7567 gamma is an.aug │ │ │ │ +00059a90: 6d65 6e74 6174 696f 6e2c 2074 6861 7420 mentation, that │ │ │ │ +00059aa0: 6973 2c20 6120 6d61 7020 6672 6f6d 2074 is, a map from t │ │ │ │ +00059ab0: 6172 6765 7420 6420 746f 204d 2e0a 0a54 arget d to M...T │ │ │ │ +00059ac0: 6865 206d 6170 2064 2069 7320 6120 7370 he map d is a sp │ │ │ │ +00059ad0: 6563 6961 6c20 6c69 6674 696e 6720 746f ecial lifting to │ │ │ │ +00059ae0: 2053 206f 6620 6120 7072 6573 656e 7461 S of a presenta │ │ │ │ +00059af0: 7469 6f6e 206f 6620 4d20 6f76 6572 2052 tion of M over R │ │ │ │ +00059b00: 2e20 546f 2065 7870 6c61 696e 0a74 6865 . To explain.the │ │ │ │ +00059b10: 2063 6f6e 7465 6e74 732c 2077 6520 696e contents, we in │ │ │ │ +00059b20: 7472 6f64 7563 6520 736f 6d65 206e 6f74 troduce some not │ │ │ │ +00059b30: 6174 696f 6e20 2866 726f 6d20 4569 7365 ation (from Eise │ │ │ │ +00059b40: 6e62 7564 2061 6e64 2050 6565 7661 2c20 nbud and Peeva, │ │ │ │ +00059b50: 224d 696e 696d 616c 0a66 7265 6520 7265 "Minimal.free re │ │ │ │ +00059b60: 736f 6c75 7469 6f6e 7320 6f76 6572 2063 solutions over c │ │ │ │ +00059b70: 6f6d 706c 6574 6520 696e 7465 7273 6563 omplete intersec │ │ │ │ +00059b80: 7469 6f6e 7322 204c 6563 7475 7265 204e tions" Lecture N │ │ │ │ +00059b90: 6f74 6573 2069 6e20 4d61 7468 656d 6174 otes in Mathemat │ │ │ │ +00059ba0: 6963 732c 0a32 3135 322e 2053 7072 696e ics,.2152. Sprin │ │ │ │ +00059bb0: 6765 722c 2043 6861 6d2c 2032 3031 362e ger, Cham, 2016. │ │ │ │ +00059bc0: 2078 2b31 3037 2070 702e 2049 5342 4e3a x+107 pp. ISBN: │ │ │ │ +00059bd0: 2039 3738 2d33 2d33 3139 2d32 3634 3336 978-3-319-26436 │ │ │ │ +00059be0: 2d33 3b0a 3937 382d 332d 3331 392d 3236 -3;.978-3-319-26 │ │ │ │ +00059bf0: 3433 372d 3029 2e0a 0a52 2869 2920 3d20 437-0)...R(i) = │ │ │ │ +00059c00: 532f 2866 665f 302c 2e2e 2c66 665f 7b69 S/(ff_0,..,ff_{i │ │ │ │ +00059c10: 2d31 7d29 2e20 4865 7265 2030 3c3d 2069 -1}). Here 0<= i │ │ │ │ +00059c20: 203c 3d20 632c 2061 6e64 2052 203d 2052 <= c, and R = R │ │ │ │ +00059c30: 2863 2920 616e 6420 5320 3d20 5228 3029 (c) and S = R(0) │ │ │ │ +00059c40: 2e0a 0a42 2869 2920 3d20 7468 6520 6d61 ...B(i) = the ma │ │ │ │ +00059c50: 7472 6978 2028 6f76 6572 2053 2920 7265 trix (over S) re │ │ │ │ +00059c60: 7072 6573 656e 7469 6e67 2064 5f69 3a20 presenting d_i: │ │ │ │ +00059c70: 425f 3128 6929 205c 746f 2042 5f30 2869 B_1(i) \to B_0(i │ │ │ │ +00059c80: 290a 0a64 2869 293a 2041 5f31 2869 2920 )..d(i): A_1(i) │ │ │ │ +00059c90: 5c74 6f20 415f 3028 6929 2074 6865 2072 \to A_0(i) the r │ │ │ │ +00059ca0: 6573 7472 6963 7469 6f6e 206f 6620 6420 estriction of d │ │ │ │ +00059cb0: 3d20 6428 6329 2e20 7768 6572 6520 4128 = d(c). where A( │ │ │ │ +00059cc0: 6929 203d 0a5c 6f70 6c75 735f 7b69 3d31 i) =.\oplus_{i=1 │ │ │ │ +00059cd0: 7d5e 7020 4228 6929 0a0a 0a0a 5468 6520 }^p B(i)....The │ │ │ │ +00059ce0: 6d61 7020 6820 6973 2061 2064 6972 6563 map h is a direc │ │ │ │ +00059cf0: 7420 7375 6d20 6f66 206d 6170 7320 7461 t sum of maps ta │ │ │ │ +00059d00: 7267 6574 2064 2870 2920 5c74 6f20 736f rget d(p) \to so │ │ │ │ +00059d10: 7572 6365 2064 2870 2920 7468 6174 2061 urce d(p) that a │ │ │ │ +00059d20: 7265 0a68 6f6d 6f74 6f70 6965 7320 666f re.homotopies fo │ │ │ │ +00059d30: 7220 6666 5f70 206f 6e20 7468 6520 7265 r ff_p on the re │ │ │ │ +00059d40: 7374 7269 6374 696f 6e20 6428 7029 3a20 striction d(p): │ │ │ │ +00059d50: 6f76 6572 2074 6865 2072 696e 6720 5223 over the ring R# │ │ │ │ +00059d60: 2870 2d31 2920 3d0a 532f 2866 6623 312e (p-1) =.S/(ff#1. │ │ │ │ +00059d70: 2e66 6623 2870 2d31 292c 2073 6f20 6428 .ff#(p-1), so d( │ │ │ │ +00059d80: 7029 202a 2068 2370 203d 2066 6623 7020 p) * h#p = ff#p │ │ │ │ +00059d90: 6d6f 6420 2866 6623 312e 2e66 6623 2870 mod (ff#1..ff#(p │ │ │ │ +00059da0: 2d31 292e 0a0a 496e 2061 6464 6974 696f -1)...In additio │ │ │ │ +00059db0: 6e2c 2068 2370 202a 2064 2870 2920 696e n, h#p * d(p) in │ │ │ │ +00059dc0: 6475 6365 7320 6666 2370 206f 6e20 4231 duces ff#p on B1 │ │ │ │ +00059dd0: 2370 206d 6f64 2028 6666 2331 2e2e 6666 #p mod (ff#1..ff │ │ │ │ +00059de0: 2328 702d 3129 2e0a 0a48 6572 6520 6973 #(p-1)...Here is │ │ │ │ +00059df0: 2061 2073 696d 706c 6520 6578 616d 706c a simple exampl │ │ │ │ +00059e00: 653a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d e:..+----------- │ │ │ │ 00059e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00059e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00059e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00059e40: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e --+.|i1 : setRan │ │ │ │ -00059e50: 646f 6d53 6565 6420 3020 2020 2020 2020 domSeed 0 │ │ │ │ -00059e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059e70: 2020 2020 2020 207c 0a7c 202d 2d20 7365 |.| -- se │ │ │ │ -00059e80: 7474 696e 6720 7261 6e64 6f6d 2073 6565 tting random see │ │ │ │ -00059e90: 6420 746f 2030 2020 2020 2020 2020 2020 d to 0 │ │ │ │ -00059ea0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00059e30: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 -------+.|i1 : s │ │ │ │ +00059e40: 6574 5261 6e64 6f6d 5365 6564 2030 2020 etRandomSeed 0 │ │ │ │ +00059e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00059e60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00059e70: 2d2d 2073 6574 7469 6e67 2072 616e 646f -- setting rando │ │ │ │ +00059e80: 6d20 7365 6564 2074 6f20 3020 2020 2020 m seed to 0 │ │ │ │ +00059e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00059ea0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00059eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059ee0: 207c 0a7c 6f31 203d 2030 2020 2020 2020 |.|o1 = 0 │ │ │ │ +00059ed0: 2020 2020 2020 7c0a 7c6f 3120 3d20 3020 |.|o1 = 0 │ │ │ │ +00059ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059f10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00059f00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00059f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00059f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00059f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00059f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00059f50: 203a 206b 6b20 3d20 5a5a 2f31 3031 2020 : kk = ZZ/101 │ │ │ │ +00059f40: 2b0a 7c69 3220 3a20 6b6b 203d 205a 5a2f +.|i2 : kk = ZZ/ │ │ │ │ +00059f50: 3130 3120 2020 2020 2020 2020 2020 2020 101 │ │ │ │ 00059f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059f80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00059f70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00059f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059fb0: 2020 2020 207c 0a7c 6f32 203d 206b 6b20 |.|o2 = kk │ │ │ │ +00059fa0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +00059fb0: 3d20 6b6b 2020 2020 2020 2020 2020 2020 = kk │ │ │ │ 00059fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059fe0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00059fd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00059fe0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00059ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a010: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0005a020: 0a7c 6f32 203a 2051 756f 7469 656e 7452 .|o2 : QuotientR │ │ │ │ -0005a030: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -0005a040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a050: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005a010: 2020 2020 7c0a 7c6f 3220 3a20 5175 6f74 |.|o2 : Quot │ │ │ │ +0005a020: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +0005a030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a040: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a080: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -0005a090: 2053 203d 206b 6b5b 612c 622c 752c 765d S = kk[a,b,u,v] │ │ │ │ +0005a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005a080: 7c69 3320 3a20 5320 3d20 6b6b 5b61 2c62 |i3 : S = kk[a,b │ │ │ │ +0005a090: 2c75 2c76 5d20 2020 2020 2020 2020 2020 ,u,v] │ │ │ │ 0005a0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0005a0c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0005a0b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0005a0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a0f0: 2020 207c 0a7c 6f33 203d 2053 2020 2020 |.|o3 = S │ │ │ │ +0005a0e0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ +0005a0f0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 0005a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a120: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0005a110: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0005a120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a150: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0005a160: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52 o3 : PolynomialR │ │ │ │ -0005a170: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -0005a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a190: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0005a150: 2020 7c0a 7c6f 3320 3a20 506f 6c79 6e6f |.|o3 : Polyno │ │ │ │ +0005a160: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +0005a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a180: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0005a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a1c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2066 -------+.|i4 : f │ │ │ │ -0005a1d0: 6620 3d20 6d61 7472 6978 2261 752c 6276 f = matrix"au,bv │ │ │ │ -0005a1e0: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ -0005a1f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0005a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0005a1c0: 3420 3a20 6666 203d 206d 6174 7269 7822 4 : ff = matrix" │ │ │ │ +0005a1d0: 6175 2c62 7622 2020 2020 2020 2020 2020 au,bv" │ │ │ │ +0005a1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a1f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0005a200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a230: 207c 0a7c 6f34 203d 207c 2061 7520 6276 |.|o4 = | au bv │ │ │ │ -0005a240: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0005a250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a260: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0005a220: 2020 2020 2020 7c0a 7c6f 3420 3d20 7c20 |.|o4 = | │ │ │ │ +0005a230: 6175 2062 7620 7c20 2020 2020 2020 2020 au bv | │ │ │ │ +0005a240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a250: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0005a260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a290: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0005a2a0: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ -0005a2b0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0005a2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a2d0: 7c0a 7c6f 3420 3a20 4d61 7472 6978 2053 |.|o4 : Matrix S │ │ │ │ -0005a2e0: 2020 3c2d 2d20 5320 2020 2020 2020 2020 <-- S │ │ │ │ -0005a2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a300: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0005a290: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0005a2a0: 3120 2020 2020 2032 2020 2020 2020 2020 1 2 │ │ │ │ +0005a2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a2c0: 2020 2020 207c 0a7c 6f34 203a 204d 6174 |.|o4 : Mat │ │ │ │ +0005a2d0: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ +0005a2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a2f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0005a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ -0005a340: 3a20 5220 3d20 532f 6964 6561 6c20 6666 : R = S/ideal ff │ │ │ │ +0005a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0005a330: 0a7c 6935 203a 2052 203d 2053 2f69 6465 .|i5 : R = S/ide │ │ │ │ +0005a340: 616c 2066 6620 2020 2020 2020 2020 2020 al ff │ │ │ │ 0005a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a360: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0005a370: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0005a360: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a3a0: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020 |.|o5 = R │ │ │ │ +0005a390: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ +0005a3a0: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 0005a3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a3d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005a3c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005a3d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005a3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a400: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0005a410: 7c6f 3520 3a20 5175 6f74 6965 6e74 5269 |o5 : QuotientRi │ │ │ │ -0005a420: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ -0005a430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a440: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0005a400: 2020 207c 0a7c 6f35 203a 2051 756f 7469 |.|o5 : Quoti │ │ │ │ +0005a410: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ +0005a420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a430: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0005a440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a470: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ -0005a480: 4d30 203d 2052 5e31 2f69 6465 616c 2261 M0 = R^1/ideal"a │ │ │ │ -0005a490: 2c62 2220 2020 2020 2020 2020 2020 2020 ,b" │ │ │ │ -0005a4a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0005a460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0005a470: 6936 203a 204d 3020 3d20 525e 312f 6964 i6 : M0 = R^1/id │ │ │ │ +0005a480: 6561 6c22 612c 6222 2020 2020 2020 2020 eal"a,b" │ │ │ │ +0005a490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a4a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0005a4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a4e0: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e |.|o6 = cokern │ │ │ │ -0005a4f0: 656c 207c 2061 2062 207c 2020 2020 2020 el | a b | │ │ │ │ -0005a500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a510: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0005a4d0: 2020 2020 2020 207c 0a7c 6f36 203d 2063 |.|o6 = c │ │ │ │ +0005a4e0: 6f6b 6572 6e65 6c20 7c20 6120 6220 7c20 okernel | a b | │ │ │ │ +0005a4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a500: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0005a510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a540: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0005a540: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0005a550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a560: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ -0005a570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a580: 207c 0a7c 6f36 203a 2052 2d6d 6f64 756c |.|o6 : R-modul │ │ │ │ -0005a590: 652c 2071 756f 7469 656e 7420 6f66 2052 e, quotient of R │ │ │ │ -0005a5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a5b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0005a560: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0005a570: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d |.|o6 : R- │ │ │ │ +0005a580: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74 module, quotient │ │ │ │ +0005a590: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ +0005a5a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0005a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ -0005a5f0: 203a 204d 203d 2068 6967 6853 797a 7967 : M = highSyzyg │ │ │ │ -0005a600: 7920 4d30 2020 2020 2020 2020 2020 2020 y M0 │ │ │ │ -0005a610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a620: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0005a5e0: 2b0a 7c69 3720 3a20 4d20 3d20 6869 6768 +.|i7 : M = high │ │ │ │ +0005a5f0: 5379 7a79 6779 204d 3020 2020 2020 2020 Syzygy M0 │ │ │ │ +0005a600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a610: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0005a620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a650: 2020 2020 207c 0a7c 6f37 203d 2063 6f6b |.|o7 = cok │ │ │ │ -0005a660: 6572 6e65 6c20 7b32 7d20 7c20 6220 2d61 ernel {2} | b -a │ │ │ │ -0005a670: 2030 2030 207c 2020 2020 2020 2020 2020 0 0 | │ │ │ │ -0005a680: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0005a690: 2020 2020 2020 2020 2020 207b 327d 207c {2} | │ │ │ │ -0005a6a0: 2030 2030 2020 6120 6220 7c20 2020 2020 0 0 a b | │ │ │ │ -0005a6b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0005a6c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0005a6d0: 7b32 7d20 7c20 3020 7620 2030 2075 207c {2} | 0 v 0 u | │ │ │ │ -0005a6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a6f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005a640: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ +0005a650: 3d20 636f 6b65 726e 656c 207b 327d 207c = cokernel {2} | │ │ │ │ +0005a660: 2062 202d 6120 3020 3020 7c20 2020 2020 b -a 0 0 | │ │ │ │ +0005a670: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0005a680: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0005a690: 7b32 7d20 7c20 3020 3020 2061 2062 207c {2} | 0 0 a b | │ │ │ │ +0005a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a6b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005a6c0: 2020 2020 207b 327d 207c 2030 2076 2020 {2} | 0 v │ │ │ │ +0005a6d0: 3020 7520 7c20 2020 2020 2020 2020 2020 0 u | │ │ │ │ +0005a6e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005a6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a720: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0005a730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a740: 2020 2020 2020 2020 3320 2020 2020 2020 3 │ │ │ │ -0005a750: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0005a760: 7c6f 3720 3a20 522d 6d6f 6475 6c65 2c20 |o7 : R-module, │ │ │ │ -0005a770: 7175 6f74 6965 6e74 206f 6620 5220 2020 quotient of R │ │ │ │ -0005a780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a790: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0005a710: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005a720: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0005a730: 2020 2020 2020 2020 2020 2020 2033 2020 3 │ │ │ │ +0005a740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a750: 2020 207c 0a7c 6f37 203a 2052 2d6d 6f64 |.|o7 : R-mod │ │ │ │ +0005a760: 756c 652c 2071 756f 7469 656e 7420 6f66 ule, quotient of │ │ │ │ +0005a770: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0005a780: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0005a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a7c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 --------+.|i8 : │ │ │ │ -0005a7d0: 4d46 203d 206d 6174 7269 7846 6163 746f MF = matrixFacto │ │ │ │ -0005a7e0: 7269 7a61 7469 6f6e 2866 662c 4d29 3b20 rization(ff,M); │ │ │ │ -0005a7f0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0005a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0005a7c0: 6938 203a 204d 4620 3d20 6d61 7472 6978 i8 : MF = matrix │ │ │ │ +0005a7d0: 4661 6374 6f72 697a 6174 696f 6e28 6666 Factorization(ff │ │ │ │ +0005a7e0: 2c4d 293b 2020 2020 2020 2020 2020 2020 ,M); │ │ │ │ +0005a7f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0005a800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a830: 2d2d 2b0a 7c69 3920 3a20 6e65 744c 6973 --+.|i9 : netLis │ │ │ │ -0005a840: 7420 4252 616e 6b73 204d 4620 2020 2020 t BRanks MF │ │ │ │ -0005a850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a860: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0005a820: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 206e -------+.|i9 : n │ │ │ │ +0005a830: 6574 4c69 7374 2042 5261 6e6b 7320 4d46 etList BRanks MF │ │ │ │ +0005a840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a850: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0005a860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a890: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0005a8a0: 2020 2020 2b2d 2b2d 2b20 2020 2020 2020 +-+-+ │ │ │ │ +0005a890: 207c 0a7c 2020 2020 202b 2d2b 2d2b 2020 |.| +-+-+ │ │ │ │ +0005a8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a8d0: 207c 0a7c 6f39 203d 207c 327c 327c 2020 |.|o9 = |2|2| │ │ │ │ +0005a8c0: 2020 2020 2020 7c0a 7c6f 3920 3d20 7c32 |.|o9 = |2 │ │ │ │ +0005a8d0: 7c32 7c20 2020 2020 2020 2020 2020 2020 |2| │ │ │ │ 0005a8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a900: 2020 2020 2020 7c0a 7c20 2020 2020 2b2d |.| +- │ │ │ │ -0005a910: 2b2d 2b20 2020 2020 2020 2020 2020 2020 +-+ │ │ │ │ +0005a8f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0005a900: 2020 202b 2d2b 2d2b 2020 2020 2020 2020 +-+-+ │ │ │ │ +0005a910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a930: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0005a940: 2020 207c 317c 327c 2020 2020 2020 2020 |1|2| │ │ │ │ +0005a930: 7c0a 7c20 2020 2020 7c31 7c32 7c20 2020 |.| |1|2| │ │ │ │ +0005a940: 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2062 207c 7c20 = |{2} | a b || │ │ │ │ +0005aa30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005aa40: 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 +-----------+ │ │ │ │ +0005aa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005aa60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005aa70: 7c6f 3130 203d 207c 7b32 7d20 7c20 6120 |o10 = |{2} | a │ │ │ │ +0005aa80: 6220 7c7c 2020 2020 2020 2020 2020 2020 b || │ │ │ │ 0005aa90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005aaa0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0005aab0: 7c20 2020 2020 207c 7b32 7d20 7c20 3020 | |{2} | 0 │ │ │ │ -0005aac0: 7520 7c7c 2020 2020 2020 2020 2020 2020 u || │ │ │ │ -0005aad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005aae0: 2020 207c 0a7c 2020 2020 2020 2b2d 2d2d |.| +--- │ │ │ │ -0005aaf0: 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020 2020 --------+ │ │ │ │ -0005ab00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ab10: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0005ab20: 207c 7b32 7d20 7c20 6220 6120 7c7c 2020 |{2} | b a || │ │ │ │ +0005aaa0: 2020 207c 0a7c 2020 2020 2020 7c7b 327d |.| |{2} │ │ │ │ +0005aab0: 207c 2030 2075 207c 7c20 2020 2020 2020 | 0 u || │ │ │ │ +0005aac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005aad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0005aae0: 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 2020 +-----------+ │ │ │ │ +0005aaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005ab00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0005ab10: 2020 2020 2020 7c7b 327d 207c 2062 2061 |{2} | b a │ │ │ │ +0005ab20: 207c 7c20 2020 2020 2020 2020 2020 2020 || │ │ │ │ 0005ab30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ab40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0005ab50: 2020 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d +--------- │ │ │ │ -0005ab60: 2d2d 2b20 2020 2020 2020 2020 2020 2020 --+ │ │ │ │ -0005ab70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ab80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0005ab40: 2020 7c0a 7c20 2020 2020 202b 2d2d 2d2d |.| +---- │ │ │ │ +0005ab50: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ +0005ab60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005ab70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0005ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ab90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005abb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ -0005abc0: 6265 7474 6920 6672 6565 5265 736f 6c75 betti freeResolu │ │ │ │ -0005abd0: 7469 6f6e 284d 2c20 4c65 6e67 7468 4c69 tion(M, LengthLi │ │ │ │ -0005abe0: 6d69 7420 3d3e 2037 2920 2020 7c0a 7c20 mit => 7) |.| │ │ │ │ +0005aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0005abb0: 3131 203a 2062 6574 7469 2066 7265 6552 11 : betti freeR │ │ │ │ +0005abc0: 6573 6f6c 7574 696f 6e28 4d2c 204c 656e esolution(M, Len │ │ │ │ +0005abd0: 6774 684c 696d 6974 203d 3e20 3729 2020 gthLimit => 7) │ │ │ │ +0005abe0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0005abf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005ac00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ac10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ac20: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0005ac30: 2030 2031 2032 2033 2034 2035 2036 2020 0 1 2 3 4 5 6 │ │ │ │ -0005ac40: 3720 2020 2020 2020 2020 2020 2020 2020 7 │ │ │ │ -0005ac50: 2020 2020 2020 7c0a 7c6f 3131 203d 2074 |.|o11 = t │ │ │ │ -0005ac60: 6f74 616c 3a20 3320 3420 3520 3620 3720 otal: 3 4 5 6 7 │ │ │ │ -0005ac70: 3820 3920 3130 2020 2020 2020 2020 2020 8 9 10 │ │ │ │ -0005ac80: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0005ac90: 2020 2020 2020 2020 323a 2033 2034 2035 2: 3 4 5 │ │ │ │ -0005aca0: 2036 2037 2038 2039 2031 3020 2020 2020 6 7 8 9 10 │ │ │ │ -0005acb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005acc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0005ac10: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0005ac20: 2020 2020 2020 3020 3120 3220 3320 3420 0 1 2 3 4 │ │ │ │ +0005ac30: 3520 3620 2037 2020 2020 2020 2020 2020 5 6 7 │ │ │ │ +0005ac40: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0005ac50: 3120 3d20 746f 7461 6c3a 2033 2034 2035 1 = total: 3 4 5 │ │ │ │ +0005ac60: 2036 2037 2038 2039 2031 3020 2020 2020 6 7 8 9 10 │ │ │ │ +0005ac70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005ac80: 7c0a 7c20 2020 2020 2020 2020 2032 3a20 |.| 2: │ │ │ │ +0005ac90: 3320 3420 3520 3620 3720 3820 3920 3130 3 4 5 6 7 8 9 10 │ │ │ │ +0005aca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005acb0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0005acc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005acd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ace0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005acf0: 2020 2020 207c 0a7c 6f31 3120 3a20 4265 |.|o11 : Be │ │ │ │ -0005ad00: 7474 6954 616c 6c79 2020 2020 2020 2020 ttiTally │ │ │ │ -0005ad10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ad20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0005ace0: 2020 2020 2020 2020 2020 7c0a 7c6f 3131 |.|o11 │ │ │ │ +0005acf0: 203a 2042 6574 7469 5461 6c6c 7920 2020 : BettiTally │ │ │ │ +0005ad00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005ad10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0005ad20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0005ad30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ad40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0005ad60: 0a7c 6931 3220 3a20 696e 6669 6e69 7465 .|i12 : infinite │ │ │ │ -0005ad70: 4265 7474 694e 756d 6265 7273 2028 4d46 BettiNumbers (MF │ │ │ │ -0005ad80: 2c37 2920 2020 2020 2020 2020 2020 2020 ,7) │ │ │ │ -0005ad90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005ad50: 2d2d 2d2d 2b0a 7c69 3132 203a 2069 6e66 ----+.|i12 : inf │ │ │ │ +0005ad60: 696e 6974 6542 6574 7469 4e75 6d62 6572 initeBettiNumber │ │ │ │ +0005ad70: 7320 284d 462c 3729 2020 2020 2020 2020 s (MF,7) │ │ │ │ +0005ad80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005ad90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005ada0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005adb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005adc0: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ -0005add0: 3d20 7b33 2c20 342c 2035 2c20 362c 2037 = {3, 4, 5, 6, 7 │ │ │ │ -0005ade0: 2c20 382c 2039 2c20 3130 7d20 2020 2020 , 8, 9, 10} │ │ │ │ -0005adf0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0005ae00: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0005adb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005adc0: 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2020 2020 2020 |.| │ │ │ │ -0005af10: 2020 2020 2020 2030 2031 2032 2020 2020 0 1 2 │ │ │ │ +0005aef0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0005af00: 2020 2020 2020 2020 2020 2020 3020 3120 0 1 │ │ │ │ +0005af10: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0005af20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005af30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0005af40: 3133 203d 2074 6f74 616c 3a20 3320 3520 13 = total: 3 5 │ │ │ │ -0005af50: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0005af60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005af70: 207c 0a7c 2020 2020 2020 2020 2020 323a |.| 2: │ │ │ │ -0005af80: 2033 2034 202e 2020 2020 2020 2020 2020 3 4 . │ │ │ │ -0005af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005afa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0005afb0: 2020 2033 3a20 2e20 3120 3220 2020 2020 3: . 1 2 │ │ │ │ +0005af30: 207c 0a7c 6f31 3320 3d20 746f 7461 6c3a |.|o13 = total: │ │ │ │ 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│ -0005b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005b040: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0005b000: 2020 2020 207c 0a7c 6f31 3320 3a20 4265 |.|o13 : Be │ │ │ │ +0005b010: 7474 6954 616c 6c79 2020 2020 2020 2020 ttiTally │ │ │ │ +0005b020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005b030: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0005b040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 ----------+.|i14 │ │ │ │ -0005b080: 203a 2066 696e 6974 6542 6574 7469 4e75 : finiteBettiNu │ │ │ │ -0005b090: 6d62 6572 7320 4d46 2020 2020 2020 2020 mbers MF │ │ │ │ -0005b0a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0005b0b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0005b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0005b070: 0a7c 6931 3420 3a20 6669 6e69 7465 4265 .|i14 : finiteBe │ │ │ │ +0005b080: 7474 694e 756d 6265 7273 204d 4620 2020 ttiNumbers MF │ │ │ │ +0005b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005b0a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005b0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005b0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005b0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005b0e0: 2020 2020 7c0a 7c6f 3134 203d 207b 332c |.|o14 = {3, │ │ │ │ -0005b0f0: 2035 2c20 327d 2020 2020 2020 2020 2020 5, 2} │ │ │ │ -0005b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005b110: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005b0d0: 2020 2020 2020 2020 207c 0a7c 6f31 3420 |.|o14 │ │ │ │ +0005b0e0: 3d20 7b33 2c20 352c 2032 7d20 2020 2020 = {3, 5, 2} │ │ │ │ +0005b0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005b100: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005b110: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005b120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005b130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005b140: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0005b150: 7c6f 3134 203a 204c 6973 7420 2020 2020 |o14 : List │ │ │ │ +0005b140: 2020 207c 0a7c 6f31 3420 3a20 4c69 7374 |.|o14 : List │ │ │ │ +0005b150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005b160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005b180: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0005b170: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0005b180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005b1b0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -0005b1c0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -0005b1d0: 2a20 2a6e 6f74 6520 6669 6e69 7465 4265 * *note finiteBe │ │ │ │ -0005b1e0: 7474 694e 756d 6265 7273 3a20 6669 6e69 ttiNumbers: fini │ │ │ │ -0005b1f0: 7465 4265 7474 694e 756d 6265 7273 2c20 teBettiNumbers, │ │ │ │ -0005b200: 2d2d 2062 6574 7469 206e 756d 6265 7273 -- betti numbers │ │ │ │ -0005b210: 206f 6620 6669 6e69 7465 0a20 2020 2072 of finite. r │ │ │ │ -0005b220: 6573 6f6c 7574 696f 6e20 636f 6d70 7574 esolution comput │ │ │ │ -0005b230: 6564 2066 726f 6d20 6120 6d61 7472 6978 ed from a matrix │ │ │ │ -0005b240: 2066 6163 746f 7269 7a61 7469 6f6e 0a20 factorization. │ │ │ │ -0005b250: 202a 202a 6e6f 7465 2069 6e66 696e 6974 * *note infinit │ │ │ │ -0005b260: 6542 6574 7469 4e75 6d62 6572 733a 2069 eBettiNumbers: i │ │ │ │ -0005b270: 6e66 696e 6974 6542 6574 7469 4e75 6d62 nfiniteBettiNumb │ │ │ │ -0005b280: 6572 732c 202d 2d20 6265 7474 6920 6e75 ers, -- betti nu │ │ │ │ -0005b290: 6d62 6572 7320 6f66 0a20 2020 2066 696e mbers of. fin │ │ │ │ -0005b2a0: 6974 6520 7265 736f 6c75 7469 6f6e 2063 ite resolution c │ │ │ │ -0005b2b0: 6f6d 7075 7465 6420 6672 6f6d 2061 206d omputed from a m │ │ │ │ -0005b2c0: 6174 7269 7820 6661 6374 6f72 697a 6174 atrix factorizat │ │ │ │ -0005b2d0: 696f 6e0a 2020 2a20 2a6e 6f74 6520 6869 ion. * *note hi │ │ │ │ -0005b2e0: 6768 5379 7a79 6779 3a20 6869 6768 5379 ghSyzygy: highSy │ │ │ │ -0005b2f0: 7a79 6779 2c20 2d2d 2052 6574 7572 6e73 zygy, -- Returns │ │ │ │ -0005b300: 2061 2073 797a 7967 7920 6d6f 6475 6c65 a syzygy module │ │ │ │ -0005b310: 206f 6e65 2062 6579 6f6e 6420 7468 650a one beyond the. │ │ │ │ -0005b320: 2020 2020 7265 6775 6c61 7269 7479 206f regularity o │ │ │ │ -0005b330: 6620 4578 7428 4d2c 6b29 0a20 202a 202a f Ext(M,k). * * │ │ │ │ -0005b340: 6e6f 7465 2062 4d61 7073 3a20 624d 6170 note bMaps: bMap │ │ │ │ -0005b350: 732c 202d 2d20 6c69 7374 2074 6865 206d s, -- list the m │ │ │ │ -0005b360: 6170 7320 2064 5f70 3a42 5f31 2870 292d aps d_p:B_1(p)- │ │ │ │ -0005b370: 2d3e 425f 3028 7029 2069 6e20 610a 2020 ->B_0(p) in a. │ │ │ │ -0005b380: 2020 6d61 7472 6978 4661 6374 6f72 697a matrixFactoriz │ │ │ │ -0005b390: 6174 696f 6e0a 2020 2a20 2a6e 6f74 6520 ation. * *note │ │ │ │ -0005b3a0: 4252 616e 6b73 3a20 4252 616e 6b73 2c20 BRanks: BRanks, │ │ │ │ -0005b3b0: 2d2d 2072 616e 6b73 206f 6620 7468 6520 -- ranks of the │ │ │ │ -0005b3c0: 6d6f 6475 6c65 7320 425f 6928 6429 2069 modules B_i(d) i │ │ │ │ -0005b3d0: 6e20 610a 2020 2020 6d61 7472 6978 4661 n a. matrixFa │ │ │ │ -0005b3e0: 6374 6f72 697a 6174 696f 6e0a 0a57 6179 ctorization..Way │ │ │ │ -0005b3f0: 7320 746f 2075 7365 206d 6174 7269 7846 s to use matrixF │ │ │ │ -0005b400: 6163 746f 7269 7a61 7469 6f6e 3a0a 3d3d actorization:.== │ │ │ │ +0005b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +0005b1b0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +0005b1c0: 3d0a 0a20 202a 202a 6e6f 7465 2066 696e =.. * *note fin │ │ │ │ +0005b1d0: 6974 6542 6574 7469 4e75 6d62 6572 733a iteBettiNumbers: │ │ │ │ +0005b1e0: 2066 696e 6974 6542 6574 7469 4e75 6d62 finiteBettiNumb │ │ │ │ +0005b1f0: 6572 732c 202d 2d20 6265 7474 6920 6e75 ers, -- betti nu │ │ │ │ +0005b200: 6d62 6572 7320 6f66 2066 696e 6974 650a mbers of finite. │ │ │ │ +0005b210: 2020 2020 7265 736f 6c75 7469 6f6e 2063 resolution c │ │ │ │ +0005b220: 6f6d 7075 7465 6420 6672 6f6d 2061 206d omputed from a m │ │ │ │ +0005b230: 6174 7269 7820 6661 6374 6f72 697a 6174 atrix factorizat │ │ │ │ +0005b240: 696f 6e0a 2020 2a20 2a6e 6f74 6520 696e ion. * *note in │ │ │ │ +0005b250: 6669 6e69 7465 4265 7474 694e 756d 6265 finiteBettiNumbe │ │ │ │ +0005b260: 7273 3a20 696e 6669 6e69 7465 4265 7474 rs: infiniteBett │ │ │ │ +0005b270: 694e 756d 6265 7273 2c20 2d2d 2062 6574 iNumbers, -- bet │ │ │ │ +0005b280: 7469 206e 756d 6265 7273 206f 660a 2020 ti numbers of. │ │ │ │ +0005b290: 2020 6669 6e69 7465 2072 6573 6f6c 7574 finite resolut │ │ │ │ +0005b2a0: 696f 6e20 636f 6d70 7574 6564 2066 726f ion computed fro │ │ │ │ +0005b2b0: 6d20 6120 6d61 7472 6978 2066 6163 746f m a matrix facto │ │ │ │ +0005b2c0: 7269 7a61 7469 6f6e 0a20 202a 202a 6e6f rization. * *no │ │ │ │ +0005b2d0: 7465 2068 6967 6853 797a 7967 793a 2068 te highSyzygy: h │ │ │ │ +0005b2e0: 6967 6853 797a 7967 792c 202d 2d20 5265 ighSyzygy, -- Re │ │ │ │ +0005b2f0: 7475 726e 7320 6120 7379 7a79 6779 206d turns a syzygy m │ │ │ │ +0005b300: 6f64 756c 6520 6f6e 6520 6265 796f 6e64 odule one beyond │ │ │ │ +0005b310: 2074 6865 0a20 2020 2072 6567 756c 6172 the. regular │ │ │ │ +0005b320: 6974 7920 6f66 2045 7874 284d 2c6b 290a ity of Ext(M,k). │ │ │ │ +0005b330: 2020 2a20 2a6e 6f74 6520 624d 6170 733a * *note bMaps: │ │ │ │ +0005b340: 2062 4d61 7073 2c20 2d2d 206c 6973 7420 bMaps, -- list │ │ │ │ +0005b350: 7468 6520 6d61 7073 2020 645f 703a 425f the maps d_p:B_ │ │ │ │ +0005b360: 3128 7029 2d2d 3e42 5f30 2870 2920 696e 1(p)-->B_0(p) in │ │ │ │ +0005b370: 2061 0a20 2020 206d 6174 7269 7846 6163 a. matrixFac │ │ │ │ +0005b380: 746f 7269 7a61 7469 6f6e 0a20 202a 202a torization. * * │ │ │ │ +0005b390: 6e6f 7465 2042 5261 6e6b 733a 2042 5261 note BRanks: BRa │ │ │ │ +0005b3a0: 6e6b 732c 202d 2d20 7261 6e6b 7320 6f66 nks, -- ranks of │ │ │ │ +0005b3b0: 2074 6865 206d 6f64 756c 6573 2042 5f69 the modules B_i │ │ │ │ +0005b3c0: 2864 2920 696e 2061 0a20 2020 206d 6174 (d) in a. mat │ │ │ │ +0005b3d0: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ +0005b3e0: 0a0a 5761 7973 2074 6f20 7573 6520 6d61 ..Ways to use ma │ │ │ │ +0005b3f0: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ +0005b400: 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d n:.============= │ │ │ │ 0005b410: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0005b420: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -0005b430: 2020 2a20 226d 6174 7269 7846 6163 746f * "matrixFacto │ │ │ │ -0005b440: 7269 7a61 7469 6f6e 284d 6174 7269 782c rization(Matrix, │ │ │ │ -0005b450: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468 Module)"..For th │ │ │ │ -0005b460: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -0005b470: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -0005b480: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -0005b490: 6520 6d61 7472 6978 4661 6374 6f72 697a e matrixFactoriz │ │ │ │ -0005b4a0: 6174 696f 6e3a 206d 6174 7269 7846 6163 ation: matrixFac │ │ │ │ -0005b4b0: 746f 7269 7a61 7469 6f6e 2c20 6973 2061 torization, is a │ │ │ │ -0005b4c0: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ -0005b4d0: 6e63 7469 6f6e 2077 6974 6820 6f70 7469 nction with opti │ │ │ │ -0005b4e0: 6f6e 733a 2028 4d61 6361 756c 6179 3244 ons: (Macaulay2D │ │ │ │ -0005b4f0: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -0005b500: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a nWithOptions,... │ │ │ │ +0005b420: 3d3d 3d0a 0a20 202a 2022 6d61 7472 6978 ===.. * "matrix │ │ │ │ +0005b430: 4661 6374 6f72 697a 6174 696f 6e28 4d61 Factorization(Ma │ │ │ │ +0005b440: 7472 6978 2c4d 6f64 756c 6529 220a 0a46 trix,Module)"..F │ │ │ │ +0005b450: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +0005b460: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +0005b470: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +0005b480: 202a 6e6f 7465 206d 6174 7269 7846 6163 *note matrixFac │ │ │ │ +0005b490: 746f 7269 7a61 7469 6f6e 3a20 6d61 7472 torization: matr │ │ │ │ +0005b4a0: 6978 4661 6374 6f72 697a 6174 696f 6e2c ixFactorization, │ │ │ │ +0005b4b0: 2069 7320 6120 2a6e 6f74 6520 6d65 7468 is a *note meth │ │ │ │ +0005b4c0: 6f64 0a66 756e 6374 696f 6e20 7769 7468 od.function with │ │ │ │ +0005b4d0: 206f 7074 696f 6e73 3a20 284d 6163 6175 options: (Macau │ │ │ │ +0005b4e0: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ +0005b4f0: 6e63 7469 6f6e 5769 7468 4f70 7469 6f6e nctionWithOption │ │ │ │ +0005b500: 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d s,...----------- │ │ │ │ 0005b510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005b530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -0005b560: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -0005b570: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -0005b580: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -0005b590: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -0005b5a0: 756c 6179 322d 312e 3236 2e30 352b 6473 ulay2-1.26.05+ds │ │ │ │ -0005b5b0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -0005b5c0: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ -0005b5d0: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ -0005b5e0: 6c75 7469 6f6e 732e 6d32 3a34 3033 333a lutions.m2:4033: │ │ │ │ -0005b5f0: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ -0005b600: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -0005b610: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ -0005b620: 204e 6f64 653a 206d 6642 6f75 6e64 2c20 Node: mfBound, │ │ │ │ -0005b630: 4e65 7874 3a20 6d6f 6475 6c65 4173 4578 Next: moduleAsEx │ │ │ │ -0005b640: 742c 2050 7265 763a 206d 6174 7269 7846 t, Prev: matrixF │ │ │ │ -0005b650: 6163 746f 7269 7a61 7469 6f6e 2c20 5570 actorization, Up │ │ │ │ -0005b660: 3a20 546f 700a 0a6d 6642 6f75 6e64 202d : Top..mfBound - │ │ │ │ -0005b670: 2d20 6465 7465 726d 696e 6573 2068 6f77 - determines how │ │ │ │ -0005b680: 2068 6967 6820 6120 7379 7a79 6779 2074 high a syzygy t │ │ │ │ -0005b690: 6f20 7461 6b65 2066 6f72 2022 6d61 7472 o take for "matr │ │ │ │ -0005b6a0: 6978 4661 6374 6f72 697a 6174 696f 6e22 ixFactorization" │ │ │ │ -0005b6b0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ +0005b550: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +0005b560: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +0005b570: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +0005b580: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +0005b590: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ +0005b5a0: 3035 2b64 732f 4d32 2f4d 6163 6175 6c61 05+ds/M2/Macaula │ │ │ │ +0005b5b0: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d y2/packages/.Com │ │ │ │ +0005b5c0: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ +0005b5d0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a nResolutions.m2: │ │ │ │ +0005b5e0: 3430 3333 3a30 2e0a 1f0a 4669 6c65 3a20 4033:0....File: │ │ │ │ +0005b5f0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +0005b600: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +0005b610: 696e 666f 2c20 4e6f 6465 3a20 6d66 426f info, Node: mfBo │ │ │ │ +0005b620: 756e 642c 204e 6578 743a 206d 6f64 756c und, Next: modul │ │ │ │ +0005b630: 6541 7345 7874 2c20 5072 6576 3a20 6d61 eAsExt, Prev: ma │ │ │ │ +0005b640: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ +0005b650: 6e2c 2055 703a 2054 6f70 0a0a 6d66 426f n, Up: Top..mfBo │ │ │ │ +0005b660: 756e 6420 2d2d 2064 6574 6572 6d69 6e65 und -- determine │ │ │ │ +0005b670: 7320 686f 7720 6869 6768 2061 2073 797a s how high a syz │ │ │ │ +0005b680: 7967 7920 746f 2074 616b 6520 666f 7220 ygy to take for │ │ │ │ +0005b690: 226d 6174 7269 7846 6163 746f 7269 7a61 "matrixFactoriza │ │ │ │ +0005b6a0: 7469 6f6e 220a 2a2a 2a2a 2a2a 2a2a 2a2a tion".********** │ │ │ │ +0005b6b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0005b6c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0005b6d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0005b6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0005b6f0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -0005b700: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -0005b710: 7020 3d20 6d66 426f 756e 6420 4d0a 2020 p = mfBound M. │ │ │ │ -0005b720: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ -0005b730: 2a20 4d2c 2061 202a 6e6f 7465 206d 6f64 * M, a *note mod │ │ │ │ -0005b740: 756c 653a 2028 4d61 6361 756c 6179 3244 ule: (Macaulay2D │ │ │ │ -0005b750: 6f63 294d 6f64 756c 652c 2c20 6f76 6572 oc)Module,, over │ │ │ │ -0005b760: 2061 2063 6f6d 706c 6574 6520 696e 7465 a complete inte │ │ │ │ -0005b770: 7273 6563 7469 6f6e 0a20 202a 204f 7574 rsection. * Out │ │ │ │ -0005b780: 7075 7473 3a0a 2020 2020 2020 2a20 702c puts:. * p, │ │ │ │ -0005b790: 2061 6e20 2a6e 6f74 6520 696e 7465 6765 an *note intege │ │ │ │ -0005b7a0: 723a 2028 4d61 6361 756c 6179 3244 6f63 r: (Macaulay2Doc │ │ │ │ -0005b7b0: 295a 5a2c 2c20 0a0a 4465 7363 7269 7074 )ZZ,, ..Descript │ │ │ │ -0005b7c0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -0005b7d0: 0a49 6620 7020 3d20 6d66 426f 756e 6420 .If p = mfBound │ │ │ │ -0005b7e0: 4d2c 2074 6865 6e20 7468 6520 702d 7468 M, then the p-th │ │ │ │ -0005b7f0: 2073 797a 7967 7920 6f66 204d 2c20 7768 syzygy of M, wh │ │ │ │ -0005b800: 6963 6820 6973 2063 6f6d 7075 7465 6420 ich is computed │ │ │ │ -0005b810: 6279 0a68 6967 6853 797a 7967 7928 4d29 by.highSyzygy(M) │ │ │ │ -0005b820: 2c20 7368 6f75 6c64 2028 7468 6973 2069 , should (this i │ │ │ │ -0005b830: 7320 6120 636f 6e6a 6563 7475 7265 2920 s a conjecture) │ │ │ │ -0005b840: 6265 2061 2022 6869 6768 2053 797a 7967 be a "high Syzyg │ │ │ │ -0005b850: 7922 2069 6e20 7468 6520 7365 6e73 650a y" in the sense. │ │ │ │ -0005b860: 7265 7175 6972 6564 2066 6f72 206d 6174 required for mat │ │ │ │ -0005b870: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -0005b880: 2e20 496e 2065 7861 6d70 6c65 732c 2074 . In examples, t │ │ │ │ -0005b890: 6865 2065 7374 696d 6174 6520 7365 656d he estimate seem │ │ │ │ -0005b8a0: 7320 7368 6172 7020 2865 7863 6570 740a s sharp (except. │ │ │ │ -0005b8b0: 7768 656e 204d 2069 7320 616c 7265 6164 when M is alread │ │ │ │ -0005b8c0: 7920 6120 6869 6768 2073 797a 7967 7929 y a high syzygy) │ │ │ │ -0005b8d0: 2e0a 0a54 6865 2061 6374 7561 6c20 666f ...The actual fo │ │ │ │ -0005b8e0: 726d 756c 6120 7573 6564 2069 733a 0a0a rmula used is:.. │ │ │ │ -0005b8f0: 6d66 426f 756e 6420 4d20 3d20 6d61 7828 mfBound M = max( │ │ │ │ -0005b900: 322a 725f 7b65 7665 6e7d 2c20 312b 322a 2*r_{even}, 1+2* │ │ │ │ -0005b910: 725f 7b6f 6464 7d29 0a0a 7768 6572 6520 r_{odd})..where │ │ │ │ -0005b920: 725f 7b65 7665 6e7d 203d 2072 6567 756c r_{even} = regul │ │ │ │ -0005b930: 6172 6974 7920 6576 656e 4578 744d 6f64 arity evenExtMod │ │ │ │ -0005b940: 756c 6520 4d20 616e 6420 725f 7b6f 6464 ule M and r_{odd │ │ │ │ -0005b950: 7d20 3d20 7265 6775 6c61 7269 7479 0a6f } = regularity.o │ │ │ │ -0005b960: 6464 4578 744d 6f64 756c 6520 4d2e 2048 ddExtModule M. H │ │ │ │ -0005b970: 6572 6520 6576 656e 4578 744d 6f64 756c ere evenExtModul │ │ │ │ -0005b980: 6520 4d20 6973 2074 6865 2065 7665 6e20 e M is the even │ │ │ │ -0005b990: 6465 6772 6565 2070 6172 7420 6f66 2045 degree part of E │ │ │ │ -0005b9a0: 7874 284d 2c20 2872 6573 6964 7565 0a63 xt(M, (residue.c │ │ │ │ -0005b9b0: 6c61 7373 2066 6965 6c64 2929 2e0a 0a53 lass field))...S │ │ │ │ -0005b9c0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ -0005b9d0: 0a0a 2020 2a20 2a6e 6f74 6520 6869 6768 .. * *note high │ │ │ │ -0005b9e0: 5379 7a79 6779 3a20 6869 6768 5379 7a79 Syzygy: highSyzy │ │ │ │ -0005b9f0: 6779 2c20 2d2d 2052 6574 7572 6e73 2061 gy, -- Returns a │ │ │ │ -0005ba00: 2073 797a 7967 7920 6d6f 6475 6c65 206f syzygy module o │ │ │ │ -0005ba10: 6e65 2062 6579 6f6e 6420 7468 650a 2020 ne beyond the. │ │ │ │ -0005ba20: 2020 7265 6775 6c61 7269 7479 206f 6620 regularity of │ │ │ │ -0005ba30: 4578 7428 4d2c 6b29 0a20 202a 202a 6e6f Ext(M,k). * *no │ │ │ │ -0005ba40: 7465 2065 7665 6e45 7874 4d6f 6475 6c65 te evenExtModule │ │ │ │ -0005ba50: 3a20 6576 656e 4578 744d 6f64 756c 652c : evenExtModule, │ │ │ │ -0005ba60: 202d 2d20 6576 656e 2070 6172 7420 6f66 -- even part of │ │ │ │ -0005ba70: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572 Ext^*(M,k) over │ │ │ │ -0005ba80: 2061 0a20 2020 2063 6f6d 706c 6574 6520 a. complete │ │ │ │ -0005ba90: 696e 7465 7273 6563 7469 6f6e 2061 7320 intersection as │ │ │ │ -0005baa0: 6d6f 6475 6c65 206f 7665 7220 4349 206f module over CI o │ │ │ │ -0005bab0: 7065 7261 746f 7220 7269 6e67 0a20 202a perator ring. * │ │ │ │ -0005bac0: 202a 6e6f 7465 206f 6464 4578 744d 6f64 *note oddExtMod │ │ │ │ -0005bad0: 756c 653a 206f 6464 4578 744d 6f64 756c ule: oddExtModul │ │ │ │ -0005bae0: 652c 202d 2d20 6f64 6420 7061 7274 206f e, -- odd part o │ │ │ │ -0005baf0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665 f Ext^*(M,k) ove │ │ │ │ -0005bb00: 7220 6120 636f 6d70 6c65 7465 0a20 2020 r a complete. │ │ │ │ -0005bb10: 2069 6e74 6572 7365 6374 696f 6e20 6173 intersection as │ │ │ │ -0005bb20: 206d 6f64 756c 6520 6f76 6572 2043 4920 module over CI │ │ │ │ -0005bb30: 6f70 6572 6174 6f72 2072 696e 670a 2020 operator ring. │ │ │ │ -0005bb40: 2a20 2a6e 6f74 6520 6d61 7472 6978 4661 * *note matrixFa │ │ │ │ -0005bb50: 6374 6f72 697a 6174 696f 6e3a 206d 6174 ctorization: mat │ │ │ │ -0005bb60: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -0005bb70: 2c20 2d2d 204d 6170 7320 696e 2061 2068 , -- Maps in a h │ │ │ │ -0005bb80: 6967 6865 720a 2020 2020 636f 6469 6d65 igher. codime │ │ │ │ -0005bb90: 6e73 696f 6e20 6d61 7472 6978 2066 6163 nsion matrix fac │ │ │ │ -0005bba0: 746f 7269 7a61 7469 6f6e 0a0a 5761 7973 torization..Ways │ │ │ │ -0005bbb0: 2074 6f20 7573 6520 6d66 426f 756e 643a to use mfBound: │ │ │ │ -0005bbc0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0005bbd0: 3d3d 3d3d 3d0a 0a20 202a 2022 6d66 426f =====.. * "mfBo │ │ │ │ -0005bbe0: 756e 6428 4d6f 6475 6c65 2922 0a0a 466f und(Module)"..Fo │ │ │ │ -0005bbf0: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -0005bc00: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0005bc10: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -0005bc20: 2a6e 6f74 6520 6d66 426f 756e 643a 206d *note mfBound: m │ │ │ │ -0005bc30: 6642 6f75 6e64 2c20 6973 2061 202a 6e6f fBound, is a *no │ │ │ │ -0005bc40: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ -0005bc50: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f on:.(Macaulay2Do │ │ │ │ -0005bc60: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ -0005bc70: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ +0005b6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +0005b6f0: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +0005b700: 2020 2020 2070 203d 206d 6642 6f75 6e64 p = mfBound │ │ │ │ +0005b710: 204d 0a20 202a 2049 6e70 7574 733a 0a20 M. * Inputs:. │ │ │ │ +0005b720: 2020 2020 202a 204d 2c20 6120 2a6e 6f74 * M, a *not │ │ │ │ +0005b730: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ +0005b740: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ +0005b750: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ +0005b760: 2069 6e74 6572 7365 6374 696f 6e0a 2020 intersection. │ │ │ │ +0005b770: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +0005b780: 202a 2070 2c20 616e 202a 6e6f 7465 2069 * p, an *note i │ │ │ │ +0005b790: 6e74 6567 6572 3a20 284d 6163 6175 6c61 nteger: (Macaula │ │ │ │ +0005b7a0: 7932 446f 6329 5a5a 2c2c 200a 0a44 6573 y2Doc)ZZ,, ..Des │ │ │ │ +0005b7b0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +0005b7c0: 3d3d 3d3d 0a0a 4966 2070 203d 206d 6642 ====..If p = mfB │ │ │ │ +0005b7d0: 6f75 6e64 204d 2c20 7468 656e 2074 6865 ound M, then the │ │ │ │ +0005b7e0: 2070 2d74 6820 7379 7a79 6779 206f 6620 p-th syzygy of │ │ │ │ +0005b7f0: 4d2c 2077 6869 6368 2069 7320 636f 6d70 M, which is comp │ │ │ │ +0005b800: 7574 6564 2062 790a 6869 6768 5379 7a79 uted by.highSyzy │ │ │ │ +0005b810: 6779 284d 292c 2073 686f 756c 6420 2874 gy(M), should (t │ │ │ │ +0005b820: 6869 7320 6973 2061 2063 6f6e 6a65 6374 his is a conject │ │ │ │ +0005b830: 7572 6529 2062 6520 6120 2268 6967 6820 ure) be a "high │ │ │ │ +0005b840: 5379 7a79 6779 2220 696e 2074 6865 2073 Syzygy" in the s │ │ │ │ +0005b850: 656e 7365 0a72 6571 7569 7265 6420 666f ense.required fo │ │ │ │ +0005b860: 7220 6d61 7472 6978 4661 6374 6f72 697a r matrixFactoriz │ │ │ │ +0005b870: 6174 696f 6e2e 2049 6e20 6578 616d 706c ation. In exampl │ │ │ │ +0005b880: 6573 2c20 7468 6520 6573 7469 6d61 7465 es, the estimate │ │ │ │ +0005b890: 2073 6565 6d73 2073 6861 7270 2028 6578 seems sharp (ex │ │ │ │ +0005b8a0: 6365 7074 0a77 6865 6e20 4d20 6973 2061 cept.when M is a │ │ │ │ +0005b8b0: 6c72 6561 6479 2061 2068 6967 6820 7379 lready a high sy │ │ │ │ +0005b8c0: 7a79 6779 292e 0a0a 5468 6520 6163 7475 zygy)...The actu │ │ │ │ +0005b8d0: 616c 2066 6f72 6d75 6c61 2075 7365 6420 al formula used │ │ │ │ +0005b8e0: 6973 3a0a 0a6d 6642 6f75 6e64 204d 203d is:..mfBound M = │ │ │ │ +0005b8f0: 206d 6178 2832 2a72 5f7b 6576 656e 7d2c max(2*r_{even}, │ │ │ │ +0005b900: 2031 2b32 2a72 5f7b 6f64 647d 290a 0a77 1+2*r_{odd})..w │ │ │ │ +0005b910: 6865 7265 2072 5f7b 6576 656e 7d20 3d20 here r_{even} = │ │ │ │ +0005b920: 7265 6775 6c61 7269 7479 2065 7665 6e45 regularity evenE │ │ │ │ +0005b930: 7874 4d6f 6475 6c65 204d 2061 6e64 2072 xtModule M and r │ │ │ │ +0005b940: 5f7b 6f64 647d 203d 2072 6567 756c 6172 _{odd} = regular │ │ │ │ +0005b950: 6974 790a 6f64 6445 7874 4d6f 6475 6c65 ity.oddExtModule │ │ │ │ +0005b960: 204d 2e20 4865 7265 2065 7665 6e45 7874 M. Here evenExt │ │ │ │ +0005b970: 4d6f 6475 6c65 204d 2069 7320 7468 6520 Module M is the │ │ │ │ +0005b980: 6576 656e 2064 6567 7265 6520 7061 7274 even degree part │ │ │ │ +0005b990: 206f 6620 4578 7428 4d2c 2028 7265 7369 of Ext(M, (resi │ │ │ │ +0005b9a0: 6475 650a 636c 6173 7320 6669 656c 6429 due.class field) │ │ │ │ +0005b9b0: 292e 0a0a 5365 6520 616c 736f 0a3d 3d3d )...See also.=== │ │ │ │ +0005b9c0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ +0005b9d0: 2068 6967 6853 797a 7967 793a 2068 6967 highSyzygy: hig │ │ │ │ +0005b9e0: 6853 797a 7967 792c 202d 2d20 5265 7475 hSyzygy, -- Retu │ │ │ │ +0005b9f0: 726e 7320 6120 7379 7a79 6779 206d 6f64 rns a syzygy mod │ │ │ │ +0005ba00: 756c 6520 6f6e 6520 6265 796f 6e64 2074 ule one beyond t │ │ │ │ +0005ba10: 6865 0a20 2020 2072 6567 756c 6172 6974 he. regularit │ │ │ │ +0005ba20: 7920 6f66 2045 7874 284d 2c6b 290a 2020 y of Ext(M,k). │ │ │ │ +0005ba30: 2a20 2a6e 6f74 6520 6576 656e 4578 744d * *note evenExtM │ │ │ │ +0005ba40: 6f64 756c 653a 2065 7665 6e45 7874 4d6f odule: evenExtMo │ │ │ │ +0005ba50: 6475 6c65 2c20 2d2d 2065 7665 6e20 7061 dule, -- even pa │ │ │ │ +0005ba60: 7274 206f 6620 4578 745e 2a28 4d2c 6b29 rt of Ext^*(M,k) │ │ │ │ +0005ba70: 206f 7665 7220 610a 2020 2020 636f 6d70 over a. comp │ │ │ │ +0005ba80: 6c65 7465 2069 6e74 6572 7365 6374 696f lete intersectio │ │ │ │ +0005ba90: 6e20 6173 206d 6f64 756c 6520 6f76 6572 n as module over │ │ │ │ +0005baa0: 2043 4920 6f70 6572 6174 6f72 2072 696e CI operator rin │ │ │ │ +0005bab0: 670a 2020 2a20 2a6e 6f74 6520 6f64 6445 g. * *note oddE │ │ │ │ +0005bac0: 7874 4d6f 6475 6c65 3a20 6f64 6445 7874 xtModule: oddExt │ │ │ │ +0005bad0: 4d6f 6475 6c65 2c20 2d2d 206f 6464 2070 Module, -- odd p │ │ │ │ +0005bae0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b art of Ext^*(M,k │ │ │ │ +0005baf0: 2920 6f76 6572 2061 2063 6f6d 706c 6574 ) over a complet │ │ │ │ +0005bb00: 650a 2020 2020 696e 7465 7273 6563 7469 e. intersecti │ │ │ │ +0005bb10: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665 on as module ove │ │ │ │ +0005bb20: 7220 4349 206f 7065 7261 746f 7220 7269 r CI operator ri │ │ │ │ +0005bb30: 6e67 0a20 202a 202a 6e6f 7465 206d 6174 ng. * *note mat │ │ │ │ +0005bb40: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ +0005bb50: 3a20 6d61 7472 6978 4661 6374 6f72 697a : matrixFactoriz │ │ │ │ +0005bb60: 6174 696f 6e2c 202d 2d20 4d61 7073 2069 ation, -- Maps i │ │ │ │ +0005bb70: 6e20 6120 6869 6768 6572 0a20 2020 2063 n a higher. c │ │ │ │ +0005bb80: 6f64 696d 656e 7369 6f6e 206d 6174 7269 odimension matri │ │ │ │ +0005bb90: 7820 6661 6374 6f72 697a 6174 696f 6e0a x factorization. │ │ │ │ +0005bba0: 0a57 6179 7320 746f 2075 7365 206d 6642 .Ways to use mfB │ │ │ │ +0005bbb0: 6f75 6e64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d ound:.========== │ │ │ │ +0005bbc0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +0005bbd0: 226d 6642 6f75 6e64 284d 6f64 756c 6529 "mfBound(Module) │ │ │ │ +0005bbe0: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ +0005bbf0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +0005bc00: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +0005bc10: 6a65 6374 202a 6e6f 7465 206d 6642 6f75 ject *note mfBou │ │ │ │ +0005bc20: 6e64 3a20 6d66 426f 756e 642c 2069 7320 nd: mfBound, is │ │ │ │ +0005bc30: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ +0005bc40: 756e 6374 696f 6e3a 0a28 4d61 6361 756c unction:.(Macaul │ │ │ │ +0005bc50: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +0005bc60: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +0005bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005bcc0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -0005bcd0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -0005bce0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -0005bcf0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -0005bd00: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -0005bd10: 352b 6473 2f4d 322f 4d61 6361 756c 6179 5+ds/M2/Macaulay │ │ │ │ -0005bd20: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ -0005bd30: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -0005bd40: 5265 736f 6c75 7469 6f6e 732e 6d32 3a33 Resolutions.m2:3 │ │ │ │ -0005bd50: 3334 393a 302e 0a1f 0a46 696c 653a 2043 349:0....File: C │ │ │ │ -0005bd60: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -0005bd70: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ -0005bd80: 6e66 6f2c 204e 6f64 653a 206d 6f64 756c nfo, Node: modul │ │ │ │ -0005bd90: 6541 7345 7874 2c20 4e65 7874 3a20 6e65 eAsExt, Next: ne │ │ │ │ -0005bda0: 7745 7874 2c20 5072 6576 3a20 6d66 426f wExt, Prev: mfBo │ │ │ │ -0005bdb0: 756e 642c 2055 703a 2054 6f70 0a0a 6d6f und, Up: Top..mo │ │ │ │ -0005bdc0: 6475 6c65 4173 4578 7420 2d2d 2046 696e duleAsExt -- Fin │ │ │ │ -0005bdd0: 6420 6120 6d6f 6475 6c65 2077 6974 6820 d a module with │ │ │ │ -0005bde0: 6769 7665 6e20 6173 796d 7074 6f74 6963 given asymptotic │ │ │ │ -0005bdf0: 2072 6573 6f6c 7574 696f 6e0a 2a2a 2a2a resolution.**** │ │ │ │ +0005bcb0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +0005bcc0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0005bcd0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0005bce0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0005bcf0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0005bd00: 2e32 362e 3035 2b64 732f 4d32 2f4d 6163 .26.05+ds/M2/Mac │ │ │ │ +0005bd10: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +0005bd20: 0a43 6f6d 706c 6574 6549 6e74 6572 7365 .CompleteInterse │ │ │ │ +0005bd30: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ +0005bd40: 2e6d 323a 3333 3439 3a30 2e0a 1f0a 4669 .m2:3349:0....Fi │ │ │ │ +0005bd50: 6c65 3a20 436f 6d70 6c65 7465 496e 7465 le: CompleteInte │ │ │ │ +0005bd60: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +0005bd70: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20 ons.info, Node: │ │ │ │ +0005bd80: 6d6f 6475 6c65 4173 4578 742c 204e 6578 moduleAsExt, Nex │ │ │ │ +0005bd90: 743a 206e 6577 4578 742c 2050 7265 763a t: newExt, Prev: │ │ │ │ +0005bda0: 206d 6642 6f75 6e64 2c20 5570 3a20 546f mfBound, Up: To │ │ │ │ +0005bdb0: 700a 0a6d 6f64 756c 6541 7345 7874 202d p..moduleAsExt - │ │ │ │ +0005bdc0: 2d20 4669 6e64 2061 206d 6f64 756c 6520 - Find a module │ │ │ │ +0005bdd0: 7769 7468 2067 6976 656e 2061 7379 6d70 with given asymp │ │ │ │ +0005bde0: 746f 7469 6320 7265 736f 6c75 7469 6f6e totic resolution │ │ │ │ +0005bdf0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ 0005be00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0005be10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0005be20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0005be30: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -0005be40: 7361 6765 3a20 0a20 2020 2020 2020 204d sage: . M │ │ │ │ -0005be50: 203d 206d 6f64 756c 6541 7345 7874 284d = moduleAsExt(M │ │ │ │ -0005be60: 4d2c 5229 0a20 202a 2049 6e70 7574 733a M,R). * Inputs: │ │ │ │ -0005be70: 0a20 2020 2020 202a 204d 2c20 6120 2a6e . * M, a *n │ │ │ │ -0005be80: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163 ote module: (Mac │ │ │ │ -0005be90: 6175 6c61 7932 446f 6329 4d6f 6475 6c65 aulay2Doc)Module │ │ │ │ -0005bea0: 2c2c 206d 6f64 756c 6520 6f76 6572 2070 ,, module over p │ │ │ │ -0005beb0: 6f6c 796e 6f6d 6961 6c20 7269 6e67 0a20 olynomial ring. │ │ │ │ -0005bec0: 2020 2020 2020 2077 6974 6820 6320 7661 with c va │ │ │ │ -0005bed0: 7269 6162 6c65 730a 2020 2020 2020 2a20 riables. * │ │ │ │ -0005bee0: 522c 2061 202a 6e6f 7465 2072 696e 673a R, a *note ring: │ │ │ │ -0005bef0: 2028 4d61 6361 756c 6179 3244 6f63 2952 (Macaulay2Doc)R │ │ │ │ -0005bf00: 696e 672c 2c20 2867 7261 6465 6429 2063 ing,, (graded) c │ │ │ │ -0005bf10: 6f6d 706c 6574 6520 696e 7465 7273 6563 omplete intersec │ │ │ │ -0005bf20: 7469 6f6e 0a20 2020 2020 2020 2072 696e tion. rin │ │ │ │ -0005bf30: 6720 6f66 2063 6f64 696d 656e 7369 6f6e g of codimension │ │ │ │ -0005bf40: 2063 2c20 656d 6265 6464 696e 6720 6469 c, embedding di │ │ │ │ -0005bf50: 6d65 6e73 696f 6e20 6e0a 2020 2a20 4f75 mension n. * Ou │ │ │ │ -0005bf60: 7470 7574 733a 0a20 2020 2020 202a 204e tputs:. * N │ │ │ │ -0005bf70: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ -0005bf80: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -0005bf90: 4d6f 6475 6c65 2c2c 206d 6f64 756c 6520 Module,, module │ │ │ │ -0005bfa0: 6f76 6572 2052 2073 7563 6820 7468 6174 over R such that │ │ │ │ -0005bfb0: 0a20 2020 2020 2020 2045 7874 5f52 284e . Ext_R(N │ │ │ │ -0005bfc0: 2c6b 2920 3d20 4d5c 6f74 696d 6573 205c ,k) = M\otimes \ │ │ │ │ -0005bfd0: 7765 6467 6528 6b5e 6e29 2069 6e20 6c61 wedge(k^n) in la │ │ │ │ -0005bfe0: 7267 6520 686f 6d6f 6c6f 6769 6361 6c20 rge homological │ │ │ │ -0005bff0: 6465 6772 6565 2e0a 0a44 6573 6372 6970 degree...Descrip │ │ │ │ -0005c000: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -0005c010: 0a0a 5468 6520 726f 7574 696e 6520 6060 ..The routine `` │ │ │ │ -0005c020: 6d6f 6475 6c65 4173 4578 7427 2720 6973 moduleAsExt'' is │ │ │ │ -0005c030: 2061 2070 6172 7469 616c 2069 6e76 6572 a partial inver │ │ │ │ -0005c040: 7365 2074 6f20 7468 6520 726f 7574 696e se to the routin │ │ │ │ -0005c050: 6520 4578 744d 6f64 756c 652c 0a63 6f6d e ExtModule,.com │ │ │ │ -0005c060: 7075 7465 6420 666f 6c6c 6f77 696e 6720 puted following │ │ │ │ -0005c070: 6964 6561 7320 6f66 2041 7672 616d 6f76 ideas of Avramov │ │ │ │ -0005c080: 2061 6e64 204a 6f72 6765 6e73 656e 3a20 and Jorgensen: │ │ │ │ -0005c090: 6769 7665 6e20 6120 6d6f 6475 6c65 2045 given a module E │ │ │ │ -0005c0a0: 206f 7665 7220 610a 706f 6c79 6e6f 6d69 over a.polynomi │ │ │ │ -0005c0b0: 616c 2072 696e 6720 6b5b 785f 312e 2e78 al ring k[x_1..x │ │ │ │ -0005c0c0: 5f63 5d2c 2069 7420 7072 6f76 6964 6573 _c], it provides │ │ │ │ -0005c0d0: 2061 206d 6f64 756c 6520 4e20 6f76 6572 a module N over │ │ │ │ -0005c0e0: 2061 2073 7065 6369 6669 6564 2070 6f6c a specified pol │ │ │ │ -0005c0f0: 796e 6f6d 6961 6c0a 7269 6e67 2069 6e20 ynomial.ring in │ │ │ │ -0005c100: 6e20 7661 7269 6162 6c65 7320 7375 6368 n variables such │ │ │ │ -0005c110: 2074 6861 7420 4578 7428 4e2c 6b29 2061 that Ext(N,k) a │ │ │ │ -0005c120: 6772 6565 7320 7769 7468 2024 4527 3d45 grees with $E'=E │ │ │ │ -0005c130: 5c6f 7469 6d65 7320 5c77 6564 6765 286b \otimes \wedge(k │ │ │ │ -0005c140: 5e6e 2924 0a61 6674 6572 2074 7275 6e63 ^n)$.after trunc │ │ │ │ -0005c150: 6174 696f 6e2e 2048 6572 6520 7468 6520 ation. Here the │ │ │ │ -0005c160: 6772 6164 696e 6720 6f6e 2045 2069 7320 grading on E is │ │ │ │ -0005c170: 7461 6b65 6e20 746f 2062 6520 6576 656e taken to be even │ │ │ │ -0005c180: 2c20 7768 696c 650a 245c 7765 6467 6528 , while.$\wedge( │ │ │ │ -0005c190: 6b5e 6e29 2420 6861 7320 6765 6e65 7261 k^n)$ has genera │ │ │ │ -0005c1a0: 746f 7273 2069 6e20 6465 6772 6565 2031 tors in degree 1 │ │ │ │ -0005c1b0: 2e20 5468 6520 726f 7574 696e 6520 6866 . The routine hf │ │ │ │ -0005c1c0: 4d6f 6475 6c65 4173 4578 7420 636f 6d70 ModuleAsExt comp │ │ │ │ -0005c1d0: 7574 6573 0a74 6865 2072 6573 756c 7469 utes.the resulti │ │ │ │ -0005c1e0: 6e67 2068 696c 6265 7274 2066 756e 6374 ng hilbert funct │ │ │ │ -0005c1f0: 696f 6e20 666f 7220 4527 2e20 5468 6973 ion for E'. This │ │ │ │ -0005c200: 2075 7365 7320 6964 6561 7320 6f66 2041 uses ideas of A │ │ │ │ -0005c210: 7672 616d 6f76 2061 6e64 0a4a 6f72 6765 vramov and.Jorge │ │ │ │ -0005c220: 6e73 656e 2e20 4e6f 7465 2074 6861 7420 nsen. Note that │ │ │ │ -0005c230: 7468 6520 6d6f 6475 6c65 2045 7874 284e the module Ext(N │ │ │ │ -0005c240: 2c6b 2920 2874 7275 6e63 6174 6564 2920 ,k) (truncated) │ │ │ │ -0005c250: 7769 6c6c 2061 7574 6f6d 6174 6963 616c will automatical │ │ │ │ -0005c260: 6c79 2062 6520 6672 6565 0a6f 7665 7220 ly be free.over │ │ │ │ -0005c270: 7468 6520 6578 7465 7269 6f72 2061 6c67 the exterior alg │ │ │ │ -0005c280: 6562 7261 2024 5c77 6564 6765 286b 5e6e ebra $\wedge(k^n │ │ │ │ -0005c290: 2924 2067 656e 6572 6174 6564 2062 7920 )$ generated by │ │ │ │ -0005c2a0: 4578 745e 3128 6b2c 6b29 3b20 6e6f 7420 Ext^1(k,k); not │ │ │ │ -0005c2b0: 6120 7479 7069 6361 6c0a 4578 7420 6d6f a typical.Ext mo │ │ │ │ -0005c2c0: 6475 6c65 2e0a 0a4d 6f72 6520 7072 6563 dule...More prec │ │ │ │ -0005c2d0: 6973 656c 793a 0a0a 5375 7070 6f73 6520 isely:..Suppose │ │ │ │ -0005c2e0: 7468 6174 2024 5220 3d20 6b5b 615f 312c that $R = k[a_1, │ │ │ │ -0005c2f0: 5c64 6f74 732c 2061 5f6e 5d2f 2866 5f31 \dots, a_n]/(f_1 │ │ │ │ -0005c300: 2c5c 646f 7473 2c66 5f63 2924 206c 6574 ,\dots,f_c)$ let │ │ │ │ -0005c310: 2024 4b4b 203d 0a6b 5b78 5f31 2c5c 646f $KK =.k[x_1,\do │ │ │ │ -0005c320: 7473 2c78 5f63 5d24 2c20 616e 6420 6c65 ts,x_c]$, and le │ │ │ │ -0005c330: 7420 245c 4c61 6d62 6461 203d 205c 7765 t $\Lambda = \we │ │ │ │ -0005c340: 6467 6520 6b5e 6e24 2e20 2445 203d 204b dge k^n$. $E = K │ │ │ │ -0005c350: 4b5c 6f74 696d 6573 5c4c 616d 6264 6124 K\otimes\Lambda$ │ │ │ │ -0005c360: 2c20 736f 0a74 6861 7420 7468 6520 6d69 , so.that the mi │ │ │ │ -0005c370: 6e69 6d61 6c20 2452 242d 6672 6565 2072 nimal $R$-free r │ │ │ │ -0005c380: 6573 6f6c 7574 696f 6e20 6f66 2024 6b24 esolution of $k$ │ │ │ │ -0005c390: 2068 6173 2075 6e64 6572 6c79 696e 6720 has underlying │ │ │ │ -0005c3a0: 6d6f 6475 6c65 2024 525c 6f74 696d 6573 module $R\otimes │ │ │ │ -0005c3b0: 0a45 5e2a 242c 2077 6865 7265 2024 455e .E^*$, where $E^ │ │ │ │ -0005c3c0: 2a24 2069 7320 7468 6520 6772 6164 6564 *$ is the graded │ │ │ │ -0005c3d0: 2076 6563 746f 7220 7370 6163 6520 6475 vector space du │ │ │ │ -0005c3e0: 616c 206f 6620 2445 242e 0a0a 4c65 7420 al of $E$...Let │ │ │ │ -0005c3f0: 4d4d 2062 6520 7468 6520 7265 7375 6c74 MM be the result │ │ │ │ -0005c400: 206f 6620 7472 756e 6361 7469 6e67 204d of truncating M │ │ │ │ -0005c410: 2061 7420 6974 7320 7265 6775 6c61 7269 at its regulari │ │ │ │ -0005c420: 7479 2061 6e64 2073 6869 6674 696e 6720 ty and shifting │ │ │ │ -0005c430: 6974 2073 6f20 7468 6174 0a69 7420 6973 it so that.it is │ │ │ │ -0005c440: 2067 656e 6572 6174 6564 2069 6e20 6465 generated in de │ │ │ │ -0005c450: 6772 6565 2030 2e20 4c65 7420 2446 2420 gree 0. Let $F$ │ │ │ │ -0005c460: 6265 2061 2024 4b4b 242d 6672 6565 2072 be a $KK$-free r │ │ │ │ -0005c470: 6573 6f6c 7574 696f 6e20 6f66 2024 4d4d esolution of $MM │ │ │ │ -0005c480: 242c 2061 6e64 0a77 7269 7465 2024 465f $, and.write $F_ │ │ │ │ -0005c490: 6920 3d20 4b4b 5c6f 7469 6d65 7320 565f i = KK\otimes V_ │ │ │ │ -0005c4a0: 692e 2420 5369 6e63 6520 6c69 6e65 6172 i.$ Since linear │ │ │ │ -0005c4b0: 2066 6f72 6d73 206f 7665 7220 244b 4b24 forms over $KK$ │ │ │ │ -0005c4c0: 2063 6f72 7265 7370 6f6e 6420 746f 2043 correspond to C │ │ │ │ -0005c4d0: 490a 6f70 6572 6174 6f72 7320 6f66 2064 I.operators of d │ │ │ │ -0005c4e0: 6567 7265 6520 2d32 206f 6e20 7468 6520 egree -2 on the │ │ │ │ -0005c4f0: 7265 736f 6c75 7469 6f6e 2047 206f 6620 resolution G of │ │ │ │ -0005c500: 6b20 6f76 6572 2052 2c20 7765 206d 6179 k over R, we may │ │ │ │ -0005c510: 2066 6f72 6d20 6120 6d61 7020 2424 0a64 form a map $$.d │ │ │ │ -0005c520: 5f31 2b64 5f32 3a20 5c73 756d 5f7b 693d _1+d_2: \sum_{i= │ │ │ │ -0005c530: 307d 5e6d 2047 5f7b 692b 317d 5c6f 7469 0}^m G_{i+1}\oti │ │ │ │ -0005c540: 6d65 7320 565f 7b6d 2d69 7d5e 2a20 5c74 mes V_{m-i}^* \t │ │ │ │ -0005c550: 6f20 5c73 756d 5f7b 693d 307d 5e6d 2047 o \sum_{i=0}^m G │ │ │ │ -0005c560: 5f69 5c6f 7469 6d65 730a 565f 7b6d 2d69 _i\otimes.V_{m-i │ │ │ │ -0005c570: 7d5e 2a20 2424 2077 6865 7265 2024 645f }^* $$ where $d_ │ │ │ │ -0005c580: 3124 2069 7320 7468 6520 6469 7265 6374 1$ is the direct │ │ │ │ -0005c590: 2073 756d 206f 6620 7468 6520 6469 6666 sum of the diff │ │ │ │ -0005c5a0: 6572 656e 7469 616c 7320 2428 475f 7b69 erentials $(G_{i │ │ │ │ -0005c5b0: 2b31 7d5c 746f 0a47 5f69 295c 6f74 696d +1}\to.G_i)\otim │ │ │ │ -0005c5c0: 6573 2056 5f69 5e2a 2420 616e 6420 2464 es V_i^*$ and $d │ │ │ │ -0005c5d0: 5f32 2420 6973 2074 6865 2064 6972 6563 _2$ is the direc │ │ │ │ -0005c5e0: 7420 7375 6d20 6f66 2074 6865 206d 6170 t sum of the map │ │ │ │ -0005c5f0: 7320 245c 7068 695f 6924 2064 6566 696e s $\phi_i$ defin │ │ │ │ -0005c600: 6564 0a66 726f 6d20 7468 6520 6469 6666 ed.from the diff │ │ │ │ -0005c610: 6572 656e 7469 616c 7320 6f66 2024 4624 erentials of $F$ │ │ │ │ -0005c620: 2062 7920 7375 6273 7469 7475 7469 6e67 by substituting │ │ │ │ -0005c630: 2043 4920 6f70 6572 6174 6f72 7320 666f CI operators fo │ │ │ │ -0005c640: 7220 6c69 6e65 6172 2066 6f72 6d73 2c0a r linear forms,. │ │ │ │ -0005c650: 245c 7068 695f 693a 2047 5f7b 692b 317d $\phi_i: G_{i+1} │ │ │ │ -0005c660: 5c6f 7469 6d65 7320 565f 6920 5c74 6f20 \otimes V_i \to │ │ │ │ -0005c670: 475f 7b69 2d31 7d5c 6f74 696d 6573 2056 G_{i-1}\otimes V │ │ │ │ -0005c680: 5f7b 692d 317d 242e 2054 6865 2073 6372 _{i-1}$. The scr │ │ │ │ -0005c690: 6970 7420 7265 7475 726e 7320 7468 650a ipt returns the. │ │ │ │ -0005c6a0: 6d6f 6475 6c65 204e 2074 6861 7420 6973 module N that is │ │ │ │ -0005c6b0: 2074 6865 2063 6f6b 6572 6e65 6c20 6f66 the cokernel of │ │ │ │ -0005c6c0: 2024 645f 312b 645f 3224 2e0a 0a54 6865 $d_1+d_2$...The │ │ │ │ -0005c6d0: 206d 6f64 756c 6520 2445 7874 5f52 284e module $Ext_R(N │ │ │ │ -0005c6e0: 2c6b 2924 2061 6772 6565 732c 2061 6674 ,k)$ agrees, aft │ │ │ │ -0005c6f0: 6572 2061 2066 6577 2073 7465 7073 2c20 er a few steps, │ │ │ │ -0005c700: 7769 7468 2074 6865 206d 6f64 756c 6520 with the module │ │ │ │ -0005c710: 6465 7269 7665 6420 6672 6f6d 0a24 4d4d derived from.$MM │ │ │ │ -0005c720: 2420 6279 2074 656e 736f 7269 6e67 2069 $ by tensoring i │ │ │ │ -0005c730: 7420 7769 7468 2024 5c4c 616d 6264 6124 t with $\Lambda$ │ │ │ │ -0005c740: 2c20 7468 6174 2069 732c 2077 6974 6820 , that is, with │ │ │ │ -0005c750: 7468 6520 6d6f 6475 6c65 c39f 2024 2420 the module.. $$ │ │ │ │ -0005c760: 4d4d 2720 3d20 5c73 756d 5f6a 0a28 4d4d MM' = \sum_j.(MM │ │ │ │ -0005c770: 2728 6a29 5c6f 7469 6d65 7320 5c4c 616d '(j)\otimes \Lam │ │ │ │ -0005c780: 6264 615f 6a29 2024 2420 736f 2074 6861 bda_j) $$ so tha │ │ │ │ -0005c790: 7420 244d 4d27 5f70 203d 2028 4d4d 5f70 t $MM'_p = (MM_p │ │ │ │ -0005c7a0: 5c6f 7469 6d65 7320 4c61 6d62 6461 5f30 \otimes Lambda_0 │ │ │ │ -0005c7b0: 2920 5c6f 706c 7573 0a28 4d4d 5f7b 702d ) \oplus.(MM_{p- │ │ │ │ -0005c7c0: 317d 5c6f 7469 6d65 7320 4c61 6d62 6461 1}\otimes Lambda │ │ │ │ -0005c7d0: 5f31 2920 5c6f 706c 7573 5c63 646f 7473 _1) \oplus\cdots │ │ │ │ -0005c7e0: 242e 0a0a 5468 6520 6675 6e63 7469 6f6e $...The function │ │ │ │ -0005c7f0: 2068 664d 6f64 756c 6541 7345 7874 2063 hfModuleAsExt c │ │ │ │ -0005c800: 6f6d 7075 7465 7320 7468 6520 4869 6c62 omputes the Hilb │ │ │ │ -0005c810: 6572 7420 6675 6e63 7469 6f6e 206f 6620 ert function of │ │ │ │ -0005c820: 4d4d 2720 6e75 6d65 7269 6361 6c6c 790a MM' numerically. │ │ │ │ -0005c830: 6672 6f6d 2074 6861 7420 6f66 204d 4d2e from that of MM. │ │ │ │ -0005c840: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +0005be20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +0005be30: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +0005be40: 2020 2020 4d20 3d20 6d6f 6475 6c65 4173 M = moduleAs │ │ │ │ +0005be50: 4578 7428 4d4d 2c52 290a 2020 2a20 496e Ext(MM,R). * In │ │ │ │ +0005be60: 7075 7473 3a0a 2020 2020 2020 2a20 4d2c puts:. * M, │ │ │ │ +0005be70: 2061 202a 6e6f 7465 206d 6f64 756c 653a a *note module: │ │ │ │ +0005be80: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +0005be90: 6f64 756c 652c 2c20 6d6f 6475 6c65 206f odule,, module o │ │ │ │ +0005bea0: 7665 7220 706f 6c79 6e6f 6d69 616c 2072 ver polynomial r │ │ │ │ +0005beb0: 696e 670a 2020 2020 2020 2020 7769 7468 ing. with │ │ │ │ +0005bec0: 2063 2076 6172 6961 626c 6573 0a20 2020 c variables. │ │ │ │ +0005bed0: 2020 202a 2052 2c20 6120 2a6e 6f74 6520 * R, a *note │ │ │ │ +0005bee0: 7269 6e67 3a20 284d 6163 6175 6c61 7932 ring: (Macaulay2 │ │ │ │ +0005bef0: 446f 6329 5269 6e67 2c2c 2028 6772 6164 Doc)Ring,, (grad │ │ │ │ +0005bf00: 6564 2920 636f 6d70 6c65 7465 2069 6e74 ed) complete int │ │ │ │ +0005bf10: 6572 7365 6374 696f 6e0a 2020 2020 2020 ersection. │ │ │ │ +0005bf20: 2020 7269 6e67 206f 6620 636f 6469 6d65 ring of codime │ │ │ │ +0005bf30: 6e73 696f 6e20 632c 2065 6d62 6564 6469 nsion c, embeddi │ │ │ │ +0005bf40: 6e67 2064 696d 656e 7369 6f6e 206e 0a20 ng dimension n. │ │ │ │ +0005bf50: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +0005bf60: 2020 2a20 4e2c 2061 202a 6e6f 7465 206d * N, a *note m │ │ │ │ +0005bf70: 6f64 756c 653a 2028 4d61 6361 756c 6179 odule: (Macaulay │ │ │ │ +0005bf80: 3244 6f63 294d 6f64 756c 652c 2c20 6d6f 2Doc)Module,, mo │ │ │ │ +0005bf90: 6475 6c65 206f 7665 7220 5220 7375 6368 dule over R such │ │ │ │ +0005bfa0: 2074 6861 740a 2020 2020 2020 2020 4578 that. Ex │ │ │ │ +0005bfb0: 745f 5228 4e2c 6b29 203d 204d 5c6f 7469 t_R(N,k) = M\oti │ │ │ │ +0005bfc0: 6d65 7320 5c77 6564 6765 286b 5e6e 2920 mes \wedge(k^n) │ │ │ │ +0005bfd0: 696e 206c 6172 6765 2068 6f6d 6f6c 6f67 in large homolog │ │ │ │ +0005bfe0: 6963 616c 2064 6567 7265 652e 0a0a 4465 ical degree...De │ │ │ │ +0005bff0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +0005c000: 3d3d 3d3d 3d0a 0a54 6865 2072 6f75 7469 =====..The routi │ │ │ │ +0005c010: 6e65 2060 606d 6f64 756c 6541 7345 7874 ne ``moduleAsExt │ │ │ │ +0005c020: 2727 2069 7320 6120 7061 7274 6961 6c20 '' is a partial │ │ │ │ +0005c030: 696e 7665 7273 6520 746f 2074 6865 2072 inverse to the r │ │ │ │ +0005c040: 6f75 7469 6e65 2045 7874 4d6f 6475 6c65 outine ExtModule │ │ │ │ +0005c050: 2c0a 636f 6d70 7574 6564 2066 6f6c 6c6f ,.computed follo │ │ │ │ +0005c060: 7769 6e67 2069 6465 6173 206f 6620 4176 wing ideas of Av │ │ │ │ +0005c070: 7261 6d6f 7620 616e 6420 4a6f 7267 656e ramov and Jorgen │ │ │ │ +0005c080: 7365 6e3a 2067 6976 656e 2061 206d 6f64 sen: given a mod │ │ │ │ +0005c090: 756c 6520 4520 6f76 6572 2061 0a70 6f6c ule E over a.pol │ │ │ │ +0005c0a0: 796e 6f6d 6961 6c20 7269 6e67 206b 5b78 ynomial ring k[x │ │ │ │ +0005c0b0: 5f31 2e2e 785f 635d 2c20 6974 2070 726f _1..x_c], it pro │ │ │ │ +0005c0c0: 7669 6465 7320 6120 6d6f 6475 6c65 204e vides a module N │ │ │ │ +0005c0d0: 206f 7665 7220 6120 7370 6563 6966 6965 over a specifie │ │ │ │ +0005c0e0: 6420 706f 6c79 6e6f 6d69 616c 0a72 696e d polynomial.rin │ │ │ │ +0005c0f0: 6720 696e 206e 2076 6172 6961 626c 6573 g in n variables │ │ │ │ +0005c100: 2073 7563 6820 7468 6174 2045 7874 284e such that Ext(N │ │ │ │ +0005c110: 2c6b 2920 6167 7265 6573 2077 6974 6820 ,k) agrees with │ │ │ │ +0005c120: 2445 273d 455c 6f74 696d 6573 205c 7765 $E'=E\otimes \we │ │ │ │ +0005c130: 6467 6528 6b5e 6e29 240a 6166 7465 7220 dge(k^n)$.after │ │ │ │ +0005c140: 7472 756e 6361 7469 6f6e 2e20 4865 7265 truncation. Here │ │ │ │ +0005c150: 2074 6865 2067 7261 6469 6e67 206f 6e20 the grading on │ │ │ │ +0005c160: 4520 6973 2074 616b 656e 2074 6f20 6265 E is taken to be │ │ │ │ +0005c170: 2065 7665 6e2c 2077 6869 6c65 0a24 5c77 even, while.$\w │ │ │ │ +0005c180: 6564 6765 286b 5e6e 2924 2068 6173 2067 edge(k^n)$ has g │ │ │ │ +0005c190: 656e 6572 6174 6f72 7320 696e 2064 6567 enerators in deg │ │ │ │ +0005c1a0: 7265 6520 312e 2054 6865 2072 6f75 7469 ree 1. The routi │ │ │ │ +0005c1b0: 6e65 2068 664d 6f64 756c 6541 7345 7874 ne hfModuleAsExt │ │ │ │ +0005c1c0: 2063 6f6d 7075 7465 730a 7468 6520 7265 computes.the re │ │ │ │ +0005c1d0: 7375 6c74 696e 6720 6869 6c62 6572 7420 sulting hilbert │ │ │ │ +0005c1e0: 6675 6e63 7469 6f6e 2066 6f72 2045 272e function for E'. │ │ │ │ +0005c1f0: 2054 6869 7320 7573 6573 2069 6465 6173 This uses ideas │ │ │ │ +0005c200: 206f 6620 4176 7261 6d6f 7620 616e 640a of Avramov and. │ │ │ │ +0005c210: 4a6f 7267 656e 7365 6e2e 204e 6f74 6520 Jorgensen. Note │ │ │ │ +0005c220: 7468 6174 2074 6865 206d 6f64 756c 6520 that the module │ │ │ │ +0005c230: 4578 7428 4e2c 6b29 2028 7472 756e 6361 Ext(N,k) (trunca │ │ │ │ +0005c240: 7465 6429 2077 696c 6c20 6175 746f 6d61 ted) will automa │ │ │ │ +0005c250: 7469 6361 6c6c 7920 6265 2066 7265 650a tically be free. │ │ │ │ +0005c260: 6f76 6572 2074 6865 2065 7874 6572 696f over the exterio │ │ │ │ +0005c270: 7220 616c 6765 6272 6120 245c 7765 6467 r algebra $\wedg │ │ │ │ +0005c280: 6528 6b5e 6e29 2420 6765 6e65 7261 7465 e(k^n)$ generate │ │ │ │ +0005c290: 6420 6279 2045 7874 5e31 286b 2c6b 293b d by Ext^1(k,k); │ │ │ │ +0005c2a0: 206e 6f74 2061 2074 7970 6963 616c 0a45 not a typical.E │ │ │ │ +0005c2b0: 7874 206d 6f64 756c 652e 0a0a 4d6f 7265 xt module...More │ │ │ │ +0005c2c0: 2070 7265 6369 7365 6c79 3a0a 0a53 7570 precisely:..Sup │ │ │ │ +0005c2d0: 706f 7365 2074 6861 7420 2452 203d 206b pose that $R = k │ │ │ │ +0005c2e0: 5b61 5f31 2c5c 646f 7473 2c20 615f 6e5d [a_1,\dots, a_n] │ │ │ │ +0005c2f0: 2f28 665f 312c 5c64 6f74 732c 665f 6329 /(f_1,\dots,f_c) │ │ │ │ +0005c300: 2420 6c65 7420 244b 4b20 3d0a 6b5b 785f $ let $KK =.k[x_ │ │ │ │ +0005c310: 312c 5c64 6f74 732c 785f 635d 242c 2061 1,\dots,x_c]$, a │ │ │ │ +0005c320: 6e64 206c 6574 2024 5c4c 616d 6264 6120 nd let $\Lambda │ │ │ │ +0005c330: 3d20 5c77 6564 6765 206b 5e6e 242e 2024 = \wedge k^n$. $ │ │ │ │ +0005c340: 4520 3d20 4b4b 5c6f 7469 6d65 735c 4c61 E = KK\otimes\La │ │ │ │ +0005c350: 6d62 6461 242c 2073 6f0a 7468 6174 2074 mbda$, so.that t │ │ │ │ +0005c360: 6865 206d 696e 696d 616c 2024 5224 2d66 he minimal $R$-f │ │ │ │ +0005c370: 7265 6520 7265 736f 6c75 7469 6f6e 206f ree resolution o │ │ │ │ +0005c380: 6620 246b 2420 6861 7320 756e 6465 726c f $k$ has underl │ │ │ │ +0005c390: 7969 6e67 206d 6f64 756c 6520 2452 5c6f ying module $R\o │ │ │ │ +0005c3a0: 7469 6d65 730a 455e 2a24 2c20 7768 6572 times.E^*$, wher │ │ │ │ +0005c3b0: 6520 2445 5e2a 2420 6973 2074 6865 2067 e $E^*$ is the g │ │ │ │ +0005c3c0: 7261 6465 6420 7665 6374 6f72 2073 7061 raded vector spa │ │ │ │ +0005c3d0: 6365 2064 7561 6c20 6f66 2024 4524 2e0a ce dual of $E$.. │ │ │ │ +0005c3e0: 0a4c 6574 204d 4d20 6265 2074 6865 2072 .Let MM be the r │ │ │ │ +0005c3f0: 6573 756c 7420 6f66 2074 7275 6e63 6174 esult of truncat │ │ │ │ +0005c400: 696e 6720 4d20 6174 2069 7473 2072 6567 ing M at its reg │ │ │ │ +0005c410: 756c 6172 6974 7920 616e 6420 7368 6966 ularity and shif │ │ │ │ +0005c420: 7469 6e67 2069 7420 736f 2074 6861 740a ting it so that. │ │ │ │ +0005c430: 6974 2069 7320 6765 6e65 7261 7465 6420 it is generated │ │ │ │ +0005c440: 696e 2064 6567 7265 6520 302e 204c 6574 in degree 0. Let │ │ │ │ +0005c450: 2024 4624 2062 6520 6120 244b 4b24 2d66 $F$ be a $KK$-f │ │ │ │ +0005c460: 7265 6520 7265 736f 6c75 7469 6f6e 206f ree resolution o │ │ │ │ +0005c470: 6620 244d 4d24 2c20 616e 640a 7772 6974 f $MM$, and.writ │ │ │ │ +0005c480: 6520 2446 5f69 203d 204b 4b5c 6f74 696d e $F_i = KK\otim │ │ │ │ +0005c490: 6573 2056 5f69 2e24 2053 696e 6365 206c es V_i.$ Since l │ │ │ │ +0005c4a0: 696e 6561 7220 666f 726d 7320 6f76 6572 inear forms over │ │ │ │ +0005c4b0: 2024 4b4b 2420 636f 7272 6573 706f 6e64 $KK$ correspond │ │ │ │ +0005c4c0: 2074 6f20 4349 0a6f 7065 7261 746f 7273 to CI.operators │ │ │ │ +0005c4d0: 206f 6620 6465 6772 6565 202d 3220 6f6e of degree -2 on │ │ │ │ +0005c4e0: 2074 6865 2072 6573 6f6c 7574 696f 6e20 the resolution │ │ │ │ +0005c4f0: 4720 6f66 206b 206f 7665 7220 522c 2077 G of k over R, w │ │ │ │ +0005c500: 6520 6d61 7920 666f 726d 2061 206d 6170 e may form a map │ │ │ │ +0005c510: 2024 240a 645f 312b 645f 323a 205c 7375 $$.d_1+d_2: \su │ │ │ │ +0005c520: 6d5f 7b69 3d30 7d5e 6d20 475f 7b69 2b31 m_{i=0}^m G_{i+1 │ │ │ │ +0005c530: 7d5c 6f74 696d 6573 2056 5f7b 6d2d 697d }\otimes V_{m-i} │ │ │ │ +0005c540: 5e2a 205c 746f 205c 7375 6d5f 7b69 3d30 ^* \to \sum_{i=0 │ │ │ │ +0005c550: 7d5e 6d20 475f 695c 6f74 696d 6573 0a56 }^m G_i\otimes.V │ │ │ │ +0005c560: 5f7b 6d2d 697d 5e2a 2024 2420 7768 6572 _{m-i}^* $$ wher │ │ │ │ +0005c570: 6520 2464 5f31 2420 6973 2074 6865 2064 e $d_1$ is the d │ │ │ │ +0005c580: 6972 6563 7420 7375 6d20 6f66 2074 6865 irect sum of the │ │ │ │ +0005c590: 2064 6966 6665 7265 6e74 6961 6c73 2024 differentials $ │ │ │ │ +0005c5a0: 2847 5f7b 692b 317d 5c74 6f0a 475f 6929 (G_{i+1}\to.G_i) │ │ │ │ +0005c5b0: 5c6f 7469 6d65 7320 565f 695e 2a24 2061 \otimes V_i^*$ a │ │ │ │ +0005c5c0: 6e64 2024 645f 3224 2069 7320 7468 6520 nd $d_2$ is the │ │ │ │ +0005c5d0: 6469 7265 6374 2073 756d 206f 6620 7468 direct sum of th │ │ │ │ +0005c5e0: 6520 6d61 7073 2024 5c70 6869 5f69 2420 e maps $\phi_i$ │ │ │ │ +0005c5f0: 6465 6669 6e65 640a 6672 6f6d 2074 6865 defined.from the │ │ │ │ +0005c600: 2064 6966 6665 7265 6e74 6961 6c73 206f differentials o │ │ │ │ +0005c610: 6620 2446 2420 6279 2073 7562 7374 6974 f $F$ by substit │ │ │ │ +0005c620: 7574 696e 6720 4349 206f 7065 7261 746f uting CI operato │ │ │ │ +0005c630: 7273 2066 6f72 206c 696e 6561 7220 666f rs for linear fo │ │ │ │ +0005c640: 726d 732c 0a24 5c70 6869 5f69 3a20 475f rms,.$\phi_i: G_ │ │ │ │ +0005c650: 7b69 2b31 7d5c 6f74 696d 6573 2056 5f69 {i+1}\otimes V_i │ │ │ │ +0005c660: 205c 746f 2047 5f7b 692d 317d 5c6f 7469 \to G_{i-1}\oti │ │ │ │ +0005c670: 6d65 7320 565f 7b69 2d31 7d24 2e20 5468 mes V_{i-1}$. Th │ │ │ │ +0005c680: 6520 7363 7269 7074 2072 6574 7572 6e73 e script returns │ │ │ │ +0005c690: 2074 6865 0a6d 6f64 756c 6520 4e20 7468 the.module N th │ │ │ │ +0005c6a0: 6174 2069 7320 7468 6520 636f 6b65 726e at is the cokern │ │ │ │ +0005c6b0: 656c 206f 6620 2464 5f31 2b64 5f32 242e el of $d_1+d_2$. │ │ │ │ +0005c6c0: 0a0a 5468 6520 6d6f 6475 6c65 2024 4578 ..The module $Ex │ │ │ │ +0005c6d0: 745f 5228 4e2c 6b29 2420 6167 7265 6573 t_R(N,k)$ agrees │ │ │ │ +0005c6e0: 2c20 6166 7465 7220 6120 6665 7720 7374 , after a few st │ │ │ │ +0005c6f0: 6570 732c 2077 6974 6820 7468 6520 6d6f eps, with the mo │ │ │ │ +0005c700: 6475 6c65 2064 6572 6976 6564 2066 726f dule derived fro │ │ │ │ +0005c710: 6d0a 244d 4d24 2062 7920 7465 6e73 6f72 m.$MM$ by tensor │ │ │ │ +0005c720: 696e 6720 6974 2077 6974 6820 245c 4c61 ing it with $\La │ │ │ │ +0005c730: 6d62 6461 242c 2074 6861 7420 6973 2c20 mbda$, that is, │ │ │ │ +0005c740: 7769 7468 2074 6865 206d 6f64 756c 65c3 with the module. │ │ │ │ +0005c750: 9f20 2424 204d 4d27 203d 205c 7375 6d5f . $$ MM' = \sum_ │ │ │ │ +0005c760: 6a0a 284d 4d27 286a 295c 6f74 696d 6573 j.(MM'(j)\otimes │ │ │ │ +0005c770: 205c 4c61 6d62 6461 5f6a 2920 2424 2073 \Lambda_j) $$ s │ │ │ │ +0005c780: 6f20 7468 6174 2024 4d4d 275f 7020 3d20 o that $MM'_p = │ │ │ │ +0005c790: 284d 4d5f 705c 6f74 696d 6573 204c 616d (MM_p\otimes Lam │ │ │ │ +0005c7a0: 6264 615f 3029 205c 6f70 6c75 730a 284d bda_0) \oplus.(M │ │ │ │ +0005c7b0: 4d5f 7b70 2d31 7d5c 6f74 696d 6573 204c M_{p-1}\otimes L │ │ │ │ +0005c7c0: 616d 6264 615f 3129 205c 6f70 6c75 735c ambda_1) \oplus\ │ │ │ │ +0005c7d0: 6364 6f74 7324 2e0a 0a54 6865 2066 756e cdots$...The fun │ │ │ │ +0005c7e0: 6374 696f 6e20 6866 4d6f 6475 6c65 4173 ction hfModuleAs │ │ │ │ +0005c7f0: 4578 7420 636f 6d70 7574 6573 2074 6865 Ext computes the │ │ │ │ +0005c800: 2048 696c 6265 7274 2066 756e 6374 696f Hilbert functio │ │ │ │ +0005c810: 6e20 6f66 204d 4d27 206e 756d 6572 6963 n of MM' numeric │ │ │ │ +0005c820: 616c 6c79 0a66 726f 6d20 7468 6174 206f ally.from that o │ │ │ │ +0005c830: 6620 4d4d 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d f MM...+-------- │ │ │ │ +0005c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005c860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005c870: 2d2d 2d2b 0a7c 6931 203a 206b 6b20 3d20 ---+.|i1 : kk = │ │ │ │ -0005c880: 5a5a 2f31 3031 3b20 2020 2020 2020 2020 ZZ/101; │ │ │ │ -0005c890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005c8a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0005c860: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +0005c870: 6b6b 203d 205a 5a2f 3130 313b 2020 2020 kk = ZZ/101; │ │ │ │ +0005c880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005c890: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0005c8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005c8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005c8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005c8d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -0005c8e0: 2053 203d 206b 6b5b 612c 622c 635d 3b20 S = kk[a,b,c]; │ │ │ │ +0005c8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005c8d0: 7c69 3220 3a20 5320 3d20 6b6b 5b61 2c62 |i2 : S = kk[a,b │ │ │ │ +0005c8e0: 2c63 5d3b 2020 2020 2020 2020 2020 2020 ,c]; │ │ │ │ 0005c8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005c900: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0005c900: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0005c910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005c920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005c930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0005c940: 0a7c 6933 203a 2066 6620 3d20 6d61 7472 .|i3 : ff = matr │ │ │ │ -0005c950: 6978 7b7b 615e 342c 2062 5e34 2c63 5e34 ix{{a^4, b^4,c^4 │ │ │ │ -0005c960: 7d7d 3b20 2020 2020 2020 2020 2020 2020 }}; │ │ │ │ -0005c970: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0005c930: 2d2d 2d2d 2b0a 7c69 3320 3a20 6666 203d ----+.|i3 : ff = │ │ │ │ +0005c940: 206d 6174 7269 787b 7b61 5e34 2c20 625e matrix{{a^4, b^ │ │ │ │ +0005c950: 342c 635e 347d 7d3b 2020 2020 2020 2020 4,c^4}}; │ │ │ │ +0005c960: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0005c970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005c980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005c990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005c9a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0005c9b0: 2020 2020 2031 2020 2020 2020 3320 2020 1 3 │ │ │ │ -0005c9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005c9d0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ -0005c9e0: 4d61 7472 6978 2053 2020 3c2d 2d20 5320 Matrix S <-- S │ │ │ │ +0005c990: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0005c9a0: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +0005c9b0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +0005c9c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0005c9d0: 6f33 203a 204d 6174 7269 7820 5320 203c o3 : Matrix S < │ │ │ │ +0005c9e0: 2d2d 2053 2020 2020 2020 2020 2020 2020 -- S │ │ │ │ 0005c9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ca00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0005ca00: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0005ca10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ca20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005ca30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0005ca40: 7c69 3420 3a20 5220 3d20 532f 6964 6561 |i4 : R = S/idea │ │ │ │ -0005ca50: 6c20 6666 3b20 2020 2020 2020 2020 2020 l ff; │ │ │ │ -0005ca60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ca70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0005ca30: 2d2d 2d2b 0a7c 6934 203a 2052 203d 2053 ---+.|i4 : R = S │ │ │ │ +0005ca40: 2f69 6465 616c 2066 663b 2020 2020 2020 /ideal ff; │ │ │ │ +0005ca50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005ca60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0005ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005caa0: 2d2d 2d2d 2b0a 7c69 3520 3a20 4f70 7320 ----+.|i5 : Ops │ │ │ │ -0005cab0: 3d20 6b6b 5b78 5f31 2c78 5f32 2c78 5f33 = kk[x_1,x_2,x_3 │ │ │ │ -0005cac0: 5d3b 2020 2020 2020 2020 2020 2020 2020 ]; │ │ │ │ -0005cad0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0005ca90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ +0005caa0: 204f 7073 203d 206b 6b5b 785f 312c 785f Ops = kk[x_1,x_ │ │ │ │ +0005cab0: 322c 785f 335d 3b20 2020 2020 2020 2020 2,x_3]; │ │ │ │ +0005cac0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0005cad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005cae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005caf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005cb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ -0005cb10: 3a20 4d4d 203d 204f 7073 5e31 2f28 785f : MM = Ops^1/(x_ │ │ │ │ -0005cb20: 312a 6964 6561 6c28 785f 325e 322c 785f 1*ideal(x_2^2,x_ │ │ │ │ -0005cb30: 3329 293b 2020 2020 2020 2020 207c 0a2b 3)); |.+ │ │ │ │ +0005caf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0005cb00: 0a7c 6936 203a 204d 4d20 3d20 4f70 735e .|i6 : MM = Ops^ │ │ │ │ +0005cb10: 312f 2878 5f31 2a69 6465 616c 2878 5f32 1/(x_1*ideal(x_2 │ │ │ │ +0005cb20: 5e32 2c78 5f33 2929 3b20 2020 2020 2020 ^2,x_3)); │ │ │ │ +0005cb30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0005cb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005cb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005cb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005cb70: 2b0a 7c69 3720 3a20 4e20 3d20 6d6f 6475 +.|i7 : N = modu │ │ │ │ -0005cb80: 6c65 4173 4578 7428 4d4d 2c52 293b 2020 leAsExt(MM,R); │ │ │ │ -0005cb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005cba0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0005cb60: 2d2d 2d2d 2d2b 0a7c 6937 203a 204e 203d -----+.|i7 : N = │ │ │ │ +0005cb70: 206d 6f64 756c 6541 7345 7874 284d 4d2c moduleAsExt(MM, │ │ │ │ +0005cb80: 5229 3b20 2020 2020 2020 2020 2020 2020 R); │ │ │ │ +0005cb90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0005cba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005cbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005cbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005cbd0: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 6265 ------+.|i8 : be │ │ │ │ -0005cbe0: 7474 6920 6672 6565 5265 736f 6c75 7469 tti freeResoluti │ │ │ │ -0005cbf0: 6f6e 2820 4e2c 204c 656e 6774 684c 696d on( N, LengthLim │ │ │ │ -0005cc00: 6974 203d 3e20 3130 297c 0a7c 2020 2020 it => 10)|.| │ │ │ │ +0005cbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ +0005cbd0: 203a 2062 6574 7469 2066 7265 6552 6573 : betti freeRes │ │ │ │ +0005cbe0: 6f6c 7574 696f 6e28 204e 2c20 4c65 6e67 olution( N, Leng │ │ │ │ +0005cbf0: 7468 4c69 6d69 7420 3d3e 2031 3029 7c0a thLimit => 10)|. │ │ │ │ +0005cc00: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005cc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005cc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005cc30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0005cc40: 2020 2020 2020 2020 2020 2020 3020 2031 0 1 │ │ │ │ -0005cc50: 2020 3220 2033 2020 3420 2035 2020 3620 2 3 4 5 6 │ │ │ │ -0005cc60: 2037 2020 3820 2039 2031 3020 2020 207c 7 8 9 10 | │ │ │ │ -0005cc70: 0a7c 6f38 203d 2074 6f74 616c 3a20 3336 .|o8 = total: 36 │ │ │ │ -0005cc80: 2032 3720 3239 2033 3120 3333 2033 3520 27 29 31 33 35 │ │ │ │ -0005cc90: 3337 2033 3920 3431 2034 3320 3435 2020 37 39 41 43 45 │ │ │ │ -0005cca0: 2020 7c0a 7c20 2020 2020 2020 202d 363a |.| -6: │ │ │ │ -0005ccb0: 2031 3820 2036 2020 2e20 202e 2020 2e20 18 6 . . . │ │ │ │ -0005ccc0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0005ccd0: 2e20 2020 207c 0a7c 2020 2020 2020 2020 . |.| │ │ │ │ -0005cce0: 2d35 3a20 202e 2020 2e20 202e 2020 2e20 -5: . . . . │ │ │ │ -0005ccf0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0005cd00: 2e20 202e 2020 2020 7c0a 7c20 2020 2020 . . |.| │ │ │ │ -0005cd10: 2020 202d 343a 2031 3820 3231 2032 3120 -4: 18 21 21 │ │ │ │ -0005cd20: 2037 2020 2e20 202e 2020 2e20 202e 2020 7 . . . . │ │ │ │ -0005cd30: 2e20 202e 2020 2e20 2020 207c 0a7c 2020 . . . |.| │ │ │ │ -0005cd40: 2020 2020 2020 2d33 3a20 202e 2020 2e20 -3: . . │ │ │ │ -0005cd50: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0005cd60: 2e20 202e 2020 2e20 202e 2020 2020 7c0a . . . . |. │ │ │ │ -0005cd70: 7c20 2020 2020 2020 202d 323a 2020 2e20 | -2: . │ │ │ │ -0005cd80: 202e 2020 3820 3234 2032 3420 2038 2020 . 8 24 24 8 │ │ │ │ -0005cd90: 2e20 202e 2020 2e20 202e 2020 2e20 2020 . . . . . │ │ │ │ -0005cda0: 207c 0a7c 2020 2020 2020 2020 2d31 3a20 |.| -1: │ │ │ │ -0005cdb0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0005cdc0: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0005cdd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0005cde0: 303a 2020 2e20 202e 2020 2e20 202e 2020 0: . . . . │ │ │ │ -0005cdf0: 3920 3237 2032 3720 2039 2020 2e20 202e 9 27 27 9 . . │ │ │ │ -0005ce00: 2020 2e20 2020 207c 0a7c 2020 2020 2020 . |.| │ │ │ │ -0005ce10: 2020 2031 3a20 202e 2020 2e20 202e 2020 1: . . . │ │ │ │ -0005ce20: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0005ce30: 2020 2e20 202e 2020 2020 7c0a 7c20 2020 . . |.| │ │ │ │ -0005ce40: 2020 2020 2020 323a 2020 2e20 202e 2020 2: . . │ │ │ │ -0005ce50: 2e20 202e 2020 2e20 202e 2031 3020 3330 . . . . 10 30 │ │ │ │ -0005ce60: 2033 3020 3130 2020 2e20 2020 207c 0a7c 30 10 . |.| │ │ │ │ -0005ce70: 2020 2020 2020 2020 2033 3a20 202e 2020 3: . │ │ │ │ -0005ce80: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0005ce90: 2020 2e20 202e 2020 2e20 202e 2020 2020 . . . . │ │ │ │ -0005cea0: 7c0a 7c20 2020 2020 2020 2020 343a 2020 |.| 4: │ │ │ │ -0005ceb0: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0005cec0: 2020 2e20 202e 2031 3120 3333 2033 3320 . . 11 33 33 │ │ │ │ -0005ced0: 2020 207c 0a7c 2020 2020 2020 2020 2035 |.| 5 │ │ │ │ -0005cee0: 3a20 202e 2020 2e20 202e 2020 2e20 202e : . . . . . │ │ │ │ -0005cef0: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ -0005cf00: 202e 2020 2020 7c0a 7c20 2020 2020 2020 . |.| │ │ │ │ -0005cf10: 2020 363a 2020 2e20 202e 2020 2e20 202e 6: . . . . │ │ │ │ -0005cf20: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ -0005cf30: 202e 2031 3220 2020 207c 0a7c 2020 2020 . 12 |.| │ │ │ │ +0005cc30: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0005cc40: 2030 2020 3120 2032 2020 3320 2034 2020 0 1 2 3 4 │ │ │ │ +0005cc50: 3520 2036 2020 3720 2038 2020 3920 3130 5 6 7 8 9 10 │ │ │ │ +0005cc60: 2020 2020 7c0a 7c6f 3820 3d20 746f 7461 |.|o8 = tota │ │ │ │ +0005cc70: 6c3a 2033 3620 3237 2032 3920 3331 2033 l: 36 27 29 31 3 │ │ │ │ +0005cc80: 3320 3335 2033 3720 3339 2034 3120 3433 3 35 37 39 41 43 │ │ │ │ +0005cc90: 2034 3520 2020 207c 0a7c 2020 2020 2020 45 |.| │ │ │ │ +0005cca0: 2020 2d36 3a20 3138 2020 3620 202e 2020 -6: 18 6 . │ │ │ │ +0005ccb0: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ +0005ccc0: 2020 2e20 202e 2020 2020 7c0a 7c20 2020 . . |.| │ │ │ │ +0005ccd0: 2020 2020 202d 353a 2020 2e20 202e 2020 -5: . . │ │ │ │ +0005cce0: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ +0005ccf0: 2020 2e20 202e 2020 2e20 2020 207c 0a7c . . . |.| │ │ │ │ +0005cd00: 2020 2020 2020 2020 2d34 3a20 3138 2032 -4: 18 2 │ │ │ │ +0005cd10: 3120 3231 2020 3720 202e 2020 2e20 202e 1 21 7 . . . │ │ │ │ +0005cd20: 2020 2e20 202e 2020 2e20 202e 2020 2020 . . . . │ │ │ │ +0005cd30: 7c0a 7c20 2020 2020 2020 202d 333a 2020 |.| -3: │ │ │ │ +0005cd40: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ +0005cd50: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0005cd60: 2020 207c 0a7c 2020 2020 2020 2020 2d32 |.| -2 │ │ │ │ +0005cd70: 3a20 202e 2020 2e20 2038 2032 3420 3234 : . . 8 24 24 │ │ │ │ +0005cd80: 2020 3820 202e 2020 2e20 202e 2020 2e20 8 . . . . │ │ │ │ +0005cd90: 202e 2020 2020 7c0a 7c20 2020 2020 2020 . |.| │ │ │ │ +0005cda0: 202d 313a 2020 2e20 202e 2020 2e20 202e -1: . . . . │ │ │ │ +0005cdb0: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0005cdc0: 202e 2020 2e20 2020 207c 0a7c 2020 2020 . . |.| │ │ │ │ +0005cdd0: 2020 2020 2030 3a20 202e 2020 2e20 202e 0: . . . │ │ │ │ +0005cde0: 2020 2e20 2039 2032 3720 3237 2020 3920 . 9 27 27 9 │ │ │ │ +0005cdf0: 202e 2020 2e20 202e 2020 2020 7c0a 7c20 . . . |.| │ │ │ │ +0005ce00: 2020 2020 2020 2020 313a 2020 2e20 202e 1: . . │ │ │ │ +0005ce10: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0005ce20: 202e 2020 2e20 202e 2020 2e20 2020 207c . . . . | │ │ │ │ +0005ce30: 0a7c 2020 2020 2020 2020 2032 3a20 202e .| 2: . │ │ │ │ +0005ce40: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0005ce50: 3130 2033 3020 3330 2031 3020 202e 2020 10 30 30 10 . │ │ │ │ +0005ce60: 2020 7c0a 7c20 2020 2020 2020 2020 333a |.| 3: │ │ │ │ +0005ce70: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0005ce80: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ +0005ce90: 2e20 2020 207c 0a7c 2020 2020 2020 2020 . |.| │ │ │ │ +0005cea0: 2034 3a20 202e 2020 2e20 202e 2020 2e20 4: . . . . │ │ │ │ +0005ceb0: 202e 2020 2e20 202e 2020 2e20 3131 2033 . . . . 11 3 │ │ │ │ +0005cec0: 3320 3333 2020 2020 7c0a 7c20 2020 2020 3 33 |.| │ │ │ │ +0005ced0: 2020 2020 353a 2020 2e20 202e 2020 2e20 5: . . . │ │ │ │ +0005cee0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ +0005cef0: 2e20 202e 2020 2e20 2020 207c 0a7c 2020 . . . |.| │ │ │ │ +0005cf00: 2020 2020 2020 2036 3a20 202e 2020 2e20 6: . . │ │ │ │ +0005cf10: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ +0005cf20: 2e20 202e 2020 2e20 3132 2020 2020 7c0a . . . 12 |. │ │ │ │ +0005cf30: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005cf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005cf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005cf60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0005cf70: 3820 3a20 4265 7474 6954 616c 6c79 2020 8 : BettiTally │ │ │ │ +0005cf60: 207c 0a7c 6f38 203a 2042 6574 7469 5461 |.|o8 : BettiTa │ │ │ │ +0005cf70: 6c6c 7920 2020 2020 2020 2020 2020 2020 lly │ │ │ │ 0005cf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005cf90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0005cfa0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0005cf90: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005cfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005cfd0: 2d2d 2b0a 7c69 3920 3a20 6866 4d6f 6475 --+.|i9 : hfModu │ │ │ │ -0005cfe0: 6c65 4173 4578 7428 3132 2c4d 4d2c 3329 leAsExt(12,MM,3) │ │ │ │ -0005cff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005d000: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0005cfc0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2068 -------+.|i9 : h │ │ │ │ +0005cfd0: 664d 6f64 756c 6541 7345 7874 2831 322c fModuleAsExt(12, │ │ │ │ +0005cfe0: 4d4d 2c33 2920 2020 2020 2020 2020 2020 MM,3) │ │ │ │ +0005cff0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0005d000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005d010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005d020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005d030: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20 |.|o9 = │ │ │ │ -0005d040: 2832 332c 2032 352c 2032 372c 2032 392c (23, 25, 27, 29, │ │ │ │ -0005d050: 2033 312c 2033 332c 2033 352c 2033 372c 31, 33, 35, 37, │ │ │ │ -0005d060: 2033 392c 2034 3129 2020 207c 0a7c 2020 39, 41) |.| │ │ │ │ +0005d020: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0005d030: 6f39 203d 2028 3233 2c20 3235 2c20 3237 o9 = (23, 25, 27 │ │ │ │ +0005d040: 2c20 3239 2c20 3331 2c20 3333 2c20 3335 , 29, 31, 33, 35 │ │ │ │ +0005d050: 2c20 3337 2c20 3339 2c20 3431 2920 2020 , 37, 39, 41) │ │ │ │ +0005d060: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0005d070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005d080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005d090: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0005d0a0: 7c6f 3920 3a20 5365 7175 656e 6365 2020 |o9 : Sequence │ │ │ │ +0005d090: 2020 207c 0a7c 6f39 203a 2053 6571 7565 |.|o9 : Seque │ │ │ │ +0005d0a0: 6e63 6520 2020 2020 2020 2020 2020 2020 nce │ │ │ │ 0005d0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005d0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005d0d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0005d0c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0005d0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005d0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005d0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005d100: 2d2d 2d2d 2b0a 0a43 6176 6561 740a 3d3d ----+..Caveat.== │ │ │ │ -0005d110: 3d3d 3d3d 0a0a 5468 6520 656c 656d 656e ====..The elemen │ │ │ │ -0005d120: 7473 2066 5f31 2e2e 665f 6320 6d75 7374 ts f_1..f_c must │ │ │ │ -0005d130: 2062 6520 686f 6d6f 6765 6e65 6f75 7320 be homogeneous │ │ │ │ -0005d140: 6f66 2074 6865 2073 616d 6520 6465 6772 of the same degr │ │ │ │ -0005d150: 6565 2e20 5468 6520 7363 7269 7074 2063 ee. The script c │ │ │ │ -0005d160: 6f75 6c64 0a62 6520 7265 7772 6974 7465 ould.be rewritte │ │ │ │ -0005d170: 6e20 746f 2061 6363 6f6d 6d6f 6461 7465 n to accommodate │ │ │ │ -0005d180: 2064 6966 6665 7265 6e74 2064 6567 7265 different degre │ │ │ │ -0005d190: 6573 2c20 6275 7420 6f6e 6c79 2062 7920 es, but only by │ │ │ │ -0005d1a0: 676f 696e 6720 746f 2074 6865 206c 6f63 going to the loc │ │ │ │ -0005d1b0: 616c 0a63 6174 6567 6f72 790a 0a53 6565 al.category..See │ │ │ │ -0005d1c0: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -0005d1d0: 2020 2a20 2a6e 6f74 6520 4578 744d 6f64 * *note ExtMod │ │ │ │ -0005d1e0: 756c 653a 2045 7874 4d6f 6475 6c65 2c20 ule: ExtModule, │ │ │ │ -0005d1f0: 2d2d 2045 7874 5e2a 284d 2c6b 2920 6f76 -- Ext^*(M,k) ov │ │ │ │ -0005d200: 6572 2061 2063 6f6d 706c 6574 6520 696e er a complete in │ │ │ │ -0005d210: 7465 7273 6563 7469 6f6e 2061 730a 2020 tersection as. │ │ │ │ -0005d220: 2020 6d6f 6475 6c65 206f 7665 7220 4349 module over CI │ │ │ │ -0005d230: 206f 7065 7261 746f 7220 7269 6e67 0a20 operator ring. │ │ │ │ -0005d240: 202a 202a 6e6f 7465 2065 7665 6e45 7874 * *note evenExt │ │ │ │ -0005d250: 4d6f 6475 6c65 3a20 6576 656e 4578 744d Module: evenExtM │ │ │ │ -0005d260: 6f64 756c 652c 202d 2d20 6576 656e 2070 odule, -- even p │ │ │ │ -0005d270: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b art of Ext^*(M,k │ │ │ │ -0005d280: 2920 6f76 6572 2061 0a20 2020 2063 6f6d ) over a. com │ │ │ │ -0005d290: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ -0005d2a0: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665 on as module ove │ │ │ │ -0005d2b0: 7220 4349 206f 7065 7261 746f 7220 7269 r CI operator ri │ │ │ │ -0005d2c0: 6e67 0a20 202a 202a 6e6f 7465 206f 6464 ng. * *note odd │ │ │ │ -0005d2d0: 4578 744d 6f64 756c 653a 206f 6464 4578 ExtModule: oddEx │ │ │ │ -0005d2e0: 744d 6f64 756c 652c 202d 2d20 6f64 6420 tModule, -- odd │ │ │ │ -0005d2f0: 7061 7274 206f 6620 4578 745e 2a28 4d2c part of Ext^*(M, │ │ │ │ -0005d300: 6b29 206f 7665 7220 6120 636f 6d70 6c65 k) over a comple │ │ │ │ -0005d310: 7465 0a20 2020 2069 6e74 6572 7365 6374 te. intersect │ │ │ │ -0005d320: 696f 6e20 6173 206d 6f64 756c 6520 6f76 ion as module ov │ │ │ │ -0005d330: 6572 2043 4920 6f70 6572 6174 6f72 2072 er CI operator r │ │ │ │ -0005d340: 696e 670a 2020 2a20 2a6e 6f74 6520 4578 ing. * *note Ex │ │ │ │ -0005d350: 744d 6f64 756c 6544 6174 613a 2045 7874 tModuleData: Ext │ │ │ │ -0005d360: 4d6f 6475 6c65 4461 7461 2c20 2d2d 2045 ModuleData, -- E │ │ │ │ -0005d370: 7665 6e20 616e 6420 6f64 6420 4578 7420 ven and odd Ext │ │ │ │ -0005d380: 6d6f 6475 6c65 7320 616e 6420 7468 6569 modules and thei │ │ │ │ -0005d390: 720a 2020 2020 7265 6775 6c61 7269 7479 r. regularity │ │ │ │ -0005d3a0: 0a20 202a 202a 6e6f 7465 2068 664d 6f64 . * *note hfMod │ │ │ │ -0005d3b0: 756c 6541 7345 7874 3a20 6866 4d6f 6475 uleAsExt: hfModu │ │ │ │ -0005d3c0: 6c65 4173 4578 742c 202d 2d20 7072 6564 leAsExt, -- pred │ │ │ │ -0005d3d0: 6963 7420 6265 7474 6920 6e75 6d62 6572 ict betti number │ │ │ │ -0005d3e0: 7320 6f66 0a20 2020 206d 6f64 756c 6541 s of. moduleA │ │ │ │ -0005d3f0: 7345 7874 284d 2c52 290a 0a57 6179 7320 sExt(M,R)..Ways │ │ │ │ -0005d400: 746f 2075 7365 206d 6f64 756c 6541 7345 to use moduleAsE │ │ │ │ -0005d410: 7874 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d xt:.============ │ │ │ │ -0005d420: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ -0005d430: 2a20 226d 6f64 756c 6541 7345 7874 284d * "moduleAsExt(M │ │ │ │ -0005d440: 6f64 756c 652c 5269 6e67 2922 0a0a 466f odule,Ring)"..Fo │ │ │ │ -0005d450: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -0005d460: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0005d470: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -0005d480: 2a6e 6f74 6520 6d6f 6475 6c65 4173 4578 *note moduleAsEx │ │ │ │ -0005d490: 743a 206d 6f64 756c 6541 7345 7874 2c20 t: moduleAsExt, │ │ │ │ -0005d4a0: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -0005d4b0: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ -0005d4c0: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -0005d4d0: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +0005d0f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 ---------+..Cave │ │ │ │ +0005d100: 6174 0a3d 3d3d 3d3d 3d0a 0a54 6865 2065 at.======..The e │ │ │ │ +0005d110: 6c65 6d65 6e74 7320 665f 312e 2e66 5f63 lements f_1..f_c │ │ │ │ +0005d120: 206d 7573 7420 6265 2068 6f6d 6f67 656e must be homogen │ │ │ │ +0005d130: 656f 7573 206f 6620 7468 6520 7361 6d65 eous of the same │ │ │ │ +0005d140: 2064 6567 7265 652e 2054 6865 2073 6372 degree. The scr │ │ │ │ +0005d150: 6970 7420 636f 756c 640a 6265 2072 6577 ipt could.be rew │ │ │ │ +0005d160: 7269 7474 656e 2074 6f20 6163 636f 6d6d ritten to accomm │ │ │ │ +0005d170: 6f64 6174 6520 6469 6666 6572 656e 7420 odate different │ │ │ │ +0005d180: 6465 6772 6565 732c 2062 7574 206f 6e6c degrees, but onl │ │ │ │ +0005d190: 7920 6279 2067 6f69 6e67 2074 6f20 7468 y by going to th │ │ │ │ +0005d1a0: 6520 6c6f 6361 6c0a 6361 7465 676f 7279 e local.category │ │ │ │ +0005d1b0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ +0005d1c0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2045 ===.. * *note E │ │ │ │ +0005d1d0: 7874 4d6f 6475 6c65 3a20 4578 744d 6f64 xtModule: ExtMod │ │ │ │ +0005d1e0: 756c 652c 202d 2d20 4578 745e 2a28 4d2c ule, -- Ext^*(M, │ │ │ │ +0005d1f0: 6b29 206f 7665 7220 6120 636f 6d70 6c65 k) over a comple │ │ │ │ +0005d200: 7465 2069 6e74 6572 7365 6374 696f 6e20 te intersection │ │ │ │ +0005d210: 6173 0a20 2020 206d 6f64 756c 6520 6f76 as. module ov │ │ │ │ +0005d220: 6572 2043 4920 6f70 6572 6174 6f72 2072 er CI operator r │ │ │ │ +0005d230: 696e 670a 2020 2a20 2a6e 6f74 6520 6576 ing. * *note ev │ │ │ │ +0005d240: 656e 4578 744d 6f64 756c 653a 2065 7665 enExtModule: eve │ │ │ │ +0005d250: 6e45 7874 4d6f 6475 6c65 2c20 2d2d 2065 nExtModule, -- e │ │ │ │ +0005d260: 7665 6e20 7061 7274 206f 6620 4578 745e ven part of Ext^ │ │ │ │ +0005d270: 2a28 4d2c 6b29 206f 7665 7220 610a 2020 *(M,k) over a. │ │ │ │ +0005d280: 2020 636f 6d70 6c65 7465 2069 6e74 6572 complete inter │ │ │ │ +0005d290: 7365 6374 696f 6e20 6173 206d 6f64 756c section as modul │ │ │ │ +0005d2a0: 6520 6f76 6572 2043 4920 6f70 6572 6174 e over CI operat │ │ │ │ +0005d2b0: 6f72 2072 696e 670a 2020 2a20 2a6e 6f74 or ring. * *not │ │ │ │ +0005d2c0: 6520 6f64 6445 7874 4d6f 6475 6c65 3a20 e oddExtModule: │ │ │ │ +0005d2d0: 6f64 6445 7874 4d6f 6475 6c65 2c20 2d2d oddExtModule, -- │ │ │ │ +0005d2e0: 206f 6464 2070 6172 7420 6f66 2045 7874 odd part of Ext │ │ │ │ +0005d2f0: 5e2a 284d 2c6b 2920 6f76 6572 2061 2063 ^*(M,k) over a c │ │ │ │ +0005d300: 6f6d 706c 6574 650a 2020 2020 696e 7465 omplete. inte │ │ │ │ +0005d310: 7273 6563 7469 6f6e 2061 7320 6d6f 6475 rsection as modu │ │ │ │ +0005d320: 6c65 206f 7665 7220 4349 206f 7065 7261 le over CI opera │ │ │ │ +0005d330: 746f 7220 7269 6e67 0a20 202a 202a 6e6f tor ring. * *no │ │ │ │ +0005d340: 7465 2045 7874 4d6f 6475 6c65 4461 7461 te ExtModuleData │ │ │ │ +0005d350: 3a20 4578 744d 6f64 756c 6544 6174 612c : ExtModuleData, │ │ │ │ +0005d360: 202d 2d20 4576 656e 2061 6e64 206f 6464 -- Even and odd │ │ │ │ +0005d370: 2045 7874 206d 6f64 756c 6573 2061 6e64 Ext modules and │ │ │ │ +0005d380: 2074 6865 6972 0a20 2020 2072 6567 756c their. regul │ │ │ │ +0005d390: 6172 6974 790a 2020 2a20 2a6e 6f74 6520 arity. * *note │ │ │ │ +0005d3a0: 6866 4d6f 6475 6c65 4173 4578 743a 2068 hfModuleAsExt: h │ │ │ │ +0005d3b0: 664d 6f64 756c 6541 7345 7874 2c20 2d2d fModuleAsExt, -- │ │ │ │ +0005d3c0: 2070 7265 6469 6374 2062 6574 7469 206e predict betti n │ │ │ │ +0005d3d0: 756d 6265 7273 206f 660a 2020 2020 6d6f umbers of. mo │ │ │ │ +0005d3e0: 6475 6c65 4173 4578 7428 4d2c 5229 0a0a duleAsExt(M,R).. │ │ │ │ +0005d3f0: 5761 7973 2074 6f20 7573 6520 6d6f 6475 Ways to use modu │ │ │ │ +0005d400: 6c65 4173 4578 743a 0a3d 3d3d 3d3d 3d3d leAsExt:.======= │ │ │ │ +0005d410: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0005d420: 3d0a 0a20 202a 2022 6d6f 6475 6c65 4173 =.. * "moduleAs │ │ │ │ +0005d430: 4578 7428 4d6f 6475 6c65 2c52 696e 6729 Ext(Module,Ring) │ │ │ │ +0005d440: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ +0005d450: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +0005d460: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +0005d470: 6a65 6374 202a 6e6f 7465 206d 6f64 756c ject *note modul │ │ │ │ +0005d480: 6541 7345 7874 3a20 6d6f 6475 6c65 4173 eAsExt: moduleAs │ │ │ │ +0005d490: 4578 742c 2069 7320 6120 2a6e 6f74 6520 Ext, is a *note │ │ │ │ +0005d4a0: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ +0005d4b0: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ +0005d4c0: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +0005d4d0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 0005d4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005d4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005d500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005d510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005d520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -0005d530: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -0005d540: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -0005d550: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -0005d560: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -0005d570: 322d 312e 3236 2e30 352b 6473 2f4d 322f 2-1.26.05+ds/M2/ │ │ │ │ -0005d580: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -0005d590: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ -0005d5a0: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -0005d5b0: 6f6e 732e 6d32 3a33 3039 363a 302e 0a1f ons.m2:3096:0... │ │ │ │ -0005d5c0: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ -0005d5d0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -0005d5e0: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ -0005d5f0: 653a 206e 6577 4578 742c 204e 6578 743a e: newExt, Next: │ │ │ │ -0005d600: 206f 6464 4578 744d 6f64 756c 652c 2050 oddExtModule, P │ │ │ │ -0005d610: 7265 763a 206d 6f64 756c 6541 7345 7874 rev: moduleAsExt │ │ │ │ -0005d620: 2c20 5570 3a20 546f 700a 0a6e 6577 4578 , Up: Top..newEx │ │ │ │ -0005d630: 7420 2d2d 2047 6c6f 6261 6c20 4578 7420 t -- Global Ext │ │ │ │ -0005d640: 666f 7220 6d6f 6475 6c65 7320 6f76 6572 for modules over │ │ │ │ -0005d650: 2061 2063 6f6d 706c 6574 6520 496e 7465 a complete Inte │ │ │ │ -0005d660: 7273 6563 7469 6f6e 0a2a 2a2a 2a2a 2a2a rsection.******* │ │ │ │ +0005d520: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +0005d530: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +0005d540: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +0005d550: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +0005d560: 6175 6c61 7932 2d31 2e32 362e 3035 2b64 aulay2-1.26.05+d │ │ │ │ +0005d570: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +0005d580: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ +0005d590: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +0005d5a0: 6f6c 7574 696f 6e73 2e6d 323a 3330 3936 olutions.m2:3096 │ │ │ │ +0005d5b0: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ +0005d5c0: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +0005d5d0: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ +0005d5e0: 2c20 4e6f 6465 3a20 6e65 7745 7874 2c20 , Node: newExt, │ │ │ │ +0005d5f0: 4e65 7874 3a20 6f64 6445 7874 4d6f 6475 Next: oddExtModu │ │ │ │ +0005d600: 6c65 2c20 5072 6576 3a20 6d6f 6475 6c65 le, Prev: module │ │ │ │ +0005d610: 4173 4578 742c 2055 703a 2054 6f70 0a0a AsExt, Up: Top.. │ │ │ │ +0005d620: 6e65 7745 7874 202d 2d20 476c 6f62 616c newExt -- Global │ │ │ │ +0005d630: 2045 7874 2066 6f72 206d 6f64 756c 6573 Ext for modules │ │ │ │ +0005d640: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ +0005d650: 2049 6e74 6572 7365 6374 696f 6e0a 2a2a Intersection.** │ │ │ │ +0005d660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0005d670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0005d680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0005d690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0005d6a0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -0005d6b0: 653a 200a 2020 2020 2020 2020 4520 3d20 e: . E = │ │ │ │ -0005d6c0: 6e65 7745 7874 284d 2c4e 290a 2020 2a20 newExt(M,N). * │ │ │ │ -0005d6d0: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -0005d6e0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c M, a *note modul │ │ │ │ -0005d6f0: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -0005d700: 294d 6f64 756c 652c 2c20 6f76 6572 2061 )Module,, over a │ │ │ │ -0005d710: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -0005d720: 6563 7469 6f6e 0a20 2020 2020 2020 2052 ection. R │ │ │ │ -0005d730: 6261 720a 2020 2020 2020 2a20 4e2c 2061 bar. * N, a │ │ │ │ -0005d740: 202a 6e6f 7465 206d 6f64 756c 653a 2028 *note module: ( │ │ │ │ -0005d750: 4d61 6361 756c 6179 3244 6f63 294d 6f64 Macaulay2Doc)Mod │ │ │ │ -0005d760: 756c 652c 2c20 6f76 6572 2052 6261 720a ule,, over Rbar. │ │ │ │ -0005d770: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e * *note Option │ │ │ │ -0005d780: 616c 2069 6e70 7574 733a 2028 4d61 6361 al inputs: (Maca │ │ │ │ -0005d790: 756c 6179 3244 6f63 2975 7369 6e67 2066 ulay2Doc)using f │ │ │ │ -0005d7a0: 756e 6374 696f 6e73 2077 6974 6820 6f70 unctions with op │ │ │ │ -0005d7b0: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a tional inputs,:. │ │ │ │ -0005d7c0: 2020 2020 2020 2a20 4368 6563 6b20 3d3e * Check => │ │ │ │ -0005d7d0: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ -0005d7e0: 6c75 6520 6661 6c73 650a 2020 2020 2020 lue false. │ │ │ │ -0005d7f0: 2a20 4772 6164 696e 6720 3d3e 202e 2e2e * Grading => ... │ │ │ │ -0005d800: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ -0005d810: 320a 2020 2020 2020 2a20 4c69 6674 203d 2. * Lift = │ │ │ │ -0005d820: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ -0005d830: 616c 7565 2066 616c 7365 0a20 2020 2020 alue false. │ │ │ │ -0005d840: 202a 2056 6172 6961 626c 6573 203d 3e20 * Variables => │ │ │ │ -0005d850: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -0005d860: 7565 2073 0a20 202a 204f 7574 7075 7473 ue s. * Outputs │ │ │ │ -0005d870: 3a0a 2020 2020 2020 2a20 452c 2061 202a :. * E, a * │ │ │ │ -0005d880: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -0005d890: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -0005d8a0: 652c 2c20 6f76 6572 2061 2072 696e 6720 e,, over a ring │ │ │ │ -0005d8b0: 5320 6d61 6465 2066 726f 6d20 7269 6e67 S made from ring │ │ │ │ -0005d8c0: 0a20 2020 2020 2020 2070 7265 7365 6e74 . present │ │ │ │ -0005d8d0: 6174 696f 6e20 5262 6172 2077 6974 6820 ation Rbar with │ │ │ │ -0005d8e0: 636f 6469 6d20 5262 6172 206e 6577 2076 codim Rbar new v │ │ │ │ -0005d8f0: 6172 6961 626c 6573 0a0a 4465 7363 7269 ariables..Descri │ │ │ │ -0005d900: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -0005d910: 3d0a 0a4c 6574 2052 6261 7220 3d20 522f =..Let Rbar = R/ │ │ │ │ -0005d920: 2866 312e 2e66 6329 2c20 6120 636f 6d70 (f1..fc), a comp │ │ │ │ -0005d930: 6c65 7465 2069 6e74 6572 7365 6374 696f lete intersectio │ │ │ │ -0005d940: 6e20 6f66 2063 6f64 696d 656e 7369 6f6e n of codimension │ │ │ │ -0005d950: 2063 2c20 616e 6420 6c65 7420 4d2c 4e20 c, and let M,N │ │ │ │ -0005d960: 6265 0a52 6261 722d 6d6f 6475 6c65 732e be.Rbar-modules. │ │ │ │ -0005d970: 2057 6520 6173 7375 6d65 2074 6861 7420 We assume that │ │ │ │ -0005d980: 7468 6520 7075 7368 466f 7277 6172 6420 the pushForward │ │ │ │ -0005d990: 6f66 204d 2074 6f20 5220 6861 7320 6669 of M to R has fi │ │ │ │ -0005d9a0: 6e69 7465 2066 7265 650a 7265 736f 6c75 nite free.resolu │ │ │ │ -0005d9b0: 7469 6f6e 2e20 5468 6520 7363 7269 7074 tion. The script │ │ │ │ -0005d9c0: 2074 6865 6e20 636f 6d70 7574 6573 2074 then computes t │ │ │ │ -0005d9d0: 6865 2074 6f74 616c 2045 7874 284d 2c4e he total Ext(M,N │ │ │ │ -0005d9e0: 2920 6173 2061 206d 6f64 756c 6520 6f76 ) as a module ov │ │ │ │ -0005d9f0: 6572 2053 203d 0a6b 6b28 735f 312e 2e73 er S =.kk(s_1..s │ │ │ │ -0005da00: 5f63 2c67 656e 7320 5229 2c20 7573 696e _c,gens R), usin │ │ │ │ -0005da10: 6720 4569 7365 6e62 7564 5368 616d 6173 g EisenbudShamas │ │ │ │ -0005da20: 6854 6f74 616c 2e0a 0a49 6620 4368 6563 hTotal...If Chec │ │ │ │ -0005da30: 6b20 3d3e 2074 7275 652c 2074 6865 6e20 k => true, then │ │ │ │ -0005da40: 7468 6520 7265 7375 6c74 2069 7320 636f the result is co │ │ │ │ -0005da50: 6d70 6172 6564 2077 6974 6820 7468 6520 mpared with the │ │ │ │ -0005da60: 6275 696c 742d 696e 2067 6c6f 6261 6c20 built-in global │ │ │ │ -0005da70: 4578 740a 7772 6974 7465 6e20 6279 2041 Ext.written by A │ │ │ │ -0005da80: 7672 616d 6f76 2061 6e64 2047 7261 7973 vramov and Grays │ │ │ │ -0005da90: 6f6e 2028 6275 7420 6e6f 7465 2074 6865 on (but note the │ │ │ │ -0005daa0: 2064 6966 6665 7265 6e63 652c 2065 7870 difference, exp │ │ │ │ -0005dab0: 6c61 696e 6564 2062 656c 6f77 292e 0a0a lained below)... │ │ │ │ -0005dac0: 4966 204c 6966 7420 3d3e 2066 616c 7365 If Lift => false │ │ │ │ -0005dad0: 2074 6865 2072 6573 756c 7420 6973 2072 the result is r │ │ │ │ -0005dae0: 6574 7572 6e65 6420 6f76 6572 2061 6e64 eturned over and │ │ │ │ -0005daf0: 2065 7874 656e 7369 6f6e 206f 6620 5262 extension of Rb │ │ │ │ -0005db00: 6172 3b20 6966 204c 6966 7420 3d3e 0a74 ar; if Lift =>.t │ │ │ │ -0005db10: 7275 6520 7468 6520 7265 7375 6c74 2069 rue the result i │ │ │ │ -0005db20: 7320 7265 7475 726e 6564 206f 7665 7220 s returned over │ │ │ │ -0005db30: 616e 6420 6578 7465 6e73 696f 6e20 6f66 and extension of │ │ │ │ -0005db40: 2052 2e0a 0a49 6620 4772 6164 696e 6720 R...If Grading │ │ │ │ -0005db50: 3d3e 2032 2c20 7468 6520 6465 6661 756c => 2, the defaul │ │ │ │ -0005db60: 742c 2074 6865 6e20 7468 6520 7265 7375 t, then the resu │ │ │ │ -0005db70: 6c74 2069 7320 6269 6772 6164 6564 2028 lt is bigraded ( │ │ │ │ -0005db80: 7468 6973 2069 7320 6e65 6365 7373 6172 this is necessar │ │ │ │ -0005db90: 790a 7768 656e 2043 6865 636b 3d3e 7472 y.when Check=>tr │ │ │ │ -0005dba0: 7565 0a0a 5468 6520 6465 6661 756c 7420 ue..The default │ │ │ │ -0005dbb0: 5661 7269 6162 6c65 7320 3d3e 2073 796d Variables => sym │ │ │ │ -0005dbc0: 626f 6c20 2273 2220 6769 7665 7320 7468 bol "s" gives th │ │ │ │ -0005dbd0: 6520 6e65 7720 7661 7269 6162 6c65 7320 e new variables │ │ │ │ -0005dbe0: 7468 6520 6e61 6d65 2073 5f69 2c0a 693d the name s_i,.i= │ │ │ │ -0005dbf0: 302e 2e63 2d31 2e20 286e 6f74 6520 7468 0..c-1. (note th │ │ │ │ -0005dc00: 6174 2074 6865 2062 7569 6c74 696e 2045 at the builtin E │ │ │ │ -0005dc10: 7874 2075 7365 7320 585f 312e 2e58 5f63 xt uses X_1..X_c │ │ │ │ -0005dc20: 2e0a 0a4f 6e20 536f 6d65 2065 7861 6d70 ...On Some examp │ │ │ │ -0005dc30: 6c65 7320 6e65 7745 7874 2069 7320 6661 les newExt is fa │ │ │ │ -0005dc40: 7374 6572 2074 6861 6e20 4578 743b 206f ster than Ext; o │ │ │ │ -0005dc50: 6e20 6f74 6865 7273 2069 7427 7320 736c n others it's sl │ │ │ │ -0005dc60: 6f77 6572 2e0a 0a41 2073 696d 706c 6520 ower...A simple │ │ │ │ -0005dc70: 6578 616d 706c 653a 2069 6620 5220 3d20 example: if R = │ │ │ │ -0005dc80: 6b5b 785f 312e 2e78 5f6e 5d20 616e 6420 k[x_1..x_n] and │ │ │ │ -0005dc90: 4920 6973 2063 6f6e 7461 696e 6564 2069 I is contained i │ │ │ │ -0005dca0: 6e20 7468 6520 6375 6265 206f 6620 7468 n the cube of th │ │ │ │ -0005dcb0: 650a 6d61 7869 6d61 6c20 6964 6561 6c2c e.maximal ideal, │ │ │ │ -0005dcc0: 2074 6865 6e20 4578 7428 6b2c 6b29 2069 then Ext(k,k) i │ │ │ │ -0005dcd0: 7320 6120 6672 6565 2053 2f28 785f 312e s a free S/(x_1. │ │ │ │ -0005dce0: 2e78 5f6e 2920 3d20 6b5b 735f 302e 2e73 .x_n) = k[s_0..s │ │ │ │ -0005dcf0: 5f28 632d 3129 5d2d 206d 6f64 756c 650a _(c-1)]- module. │ │ │ │ -0005dd00: 7769 7468 2062 696e 6f6d 6961 6c28 6e2c with binomial(n, │ │ │ │ -0005dd10: 6929 2067 656e 6572 6174 6f72 7320 696e i) generators in │ │ │ │ -0005dd20: 2064 6567 7265 6520 690a 0a2b 2d2d 2d2d degree i..+---- │ │ │ │ +0005d690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +0005d6a0: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +0005d6b0: 2045 203d 206e 6577 4578 7428 4d2c 4e29 E = newExt(M,N) │ │ │ │ +0005d6c0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ +0005d6d0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520 * M, a *note │ │ │ │ +0005d6e0: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61 module: (Macaula │ │ │ │ +0005d6f0: 7932 446f 6329 4d6f 6475 6c65 2c2c 206f y2Doc)Module,, o │ │ │ │ +0005d700: 7665 7220 6120 636f 6d70 6c65 7465 2069 ver a complete i │ │ │ │ +0005d710: 6e74 6572 7365 6374 696f 6e0a 2020 2020 ntersection. │ │ │ │ +0005d720: 2020 2020 5262 6172 0a20 2020 2020 202a Rbar. * │ │ │ │ +0005d730: 204e 2c20 6120 2a6e 6f74 6520 6d6f 6475 N, a *note modu │ │ │ │ +0005d740: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ +0005d750: 6329 4d6f 6475 6c65 2c2c 206f 7665 7220 c)Module,, over │ │ │ │ +0005d760: 5262 6172 0a20 202a 202a 6e6f 7465 204f Rbar. * *note O │ │ │ │ +0005d770: 7074 696f 6e61 6c20 696e 7075 7473 3a20 ptional inputs: │ │ │ │ +0005d780: 284d 6163 6175 6c61 7932 446f 6329 7573 (Macaulay2Doc)us │ │ │ │ +0005d790: 696e 6720 6675 6e63 7469 6f6e 7320 7769 ing functions wi │ │ │ │ +0005d7a0: 7468 206f 7074 696f 6e61 6c20 696e 7075 th optional inpu │ │ │ │ +0005d7b0: 7473 2c3a 0a20 2020 2020 202a 2043 6865 ts,:. * Che │ │ │ │ +0005d7c0: 636b 203d 3e20 2e2e 2e2c 2064 6566 6175 ck => ..., defau │ │ │ │ +0005d7d0: 6c74 2076 616c 7565 2066 616c 7365 0a20 lt value false. │ │ │ │ +0005d7e0: 2020 2020 202a 2047 7261 6469 6e67 203d * Grading = │ │ │ │ +0005d7f0: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ +0005d800: 616c 7565 2032 0a20 2020 2020 202a 204c alue 2. * L │ │ │ │ +0005d810: 6966 7420 3d3e 202e 2e2e 2c20 6465 6661 ift => ..., defa │ │ │ │ +0005d820: 756c 7420 7661 6c75 6520 6661 6c73 650a ult value false. │ │ │ │ +0005d830: 2020 2020 2020 2a20 5661 7269 6162 6c65 * Variable │ │ │ │ +0005d840: 7320 3d3e 202e 2e2e 2c20 6465 6661 756c s => ..., defaul │ │ │ │ +0005d850: 7420 7661 6c75 6520 730a 2020 2a20 4f75 t value s. * Ou │ │ │ │ +0005d860: 7470 7574 733a 0a20 2020 2020 202a 2045 tputs:. * E │ │ │ │ +0005d870: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +0005d880: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0005d890: 4d6f 6475 6c65 2c2c 206f 7665 7220 6120 Module,, over a │ │ │ │ +0005d8a0: 7269 6e67 2053 206d 6164 6520 6672 6f6d ring S made from │ │ │ │ +0005d8b0: 2072 696e 670a 2020 2020 2020 2020 7072 ring. pr │ │ │ │ +0005d8c0: 6573 656e 7461 7469 6f6e 2052 6261 7220 esentation Rbar │ │ │ │ +0005d8d0: 7769 7468 2063 6f64 696d 2052 6261 7220 with codim Rbar │ │ │ │ +0005d8e0: 6e65 7720 7661 7269 6162 6c65 730a 0a44 new variables..D │ │ │ │ +0005d8f0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +0005d900: 3d3d 3d3d 3d3d 0a0a 4c65 7420 5262 6172 ======..Let Rbar │ │ │ │ +0005d910: 203d 2052 2f28 6631 2e2e 6663 292c 2061 = R/(f1..fc), a │ │ │ │ +0005d920: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ +0005d930: 6563 7469 6f6e 206f 6620 636f 6469 6d65 ection of codime │ │ │ │ +0005d940: 6e73 696f 6e20 632c 2061 6e64 206c 6574 nsion c, and let │ │ │ │ +0005d950: 204d 2c4e 2062 650a 5262 6172 2d6d 6f64 M,N be.Rbar-mod │ │ │ │ +0005d960: 756c 6573 2e20 5765 2061 7373 756d 6520 ules. We assume │ │ │ │ +0005d970: 7468 6174 2074 6865 2070 7573 6846 6f72 that the pushFor │ │ │ │ +0005d980: 7761 7264 206f 6620 4d20 746f 2052 2068 ward of M to R h │ │ │ │ +0005d990: 6173 2066 696e 6974 6520 6672 6565 0a72 as finite free.r │ │ │ │ +0005d9a0: 6573 6f6c 7574 696f 6e2e 2054 6865 2073 esolution. The s │ │ │ │ +0005d9b0: 6372 6970 7420 7468 656e 2063 6f6d 7075 cript then compu │ │ │ │ +0005d9c0: 7465 7320 7468 6520 746f 7461 6c20 4578 tes the total Ex │ │ │ │ +0005d9d0: 7428 4d2c 4e29 2061 7320 6120 6d6f 6475 t(M,N) as a modu │ │ │ │ +0005d9e0: 6c65 206f 7665 7220 5320 3d0a 6b6b 2873 le over S =.kk(s │ │ │ │ +0005d9f0: 5f31 2e2e 735f 632c 6765 6e73 2052 292c _1..s_c,gens R), │ │ │ │ +0005da00: 2075 7369 6e67 2045 6973 656e 6275 6453 using EisenbudS │ │ │ │ +0005da10: 6861 6d61 7368 546f 7461 6c2e 0a0a 4966 hamashTotal...If │ │ │ │ +0005da20: 2043 6865 636b 203d 3e20 7472 7565 2c20 Check => true, │ │ │ │ +0005da30: 7468 656e 2074 6865 2072 6573 756c 7420 then the result │ │ │ │ +0005da40: 6973 2063 6f6d 7061 7265 6420 7769 7468 is compared with │ │ │ │ +0005da50: 2074 6865 2062 7569 6c74 2d69 6e20 676c the built-in gl │ │ │ │ +0005da60: 6f62 616c 2045 7874 0a77 7269 7474 656e obal Ext.written │ │ │ │ +0005da70: 2062 7920 4176 7261 6d6f 7620 616e 6420 by Avramov and │ │ │ │ +0005da80: 4772 6179 736f 6e20 2862 7574 206e 6f74 Grayson (but not │ │ │ │ +0005da90: 6520 7468 6520 6469 6666 6572 656e 6365 e the difference │ │ │ │ +0005daa0: 2c20 6578 706c 6169 6e65 6420 6265 6c6f , explained belo │ │ │ │ +0005dab0: 7729 2e0a 0a49 6620 4c69 6674 203d 3e20 w)...If Lift => │ │ │ │ +0005dac0: 6661 6c73 6520 7468 6520 7265 7375 6c74 false the result │ │ │ │ +0005dad0: 2069 7320 7265 7475 726e 6564 206f 7665 is returned ove │ │ │ │ +0005dae0: 7220 616e 6420 6578 7465 6e73 696f 6e20 r and extension │ │ │ │ +0005daf0: 6f66 2052 6261 723b 2069 6620 4c69 6674 of Rbar; if Lift │ │ │ │ +0005db00: 203d 3e0a 7472 7565 2074 6865 2072 6573 =>.true the res │ │ │ │ +0005db10: 756c 7420 6973 2072 6574 7572 6e65 6420 ult is returned │ │ │ │ +0005db20: 6f76 6572 2061 6e64 2065 7874 656e 7369 over and extensi │ │ │ │ +0005db30: 6f6e 206f 6620 522e 0a0a 4966 2047 7261 on of R...If Gra │ │ │ │ +0005db40: 6469 6e67 203d 3e20 322c 2074 6865 2064 ding => 2, the d │ │ │ │ +0005db50: 6566 6175 6c74 2c20 7468 656e 2074 6865 efault, then the │ │ │ │ +0005db60: 2072 6573 756c 7420 6973 2062 6967 7261 result is bigra │ │ │ │ +0005db70: 6465 6420 2874 6869 7320 6973 206e 6563 ded (this is nec │ │ │ │ +0005db80: 6573 7361 7279 0a77 6865 6e20 4368 6563 essary.when Chec │ │ │ │ +0005db90: 6b3d 3e74 7275 650a 0a54 6865 2064 6566 k=>true..The def │ │ │ │ +0005dba0: 6175 6c74 2056 6172 6961 626c 6573 203d ault Variables = │ │ │ │ +0005dbb0: 3e20 7379 6d62 6f6c 2022 7322 2067 6976 > symbol "s" giv │ │ │ │ +0005dbc0: 6573 2074 6865 206e 6577 2076 6172 6961 es the new varia │ │ │ │ +0005dbd0: 626c 6573 2074 6865 206e 616d 6520 735f bles the name s_ │ │ │ │ +0005dbe0: 692c 0a69 3d30 2e2e 632d 312e 2028 6e6f i,.i=0..c-1. (no │ │ │ │ +0005dbf0: 7465 2074 6861 7420 7468 6520 6275 696c te that the buil │ │ │ │ +0005dc00: 7469 6e20 4578 7420 7573 6573 2058 5f31 tin Ext uses X_1 │ │ │ │ +0005dc10: 2e2e 585f 632e 0a0a 4f6e 2053 6f6d 6520 ..X_c...On Some │ │ │ │ +0005dc20: 6578 616d 706c 6573 206e 6577 4578 7420 examples newExt │ │ │ │ +0005dc30: 6973 2066 6173 7465 7220 7468 616e 2045 is faster than E │ │ │ │ +0005dc40: 7874 3b20 6f6e 206f 7468 6572 7320 6974 xt; on others it │ │ │ │ +0005dc50: 2773 2073 6c6f 7765 722e 0a0a 4120 7369 's slower...A si │ │ │ │ +0005dc60: 6d70 6c65 2065 7861 6d70 6c65 3a20 6966 mple example: if │ │ │ │ +0005dc70: 2052 203d 206b 5b78 5f31 2e2e 785f 6e5d R = k[x_1..x_n] │ │ │ │ +0005dc80: 2061 6e64 2049 2069 7320 636f 6e74 6169 and I is contai │ │ │ │ +0005dc90: 6e65 6420 696e 2074 6865 2063 7562 6520 ned in the cube │ │ │ │ +0005dca0: 6f66 2074 6865 0a6d 6178 696d 616c 2069 of the.maximal i │ │ │ │ +0005dcb0: 6465 616c 2c20 7468 656e 2045 7874 286b deal, then Ext(k │ │ │ │ +0005dcc0: 2c6b 2920 6973 2061 2066 7265 6520 532f ,k) is a free S/ │ │ │ │ +0005dcd0: 2878 5f31 2e2e 785f 6e29 203d 206b 5b73 (x_1..x_n) = k[s │ │ │ │ +0005dce0: 5f30 2e2e 735f 2863 2d31 295d 2d20 6d6f _0..s_(c-1)]- mo │ │ │ │ +0005dcf0: 6475 6c65 0a77 6974 6820 6269 6e6f 6d69 dule.with binomi │ │ │ │ +0005dd00: 616c 286e 2c69 2920 6765 6e65 7261 746f al(n,i) generato │ │ │ │ +0005dd10: 7273 2069 6e20 6465 6772 6565 2069 0a0a rs in degree i.. │ │ │ │ +0005dd20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0005dd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005dd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005dd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005dd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005dd70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ -0005dd80: 206e 203d 2033 3b63 3d32 3b20 2020 2020 n = 3;c=2; │ │ │ │ +0005dd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005dd70: 7c69 3120 3a20 6e20 3d20 333b 633d 323b |i1 : n = 3;c=2; │ │ │ │ +0005dd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005dd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005dda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ddb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ddc0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005ddb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005ddc0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0005ddd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005dde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ddf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005de00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005de10: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -0005de20: 2052 203d 205a 5a2f 3130 315b 785f 302e R = ZZ/101[x_0. │ │ │ │ -0005de30: 2e78 5f28 6e2d 3129 5d20 2020 2020 2020 .x_(n-1)] │ │ │ │ +0005de00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005de10: 7c69 3320 3a20 5220 3d20 5a5a 2f31 3031 |i3 : R = ZZ/101 │ │ │ │ +0005de20: 5b78 5f30 2e2e 785f 286e 2d31 295d 2020 [x_0..x_(n-1)] │ │ │ │ +0005de30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005de40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005de50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005de60: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005de50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005de60: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005de70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005de80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005de90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005dea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005deb0: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ -0005dec0: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0005dea0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005deb0: 7c6f 3320 3d20 5220 2020 2020 2020 2020 |o3 = R │ │ │ │ +0005dec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005ded0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005dee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005def0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005df00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005def0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005df00: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005df10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005df30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005df40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005df50: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ -0005df60: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ +0005df40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005df50: 7c6f 3320 3a20 506f 6c79 6e6f 6d69 616c |o3 : Polynomial │ │ │ │ +0005df60: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ 0005df70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005df80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005df90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005dfa0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005df90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005dfa0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0005dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005dfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005dfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005dfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005dff0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ -0005e000: 2052 6261 7220 3d20 522f 2869 6465 616c Rbar = R/(ideal │ │ │ │ -0005e010: 2061 7070 6c79 2863 2c20 692d 3e20 525f apply(c, i-> R_ │ │ │ │ -0005e020: 695e 3329 2920 2020 2020 2020 2020 2020 i^3)) │ │ │ │ -0005e030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e040: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005dfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005dff0: 7c69 3420 3a20 5262 6172 203d 2052 2f28 |i4 : Rbar = R/( │ │ │ │ +0005e000: 6964 6561 6c20 6170 706c 7928 632c 2069 ideal apply(c, i │ │ │ │ +0005e010: 2d3e 2052 5f69 5e33 2929 2020 2020 2020 -> R_i^3)) │ │ │ │ +0005e020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005e030: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e040: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005e050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e090: 2020 2020 2020 2020 207c 0a7c 6f34 203d |.|o4 = │ │ │ │ -0005e0a0: 2052 6261 7220 2020 2020 2020 2020 2020 Rbar │ │ │ │ +0005e080: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e090: 7c6f 3420 3d20 5262 6172 2020 2020 2020 |o4 = Rbar │ │ │ │ +0005e0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e0e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005e0d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e0e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005e0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e130: 2020 2020 2020 2020 207c 0a7c 6f34 203a |.|o4 : │ │ │ │ -0005e140: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +0005e120: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e130: 7c6f 3420 3a20 5175 6f74 6965 6e74 5269 |o4 : QuotientRi │ │ │ │ +0005e140: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ 0005e150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e180: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005e170: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e180: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0005e190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005e1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005e1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005e1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005e1d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ -0005e1e0: 204d 6261 7220 3d20 4e62 6172 203d 2063 Mbar = Nbar = c │ │ │ │ -0005e1f0: 6f6b 6572 2076 6172 7320 5262 6172 2020 oker vars Rbar │ │ │ │ +0005e1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005e1d0: 7c69 3520 3a20 4d62 6172 203d 204e 6261 |i5 : Mbar = Nba │ │ │ │ +0005e1e0: 7220 3d20 636f 6b65 7220 7661 7273 2052 r = coker vars R │ │ │ │ +0005e1f0: 6261 7220 2020 2020 2020 2020 2020 2020 bar │ │ │ │ 0005e200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e220: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005e210: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e220: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005e230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e270: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ -0005e280: 2063 6f6b 6572 6e65 6c20 7c20 785f 3020 cokernel | x_0 │ │ │ │ -0005e290: 785f 3120 785f 3220 7c20 2020 2020 2020 x_1 x_2 | │ │ │ │ +0005e260: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e270: 7c6f 3520 3d20 636f 6b65 726e 656c 207c |o5 = cokernel | │ │ │ │ +0005e280: 2078 5f30 2078 5f31 2078 5f32 207c 2020 x_0 x_1 x_2 | │ │ │ │ +0005e290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e2c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005e2b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e2c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005e2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e310: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005e300: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e310: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005e320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e330: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ +0005e330: 2020 2031 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 0005e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e360: 2020 2020 2020 2020 207c 0a7c 6f35 203a |.|o5 : │ │ │ │ -0005e370: 2052 6261 722d 6d6f 6475 6c65 2c20 7175 Rbar-module, qu │ │ │ │ -0005e380: 6f74 6965 6e74 206f 6620 5262 6172 2020 otient of Rbar │ │ │ │ +0005e350: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e360: 7c6f 3520 3a20 5262 6172 2d6d 6f64 756c |o5 : Rbar-modul │ │ │ │ +0005e370: 652c 2071 756f 7469 656e 7420 6f66 2052 e, quotient of R │ │ │ │ +0005e380: 6261 7220 2020 2020 2020 2020 2020 2020 bar │ │ │ │ 0005e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e3b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005e3a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e3b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0005e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005e400: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ -0005e410: 2045 203d 206e 6577 4578 7428 4d62 6172 E = newExt(Mbar │ │ │ │ -0005e420: 2c4e 6261 7229 2020 2020 2020 2020 2020 ,Nbar) │ │ │ │ +0005e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005e400: 7c69 3620 3a20 4520 3d20 6e65 7745 7874 |i6 : E = newExt │ │ │ │ +0005e410: 284d 6261 722c 4e62 6172 2920 2020 2020 (Mbar,Nbar) │ │ │ │ +0005e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e450: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005e440: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e450: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e4a0: 2020 2020 2020 2020 207c 0a7c 6f36 203d |.|o6 = │ │ │ │ -0005e4b0: 2063 6f6b 6572 6e65 6c20 7b30 2c20 307d cokernel {0, 0} │ │ │ │ -0005e4c0: 2020 207c 2078 5f32 2078 5f31 2078 5f30 | x_2 x_1 x_0 │ │ │ │ -0005e4d0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e4e0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e4f0: 2030 2020 2030 2020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ -0005e500: 2020 2020 2020 2020 2020 7b2d 322c 202d {-2, - │ │ │ │ -0005e510: 327d 207c 2030 2020 2030 2020 2030 2020 2} | 0 0 0 │ │ │ │ -0005e520: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e530: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e540: 2030 2020 2078 5f32 207c 0a7c 2020 2020 0 x_2 |.| │ │ │ │ -0005e550: 2020 2020 2020 2020 2020 7b2d 322c 202d {-2, - │ │ │ │ -0005e560: 327d 207c 2030 2020 2030 2020 2030 2020 2} | 0 0 0 │ │ │ │ -0005e570: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e580: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e590: 2030 2020 2030 2020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ -0005e5a0: 2020 2020 2020 2020 2020 7b2d 322c 202d {-2, - │ │ │ │ -0005e5b0: 327d 207c 2030 2020 2030 2020 2030 2020 2} | 0 0 0 │ │ │ │ -0005e5c0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e5d0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e5e0: 2030 2020 2030 2020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ -0005e5f0: 2020 2020 2020 2020 2020 7b2d 312c 202d {-1, - │ │ │ │ -0005e600: 317d 207c 2030 2020 2030 2020 2030 2020 1} | 0 0 0 │ │ │ │ -0005e610: 2078 5f32 2078 5f31 2078 5f30 2030 2020 x_2 x_1 x_0 0 │ │ │ │ -0005e620: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e630: 2030 2020 2030 2020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ -0005e640: 2020 2020 2020 2020 2020 7b2d 312c 202d {-1, - │ │ │ │ -0005e650: 317d 207c 2030 2020 2030 2020 2030 2020 1} | 0 0 0 │ │ │ │ -0005e660: 2030 2020 2030 2020 2030 2020 2078 5f32 0 0 0 x_2 │ │ │ │ -0005e670: 2078 5f31 2078 5f30 2030 2020 2030 2020 x_1 x_0 0 0 │ │ │ │ -0005e680: 2030 2020 2030 2020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ -0005e690: 2020 2020 2020 2020 2020 7b2d 312c 202d {-1, - │ │ │ │ -0005e6a0: 317d 207c 2030 2020 2030 2020 2030 2020 1} | 0 0 0 │ │ │ │ -0005e6b0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e6c0: 2030 2020 2030 2020 2078 5f32 2078 5f31 0 0 x_2 x_1 │ │ │ │ -0005e6d0: 2078 5f30 2030 2020 207c 0a7c 2020 2020 x_0 0 |.| │ │ │ │ -0005e6e0: 2020 2020 2020 2020 2020 7b2d 332c 202d {-3, - │ │ │ │ -0005e6f0: 337d 207c 2030 2020 2030 2020 2030 2020 3} | 0 0 0 │ │ │ │ -0005e700: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e710: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e720: 2030 2020 2030 2020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ +0005e490: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e4a0: 7c6f 3620 3d20 636f 6b65 726e 656c 207b |o6 = cokernel { │ │ │ │ +0005e4b0: 302c 2030 7d20 2020 7c20 785f 3220 785f 0, 0} | x_2 x_ │ │ │ │ +0005e4c0: 3120 785f 3020 3020 2020 3020 2020 3020 1 x_0 0 0 0 │ │ │ │ +0005e4d0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +0005e4e0: 2020 3020 2020 3020 2020 3020 2020 7c0a 0 0 0 |. │ │ │ │ +0005e4f0: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ +0005e500: 2d32 2c20 2d32 7d20 7c20 3020 2020 3020 -2, -2} | 0 0 │ │ │ │ +0005e510: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +0005e520: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +0005e530: 2020 3020 2020 3020 2020 785f 3220 7c0a 0 0 x_2 |. │ │ │ │ +0005e540: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ +0005e550: 2d32 2c20 2d32 7d20 7c20 3020 2020 3020 -2, -2} | 0 0 │ │ │ │ +0005e560: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +0005e570: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +0005e580: 2020 3020 2020 3020 2020 3020 2020 7c0a 0 0 0 |. │ │ │ │ +0005e590: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ +0005e5a0: 2d32 2c20 2d32 7d20 7c20 3020 2020 3020 -2, -2} | 0 0 │ │ │ │ +0005e5b0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +0005e5c0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +0005e5d0: 2020 3020 2020 3020 2020 3020 2020 7c0a 0 0 0 |. │ │ │ │ +0005e5e0: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ +0005e5f0: 2d31 2c20 2d31 7d20 7c20 3020 2020 3020 -1, -1} | 0 0 │ │ │ │ +0005e600: 2020 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2020 2020 2020 2020 │ │ │ │ -0005f320: 2020 3020 2020 3120 2020 2020 2020 2020 0 1 │ │ │ │ +0005f2f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005f300: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0005f310: 2020 2020 2020 2030 2020 2031 2020 2020 0 1 │ │ │ │ +0005f320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f350: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005f340: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005f350: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0005f360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f3a0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 416e 2065 ---------+..An e │ │ │ │ -0005f3b0: 7861 6d70 6c65 2077 6865 7265 2074 6865 xample where the │ │ │ │ -0005f3c0: 2062 7569 6c74 2d69 6e20 676c 6f62 616c built-in global │ │ │ │ -0005f3d0: 2045 7874 2069 7320 6861 7264 2074 6f20 Ext is hard to │ │ │ │ -0005f3e0: 636f 6d70 6172 6520 6469 7265 6374 6c79 compare directly │ │ │ │ -0005f3f0: 2077 6974 6820 6f75 720a 6d65 7468 6f64 with our.method │ │ │ │ -0005f400: 206f 6620 636f 6d70 7574 6174 696f 6e3a of computation: │ │ │ │ -0005f410: 2049 202a 6775 6573 732a 2074 6861 7420 I *guess* that │ │ │ │ -0005f420: 7468 6520 7369 676e 2063 686f 6963 6573 the sign choices │ │ │ │ -0005f430: 2069 6e20 7468 6520 6275 696c 742d 696e in the built-in │ │ │ │ -0005f440: 2061 6d6f 756e 740a 6573 7365 6e74 6961 amount.essentia │ │ │ │ -0005f450: 6c6c 7920 746f 2061 2063 6861 6e67 6520 lly to a change │ │ │ │ -0005f460: 6f66 2076 6172 6961 626c 6520 696e 2074 of variable in t │ │ │ │ -0005f470: 6865 206e 6577 2076 6172 6961 626c 6573 he new variables │ │ │ │ -0005f480: 2c20 616e 6420 7370 6f69 6c20 616e 2065 , and spoil an e │ │ │ │ -0005f490: 6173 790a 636f 6d70 6172 6973 6f6e 2e20 asy.comparison. │ │ │ │ -0005f4a0: 4275 7420 666f 7220 6578 616d 706c 6520 But for example │ │ │ │ -0005f4b0: 7468 6520 6269 2d67 7261 6465 6420 4265 the bi-graded Be │ │ │ │ -0005f4c0: 7474 6920 6e75 6d62 6572 7320 6172 6520 tti numbers are │ │ │ │ -0005f4d0: 6571 7561 6c2e 2074 6869 7320 7365 656d equal. this seem │ │ │ │ -0005f4e0: 730a 746f 2073 7461 7274 2077 6974 6820 s.to start with │ │ │ │ -0005f4f0: 633d 332e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d c=3...+--------- │ │ │ │ +0005f390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005f3a0: 0a41 6e20 6578 616d 706c 6520 7768 6572 .An example wher │ │ │ │ +0005f3b0: 6520 7468 6520 6275 696c 742d 696e 2067 e the built-in g │ │ │ │ +0005f3c0: 6c6f 6261 6c20 4578 7420 6973 2068 6172 lobal Ext is har │ │ │ │ +0005f3d0: 6420 746f 2063 6f6d 7061 7265 2064 6972 d to compare dir │ │ │ │ +0005f3e0: 6563 746c 7920 7769 7468 206f 7572 0a6d ectly with our.m │ │ │ │ +0005f3f0: 6574 686f 6420 6f66 2063 6f6d 7075 7461 ethod of computa │ │ │ │ +0005f400: 7469 6f6e 3a20 4920 2a67 7565 7373 2a20 tion: I *guess* │ │ │ │ +0005f410: 7468 6174 2074 6865 2073 6967 6e20 6368 that the sign ch │ │ │ │ +0005f420: 6f69 6365 7320 696e 2074 6865 2062 7569 oices in the bui │ │ │ │ +0005f430: 6c74 2d69 6e20 616d 6f75 6e74 0a65 7373 lt-in amount.ess │ │ │ │ +0005f440: 656e 7469 616c 6c79 2074 6f20 6120 6368 entially to a ch │ │ │ │ +0005f450: 616e 6765 206f 6620 7661 7269 6162 6c65 ange of variable │ │ │ │ +0005f460: 2069 6e20 7468 6520 6e65 7720 7661 7269 in the new vari │ │ │ │ +0005f470: 6162 6c65 732c 2061 6e64 2073 706f 696c ables, and spoil │ │ │ │ +0005f480: 2061 6e20 6561 7379 0a63 6f6d 7061 7269 an easy.compari │ │ │ │ +0005f490: 736f 6e2e 2042 7574 2066 6f72 2065 7861 son. But for exa │ │ │ │ +0005f4a0: 6d70 6c65 2074 6865 2062 692d 6772 6164 mple the bi-grad │ │ │ │ +0005f4b0: 6564 2042 6574 7469 206e 756d 6265 7273 ed Betti numbers │ │ │ │ +0005f4c0: 2061 7265 2065 7175 616c 2e20 7468 6973 are equal. this │ │ │ │ +0005f4d0: 2073 6565 6d73 0a74 6f20 7374 6172 7420 seems.to start │ │ │ │ +0005f4e0: 7769 7468 2063 3d33 2e0a 0a2b 2d2d 2d2d with c=3...+---- │ │ │ │ +0005f4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f540: 2d2d 2d2d 2b0a 7c69 3920 3a20 7365 7452 ----+.|i9 : setR │ │ │ │ -0005f550: 616e 646f 6d53 6565 6420 3020 2020 2020 andomSeed 0 │ │ │ │ +0005f530: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a ---------+.|i9 : │ │ │ │ +0005f540: 2073 6574 5261 6e64 6f6d 5365 6564 2030 setRandomSeed 0 │ │ │ │ +0005f550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f590: 2020 2020 7c0a 7c20 2d2d 2073 6574 7469 |.| -- setti │ │ │ │ -0005f5a0: 6e67 2072 616e 646f 6d20 7365 6564 2074 ng random seed t │ │ │ │ -0005f5b0: 6f20 3020 2020 2020 2020 2020 2020 2020 o 0 │ │ │ │ +0005f580: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ +0005f590: 7365 7474 696e 6720 7261 6e64 6f6d 2073 setting random s │ │ │ │ +0005f5a0: 6565 6420 746f 2030 2020 2020 2020 2020 eed to 0 │ │ │ │ +0005f5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f5e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005f5d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005f5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f630: 2020 2020 7c0a 7c6f 3920 3d20 3020 2020 |.|o9 = 0 │ │ │ │ +0005f620: 2020 2020 2020 2020 207c 0a7c 6f39 203d |.|o9 = │ │ │ │ +0005f630: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ 0005f640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f680: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005f670: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005f680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f6d0: 2d2d 2d2d 2b0a 7c69 3130 203a 206e 203d ----+.|i10 : n = │ │ │ │ -0005f6e0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +0005f6c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 ---------+.|i10 │ │ │ │ +0005f6d0: 3a20 6e20 3d20 3320 2020 2020 2020 2020 : n = 3 │ │ │ │ +0005f6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f720: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005f710: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005f720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f770: 2020 2020 7c0a 7c6f 3130 203d 2033 2020 |.|o10 = 3 │ │ │ │ +0005f760: 2020 2020 2020 2020 207c 0a7c 6f31 3020 |.|o10 │ │ │ │ +0005f770: 3d20 3320 2020 2020 2020 2020 2020 2020 = 3 │ │ │ │ 0005f780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f7c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005f7b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005f7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f810: 2d2d 2d2d 2b0a 7c69 3131 203a 2063 203d ----+.|i11 : c = │ │ │ │ -0005f820: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +0005f800: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 ---------+.|i11 │ │ │ │ +0005f810: 3a20 6320 3d20 3320 2020 2020 2020 2020 : c = 3 │ │ │ │ +0005f820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f860: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005f850: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005f860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f8b0: 2020 2020 7c0a 7c6f 3131 203d 2033 2020 |.|o11 = 3 │ │ │ │ +0005f8a0: 2020 2020 2020 2020 207c 0a7c 6f31 3120 |.|o11 │ │ │ │ +0005f8b0: 3d20 3320 2020 2020 2020 2020 2020 2020 = 3 │ │ │ │ 0005f8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f900: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005f8f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f950: 2d2d 2d2d 2b0a 7c69 3132 203a 206b 6b20 ----+.|i12 : kk │ │ │ │ -0005f960: 3d20 5a5a 2f31 3031 2020 2020 2020 2020 = ZZ/101 │ │ │ │ +0005f940: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 ---------+.|i12 │ │ │ │ +0005f950: 3a20 6b6b 203d 205a 5a2f 3130 3120 2020 : kk = ZZ/101 │ │ │ │ +0005f960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f9a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005f990: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005f9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f9f0: 2020 2020 7c0a 7c6f 3132 203d 206b 6b20 |.|o12 = kk │ │ │ │ +0005f9e0: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ +0005f9f0: 3d20 6b6b 2020 2020 2020 2020 2020 2020 = kk │ │ │ │ 0005fa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fa30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fa40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005fa30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005fa40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fa90: 2020 2020 7c0a 7c6f 3132 203a 2051 756f |.|o12 : Quo │ │ │ │ -0005faa0: 7469 656e 7452 696e 6720 2020 2020 2020 tientRing │ │ │ │ +0005fa80: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ +0005fa90: 3a20 5175 6f74 6965 6e74 5269 6e67 2020 : QuotientRing │ │ │ │ +0005faa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fae0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005fad0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005fae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005faf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005fb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005fb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005fb30: 2d2d 2d2d 2b0a 7c69 3133 203a 2052 203d ----+.|i13 : R = │ │ │ │ -0005fb40: 206b 6b5b 785f 302e 2e78 5f28 6e2d 3129 kk[x_0..x_(n-1) │ │ │ │ -0005fb50: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +0005fb20: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 ---------+.|i13 │ │ │ │ +0005fb30: 3a20 5220 3d20 6b6b 5b78 5f30 2e2e 785f : R = kk[x_0..x_ │ │ │ │ +0005fb40: 286e 2d31 295d 2020 2020 2020 2020 2020 (n-1)] │ │ │ │ +0005fb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fb80: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005fb70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005fb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fbd0: 2020 2020 7c0a 7c6f 3133 203d 2052 2020 |.|o13 = R │ │ │ │ +0005fbc0: 2020 2020 2020 2020 207c 0a7c 6f31 3320 |.|o13 │ │ │ │ +0005fbd0: 3d20 5220 2020 2020 2020 2020 2020 2020 = R │ │ │ │ 0005fbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fc20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005fc10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005fc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fc70: 2020 2020 7c0a 7c6f 3133 203a 2050 6f6c |.|o13 : Pol │ │ │ │ -0005fc80: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ +0005fc60: 2020 2020 2020 2020 207c 0a7c 6f31 3320 |.|o13 │ │ │ │ +0005fc70: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +0005fc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fcc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005fcb0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005fcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005fcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005fce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005fd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005fd10: 2d2d 2d2d 2b0a 7c69 3134 203a 2049 203d ----+.|i14 : I = │ │ │ │ -0005fd20: 2069 6465 616c 2061 7070 6c79 2863 2c20 ideal apply(c, │ │ │ │ -0005fd30: 692d 3e52 5f69 5e32 2920 2020 2020 2020 i->R_i^2) │ │ │ │ +0005fd00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 ---------+.|i14 │ │ │ │ +0005fd10: 3a20 4920 3d20 6964 6561 6c20 6170 706c : I = ideal appl │ │ │ │ +0005fd20: 7928 632c 2069 2d3e 525f 695e 3229 2020 y(c, i->R_i^2) │ │ │ │ +0005fd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fd60: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005fd50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005fd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fdb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0005fdc0: 2020 2020 2032 2020 2032 2020 2032 2020 2 2 2 │ │ │ │ +0005fda0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005fdb0: 2020 2020 2020 2020 2020 3220 2020 3220 2 2 │ │ │ │ +0005fdc0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0005fdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fe00: 2020 2020 7c0a 7c6f 3134 203d 2069 6465 |.|o14 = ide │ │ │ │ -0005fe10: 616c 2028 7820 2c20 7820 2c20 7820 2920 al (x , x , x ) │ │ │ │ +0005fdf0: 2020 2020 2020 2020 207c 0a7c 6f31 3420 |.|o14 │ │ │ │ +0005fe00: 3d20 6964 6561 6c20 2878 202c 2078 202c = ideal (x , x , │ │ │ │ +0005fe10: 2078 2029 2020 2020 2020 2020 2020 2020 x ) │ │ │ │ 0005fe20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fe30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fe40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fe50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0005fe60: 2020 2020 2030 2020 2031 2020 2032 2020 0 1 2 │ │ │ │ +0005fe40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005fe50: 2020 2020 2020 2020 2020 3020 2020 3120 0 1 │ │ │ │ +0005fe60: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0005fe70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fe80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fe90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fea0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005fe90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005fea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005feb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fef0: 2020 2020 7c0a 7c6f 3134 203a 2049 6465 |.|o14 : Ide │ │ │ │ -0005ff00: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ +0005fee0: 2020 2020 2020 2020 207c 0a7c 6f31 3420 |.|o14 │ │ │ │ +0005fef0: 3a20 4964 6561 6c20 6f66 2052 2020 2020 : Ideal of R │ │ │ │ +0005ff00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005ff10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005ff20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ff30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ff40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005ff30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005ff40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ff50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ff60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ff70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005ff80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005ff90: 2d2d 2d2d 2b0a 7c69 3135 203a 2066 6620 ----+.|i15 : ff │ │ │ │ -0005ffa0: 3d20 6765 6e73 2049 2020 2020 2020 2020 = gens I │ │ │ │ +0005ff80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 ---------+.|i15 │ │ │ │ +0005ff90: 3a20 6666 203d 2067 656e 7320 4920 2020 : ff = gens I │ │ │ │ +0005ffa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005ffb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005ffc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ffd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ffe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005ffd0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005ffe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060030: 2020 2020 7c0a 7c6f 3135 203d 207c 2078 |.|o15 = | x │ │ │ │ -00060040: 5f30 5e32 2078 5f31 5e32 2078 5f32 5e32 _0^2 x_1^2 x_2^2 │ │ │ │ -00060050: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00060020: 2020 2020 2020 2020 207c 0a7c 6f31 3520 |.|o15 │ │ │ │ +00060030: 3d20 7c20 785f 305e 3220 785f 315e 3220 = | x_0^2 x_1^2 │ │ │ │ +00060040: 785f 325e 3220 7c20 2020 2020 2020 2020 x_2^2 | │ │ │ │ +00060050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060080: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00060070: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000600a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000600b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000600c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000600d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000600e0: 2020 2020 2031 2020 2020 2020 3320 2020 1 3 │ │ │ │ +000600c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000600d0: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +000600e0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 000600f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060120: 2020 2020 7c0a 7c6f 3135 203a 204d 6174 |.|o15 : Mat │ │ │ │ -00060130: 7269 7820 5220 203c 2d2d 2052 2020 2020 rix R <-- R │ │ │ │ +00060110: 2020 2020 2020 2020 207c 0a7c 6f31 3520 |.|o15 │ │ │ │ +00060120: 3a20 4d61 7472 6978 2052 2020 3c2d 2d20 : Matrix R <-- │ │ │ │ +00060130: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 00060140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060170: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00060160: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00060170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000601a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000601b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000601c0: 2d2d 2d2d 2b0a 7c69 3136 203a 2052 6261 ----+.|i16 : Rba │ │ │ │ -000601d0: 7220 3d20 522f 4920 2020 2020 2020 2020 r = R/I │ │ │ │ +000601b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 ---------+.|i16 │ │ │ │ +000601c0: 3a20 5262 6172 203d 2052 2f49 2020 2020 : Rbar = R/I │ │ │ │ +000601d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000601e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000601f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060210: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00060200: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060260: 2020 2020 7c0a 7c6f 3136 203d 2052 6261 |.|o16 = Rba │ │ │ │ -00060270: 7220 2020 2020 2020 2020 2020 2020 2020 r │ │ │ │ +00060250: 2020 2020 2020 2020 207c 0a7c 6f31 3620 |.|o16 │ │ │ │ +00060260: 3d20 5262 6172 2020 2020 2020 2020 2020 = Rbar │ │ │ │ +00060270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000602a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000602b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000602a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000602b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000602c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000602d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000602e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000602f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060300: 2020 2020 7c0a 7c6f 3136 203a 2051 756f |.|o16 : Quo │ │ │ │ -00060310: 7469 656e 7452 696e 6720 2020 2020 2020 tientRing │ │ │ │ +000602f0: 2020 2020 2020 2020 207c 0a7c 6f31 3620 |.|o16 │ │ │ │ +00060300: 3a20 5175 6f74 6965 6e74 5269 6e67 2020 : QuotientRing │ │ │ │ +00060310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060350: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00060340: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00060350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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00060420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060440: 2020 2020 7c0a 7c6f 3137 203d 206d 6170 |.|o17 = map │ │ │ │ -00060450: 2028 5262 6172 2c20 522c 207b 7820 2c20 (Rbar, R, {x , │ │ │ │ -00060460: 7820 2c20 7820 7d29 2020 2020 2020 2020 x , x }) │ │ │ │ +00060430: 2020 2020 2020 2020 207c 0a7c 6f31 3720 |.|o17 │ │ │ │ +00060440: 3d20 6d61 7020 2852 6261 722c 2052 2c20 = map (Rbar, R, │ │ │ │ +00060450: 7b78 202c 2078 202c 2078 207d 2920 2020 {x , x , x }) │ │ │ │ +00060460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060490: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000604a0: 2020 2020 2020 2020 2020 2020 2030 2020 0 │ │ │ │ -000604b0: 2031 2020 2032 2020 2020 2020 2020 2020 1 2 │ │ │ │ +00060480: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000604a0: 2020 3020 2020 3120 2020 3220 2020 2020 0 1 2 │ │ │ │ +000604b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000604c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000604d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000604e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000604d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000604e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000604f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060530: 2020 2020 7c0a 7c6f 3137 203a 2052 696e |.|o17 : Rin │ │ │ │ -00060540: 674d 6170 2052 6261 7220 3c2d 2d20 5220 gMap Rbar <-- R │ │ │ │ +00060520: 2020 2020 2020 2020 207c 0a7c 6f31 3720 |.|o17 │ │ │ │ +00060530: 3a20 5269 6e67 4d61 7020 5262 6172 203c : RingMap Rbar < │ │ │ │ +00060540: 2d2d 2052 2020 2020 2020 2020 2020 2020 -- R │ │ │ │ 00060550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060580: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00060570: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00060580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000605a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000605b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000605c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000605d0: 2d2d 2d2d 2b0a 7c69 3138 203a 204b 203d ----+.|i18 : K = │ │ │ │ -000605e0: 2063 6f6b 6572 2076 6172 7320 5262 6172 coker vars Rbar │ │ │ │ +000605c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3820 ---------+.|i18 │ │ │ │ +000605d0: 3a20 4b20 3d20 636f 6b65 7220 7661 7273 : K = coker vars │ │ │ │ +000605e0: 2052 6261 7220 2020 2020 2020 2020 2020 Rbar │ │ │ │ 000605f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060620: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00060610: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060670: 2020 2020 7c0a 7c6f 3138 203d 2063 6f6b |.|o18 = cok │ │ │ │ -00060680: 6572 6e65 6c20 7c20 785f 3020 785f 3120 ernel | x_0 x_1 │ │ │ │ -00060690: 785f 3220 7c20 2020 2020 2020 2020 2020 x_2 | │ │ │ │ +00060660: 2020 2020 2020 2020 207c 0a7c 6f31 3820 |.|o18 │ │ │ │ +00060670: 3d20 636f 6b65 726e 656c 207c 2078 5f30 = cokernel | x_0 │ │ │ │ +00060680: 2078 5f31 2078 5f32 207c 2020 2020 2020 x_1 x_2 | │ │ │ │ +00060690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000606a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000606b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000606c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000606b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000606c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000606d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000606e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000606f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060710: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00060720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060730: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +00060700: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00060720: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ +00060730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060760: 2020 2020 7c0a 7c6f 3138 203a 2052 6261 |.|o18 : Rba │ │ │ │ -00060770: 722d 6d6f 6475 6c65 2c20 7175 6f74 6965 r-module, quotie │ │ │ │ -00060780: 6e74 206f 6620 5262 6172 2020 2020 2020 nt of Rbar │ │ │ │ +00060750: 2020 2020 2020 2020 207c 0a7c 6f31 3820 |.|o18 │ │ │ │ +00060760: 3a20 5262 6172 2d6d 6f64 756c 652c 2071 : Rbar-module, q │ │ │ │ +00060770: 756f 7469 656e 7420 6f66 2052 6261 7220 uotient of Rbar │ │ │ │ +00060780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000607a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000607b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000607a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000607b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000607c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000607d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000607e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000607f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00060800: 2d2d 2d2d 2b0a 7c69 3139 203a 204d 6261 ----+.|i19 : Mba │ │ │ │ -00060810: 7220 3d20 7072 756e 6520 636f 6b65 7220 r = prune coker │ │ │ │ -00060820: 7261 6e64 6f6d 2852 6261 725e 322c 2052 random(Rbar^2, R │ │ │ │ -00060830: 6261 725e 7b2d 322c 2d32 7d29 2020 2020 bar^{-2,-2}) │ │ │ │ -00060840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060850: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000607f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920 ---------+.|i19 │ │ │ │ +00060800: 3a20 4d62 6172 203d 2070 7275 6e65 2063 : Mbar = prune c │ │ │ │ +00060810: 6f6b 6572 2072 616e 646f 6d28 5262 6172 oker random(Rbar │ │ │ │ +00060820: 5e32 2c20 5262 6172 5e7b 2d32 2c2d 327d ^2, Rbar^{-2,-2} │ │ │ │ +00060830: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00060840: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000608a0: 2020 2020 7c0a 7c6f 3139 203d 2063 6f6b |.|o19 = cok │ │ │ │ -000608b0: 6572 6e65 6c20 7c20 785f 3078 5f31 2b31 ernel | x_0x_1+1 │ │ │ │ -000608c0: 3578 5f30 785f 322b 3338 785f 3178 5f32 5x_0x_2+38x_1x_2 │ │ │ │ -000608d0: 2034 3578 5f30 785f 322b 3239 785f 3178 45x_0x_2+29x_1x │ │ │ │ -000608e0: 5f32 2020 2020 2020 2020 7c20 2020 2020 _2 | │ │ │ │ -000608f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00060900: 2020 2020 2020 7c20 3335 785f 3078 5f32 | 35x_0x_2 │ │ │ │ -00060910: 2d33 3078 5f31 785f 3220 2020 2020 2020 -30x_1x_2 │ │ │ │ -00060920: 2078 5f30 785f 312d 3130 785f 3078 5f32 x_0x_1-10x_0x_2 │ │ │ │ -00060930: 2d32 3278 5f31 785f 3220 7c20 2020 2020 -22x_1x_2 | │ │ │ │ -00060940: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00060890: 2020 2020 2020 2020 207c 0a7c 6f31 3920 |.|o19 │ │ │ │ +000608a0: 3d20 636f 6b65 726e 656c 207c 2078 5f30 = cokernel | x_0 │ │ │ │ +000608b0: 785f 312b 3135 785f 3078 5f32 2b33 3878 x_1+15x_0x_2+38x │ │ │ │ +000608c0: 5f31 785f 3220 3435 785f 3078 5f32 2b32 _1x_2 45x_0x_2+2 │ │ │ │ +000608d0: 3978 5f31 785f 3220 2020 2020 2020 207c 9x_1x_2 | │ │ │ │ +000608e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000608f0: 2020 2020 2020 2020 2020 207c 2033 3578 | 35x │ │ │ │ +00060900: 5f30 785f 322d 3330 785f 3178 5f32 2020 _0x_2-30x_1x_2 │ │ │ │ +00060910: 2020 2020 2020 785f 3078 5f31 2d31 3078 x_0x_1-10x │ │ │ │ +00060920: 5f30 785f 322d 3232 785f 3178 5f32 207c _0x_2-22x_1x_2 | │ │ │ │ +00060930: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060990: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000609a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000609b0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00060980: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000609a0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +000609b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000609c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000609d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000609e0: 2020 2020 7c0a 7c6f 3139 203a 2052 6261 |.|o19 : Rba │ │ │ │ -000609f0: 722d 6d6f 6475 6c65 2c20 7175 6f74 6965 r-module, quotie │ │ │ │ -00060a00: 6e74 206f 6620 5262 6172 2020 2020 2020 nt of Rbar │ │ │ │ +000609d0: 2020 2020 2020 2020 207c 0a7c 6f31 3920 |.|o19 │ │ │ │ +000609e0: 3a20 5262 6172 2d6d 6f64 756c 652c 2071 : Rbar-module, q │ │ │ │ +000609f0: 756f 7469 656e 7420 6f66 2052 6261 7220 uotient of Rbar │ │ │ │ +00060a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060a30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00060a20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00060a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00060a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00060a80: 2d2d 2d2d 2b0a 7c69 3230 203a 2045 5320 ----+.|i20 : ES │ │ │ │ -00060a90: 3d20 6e65 7745 7874 284d 6261 722c 4b2c = newExt(Mbar,K, │ │ │ │ -00060aa0: 4c69 6674 203d 3e20 7472 7565 2920 2020 Lift => true) │ │ │ │ +00060a70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020 ---------+.|i20 │ │ │ │ +00060a80: 3a20 4553 203d 206e 6577 4578 7428 4d62 : ES = newExt(Mb │ │ │ │ +00060a90: 6172 2c4b 2c4c 6966 7420 3d3e 2074 7275 ar,K,Lift => tru │ │ │ │ +00060aa0: 6529 2020 2020 2020 2020 2020 2020 2020 e) │ │ │ │ 00060ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060ad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00060ac0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060b20: 2020 2020 7c0a 7c6f 3230 203d 2063 6f6b |.|o20 = cok │ │ │ │ -00060b30: 6572 6e65 6c20 7b30 2c20 307d 2020 207c ernel {0, 0} | │ │ │ │ -00060b40: 2078 5f32 2078 5f31 2078 5f30 2030 2020 x_2 x_1 x_0 0 │ │ │ │ -00060b50: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060b60: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060b70: 2073 5f32 7c0a 7c20 2020 2020 2020 2020 s_2|.| │ │ │ │ -00060b80: 2020 2020 2020 7b30 2c20 307d 2020 207c {0, 0} | │ │ │ │ -00060b90: 2030 2020 2030 2020 2030 2020 2078 5f32 0 0 0 x_2 │ │ │ │ -00060ba0: 2078 5f31 2078 5f30 2030 2020 2030 2020 x_1 x_0 0 0 │ │ │ │ -00060bb0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060bc0: 2030 2020 7c0a 7c20 2020 2020 2020 2020 0 |.| │ │ │ │ -00060bd0: 2020 2020 2020 7b2d 322c 202d 337d 207c {-2, -3} | │ │ │ │ -00060be0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060bf0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060c00: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060c10: 2030 2020 7c0a 7c20 2020 2020 2020 2020 0 |.| │ │ │ │ -00060c20: 2020 2020 2020 7b2d 322c 202d 337d 207c {-2, -3} | │ │ │ │ -00060c30: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060c40: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060c50: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060c60: 2030 2020 7c0a 7c20 2020 2020 2020 2020 0 |.| │ │ │ │ -00060c70: 2020 2020 2020 7b2d 322c 202d 337d 207c {-2, -3} | │ │ │ │ -00060c80: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060c90: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060ca0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060cb0: 2030 2020 7c0a 7c20 2020 2020 2020 2020 0 |.| │ │ │ │ -00060cc0: 2020 2020 2020 7b2d 322c 202d 337d 207c {-2, -3} | │ │ │ │ -00060cd0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060ce0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060cf0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060d00: 2030 2020 7c0a 7c20 2020 2020 2020 2020 0 |.| │ │ │ │ -00060d10: 2020 2020 2020 7b2d 312c 202d 327d 207c {-1, -2} | │ │ │ │ -00060d20: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060d30: 2030 2020 2030 2020 2078 5f32 2078 5f31 0 0 x_2 x_1 │ │ │ │ -00060d40: 2078 5f30 2030 2020 2030 2020 2030 2020 x_0 0 0 0 │ │ │ │ -00060d50: 2030 2020 7c0a 7c20 2020 2020 2020 2020 0 |.| │ │ │ │ -00060d60: 2020 2020 2020 7b2d 312c 202d 327d 207c {-1, -2} | │ │ │ │ -00060d70: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060d80: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060d90: 2030 2020 2078 5f32 2078 5f31 2078 5f30 0 x_2 x_1 x_0 │ │ │ │ -00060da0: 2030 2020 7c0a 7c20 2020 2020 2020 2020 0 |.| │ │ │ │ +00060b10: 2020 2020 2020 2020 207c 0a7c 6f32 3020 |.|o20 │ │ │ │ +00060b20: 3d20 636f 6b65 726e 656c 207b 302c 2030 = cokernel {0, 0 │ │ │ │ +00060b30: 7d20 2020 7c20 785f 3220 785f 3120 785f } | x_2 x_1 x_ │ │ │ │ +00060b40: 3020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 0 │ │ │ │ +00060b50: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060b60: 2020 3020 2020 735f 327c 0a7c 2020 2020 0 s_2|.| │ │ │ │ +00060b70: 2020 2020 2020 2020 2020 207b 302c 2030 {0, 0 │ │ │ │ +00060b80: 7d20 2020 7c20 3020 2020 3020 2020 3020 } | 0 0 0 │ │ │ │ +00060b90: 2020 785f 3220 785f 3120 785f 3020 3020 x_2 x_1 x_0 0 │ │ │ │ +00060ba0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060bb0: 2020 3020 2020 3020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ +00060bc0: 2020 2020 2020 2020 2020 207b 2d32 2c20 {-2, │ │ │ │ +00060bd0: 2d33 7d20 7c20 3020 2020 3020 2020 3020 -3} | 0 0 0 │ │ │ │ +00060be0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060bf0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060c00: 2020 3020 2020 3020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ +00060c10: 2020 2020 2020 2020 2020 207b 2d32 2c20 {-2, │ │ │ │ +00060c20: 2d33 7d20 7c20 3020 2020 3020 2020 3020 -3} | 0 0 0 │ │ │ │ +00060c30: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060c40: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060c50: 2020 3020 2020 3020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ +00060c60: 2020 2020 2020 2020 2020 207b 2d32 2c20 {-2, │ │ │ │ +00060c70: 2d33 7d20 7c20 3020 2020 3020 2020 3020 -3} | 0 0 0 │ │ │ │ +00060c80: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060c90: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060ca0: 2020 3020 2020 3020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ +00060cb0: 2020 2020 2020 2020 2020 207b 2d32 2c20 {-2, │ │ │ │ +00060cc0: 2d33 7d20 7c20 3020 2020 3020 2020 3020 -3} | 0 0 0 │ │ │ │ +00060cd0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060ce0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060cf0: 2020 3020 2020 3020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ +00060d00: 2020 2020 2020 2020 2020 207b 2d31 2c20 {-1, │ │ │ │ +00060d10: 2d32 7d20 7c20 3020 2020 3020 2020 3020 -2} | 0 0 0 │ │ │ │ +00060d20: 2020 3020 2020 3020 2020 3020 2020 785f 0 0 0 x_ │ │ │ │ +00060d30: 3220 785f 3120 785f 3020 3020 2020 3020 2 x_1 x_0 0 0 │ │ │ │ +00060d40: 2020 3020 2020 3020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ +00060d50: 2020 2020 2020 2020 2020 207b 2d31 2c20 {-1, │ │ │ │ +00060d60: 2d32 7d20 7c20 3020 2020 3020 2020 3020 -2} | 0 0 0 │ │ │ │ +00060d70: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060d80: 2020 3020 2020 3020 2020 785f 3220 785f 0 0 x_2 x_ │ │ │ │ +00060d90: 3120 785f 3020 3020 207c 0a7c 2020 2020 1 x_0 0 |.| │ │ │ │ +00060da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060df0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00060de0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060e30: 2020 2020 2020 2020 3820 2020 2020 2020 8 │ │ │ │ -00060e40: 2020 2020 7c0a 7c6f 3230 203a 206b 6b5b |.|o20 : kk[ │ │ │ │ -00060e50: 7320 2e2e 7320 2c20 7820 2e2e 7820 5d2d s ..s , x ..x ]- │ │ │ │ -00060e60: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74 module, quotient │ │ │ │ -00060e70: 206f 6620 286b 6b5b 7320 2e2e 7320 2c20 of (kk[s ..s , │ │ │ │ -00060e80: 7820 2e2e 7820 5d29 2020 2020 2020 2020 x ..x ]) │ │ │ │ -00060e90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00060ea0: 2030 2020 2032 2020 2030 2020 2032 2020 0 2 0 2 │ │ │ │ -00060eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060ec0: 2020 2020 2020 2020 2030 2020 2032 2020 0 2 │ │ │ │ -00060ed0: 2030 2020 2032 2020 2020 2020 2020 2020 0 2 │ │ │ │ -00060ee0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d |.|--------- │ │ │ │ +00060e20: 2020 2020 2020 2020 2020 2020 2038 2020 8 │ │ │ │ +00060e30: 2020 2020 2020 2020 207c 0a7c 6f32 3020 |.|o20 │ │ │ │ +00060e40: 3a20 6b6b 5b73 202e 2e73 202c 2078 202e : kk[s ..s , x . │ │ │ │ +00060e50: 2e78 205d 2d6d 6f64 756c 652c 2071 756f .x ]-module, quo │ │ │ │ +00060e60: 7469 656e 7420 6f66 2028 6b6b 5b73 202e tient of (kk[s . │ │ │ │ +00060e70: 2e73 202c 2078 202e 2e78 205d 2920 2020 .s , x ..x ]) │ │ │ │ +00060e80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060e90: 2020 2020 2020 3020 2020 3220 2020 3020 0 2 0 │ │ │ │ +00060ea0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00060eb0: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ +00060ec0: 2020 3220 2020 3020 2020 3220 2020 2020 2 0 2 │ │ │ │ +00060ed0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d |.|---- │ │ │ │ +00060ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00060f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00060f30: 2d2d 2d2d 7c0a 7c73 5f31 2073 5f30 2030 ----|.|s_1 s_0 0 │ │ │ │ -00060f40: 2020 2030 2020 2030 2020 2030 2020 2030 0 0 0 0 │ │ │ │ -00060f50: 2020 2030 2020 2030 2020 2030 2020 2030 0 0 0 0 │ │ │ │ -00060f60: 2020 2030 2020 2030 2020 2030 2020 2030 0 0 0 0 │ │ │ │ -00060f70: 2020 2030 2020 2030 2020 2020 2020 2020 0 0 │ │ │ │ -00060f80: 2020 2020 7c0a 7c30 2020 2030 2020 2073 |.|0 0 s │ │ │ │ -00060f90: 5f32 2073 5f31 2073 5f30 2030 2020 2030 _2 s_1 s_0 0 0 │ │ │ │ -00060fa0: 2020 2030 2020 2030 2020 2030 2020 2030 0 0 0 0 │ │ │ │ -00060fb0: 2020 2030 2020 2030 2020 2030 2020 2030 0 0 0 0 │ │ │ │ -00060fc0: 2020 2030 2020 2030 2020 2020 2020 2020 0 0 │ │ 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2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061a70: 2d2d 2d2d 2b0a 7c69 3231 203a 2053 203d ----+.|i21 : S = │ │ │ │ -00061a80: 2072 696e 6720 4553 2020 2020 2020 2020 ring ES │ │ │ │ +00061a60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 ---------+.|i21 │ │ │ │ +00061a70: 3a20 5320 3d20 7269 6e67 2045 5320 2020 : S = ring ES │ │ │ │ +00061a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061ac0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00061ab0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00061ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061b10: 2020 2020 7c0a 7c6f 3231 203d 2053 2020 |.|o21 = S │ │ │ │ +00061b00: 2020 2020 2020 2020 207c 0a7c 6f32 3120 |.|o21 │ │ │ │ +00061b10: 3d20 5320 2020 2020 2020 2020 2020 2020 = S │ │ │ │ 00061b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061b60: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00061b50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00061b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061bb0: 2020 2020 7c0a 7c6f 3231 203a 2050 6f6c |.|o21 : Pol │ │ │ │ -00061bc0: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ +00061ba0: 2020 2020 2020 2020 207c 0a7c 6f32 3120 |.|o21 │ │ │ │ +00061bb0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +00061bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061c00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00061bf0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00061c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061c50: 2d2d 2d2d 2b0a 0a63 6f6d 7061 7265 2077 ----+..compare w │ │ │ │ -00061c60: 6974 6820 7468 6520 6275 696c 742d 696e ith the built-in │ │ │ │ -00061c70: 2045 7874 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d Ext..+--------- │ │ │ │ +00061c40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 636f 6d70 ---------+..comp │ │ │ │ +00061c50: 6172 6520 7769 7468 2074 6865 2062 7569 are with the bui │ │ │ │ +00061c60: 6c74 2d69 6e20 4578 740a 0a2b 2d2d 2d2d lt-in Ext..+---- │ │ │ │ +00061c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061cb0: 2b0a 7c69 3232 203a 2045 4520 3d20 4578 +.|i22 : EE = Ex │ │ │ │ -00061cc0: 7428 4d62 6172 2c4b 293b 2020 2020 2020 t(Mbar,K); │ │ │ │ +00061ca0: 2d2d 2d2d 2d2b 0a7c 6932 3220 3a20 4545 -----+.|i22 : EE │ │ │ │ +00061cb0: 203d 2045 7874 284d 6261 722c 4b29 3b20 = Ext(Mbar,K); │ │ │ │ +00061cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061ce0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00061ce0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00061cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061d20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a --------+.|i23 : │ │ │ │ -00061d30: 2053 2720 3d20 7269 6e67 2045 4520 2d2d S' = ring EE -- │ │ │ │ -00061d40: 206e 6f74 6520 7468 6174 2053 2720 6973 note that S' is │ │ │ │ -00061d50: 2074 6865 2070 6f6c 796e 6f6d 6961 6c20 the polynomial │ │ │ │ -00061d60: 7269 6e67 7c0a 7c20 2020 2020 2020 2020 ring|.| │ │ │ │ +00061d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00061d20: 6932 3320 3a20 5327 203d 2072 696e 6720 i23 : S' = ring │ │ │ │ +00061d30: 4545 202d 2d20 6e6f 7465 2074 6861 7420 EE -- note that │ │ │ │ +00061d40: 5327 2069 7320 7468 6520 706f 6c79 6e6f S' is the polyno │ │ │ │ +00061d50: 6d69 616c 2072 696e 677c 0a7c 2020 2020 mial ring|.| │ │ │ │ +00061d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061da0: 7c0a 7c6f 3233 203d 2053 2720 2020 2020 |.|o23 = S' │ │ │ │ +00061d90: 2020 2020 207c 0a7c 6f32 3320 3d20 5327 |.|o23 = S' │ │ │ │ +00061da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061dd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00061dd0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00061de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061e10: 2020 2020 2020 2020 7c0a 7c6f 3233 203a |.|o23 : │ │ │ │ -00061e20: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ +00061e00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00061e10: 6f32 3320 3a20 506f 6c79 6e6f 6d69 616c o23 : Polynomial │ │ │ │ +00061e20: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ 00061e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061e50: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00061e40: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00061e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061e90: 2b0a 0a54 6865 2074 776f 2076 6572 7369 +..The two versi │ │ │ │ -00061ea0: 6f6e 7320 6f66 2045 7874 2061 7070 6561 ons of Ext appea │ │ │ │ -00061eb0: 7220 746f 2062 6520 7468 6520 7361 6d65 r to be the same │ │ │ │ -00061ec0: 2075 7020 746f 2063 6861 6e67 6520 6f66 up to change of │ │ │ │ -00061ed0: 2076 6172 6961 626c 6573 3a0a 0a2b 2d2d variables:..+-- │ │ │ │ +00061e80: 2d2d 2d2d 2d2b 0a0a 5468 6520 7477 6f20 -----+..The two │ │ │ │ +00061e90: 7665 7273 696f 6e73 206f 6620 4578 7420 versions of Ext │ │ │ │ +00061ea0: 6170 7065 6172 2074 6f20 6265 2074 6865 appear to be the │ │ │ │ +00061eb0: 2073 616d 6520 7570 2074 6f20 6368 616e same up to chan │ │ │ │ +00061ec0: 6765 206f 6620 7661 7269 6162 6c65 733a ge of variables: │ │ │ │ +00061ed0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ 00061ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00061f20: 6932 3420 3a20 4120 3d20 6672 6565 5265 i24 : A = freeRe │ │ │ │ -00061f30: 736f 6c75 7469 6f6e 2045 5320 2020 2020 solution ES │ │ │ │ +00061f10: 2d2d 2b0a 7c69 3234 203a 2041 203d 2066 --+.|i24 : A = f │ │ │ │ +00061f20: 7265 6552 6573 6f6c 7574 696f 6e20 4553 reeResolution ES │ │ │ │ +00061f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061f50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00061f60: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00061f50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00061f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061fa0: 207c 0a7c 2020 2020 2020 2038 2020 2020 |.| 8 │ │ │ │ -00061fb0: 2020 3336 2020 2020 2020 3636 2020 2020 36 66 │ │ │ │ -00061fc0: 2020 3634 2020 2020 2020 3336 2020 2020 64 36 │ │ │ │ -00061fd0: 2020 3132 2020 2020 2020 3220 2020 2020 12 2 │ │ │ │ -00061fe0: 2020 207c 0a7c 6f32 3420 3d20 5320 203c |.|o24 = S < │ │ │ │ -00061ff0: 2d2d 2053 2020 203c 2d2d 2053 2020 203c -- S <-- S < │ │ │ │ -00062000: 2d2d 2053 2020 203c 2d2d 2053 2020 203c -- S <-- S < │ │ │ │ -00062010: 2d2d 2053 2020 203c 2d2d 2053 2020 2020 -- S <-- S │ │ │ │ -00062020: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00061f90: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00061fa0: 3820 2020 2020 2033 3620 2020 2020 2036 8 36 6 │ │ │ │ +00061fb0: 3620 2020 2020 2036 3420 2020 2020 2033 6 64 3 │ │ │ │ +00061fc0: 3620 2020 2020 2031 3220 2020 2020 2032 6 12 2 │ │ │ │ +00061fd0: 2020 2020 2020 2020 7c0a 7c6f 3234 203d |.|o24 = │ │ │ │ +00061fe0: 2053 2020 3c2d 2d20 5320 2020 3c2d 2d20 S <-- S <-- │ │ │ │ +00061ff0: 5320 2020 3c2d 2d20 5320 2020 3c2d 2d20 S <-- S <-- │ │ │ │ +00062000: 5320 2020 3c2d 2d20 5320 2020 3c2d 2d20 S <-- S <-- │ │ │ │ +00062010: 5320 2020 2020 2020 2020 7c0a 7c20 2020 S |.| │ │ │ │ +00062020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062060: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00062070: 3020 2020 2020 2031 2020 2020 2020 2032 0 1 2 │ │ │ │ -00062080: 2020 2020 2020 2033 2020 2020 2020 2034 3 4 │ │ │ │ -00062090: 2020 2020 2020 2035 2020 2020 2020 2036 5 6 │ │ │ │ -000620a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00062050: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00062060: 2020 2020 2030 2020 2020 2020 3120 2020 0 1 │ │ │ │ +00062070: 2020 2020 3220 2020 2020 2020 3320 2020 2 3 │ │ │ │ +00062080: 2020 2020 3420 2020 2020 2020 3520 2020 4 5 │ │ │ │ +00062090: 2020 2020 3620 2020 2020 2020 2020 7c0a 6 |. │ │ │ │ +000620a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000620b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000620c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000620d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000620e0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -000620f0: 3420 3a20 436f 6d70 6c65 7820 2020 2020 4 : Complex │ │ │ │ +000620e0: 7c0a 7c6f 3234 203a 2043 6f6d 706c 6578 |.|o24 : Complex │ │ │ │ +000620f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062120: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00062120: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00062130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00062170: 0a7c 6932 3520 3a20 4220 3d20 6672 6565 .|i25 : B = free │ │ │ │ -00062180: 5265 736f 6c75 7469 6f6e 2045 4520 2020 Resolution EE │ │ │ │ +00062160: 2d2d 2d2d 2b0a 7c69 3235 203a 2042 203d ----+.|i25 : B = │ │ │ │ +00062170: 2066 7265 6552 6573 6f6c 7574 696f 6e20 freeResolution │ │ │ │ +00062180: 4545 2020 2020 2020 2020 2020 2020 2020 EE │ │ │ │ 00062190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000621a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000621b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000621a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000621b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000621c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000621d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000621e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000621f0: 2020 207c 0a7c 2020 2020 2020 2020 3820 |.| 8 │ │ │ │ -00062200: 2020 2020 2020 3336 2020 2020 2020 2036 36 6 │ │ │ │ -00062210: 3620 2020 2020 2020 3634 2020 2020 2020 6 64 │ │ │ │ -00062220: 2033 3620 2020 2020 2020 3132 2020 2020 36 12 │ │ │ │ -00062230: 2020 2032 207c 0a7c 6f32 3520 3d20 5327 2 |.|o25 = S' │ │ │ │ -00062240: 2020 3c2d 2d20 5327 2020 203c 2d2d 2053 <-- S' <-- S │ │ │ │ -00062250: 2720 2020 3c2d 2d20 5327 2020 203c 2d2d ' <-- S' <-- │ │ │ │ -00062260: 2053 2720 2020 3c2d 2d20 5327 2020 203c S' <-- S' < │ │ │ │ -00062270: 2d2d 2053 2720 207c 0a7c 2020 2020 2020 -- S' |.| │ │ │ │ +000621e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000621f0: 2020 2038 2020 2020 2020 2033 3620 2020 8 36 │ │ │ │ +00062200: 2020 2020 3636 2020 2020 2020 2036 3420 66 64 │ │ │ │ +00062210: 2020 2020 2020 3336 2020 2020 2020 2031 36 1 │ │ │ │ +00062220: 3220 2020 2020 2020 3220 7c0a 7c6f 3235 2 2 |.|o25 │ │ │ │ +00062230: 203d 2053 2720 203c 2d2d 2053 2720 2020 = S' <-- S' │ │ │ │ +00062240: 3c2d 2d20 5327 2020 203c 2d2d 2053 2720 <-- S' <-- S' │ │ │ │ +00062250: 2020 3c2d 2d20 5327 2020 203c 2d2d 2053 <-- S' <-- S │ │ │ │ +00062260: 2720 2020 3c2d 2d20 5327 2020 7c0a 7c20 ' <-- S' |.| │ │ │ │ +00062270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000622a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000622b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -000622c0: 2020 3020 2020 2020 2020 3120 2020 2020 0 1 │ │ │ │ -000622d0: 2020 2032 2020 2020 2020 2020 3320 2020 2 3 │ │ │ │ -000622e0: 2020 2020 2034 2020 2020 2020 2020 3520 4 5 │ │ │ │ -000622f0: 2020 2020 2020 2036 2020 207c 0a7c 2020 6 |.| │ │ │ │ +000622a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000622b0: 7c20 2020 2020 2030 2020 2020 2020 2031 | 0 1 │ │ │ │ +000622c0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +000622d0: 2033 2020 2020 2020 2020 3420 2020 2020 3 4 │ │ │ │ +000622e0: 2020 2035 2020 2020 2020 2020 3620 2020 5 6 │ │ │ │ +000622f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00062300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062330: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00062340: 6f32 3520 3a20 436f 6d70 6c65 7820 2020 o25 : Complex │ │ │ │ +00062330: 2020 7c0a 7c6f 3235 203a 2043 6f6d 706c |.|o25 : Compl │ │ │ │ +00062340: 6578 2020 2020 2020 2020 2020 2020 2020 ex │ │ │ │ 00062350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062370: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00062380: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00062370: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00062380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000623a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000623b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000623c0: 2d2b 0a7c 6932 3620 3a20 616c 6c28 6c65 -+.|i26 : all(le │ │ │ │ -000623d0: 6e67 7468 2041 2b31 2c20 692d 3e20 736f ngth A+1, i-> so │ │ │ │ -000623e0: 7274 2064 6567 7265 6573 2041 5f69 203d rt degrees A_i = │ │ │ │ -000623f0: 3d20 736f 7274 2064 6567 7265 6573 2042 = sort degrees B │ │ │ │ -00062400: 5f69 297c 0a7c 2020 2020 2020 2020 2020 _i)|.| │ │ │ │ +000623b0: 2d2d 2d2d 2d2d 2b0a 7c69 3236 203a 2061 ------+.|i26 : a │ │ │ │ +000623c0: 6c6c 286c 656e 6774 6820 412b 312c 2069 ll(length A+1, i │ │ │ │ +000623d0: 2d3e 2073 6f72 7420 6465 6772 6565 7320 -> sort degrees │ │ │ │ +000623e0: 415f 6920 3d3d 2073 6f72 7420 6465 6772 A_i == sort degr │ │ │ │ +000623f0: 6565 7320 425f 6929 7c0a 7c20 2020 2020 ees B_i)|.| │ │ │ │ +00062400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062440: 2020 2020 207c 0a7c 6f32 3620 3d20 7472 |.|o26 = tr │ │ │ │ -00062450: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ +00062430: 2020 2020 2020 2020 2020 7c0a 7c6f 3236 |.|o26 │ │ │ │ +00062440: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ +00062450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062480: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00062470: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00062480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000624a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000624b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000624c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 6275 7420 ---------+..but │ │ │ │ -000624d0: 7468 6579 2068 6176 6520 6170 7061 7265 they have appare │ │ │ │ -000624e0: 6e74 6c79 2064 6966 6665 7265 6e74 2061 ntly different a │ │ │ │ -000624f0: 6e6e 6968 696c 6174 6f72 730a 0a2b 2d2d nnihilators..+-- │ │ │ │ +000624b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000624c0: 0a62 7574 2074 6865 7920 6861 7665 2061 .but they have a │ │ │ │ +000624d0: 7070 6172 656e 746c 7920 6469 6666 6572 pparently differ │ │ │ │ +000624e0: 656e 7420 616e 6e69 6869 6c61 746f 7273 ent annihilators │ │ │ │ +000624f0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ 00062500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062540: 2d2d 2d2b 0a7c 6932 3720 3a20 616e 6e20 ---+.|i27 : ann │ │ │ │ -00062550: 4545 2020 2020 2020 2020 2020 2020 2020 EE │ │ │ │ +00062530: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3237 203a --------+.|i27 : │ │ │ │ +00062540: 2061 6e6e 2045 4520 2020 2020 2020 2020 ann EE │ │ │ │ +00062550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062580: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00062580: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00062590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000625a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000625b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000625c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000625d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -000625e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000625f0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00062600: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00062610: 2020 2020 2020 2020 2032 207c 0a7c 6f32 2 |.|o2 │ │ │ │ -00062620: 3720 3d20 6964 6561 6c20 2878 202c 2078 7 = ideal (x , x │ │ │ │ -00062630: 202c 2078 202c 2058 2020 2b20 3431 5820 , x , X + 41X │ │ │ │ -00062640: 5820 202d 2033 3758 2020 2d20 3134 5820 X - 37X - 14X │ │ │ │ -00062650: 5820 202d 2032 3958 2058 2020 2b20 3435 X - 29X X + 45 │ │ │ │ -00062660: 5820 297c 0a7c 2020 2020 2020 2020 2020 X )|.| │ │ │ │ -00062670: 2020 2020 3220 2020 3120 2020 3020 2020 2 1 0 │ │ │ │ -00062680: 3120 2020 2020 2031 2032 2020 2020 2020 1 1 2 │ │ │ │ -00062690: 3220 2020 2020 2031 2033 2020 2020 2020 2 1 3 │ │ │ │ -000626a0: 3220 3320 2020 2020 2033 207c 0a7c 2020 2 3 3 |.| │ │ │ │ +000625c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000625d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000625e0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +000625f0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00062600: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +00062610: 7c0a 7c6f 3237 203d 2069 6465 616c 2028 |.|o27 = ideal ( │ │ │ │ +00062620: 7820 2c20 7820 2c20 7820 2c20 5820 202b x , x , x , X + │ │ │ │ +00062630: 2034 3158 2058 2020 2d20 3337 5820 202d 41X X - 37X - │ │ │ │ +00062640: 2031 3458 2058 2020 2d20 3239 5820 5820 14X X - 29X X │ │ │ │ +00062650: 202b 2034 3558 2029 7c0a 7c20 2020 2020 + 45X )|.| │ │ │ │ +00062660: 2020 2020 2020 2020 2032 2020 2031 2020 2 1 │ │ │ │ +00062670: 2030 2020 2031 2020 2020 2020 3120 3220 0 1 1 2 │ │ │ │ +00062680: 2020 2020 2032 2020 2020 2020 3120 3320 2 1 3 │ │ │ │ +00062690: 2020 2020 2032 2033 2020 2020 2020 3320 2 3 3 │ │ │ │ +000626a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000626b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000626c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000626d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000626e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000626f0: 2020 207c 0a7c 6f32 3720 3a20 4964 6561 |.|o27 : Idea │ │ │ │ -00062700: 6c20 6f66 2053 2720 2020 2020 2020 2020 l of S' │ │ │ │ +000626e0: 2020 2020 2020 2020 7c0a 7c6f 3237 203a |.|o27 : │ │ │ │ +000626f0: 2049 6465 616c 206f 6620 5327 2020 2020 Ideal of S' │ │ │ │ +00062700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062730: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00062730: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00062740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062780: 2d2d 2d2b 0a7c 6932 3820 3a20 616e 6e20 ---+.|i28 : ann │ │ │ │ -00062790: 4553 2020 2020 2020 2020 2020 2020 2020 ES │ │ │ │ +00062770: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3238 203a --------+.|i28 : │ │ │ │ +00062780: 2061 6e6e 2045 5320 2020 2020 2020 2020 ann ES │ │ │ │ +00062790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000627a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000627b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000627c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000627c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000627d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000627e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000627f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062810: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00062820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062830: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00062840: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00062850: 2020 2020 2020 2020 3220 207c 0a7c 6f32 2 |.|o2 │ │ │ │ -00062860: 3820 3d20 6964 6561 6c20 2878 202c 2078 8 = ideal (x , x │ │ │ │ -00062870: 202c 2078 202c 2073 2020 2b20 3432 7320 , x , s + 42s │ │ │ │ -00062880: 7320 202d 2033 3073 2020 2d20 3235 7320 s - 30s - 25s │ │ │ │ -00062890: 7320 202d 2033 3573 2073 2020 2b20 3973 s - 35s s + 9s │ │ │ │ -000628a0: 2029 207c 0a7c 2020 2020 2020 2020 2020 ) |.| │ │ │ │ -000628b0: 2020 2020 3220 2020 3120 2020 3020 2020 2 1 0 │ │ │ │ -000628c0: 3020 2020 2020 2030 2031 2020 2020 2020 0 0 1 │ │ │ │ -000628d0: 3120 2020 2020 2030 2032 2020 2020 2020 1 0 2 │ │ │ │ -000628e0: 3120 3220 2020 2020 3220 207c 0a7c 2020 1 2 2 |.| │ │ │ │ +00062800: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00062810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00062820: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00062830: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00062840: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00062850: 7c0a 7c6f 3238 203d 2069 6465 616c 2028 |.|o28 = ideal ( │ │ │ │ +00062860: 7820 2c20 7820 2c20 7820 2c20 7320 202b x , x , x , s + │ │ │ │ +00062870: 2034 3273 2073 2020 2d20 3330 7320 202d 42s s - 30s - │ │ │ │ +00062880: 2032 3573 2073 2020 2d20 3335 7320 7320 25s s - 35s s │ │ │ │ +00062890: 202b 2039 7320 2920 7c0a 7c20 2020 2020 + 9s ) |.| │ │ │ │ +000628a0: 2020 2020 2020 2020 2032 2020 2031 2020 2 1 │ │ │ │ +000628b0: 2030 2020 2030 2020 2020 2020 3020 3120 0 0 0 1 │ │ │ │ +000628c0: 2020 2020 2031 2020 2020 2020 3020 3220 1 0 2 │ │ │ │ +000628d0: 2020 2020 2031 2032 2020 2020 2032 2020 1 2 2 │ │ │ │ +000628e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000628f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062930: 2020 207c 0a7c 6f32 3820 3a20 4964 6561 |.|o28 : Idea │ │ │ │ -00062940: 6c20 6f66 2053 2020 2020 2020 2020 2020 l of S │ │ │ │ +00062920: 2020 2020 2020 2020 7c0a 7c6f 3238 203a |.|o28 : │ │ │ │ +00062930: 2049 6465 616c 206f 6620 5320 2020 2020 Ideal of S │ │ │ │ +00062940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062970: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00062970: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00062980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000629a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000629b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000629c0: 2d2d 2d2b 0a0a 616e 6420 696e 2066 6163 ---+..and in fac │ │ │ │ -000629d0: 7420 7468 6579 2061 7265 206e 6f74 2069 t they are not i │ │ │ │ -000629e0: 736f 6d6f 7270 6869 633a 0a0a 2b2d 2d2d somorphic:..+--- │ │ │ │ +000629b0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a61 6e64 2069 --------+..and i │ │ │ │ +000629c0: 6e20 6661 6374 2074 6865 7920 6172 6520 n fact they are │ │ │ │ +000629d0: 6e6f 7420 6973 6f6d 6f72 7068 6963 3a0a not isomorphic:. │ │ │ │ +000629e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 000629f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3239 ----------+.|i29 │ │ │ │ -00062a40: 203a 2045 4574 6f45 5320 3d20 6d61 7028 : EEtoES = map( │ │ │ │ -00062a50: 7269 6e67 2045 532c 7269 6e67 2045 452c ring ES,ring EE, │ │ │ │ -00062a60: 2067 656e 7320 7269 6e67 2045 5329 2020 gens ring ES) │ │ │ │ -00062a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062a80: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00062a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00062a30: 0a7c 6932 3920 3a20 4545 746f 4553 203d .|i29 : EEtoES = │ │ │ │ +00062a40: 206d 6170 2872 696e 6720 4553 2c72 696e map(ring ES,rin │ │ │ │ +00062a50: 6720 4545 2c20 6765 6e73 2072 696e 6720 g EE, gens ring │ │ │ │ +00062a60: 4553 2920 2020 2020 2020 2020 2020 2020 ES) │ │ │ │ +00062a70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00062a80: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00062a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062ad0: 2020 2020 2020 2020 2020 7c0a 7c6f 3239 |.|o29 │ │ │ │ -00062ae0: 203d 206d 6170 2028 532c 2053 272c 207b = map (S, S', { │ │ │ │ -00062af0: 7320 2c20 7320 2c20 7320 2c20 7820 2c20 s , s , s , x , │ │ │ │ -00062b00: 7820 2c20 7820 7d29 2020 2020 2020 2020 x , x }) │ │ │ │ -00062b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062b20: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00062b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062b40: 2030 2020 2031 2020 2032 2020 2030 2020 0 1 2 0 │ │ │ │ -00062b50: 2031 2020 2032 2020 2020 2020 2020 2020 1 2 │ │ │ │ -00062b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062b70: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00062ac0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00062ad0: 0a7c 6f32 3920 3d20 6d61 7020 2853 2c20 .|o29 = map (S, │ │ │ │ +00062ae0: 5327 2c20 7b73 202c 2073 202c 2073 202c S', {s , s , s , │ │ │ │ +00062af0: 2078 202c 2078 202c 2078 207d 2920 2020 x , x , x }) │ │ │ │ +00062b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00062b10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00062b20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00062b30: 2020 2020 2020 3020 2020 3120 2020 3220 0 1 2 │ │ │ │ +00062b40: 2020 3020 2020 3120 2020 3220 2020 2020 0 1 2 │ │ │ │ +00062b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00062b60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00062b70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00062b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062bc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3239 |.|o29 │ │ │ │ -00062bd0: 203a 2052 696e 674d 6170 2053 203c 2d2d : RingMap S <-- │ │ │ │ -00062be0: 2053 2720 2020 2020 2020 2020 2020 2020 S' │ │ │ │ +00062bb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00062bc0: 0a7c 6f32 3920 3a20 5269 6e67 4d61 7020 .|o29 : RingMap │ │ │ │ +00062bd0: 5320 3c2d 2d20 5327 2020 2020 2020 2020 S <-- S' │ │ │ │ +00062be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062c10: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00062c00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00062c10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00062c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3330 ----------+.|i30 │ │ │ │ -00062c70: 203a 2045 4527 203d 2063 6f6b 6572 2045 : EE' = coker E │ │ │ │ -00062c80: 4574 6f45 5320 7072 6573 656e 7461 7469 EtoES presentati │ │ │ │ -00062c90: 6f6e 2045 4520 2020 2020 2020 2020 2020 on EE │ │ │ │ -00062ca0: 2020 2020 2020 2020 2020 2020 2020 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2020 2020 .|0 │ │ │ │ 00063ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063af0: 2020 2020 2020 2020 2020 2020 2020 7c20 | │ │ │ │ -00063b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063b10: 2020 2020 2020 2020 2020 7c0a 7c30 2020 |.|0 │ │ │ │ +00063af0: 2020 207c 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00063b00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063b10: 0a7c 3020 2020 2020 2020 2020 2020 2020 .|0 │ │ │ │ 00063b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063b40: 2020 2020 2020 2020 2020 2020 2020 7c20 | │ │ │ │ -00063b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063b60: 2020 2020 2020 2020 2020 7c0a 7c30 2020 |.|0 │ │ │ │ +00063b40: 2020 207c 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00063b50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063b60: 0a7c 3020 2020 2020 2020 2020 2020 2020 .|0 │ │ │ │ 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2020 S); │ │ │ │ +00063bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00063c00: 0a7c 6933 3120 3a20 4820 3d20 486f 6d28 .|i31 : H = Hom( │ │ │ │ +00063c10: 4545 272c 4553 293b 2020 2020 2020 2020 EE',ES); │ │ │ │ +00063c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063c50: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00063c40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063c50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00063c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00063c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00063c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00063c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00063ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3332 ----------+.|i32 │ │ │ │ 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2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063d40: 2020 2020 2020 2020 2020 7c0a 7c6f 3332 |.|o32 │ │ │ │ -00063d50: 203d 207b 382c 2039 2c20 3130 2c20 3131 = {8, 9, 10, 11 │ │ │ │ -00063d60: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +00063d30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063d40: 0a7c 6f33 3220 3d20 7b38 2c20 392c 2031 .|o32 = {8, 9, 1 │ │ │ │ +00063d50: 302c 2031 317d 2020 2020 2020 2020 2020 0, 11} │ │ │ │ +00063d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063d90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00063d80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063d90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00063da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063de0: 2020 2020 2020 2020 2020 7c0a 7c6f 3332 |.|o32 │ │ │ │ -00063df0: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +00063dd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063de0: 0a7c 6f33 3220 3a20 4c69 7374 2020 2020 .|o32 : List │ │ │ │ +00063df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063e30: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00063e20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063e30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00063e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00063e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00063e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00063e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00063e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3333 ----------+.|i33 │ │ │ │ -00063e90: 203a 2066 203d 2073 756d 2851 2c20 702d : f = sum(Q, p- │ │ │ │ -00063ea0: 3e20 7261 6e64 6f6d 2028 535e 312c 2053 > random (S^1, S │ │ │ │ -00063eb0: 5e31 292a 2a68 6f6d 6f6d 6f72 7068 6973 ^1)**homomorphis │ │ │ │ -00063ec0: 6d20 485f 7b70 7d29 2020 2020 2020 2020 m H_{p}) │ │ │ │ -00063ed0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00063e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00063e80: 0a7c 6933 3320 3a20 6620 3d20 7375 6d28 .|i33 : f = sum( │ │ │ │ +00063e90: 512c 2070 2d3e 2072 616e 646f 6d20 2853 Q, p-> random (S │ │ │ │ +00063ea0: 5e31 2c20 535e 3129 2a2a 686f 6d6f 6d6f ^1, S^1)**homomo │ │ │ │ +00063eb0: 7270 6869 736d 2048 5f7b 707d 2920 2020 rphism H_{p}) │ │ │ │ +00063ec0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063ed0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00063ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063f20: 2020 2020 2020 2020 2020 7c0a 7c6f 3333 |.|o33 │ │ │ │ -00063f30: 203d 207b 302c 2030 7d20 2020 7c20 2d33 = {0, 0} | -3 │ │ │ │ -00063f40: 3820 3339 2030 2030 2030 2030 2030 2030 8 39 0 0 0 0 0 0 │ │ │ │ -00063f50: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00063f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063f70: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00063f80: 2020 207b 302c 2030 7d20 2020 7c20 2d31 {0, 0} | -1 │ │ │ │ -00063f90: 3620 3231 2030 2030 2030 2030 2030 2030 6 21 0 0 0 0 0 0 │ │ │ │ -00063fa0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00063fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063fc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00063fd0: 2020 207b 2d32 2c20 2d33 7d20 7c20 3020 {-2, -3} | 0 │ │ │ │ -00063fe0: 2020 3020 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -00063ff0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00064000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064010: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00064020: 2020 207b 2d32 2c20 2d33 7d20 7c20 3020 {-2, -3} | 0 │ │ │ │ -00064030: 2020 3020 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -00064040: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00064050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064060: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00064070: 2020 207b 2d32 2c20 2d33 7d20 7c20 3020 {-2, -3} | 0 │ │ │ │ -00064080: 2020 3020 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -00064090: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000640a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000640b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -000640c0: 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2020 2020 2020 2020 207c | │ │ │ │ +00063f20: 0a7c 6f33 3320 3d20 7b30 2c20 307d 2020 .|o33 = {0, 0} │ │ │ │ +00063f30: 207c 202d 3338 2033 3920 3020 3020 3020 | -38 39 0 0 0 │ │ │ │ +00063f40: 3020 3020 3020 7c20 2020 2020 2020 2020 0 0 0 | │ │ │ │ +00063f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00063f60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063f70: 0a7c 2020 2020 2020 7b30 2c20 307d 2020 .| {0, 0} │ │ │ │ +00063f80: 207c 202d 3136 2032 3120 3020 3020 3020 | -16 21 0 0 0 │ │ │ │ +00063f90: 3020 3020 3020 7c20 2020 2020 2020 2020 0 0 0 | │ │ │ │ +00063fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00063fb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063fc0: 0a7c 2020 2020 2020 7b2d 322c 202d 337d .| {-2, -3} │ │ │ │ +00063fd0: 207c 2030 2020 2030 2020 3020 3020 3020 | 0 0 0 0 0 │ │ │ │ +00063fe0: 3020 3020 3020 7c20 2020 2020 2020 2020 0 0 0 | │ │ │ │ +00063ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00064000: 2020 2020 2020 2020 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2020 2020 7c0a 7c6f 3333 |.|o33 │ │ │ │ -00064200: 203a 204d 6174 7269 7820 4553 203c 2d2d : Matrix ES <-- │ │ │ │ -00064210: 2045 4527 2020 2020 2020 2020 2020 2020 EE' │ │ │ │ +000641e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000641f0: 0a7c 6f33 3320 3a20 4d61 7472 6978 2045 .|o33 : Matrix E │ │ │ │ +00064200: 5320 3c2d 2d20 4545 2720 2020 2020 2020 S <-- EE' │ │ │ │ +00064210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064240: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00064230: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00064240: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00064250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064290: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6620 ----------+..If │ │ │ │ -000642a0: 4545 2061 6e64 2045 5320 7765 7265 2069 EE and ES were i │ │ │ │ -000642b0: 736f 6d6f 7270 6869 632c 2077 6520 776f somorphic, we wo │ │ │ │ -000642c0: 756c 6420 6578 7065 6374 2063 6f6b 6572 uld expect coker │ │ │ │ -000642d0: 2066 2074 6f20 6265 2030 2c20 616e 6420 f to be 0, and │ │ │ │ -000642e0: 6974 2773 206e 6f74 2e0a 7072 756e 6520 it's not..prune │ │ │ │ -000642f0: 636f 6b65 7220 660a 0a53 6565 2061 6c73 coker f..See als │ │ │ │ -00064300: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ -00064310: 2a6e 6f74 6520 4578 743a 2028 4d61 6361 *note Ext: (Maca │ │ │ │ -00064320: 756c 6179 3244 6f63 2945 7874 2c20 2d2d ulay2Doc)Ext, -- │ │ │ │ -00064330: 2063 6f6d 7075 7465 2061 6e20 4578 7420 compute an Ext │ │ │ │ -00064340: 6d6f 6475 6c65 0a20 202a 202a 6e6f 7465 module. * *note │ │ │ │ -00064350: 2045 6973 656e 6275 6453 6861 6d61 7368 EisenbudShamash │ │ │ │ -00064360: 546f 7461 6c3a 2045 6973 656e 6275 6453 Total: EisenbudS │ │ │ │ -00064370: 6861 6d61 7368 546f 7461 6c2c 202d 2d20 hamashTotal, -- │ │ │ │ -00064380: 5072 6563 7572 736f 7220 636f 6d70 6c65 Precursor comple │ │ │ │ -00064390: 7820 6f66 0a20 2020 2074 6f74 616c 2045 x of. total E │ │ │ │ -000643a0: 7874 0a0a 5761 7973 2074 6f20 7573 6520 xt..Ways to use │ │ │ │ -000643b0: 6e65 7745 7874 3a0a 3d3d 3d3d 3d3d 3d3d newExt:.======== │ │ │ │ -000643c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -000643d0: 2022 6e65 7745 7874 284d 6f64 756c 652c "newExt(Module, │ │ │ │ -000643e0: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468 Module)"..For th │ │ │ │ -000643f0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -00064400: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00064410: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -00064420: 6520 6e65 7745 7874 3a20 6e65 7745 7874 e newExt: newExt │ │ │ │ -00064430: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -00064440: 686f 6420 6675 6e63 7469 6f6e 2077 6974 hod function wit │ │ │ │ -00064450: 6820 6f70 7469 6f6e 733a 0a28 4d61 6361 h options:.(Maca │ │ │ │ -00064460: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ -00064470: 756e 6374 696f 6e57 6974 684f 7074 696f unctionWithOptio │ │ │ │ -00064480: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ns,...---------- │ │ │ │ +00064280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00064290: 0a0a 4966 2045 4520 616e 6420 4553 2077 ..If EE and ES w │ │ │ │ +000642a0: 6572 6520 6973 6f6d 6f72 7068 6963 2c20 ere isomorphic, │ │ │ │ +000642b0: 7765 2077 6f75 6c64 2065 7870 6563 7420 we would expect │ │ │ │ +000642c0: 636f 6b65 7220 6620 746f 2062 6520 302c coker f to be 0, │ │ │ │ +000642d0: 2061 6e64 2069 7427 7320 6e6f 742e 0a70 and it's not..p │ │ │ │ +000642e0: 7275 6e65 2063 6f6b 6572 2066 0a0a 5365 rune coker f..Se │ │ │ │ +000642f0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ +00064300: 0a20 202a 202a 6e6f 7465 2045 7874 3a20 . * *note Ext: │ │ │ │ +00064310: 284d 6163 6175 6c61 7932 446f 6329 4578 (Macaulay2Doc)Ex │ │ │ │ +00064320: 742c 202d 2d20 636f 6d70 7574 6520 616e t, -- compute an │ │ │ │ +00064330: 2045 7874 206d 6f64 756c 650a 2020 2a20 Ext module. * │ │ │ │ +00064340: 2a6e 6f74 6520 4569 7365 6e62 7564 5368 *note EisenbudSh │ │ │ │ +00064350: 616d 6173 6854 6f74 616c 3a20 4569 7365 amashTotal: Eise │ │ │ │ +00064360: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ +00064370: 2c20 2d2d 2050 7265 6375 7273 6f72 2063 , -- Precursor c │ │ │ │ +00064380: 6f6d 706c 6578 206f 660a 2020 2020 746f omplex of. to │ │ │ │ +00064390: 7461 6c20 4578 740a 0a57 6179 7320 746f tal Ext..Ways to │ │ │ │ +000643a0: 2075 7365 206e 6577 4578 743a 0a3d 3d3d use newExt:.=== │ │ │ │ +000643b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000643c0: 0a0a 2020 2a20 226e 6577 4578 7428 4d6f .. * "newExt(Mo │ │ │ │ +000643d0: 6475 6c65 2c4d 6f64 756c 6529 220a 0a46 dule,Module)"..F │ │ │ │ +000643e0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +000643f0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +00064400: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +00064410: 202a 6e6f 7465 206e 6577 4578 743a 206e *note newExt: n │ │ │ │ +00064420: 6577 4578 742c 2069 7320 6120 2a6e 6f74 ewExt, is a *not │ │ │ │ +00064430: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +00064440: 6e20 7769 7468 206f 7074 696f 6e73 3a0a n with options:. │ │ │ │ +00064450: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +00064460: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ +00064470: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ +00064480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000644a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000644b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000644c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000644d0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -000644e0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -000644f0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -00064500: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -00064510: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -00064520: 2e30 352b 6473 2f4d 322f 4d61 6361 756c .05+ds/M2/Macaul │ │ │ │ -00064530: 6179 322f 7061 636b 6167 6573 2f0a 436f ay2/packages/.Co │ │ │ │ -00064540: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -00064550: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32 onResolutions.m2 │ │ │ │ -00064560: 3a32 3536 343a 302e 0a1f 0a46 696c 653a :2564:0....File: │ │ │ │ -00064570: 2043 6f6d 706c 6574 6549 6e74 6572 7365 CompleteInterse │ │ │ │ -00064580: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ -00064590: 2e69 6e66 6f2c 204e 6f64 653a 206f 6464 .info, Node: odd │ │ │ │ -000645a0: 4578 744d 6f64 756c 652c 204e 6578 743a ExtModule, Next: │ │ │ │ -000645b0: 204f 7074 696d 6973 6d2c 2050 7265 763a Optimism, Prev: │ │ │ │ -000645c0: 206e 6577 4578 742c 2055 703a 2054 6f70 newExt, Up: Top │ │ │ │ -000645d0: 0a0a 6f64 6445 7874 4d6f 6475 6c65 202d ..oddExtModule - │ │ │ │ -000645e0: 2d20 6f64 6420 7061 7274 206f 6620 4578 - odd part of Ex │ │ │ │ -000645f0: 745e 2a28 4d2c 6b29 206f 7665 7220 6120 t^*(M,k) over a │ │ │ │ -00064600: 636f 6d70 6c65 7465 2069 6e74 6572 7365 complete interse │ │ │ │ -00064610: 6374 696f 6e20 6173 206d 6f64 756c 6520 ction as module │ │ │ │ -00064620: 6f76 6572 2043 4920 6f70 6572 6174 6f72 over CI operator │ │ │ │ -00064630: 2072 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a ring.********** │ │ │ │ +000644c0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +000644d0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +000644e0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +000644f0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +00064500: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +00064510: 2d31 2e32 362e 3035 2b64 732f 4d32 2f4d -1.26.05+ds/M2/M │ │ │ │ +00064520: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +00064530: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ +00064540: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +00064550: 6e73 2e6d 323a 3235 3634 3a30 2e0a 1f0a ns.m2:2564:0.... │ │ │ │ +00064560: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ +00064570: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ +00064580: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ +00064590: 3a20 6f64 6445 7874 4d6f 6475 6c65 2c20 : oddExtModule, │ │ │ │ +000645a0: 4e65 7874 3a20 4f70 7469 6d69 736d 2c20 Next: Optimism, │ │ │ │ +000645b0: 5072 6576 3a20 6e65 7745 7874 2c20 5570 Prev: newExt, Up │ │ │ │ +000645c0: 3a20 546f 700a 0a6f 6464 4578 744d 6f64 : Top..oddExtMod │ │ │ │ +000645d0: 756c 6520 2d2d 206f 6464 2070 6172 7420 ule -- odd part │ │ │ │ +000645e0: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76 of Ext^*(M,k) ov │ │ │ │ +000645f0: 6572 2061 2063 6f6d 706c 6574 6520 696e er a complete in │ │ │ │ +00064600: 7465 7273 6563 7469 6f6e 2061 7320 6d6f tersection as mo │ │ │ │ +00064610: 6475 6c65 206f 7665 7220 4349 206f 7065 dule over CI ope │ │ │ │ +00064620: 7261 746f 7220 7269 6e67 0a2a 2a2a 2a2a rator ring.***** │ │ │ │ +00064630: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00064640: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00064650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00064660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00064670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00064680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00064690: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -000646a0: 7361 6765 3a20 0a20 2020 2020 2020 2045 sage: . E │ │ │ │ -000646b0: 203d 206f 6464 4578 744d 6f64 756c 6520 = oddExtModule │ │ │ │ -000646c0: 4d0a 2020 2a20 496e 7075 7473 3a0a 2020 M. * Inputs:. │ │ │ │ -000646d0: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ -000646e0: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ -000646f0: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ -00064700: 6f76 6572 2061 2063 6f6d 706c 6574 6520 over a complete │ │ │ │ -00064710: 696e 7465 7273 6563 7469 6f6e 0a20 2020 intersection. │ │ │ │ -00064720: 2020 2020 2072 696e 670a 2020 2a20 2a6e ring. * *n │ │ │ │ -00064730: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70 ote Optional inp │ │ │ │ -00064740: 7574 733a 2028 4d61 6361 756c 6179 3244 uts: (Macaulay2D │ │ │ │ -00064750: 6f63 2975 7369 6e67 2066 756e 6374 696f oc)using functio │ │ │ │ -00064760: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ -00064770: 2069 6e70 7574 732c 3a0a 2020 2020 2020 inputs,:. │ │ │ │ -00064780: 2a20 4f75 7452 696e 6720 3d3e 202e 2e2e * OutRing => ... │ │ │ │ -00064790: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ -000647a0: 300a 2020 2a20 4f75 7470 7574 733a 0a20 0. * Outputs:. │ │ │ │ -000647b0: 2020 2020 202a 2045 2c20 6120 2a6e 6f74 * E, a *not │ │ │ │ -000647c0: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ -000647d0: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ -000647e0: 206f 7665 7220 6120 706f 6c79 6e6f 6d69 over a polynomi │ │ │ │ -000647f0: 616c 2072 696e 6720 7769 7468 0a20 2020 al ring with. │ │ │ │ -00064800: 2020 2020 2067 656e 7320 696e 2064 6567 gens in deg │ │ │ │ -00064810: 7265 6520 310a 0a44 6573 6372 6970 7469 ree 1..Descripti │ │ │ │ -00064820: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00064830: 4578 7472 6163 7473 2074 6865 206f 6464 Extracts the odd │ │ │ │ -00064840: 2064 6567 7265 6520 7061 7274 2066 726f degree part fro │ │ │ │ -00064850: 6d20 4578 744d 6f64 756c 6520 4d2e 2049 m ExtModule M. I │ │ │ │ -00064860: 6620 7468 6520 6f70 7469 6f6e 616c 2061 f the optional a │ │ │ │ -00064870: 7267 756d 656e 7420 4f75 7452 696e 670a rgument OutRing. │ │ │ │ -00064880: 3d3e 2054 2069 7320 6769 7665 6e2c 2061 => T is given, a │ │ │ │ -00064890: 6e64 2063 6c61 7373 2054 203d 3d3d 2050 nd class T === P │ │ │ │ -000648a0: 6f6c 796e 6f6d 6961 6c52 696e 672c 2074 olynomialRing, t │ │ │ │ -000648b0: 6865 6e20 7468 6520 6f75 7470 7574 2077 hen the output w │ │ │ │ -000648c0: 696c 6c20 6265 2061 206d 6f64 756c 650a ill be a module. │ │ │ │ -000648d0: 6f76 6572 2054 2e0a 0a2b 2d2d 2d2d 2d2d over T...+------ │ │ │ │ +00064680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +00064690: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +000646a0: 2020 2020 4520 3d20 6f64 6445 7874 4d6f E = oddExtMo │ │ │ │ +000646b0: 6475 6c65 204d 0a20 202a 2049 6e70 7574 dule M. * Input │ │ │ │ +000646c0: 733a 0a20 2020 2020 202a 204d 2c20 6120 s:. * M, a │ │ │ │ +000646d0: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ +000646e0: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ +000646f0: 6c65 2c2c 206f 7665 7220 6120 636f 6d70 le,, over a comp │ │ │ │ +00064700: 6c65 7465 2069 6e74 6572 7365 6374 696f lete intersectio │ │ │ │ +00064710: 6e0a 2020 2020 2020 2020 7269 6e67 0a20 n. ring. │ │ │ │ +00064720: 202a 202a 6e6f 7465 204f 7074 696f 6e61 * *note Optiona │ │ │ │ +00064730: 6c20 696e 7075 7473 3a20 284d 6163 6175 l inputs: (Macau │ │ │ │ +00064740: 6c61 7932 446f 6329 7573 696e 6720 6675 lay2Doc)using fu │ │ │ │ +00064750: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ +00064760: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20 ional inputs,:. │ │ │ │ +00064770: 2020 2020 202a 204f 7574 5269 6e67 203d * OutRing = │ │ │ │ +00064780: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ +00064790: 616c 7565 2030 0a20 202a 204f 7574 7075 alue 0. * Outpu │ │ │ │ +000647a0: 7473 3a0a 2020 2020 2020 2a20 452c 2061 ts:. * E, a │ │ │ │ +000647b0: 202a 6e6f 7465 206d 6f64 756c 653a 2028 *note module: ( │ │ │ │ +000647c0: 4d61 6361 756c 6179 3244 6f63 294d 6f64 Macaulay2Doc)Mod │ │ │ │ +000647d0: 756c 652c 2c20 6f76 6572 2061 2070 6f6c ule,, over a pol │ │ │ │ +000647e0: 796e 6f6d 6961 6c20 7269 6e67 2077 6974 ynomial ring wit │ │ │ │ +000647f0: 680a 2020 2020 2020 2020 6765 6e73 2069 h. gens i │ │ │ │ +00064800: 6e20 6465 6772 6565 2031 0a0a 4465 7363 n degree 1..Desc │ │ │ │ +00064810: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +00064820: 3d3d 3d0a 0a45 7874 7261 6374 7320 7468 ===..Extracts th │ │ │ │ +00064830: 6520 6f64 6420 6465 6772 6565 2070 6172 e odd degree par │ │ │ │ +00064840: 7420 6672 6f6d 2045 7874 4d6f 6475 6c65 t from ExtModule │ │ │ │ +00064850: 204d 2e20 4966 2074 6865 206f 7074 696f M. If the optio │ │ │ │ +00064860: 6e61 6c20 6172 6775 6d65 6e74 204f 7574 nal argument Out │ │ │ │ +00064870: 5269 6e67 0a3d 3e20 5420 6973 2067 6976 Ring.=> T is giv │ │ │ │ +00064880: 656e 2c20 616e 6420 636c 6173 7320 5420 en, and class T │ │ │ │ +00064890: 3d3d 3d20 506f 6c79 6e6f 6d69 616c 5269 === PolynomialRi │ │ │ │ +000648a0: 6e67 2c20 7468 656e 2074 6865 206f 7574 ng, then the out │ │ │ │ +000648b0: 7075 7420 7769 6c6c 2062 6520 6120 6d6f put will be a mo │ │ │ │ +000648c0: 6475 6c65 0a6f 7665 7220 542e 0a0a 2b2d dule.over T...+- │ │ │ │ +000648d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000648e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000648f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064910: 2d2d 2d2b 0a7c 6931 203a 206b 6b3d 205a ---+.|i1 : kk= Z │ │ │ │ -00064920: 5a2f 3130 3120 2020 2020 2020 2020 2020 Z/101 │ │ │ │ +00064900: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +00064910: 6b6b 3d20 5a5a 2f31 3031 2020 2020 2020 kk= ZZ/101 │ │ │ │ +00064920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064940: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00064950: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00064940: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00064950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064980: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -00064990: 203d 206b 6b20 2020 2020 2020 2020 2020 = kk │ │ │ │ +00064980: 7c0a 7c6f 3120 3d20 6b6b 2020 2020 2020 |.|o1 = kk │ │ │ │ +00064990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000649a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000649b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000649c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000649b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000649c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000649d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000649e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000649f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064a00: 2020 207c 0a7c 6f31 203a 2051 756f 7469 |.|o1 : Quoti │ │ │ │ -00064a10: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ +000649f0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ +00064a00: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +00064a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064a30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00064a40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00064a30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00064a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00064a80: 203a 2053 203d 206b 6b5b 782c 792c 7a5d : S = kk[x,y,z] │ │ │ │ +00064a70: 2b0a 7c69 3220 3a20 5320 3d20 6b6b 5b78 +.|i2 : S = kk[x │ │ │ │ +00064a80: 2c79 2c7a 5d20 2020 2020 2020 2020 2020 ,y,z] │ │ │ │ 00064a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064ab0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00064aa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00064ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064af0: 2020 207c 0a7c 6f32 203d 2053 2020 2020 |.|o2 = S │ │ │ │ +00064ae0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ +00064af0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 00064b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064b20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00064b30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00064b20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00064b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064b60: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00064b70: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ -00064b80: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ -00064b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064ba0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00064b60: 7c0a 7c6f 3220 3a20 506f 6c79 6e6f 6d69 |.|o2 : Polynomi │ │ │ │ +00064b70: 616c 5269 6e67 2020 2020 2020 2020 2020 alRing │ │ │ │ +00064b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00064b90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00064ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064be0: 2d2d 2d2b 0a7c 6933 203a 2049 3220 3d20 ---+.|i3 : I2 = │ │ │ │ -00064bf0: 6964 6561 6c22 7833 2c79 7a22 2020 2020 ideal"x3,yz" │ │ │ │ +00064bd0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ +00064be0: 4932 203d 2069 6465 616c 2278 332c 797a I2 = ideal"x3,yz │ │ │ │ +00064bf0: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ 00064c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064c10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00064c20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00064c10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00064c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064c50: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00064c60: 2020 2020 2020 2020 2020 2033 2020 2020 3 │ │ │ │ +00064c50: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00064c60: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00064c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064c90: 2020 2020 2020 207c 0a7c 6f33 203d 2069 |.|o3 = i │ │ │ │ -00064ca0: 6465 616c 2028 7820 2c20 792a 7a29 2020 deal (x , y*z) │ │ │ │ +00064c80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00064c90: 3320 3d20 6964 6561 6c20 2878 202c 2079 3 = ideal (x , y │ │ │ │ +00064ca0: 2a7a 2920 2020 2020 2020 2020 2020 2020 *z) │ │ │ │ 00064cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064cd0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00064cc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00064cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064d00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00064d10: 0a7c 6f33 203a 2049 6465 616c 206f 6620 .|o3 : Ideal of │ │ │ │ -00064d20: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +00064d00: 2020 2020 7c0a 7c6f 3320 3a20 4964 6561 |.|o3 : Idea │ │ │ │ +00064d10: 6c20 6f66 2053 2020 2020 2020 2020 2020 l of S │ │ │ │ +00064d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064d40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00064d40: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00064d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064d80: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 -------+.|i4 : R │ │ │ │ -00064d90: 3220 3d20 532f 4932 2020 2020 2020 2020 2 = S/I2 │ │ │ │ +00064d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00064d80: 3420 3a20 5232 203d 2053 2f49 3220 2020 4 : R2 = S/I2 │ │ │ │ +00064d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064dc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00064db0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00064dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064df0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00064e00: 0a7c 6f34 203d 2052 3220 2020 2020 2020 .|o4 = R2 │ │ │ │ +00064df0: 2020 2020 7c0a 7c6f 3420 3d20 5232 2020 |.|o4 = R2 │ │ │ │ +00064e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064e30: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00064e30: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00064e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064e70: 2020 2020 2020 207c 0a7c 6f34 203a 2051 |.|o4 : Q │ │ │ │ -00064e80: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +00064e60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00064e70: 3420 3a20 5175 6f74 6965 6e74 5269 6e67 4 : QuotientRing │ │ │ │ +00064e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064eb0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00064ea0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00064eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00064ef0: 0a7c 6935 203a 204d 3220 3d20 5232 5e31 .|i5 : M2 = R2^1 │ │ │ │ -00064f00: 2f69 6465 616c 2278 322c 792c 7a22 2020 /ideal"x2,y,z" │ │ │ │ +00064ee0: 2d2d 2d2d 2b0a 7c69 3520 3a20 4d32 203d ----+.|i5 : M2 = │ │ │ │ +00064ef0: 2052 325e 312f 6964 6561 6c22 7832 2c79 R2^1/ideal"x2,y │ │ │ │ +00064f00: 2c7a 2220 2020 2020 2020 2020 2020 2020 ,z" │ │ │ │ 00064f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064f20: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00064f20: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00064f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064f60: 2020 2020 2020 207c 0a7c 6f35 203d 2063 |.|o5 = c │ │ │ │ -00064f70: 6f6b 6572 6e65 6c20 7c20 7832 2079 207a okernel | x2 y z │ │ │ │ -00064f80: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00064f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064fa0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00064f50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00064f60: 3520 3d20 636f 6b65 726e 656c 207c 2078 5 = cokernel | x │ │ │ │ +00064f70: 3220 7920 7a20 7c20 2020 2020 2020 2020 2 y z | │ │ │ │ +00064f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00064f90: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00064fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064fd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00064fe0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00064ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065000: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -00065010: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ -00065020: 203a 2052 322d 6d6f 6475 6c65 2c20 7175 : R2-module, qu │ │ │ │ -00065030: 6f74 6965 6e74 206f 6620 5232 2020 2020 otient of R2 │ │ │ │ -00065040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065050: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00064fd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00064fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00064ff0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +00065000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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|.|o │ │ │ │ +00065320: 3620 3a20 4265 7474 6954 616c 6c79 2020 6 : BettiTally │ │ │ │ +00065330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065360: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00065350: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00065360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00065370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00065380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00065390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000653a0: 0a7c 6937 203a 2045 203d 2045 7874 4d6f .|i7 : E = ExtMo │ │ │ │ -000653b0: 6475 6c65 204d 3220 2020 2020 2020 2020 dule M2 │ │ │ │ +00065390: 2d2d 2d2d 2b0a 7c69 3720 3a20 4520 3d20 ----+.|i7 : E = │ │ │ │ +000653a0: 4578 744d 6f64 756c 6520 4d32 2020 2020 ExtModule M2 │ │ │ │ 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+000657a0: 2020 2020 3420 2020 2020 2020 2020 2020 4 │ │ │ │ 000657b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000657c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000657d0: 2020 2020 2020 207c 0a7c 6f39 203d 2028 |.|o9 = ( │ │ │ │ -000657e0: 6b6b 5b58 202e 2e58 205d 2920 2020 2020 kk[X ..X ]) │ │ │ │ +000657c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000657d0: 3920 3d20 286b 6b5b 5820 2e2e 5820 5d29 9 = (kk[X ..X ]) │ │ │ │ +000657e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000657f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065810: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00065820: 3020 2020 3120 2020 2020 2020 2020 2020 0 1 │ │ │ │ +00065800: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00065810: 2020 2020 2030 2020 2031 2020 2020 2020 0 1 │ │ │ │ +00065820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065840: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00065850: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00065840: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00065850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065880: 2020 2020 2020 2020 2020 207c 0a7c 6f39 |.|o9 │ │ │ │ -00065890: 203a 206b 6b5b 5820 2e2e 5820 5d2d 6d6f : kk[X ..X ]-mo │ │ │ │ -000658a0: 6475 6c65 2c20 6672 6565 2c20 6465 6772 dule, free, degr │ │ │ │ -000658b0: 6565 7320 7b33 3a30 2c20 317d 2020 2020 ees {3:0, 1} │ │ │ │ -000658c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000658d0: 2020 2030 2020 2031 2020 2020 2020 2020 0 1 │ │ │ │ +00065880: 7c0a 7c6f 3920 3a20 6b6b 5b58 202e 2e58 |.|o9 : kk[X ..X │ │ │ │ +00065890: 205d 2d6d 6f64 756c 652c 2066 7265 652c ]-module, free, │ │ │ │ +000658a0: 2064 6567 7265 6573 207b 333a 302c 2031 degrees {3:0, 1 │ │ │ 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6464 2929 2020 207c 0a7c 2020 , Eodd)) |.| │ │ │ │ +00065930: 2d2d 2d2d 2b0a 7c69 3130 203a 2061 7070 ----+.|i10 : app │ │ │ │ +00065940: 6c79 2874 6f4c 6973 7428 302e 2e35 292c ly(toList(0..5), │ │ │ │ +00065950: 2069 2d3e 6869 6c62 6572 7446 756e 6374 i->hilbertFunct │ │ │ │ +00065960: 696f 6e28 692c 2045 6f64 6429 2920 2020 ion(i, Eodd)) │ │ │ │ +00065970: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00065980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000659a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000659b0: 2020 2020 2020 207c 0a7c 6f31 3020 3d20 |.|o10 = │ │ │ │ -000659c0: 7b33 2c20 372c 2031 312c 2031 352c 2031 {3, 7, 11, 15, 1 │ │ │ │ -000659d0: 392c 2032 337d 2020 2020 2020 2020 2020 9, 23} │ │ │ │ -000659e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000659f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000659a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000659b0: 3130 203d 207b 332c 2037 2c20 3131 2c20 10 = {3, 7, 11, │ │ │ │ +000659c0: 3135 2c20 3139 2c20 3233 7d20 2020 2020 15, 19, 23} │ │ │ │ +000659d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000659e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000659f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065a20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00065a30: 0a7c 6f31 3020 3a20 4c69 7374 2020 2020 .|o10 : List │ │ │ │ +00065a20: 2020 2020 7c0a 7c6f 3130 203a 204c 6973 |.|o10 : Lis │ │ │ │ +00065a30: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ 00065a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065a60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00065a60: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00065a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00065a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00065a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00065aa0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ -00065ab0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -00065ac0: 202a 6e6f 7465 2045 7874 4d6f 6475 6c65 *note ExtModule │ │ │ │ -00065ad0: 3a20 4578 744d 6f64 756c 652c 202d 2d20 : ExtModule, -- │ │ │ │ -00065ae0: 4578 745e 2a28 4d2c 6b29 206f 7665 7220 Ext^*(M,k) over │ │ │ │ -00065af0: 6120 636f 6d70 6c65 7465 2069 6e74 6572 a complete inter │ │ │ │ -00065b00: 7365 6374 696f 6e20 6173 0a20 2020 206d section as. m │ │ │ │ -00065b10: 6f64 756c 6520 6f76 6572 2043 4920 6f70 odule over CI op │ │ │ │ -00065b20: 6572 6174 6f72 2072 696e 670a 2020 2a20 erator ring. * │ │ │ │ -00065b30: 2a6e 6f74 6520 6576 656e 4578 744d 6f64 *note evenExtMod │ │ │ │ -00065b40: 756c 653a 2065 7665 6e45 7874 4d6f 6475 ule: evenExtModu │ │ │ │ -00065b50: 6c65 2c20 2d2d 2065 7665 6e20 7061 7274 le, -- even part │ │ │ │ -00065b60: 206f 6620 4578 745e 2a28 4d2c 6b29 206f of Ext^*(M,k) o │ │ │ │ -00065b70: 7665 7220 610a 2020 2020 636f 6d70 6c65 ver a. comple │ │ │ │ -00065b80: 7465 2069 6e74 6572 7365 6374 696f 6e20 te intersection │ │ │ │ -00065b90: 6173 206d 6f64 756c 6520 6f76 6572 2043 as module over C │ │ │ │ -00065ba0: 4920 6f70 6572 6174 6f72 2072 696e 670a I operator ring. │ │ │ │ -00065bb0: 2020 2a20 2a6e 6f74 6520 4f75 7452 696e * *note OutRin │ │ │ │ -00065bc0: 673a 204f 7574 5269 6e67 2c20 2d2d 204f g: OutRing, -- O │ │ │ │ -00065bd0: 7074 696f 6e20 616c 6c6f 7769 6e67 2073 ption allowing s │ │ │ │ -00065be0: 7065 6369 6669 6361 7469 6f6e 206f 6620 pecification of │ │ │ │ -00065bf0: 7468 6520 7269 6e67 206f 7665 720a 2020 the ring over. │ │ │ │ -00065c00: 2020 7768 6963 6820 7468 6520 6f75 7470 which the outp │ │ │ │ -00065c10: 7574 2069 7320 6465 6669 6e65 640a 0a57 ut is defined..W │ │ │ │ -00065c20: 6179 7320 746f 2075 7365 206f 6464 4578 ays to use oddEx │ │ │ │ -00065c30: 744d 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d tModule:.======= │ │ │ │ -00065c40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00065c50: 3d3d 0a0a 2020 2a20 226f 6464 4578 744d ==.. * "oddExtM │ │ │ │ -00065c60: 6f64 756c 6528 4d6f 6475 6c65 2922 0a0a odule(Module)".. │ │ │ │ -00065c70: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -00065c80: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -00065c90: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -00065ca0: 7420 2a6e 6f74 6520 6f64 6445 7874 4d6f t *note oddExtMo │ │ │ │ -00065cb0: 6475 6c65 3a20 6f64 6445 7874 4d6f 6475 dule: oddExtModu │ │ │ │ -00065cc0: 6c65 2c20 6973 2061 202a 6e6f 7465 206d le, is a *note m │ │ │ │ -00065cd0: 6574 686f 6420 6675 6e63 7469 6f6e 2077 ethod function w │ │ │ │ -00065ce0: 6974 680a 6f70 7469 6f6e 733a 2028 4d61 ith.options: (Ma │ │ │ │ -00065cf0: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ -00065d00: 6446 756e 6374 696f 6e57 6974 684f 7074 dFunctionWithOpt │ │ │ │ -00065d10: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d ions,...-------- │ │ │ │ +00065a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 ------------+..S │ │ │ │ +00065aa0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +00065ab0: 0a0a 2020 2a20 2a6e 6f74 6520 4578 744d .. * *note ExtM │ │ │ │ +00065ac0: 6f64 756c 653a 2045 7874 4d6f 6475 6c65 odule: ExtModule │ │ │ │ +00065ad0: 2c20 2d2d 2045 7874 5e2a 284d 2c6b 2920 , -- Ext^*(M,k) │ │ │ │ +00065ae0: 6f76 6572 2061 2063 6f6d 706c 6574 6520 over a complete │ │ │ │ +00065af0: 696e 7465 7273 6563 7469 6f6e 2061 730a intersection as. │ │ │ │ +00065b00: 2020 2020 6d6f 6475 6c65 206f 7665 7220 module over │ │ │ │ +00065b10: 4349 206f 7065 7261 746f 7220 7269 6e67 CI operator ring │ │ │ │ +00065b20: 0a20 202a 202a 6e6f 7465 2065 7665 6e45 . * *note evenE │ │ │ │ +00065b30: 7874 4d6f 6475 6c65 3a20 6576 656e 4578 xtModule: evenEx │ │ │ │ +00065b40: 744d 6f64 756c 652c 202d 2d20 6576 656e tModule, -- even │ │ │ │ +00065b50: 2070 6172 7420 6f66 2045 7874 5e2a 284d part of Ext^*(M │ │ │ │ +00065b60: 2c6b 2920 6f76 6572 2061 0a20 2020 2063 ,k) over a. c │ │ │ │ +00065b70: 6f6d 706c 6574 6520 696e 7465 7273 6563 omplete intersec │ │ │ │ +00065b80: 7469 6f6e 2061 7320 6d6f 6475 6c65 206f tion as module o │ │ │ │ +00065b90: 7665 7220 4349 206f 7065 7261 746f 7220 ver CI operator │ │ │ │ +00065ba0: 7269 6e67 0a20 202a 202a 6e6f 7465 204f ring. * *note O │ │ │ │ +00065bb0: 7574 5269 6e67 3a20 4f75 7452 696e 672c utRing: OutRing, │ │ │ │ +00065bc0: 202d 2d20 4f70 7469 6f6e 2061 6c6c 6f77 -- Option allow │ │ │ │ +00065bd0: 696e 6720 7370 6563 6966 6963 6174 696f ing specificatio │ │ │ │ +00065be0: 6e20 6f66 2074 6865 2072 696e 6720 6f76 n of the ring ov │ │ │ │ +00065bf0: 6572 0a20 2020 2077 6869 6368 2074 6865 er. which the │ │ │ │ +00065c00: 206f 7574 7075 7420 6973 2064 6566 696e output is defin │ │ │ │ +00065c10: 6564 0a0a 5761 7973 2074 6f20 7573 6520 ed..Ways to use │ │ │ │ +00065c20: 6f64 6445 7874 4d6f 6475 6c65 3a0a 3d3d oddExtModule:.== │ │ │ │ +00065c30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00065c40: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6f64 =======.. * "od │ │ │ │ +00065c50: 6445 7874 4d6f 6475 6c65 284d 6f64 756c dExtModule(Modul │ │ │ │ +00065c60: 6529 220a 0a46 6f72 2074 6865 2070 726f e)"..For the pro │ │ │ │ +00065c70: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +00065c80: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +00065c90: 6f62 6a65 6374 202a 6e6f 7465 206f 6464 object *note odd │ │ │ │ +00065ca0: 4578 744d 6f64 756c 653a 206f 6464 4578 ExtModule: oddEx │ │ │ │ +00065cb0: 744d 6f64 756c 652c 2069 7320 6120 2a6e tModule, is a *n │ │ │ │ +00065cc0: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ +00065cd0: 696f 6e20 7769 7468 0a6f 7074 696f 6e73 ion with.options │ │ │ │ +00065ce0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00065cf0: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769 MethodFunctionWi │ │ │ │ +00065d00: 7468 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d thOptions,...--- │ │ │ │ +00065d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00065d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00065d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00065d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00065d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00065d60: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -00065d70: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -00065d80: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -00065d90: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -00065da0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -00065db0: 3236 2e30 352b 6473 2f4d 322f 4d61 6361 26.05+ds/M2/Maca │ │ │ │ -00065dc0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ -00065dd0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -00065de0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -00065df0: 6d32 3a33 3637 393a 302e 0a1f 0a46 696c m2:3679:0....Fil │ │ │ │ -00065e00: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ -00065e10: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -00065e20: 6e73 2e69 6e66 6f2c 204e 6f64 653a 204f ns.info, Node: O │ │ │ │ -00065e30: 7074 696d 6973 6d2c 204e 6578 743a 204f ptimism, Next: O │ │ │ │ -00065e40: 7574 5269 6e67 2c20 5072 6576 3a20 6f64 utRing, Prev: od │ │ │ │ -00065e50: 6445 7874 4d6f 6475 6c65 2c20 5570 3a20 dExtModule, Up: │ │ │ │ -00065e60: 546f 700a 0a4f 7074 696d 6973 6d20 2d2d Top..Optimism -- │ │ │ │ -00065e70: 204f 7074 696f 6e20 746f 2068 6967 6853 Option to highS │ │ │ │ -00065e80: 797a 7967 790a 2a2a 2a2a 2a2a 2a2a 2a2a yzygy.********** │ │ │ │ -00065e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00065ea0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -00065eb0: 653a 200a 2020 2020 2020 2020 6869 6768 e: . high │ │ │ │ -00065ec0: 5379 7a79 6779 284d 2c20 4f70 7469 6d69 Syzygy(M, Optimi │ │ │ │ -00065ed0: 736d 203d 3e20 3129 0a20 202a 2049 6e70 sm => 1). * Inp │ │ │ │ -00065ee0: 7574 733a 0a20 2020 2020 202a 204f 7074 uts:. * Opt │ │ │ │ -00065ef0: 696d 6973 6d2c 2061 6e20 2a6e 6f74 6520 imism, an *note │ │ │ │ -00065f00: 696e 7465 6765 723a 2028 4d61 6361 756c integer: (Macaul │ │ │ │ -00065f10: 6179 3244 6f63 295a 5a2c 2c20 0a0a 4465 ay2Doc)ZZ,, ..De │ │ │ │ -00065f20: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ -00065f30: 3d3d 3d3d 3d0a 0a49 6620 6869 6768 5379 =====..If highSy │ │ │ │ -00065f40: 7a79 6779 284d 2920 6368 6f6f 7365 7320 zygy(M) chooses │ │ │ │ -00065f50: 7468 6520 702d 7468 2073 797a 7967 792c the p-th syzygy, │ │ │ │ -00065f60: 2074 6865 6e20 6869 6768 5379 7a79 6779 then highSyzygy │ │ │ │ -00065f70: 284d 2c4f 7074 696d 6973 6d3d 3e72 290a (M,Optimism=>r). │ │ │ │ -00065f80: 6368 6f6f 7365 7320 7468 6520 2870 2d72 chooses the (p-r │ │ │ │ -00065f90: 292d 7468 2073 797a 7967 792e 2028 506f )-th syzygy. (Po │ │ │ │ -00065fa0: 7369 7469 7665 204f 7074 696d 6973 6d20 sitive Optimism │ │ │ │ -00065fb0: 6368 6f6f 7365 7320 6120 6c6f 7765 7220 chooses a lower │ │ │ │ -00065fc0: 2268 6967 6822 2073 797a 7967 792c 0a6e "high" syzygy,.n │ │ │ │ -00065fd0: 6567 6174 6976 6520 4f70 7469 6d69 736d egative Optimism │ │ │ │ -00065fe0: 2061 2068 6967 6865 7220 2268 6967 6822 a higher "high" │ │ │ │ -00065ff0: 2073 797a 7967 792e 0a0a 4361 7665 6174 syzygy...Caveat │ │ │ │ -00066000: 0a3d 3d3d 3d3d 3d0a 0a41 7265 2074 6865 .======..Are the │ │ │ │ -00066010: 7265 2063 6173 6573 2077 6865 6e20 706f re cases when po │ │ │ │ -00066020: 7369 7469 7665 204f 7074 696d 6973 6d20 sitive Optimism │ │ │ │ -00066030: 6973 206a 7573 7469 6669 6564 3f0a 0a53 is justified?..S │ │ │ │ -00066040: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ -00066050: 0a0a 2020 2a20 2a6e 6f74 6520 6d66 426f .. * *note mfBo │ │ │ │ -00066060: 756e 643a 206d 6642 6f75 6e64 2c20 2d2d und: mfBound, -- │ │ │ │ -00066070: 2064 6574 6572 6d69 6e65 7320 686f 7720 determines how │ │ │ │ -00066080: 6869 6768 2061 2073 797a 7967 7920 746f high a syzygy to │ │ │ │ -00066090: 2074 616b 6520 666f 720a 2020 2020 226d take for. "m │ │ │ │ -000660a0: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ -000660b0: 6f6e 220a 2020 2a20 2a6e 6f74 6520 6869 on". * *note hi │ │ │ │ -000660c0: 6768 5379 7a79 6779 3a20 6869 6768 5379 ghSyzygy: highSy │ │ │ │ -000660d0: 7a79 6779 2c20 2d2d 2052 6574 7572 6e73 zygy, -- Returns │ │ │ │ -000660e0: 2061 2073 797a 7967 7920 6d6f 6475 6c65 a syzygy module │ │ │ │ -000660f0: 206f 6e65 2062 6579 6f6e 6420 7468 650a one beyond the. │ │ │ │ -00066100: 2020 2020 7265 6775 6c61 7269 7479 206f regularity o │ │ │ │ -00066110: 6620 4578 7428 4d2c 6b29 0a0a 4675 6e63 f Ext(M,k)..Func │ │ │ │ -00066120: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ -00066130: 6e61 6c20 6172 6775 6d65 6e74 206e 616d nal argument nam │ │ │ │ -00066140: 6564 204f 7074 696d 6973 6d3a 0a3d 3d3d ed Optimism:.=== │ │ │ │ +00065d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +00065d60: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +00065d70: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +00065d80: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +00065d90: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +00065da0: 7932 2d31 2e32 362e 3035 2b64 732f 4d32 y2-1.26.05+ds/M2 │ │ │ │ +00065db0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +00065dc0: 6765 732f 0a43 6f6d 706c 6574 6549 6e74 ges/.CompleteInt │ │ │ │ +00065dd0: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00065de0: 696f 6e73 2e6d 323a 3336 3739 3a30 2e0a ions.m2:3679:0.. │ │ │ │ +00065df0: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465 ..File: Complete │ │ │ │ +00065e00: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +00065e10: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f lutions.info, No │ │ │ │ +00065e20: 6465 3a20 4f70 7469 6d69 736d 2c20 4e65 de: Optimism, Ne │ │ │ │ +00065e30: 7874 3a20 4f75 7452 696e 672c 2050 7265 xt: OutRing, Pre │ │ │ │ +00065e40: 763a 206f 6464 4578 744d 6f64 756c 652c v: oddExtModule, │ │ │ │ +00065e50: 2055 703a 2054 6f70 0a0a 4f70 7469 6d69 Up: Top..Optimi │ │ │ │ +00065e60: 736d 202d 2d20 4f70 7469 6f6e 2074 6f20 sm -- Option to │ │ │ │ +00065e70: 6869 6768 5379 7a79 6779 0a2a 2a2a 2a2a highSyzygy.***** │ │ │ │ +00065e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00065e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00065ea0: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +00065eb0: 2068 6967 6853 797a 7967 7928 4d2c 204f highSyzygy(M, O │ │ │ │ +00065ec0: 7074 696d 6973 6d20 3d3e 2031 290a 2020 ptimism => 1). │ │ │ │ +00065ed0: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ +00065ee0: 2a20 4f70 7469 6d69 736d 2c20 616e 202a * Optimism, an * │ │ │ │ +00065ef0: 6e6f 7465 2069 6e74 6567 6572 3a20 284d note integer: (M │ │ │ │ +00065f00: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c acaulay2Doc)ZZ,, │ │ │ │ +00065f10: 200a 0a44 6573 6372 6970 7469 6f6e 0a3d ..Description.= │ │ │ │ +00065f20: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966 2068 ==========..If h │ │ │ │ +00065f30: 6967 6853 797a 7967 7928 4d29 2063 686f ighSyzygy(M) cho │ │ │ │ +00065f40: 6f73 6573 2074 6865 2070 2d74 6820 7379 oses the p-th sy │ │ │ │ +00065f50: 7a79 6779 2c20 7468 656e 2068 6967 6853 zygy, then highS │ │ │ │ +00065f60: 797a 7967 7928 4d2c 4f70 7469 6d69 736d yzygy(M,Optimism │ │ │ │ +00065f70: 3d3e 7229 0a63 686f 6f73 6573 2074 6865 =>r).chooses the │ │ │ │ +00065f80: 2028 702d 7229 2d74 6820 7379 7a79 6779 (p-r)-th syzygy │ │ │ │ +00065f90: 2e20 2850 6f73 6974 6976 6520 4f70 7469 . (Positive Opti │ │ │ │ +00065fa0: 6d69 736d 2063 686f 6f73 6573 2061 206c mism chooses a l │ │ │ │ +00065fb0: 6f77 6572 2022 6869 6768 2220 7379 7a79 ower "high" syzy │ │ │ │ +00065fc0: 6779 2c0a 6e65 6761 7469 7665 204f 7074 gy,.negative Opt │ │ │ │ +00065fd0: 696d 6973 6d20 6120 6869 6768 6572 2022 imism a higher " │ │ │ │ +00065fe0: 6869 6768 2220 7379 7a79 6779 2e0a 0a43 high" syzygy...C │ │ │ │ +00065ff0: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 4172 aveat.======..Ar │ │ │ │ +00066000: 6520 7468 6572 6520 6361 7365 7320 7768 e there cases wh │ │ │ │ +00066010: 656e 2070 6f73 6974 6976 6520 4f70 7469 en positive Opti │ │ │ │ +00066020: 6d69 736d 2069 7320 6a75 7374 6966 6965 mism is justifie │ │ │ │ +00066030: 643f 0a0a 5365 6520 616c 736f 0a3d 3d3d d?..See also.=== │ │ │ │ +00066040: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ +00066050: 206d 6642 6f75 6e64 3a20 6d66 426f 756e mfBound: mfBoun │ │ │ │ +00066060: 642c 202d 2d20 6465 7465 726d 696e 6573 d, -- determines │ │ │ │ +00066070: 2068 6f77 2068 6967 6820 6120 7379 7a79 how high a syzy │ │ │ │ +00066080: 6779 2074 6f20 7461 6b65 2066 6f72 0a20 gy to take for. │ │ │ │ +00066090: 2020 2022 6d61 7472 6978 4661 6374 6f72 "matrixFactor │ │ │ │ +000660a0: 697a 6174 696f 6e22 0a20 202a 202a 6e6f ization". * *no │ │ │ │ +000660b0: 7465 2068 6967 6853 797a 7967 793a 2068 te highSyzygy: h │ │ │ │ +000660c0: 6967 6853 797a 7967 792c 202d 2d20 5265 ighSyzygy, -- Re │ │ │ │ +000660d0: 7475 726e 7320 6120 7379 7a79 6779 206d turns a syzygy m │ │ │ │ +000660e0: 6f64 756c 6520 6f6e 6520 6265 796f 6e64 odule one beyond │ │ │ │ +000660f0: 2074 6865 0a20 2020 2072 6567 756c 6172 the. regular │ │ │ │ +00066100: 6974 7920 6f66 2045 7874 284d 2c6b 290a ity of Ext(M,k). │ │ │ │ +00066110: 0a46 756e 6374 696f 6e73 2077 6974 6820 .Functions with │ │ │ │ +00066120: 6f70 7469 6f6e 616c 2061 7267 756d 656e optional argumen │ │ │ │ +00066130: 7420 6e61 6d65 6420 4f70 7469 6d69 736d t named Optimism │ │ │ │ +00066140: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ 00066150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00066160: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00066170: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -00066180: 202a 2022 6869 6768 5379 7a79 6779 282e * "highSyzygy(. │ │ │ │ -00066190: 2e2e 2c4f 7074 696d 6973 6d3d 3e2e 2e2e ..,Optimism=>... │ │ │ │ -000661a0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ -000661b0: 6869 6768 5379 7a79 6779 3a20 6869 6768 highSyzygy: high │ │ │ │ -000661c0: 5379 7a79 6779 2c20 2d2d 0a20 2020 2052 Syzygy, --. R │ │ │ │ -000661d0: 6574 7572 6e73 2061 2073 797a 7967 7920 eturns a syzygy │ │ │ │ -000661e0: 6d6f 6475 6c65 206f 6e65 2062 6579 6f6e module one beyon │ │ │ │ -000661f0: 6420 7468 6520 7265 6775 6c61 7269 7479 d the regularity │ │ │ │ -00066200: 206f 6620 4578 7428 4d2c 6b29 0a20 202a of Ext(M,k). * │ │ │ │ -00066210: 2022 7477 6f4d 6f6e 6f6d 6961 6c73 282e "twoMonomials(. │ │ │ │ -00066220: 2e2e 2c4f 7074 696d 6973 6d3d 3e2e 2e2e ..,Optimism=>... │ │ │ │ -00066230: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ -00066240: 7477 6f4d 6f6e 6f6d 6961 6c73 3a20 7477 twoMonomials: tw │ │ │ │ -00066250: 6f4d 6f6e 6f6d 6961 6c73 2c0a 2020 2020 oMonomials,. │ │ │ │ -00066260: 2d2d 2074 616c 6c79 2074 6865 2073 6571 -- tally the seq │ │ │ │ -00066270: 7565 6e63 6573 206f 6620 4252 616e 6b73 uences of BRanks │ │ │ │ -00066280: 2066 6f72 2063 6572 7461 696e 2065 7861 for certain exa │ │ │ │ -00066290: 6d70 6c65 730a 0a46 6f72 2074 6865 2070 mples..For the p │ │ │ │ -000662a0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -000662b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -000662c0: 6520 6f62 6a65 6374 202a 6e6f 7465 204f e object *note O │ │ │ │ -000662d0: 7074 696d 6973 6d3a 204f 7074 696d 6973 ptimism: Optimis │ │ │ │ -000662e0: 6d2c 2069 7320 6120 2a6e 6f74 6520 7379 m, is a *note sy │ │ │ │ -000662f0: 6d62 6f6c 3a20 284d 6163 6175 6c61 7932 mbol: (Macaulay2 │ │ │ │ -00066300: 446f 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d Doc)Symbol,...-- │ │ │ │ +00066170: 3d3d 0a0a 2020 2a20 2268 6967 6853 797a ==.. * "highSyz │ │ │ │ +00066180: 7967 7928 2e2e 2e2c 4f70 7469 6d69 736d ygy(...,Optimism │ │ │ │ +00066190: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a =>...)" -- see * │ │ │ │ +000661a0: 6e6f 7465 2068 6967 6853 797a 7967 793a note highSyzygy: │ │ │ │ +000661b0: 2068 6967 6853 797a 7967 792c 202d 2d0a highSyzygy, --. │ │ │ │ +000661c0: 2020 2020 5265 7475 726e 7320 6120 7379 Returns a sy │ │ │ │ +000661d0: 7a79 6779 206d 6f64 756c 6520 6f6e 6520 zygy module one │ │ │ │ +000661e0: 6265 796f 6e64 2074 6865 2072 6567 756c beyond the regul │ │ │ │ +000661f0: 6172 6974 7920 6f66 2045 7874 284d 2c6b arity of Ext(M,k │ │ │ │ +00066200: 290a 2020 2a20 2274 776f 4d6f 6e6f 6d69 ). * "twoMonomi │ │ │ │ +00066210: 616c 7328 2e2e 2e2c 4f70 7469 6d69 736d als(...,Optimism │ │ │ │ +00066220: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a =>...)" -- see * │ │ │ │ +00066230: 6e6f 7465 2074 776f 4d6f 6e6f 6d69 616c note twoMonomial │ │ │ │ +00066240: 733a 2074 776f 4d6f 6e6f 6d69 616c 732c s: twoMonomials, │ │ │ │ +00066250: 0a20 2020 202d 2d20 7461 6c6c 7920 7468 . -- tally th │ │ │ │ +00066260: 6520 7365 7175 656e 6365 7320 6f66 2042 e sequences of B │ │ │ │ +00066270: 5261 6e6b 7320 666f 7220 6365 7274 6169 Ranks for certai │ │ │ │ +00066280: 6e20 6578 616d 706c 6573 0a0a 466f 7220 n examples..For │ │ │ │ +00066290: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +000662a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000662b0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +000662c0: 6f74 6520 4f70 7469 6d69 736d 3a20 4f70 ote Optimism: Op │ │ │ │ +000662d0: 7469 6d69 736d 2c20 6973 2061 202a 6e6f timism, is a *no │ │ │ │ +000662e0: 7465 2073 796d 626f 6c3a 2028 4d61 6361 te symbol: (Maca │ │ │ │ +000662f0: 756c 6179 3244 6f63 2953 796d 626f 6c2c ulay2Doc)Symbol, │ │ │ │ +00066300: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 00066310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00066350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -00066360: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -00066370: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -00066380: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -00066390: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -000663a0: 6179 322d 312e 3236 2e30 352b 6473 2f4d ay2-1.26.05+ds/M │ │ │ │ -000663b0: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -000663c0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ -000663d0: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -000663e0: 7469 6f6e 732e 6d32 3a33 3136 353a 302e tions.m2:3165:0. │ │ │ │ -000663f0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ -00066400: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -00066410: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ -00066420: 6f64 653a 204f 7574 5269 6e67 2c20 4e65 ode: OutRing, Ne │ │ │ │ -00066430: 7874 3a20 7073 694d 6170 732c 2050 7265 xt: psiMaps, Pre │ │ │ │ -00066440: 763a 204f 7074 696d 6973 6d2c 2055 703a v: Optimism, Up: │ │ │ │ -00066450: 2054 6f70 0a0a 4f75 7452 696e 6720 2d2d Top..OutRing -- │ │ │ │ -00066460: 204f 7074 696f 6e20 616c 6c6f 7769 6e67 Option allowing │ │ │ │ -00066470: 2073 7065 6369 6669 6361 7469 6f6e 206f specification o │ │ │ │ -00066480: 6620 7468 6520 7269 6e67 206f 7665 7220 f the ring over │ │ │ │ -00066490: 7768 6963 6820 7468 6520 6f75 7470 7574 which the output │ │ │ │ -000664a0: 2069 7320 6465 6669 6e65 640a 2a2a 2a2a is defined.**** │ │ │ │ +00066350: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +00066360: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +00066370: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +00066380: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +00066390: 6163 6175 6c61 7932 2d31 2e32 362e 3035 acaulay2-1.26.05 │ │ │ │ +000663a0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +000663b0: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ +000663c0: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +000663d0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3331 esolutions.m2:31 │ │ │ │ +000663e0: 3635 3a30 2e0a 1f0a 4669 6c65 3a20 436f 65:0....File: Co │ │ │ │ +000663f0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +00066400: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ +00066410: 666f 2c20 4e6f 6465 3a20 4f75 7452 696e fo, Node: OutRin │ │ │ │ +00066420: 672c 204e 6578 743a 2070 7369 4d61 7073 g, Next: psiMaps │ │ │ │ +00066430: 2c20 5072 6576 3a20 4f70 7469 6d69 736d , Prev: Optimism │ │ │ │ +00066440: 2c20 5570 3a20 546f 700a 0a4f 7574 5269 , Up: Top..OutRi │ │ │ │ +00066450: 6e67 202d 2d20 4f70 7469 6f6e 2061 6c6c ng -- Option all │ │ │ │ +00066460: 6f77 696e 6720 7370 6563 6966 6963 6174 owing specificat │ │ │ │ +00066470: 696f 6e20 6f66 2074 6865 2072 696e 6720 ion of the ring │ │ │ │ +00066480: 6f76 6572 2077 6869 6368 2074 6865 206f over which the o │ │ │ │ +00066490: 7574 7075 7420 6973 2064 6566 696e 6564 utput is defined │ │ │ │ +000664a0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ 000664b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000664c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000664d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000664e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000664f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00066500: 2a0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d *..See also.==== │ │ │ │ -00066510: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -00066520: 6576 656e 4578 744d 6f64 756c 653a 2065 evenExtModule: e │ │ │ │ -00066530: 7665 6e45 7874 4d6f 6475 6c65 2c20 2d2d venExtModule, -- │ │ │ │ -00066540: 2065 7665 6e20 7061 7274 206f 6620 4578 even part of Ex │ │ │ │ -00066550: 745e 2a28 4d2c 6b29 206f 7665 7220 610a t^*(M,k) over a. │ │ │ │ -00066560: 2020 2020 636f 6d70 6c65 7465 2069 6e74 complete int │ │ │ │ -00066570: 6572 7365 6374 696f 6e20 6173 206d 6f64 ersection as mod │ │ │ │ -00066580: 756c 6520 6f76 6572 2043 4920 6f70 6572 ule over CI oper │ │ │ │ -00066590: 6174 6f72 2072 696e 670a 2020 2a20 2a6e ator ring. * *n │ │ │ │ -000665a0: 6f74 6520 6f64 6445 7874 4d6f 6475 6c65 ote oddExtModule │ │ │ │ -000665b0: 3a20 6f64 6445 7874 4d6f 6475 6c65 2c20 : oddExtModule, │ │ │ │ -000665c0: 2d2d 206f 6464 2070 6172 7420 6f66 2045 -- odd part of E │ │ │ │ -000665d0: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061 xt^*(M,k) over a │ │ │ │ -000665e0: 2063 6f6d 706c 6574 650a 2020 2020 696e complete. in │ │ │ │ -000665f0: 7465 7273 6563 7469 6f6e 2061 7320 6d6f tersection as mo │ │ │ │ -00066600: 6475 6c65 206f 7665 7220 4349 206f 7065 dule over CI ope │ │ │ │ -00066610: 7261 746f 7220 7269 6e67 0a0a 4675 6e63 rator ring..Func │ │ │ │ -00066620: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ -00066630: 6e61 6c20 6172 6775 6d65 6e74 206e 616d nal argument nam │ │ │ │ -00066640: 6564 204f 7574 5269 6e67 3a0a 3d3d 3d3d ed OutRing:.==== │ │ │ │ +000664f0: 2a2a 2a2a 2a2a 0a0a 5365 6520 616c 736f ******..See also │ │ │ │ +00066500: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +00066510: 6e6f 7465 2065 7665 6e45 7874 4d6f 6475 note evenExtModu │ │ │ │ +00066520: 6c65 3a20 6576 656e 4578 744d 6f64 756c le: evenExtModul │ │ │ │ +00066530: 652c 202d 2d20 6576 656e 2070 6172 7420 e, -- even part │ │ │ │ +00066540: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76 of Ext^*(M,k) ov │ │ │ │ +00066550: 6572 2061 0a20 2020 2063 6f6d 706c 6574 er a. complet │ │ │ │ +00066560: 6520 696e 7465 7273 6563 7469 6f6e 2061 e intersection a │ │ │ │ +00066570: 7320 6d6f 6475 6c65 206f 7665 7220 4349 s module over CI │ │ │ │ +00066580: 206f 7065 7261 746f 7220 7269 6e67 0a20 operator ring. │ │ │ │ +00066590: 202a 202a 6e6f 7465 206f 6464 4578 744d * *note oddExtM │ │ │ │ +000665a0: 6f64 756c 653a 206f 6464 4578 744d 6f64 odule: oddExtMod │ │ │ │ +000665b0: 756c 652c 202d 2d20 6f64 6420 7061 7274 ule, -- odd part │ │ │ │ +000665c0: 206f 6620 4578 745e 2a28 4d2c 6b29 206f of Ext^*(M,k) o │ │ │ │ +000665d0: 7665 7220 6120 636f 6d70 6c65 7465 0a20 ver a complete. │ │ │ │ +000665e0: 2020 2069 6e74 6572 7365 6374 696f 6e20 intersection │ │ │ │ +000665f0: 6173 206d 6f64 756c 6520 6f76 6572 2043 as module over C │ │ │ │ +00066600: 4920 6f70 6572 6174 6f72 2072 696e 670a I operator ring. │ │ │ │ +00066610: 0a46 756e 6374 696f 6e73 2077 6974 6820 .Functions with │ │ │ │ +00066620: 6f70 7469 6f6e 616c 2061 7267 756d 656e optional argumen │ │ │ │ +00066630: 7420 6e61 6d65 6420 4f75 7452 696e 673a t named OutRing: │ │ │ │ +00066640: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ 00066650: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00066660: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00066670: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00066680: 2022 6576 656e 4578 744d 6f64 756c 6528 "evenExtModule( │ │ │ │ -00066690: 2e2e 2e2c 4f75 7452 696e 673d 3e2e 2e2e ...,OutRing=>... │ │ │ │ -000666a0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ -000666b0: 6576 656e 4578 744d 6f64 756c 653a 0a20 evenExtModule:. │ │ │ │ -000666c0: 2020 2065 7665 6e45 7874 4d6f 6475 6c65 evenExtModule │ │ │ │ -000666d0: 2c20 2d2d 2065 7665 6e20 7061 7274 206f , -- even part o │ │ │ │ -000666e0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665 f Ext^*(M,k) ove │ │ │ │ -000666f0: 7220 6120 636f 6d70 6c65 7465 2069 6e74 r a complete int │ │ │ │ -00066700: 6572 7365 6374 696f 6e20 6173 0a20 2020 ersection as. │ │ │ │ -00066710: 206d 6f64 756c 6520 6f76 6572 2043 4920 module over CI │ │ │ │ -00066720: 6f70 6572 6174 6f72 2072 696e 670a 2020 operator ring. │ │ │ │ -00066730: 2a20 226f 6464 4578 744d 6f64 756c 6528 * "oddExtModule( │ │ │ │ -00066740: 2e2e 2e2c 4f75 7452 696e 673d 3e2e 2e2e ...,OutRing=>... │ │ │ │ -00066750: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ -00066760: 6f64 6445 7874 4d6f 6475 6c65 3a20 6f64 oddExtModule: od │ │ │ │ -00066770: 6445 7874 4d6f 6475 6c65 2c0a 2020 2020 dExtModule,. │ │ │ │ -00066780: 2d2d 206f 6464 2070 6172 7420 6f66 2045 -- odd part of E │ │ │ │ -00066790: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061 xt^*(M,k) over a │ │ │ │ -000667a0: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -000667b0: 6563 7469 6f6e 2061 7320 6d6f 6475 6c65 ection as module │ │ │ │ -000667c0: 206f 7665 7220 4349 0a20 2020 206f 7065 over CI. ope │ │ │ │ -000667d0: 7261 746f 7220 7269 6e67 0a0a 466f 7220 rator ring..For │ │ │ │ -000667e0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -000667f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00066800: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -00066810: 6f74 6520 4f75 7452 696e 673a 204f 7574 ote OutRing: Out │ │ │ │ -00066820: 5269 6e67 2c20 6973 2061 202a 6e6f 7465 Ring, is a *note │ │ │ │ -00066830: 2073 796d 626f 6c3a 2028 4d61 6361 756c symbol: (Macaul │ │ │ │ -00066840: 6179 3244 6f63 2953 796d 626f 6c2c 2e0a ay2Doc)Symbol,.. │ │ │ │ -00066850: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +00066670: 0a0a 2020 2a20 2265 7665 6e45 7874 4d6f .. * "evenExtMo │ │ │ │ +00066680: 6475 6c65 282e 2e2e 2c4f 7574 5269 6e67 dule(...,OutRing │ │ │ │ +00066690: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a =>...)" -- see * │ │ │ │ +000666a0: 6e6f 7465 2065 7665 6e45 7874 4d6f 6475 note evenExtModu │ │ │ │ +000666b0: 6c65 3a0a 2020 2020 6576 656e 4578 744d le:. evenExtM │ │ │ │ +000666c0: 6f64 756c 652c 202d 2d20 6576 656e 2070 odule, -- even p │ │ │ │ +000666d0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b art of Ext^*(M,k │ │ │ │ +000666e0: 2920 6f76 6572 2061 2063 6f6d 706c 6574 ) over a complet │ │ │ │ +000666f0: 6520 696e 7465 7273 6563 7469 6f6e 2061 e intersection a │ │ │ │ +00066700: 730a 2020 2020 6d6f 6475 6c65 206f 7665 s. module ove │ │ │ │ +00066710: 7220 4349 206f 7065 7261 746f 7220 7269 r CI operator ri │ │ │ │ +00066720: 6e67 0a20 202a 2022 6f64 6445 7874 4d6f ng. * "oddExtMo │ │ │ │ +00066730: 6475 6c65 282e 2e2e 2c4f 7574 5269 6e67 dule(...,OutRing │ │ │ │ +00066740: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a =>...)" -- see * │ │ │ │ +00066750: 6e6f 7465 206f 6464 4578 744d 6f64 756c note oddExtModul │ │ │ │ +00066760: 653a 206f 6464 4578 744d 6f64 756c 652c e: oddExtModule, │ │ │ │ +00066770: 0a20 2020 202d 2d20 6f64 6420 7061 7274 . -- odd part │ │ │ │ +00066780: 206f 6620 4578 745e 2a28 4d2c 6b29 206f of Ext^*(M,k) o │ │ │ │ +00066790: 7665 7220 6120 636f 6d70 6c65 7465 2069 ver a complete i │ │ │ │ +000667a0: 6e74 6572 7365 6374 696f 6e20 6173 206d ntersection as m │ │ │ │ +000667b0: 6f64 756c 6520 6f76 6572 2043 490a 2020 odule over CI. │ │ │ │ +000667c0: 2020 6f70 6572 6174 6f72 2072 696e 670a operator ring. │ │ │ │ +000667d0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +000667e0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +000667f0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +00066800: 6374 202a 6e6f 7465 204f 7574 5269 6e67 ct *note OutRing │ │ │ │ +00066810: 3a20 4f75 7452 696e 672c 2069 7320 6120 : OutRing, is a │ │ │ │ +00066820: 2a6e 6f74 6520 7379 6d62 6f6c 3a20 284d *note symbol: (M │ │ │ │ +00066830: 6163 6175 6c61 7932 446f 6329 5379 6d62 acaulay2Doc)Symb │ │ │ │ +00066840: 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ol,...---------- │ │ │ │ +00066850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00066890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000668a0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -000668b0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -000668c0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -000668d0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -000668e0: 6175 6c61 7932 2d31 2e32 362e 3035 2b64 aulay2-1.26.05+d │ │ │ │ -000668f0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -00066900: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ -00066910: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -00066920: 6f6c 7574 696f 6e73 2e6d 323a 3336 3035 olutions.m2:3605 │ │ │ │ -00066930: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ -00066940: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -00066950: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ -00066960: 2c20 4e6f 6465 3a20 7073 694d 6170 732c , Node: psiMaps, │ │ │ │ -00066970: 204e 6578 743a 2072 6567 756c 6172 6974 Next: regularit │ │ │ │ -00066980: 7953 6571 7565 6e63 652c 2050 7265 763a ySequence, Prev: │ │ │ │ -00066990: 204f 7574 5269 6e67 2c20 5570 3a20 546f OutRing, Up: To │ │ │ │ -000669a0: 700a 0a70 7369 4d61 7073 202d 2d20 6c69 p..psiMaps -- li │ │ │ │ -000669b0: 7374 2074 6865 206d 6170 7320 2070 7369 st the maps psi │ │ │ │ -000669c0: 2870 293a 2042 5f31 2870 2920 2d2d 3e20 (p): B_1(p) --> │ │ │ │ -000669d0: 415f 3028 702d 3129 2069 6e20 6120 6d61 A_0(p-1) in a ma │ │ │ │ -000669e0: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ -000669f0: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a n.************** │ │ │ │ +00066890: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +000668a0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +000668b0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +000668c0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +000668d0: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +000668e0: 2e30 352b 6473 2f4d 322f 4d61 6361 756c .05+ds/M2/Macaul │ │ │ │ +000668f0: 6179 322f 7061 636b 6167 6573 2f0a 436f ay2/packages/.Co │ │ │ │ +00066900: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +00066910: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32 onResolutions.m2 │ │ │ │ +00066920: 3a33 3630 353a 302e 0a1f 0a46 696c 653a :3605:0....File: │ │ │ │ +00066930: 2043 6f6d 706c 6574 6549 6e74 6572 7365 CompleteInterse │ │ │ │ +00066940: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ +00066950: 2e69 6e66 6f2c 204e 6f64 653a 2070 7369 .info, Node: psi │ │ │ │ +00066960: 4d61 7073 2c20 4e65 7874 3a20 7265 6775 Maps, Next: regu │ │ │ │ +00066970: 6c61 7269 7479 5365 7175 656e 6365 2c20 laritySequence, │ │ │ │ +00066980: 5072 6576 3a20 4f75 7452 696e 672c 2055 Prev: OutRing, U │ │ │ │ +00066990: 703a 2054 6f70 0a0a 7073 694d 6170 7320 p: Top..psiMaps │ │ │ │ +000669a0: 2d2d 206c 6973 7420 7468 6520 6d61 7073 -- list the maps │ │ │ │ +000669b0: 2020 7073 6928 7029 3a20 425f 3128 7029 psi(p): B_1(p) │ │ │ │ +000669c0: 202d 2d3e 2041 5f30 2870 2d31 2920 696e --> A_0(p-1) in │ │ │ │ +000669d0: 2061 206d 6174 7269 7846 6163 746f 7269 a matrixFactori │ │ │ │ +000669e0: 7a61 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a zation.********* │ │ │ │ +000669f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00066a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00066a10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00066a20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00066a30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00066a40: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ -00066a50: 2020 2020 2020 7073 6d61 7073 203d 2070 psmaps = p │ │ │ │ -00066a60: 7369 4d61 7073 206d 660a 2020 2a20 496e siMaps mf. * In │ │ │ │ -00066a70: 7075 7473 3a0a 2020 2020 2020 2a20 6d66 puts:. * mf │ │ │ │ -00066a80: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ -00066a90: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ -00066aa0: 7374 2c2c 206f 7574 7075 7420 6f66 2061 st,, output of a │ │ │ │ -00066ab0: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ -00066ac0: 7469 6f6e 0a20 2020 2020 2020 2063 6f6d tion. com │ │ │ │ -00066ad0: 7075 7461 7469 6f6e 0a20 202a 204f 7574 putation. * Out │ │ │ │ -00066ae0: 7075 7473 3a0a 2020 2020 2020 2a20 7073 puts:. * ps │ │ │ │ -00066af0: 6d61 7073 2c20 6120 2a6e 6f74 6520 6c69 maps, a *note li │ │ │ │ -00066b00: 7374 3a20 284d 6163 6175 6c61 7932 446f st: (Macaulay2Do │ │ │ │ -00066b10: 6329 4c69 7374 2c2c 206c 6973 7420 6d61 c)List,, list ma │ │ │ │ -00066b20: 7472 6963 6573 2024 645f 703a 0a20 2020 trices $d_p:. │ │ │ │ -00066b30: 2020 2020 2042 5f31 2870 295c 746f 2041 B_1(p)\to A │ │ │ │ -00066b40: 5f30 2870 2d31 2924 0a0a 4465 7363 7269 _0(p-1)$..Descri │ │ │ │ -00066b50: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -00066b60: 3d0a 0a53 6565 2074 6865 2064 6f63 756d =..See the docum │ │ │ │ -00066b70: 656e 7461 7469 6f6e 2066 6f72 206d 6174 entation for mat │ │ │ │ -00066b80: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -00066b90: 2066 6f72 2061 6e20 6578 616d 706c 652e for an example. │ │ │ │ -00066ba0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -00066bb0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d ===.. * *note m │ │ │ │ -00066bc0: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ -00066bd0: 6f6e 3a20 6d61 7472 6978 4661 6374 6f72 on: matrixFactor │ │ │ │ -00066be0: 697a 6174 696f 6e2c 202d 2d20 4d61 7073 ization, -- Maps │ │ │ │ -00066bf0: 2069 6e20 6120 6869 6768 6572 0a20 2020 in a higher. │ │ │ │ -00066c00: 2063 6f64 696d 656e 7369 6f6e 206d 6174 codimension mat │ │ │ │ -00066c10: 7269 7820 6661 6374 6f72 697a 6174 696f rix factorizatio │ │ │ │ -00066c20: 6e0a 2020 2a20 2a6e 6f74 6520 4252 616e n. * *note BRan │ │ │ │ -00066c30: 6b73 3a20 4252 616e 6b73 2c20 2d2d 2072 ks: BRanks, -- r │ │ │ │ -00066c40: 616e 6b73 206f 6620 7468 6520 6d6f 6475 anks of the modu │ │ │ │ -00066c50: 6c65 7320 425f 6928 6429 2069 6e20 610a les B_i(d) in a. │ │ │ │ -00066c60: 2020 2020 6d61 7472 6978 4661 6374 6f72 matrixFactor │ │ │ │ -00066c70: 697a 6174 696f 6e0a 2020 2a20 2a6e 6f74 ization. * *not │ │ │ │ -00066c80: 6520 624d 6170 733a 2062 4d61 7073 2c20 e bMaps: bMaps, │ │ │ │ -00066c90: 2d2d 206c 6973 7420 7468 6520 6d61 7073 -- list the maps │ │ │ │ -00066ca0: 2020 645f 703a 425f 3128 7029 2d2d 3e42 d_p:B_1(p)-->B │ │ │ │ -00066cb0: 5f30 2870 2920 696e 2061 0a20 2020 206d _0(p) in a. m │ │ │ │ -00066cc0: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ -00066cd0: 6f6e 0a20 202a 202a 6e6f 7465 2064 4d61 on. * *note dMa │ │ │ │ -00066ce0: 7073 3a20 644d 6170 732c 202d 2d20 6c69 ps: dMaps, -- li │ │ │ │ -00066cf0: 7374 2074 6865 206d 6170 7320 2064 2870 st the maps d(p │ │ │ │ -00066d00: 293a 415f 3128 7029 2d2d 3e20 415f 3028 ):A_1(p)--> A_0( │ │ │ │ -00066d10: 7029 2069 6e20 610a 2020 2020 6d61 7472 p) in a. matr │ │ │ │ -00066d20: 6978 4661 6374 6f72 697a 6174 696f 6e0a ixFactorization. │ │ │ │ -00066d30: 2020 2a20 2a6e 6f74 6520 684d 6170 733a * *note hMaps: │ │ │ │ -00066d40: 2068 4d61 7073 2c20 2d2d 206c 6973 7420 hMaps, -- list │ │ │ │ -00066d50: 7468 6520 6d61 7073 2020 6828 7029 3a20 the maps h(p): │ │ │ │ -00066d60: 415f 3028 7029 2d2d 3e20 415f 3128 7029 A_0(p)--> A_1(p) │ │ │ │ -00066d70: 2069 6e20 610a 2020 2020 6d61 7472 6978 in a. matrix │ │ │ │ -00066d80: 4661 6374 6f72 697a 6174 696f 6e0a 0a57 Factorization..W │ │ │ │ -00066d90: 6179 7320 746f 2075 7365 2070 7369 4d61 ays to use psiMa │ │ │ │ -00066da0: 7073 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ps:.============ │ │ │ │ -00066db0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2270 ========.. * "p │ │ │ │ -00066dc0: 7369 4d61 7073 284c 6973 7429 220a 0a46 siMaps(List)"..F │ │ │ │ -00066dd0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00066de0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00066df0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00066e00: 202a 6e6f 7465 2070 7369 4d61 7073 3a20 *note psiMaps: │ │ │ │ -00066e10: 7073 694d 6170 732c 2069 7320 6120 2a6e psiMaps, is a *n │ │ │ │ -00066e20: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ -00066e30: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ -00066e40: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00066e50: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +00066a30: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ +00066a40: 3a20 0a20 2020 2020 2020 2070 736d 6170 : . psmap │ │ │ │ +00066a50: 7320 3d20 7073 694d 6170 7320 6d66 0a20 s = psiMaps mf. │ │ │ │ +00066a60: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +00066a70: 202a 206d 662c 2061 202a 6e6f 7465 206c * mf, a *note l │ │ │ │ +00066a80: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ +00066a90: 6f63 294c 6973 742c 2c20 6f75 7470 7574 oc)List,, output │ │ │ │ +00066aa0: 206f 6620 6120 6d61 7472 6978 4661 6374 of a matrixFact │ │ │ │ +00066ab0: 6f72 697a 6174 696f 6e0a 2020 2020 2020 orization. │ │ │ │ +00066ac0: 2020 636f 6d70 7574 6174 696f 6e0a 2020 computation. │ │ │ │ +00066ad0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +00066ae0: 202a 2070 736d 6170 732c 2061 202a 6e6f * psmaps, a *no │ │ │ │ +00066af0: 7465 206c 6973 743a 2028 4d61 6361 756c te list: (Macaul │ │ │ │ +00066b00: 6179 3244 6f63 294c 6973 742c 2c20 6c69 ay2Doc)List,, li │ │ │ │ +00066b10: 7374 206d 6174 7269 6365 7320 2464 5f70 st matrices $d_p │ │ │ │ +00066b20: 3a0a 2020 2020 2020 2020 425f 3128 7029 :. B_1(p) │ │ │ │ +00066b30: 5c74 6f20 415f 3028 702d 3129 240a 0a44 \to A_0(p-1)$..D │ │ │ │ +00066b40: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +00066b50: 3d3d 3d3d 3d3d 0a0a 5365 6520 7468 6520 ======..See the │ │ │ │ +00066b60: 646f 6375 6d65 6e74 6174 696f 6e20 666f documentation fo │ │ │ │ +00066b70: 7220 6d61 7472 6978 4661 6374 6f72 697a r matrixFactoriz │ │ │ │ +00066b80: 6174 696f 6e20 666f 7220 616e 2065 7861 ation for an exa │ │ │ │ +00066b90: 6d70 6c65 2e0a 0a53 6565 2061 6c73 6f0a mple...See also. │ │ │ │ +00066ba0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +00066bb0: 6f74 6520 6d61 7472 6978 4661 6374 6f72 ote matrixFactor │ │ │ │ +00066bc0: 697a 6174 696f 6e3a 206d 6174 7269 7846 ization: matrixF │ │ │ │ +00066bd0: 6163 746f 7269 7a61 7469 6f6e 2c20 2d2d actorization, -- │ │ │ │ +00066be0: 204d 6170 7320 696e 2061 2068 6967 6865 Maps in a highe │ │ │ │ +00066bf0: 720a 2020 2020 636f 6469 6d65 6e73 696f r. codimensio │ │ │ │ +00066c00: 6e20 6d61 7472 6978 2066 6163 746f 7269 n matrix factori │ │ │ │ +00066c10: 7a61 7469 6f6e 0a20 202a 202a 6e6f 7465 zation. * *note │ │ │ │ +00066c20: 2042 5261 6e6b 733a 2042 5261 6e6b 732c BRanks: BRanks, │ │ │ │ +00066c30: 202d 2d20 7261 6e6b 7320 6f66 2074 6865 -- ranks of the │ │ │ │ +00066c40: 206d 6f64 756c 6573 2042 5f69 2864 2920 modules B_i(d) │ │ │ │ +00066c50: 696e 2061 0a20 2020 206d 6174 7269 7846 in a. matrixF │ │ │ │ +00066c60: 6163 746f 7269 7a61 7469 6f6e 0a20 202a actorization. * │ │ │ │ +00066c70: 202a 6e6f 7465 2062 4d61 7073 3a20 624d *note bMaps: bM │ │ │ │ +00066c80: 6170 732c 202d 2d20 6c69 7374 2074 6865 aps, -- list the │ │ │ │ +00066c90: 206d 6170 7320 2064 5f70 3a42 5f31 2870 maps d_p:B_1(p │ │ │ │ +00066ca0: 292d 2d3e 425f 3028 7029 2069 6e20 610a )-->B_0(p) in a. │ │ │ │ +00066cb0: 2020 2020 6d61 7472 6978 4661 6374 6f72 matrixFactor │ │ │ │ +00066cc0: 697a 6174 696f 6e0a 2020 2a20 2a6e 6f74 ization. * *not │ │ │ │ +00066cd0: 6520 644d 6170 733a 2064 4d61 7073 2c20 e dMaps: dMaps, │ │ │ │ +00066ce0: 2d2d 206c 6973 7420 7468 6520 6d61 7073 -- list the maps │ │ │ │ +00066cf0: 2020 6428 7029 3a41 5f31 2870 292d 2d3e d(p):A_1(p)--> │ │ │ │ +00066d00: 2041 5f30 2870 2920 696e 2061 0a20 2020 A_0(p) in a. │ │ │ │ +00066d10: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ +00066d20: 7469 6f6e 0a20 202a 202a 6e6f 7465 2068 tion. * *note h │ │ │ │ +00066d30: 4d61 7073 3a20 684d 6170 732c 202d 2d20 Maps: hMaps, -- │ │ │ │ +00066d40: 6c69 7374 2074 6865 206d 6170 7320 2068 list the maps h │ │ │ │ +00066d50: 2870 293a 2041 5f30 2870 292d 2d3e 2041 (p): A_0(p)--> A │ │ │ │ +00066d60: 5f31 2870 2920 696e 2061 0a20 2020 206d _1(p) in a. m │ │ │ │ +00066d70: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ +00066d80: 6f6e 0a0a 5761 7973 2074 6f20 7573 6520 on..Ways to use │ │ │ │ +00066d90: 7073 694d 6170 733a 0a3d 3d3d 3d3d 3d3d psiMaps:.======= │ │ │ │ +00066da0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +00066db0: 202a 2022 7073 694d 6170 7328 4c69 7374 * "psiMaps(List │ │ │ │ +00066dc0: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ +00066dd0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +00066de0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +00066df0: 626a 6563 7420 2a6e 6f74 6520 7073 694d bject *note psiM │ │ │ │ +00066e00: 6170 733a 2070 7369 4d61 7073 2c20 6973 aps: psiMaps, is │ │ │ │ +00066e10: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ +00066e20: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ +00066e30: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ +00066e40: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +00066e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00066e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00066ea0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00066eb0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00066ec0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00066ed0: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00066ee0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -00066ef0: 3035 2b64 732f 4d32 2f4d 6163 6175 6c61 05+ds/M2/Macaula │ │ │ │ -00066f00: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d y2/packages/.Com │ │ │ │ -00066f10: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ -00066f20: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a nResolutions.m2: │ │ │ │ -00066f30: 3434 3832 3a30 2e0a 1f0a 4669 6c65 3a20 4482:0....File: │ │ │ │ -00066f40: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -00066f50: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -00066f60: 696e 666f 2c20 4e6f 6465 3a20 7265 6775 info, Node: regu │ │ │ │ -00066f70: 6c61 7269 7479 5365 7175 656e 6365 2c20 laritySequence, │ │ │ │ -00066f80: 4e65 7874 3a20 5332 2c20 5072 6576 3a20 Next: S2, Prev: │ │ │ │ -00066f90: 7073 694d 6170 732c 2055 703a 2054 6f70 psiMaps, Up: Top │ │ │ │ -00066fa0: 0a0a 7265 6775 6c61 7269 7479 5365 7175 ..regularitySequ │ │ │ │ -00066fb0: 656e 6365 202d 2d20 7265 6775 6c61 7269 ence -- regulari │ │ │ │ -00066fc0: 7479 206f 6620 4578 7420 6d6f 6475 6c65 ty of Ext module │ │ │ │ -00066fd0: 7320 666f 7220 6120 7365 7175 656e 6365 s for a sequence │ │ │ │ -00066fe0: 206f 6620 4d43 4d20 6170 7072 6f78 696d of MCM approxim │ │ │ │ -00066ff0: 6174 696f 6e73 0a2a 2a2a 2a2a 2a2a 2a2a ations.********* │ │ │ │ +00066e90: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +00066ea0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +00066eb0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +00066ec0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +00066ed0: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +00066ee0: 312e 3236 2e30 352b 6473 2f4d 322f 4d61 1.26.05+ds/M2/Ma │ │ │ │ +00066ef0: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +00066f00: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ +00066f10: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ +00066f20: 732e 6d32 3a34 3438 323a 302e 0a1f 0a46 s.m2:4482:0....F │ │ │ │ +00066f30: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ +00066f40: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00066f50: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ +00066f60: 2072 6567 756c 6172 6974 7953 6571 7565 regularitySeque │ │ │ │ +00066f70: 6e63 652c 204e 6578 743a 2053 322c 2050 nce, Next: S2, P │ │ │ │ +00066f80: 7265 763a 2070 7369 4d61 7073 2c20 5570 rev: psiMaps, Up │ │ │ │ +00066f90: 3a20 546f 700a 0a72 6567 756c 6172 6974 : Top..regularit │ │ │ │ +00066fa0: 7953 6571 7565 6e63 6520 2d2d 2072 6567 ySequence -- reg │ │ │ │ +00066fb0: 756c 6172 6974 7920 6f66 2045 7874 206d ularity of Ext m │ │ │ │ +00066fc0: 6f64 756c 6573 2066 6f72 2061 2073 6571 odules for a seq │ │ │ │ +00066fd0: 7565 6e63 6520 6f66 204d 434d 2061 7070 uence of MCM app │ │ │ │ +00066fe0: 726f 7869 6d61 7469 6f6e 730a 2a2a 2a2a roximations.**** │ │ │ │ +00066ff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00067000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00067010: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00067020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00067030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00067040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -00067050: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -00067060: 204c 203d 2072 6567 756c 6172 6974 7953 L = regularityS │ │ │ │ -00067070: 6571 7565 6e63 6520 2852 2c4d 290a 2020 equence (R,M). │ │ │ │ -00067080: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ -00067090: 2a20 522c 2061 202a 6e6f 7465 206c 6973 * R, a *note lis │ │ │ │ -000670a0: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ -000670b0: 294c 6973 742c 2c20 6c69 7374 206f 6620 )List,, list of │ │ │ │ -000670c0: 7269 6e67 7320 525f 6920 3d0a 2020 2020 rings R_i =. │ │ │ │ -000670d0: 2020 2020 532f 2866 5f30 2e2e 665f 7b28 S/(f_0..f_{( │ │ │ │ -000670e0: 692d 3129 7d29 2c20 636f 6d70 6c65 7465 i-1)}), complete │ │ │ │ -000670f0: 2069 6e74 6572 7365 6374 696f 6e73 0a20 intersections. │ │ │ │ -00067100: 2020 2020 202a 204d 2c20 6120 2a6e 6f74 * M, a *not │ │ │ │ -00067110: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ -00067120: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ -00067130: 206d 6f64 756c 6520 6f76 6572 2052 5f63 module over R_c │ │ │ │ -00067140: 2077 6865 7265 2063 203d 0a20 2020 2020 where c =. │ │ │ │ -00067150: 2020 206c 656e 6774 6820 5220 2d20 312e length R - 1. │ │ │ │ -00067160: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -00067170: 2020 2020 2a20 4c2c 2061 202a 6e6f 7465 * L, a *note │ │ │ │ -00067180: 206c 6973 743a 2028 4d61 6361 756c 6179 list: (Macaulay │ │ │ │ -00067190: 3244 6f63 294c 6973 742c 2c20 4c69 7374 2Doc)List,, List │ │ │ │ -000671a0: 206f 6620 7061 6972 7320 7b72 6567 756c of pairs {regul │ │ │ │ -000671b0: 6172 6974 790a 2020 2020 2020 2020 6576 arity. ev │ │ │ │ -000671c0: 656e 4578 744d 6f64 756c 6520 4d5f 692c enExtModule M_i, │ │ │ │ -000671d0: 2072 6567 756c 6172 6974 7920 6f64 6445 regularity oddE │ │ │ │ -000671e0: 7874 4d6f 6475 6c65 204d 5f69 290a 0a44 xtModule M_i)..D │ │ │ │ -000671f0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -00067200: 3d3d 3d3d 3d3d 0a0a 436f 6d70 7574 6573 ======..Computes │ │ │ │ -00067210: 2074 6865 206e 6f6e 2d66 7265 6520 7061 the non-free pa │ │ │ │ -00067220: 7274 7320 4d5f 6920 6f66 2074 6865 204d rts M_i of the M │ │ │ │ -00067230: 434d 2061 7070 726f 7869 6d61 7469 6f6e CM approximation │ │ │ │ -00067240: 2074 6f20 4d20 6f76 6572 2052 5f69 2c0a to M over R_i,. │ │ │ │ -00067250: 7374 6f70 7069 6e67 2077 6865 6e20 4d5f stopping when M_ │ │ │ │ -00067260: 6920 6265 636f 6d65 7320 6672 6565 2c20 i becomes free, │ │ │ │ -00067270: 616e 6420 7265 7475 726e 7320 7468 6520 and returns the │ │ │ │ -00067280: 6c69 7374 2077 686f 7365 2065 6c65 6d65 list whose eleme │ │ │ │ -00067290: 6e74 7320 6172 6520 7468 650a 7061 6972 nts are the.pair │ │ │ │ -000672a0: 7320 6f66 2072 6567 756c 6172 6974 6965 s of regularitie │ │ │ │ -000672b0: 732c 2073 7461 7274 696e 6720 7769 7468 s, starting with │ │ │ │ -000672c0: 204d 5f7b 2863 2d31 297d 204e 6f74 6520 M_{(c-1)} Note │ │ │ │ -000672d0: 7468 6174 2074 6865 2066 6972 7374 2070 that the first p │ │ │ │ -000672e0: 6169 7220 6973 2066 6f72 0a74 6865 0a0a air is for.the.. │ │ │ │ -000672f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00067040: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +00067050: 2020 2020 2020 4c20 3d20 7265 6775 6c61 L = regula │ │ │ │ +00067060: 7269 7479 5365 7175 656e 6365 2028 522c ritySequence (R, │ │ │ │ +00067070: 4d29 0a20 202a 2049 6e70 7574 733a 0a20 M). * Inputs:. │ │ │ │ +00067080: 2020 2020 202a 2052 2c20 6120 2a6e 6f74 * R, a *not │ │ │ │ +00067090: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ +000670a0: 7932 446f 6329 4c69 7374 2c2c 206c 6973 y2Doc)List,, lis │ │ │ │ +000670b0: 7420 6f66 2072 696e 6773 2052 5f69 203d t of rings R_i = │ │ │ │ +000670c0: 0a20 2020 2020 2020 2053 2f28 665f 302e . S/(f_0. │ │ │ │ +000670d0: 2e66 5f7b 2869 2d31 297d 292c 2063 6f6d .f_{(i-1)}), com │ │ │ │ +000670e0: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ +000670f0: 6f6e 730a 2020 2020 2020 2a20 4d2c 2061 ons. * M, a │ │ │ │ +00067100: 202a 6e6f 7465 206d 6f64 756c 653a 2028 *note module: ( │ │ │ │ +00067110: 4d61 6361 756c 6179 3244 6f63 294d 6f64 Macaulay2Doc)Mod │ │ │ │ +00067120: 756c 652c 2c20 6d6f 6475 6c65 206f 7665 ule,, module ove │ │ │ │ +00067130: 7220 525f 6320 7768 6572 6520 6320 3d0a r R_c where c =. │ │ │ │ +00067140: 2020 2020 2020 2020 6c65 6e67 7468 2052 length R │ │ │ │ +00067150: 202d 2031 2e0a 2020 2a20 4f75 7470 7574 - 1.. * Output │ │ │ │ +00067160: 733a 0a20 2020 2020 202a 204c 2c20 6120 s:. * L, a │ │ │ │ +00067170: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ +00067180: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ +00067190: 204c 6973 7420 6f66 2070 6169 7273 207b List of pairs { │ │ │ │ +000671a0: 7265 6775 6c61 7269 7479 0a20 2020 2020 regularity. │ │ │ │ +000671b0: 2020 2065 7665 6e45 7874 4d6f 6475 6c65 evenExtModule │ │ │ │ +000671c0: 204d 5f69 2c20 7265 6775 6c61 7269 7479 M_i, regularity │ │ │ │ +000671d0: 206f 6464 4578 744d 6f64 756c 6520 4d5f oddExtModule M_ │ │ │ │ +000671e0: 6929 0a0a 4465 7363 7269 7074 696f 6e0a i)..Description. │ │ │ │ +000671f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a43 6f6d ===========..Com │ │ │ │ +00067200: 7075 7465 7320 7468 6520 6e6f 6e2d 6672 putes the non-fr │ │ │ │ +00067210: 6565 2070 6172 7473 204d 5f69 206f 6620 ee parts M_i of │ │ │ │ +00067220: 7468 6520 4d43 4d20 6170 7072 6f78 696d the MCM approxim │ │ │ │ +00067230: 6174 696f 6e20 746f 204d 206f 7665 7220 ation to M over │ │ │ │ +00067240: 525f 692c 0a73 746f 7070 696e 6720 7768 R_i,.stopping wh │ │ │ │ +00067250: 656e 204d 5f69 2062 6563 6f6d 6573 2066 en M_i becomes f │ │ │ │ +00067260: 7265 652c 2061 6e64 2072 6574 7572 6e73 ree, and returns │ │ │ │ +00067270: 2074 6865 206c 6973 7420 7768 6f73 6520 the list whose │ │ │ │ +00067280: 656c 656d 656e 7473 2061 7265 2074 6865 elements are the │ │ │ │ +00067290: 0a70 6169 7273 206f 6620 7265 6775 6c61 .pairs of regula │ │ │ │ +000672a0: 7269 7469 6573 2c20 7374 6172 7469 6e67 rities, starting │ │ │ │ +000672b0: 2077 6974 6820 4d5f 7b28 632d 3129 7d20 with M_{(c-1)} │ │ │ │ +000672c0: 4e6f 7465 2074 6861 7420 7468 6520 6669 Note that the fi │ │ │ │ +000672d0: 7273 7420 7061 6972 2069 7320 666f 720a rst pair is for. │ │ │ │ +000672e0: 7468 650a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d the..+---------- │ │ │ │ +000672f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00067330: 0a7c 6931 203a 2063 203d 2033 3b64 3d32 .|i1 : c = 3;d=2 │ │ │ │ +00067320: 2d2d 2d2d 2b0a 7c69 3120 3a20 6320 3d20 ----+.|i1 : c = │ │ │ │ +00067330: 333b 643d 3220 2020 2020 2020 2020 2020 3;d=2 │ │ │ │ 00067340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067370: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00067360: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00067370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000673a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000673b0: 207c 0a7c 6f32 203d 2032 2020 2020 2020 |.|o2 = 2 │ │ │ │ +000673a0: 2020 2020 2020 7c0a 7c6f 3220 3d20 3220 |.|o2 = 2 │ │ │ │ +000673b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000673c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000673d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000673e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000673f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000673e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000673f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067430: 2d2d 2d2b 0a7c 6933 203a 2052 203d 2073 ---+.|i3 : R = s │ │ │ │ -00067440: 6574 7570 5269 6e67 7328 632c 6429 3b20 etupRings(c,d); │ │ │ │ +00067420: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ +00067430: 5220 3d20 7365 7475 7052 696e 6773 2863 R = setupRings(c │ │ │ │ +00067440: 2c64 293b 2020 2020 2020 2020 2020 2020 ,d); │ │ │ │ 00067450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067470: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00067460: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00067470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000674a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000674b0: 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 6320 -----+.|i4 : Rc │ │ │ │ -000674c0: 3d20 525f 6320 2020 2020 2020 2020 2020 = R_c │ │ │ │ +000674a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ +000674b0: 3a20 5263 203d 2052 5f63 2020 2020 2020 : Rc = R_c │ │ │ │ +000674c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000674d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000674e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000674f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000674e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000674f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067530: 2020 2020 2020 207c 0a7c 6f34 203d 2052 |.|o4 = R │ │ │ │ -00067540: 6320 2020 2020 2020 2020 2020 2020 2020 c │ │ │ │ +00067520: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00067530: 3420 3d20 5263 2020 2020 2020 2020 2020 4 = Rc │ │ │ │ +00067540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067570: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00067560: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00067570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000675a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000675b0: 2020 2020 2020 2020 207c 0a7c 6f34 203a |.|o4 : │ │ │ │ -000675c0: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +000675a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000675b0: 7c6f 3420 3a20 5175 6f74 6965 6e74 5269 |o4 : QuotientRi │ │ │ │ +000675c0: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ 000675d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000675e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000675f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +000675e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000675f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00067600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ -00067640: 203a 204d 203d 2063 6f6b 6572 206d 6174 : M = coker mat │ │ │ │ -00067650: 7269 787b 7b52 635f 302c 5263 5f31 2c52 rix{{Rc_0,Rc_1,R │ │ │ │ -00067660: 635f 327d 2c7b 5263 5f31 2c52 635f 322c c_2},{Rc_1,Rc_2, │ │ │ │ -00067670: 5263 5f30 7d7d 2020 2020 2020 7c0a 7c20 Rc_0}} |.| │ │ │ │ +00067630: 2b0a 7c69 3520 3a20 4d20 3d20 636f 6b65 +.|i5 : M = coke │ │ │ │ +00067640: 7220 6d61 7472 6978 7b7b 5263 5f30 2c52 r matrix{{Rc_0,R │ │ │ │ +00067650: 635f 312c 5263 5f32 7d2c 7b52 635f 312c c_1,Rc_2},{Rc_1, │ │ │ │ +00067660: 5263 5f32 2c52 635f 307d 7d20 2020 2020 Rc_2,Rc_0}} │ │ │ │ +00067670: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00067680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000676a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000676b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000676c0: 6f35 203d 2063 6f6b 6572 6e65 6c20 7c20 o5 = cokernel | │ │ │ │ -000676d0: 785f 3020 785f 3120 785f 3220 7c20 2020 x_0 x_1 x_2 | │ │ │ │ +000676b0: 2020 7c0a 7c6f 3520 3d20 636f 6b65 726e |.|o5 = cokern │ │ │ │ +000676c0: 656c 207c 2078 5f30 2078 5f31 2078 5f32 el | x_0 x_1 x_2 │ │ │ │ +000676d0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000676e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000676f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00067700: 7c20 2020 2020 2020 2020 2020 2020 207c | | │ │ │ │ -00067710: 2078 5f31 2078 5f32 2078 5f30 207c 2020 x_1 x_2 x_0 | │ │ │ │ +000676f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00067700: 2020 2020 7c20 785f 3120 785f 3220 785f | x_1 x_2 x_ │ │ │ │ +00067710: 3020 7c20 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ 00067720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067730: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00067740: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00067730: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00067740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067780: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00067790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000677a0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000677b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000677c0: 207c 0a7c 6f35 203a 2052 632d 6d6f 6475 |.|o5 : Rc-modu │ │ │ │ -000677d0: 6c65 2c20 7175 6f74 6965 6e74 206f 6620 le, quotient of │ │ │ │ -000677e0: 5263 2020 2020 2020 2020 2020 2020 2020 Rc │ │ │ │ -000677f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067800: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00067770: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00067780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00067790: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +000677a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000677b0: 2020 2020 2020 7c0a 7c6f 3520 3a20 5263 |.|o5 : Rc │ │ │ │ +000677c0: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ +000677d0: 7420 6f66 2052 6320 2020 2020 2020 2020 t of Rc │ │ │ │ +000677e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000677f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00067800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067840: 2d2d 2d2b 0a7c 6936 203a 2072 6567 756c ---+.|i6 : regul │ │ │ │ -00067850: 6172 6974 7953 6571 7565 6e63 6528 522c aritySequence(R, │ │ │ │ -00067860: 4d29 2020 2020 2020 2020 2020 2020 2020 M) │ │ │ │ -00067870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067880: 2020 2020 7c0a 7c72 6567 2065 7665 6e20 |.|reg even │ │ │ │ -00067890: 6578 742c 2073 6f63 2064 6567 7320 6576 ext, soc degs ev │ │ │ │ -000678a0: 656e 2065 7874 2c20 7265 6720 6f64 6420 en ext, reg odd │ │ │ │ -000678b0: 6578 742c 2073 6f63 2064 6567 7320 6f64 ext, soc degs od │ │ │ │ -000678c0: 6420 6578 747c 0a7c 2020 2020 2020 2020 d ext|.| │ │ │ │ +00067830: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ +00067840: 7265 6775 6c61 7269 7479 5365 7175 656e regularitySequen │ │ │ │ +00067850: 6365 2852 2c4d 2920 2020 2020 2020 2020 ce(R,M) │ │ │ │ +00067860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00067870: 2020 2020 2020 2020 207c 0a7c 7265 6720 |.|reg │ │ │ │ +00067880: 6576 656e 2065 7874 2c20 736f 6320 6465 even ext, soc de │ │ │ │ +00067890: 6773 2065 7665 6e20 6578 742c 2072 6567 gs even ext, reg │ │ │ │ +000678a0: 206f 6464 2065 7874 2c20 736f 6320 6465 odd ext, soc de │ │ │ │ +000678b0: 6773 206f 6464 2065 7874 7c0a 7c20 2020 gs odd ext|.| │ │ │ │ +000678c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000678d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000678e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000678f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067900: 2020 2020 2020 7c0a 7c7b 332c 207b 312c |.|{3, {1, │ │ │ │ -00067910: 2031 2c20 317d 2c20 322c 207b 312c 2031 1, 1}, 2, {1, 1 │ │ │ │ -00067920: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ -00067930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067940: 2020 2020 2020 207c 0a7c 7b32 2c20 7b30 |.|{2, {0 │ │ │ │ -00067950: 2c20 302c 2030 2c20 317d 2c20 322c 207b , 0, 0, 1}, 2, { │ │ │ │ -00067960: 302c 2030 2c20 307d 7d20 2020 2020 2020 0, 0, 0}} │ │ │ │ -00067970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067980: 2020 2020 2020 2020 7c0a 7c7b 302c 207b |.|{0, { │ │ │ │ -00067990: 7d2c 2030 2c20 7b7d 7d20 2020 2020 2020 }, 0, {}} │ │ │ │ +000678f0: 2020 2020 2020 2020 2020 207c 0a7c 7b33 |.|{3 │ │ │ │ +00067900: 2c20 7b31 2c20 312c 2031 7d2c 2032 2c20 , {1, 1, 1}, 2, │ │ │ │ +00067910: 7b31 2c20 317d 7d20 2020 2020 2020 2020 {1, 1}} │ │ │ │ +00067920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00067930: 2020 2020 2020 2020 2020 2020 7c0a 7c7b |.|{ │ │ │ │ +00067940: 322c 207b 302c 2030 2c20 302c 2031 7d2c 2, {0, 0, 0, 1}, │ │ │ │ +00067950: 2032 2c20 7b30 2c20 302c 2030 7d7d 2020 2, {0, 0, 0}} │ │ │ │ +00067960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00067970: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00067980: 7b30 2c20 7b7d 2c20 302c 207b 7d7d 2020 {0, {}, 0, {}} │ │ │ │ +00067990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000679a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000679b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000679c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000679b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000679c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000679d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000679e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000679f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 ----------+..See │ │ │ │ -00067a10: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -00067a20: 2020 2a20 2a6e 6f74 6520 6170 7072 6f78 * *note approx │ │ │ │ -00067a30: 696d 6174 696f 6e3a 2028 4d43 4d41 7070 imation: (MCMApp │ │ │ │ -00067a40: 726f 7869 6d61 7469 6f6e 7329 6170 7072 roximations)appr │ │ │ │ -00067a50: 6f78 696d 6174 696f 6e2c 202d 2d20 7265 oximation, -- re │ │ │ │ -00067a60: 7475 726e 7320 7061 6972 206f 660a 2020 turns pair of. │ │ │ │ -00067a70: 2020 636f 6d70 6f6e 656e 7473 206f 6620 components of │ │ │ │ -00067a80: 7468 6520 6d61 7020 6672 6f6d 2074 6865 the map from the │ │ │ │ -00067a90: 204d 434d 2061 7070 726f 7869 6d61 7469 MCM approximati │ │ │ │ -00067aa0: 6f6e 0a20 202a 202a 6e6f 7465 2061 7573 on. * *note aus │ │ │ │ -00067ab0: 6c61 6e64 6572 496e 7661 7269 616e 743a landerInvariant: │ │ │ │ -00067ac0: 2028 4d43 4d41 7070 726f 7869 6d61 7469 (MCMApproximati │ │ │ │ -00067ad0: 6f6e 7329 6175 736c 616e 6465 7249 6e76 ons)auslanderInv │ │ │ │ -00067ae0: 6172 6961 6e74 2c20 2d2d 0a20 2020 206d ariant, --. m │ │ │ │ -00067af0: 6561 7375 7265 7320 6661 696c 7572 6520 easures failure │ │ │ │ -00067b00: 6f66 2073 7572 6a65 6374 6976 6974 7920 of surjectivity │ │ │ │ -00067b10: 6f66 2074 6865 2065 7373 656e 7469 616c of the essential │ │ │ │ -00067b20: 204d 434d 2061 7070 726f 7869 6d61 7469 MCM approximati │ │ │ │ -00067b30: 6f6e 0a0a 5761 7973 2074 6f20 7573 6520 on..Ways to use │ │ │ │ -00067b40: 7265 6775 6c61 7269 7479 5365 7175 656e regularitySequen │ │ │ │ -00067b50: 6365 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ce:.============ │ │ │ │ -00067b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00067b70: 3d3d 3d0a 0a20 202a 2022 7265 6775 6c61 ===.. * "regula │ │ │ │ -00067b80: 7269 7479 5365 7175 656e 6365 284c 6973 ritySequence(Lis │ │ │ │ -00067b90: 742c 4d6f 6475 6c65 2922 0a0a 466f 7220 t,Module)"..For │ │ │ │ -00067ba0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -00067bb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00067bc0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -00067bd0: 6f74 6520 7265 6775 6c61 7269 7479 5365 ote regularitySe │ │ │ │ -00067be0: 7175 656e 6365 3a20 7265 6775 6c61 7269 quence: regulari │ │ │ │ -00067bf0: 7479 5365 7175 656e 6365 2c20 6973 2061 tySequence, is a │ │ │ │ -00067c00: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ -00067c10: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61 nction: (Macaula │ │ │ │ -00067c20: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ -00067c30: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ +000679f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00067a00: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ +00067a10: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2061 ===.. * *note a │ │ │ │ +00067a20: 7070 726f 7869 6d61 7469 6f6e 3a20 284d pproximation: (M │ │ │ │ +00067a30: 434d 4170 7072 6f78 696d 6174 696f 6e73 CMApproximations │ │ │ │ +00067a40: 2961 7070 726f 7869 6d61 7469 6f6e 2c20 )approximation, │ │ │ │ +00067a50: 2d2d 2072 6574 7572 6e73 2070 6169 7220 -- returns pair │ │ │ │ +00067a60: 6f66 0a20 2020 2063 6f6d 706f 6e65 6e74 of. component │ │ │ │ +00067a70: 7320 6f66 2074 6865 206d 6170 2066 726f s of the map fro │ │ │ │ +00067a80: 6d20 7468 6520 4d43 4d20 6170 7072 6f78 m the MCM approx │ │ │ │ +00067a90: 696d 6174 696f 6e0a 2020 2a20 2a6e 6f74 imation. * *not │ │ │ │ +00067aa0: 6520 6175 736c 616e 6465 7249 6e76 6172 e auslanderInvar │ │ │ │ +00067ab0: 6961 6e74 3a20 284d 434d 4170 7072 6f78 iant: (MCMApprox │ │ │ │ +00067ac0: 696d 6174 696f 6e73 2961 7573 6c61 6e64 imations)ausland │ │ │ │ +00067ad0: 6572 496e 7661 7269 616e 742c 202d 2d0a erInvariant, --. │ │ │ │ +00067ae0: 2020 2020 6d65 6173 7572 6573 2066 6169 measures fai │ │ │ │ +00067af0: 6c75 7265 206f 6620 7375 726a 6563 7469 lure of surjecti │ │ │ │ +00067b00: 7669 7479 206f 6620 7468 6520 6573 7365 vity of the esse │ │ │ │ +00067b10: 6e74 6961 6c20 4d43 4d20 6170 7072 6f78 ntial MCM approx │ │ │ │ +00067b20: 696d 6174 696f 6e0a 0a57 6179 7320 746f imation..Ways to │ │ │ │ +00067b30: 2075 7365 2072 6567 756c 6172 6974 7953 use regularityS │ │ │ │ +00067b40: 6571 7565 6e63 653a 0a3d 3d3d 3d3d 3d3d equence:.======= │ │ │ │ +00067b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00067b60: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2272 ========.. * "r │ │ │ │ +00067b70: 6567 756c 6172 6974 7953 6571 7565 6e63 egularitySequenc │ │ │ │ +00067b80: 6528 4c69 7374 2c4d 6f64 756c 6529 220a e(List,Module)". │ │ │ │ +00067b90: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +00067ba0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +00067bb0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +00067bc0: 6374 202a 6e6f 7465 2072 6567 756c 6172 ct *note regular │ │ │ │ +00067bd0: 6974 7953 6571 7565 6e63 653a 2072 6567 itySequence: reg │ │ │ │ +00067be0: 756c 6172 6974 7953 6571 7565 6e63 652c ularitySequence, │ │ │ │ +00067bf0: 2069 7320 6120 2a6e 6f74 6520 6d65 7468 is a *note meth │ │ │ │ +00067c00: 6f64 0a66 756e 6374 696f 6e3a 2028 4d61 od.function: (Ma │ │ │ │ +00067c10: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ +00067c20: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d dFunction,...--- │ │ │ │ +00067c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067c80: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -00067c90: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -00067ca0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -00067cb0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -00067cc0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -00067cd0: 3236 2e30 352b 6473 2f4d 322f 4d61 6361 26.05+ds/M2/Maca │ │ │ │ -00067ce0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ -00067cf0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -00067d00: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -00067d10: 6d32 3a32 3630 303a 302e 0a1f 0a46 696c m2:2600:0....Fil │ │ │ │ -00067d20: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ -00067d30: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -00067d40: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2053 ns.info, Node: S │ │ │ │ -00067d50: 322c 204e 6578 743a 2053 6861 6d61 7368 2, Next: Shamash │ │ │ │ -00067d60: 2c20 5072 6576 3a20 7265 6775 6c61 7269 , Prev: regulari │ │ │ │ -00067d70: 7479 5365 7175 656e 6365 2c20 5570 3a20 tySequence, Up: │ │ │ │ -00067d80: 546f 700a 0a53 3220 2d2d 2055 6e69 7665 Top..S2 -- Unive │ │ │ │ -00067d90: 7273 616c 206d 6170 2074 6f20 6120 6d6f rsal map to a mo │ │ │ │ -00067da0: 6475 6c65 2073 6174 6973 6679 696e 6720 dule satisfying │ │ │ │ -00067db0: 5365 7272 6527 7320 636f 6e64 6974 696f Serre's conditio │ │ │ │ -00067dc0: 6e20 5332 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a n S2.*********** │ │ │ │ +00067c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +00067c80: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +00067c90: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +00067ca0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +00067cb0: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +00067cc0: 7932 2d31 2e32 362e 3035 2b64 732f 4d32 y2-1.26.05+ds/M2 │ │ │ │ +00067cd0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +00067ce0: 6765 732f 0a43 6f6d 706c 6574 6549 6e74 ges/.CompleteInt │ │ │ │ +00067cf0: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00067d00: 696f 6e73 2e6d 323a 3236 3030 3a30 2e0a ions.m2:2600:0.. │ │ │ │ +00067d10: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465 ..File: Complete │ │ │ │ +00067d20: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +00067d30: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f lutions.info, No │ │ │ │ +00067d40: 6465 3a20 5332 2c20 4e65 7874 3a20 5368 de: S2, Next: Sh │ │ │ │ +00067d50: 616d 6173 682c 2050 7265 763a 2072 6567 amash, Prev: reg │ │ │ │ +00067d60: 756c 6172 6974 7953 6571 7565 6e63 652c ularitySequence, │ │ │ │ +00067d70: 2055 703a 2054 6f70 0a0a 5332 202d 2d20 Up: Top..S2 -- │ │ │ │ +00067d80: 556e 6976 6572 7361 6c20 6d61 7020 746f Universal map to │ │ │ │ +00067d90: 2061 206d 6f64 756c 6520 7361 7469 7366 a module satisf │ │ │ │ +00067da0: 7969 6e67 2053 6572 7265 2773 2063 6f6e ying Serre's con │ │ │ │ +00067db0: 6469 7469 6f6e 2053 320a 2a2a 2a2a 2a2a dition S2.****** │ │ │ │ +00067dc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00067dd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00067de0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00067df0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00067e00: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -00067e10: 200a 2020 2020 2020 2020 6620 3d20 5332 . f = S2 │ │ │ │ -00067e20: 2862 2c4d 290a 2020 2a20 496e 7075 7473 (b,M). * Inputs │ │ │ │ -00067e30: 3a0a 2020 2020 2020 2a20 622c 2061 6e20 :. * b, an │ │ │ │ -00067e40: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ -00067e50: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ -00067e60: 2c20 6465 6772 6565 2062 6f75 6e64 2074 , degree bound t │ │ │ │ -00067e70: 6f20 7768 6963 6820 746f 2063 6172 7279 o which to carry │ │ │ │ -00067e80: 0a20 2020 2020 2020 2074 6865 2063 6f6d . the com │ │ │ │ -00067e90: 7075 7461 7469 6f6e 0a20 2020 2020 202a putation. * │ │ │ │ -00067ea0: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -00067eb0: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -00067ec0: 6329 4d6f 6475 6c65 2c2c 200a 2020 2a20 c)Module,, . * │ │ │ │ -00067ed0: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -00067ee0: 2066 2c20 6120 2a6e 6f74 6520 6d61 7472 f, a *note matr │ │ │ │ -00067ef0: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ -00067f00: 6329 4d61 7472 6978 2c2c 2064 6566 696e c)Matrix,, defin │ │ │ │ -00067f10: 696e 6720 6120 6d61 7020 4d2d 2d3e 4d27 ing a map M-->M' │ │ │ │ -00067f20: 2074 6861 740a 2020 2020 2020 2020 6167 that. ag │ │ │ │ -00067f30: 7265 6573 2077 6974 6820 7468 6520 5332 rees with the S2 │ │ │ │ -00067f40: 2d69 6669 6361 7469 6f6e 206f 6620 4d20 -ification of M │ │ │ │ -00067f50: 696e 2064 6567 7265 6573 2024 5c67 6571 in degrees $\geq │ │ │ │ -00067f60: 2062 240a 0a44 6573 6372 6970 7469 6f6e b$..Description │ │ │ │ -00067f70: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966 .===========..If │ │ │ │ -00067f80: 204d 2069 7320 6120 6772 6164 6564 206d M is a graded m │ │ │ │ -00067f90: 6f64 756c 6520 6f76 6572 2061 2072 696e odule over a rin │ │ │ │ -00067fa0: 6720 532c 2074 6865 6e20 7468 6520 5332 g S, then the S2 │ │ │ │ -00067fb0: 2d69 6669 6361 7469 6f6e 206f 6620 4d20 -ification of M │ │ │ │ -00067fc0: 6973 205c 7375 6d5f 7b64 0a5c 696e 205a is \sum_{d.\in Z │ │ │ │ -00067fd0: 5a7d 2048 5e30 2828 7368 6561 6620 4d29 Z} H^0((sheaf M) │ │ │ │ -00067fe0: 2864 2929 2c20 7768 6963 6820 6d61 7920 (d)), which may │ │ │ │ -00067ff0: 6265 2063 6f6d 7075 7465 6420 6173 206c be computed as l │ │ │ │ -00068000: 696d 5f7b 642d 3e5c 696e 6674 797d 2048 im_{d->\infty} H │ │ │ │ -00068010: 6f6d 2849 5f64 2c4d 292c 0a77 6865 7265 om(I_d,M),.where │ │ │ │ -00068020: 2049 5f64 2069 7320 616e 7920 7365 7175 I_d is any sequ │ │ │ │ -00068030: 656e 6365 206f 6620 6964 6561 6c73 2063 ence of ideals c │ │ │ │ -00068040: 6f6e 7461 696e 6564 2069 6e20 6869 6768 ontained in high │ │ │ │ -00068050: 6572 2061 6e64 2068 6967 6865 7220 706f er and higher po │ │ │ │ -00068060: 7765 7273 206f 660a 535f 2b2e 2054 6865 wers of.S_+. The │ │ │ │ -00068070: 7265 2069 7320 6120 6e61 7475 7261 6c20 re is a natural │ │ │ │ -00068080: 7265 7374 7269 6374 696f 6e20 6d61 7020 restriction map │ │ │ │ -00068090: 663a 204d 203d 2048 6f6d 2853 2c4d 2920 f: M = Hom(S,M) │ │ │ │ -000680a0: 5c74 6f20 486f 6d28 495f 642c 4d29 2e20 \to Hom(I_d,M). │ │ │ │ -000680b0: 5765 0a63 6f6d 7075 7465 2061 6c6c 2074 We.compute all t │ │ │ │ -000680c0: 6869 7320 7573 696e 6720 7468 6520 6964 his using the id │ │ │ │ -000680d0: 6561 6c73 2049 5f64 2067 656e 6572 6174 eals I_d generat │ │ │ │ -000680e0: 6564 2062 7920 7468 6520 642d 7468 2070 ed by the d-th p │ │ │ │ -000680f0: 6f77 6572 7320 6f66 2074 6865 0a76 6172 owers of the.var │ │ │ │ -00068100: 6961 626c 6573 2069 6e20 532e 0a0a 5369 iables in S...Si │ │ │ │ -00068110: 6e63 6520 7468 6520 7265 7375 6c74 206d nce the result m │ │ │ │ -00068120: 6179 206e 6f74 2062 6520 6669 6e69 7465 ay not be finite │ │ │ │ -00068130: 6c79 2067 656e 6572 6174 6564 2028 7468 ly generated (th │ │ │ │ -00068140: 6973 2068 6170 7065 6e73 2069 6620 616e is happens if an │ │ │ │ -00068150: 6420 6f6e 6c79 2069 6620 4d0a 6861 7320 d only if M.has │ │ │ │ -00068160: 616e 2061 7373 6f63 6961 7465 6420 7072 an associated pr │ │ │ │ -00068170: 696d 6520 6f66 2064 696d 656e 7369 6f6e ime of dimension │ │ │ │ -00068180: 2031 292c 2077 6520 636f 6d70 7574 6520 1), we compute │ │ │ │ -00068190: 6f6e 6c79 2075 7020 746f 2061 2073 7065 only up to a spe │ │ │ │ -000681a0: 6369 6669 6564 0a64 6567 7265 6520 626f cified.degree bo │ │ │ │ -000681b0: 756e 6420 622e 2046 6f72 2074 6865 2072 und b. For the r │ │ │ │ -000681c0: 6573 756c 7420 746f 2062 6520 636f 7272 esult to be corr │ │ │ │ -000681d0: 6563 7420 646f 776e 2074 6f20 6465 6772 ect down to degr │ │ │ │ -000681e0: 6565 2062 2c20 6974 2069 7320 7375 6666 ee b, it is suff │ │ │ │ -000681f0: 6963 6965 6e74 0a74 6f20 636f 6d70 7574 icient.to comput │ │ │ │ -00068200: 6520 486f 6d28 492c 4d29 2077 6865 7265 e Hom(I,M) where │ │ │ │ -00068210: 2049 205c 7375 6273 6574 2028 535f 2b29 I \subset (S_+) │ │ │ │ -00068220: 5e7b 722d 627d 2e0a 0a2b 2d2d 2d2d 2d2d ^{r-b}...+------ │ │ │ │ +00067df0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +00067e00: 7361 6765 3a20 0a20 2020 2020 2020 2066 sage: . f │ │ │ │ +00067e10: 203d 2053 3228 622c 4d29 0a20 202a 2049 = S2(b,M). * I │ │ │ │ +00067e20: 6e70 7574 733a 0a20 2020 2020 202a 2062 nputs:. * b │ │ │ │ +00067e30: 2c20 616e 202a 6e6f 7465 2069 6e74 6567 , an *note integ │ │ │ │ +00067e40: 6572 3a20 284d 6163 6175 6c61 7932 446f er: (Macaulay2Do │ │ │ │ +00067e50: 6329 5a5a 2c2c 2064 6567 7265 6520 626f c)ZZ,, degree bo │ │ │ │ +00067e60: 756e 6420 746f 2077 6869 6368 2074 6f20 und to which to │ │ │ │ +00067e70: 6361 7272 790a 2020 2020 2020 2020 7468 carry. th │ │ │ │ +00067e80: 6520 636f 6d70 7574 6174 696f 6e0a 2020 e computation. │ │ │ │ +00067e90: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +00067ea0: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +00067eb0: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +00067ec0: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +00067ed0: 2020 2020 2a20 662c 2061 202a 6e6f 7465 * f, a *note │ │ │ │ +00067ee0: 206d 6174 7269 783a 2028 4d61 6361 756c matrix: (Macaul │ │ │ │ +00067ef0: 6179 3244 6f63 294d 6174 7269 782c 2c20 ay2Doc)Matrix,, │ │ │ │ +00067f00: 6465 6669 6e69 6e67 2061 206d 6170 204d defining a map M │ │ │ │ +00067f10: 2d2d 3e4d 2720 7468 6174 0a20 2020 2020 -->M' that. │ │ │ │ +00067f20: 2020 2061 6772 6565 7320 7769 7468 2074 agrees with t │ │ │ │ +00067f30: 6865 2053 322d 6966 6963 6174 696f 6e20 he S2-ification │ │ │ │ +00067f40: 6f66 204d 2069 6e20 6465 6772 6565 7320 of M in degrees │ │ │ │ +00067f50: 245c 6765 7120 6224 0a0a 4465 7363 7269 $\geq b$..Descri │ │ │ │ +00067f60: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ +00067f70: 3d0a 0a49 6620 4d20 6973 2061 2067 7261 =..If M is a gra │ │ │ │ +00067f80: 6465 6420 6d6f 6475 6c65 206f 7665 7220 ded module over │ │ │ │ +00067f90: 6120 7269 6e67 2053 2c20 7468 656e 2074 a ring S, then t │ │ │ │ +00067fa0: 6865 2053 322d 6966 6963 6174 696f 6e20 he S2-ification │ │ │ │ +00067fb0: 6f66 204d 2069 7320 5c73 756d 5f7b 640a of M is \sum_{d. │ │ │ │ +00067fc0: 5c69 6e20 5a5a 7d20 485e 3028 2873 6865 \in ZZ} H^0((she │ │ │ │ +00067fd0: 6166 204d 2928 6429 292c 2077 6869 6368 af M)(d)), which │ │ │ │ +00067fe0: 206d 6179 2062 6520 636f 6d70 7574 6564 may be computed │ │ │ │ +00067ff0: 2061 7320 6c69 6d5f 7b64 2d3e 5c69 6e66 as lim_{d->\inf │ │ │ │ +00068000: 7479 7d20 486f 6d28 495f 642c 4d29 2c0a ty} Hom(I_d,M),. │ │ │ │ +00068010: 7768 6572 6520 495f 6420 6973 2061 6e79 where I_d is any │ │ │ │ +00068020: 2073 6571 7565 6e63 6520 6f66 2069 6465 sequence of ide │ │ │ │ +00068030: 616c 7320 636f 6e74 6169 6e65 6420 696e als contained in │ │ │ │ +00068040: 2068 6967 6865 7220 616e 6420 6869 6768 higher and high │ │ │ │ +00068050: 6572 2070 6f77 6572 7320 6f66 0a53 5f2b er powers of.S_+ │ │ │ │ +00068060: 2e20 5468 6572 6520 6973 2061 206e 6174 . There is a nat │ │ │ │ +00068070: 7572 616c 2072 6573 7472 6963 7469 6f6e ural restriction │ │ │ │ +00068080: 206d 6170 2066 3a20 4d20 3d20 486f 6d28 map f: M = Hom( │ │ │ │ +00068090: 532c 4d29 205c 746f 2048 6f6d 2849 5f64 S,M) \to Hom(I_d │ │ │ │ +000680a0: 2c4d 292e 2057 650a 636f 6d70 7574 6520 ,M). We.compute │ │ │ │ +000680b0: 616c 6c20 7468 6973 2075 7369 6e67 2074 all this using t │ │ │ │ +000680c0: 6865 2069 6465 616c 7320 495f 6420 6765 he ideals I_d ge │ │ │ │ +000680d0: 6e65 7261 7465 6420 6279 2074 6865 2064 nerated by the d │ │ │ │ +000680e0: 2d74 6820 706f 7765 7273 206f 6620 7468 -th powers of th │ │ │ │ +000680f0: 650a 7661 7269 6162 6c65 7320 696e 2053 e.variables in S │ │ │ │ +00068100: 2e0a 0a53 696e 6365 2074 6865 2072 6573 ...Since the res │ │ │ │ +00068110: 756c 7420 6d61 7920 6e6f 7420 6265 2066 ult may not be f │ │ │ │ +00068120: 696e 6974 656c 7920 6765 6e65 7261 7465 initely generate │ │ │ │ +00068130: 6420 2874 6869 7320 6861 7070 656e 7320 d (this happens │ │ │ │ +00068140: 6966 2061 6e64 206f 6e6c 7920 6966 204d if and only if M │ │ │ │ +00068150: 0a68 6173 2061 6e20 6173 736f 6369 6174 .has an associat │ │ │ │ +00068160: 6564 2070 7269 6d65 206f 6620 6469 6d65 ed prime of dime │ │ │ │ +00068170: 6e73 696f 6e20 3129 2c20 7765 2063 6f6d nsion 1), we com │ │ │ │ +00068180: 7075 7465 206f 6e6c 7920 7570 2074 6f20 pute only up to │ │ │ │ +00068190: 6120 7370 6563 6966 6965 640a 6465 6772 a specified.degr │ │ │ │ +000681a0: 6565 2062 6f75 6e64 2062 2e20 466f 7220 ee bound b. For │ │ │ │ +000681b0: 7468 6520 7265 7375 6c74 2074 6f20 6265 the result to be │ │ │ │ +000681c0: 2063 6f72 7265 6374 2064 6f77 6e20 746f correct down to │ │ │ │ +000681d0: 2064 6567 7265 6520 622c 2069 7420 6973 degree b, it is │ │ │ │ +000681e0: 2073 7566 6669 6369 656e 740a 746f 2063 sufficient.to c │ │ │ │ +000681f0: 6f6d 7075 7465 2048 6f6d 2849 2c4d 2920 ompute Hom(I,M) │ │ │ │ +00068200: 7768 6572 6520 4920 5c73 7562 7365 7420 where I \subset │ │ │ │ +00068210: 2853 5f2b 295e 7b72 2d62 7d2e 0a0a 2b2d (S_+)^{r-b}...+- │ │ │ │ +00068220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068270: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b -------+.|i1 : k │ │ │ │ -00068280: 6b3d 5a5a 2f31 3031 2020 2020 2020 2020 k=ZZ/101 │ │ │ │ +00068260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068270: 3120 3a20 6b6b 3d5a 5a2f 3130 3120 2020 1 : kk=ZZ/101 │ │ │ │ +00068280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000682a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000682b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000682c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000682b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000682c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000682d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000682e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000682f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068310: 2020 2020 2020 207c 0a7c 6f31 203d 206b |.|o1 = k │ │ │ │ -00068320: 6b20 2020 2020 2020 2020 2020 2020 2020 k │ │ │ │ +00068300: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068310: 3120 3d20 6b6b 2020 2020 2020 2020 2020 1 = kk │ │ │ │ +00068320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068360: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068350: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000683a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000683b0: 2020 2020 2020 207c 0a7c 6f31 203a 2051 |.|o1 : Q │ │ │ │ -000683c0: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +000683a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000683b0: 3120 3a20 5175 6f74 6965 6e74 5269 6e67 1 : QuotientRing │ │ │ │ +000683c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000683d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000683e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000683f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068400: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000683f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00068400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068450: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 -------+.|i2 : S │ │ │ │ -00068460: 203d 206b 6b5b 612c 622c 632c 645d 2020 = kk[a,b,c,d] │ │ │ │ +00068440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068450: 3220 3a20 5320 3d20 6b6b 5b61 2c62 2c63 2 : S = kk[a,b,c │ │ │ │ +00068460: 2c64 5d20 2020 2020 2020 2020 2020 2020 ,d] │ │ │ │ 00068470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000684a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068490: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000684a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000684b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000684c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000684d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000684e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000684f0: 2020 2020 2020 207c 0a7c 6f32 203d 2053 |.|o2 = S │ │ │ │ +000684e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000684f0: 3220 3d20 5320 2020 2020 2020 2020 2020 2 = S │ │ │ │ 00068500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068540: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068530: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068590: 2020 2020 2020 207c 0a7c 6f32 203a 2050 |.|o2 : P │ │ │ │ -000685a0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ +00068580: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068590: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269 2 : PolynomialRi │ │ │ │ +000685a0: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ 000685b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000685c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000685d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000685e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000685d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000685e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000685f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068630: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 204d -------+.|i3 : M │ │ │ │ -00068640: 203d 2074 7275 6e63 6174 6528 332c 535e = truncate(3,S^ │ │ │ │ -00068650: 3129 2020 2020 2020 2020 2020 2020 2020 1) │ │ │ │ +00068620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068630: 3320 3a20 4d20 3d20 7472 756e 6361 7465 3 : M = truncate │ │ │ │ +00068640: 2833 2c53 5e31 2920 2020 2020 2020 2020 (3,S^1) │ │ │ │ +00068650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068680: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068670: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000686a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000686b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000686c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000686d0: 2020 2020 2020 207c 0a7c 6f33 203d 2069 |.|o3 = i │ │ │ │ -000686e0: 6d61 6765 207c 2064 3320 6364 3220 6264 mage | d3 cd2 bd │ │ │ │ -000686f0: 3220 6164 3220 6332 6420 6263 6420 6163 2 ad2 c2d bcd ac │ │ │ │ -00068700: 6420 6232 6420 6162 6420 6132 6420 6333 d b2d abd a2d c3 │ │ │ │ -00068710: 2062 6332 2061 6332 2062 3263 2061 6263 bc2 ac2 b2c abc │ │ │ │ -00068720: 2061 3263 2062 337c 0a7c 2020 2020 2020 a2c b3|.| │ │ │ │ +000686c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000686d0: 3320 3d20 696d 6167 6520 7c20 6433 2063 3 = image | d3 c │ │ │ │ +000686e0: 6432 2062 6432 2061 6432 2063 3264 2062 d2 bd2 ad2 c2d b │ │ │ │ +000686f0: 6364 2061 6364 2062 3264 2061 6264 2061 cd acd b2d abd a │ │ │ │ +00068700: 3264 2063 3320 6263 3220 6163 3220 6232 2d c3 bc2 ac2 b2 │ │ │ │ +00068710: 6320 6162 6320 6132 6320 6233 7c0a 7c20 c abc a2c b3|.| │ │ │ │ +00068720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068770: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068790: 2020 2020 2020 2031 2020 2020 2020 2020 1 │ │ │ │ +00068760: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00068780: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ +00068790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000687a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000687b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000687c0: 2020 2020 2020 207c 0a7c 6f33 203a 2053 |.|o3 : S │ │ │ │ -000687d0: 2d6d 6f64 756c 652c 2073 7562 6d6f 6475 -module, submodu │ │ │ │ -000687e0: 6c65 206f 6620 5320 2020 2020 2020 2020 le of S │ │ │ │ +000687b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000687c0: 3320 3a20 532d 6d6f 6475 6c65 2c20 7375 3 : S-module, su │ │ │ │ +000687d0: 626d 6f64 756c 6520 6f66 2053 2020 2020 bmodule of S │ │ │ │ +000687e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000687f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068810: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d |.|------ │ │ │ │ +00068800: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ +00068810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068860: 2d2d 2d2d 2d2d 2d7c 0a7c 6162 3220 6132 -------|.|ab2 a2 │ │ │ │ -00068870: 6220 6133 207c 2020 2020 2020 2020 2020 b a3 | │ │ │ │ +00068850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c61 ------------|.|a │ │ │ │ +00068860: 6232 2061 3262 2061 3320 7c20 2020 2020 b2 a2b a3 | │ │ │ │ +00068870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000688a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000688b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000688a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000688b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000688c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000688d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000688e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000688f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068900: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2062 -------+.|i4 : b │ │ │ │ -00068910: 6574 7469 206d 6174 7269 7820 5332 2830 etti matrix S2(0 │ │ │ │ -00068920: 2c4d 2920 2020 2020 2020 2020 2020 2020 ,M) │ │ │ │ +000688f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068900: 3420 3a20 6265 7474 6920 6d61 7472 6978 4 : betti matrix │ │ │ │ +00068910: 2053 3228 302c 4d29 2020 2020 2020 2020 S2(0,M) │ │ │ │ +00068920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068950: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068940: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000689a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000689b0: 2020 2020 2020 3020 2031 2020 2020 2020 0 1 │ │ │ │ +00068990: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000689a0: 2020 2020 2020 2020 2020 2030 2020 3120 0 1 │ │ │ │ +000689b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000689c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000689d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000689e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000689f0: 2020 2020 2020 207c 0a7c 6f34 203d 2074 |.|o4 = t │ │ │ │ -00068a00: 6f74 616c 3a20 3120 3230 2020 2020 2020 otal: 1 20 │ │ │ │ +000689e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000689f0: 3420 3d20 746f 7461 6c3a 2031 2032 3020 4 = total: 1 20 │ │ │ │ +00068a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068a40: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068a50: 2020 2030 3a20 3120 202e 2020 2020 2020 0: 1 . │ │ │ │ +00068a30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068a40: 2020 2020 2020 2020 303a 2031 2020 2e20 0: 1 . │ │ │ │ +00068a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068a90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068aa0: 2020 2031 3a20 2e20 202e 2020 2020 2020 1: . . │ │ │ │ +00068a80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068a90: 2020 2020 2020 2020 313a 202e 2020 2e20 1: . . │ │ │ │ +00068aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068ae0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068af0: 2020 2032 3a20 2e20 3230 2020 2020 2020 2: . 20 │ │ │ │ +00068ad0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068ae0: 2020 2020 2020 2020 323a 202e 2032 3020 2: . 20 │ │ │ │ +00068af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068b30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068b20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068b80: 2020 2020 2020 207c 0a7c 6f34 203a 2042 |.|o4 : B │ │ │ │ -00068b90: 6574 7469 5461 6c6c 7920 2020 2020 2020 ettiTally │ │ │ │ +00068b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068b80: 3420 3a20 4265 7474 6954 616c 6c79 2020 4 : BettiTally │ │ │ │ +00068b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068bd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00068bc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00068bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068c20: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2062 -------+.|i5 : b │ │ │ │ -00068c30: 6574 7469 206d 6174 7269 7820 5332 2831 etti matrix S2(1 │ │ │ │ -00068c40: 2c4d 2920 2020 2020 2020 2020 2020 2020 ,M) │ │ │ │ +00068c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068c20: 3520 3a20 6265 7474 6920 6d61 7472 6978 5 : betti matrix │ │ │ │ +00068c30: 2053 3228 312c 4d29 2020 2020 2020 2020 S2(1,M) │ │ │ │ +00068c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068c70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068c60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068cc0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068cd0: 2020 2020 2020 3020 2031 2020 2020 2020 0 1 │ │ │ │ +00068cb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068cc0: 2020 2020 2020 2020 2020 2030 2020 3120 0 1 │ │ │ │ +00068cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068d10: 2020 2020 2020 207c 0a7c 6f35 203d 2074 |.|o5 = t │ │ │ │ -00068d20: 6f74 616c 3a20 3120 3230 2020 2020 2020 otal: 1 20 │ │ │ │ +00068d00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068d10: 3520 3d20 746f 7461 6c3a 2031 2032 3020 5 = total: 1 20 │ │ │ │ +00068d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068d60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068d70: 2020 2030 3a20 3120 202e 2020 2020 2020 0: 1 . │ │ │ │ +00068d50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068d60: 2020 2020 2020 2020 303a 2031 2020 2e20 0: 1 . │ │ │ │ +00068d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068db0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068dc0: 2020 2031 3a20 2e20 202e 2020 2020 2020 1: . . │ │ │ │ +00068da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068db0: 2020 2020 2020 2020 313a 202e 2020 2e20 1: . . │ │ │ │ +00068dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068e00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068e10: 2020 2032 3a20 2e20 3230 2020 2020 2020 2: . 20 │ │ │ │ +00068df0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068e00: 2020 2020 2020 2020 323a 202e 2032 3020 2: . 20 │ │ │ │ +00068e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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+00068ee0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00068ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068f40: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204d -------+.|i6 : M │ │ │ │ -00068f50: 203d 2053 5e31 2f69 6e74 6572 7365 6374 = S^1/intersect │ │ │ │ -00068f60: 2869 6465 616c 2261 2c62 2c63 222c 2069 (ideal"a,b,c", i │ │ │ │ -00068f70: 6465 616c 2262 2c63 2c64 222c 6964 6561 deal"b,c,d",idea │ │ │ │ -00068f80: 6c22 632c 642c 6122 2c69 6465 616c 2264 l"c,d,a",ideal"d │ │ │ │ -00068f90: 2c61 2c62 2229 207c 0a7c 2020 2020 2020 ,a,b") |.| │ │ │ │ +00068f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068f40: 3620 3a20 4d20 3d20 535e 312f 696e 7465 6 : M = S^1/inte │ │ │ │ +00068f50: 7273 6563 7428 6964 6561 6c22 612c 622c rsect(ideal"a,b, │ │ │ │ +00068f60: 6322 2c20 6964 6561 6c22 622c 632c 6422 c", ideal"b,c,d" │ │ │ │ +00068f70: 2c69 6465 616c 2263 2c64 2c61 222c 6964 ,ideal"c,d,a",id │ │ │ │ +00068f80: 6561 6c22 642c 612c 6222 2920 7c0a 7c20 eal"d,a,b") |.| │ │ │ │ +00068f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068fe0: 2020 2020 2020 207c 0a7c 6f36 203d 2063 |.|o6 = c │ │ │ │ -00068ff0: 6f6b 6572 6e65 6c20 7c20 6364 2062 6420 okernel | cd bd │ │ │ │ -00069000: 6164 2062 6320 6163 2061 6220 7c20 2020 ad bc ac ab | │ │ │ │ +00068fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068fe0: 3620 3d20 636f 6b65 726e 656c 207c 2063 6 = cokernel | c │ │ │ │ +00068ff0: 6420 6264 2061 6420 6263 2061 6320 6162 d bd ad bc ac ab │ │ │ │ +00069000: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00069010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069030: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00069020: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069080: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00069090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000690a0: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +00069070: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069090: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +000690a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000690b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000690c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000690d0: 2020 2020 2020 207c 0a7c 6f36 203a 2053 |.|o6 : S │ │ │ │ -000690e0: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ -000690f0: 7420 6f66 2053 2020 2020 2020 2020 2020 t of S │ │ │ │ +000690c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000690d0: 3620 3a20 532d 6d6f 6475 6c65 2c20 7175 6 : S-module, qu │ │ │ │ +000690e0: 6f74 6965 6e74 206f 6620 5320 2020 2020 otient of S │ │ │ │ +000690f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069120: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069110: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069170: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2070 -------+.|i7 : p │ │ │ │ -00069180: 7275 6e65 2073 6f75 7263 6520 5332 2830 rune source S2(0 │ │ │ │ -00069190: 2c4d 2920 2020 2020 2020 2020 2020 2020 ,M) │ │ │ │ +00069160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00069170: 3720 3a20 7072 756e 6520 736f 7572 6365 7 : prune source │ │ │ │ +00069180: 2053 3228 302c 4d29 2020 2020 2020 2020 S2(0,M) │ │ │ │ +00069190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000691a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000691b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000691c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 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0a7c 2020 2020 2020 |.| │ │ │ │ +00069250: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000692a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000692b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000692c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000692d0: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +000692a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000692b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000692c0: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +000692d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000692e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000692f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069300: 2020 2020 2020 207c 0a7c 6f37 203a 2053 |.|o7 : S │ │ │ │ -00069310: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ -00069320: 7420 6f66 2053 2020 2020 2020 2020 2020 t of S │ │ │ │ +000692f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00069300: 3720 3a20 532d 6d6f 6475 6c65 2c20 7175 7 : S-module, qu │ │ │ │ +00069310: 6f74 6965 6e74 206f 6620 5320 2020 2020 otient of S │ │ │ │ +00069320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069350: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069340: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000693a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2070 -------+.|i8 : p │ │ │ │ -000693b0: 7275 6e65 2074 6172 6765 7420 5332 2830 rune target S2(0 │ │ │ │ -000693c0: 2c4d 2920 2020 2020 2020 2020 2020 2020 ,M) │ │ │ │ +00069390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000693a0: 3820 3a20 7072 756e 6520 7461 7267 6574 8 : prune target │ │ │ │ +000693b0: 2053 3228 302c 4d29 2020 2020 2020 2020 S2(0,M) │ │ │ │ +000693c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000693d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000693e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000693f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000693e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000693f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069440: 2020 2020 2020 207c 0a7c 6f38 203d 2063 |.|o8 = c │ │ │ │ -00069450: 6f6b 6572 6e65 6c20 7b2d 317d 207c 2064 okernel {-1} | d │ │ │ │ -00069460: 2063 2062 2030 2030 2030 2030 2030 2030 c b 0 0 0 0 0 0 │ │ │ │ -00069470: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ -00069480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069490: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000694a0: 2020 2020 2020 2020 7b2d 317d 207c 2030 {-1} | 0 │ │ │ │ -000694b0: 2030 2030 2064 2063 2061 2030 2030 2030 0 0 d c a 0 0 0 │ │ │ │ -000694c0: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ -000694d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000694e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000694f0: 2020 2020 2020 2020 7b2d 317d 207c 2030 {-1} | 0 │ │ │ │ -00069500: 2030 2030 2030 2030 2030 2064 2062 2061 0 0 0 0 0 d b a │ │ │ │ -00069510: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ -00069520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069530: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00069540: 2020 2020 2020 2020 7b2d 317d 207c 2030 {-1} | 0 │ │ │ │ -00069550: 2030 2030 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 0 │ │ │ │ -00069560: 2063 2062 2061 207c 2020 2020 2020 2020 c b a | │ │ │ │ -00069570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069580: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00069430: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00069440: 3820 3d20 636f 6b65 726e 656c 207b 2d31 8 = cokernel {-1 │ │ │ │ +00069450: 7d20 7c20 6420 6320 6220 3020 3020 3020 } | d c b 0 0 0 │ │ │ │ +00069460: 3020 3020 3020 3020 3020 3020 7c20 2020 0 0 0 0 0 0 | │ │ │ │ +00069470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069480: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069490: 2020 2020 2020 2020 2020 2020 207b 2d31 {-1 │ │ │ │ +000694a0: 7d20 7c20 3020 3020 3020 6420 6320 6120 } | 0 0 0 d c a │ │ │ │ +000694b0: 3020 3020 3020 3020 3020 3020 7c20 2020 0 0 0 0 0 0 | │ │ │ │ +000694c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000694d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000694e0: 2020 2020 2020 2020 2020 2020 207b 2d31 {-1 │ │ │ │ +000694f0: 7d20 7c20 3020 3020 3020 3020 3020 3020 } | 0 0 0 0 0 0 │ │ │ │ +00069500: 6420 6220 6120 3020 3020 3020 7c20 2020 d b a 0 0 0 | │ │ │ │ +00069510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069520: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069530: 2020 2020 2020 2020 2020 2020 207b 2d31 {-1 │ │ │ │ +00069540: 7d20 7c20 3020 3020 3020 3020 3020 3020 } | 0 0 0 0 0 0 │ │ │ │ +00069550: 3020 3020 3020 6320 6220 6120 7c20 2020 0 0 0 c b a | │ │ │ │ +00069560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069570: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000695a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000695b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000695c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000695d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000695e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000695f0: 2020 2020 2020 3420 2020 2020 2020 2020 4 │ │ │ │ +000695c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000695d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000695e0: 2020 2020 2020 2020 2020 2034 2020 2020 4 │ │ │ │ +000695f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069620: 2020 2020 2020 207c 0a7c 6f38 203a 2053 |.|o8 : S │ │ │ │ -00069630: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ -00069640: 7420 6f66 2053 2020 2020 2020 2020 2020 t of S │ │ │ │ +00069610: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00069620: 3820 3a20 532d 6d6f 6475 6c65 2c20 7175 8 : S-module, qu │ │ │ │ +00069630: 6f74 6965 6e74 206f 6620 5320 2020 2020 otient of S │ │ │ │ +00069640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069670: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069660: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000696a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000696b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000696c0: 2d2d 2d2d 2d2d 2d2b 0a0a 4174 206f 6e65 -------+..At one │ │ │ │ -000696d0: 2074 696d 6520 4445 2068 6f70 6564 2074 time DE hoped t │ │ │ │ -000696e0: 6861 742c 2069 6620 4d20 7765 7265 2061 hat, if M were a │ │ │ │ -000696f0: 206d 6f64 756c 6520 6f76 6572 2074 6865 module over the │ │ │ │ -00069700: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -00069710: 6563 7469 6f6e 2052 0a77 6974 6820 7265 ection R.with re │ │ │ │ -00069720: 7369 6475 6520 6669 656c 6420 6b2c 2074 sidue field k, t │ │ │ │ -00069730: 6865 6e20 7468 6520 6e61 7475 7261 6c20 hen the natural │ │ │ │ -00069740: 6d61 7020 6672 6f6d 2022 636f 6d70 6c65 map from "comple │ │ │ │ -00069750: 7465 2220 4578 7420 6d6f 6475 6c65 2022 te" Ext module " │ │ │ │ -00069760: 2877 6964 6568 6174 0a45 7874 295f 5228 (widehat.Ext)_R( │ │ │ │ -00069770: 4d2c 6b29 2220 746f 2074 6865 2053 322d M,k)" to the S2- │ │ │ │ -00069780: 6966 6963 6174 696f 6e20 6f66 2045 7874 ification of Ext │ │ │ │ -00069790: 5f52 284d 2c6b 2920 776f 756c 6420 6265 _R(M,k) would be │ │ │ │ -000697a0: 2073 7572 6a65 6374 6976 653b 0a65 7175 surjective;.equ │ │ │ │ -000697b0: 6976 616c 656e 746c 792c 2069 6620 4e20 ivalently, if N │ │ │ │ -000697c0: 7765 7265 2061 2073 7566 6669 6369 656e were a sufficien │ │ │ │ -000697d0: 746c 7920 6e65 6761 7469 7665 2073 797a tly negative syz │ │ │ │ -000697e0: 7967 7920 6f66 204d 2c20 7468 656e 2074 ygy of M, then t │ │ │ │ -000697f0: 6865 2066 6972 7374 0a6c 6f63 616c 2063 he first.local c │ │ │ │ -00069800: 6f68 6f6d 6f6c 6f67 7920 6d6f 6475 6c65 ohomology module │ │ │ │ -00069810: 206f 6620 4578 745f 5228 4d2c 6b29 2077 of Ext_R(M,k) w │ │ │ │ -00069820: 6f75 6c64 2062 6520 7a65 726f 2e20 5468 ould be zero. Th │ │ │ │ -00069830: 6973 2069 7320 6661 6c73 652c 2061 7320 is is false, as │ │ │ │ -00069840: 7368 6f77 6e20 6279 0a74 6865 2066 6f6c shown by.the fol │ │ │ │ -00069850: 6c6f 7769 6e67 2065 7861 6d70 6c65 3a0a lowing example:. │ │ │ │ -00069860: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000696b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a41 ------------+..A │ │ │ │ +000696c0: 7420 6f6e 6520 7469 6d65 2044 4520 686f t one time DE ho │ │ │ │ +000696d0: 7065 6420 7468 6174 2c20 6966 204d 2077 ped that, if M w │ │ │ │ +000696e0: 6572 6520 6120 6d6f 6475 6c65 206f 7665 ere a module ove │ │ │ │ +000696f0: 7220 7468 6520 636f 6d70 6c65 7465 2069 r the complete i │ │ │ │ +00069700: 6e74 6572 7365 6374 696f 6e20 520a 7769 ntersection R.wi │ │ │ │ +00069710: 7468 2072 6573 6964 7565 2066 6965 6c64 th residue field │ │ │ │ +00069720: 206b 2c20 7468 656e 2074 6865 206e 6174 k, then the nat │ │ │ │ +00069730: 7572 616c 206d 6170 2066 726f 6d20 2263 ural map from "c │ │ │ │ +00069740: 6f6d 706c 6574 6522 2045 7874 206d 6f64 omplete" Ext mod │ │ │ │ +00069750: 756c 6520 2228 7769 6465 6861 740a 4578 ule "(widehat.Ex │ │ │ │ +00069760: 7429 5f52 284d 2c6b 2922 2074 6f20 7468 t)_R(M,k)" to th │ │ │ │ +00069770: 6520 5332 2d69 6669 6361 7469 6f6e 206f e S2-ification o │ │ │ │ +00069780: 6620 4578 745f 5228 4d2c 6b29 2077 6f75 f Ext_R(M,k) wou │ │ │ │ +00069790: 6c64 2062 6520 7375 726a 6563 7469 7665 ld be surjective │ │ │ │ +000697a0: 3b0a 6571 7569 7661 6c65 6e74 6c79 2c20 ;.equivalently, │ │ │ │ +000697b0: 6966 204e 2077 6572 6520 6120 7375 6666 if N were a suff │ │ │ │ +000697c0: 6963 6965 6e74 6c79 206e 6567 6174 6976 iciently negativ │ │ │ │ +000697d0: 6520 7379 7a79 6779 206f 6620 4d2c 2074 e syzygy of M, t │ │ │ │ +000697e0: 6865 6e20 7468 6520 6669 7273 740a 6c6f hen the first.lo │ │ │ │ +000697f0: 6361 6c20 636f 686f 6d6f 6c6f 6779 206d cal cohomology m │ │ │ │ +00069800: 6f64 756c 6520 6f66 2045 7874 5f52 284d odule of Ext_R(M │ │ │ │ +00069810: 2c6b 2920 776f 756c 6420 6265 207a 6572 ,k) would be zer │ │ │ │ +00069820: 6f2e 2054 6869 7320 6973 2066 616c 7365 o. This is false │ │ │ │ +00069830: 2c20 6173 2073 686f 776e 2062 790a 7468 , as shown by.th │ │ │ │ +00069840: 6520 666f 6c6c 6f77 696e 6720 6578 616d e following exam │ │ │ │ +00069850: 706c 653a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ple:..+--------- │ │ │ │ +00069860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069890: 2d2d 2d2b 0a7c 6939 203a 2053 203d 205a ---+.|i9 : S = Z │ │ │ │ -000698a0: 5a2f 3130 315b 785f 302e 2e78 5f32 5d3b Z/101[x_0..x_2]; │ │ │ │ -000698b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000698c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069880: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 --------+.|i9 : │ │ │ │ +00069890: 5320 3d20 5a5a 2f31 3031 5b78 5f30 2e2e S = ZZ/101[x_0.. │ │ │ │ +000698a0: 785f 325d 3b20 2020 2020 2020 2020 2020 x_2]; │ │ │ │ +000698b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000698c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000698d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000698e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000698f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069900: 3020 3a20 6666 203d 2061 7070 6c79 2833 0 : ff = apply(3 │ │ │ │ -00069910: 2c20 692d 3e78 5f69 5e32 293b 2020 2020 , i->x_i^2); │ │ │ │ -00069920: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00069930: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000698f0: 2b0a 7c69 3130 203a 2066 6620 3d20 6170 +.|i10 : ff = ap │ │ │ │ +00069900: 706c 7928 332c 2069 2d3e 785f 695e 3229 ply(3, i->x_i^2) │ │ │ │ +00069910: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +00069920: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069960: 2d2d 2d2b 0a7c 6931 3120 3a20 5220 3d20 ---+.|i11 : R = │ │ │ │ -00069970: 532f 6964 6561 6c20 6666 3b20 2020 2020 S/ideal ff; │ │ │ │ -00069980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069990: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069950: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a --------+.|i11 : │ │ │ │ +00069960: 2052 203d 2053 2f69 6465 616c 2066 663b R = S/ideal ff; │ │ │ │ +00069970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069980: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000699a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000699b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000699c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -000699d0: 3220 3a20 4d20 3d20 636f 6b65 726e 656c 2 : M = cokernel │ │ │ │ -000699e0: 206d 6174 7269 7820 7b7b 785f 302c 2078 matrix {{x_0, x │ │ │ │ -000699f0: 5f31 2a78 5f32 7d7d 3b20 2020 2020 207c _1*x_2}}; | │ │ │ │ -00069a00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000699c0: 2b0a 7c69 3132 203a 204d 203d 2063 6f6b +.|i12 : M = cok │ │ │ │ +000699d0: 6572 6e65 6c20 6d61 7472 6978 207b 7b78 ernel matrix {{x │ │ │ │ +000699e0: 5f30 2c20 785f 312a 785f 327d 7d3b 2020 _0, x_1*x_2}}; │ │ │ │ +000699f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069a30: 2d2d 2d2b 0a7c 6931 3320 3a20 6220 3d20 ---+.|i13 : b = │ │ │ │ -00069a40: 353b 2020 2020 2020 2020 2020 2020 2020 5; │ │ │ │ -00069a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069a60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069a20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a --------+.|i13 : │ │ │ │ +00069a30: 2062 203d 2035 3b20 2020 2020 2020 2020 b = 5; │ │ │ │ +00069a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069a50: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069aa0: 3420 3a20 4d62 203d 2070 7275 6e65 2073 4 : Mb = prune s │ │ │ │ -00069ab0: 797a 7967 794d 6f64 756c 6528 2d62 2c4d yzygyModule(-b,M │ │ │ │ -00069ac0: 293b 2020 2020 2020 2020 2020 2020 207c ); | │ │ │ │ -00069ad0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00069a90: 2b0a 7c69 3134 203a 204d 6220 3d20 7072 +.|i14 : Mb = pr │ │ │ │ +00069aa0: 756e 6520 7379 7a79 6779 4d6f 6475 6c65 une syzygyModule │ │ │ │ +00069ab0: 282d 622c 4d29 3b20 2020 2020 2020 2020 (-b,M); │ │ │ │ +00069ac0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069b00: 2d2d 2d2b 0a7c 6931 3520 3a20 4520 3d20 ---+.|i15 : E = │ │ │ │ -00069b10: 7072 756e 6520 6576 656e 4578 744d 6f64 prune evenExtMod │ │ │ │ -00069b20: 756c 6520 4d62 3b20 2020 2020 2020 2020 ule Mb; │ │ │ │ -00069b30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069af0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a --------+.|i15 : │ │ │ │ +00069b00: 2045 203d 2070 7275 6e65 2065 7665 6e45 E = prune evenE │ │ │ │ +00069b10: 7874 4d6f 6475 6c65 204d 623b 2020 2020 xtModule Mb; │ │ │ │ +00069b20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069b70: 3620 3a20 5332 6d61 7020 3d20 5332 2830 6 : S2map = S2(0 │ │ │ │ -00069b80: 2c45 293b 2020 2020 2020 2020 2020 2020 ,E); │ │ │ │ -00069b90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00069ba0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00069b60: 2b0a 7c69 3136 203a 2053 326d 6170 203d +.|i16 : S2map = │ │ │ │ +00069b70: 2053 3228 302c 4529 3b20 2020 2020 2020 S2(0,E); │ │ │ │ +00069b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069b90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00069ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069bd0: 2020 207c 0a7c 6f31 3620 3a20 4d61 7472 |.|o16 : Matr │ │ │ │ -00069be0: 6978 2020 2020 2020 2020 2020 2020 2020 ix │ │ │ │ -00069bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069c00: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069bc0: 2020 2020 2020 2020 7c0a 7c6f 3136 203a |.|o16 : │ │ │ │ +00069bd0: 204d 6174 7269 7820 2020 2020 2020 2020 Matrix │ │ │ │ +00069be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069bf0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069c40: 3720 3a20 5345 203d 2070 7275 6e65 2074 7 : SE = prune t │ │ │ │ -00069c50: 6172 6765 7420 5332 6d61 703b 2020 2020 arget S2map; │ │ │ │ -00069c60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00069c70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00069c30: 2b0a 7c69 3137 203a 2053 4520 3d20 7072 +.|i17 : SE = pr │ │ │ │ +00069c40: 756e 6520 7461 7267 6574 2053 326d 6170 une target S2map │ │ │ │ +00069c50: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +00069c60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069ca0: 2d2d 2d2b 0a7c 6931 3820 3a20 6578 7472 ---+.|i18 : extr │ │ │ │ -00069cb0: 6120 3d20 7072 756e 6520 636f 6b65 7220 a = prune coker │ │ │ │ -00069cc0: 5332 6d61 703b 2020 2020 2020 2020 2020 S2map; │ │ │ │ -00069cd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069c90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a --------+.|i18 : │ │ │ │ +00069ca0: 2065 7874 7261 203d 2070 7275 6e65 2063 extra = prune c │ │ │ │ +00069cb0: 6f6b 6572 2053 326d 6170 3b20 2020 2020 oker S2map; │ │ │ │ +00069cc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069d10: 3920 3a20 4b45 203d 2070 7275 6e65 206b 9 : KE = prune k │ │ │ │ -00069d20: 6572 2053 326d 6170 3b20 2020 2020 2020 er S2map; │ │ │ │ -00069d30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00069d40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00069d00: 2b0a 7c69 3139 203a 204b 4520 3d20 7072 +.|i19 : KE = pr │ │ │ │ +00069d10: 756e 6520 6b65 7220 5332 6d61 703b 2020 une ker S2map; │ │ │ │ +00069d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069d30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069d70: 2d2d 2d2b 0a7c 6932 3020 3a20 6265 7474 ---+.|i20 : bett │ │ │ │ -00069d80: 6920 6672 6565 5265 736f 6c75 7469 6f6e i freeResolution │ │ │ │ -00069d90: 284d 622c 204c 656e 6774 684c 696d 6974 (Mb, LengthLimit │ │ │ │ -00069da0: 203d 3e20 3130 297c 0a7c 2020 2020 2020 => 10)|.| │ │ │ │ +00069d60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a --------+.|i20 : │ │ │ │ +00069d70: 2062 6574 7469 2066 7265 6552 6573 6f6c betti freeResol │ │ │ │ +00069d80: 7574 696f 6e28 4d62 2c20 4c65 6e67 7468 ution(Mb, Length │ │ │ │ +00069d90: 4c69 6d69 7420 3d3e 2031 3029 7c0a 7c20 Limit => 10)|.| │ │ │ │ +00069da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069dd0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00069de0: 2020 2020 2020 2020 2020 2020 3020 2031 0 1 │ │ │ │ -00069df0: 2032 2033 2034 2035 2036 2037 2038 2020 2 3 4 5 6 7 8 │ │ │ │ -00069e00: 3920 3130 2020 2020 2020 2020 2020 207c 9 10 | │ │ │ │ -00069e10: 0a7c 6f32 3020 3d20 746f 7461 6c3a 2032 .|o20 = total: 2 │ │ │ │ -00069e20: 3020 3134 2039 2035 2032 2031 2032 2034 0 14 9 5 2 1 2 4 │ │ │ │ -00069e30: 2037 2031 3120 3136 2020 2020 2020 2020 7 11 16 │ │ │ │ -00069e40: 2020 207c 0a7c 2020 2020 2020 2020 202d |.| - │ │ │ │ -00069e50: 363a 2032 3020 3134 2039 2035 2032 202e 6: 20 14 9 5 2 . │ │ │ │ -00069e60: 202e 202e 202e 2020 2e20 202e 2020 2020 . . . . . │ │ │ │ -00069e70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00069e80: 2020 202d 353a 2020 2e20 202e 202e 202e -5: . . . . │ │ │ │ -00069e90: 202e 2031 2031 2031 2031 2020 3120 2031 . 1 1 1 1 1 1 │ │ │ │ -00069ea0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00069eb0: 2020 2020 2020 202d 343a 2020 2e20 202e -4: . . │ │ │ │ -00069ec0: 202e 202e 202e 202e 2031 2033 2036 2031 . . . . 1 3 6 1 │ │ │ │ -00069ed0: 3020 3135 2020 2020 2020 2020 2020 207c 0 15 | │ │ │ │ -00069ee0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00069dd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00069de0: 2030 2020 3120 3220 3320 3420 3520 3620 0 1 2 3 4 5 6 │ │ │ │ +00069df0: 3720 3820 2039 2031 3020 2020 2020 2020 7 8 9 10 │ │ │ │ +00069e00: 2020 2020 7c0a 7c6f 3230 203d 2074 6f74 |.|o20 = tot │ │ │ │ +00069e10: 616c 3a20 3230 2031 3420 3920 3520 3220 al: 20 14 9 5 2 │ │ │ │ +00069e20: 3120 3220 3420 3720 3131 2031 3620 2020 1 2 4 7 11 16 │ │ │ │ +00069e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00069e40: 2020 2020 2d36 3a20 3230 2031 3420 3920 -6: 20 14 9 │ │ │ │ +00069e50: 3520 3220 2e20 2e20 2e20 2e20 202e 2020 5 2 . . . . . │ │ │ │ +00069e60: 2e20 2020 2020 2020 2020 2020 7c0a 7c20 . |.| │ │ │ │ +00069e70: 2020 2020 2020 2020 2d35 3a20 202e 2020 -5: . │ │ │ │ +00069e80: 2e20 2e20 2e20 2e20 3120 3120 3120 3120 . . . . 1 1 1 1 │ │ │ │ +00069e90: 2031 2020 3120 2020 2020 2020 2020 2020 1 1 │ │ │ │ +00069ea0: 7c0a 7c20 2020 2020 2020 2020 2d34 3a20 |.| -4: │ │ │ │ +00069eb0: 202e 2020 2e20 2e20 2e20 2e20 2e20 3120 . . . . . . 1 │ │ │ │ +00069ec0: 3320 3620 3130 2031 3520 2020 2020 2020 3 6 10 15 │ │ │ │ +00069ed0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00069ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069f10: 2020 207c 0a7c 6f32 3020 3a20 4265 7474 |.|o20 : Bett │ │ │ │ -00069f20: 6954 616c 6c79 2020 2020 2020 2020 2020 iTally │ │ │ │ -00069f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069f40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069f00: 2020 2020 2020 2020 7c0a 7c6f 3230 203a |.|o20 : │ │ │ │ +00069f10: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +00069f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069f30: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00069f80: 3120 3a20 6170 706c 7920 2835 2c20 692d 1 : apply (5, i- │ │ │ │ -00069f90: 3e20 6869 6c62 6572 7446 756e 6374 696f > hilbertFunctio │ │ │ │ -00069fa0: 6e28 692c 204b 4529 2920 2020 2020 207c n(i, KE)) | │ │ │ │ -00069fb0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00069f70: 2b0a 7c69 3231 203a 2061 7070 6c79 2028 +.|i21 : apply ( │ │ │ │ +00069f80: 352c 2069 2d3e 2068 696c 6265 7274 4675 5, i-> hilbertFu │ │ │ │ +00069f90: 6e63 7469 6f6e 2869 2c20 4b45 2929 2020 nction(i, KE)) │ │ │ │ +00069fa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00069fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069fe0: 2020 207c 0a7c 6f32 3120 3d20 7b32 302c |.|o21 = {20, │ │ │ │ -00069ff0: 2039 2c20 322c 2030 2c20 307d 2020 2020 9, 2, 0, 0} │ │ │ │ -0006a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a010: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00069fd0: 2020 2020 2020 2020 7c0a 7c6f 3231 203d |.|o21 = │ │ │ │ +00069fe0: 207b 3230 2c20 392c 2032 2c20 302c 2030 {20, 9, 2, 0, 0 │ │ │ │ +00069ff0: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0006a000: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006a010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a040: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006a050: 3120 3a20 4c69 7374 2020 2020 2020 2020 1 : List │ │ │ │ +0006a040: 7c0a 7c6f 3231 203a 204c 6973 7420 2020 |.|o21 : List │ │ │ │ +0006a050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a070: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006a080: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0006a070: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0006a080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a0b0: 2d2d 2d2b 0a7c 6932 3220 3a20 6170 706c ---+.|i22 : appl │ │ │ │ -0006a0c0: 7920 2835 2c20 692d 3e20 6869 6c62 6572 y (5, i-> hilber │ │ │ │ -0006a0d0: 7446 756e 6374 696f 6e28 692c 2045 2929 tFunction(i, E)) │ │ │ │ -0006a0e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006a0a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a --------+.|i22 : │ │ │ │ +0006a0b0: 2061 7070 6c79 2028 352c 2069 2d3e 2068 apply (5, i-> h │ │ │ │ +0006a0c0: 696c 6265 7274 4675 6e63 7469 6f6e 2869 ilbertFunction(i │ │ │ │ +0006a0d0: 2c20 4529 2920 2020 2020 2020 7c0a 7c20 , E)) |.| │ │ │ │ +0006a0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a110: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006a120: 3220 3d20 7b32 302c 2039 2c20 322c 2032 2 = {20, 9, 2, 2 │ │ │ │ -0006a130: 2c20 377d 2020 2020 2020 2020 2020 2020 , 7} │ │ │ │ -0006a140: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006a150: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006a110: 7c0a 7c6f 3232 203d 207b 3230 2c20 392c |.|o22 = {20, 9, │ │ │ │ +0006a120: 2032 2c20 322c 2037 7d20 2020 2020 2020 2, 2, 7} │ │ │ │ +0006a130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006a140: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a180: 2020 207c 0a7c 6f32 3220 3a20 4c69 7374 |.|o22 : List │ │ │ │ +0006a170: 2020 2020 2020 2020 7c0a 7c6f 3232 203a |.|o22 : │ │ │ │ +0006a180: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ 0006a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a1b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0006a1a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0006a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -0006a1f0: 3320 3a20 6170 706c 7920 2835 2c20 692d 3 : apply (5, i- │ │ │ │ -0006a200: 3e20 6869 6c62 6572 7446 756e 6374 696f > hilbertFunctio │ │ │ │ -0006a210: 6e28 692c 2053 4529 2920 2020 2020 207c n(i, SE)) | │ │ │ │ -0006a220: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006a1e0: 2b0a 7c69 3233 203a 2061 7070 6c79 2028 +.|i23 : apply ( │ │ │ │ +0006a1f0: 352c 2069 2d3e 2068 696c 6265 7274 4675 5, i-> hilbertFu │ │ │ │ +0006a200: 6e63 7469 6f6e 2869 2c20 5345 2929 2020 nction(i, SE)) │ │ │ │ +0006a210: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006a220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a250: 2020 207c 0a7c 6f32 3320 3d20 7b31 2c20 |.|o23 = {1, │ │ │ │ -0006a260: 312c 2031 2c20 322c 2037 7d20 2020 2020 1, 1, 2, 7} │ │ │ │ -0006a270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a280: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006a240: 2020 2020 2020 2020 7c0a 7c6f 3233 203d |.|o23 = │ │ │ │ +0006a250: 207b 312c 2031 2c20 312c 2032 2c20 377d {1, 1, 1, 2, 7} │ │ │ │ +0006a260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006a270: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006a280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a2b0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006a2c0: 3320 3a20 4c69 7374 2020 2020 2020 2020 3 : List │ │ │ │ +0006a2b0: 7c0a 7c6f 3233 203a 204c 6973 7420 2020 |.|o23 : List │ │ │ │ +0006a2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a2e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006a2f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0006a2e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0006a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a320: 2d2d 2d2b 0a7c 6932 3420 3a20 6170 706c ---+.|i24 : appl │ │ │ │ -0006a330: 7920 2835 2c20 692d 3e20 6869 6c62 6572 y (5, i-> hilber │ │ │ │ -0006a340: 7446 756e 6374 696f 6e28 692c 2065 7874 tFunction(i, ext │ │ │ │ -0006a350: 7261 2929 2020 207c 0a7c 2020 2020 2020 ra)) |.| │ │ │ │ +0006a310: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3234 203a --------+.|i24 : │ │ │ │ +0006a320: 2061 7070 6c79 2028 352c 2069 2d3e 2068 apply (5, i-> h │ │ │ │ +0006a330: 696c 6265 7274 4675 6e63 7469 6f6e 2869 ilbertFunction(i │ │ │ │ +0006a340: 2c20 6578 7472 6129 2920 2020 7c0a 7c20 , extra)) |.| │ │ │ │ +0006a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a380: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006a390: 3420 3d20 7b31 2c20 312c 2031 2c20 302c 4 = {1, 1, 1, 0, │ │ │ │ -0006a3a0: 2030 7d20 2020 2020 2020 2020 2020 2020 0} │ │ │ │ -0006a3b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006a3c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006a380: 7c0a 7c6f 3234 203d 207b 312c 2031 2c20 |.|o24 = {1, 1, │ │ │ │ +0006a390: 312c 2030 2c20 307d 2020 2020 2020 2020 1, 0, 0} │ │ │ │ +0006a3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006a3b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006a3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a3f0: 2020 207c 0a7c 6f32 3420 3a20 4c69 7374 |.|o24 : List │ │ │ │ +0006a3e0: 2020 2020 2020 2020 7c0a 7c6f 3234 203a |.|o24 : │ │ │ │ +0006a3f0: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ 0006a400: 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546f 702c 202d 2d20 4265 (BGG)Top, -- Be │ │ │ │ +0006a650: 726e 7374 6569 6e2d 4765 6c27 6661 6e64 rnstein-Gel'fand │ │ │ │ +0006a660: 2d47 656c 2766 616e 6420 636f 7272 6573 -Gel'fand corres │ │ │ │ +0006a670: 706f 6e64 656e 6365 0a20 202a 202a 6e6f pondence. * *no │ │ │ │ +0006a680: 7465 2063 6f68 6f6d 6f6c 6f67 793a 2028 te cohomology: ( │ │ │ │ +0006a690: 4d61 6361 756c 6179 3244 6f63 2963 6f68 Macaulay2Doc)coh │ │ │ │ +0006a6a0: 6f6d 6f6c 6f67 792c 202d 2d20 6765 6e65 omology, -- gene │ │ │ │ +0006a6b0: 7261 6c20 636f 686f 6d6f 6c6f 6779 2066 ral cohomology f │ │ │ │ +0006a6c0: 756e 6374 6f72 0a20 202a 2048 485e 5a5a unctor. * HH^ZZ │ │ │ │ +0006a6d0: 2053 756d 4f66 5477 6973 7473 2028 6d69 SumOfTwists (mi │ │ │ │ +0006a6e0: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ +0006a6f0: 696f 6e29 0a0a 5761 7973 2074 6f20 7573 ion)..Ways to us │ │ │ │ +0006a700: 6520 5332 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d e S2:.========== │ │ │ │ +0006a710: 3d3d 3d3d 3d0a 0a20 202a 2022 5332 285a =====.. * "S2(Z │ │ │ │ +0006a720: 5a2c 4d6f 6475 6c65 2922 0a0a 466f 7220 Z,Module)"..For │ │ │ │ +0006a730: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +0006a740: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006a750: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +0006a760: 6f74 6520 5332 3a20 5332 2c20 6973 2061 ote S2: S2, is a │ │ │ │ +0006a770: 202a 6e6f 7465 206d 6574 686f 6420 6675 *note method fu │ │ │ │ +0006a780: 6e63 7469 6f6e 3a0a 284d 6163 6175 6c61 nction:.(Macaula │ │ │ │ +0006a790: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ +0006a7a0: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ +0006a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a800: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -0006a810: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -0006a820: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -0006a830: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -0006a840: 6163 6175 6c61 7932 2d31 2e32 362e 3035 acaulay2-1.26.05 │ │ │ │ -0006a850: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0006a860: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ -0006a870: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -0006a880: 6573 6f6c 7574 696f 6e73 2e6d 323a 3338 esolutions.m2:38 │ │ │ │ -0006a890: 3833 3a30 2e0a 1f0a 4669 6c65 3a20 436f 83:0....File: Co │ │ │ │ -0006a8a0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -0006a8b0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e 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ulay2/packages/. │ │ │ │ +0006a860: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +0006a870: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +0006a880: 6d32 3a33 3838 333a 302e 0a1f 0a46 696c m2:3883:0....Fil │ │ │ │ +0006a890: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ +0006a8a0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +0006a8b0: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2053 ns.info, Node: S │ │ │ │ +0006a8c0: 6861 6d61 7368 2c20 4e65 7874 3a20 7370 hamash, Next: sp │ │ │ │ +0006a8d0: 6c69 7474 696e 6773 2c20 5072 6576 3a20 littings, Prev: │ │ │ │ +0006a8e0: 5332 2c20 5570 3a20 546f 700a 0a53 6861 S2, Up: Top..Sha │ │ │ │ +0006a8f0: 6d61 7368 202d 2d20 436f 6d70 7574 6573 mash -- Computes │ │ │ │ +0006a900: 2074 6865 2053 6861 6d61 7368 2043 6f6d the Shamash Com │ │ │ │ +0006a910: 706c 6578 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a plex.*********** │ │ │ │ 0006a920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006a930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006a940: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ -0006a950: 6765 3a20 0a20 2020 2020 2020 2046 4620 ge: . FF │ │ │ │ -0006a960: 3d20 5368 616d 6173 6828 6666 2c46 2c6c = Shamash(ff,F,l │ │ │ │ -0006a970: 656e 290a 2020 2020 2020 2020 4646 203d en). FF = │ │ │ │ -0006a980: 2053 6861 6d61 7368 2852 6261 722c 462c Shamash(Rbar,F, │ │ │ │ -0006a990: 6c65 6e29 0a20 202a 2049 6e70 7574 733a len). * Inputs: │ │ │ │ -0006a9a0: 0a20 2020 2020 202a 2066 662c 2061 202a . * ff, a * │ │ │ │ -0006a9b0: 6e6f 7465 206d 6174 7269 783a 2028 4d61 note matrix: (Ma │ │ │ │ -0006a9c0: 6361 756c 6179 3244 6f63 294d 6174 7269 caulay2Doc)Matri │ │ │ │ -0006a9d0: 782c 2c20 3120 7820 3120 4d61 7472 6978 x,, 1 x 1 Matrix │ │ │ │ -0006a9e0: 206f 7665 7220 7269 6e67 2046 2e0a 2020 over ring F.. │ │ │ │ -0006a9f0: 2020 2020 2a20 5262 6172 2c20 6120 2a6e * Rbar, a *n │ │ │ │ -0006aa00: 6f74 6520 7269 6e67 3a20 284d 6163 6175 ote ring: (Macau │ │ │ │ -0006aa10: 6c61 7932 446f 6329 5269 6e67 2c2c 2072 lay2Doc)Ring,, r │ │ │ │ -0006aa20: 696e 6720 4620 6d6f 6420 6964 6561 6c20 ing F mod ideal │ │ │ │ -0006aa30: 6666 0a20 2020 2020 202a 2046 2c20 6120 ff. * F, a │ │ │ │ -0006aa40: 2a6e 6f74 6520 636f 6d70 6c65 783a 2028 *note complex: ( │ │ │ │ -0006aa50: 436f 6d70 6c65 7865 7329 436f 6d70 6c65 Complexes)Comple │ │ │ │ -0006aa60: 782c 2c20 6465 6669 6e65 6420 6f76 6572 x,, defined over │ │ │ │ -0006aa70: 2072 696e 6720 6666 0a20 2020 2020 202a ring ff. * │ │ │ │ -0006aa80: 206c 656e 2c20 616e 202a 6e6f 7465 2069 len, an *note i │ │ │ │ -0006aa90: 6e74 6567 6572 3a20 284d 6163 6175 6c61 nteger: (Macaula │ │ │ │ -0006aaa0: 7932 446f 6329 5a5a 2c2c 200a 2020 2a20 y2Doc)ZZ,, . * │ │ │ │ -0006aab0: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -0006aac0: 2046 462c 2061 202a 6e6f 7465 2063 6f6d FF, a *note com │ │ │ │ -0006aad0: 706c 6578 3a20 2843 6f6d 706c 6578 6573 plex: (Complexes │ │ │ │ -0006aae0: 2943 6f6d 706c 6578 2c2c 2063 6861 696e )Complex,, chain │ │ │ │ -0006aaf0: 2063 6f6d 706c 6578 206f 7665 7220 2872 complex over (r │ │ │ │ -0006ab00: 696e 670a 2020 2020 2020 2020 4629 2f28 ing. F)/( │ │ │ │ -0006ab10: 6964 6561 6c20 6666 290a 0a44 6573 6372 ideal ff)..Descr │ │ │ │ -0006ab20: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -0006ab30: 3d3d 0a0a 4c65 7420 5220 3d20 7269 6e67 ==..Let R = ring │ │ │ │ -0006ab40: 2046 203d 2072 696e 6720 6666 2c20 616e F = ring ff, an │ │ │ │ -0006ab50: 6420 5262 6172 203d 2052 2f28 6964 6561 d Rbar = R/(idea │ │ │ │ -0006ab60: 6c20 6629 2c20 7768 6572 6520 6666 203d l f), where ff = │ │ │ │ -0006ab70: 206d 6174 7269 787b 7b66 7d7d 2069 7320 matrix{{f}} is │ │ │ │ -0006ab80: 610a 3178 3120 6d61 7472 6978 2077 686f a.1x1 matrix who │ │ │ │ -0006ab90: 7365 2065 6e74 7279 2069 7320 6120 6e6f se entry is a no │ │ │ │ -0006aba0: 6e7a 6572 6f64 6976 6973 6f72 2069 6e20 nzerodivisor in │ │ │ │ -0006abb0: 522e 2054 6865 2063 6f6d 706c 6578 2046 R. The complex F │ │ │ │ -0006abc0: 2073 686f 756c 6420 6164 6d69 7420 610a should admit a. │ │ │ │ -0006abd0: 7379 7374 656d 206f 6620 6869 6768 6572 system of higher │ │ │ │ -0006abe0: 2068 6f6d 6f74 6f70 6965 7320 666f 7220 homotopies for │ │ │ │ -0006abf0: 7468 6520 656e 7472 7920 6f66 2066 662c the entry of ff, │ │ │ │ -0006ac00: 2072 6574 7572 6e65 6420 6279 2074 6865 returned by the │ │ │ │ -0006ac10: 2063 616c 6c0a 6d61 6b65 486f 6d6f 746f call.makeHomoto │ │ │ │ -0006ac20: 7069 6573 2866 662c 4629 2e0a 0a54 6865 pies(ff,F)...The │ │ │ │ -0006ac30: 2063 6f6d 706c 6578 2046 4620 6861 7320 complex FF has │ │ │ │ -0006ac40: 7465 726d 730a 0a46 465f 7b32 2a69 7d20 terms..FF_{2*i} │ │ │ │ -0006ac50: 3d20 5262 6172 2a2a 2846 5f30 202b 2b20 = Rbar**(F_0 ++ │ │ │ │ -0006ac60: 465f 3220 2b2b 202e 2e20 2b2b 2046 5f69 F_2 ++ .. ++ F_i │ │ │ │ -0006ac70: 290a 0a46 465f 7b32 2a69 2b31 7d20 3d20 )..FF_{2*i+1} = │ │ │ │ -0006ac80: 5262 6172 2a2a 2846 5f31 202b 2b20 465f Rbar**(F_1 ++ F_ │ │ │ │ -0006ac90: 3320 2b2b 2e2e 2b2b 465f 7b32 2a69 2b31 3 ++..++F_{2*i+1 │ │ │ │ -0006aca0: 7d29 0a0a 616e 6420 6d61 7073 206d 6164 })..and maps mad │ │ │ │ -0006acb0: 6520 6672 6f6d 2074 6865 2068 6967 6865 e from the highe │ │ │ │ -0006acc0: 7220 686f 6d6f 746f 7069 6573 2e0a 0a46 r homotopies...F │ │ │ │ -0006acd0: 6f72 2074 6865 2063 6173 6520 6f66 2061 or the case of a │ │ │ │ -0006ace0: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -0006acf0: 6563 7469 6f6e 206f 6620 6869 6768 6572 ection of higher │ │ │ │ -0006ad00: 2063 6f64 696d 656e 7369 6f6e 2c20 6f72 codimension, or │ │ │ │ -0006ad10: 2074 6f20 7365 6520 7468 650a 636f 6d70 to see the.comp │ │ │ │ -0006ad20: 6f6e 656e 7473 206f 6620 7468 6520 7265 onents of the re │ │ │ │ -0006ad30: 736f 6c75 7469 6f6e 2061 7320 7375 6d6d solution as summ │ │ │ │ -0006ad40: 616e 6473 206f 6620 4646 5f6a 2c20 7573 ands of FF_j, us │ │ │ │ -0006ad50: 6520 7468 6520 726f 7574 696e 650a 4569 e the routine.Ei │ │ │ │ -0006ad60: 7365 6e62 7564 5368 616d 6173 6820 696e senbudShamash in │ │ │ │ -0006ad70: 7374 6561 642e 0a0a 2b2d 2d2d 2d2d 2d2d stead...+------- │ │ │ │ +0006a930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ +0006a940: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ +0006a950: 2020 4646 203d 2053 6861 6d61 7368 2866 FF = Shamash(f │ │ │ │ +0006a960: 662c 462c 6c65 6e29 0a20 2020 2020 2020 f,F,len). │ │ │ │ +0006a970: 2046 4620 3d20 5368 616d 6173 6828 5262 FF = Shamash(Rb │ │ │ │ +0006a980: 6172 2c46 2c6c 656e 290a 2020 2a20 496e ar,F,len). * In │ │ │ │ +0006a990: 7075 7473 3a0a 2020 2020 2020 2a20 6666 puts:. * ff │ │ │ │ +0006a9a0: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978 , a *note matrix │ │ │ │ +0006a9b0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0006a9c0: 4d61 7472 6978 2c2c 2031 2078 2031 204d Matrix,, 1 x 1 M │ │ │ │ +0006a9d0: 6174 7269 7820 6f76 6572 2072 696e 6720 atrix over ring │ │ │ │ +0006a9e0: 462e 0a20 2020 2020 202a 2052 6261 722c F.. * Rbar, │ │ │ │ +0006a9f0: 2061 202a 6e6f 7465 2072 696e 673a 2028 a *note ring: ( │ │ │ │ +0006aa00: 4d61 6361 756c 6179 3244 6f63 2952 696e Macaulay2Doc)Rin │ │ │ │ +0006aa10: 672c 2c20 7269 6e67 2046 206d 6f64 2069 g,, ring F mod i │ │ │ │ +0006aa20: 6465 616c 2066 660a 2020 2020 2020 2a20 deal ff. * │ │ │ │ +0006aa30: 462c 2061 202a 6e6f 7465 2063 6f6d 706c F, a *note compl │ │ │ │ +0006aa40: 6578 3a20 2843 6f6d 706c 6578 6573 2943 ex: (Complexes)C │ │ │ │ +0006aa50: 6f6d 706c 6578 2c2c 2064 6566 696e 6564 omplex,, defined │ │ │ │ +0006aa60: 206f 7665 7220 7269 6e67 2066 660a 2020 over ring ff. │ │ │ │ +0006aa70: 2020 2020 2a20 6c65 6e2c 2061 6e20 2a6e * len, an *n │ │ │ │ +0006aa80: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ +0006aa90: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ +0006aaa0: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +0006aab0: 2020 2020 2a20 4646 2c20 6120 2a6e 6f74 * FF, a *not │ │ │ │ +0006aac0: 6520 636f 6d70 6c65 783a 2028 436f 6d70 e complex: (Comp │ │ │ │ +0006aad0: 6c65 7865 7329 436f 6d70 6c65 782c 2c20 lexes)Complex,, │ │ │ │ +0006aae0: 6368 6169 6e20 636f 6d70 6c65 7820 6f76 chain complex ov │ │ │ │ +0006aaf0: 6572 2028 7269 6e67 0a20 2020 2020 2020 er (ring. │ │ │ │ +0006ab00: 2046 292f 2869 6465 616c 2066 6629 0a0a F)/(ideal ff).. │ │ │ │ +0006ab10: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +0006ab20: 3d3d 3d3d 3d3d 3d0a 0a4c 6574 2052 203d =======..Let R = │ │ │ │ +0006ab30: 2072 696e 6720 4620 3d20 7269 6e67 2066 ring F = ring f │ │ │ │ +0006ab40: 662c 2061 6e64 2052 6261 7220 3d20 522f f, and Rbar = R/ │ │ │ │ +0006ab50: 2869 6465 616c 2066 292c 2077 6865 7265 (ideal f), where │ │ │ │ +0006ab60: 2066 6620 3d20 6d61 7472 6978 7b7b 667d ff = matrix{{f} │ │ │ │ +0006ab70: 7d20 6973 2061 0a31 7831 206d 6174 7269 } is a.1x1 matri │ │ │ │ +0006ab80: 7820 7768 6f73 6520 656e 7472 7920 6973 x whose entry is │ │ │ │ +0006ab90: 2061 206e 6f6e 7a65 726f 6469 7669 736f a nonzerodiviso │ │ │ │ +0006aba0: 7220 696e 2052 2e20 5468 6520 636f 6d70 r in R. The comp │ │ │ │ +0006abb0: 6c65 7820 4620 7368 6f75 6c64 2061 646d lex F should adm │ │ │ │ +0006abc0: 6974 2061 0a73 7973 7465 6d20 6f66 2068 it a.system of h │ │ │ │ +0006abd0: 6967 6865 7220 686f 6d6f 746f 7069 6573 igher homotopies │ │ │ │ +0006abe0: 2066 6f72 2074 6865 2065 6e74 7279 206f for the entry o │ │ │ │ +0006abf0: 6620 6666 2c20 7265 7475 726e 6564 2062 f ff, returned b │ │ │ │ +0006ac00: 7920 7468 6520 6361 6c6c 0a6d 616b 6548 y the call.makeH │ │ │ │ +0006ac10: 6f6d 6f74 6f70 6965 7328 6666 2c46 292e omotopies(ff,F). │ │ │ │ +0006ac20: 0a0a 5468 6520 636f 6d70 6c65 7820 4646 ..The complex FF │ │ │ │ +0006ac30: 2068 6173 2074 6572 6d73 0a0a 4646 5f7b has terms..FF_{ │ │ │ │ +0006ac40: 322a 697d 203d 2052 6261 722a 2a28 465f 2*i} = Rbar**(F_ │ │ │ │ +0006ac50: 3020 2b2b 2046 5f32 202b 2b20 2e2e 202b 0 ++ F_2 ++ .. + │ │ │ │ +0006ac60: 2b20 465f 6929 0a0a 4646 5f7b 322a 692b + F_i)..FF_{2*i+ │ │ │ │ +0006ac70: 317d 203d 2052 6261 722a 2a28 465f 3120 1} = Rbar**(F_1 │ │ │ │ +0006ac80: 2b2b 2046 5f33 202b 2b2e 2e2b 2b46 5f7b ++ F_3 ++..++F_{ │ │ │ │ +0006ac90: 322a 692b 317d 290a 0a61 6e64 206d 6170 2*i+1})..and map │ │ │ │ +0006aca0: 7320 6d61 6465 2066 726f 6d20 7468 6520 s made from the │ │ │ │ +0006acb0: 6869 6768 6572 2068 6f6d 6f74 6f70 6965 higher homotopie │ │ │ │ +0006acc0: 732e 0a0a 466f 7220 7468 6520 6361 7365 s...For the case │ │ │ │ +0006acd0: 206f 6620 6120 636f 6d70 6c65 7465 2069 of a complete i │ │ │ │ +0006ace0: 6e74 6572 7365 6374 696f 6e20 6f66 2068 ntersection of h │ │ │ │ +0006acf0: 6967 6865 7220 636f 6469 6d65 6e73 696f igher codimensio │ │ │ │ +0006ad00: 6e2c 206f 7220 746f 2073 6565 2074 6865 n, or to see the │ │ │ │ +0006ad10: 0a63 6f6d 706f 6e65 6e74 7320 6f66 2074 .components of t │ │ │ │ +0006ad20: 6865 2072 6573 6f6c 7574 696f 6e20 6173 he resolution as │ │ │ │ +0006ad30: 2073 756d 6d61 6e64 7320 6f66 2046 465f summands of FF_ │ │ │ │ +0006ad40: 6a2c 2075 7365 2074 6865 2072 6f75 7469 j, use the routi │ │ │ │ +0006ad50: 6e65 0a45 6973 656e 6275 6453 6861 6d61 ne.EisenbudShama │ │ │ │ +0006ad60: 7368 2069 6e73 7465 6164 2e0a 0a2b 2d2d sh instead...+-- │ │ │ │ +0006ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ad80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0006adb0: 6931 203a 2053 203d 205a 5a2f 3130 315b i1 : S = ZZ/101[ │ │ │ │ -0006adc0: 782c 792c 7a5d 2020 2020 2020 2020 2020 x,y,z] │ │ │ │ -0006add0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ade0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006ada0: 2d2d 2b0a 7c69 3120 3a20 5320 3d20 5a5a --+.|i1 : S = ZZ │ │ │ │ +0006adb0: 2f31 3031 5b78 2c79 2c7a 5d20 2020 2020 /101[x,y,z] │ │ │ │ +0006adc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006add0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006ade0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006adf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ae00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae10: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0006ae20: 203d 2053 2020 2020 2020 2020 2020 2020 = S │ │ │ │ +0006ae10: 7c0a 7c6f 3120 3d20 5320 2020 2020 2020 |.|o1 = S │ │ │ │ +0006ae20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ae30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae50: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006ae40: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006ae50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ae60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae80: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ -0006ae90: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ +0006ae70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006ae80: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c |o1 : Polynomial │ │ │ │ +0006ae90: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ 0006aea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006aeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006aec0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0006aeb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0006aec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006aed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006aef0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052 -------+.|i2 : R │ │ │ │ -0006af00: 203d 2053 2f69 6465 616c 2278 332c 7933 = S/ideal"x3,y3 │ │ │ │ -0006af10: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ -0006af20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006af30: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0006aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0006aef0: 3220 3a20 5220 3d20 532f 6964 6561 6c22 2 : R = S/ideal" │ │ │ │ +0006af00: 7833 2c79 3322 2020 2020 2020 2020 2020 x3,y3" │ │ │ │ +0006af10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006af20: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006af30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006af40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006af50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006af60: 2020 2020 207c 0a7c 6f32 203d 2052 2020 |.|o2 = R │ │ │ │ +0006af50: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0006af60: 3d20 5220 2020 2020 2020 2020 2020 2020 = R │ │ │ │ 0006af70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006af80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006af90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006af90: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006afc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006afd0: 2020 207c 0a7c 6f32 203a 2051 756f 7469 |.|o2 : Quoti │ │ │ │ -0006afe0: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ -0006aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b000: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0006afc0: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ +0006afd0: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +0006afe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006aff0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006b000: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0006b010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b040: 2d2b 0a7c 6933 203a 204d 203d 2052 5e31 -+.|i3 : M = R^1 │ │ │ │ -0006b050: 2f69 6465 616c 2878 2c79 2c7a 2920 2020 /ideal(x,y,z) │ │ │ │ -0006b060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b070: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006b030: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 4d20 ------+.|i3 : M │ │ │ │ +0006b040: 3d20 525e 312f 6964 6561 6c28 782c 792c = R^1/ideal(x,y, │ │ │ │ +0006b050: 7a29 2020 2020 2020 2020 2020 2020 2020 z) │ │ │ │ +0006b060: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b0a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006b0b0: 0a7c 6f33 203d 2063 6f6b 6572 6e65 6c20 .|o3 = cokernel │ │ │ │ -0006b0c0: 7c20 7820 7920 7a20 7c20 2020 2020 2020 | x y z | │ │ │ │ -0006b0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b0e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006b0a0: 2020 2020 7c0a 7c6f 3320 3d20 636f 6b65 |.|o3 = coke │ │ │ │ +0006b0b0: 726e 656c 207c 2078 2079 207a 207c 2020 rnel | x y z | │ │ │ │ +0006b0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b0d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006b0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b110: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b110: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006b120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b130: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ -0006b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b150: 2020 2020 7c0a 7c6f 3320 3a20 522d 6d6f |.|o3 : R-mo │ │ │ │ -0006b160: 6475 6c65 2c20 7175 6f74 6965 6e74 206f dule, quotient o │ │ │ │ -0006b170: 6620 5220 2020 2020 2020 2020 2020 2020 f R │ │ │ │ -0006b180: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0006b130: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0006b140: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ +0006b150: 2052 2d6d 6f64 756c 652c 2071 756f 7469 R-module, quoti │ │ │ │ +0006b160: 656e 7420 6f66 2052 2020 2020 2020 2020 ent of R │ │ │ │ +0006b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b180: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0006b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b1c0: 2d2d 2b0a 7c69 3420 3a20 4620 3d20 6672 --+.|i4 : F = fr │ │ │ │ -0006b1d0: 6565 5265 736f 6c75 7469 6f6e 284d 2c20 eeResolution(M, │ │ │ │ -0006b1e0: 4c65 6e67 7468 4c69 6d69 7420 3d3e 2034 LengthLimit => 4 │ │ │ │ -0006b1f0: 2920 2020 2020 2020 207c 0a7c 2020 2020 ) |.| │ │ │ │ +0006b1b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2046 -------+.|i4 : F │ │ │ │ +0006b1c0: 203d 2066 7265 6552 6573 6f6c 7574 696f = freeResolutio │ │ │ │ +0006b1d0: 6e28 4d2c 204c 656e 6774 684c 696d 6974 n(M, LengthLimit │ │ │ │ +0006b1e0: 203d 3e20 3429 2020 2020 2020 2020 7c0a => 4) |. │ │ │ │ +0006b1f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006b200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b230: 7c0a 7c20 2020 2020 2031 2020 2020 2020 |.| 1 │ │ │ │ -0006b240: 3320 2020 2020 2035 2020 2020 2020 3720 3 5 7 │ │ │ │ -0006b250: 2020 2020 2039 2020 2020 2020 2020 2020 9 │ │ │ │ -0006b260: 2020 2020 2020 207c 0a7c 6f34 203d 2052 |.|o4 = R │ │ │ │ -0006b270: 2020 3c2d 2d20 5220 203c 2d2d 2052 2020 <-- R <-- R │ │ │ │ -0006b280: 3c2d 2d20 5220 203c 2d2d 2052 2020 2020 <-- R <-- R │ │ │ │ -0006b290: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006b2a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0006b220: 2020 2020 207c 0a7c 2020 2020 2020 3120 |.| 1 │ │ │ │ +0006b230: 2020 2020 2033 2020 2020 2020 3520 2020 3 5 │ │ │ │ +0006b240: 2020 2037 2020 2020 2020 3920 2020 2020 7 9 │ │ │ │ +0006b250: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0006b260: 3420 3d20 5220 203c 2d2d 2052 2020 3c2d 4 = R <-- R <- │ │ │ │ +0006b270: 2d20 5220 203c 2d2d 2052 2020 3c2d 2d20 - R <-- R <-- │ │ │ │ +0006b280: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0006b290: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006b2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b2d0: 2020 2020 207c 0a7c 2020 2020 2030 2020 |.| 0 │ │ │ │ -0006b2e0: 2020 2020 3120 2020 2020 2032 2020 2020 1 2 │ │ │ │ -0006b2f0: 2020 3320 2020 2020 2034 2020 2020 2020 3 4 │ │ │ │ -0006b300: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006b2c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006b2d0: 2020 3020 2020 2020 2031 2020 2020 2020 0 1 │ │ │ │ +0006b2e0: 3220 2020 2020 2033 2020 2020 2020 3420 2 3 4 │ │ │ │ +0006b2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b300: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b340: 2020 207c 0a7c 6f34 203a 2043 6f6d 706c |.|o4 : Compl │ │ │ │ -0006b350: 6578 2020 2020 2020 2020 2020 2020 2020 ex │ │ │ │ -0006b360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b370: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0006b330: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ +0006b340: 436f 6d70 6c65 7820 2020 2020 2020 2020 Complex │ │ │ │ +0006b350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b360: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006b370: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0006b380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b3b0: 2d2b 0a7c 6935 203a 2066 6620 3d20 6d61 -+.|i5 : ff = ma │ │ │ │ -0006b3c0: 7472 6978 7b7b 7a5e 337d 7d20 2020 2020 trix{{z^3}} │ │ │ │ -0006b3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b3e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006b3a0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 6666 ------+.|i5 : ff │ │ │ │ +0006b3b0: 203d 206d 6174 7269 787b 7b7a 5e33 7d7d = matrix{{z^3}} │ │ │ │ +0006b3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b3d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b410: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006b420: 0a7c 6f35 203d 207c 207a 3320 7c20 2020 .|o5 = | z3 | │ │ │ │ +0006b410: 2020 2020 7c0a 7c6f 3520 3d20 7c20 7a33 |.|o5 = | z3 │ │ │ │ +0006b420: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b450: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006b440: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006b450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b480: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0006b490: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ -0006b4a0: 2020 2020 3120 2020 2020 2020 2020 2020 1 │ │ │ │ -0006b4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b4c0: 2020 2020 7c0a 7c6f 3520 3a20 4d61 7472 |.|o5 : Matr │ │ │ │ -0006b4d0: 6978 2052 2020 3c2d 2d20 5220 2020 2020 ix R <-- R │ │ │ │ +0006b480: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006b490: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ +0006b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b4b0: 2020 2020 2020 2020 207c 0a7c 6f35 203a |.|o5 : │ │ │ │ +0006b4c0: 204d 6174 7269 7820 5220 203c 2d2d 2052 Matrix R <-- R │ │ │ │ +0006b4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b4f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0006b4f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0006b500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b530: 2d2d 2b0a 7c69 3620 3a20 5231 203d 2052 --+.|i6 : R1 = R │ │ │ │ -0006b540: 2f69 6465 616c 2066 6620 2020 2020 2020 /ideal ff │ │ │ │ -0006b550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b560: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006b520: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2052 -------+.|i6 : R │ │ │ │ +0006b530: 3120 3d20 522f 6964 6561 6c20 6666 2020 1 = R/ideal ff │ │ │ │ +0006b540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b550: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006b560: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006b570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b5a0: 7c0a 7c6f 3620 3d20 5231 2020 2020 2020 |.|o6 = R1 │ │ │ │ +0006b590: 2020 2020 207c 0a7c 6f36 203d 2052 3120 |.|o6 = R1 │ │ │ │ +0006b5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b5d0: 2020 2020 2020 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---------------- │ │ │ │ 0006b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0006b680: 3720 3a20 6265 7474 6920 4620 2020 2020 7 : betti F │ │ │ │ +0006b670: 2d2b 0a7c 6937 203a 2062 6574 7469 2046 -+.|i7 : betti F │ │ │ │ +0006b680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b6b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006b6a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006b6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b6e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0006b6f0: 2020 2020 2020 2020 2030 2031 2032 2033 0 1 2 3 │ │ │ │ -0006b700: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -0006b710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b720: 207c 0a7c 6f37 203d 2074 6f74 616c 3a20 |.|o7 = total: │ │ │ │ -0006b730: 3120 3320 3520 3720 3920 2020 2020 2020 1 3 5 7 9 │ │ │ │ -0006b740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b750: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006b760: 2020 2020 303a 2031 2033 2033 2031 202e 0: 1 3 3 1 . │ │ │ │ +0006b6d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006b6e0: 0a7c 2020 2020 2020 2020 2020 2020 3020 .| 0 │ │ │ │ +0006b6f0: 3120 3220 3320 3420 2020 2020 2020 2020 1 2 3 4 │ │ │ │ +0006b700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b710: 2020 2020 2020 7c0a 7c6f 3720 3d20 746f |.|o7 = to │ │ │ │ +0006b720: 7461 6c3a 2031 2033 2035 2037 2039 2020 tal: 1 3 5 7 9 │ │ │ │ +0006b730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b740: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b750: 2020 2020 2020 2020 2030 3a20 3120 3320 0: 1 3 │ │ │ │ +0006b760: 3320 3120 2e20 2020 2020 2020 2020 2020 3 1 . │ │ │ │ 0006b770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b780: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006b790: 0a7c 2020 2020 2020 2020 2031 3a20 2e20 .| 1: . │ │ │ │ -0006b7a0: 2e20 3220 3620 3620 2020 2020 2020 2020 . 2 6 6 │ │ │ │ -0006b7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b7c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0006b7d0: 2020 323a 202e 202e 202e 202e 2033 2020 2: . . . . 3 │ │ │ │ +0006b780: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006b790: 313a 202e 202e 2032 2036 2036 2020 2020 1: . . 2 6 6 │ │ │ │ +0006b7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b7b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006b7c0: 2020 2020 2020 2032 3a20 2e20 2e20 2e20 2: . . . │ │ │ │ +0006b7d0: 2e20 3320 2020 2020 2020 2020 2020 2020 . 3 │ │ │ │ 0006b7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b7f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b7f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006b800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b830: 2020 2020 7c0a 7c6f 3720 3a20 4265 7474 |.|o7 : Bett │ │ │ │ -0006b840: 6954 616c 6c79 2020 2020 2020 2020 2020 iTally │ │ │ │ +0006b820: 2020 2020 2020 2020 207c 0a7c 6f37 203a |.|o7 : │ │ │ │ +0006b830: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +0006b840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b860: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0006b860: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0006b870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b8a0: 2d2d 2b0a 7c69 3820 3a20 4646 203d 2053 --+.|i8 : FF = S │ │ │ │ -0006b8b0: 6861 6d61 7368 2866 662c 462c 3429 2020 hamash(ff,F,4) │ │ │ │ -0006b8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b8d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006b890: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2046 -------+.|i8 : F │ │ │ │ +0006b8a0: 4620 3d20 5368 616d 6173 6828 6666 2c46 F = Shamash(ff,F │ │ │ │ +0006b8b0: 2c34 2920 2020 2020 2020 2020 2020 2020 ,4) │ │ │ │ +0006b8c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006b8d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006b8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b910: 7c0a 7c20 2020 2020 2f20 525c 3120 2020 |.| / R\1 │ │ │ │ -0006b920: 2020 2f20 525c 3320 2020 2020 2f20 525c / R\3 / R\ │ │ │ │ -0006b930: 3620 2020 2020 2f20 525c 3130 2020 2020 6 / R\10 │ │ │ │ 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2020 2f20 525c 3135 7c0a 7c6f 0 / R\15|.|o │ │ │ │ +0006b940: 3820 3d20 7c2d 2d7c 2020 3c2d 2d20 7c2d 8 = |--| <-- |- │ │ │ │ +0006b950: 2d7c 2020 3c2d 2d20 7c2d 2d7c 2020 3c2d -| <-- |--| <- │ │ │ │ +0006b960: 2d20 7c2d 2d7c 2020 203c 2d2d 207c 2d2d - |--| <-- |-- │ │ │ │ +0006b970: 7c20 207c 0a7c 2020 2020 207c 2033 7c20 | |.| | 3| │ │ │ │ +0006b980: 2020 2020 207c 2033 7c20 2020 2020 207c | 3| | │ │ │ │ +0006b990: 2033 7c20 2020 2020 207c 2033 7c20 2020 3| | 3| │ │ │ │ +0006b9a0: 2020 2020 7c20 337c 2020 7c0a 7c20 2020 | 3| |.| │ │ │ │ +0006b9b0: 2020 5c7a 202f 2020 2020 2020 5c7a 202f \z / \z / │ │ │ │ +0006b9c0: 2020 2020 2020 5c7a 202f 2020 2020 2020 \z / │ │ │ │ +0006b9d0: 5c7a 202f 2020 2020 2020 205c 7a20 2f20 \z / \z / │ │ │ │ +0006b9e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006b9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ba10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ba20: 2020 207c 0a7c 2020 2020 2030 2020 2020 |.| 0 │ │ │ │ -0006ba30: 2020 2020 2031 2020 2020 2020 2020 2032 1 2 │ │ │ │ -0006ba40: 2020 2020 2020 2020 2033 2020 2020 2020 3 │ │ │ │ -0006ba50: 2020 2020 3420 2020 2020 7c0a 7c20 2020 4 |.| │ │ │ │ +0006ba10: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006ba20: 3020 2020 2020 2020 2020 3120 2020 2020 0 1 │ │ │ │ +0006ba30: 2020 2020 3220 2020 2020 2020 2020 3320 2 3 │ │ │ │ +0006ba40: 2020 2020 2020 2020 2034 2020 2020 207c 4 | │ │ │ │ +0006ba50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0006ba60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ba70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ba80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ba90: 207c 0a7c 6f38 203a 2043 6f6d 706c 6578 |.|o8 : Complex │ │ │ │ +0006ba80: 2020 2020 2020 7c0a 7c6f 3820 3a20 436f |.|o8 : Co │ │ │ │ +0006ba90: 6d70 6c65 7820 2020 2020 2020 2020 2020 mplex │ │ │ │ 0006baa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bac0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0006bab0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0006bac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006bad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006bae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006baf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0006bb00: 0a7c 6939 203a 2047 4720 3d20 5368 616d .|i9 : GG = Sham │ │ │ │ -0006bb10: 6173 6828 5231 2c46 2c34 2920 2020 2020 ash(R1,F,4) │ │ │ │ -0006bb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bb30: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006baf0: 2d2d 2d2d 2b0a 7c69 3920 3a20 4747 203d ----+.|i9 : GG = │ │ │ │ +0006bb00: 2053 6861 6d61 7368 2852 312c 462c 3429 Shamash(R1,F,4) │ │ │ │ +0006bb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006bb20: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006bb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bb60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0006bb70: 2020 2020 2020 2031 2020 2020 2020 2033 1 3 │ │ │ │ -0006bb80: 2020 2020 2020 2036 2020 2020 2020 2031 6 1 │ │ │ │ -0006bb90: 3020 2020 2020 2020 3135 2020 2020 2020 0 15 │ │ │ │ -0006bba0: 2020 2020 7c0a 7c6f 3920 3d20 5231 2020 |.|o9 = R1 │ │ │ │ -0006bbb0: 3c2d 2d20 5231 2020 3c2d 2d20 5231 2020 <-- R1 <-- R1 │ │ │ │ -0006bbc0: 3c2d 2d20 5231 2020 203c 2d2d 2052 3120 <-- R1 <-- R1 │ │ │ │ -0006bbd0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006bb60: 2020 7c0a 7c20 2020 2020 2020 3120 2020 |.| 1 │ │ │ │ +0006bb70: 2020 2020 3320 2020 2020 2020 3620 2020 3 6 │ │ │ │ +0006bb80: 2020 2020 3130 2020 2020 2020 2031 3520 10 15 │ │ │ │ +0006bb90: 2020 2020 2020 2020 207c 0a7c 6f39 203d |.|o9 = │ │ │ │ +0006bba0: 2052 3120 203c 2d2d 2052 3120 203c 2d2d R1 <-- R1 <-- │ │ │ │ +0006bbb0: 2052 3120 203c 2d2d 2052 3120 2020 3c2d R1 <-- R1 <- │ │ │ │ +0006bbc0: 2d20 5231 2020 2020 2020 2020 2020 2020 - R1 │ │ │ │ +0006bbd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bc10: 2020 7c0a 7c20 2020 2020 3020 2020 2020 |.| 0 │ │ │ │ -0006bc20: 2020 3120 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ -0006bc30: 2020 3320 2020 2020 2020 2034 2020 2020 3 4 │ │ │ │ -0006bc40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006bc00: 2020 2020 2020 207c 0a7c 2020 2020 2030 |.| 0 │ │ │ │ +0006bc10: 2020 2020 2020 2031 2020 2020 2020 2032 1 2 │ │ │ │ +0006bc20: 2020 2020 2020 2033 2020 2020 2020 2020 3 │ │ │ │ +0006bc30: 3420 2020 2020 2020 2020 2020 2020 7c0a 4 |. │ │ │ │ +0006bc40: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006bc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bc80: 7c0a 7c6f 3920 3a20 436f 6d70 6c65 7820 |.|o9 : Complex │ │ │ │ +0006bc70: 2020 2020 207c 0a7c 6f39 203a 2043 6f6d |.|o9 : Com │ │ │ │ +0006bc80: 706c 6578 2020 2020 2020 2020 2020 2020 plex │ │ │ │ 0006bc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bcb0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0006bca0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0006bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006bce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0006bcf0: 7c69 3130 203a 2062 6574 7469 2046 4620 |i10 : betti FF │ │ │ │ +0006bce0: 2d2d 2d2b 0a7c 6931 3020 3a20 6265 7474 ---+.|i10 : bett │ │ │ │ +0006bcf0: 6920 4646 2020 2020 2020 2020 2020 2020 i FF │ │ │ │ 0006bd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bd20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006bd10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006bd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bd50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0006bd60: 2020 2020 2020 2020 2020 2020 3020 3120 0 1 │ │ │ │ -0006bd70: 3220 2033 2020 3420 2020 2020 2020 2020 2 3 4 │ │ │ │ -0006bd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bd90: 2020 207c 0a7c 6f31 3020 3d20 746f 7461 |.|o10 = tota │ │ │ │ -0006bda0: 6c3a 2031 2033 2036 2031 3020 3135 2020 l: 1 3 6 10 15 │ │ │ │ -0006bdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bdc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0006bdd0: 2020 2020 2020 2030 3a20 3120 3320 3320 0: 1 3 3 │ │ │ │ -0006bde0: 2031 2020 2e20 2020 2020 2020 2020 2020 1 . │ │ │ │ -0006bdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006be00: 207c 0a7c 2020 2020 2020 2020 2020 313a |.| 1: │ │ │ │ -0006be10: 202e 202e 2033 2020 3920 2039 2020 2020 . . 3 9 9 │ │ │ │ -0006be20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006be30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006be40: 2020 2020 2032 3a20 2e20 2e20 2e20 202e 2: . . . . │ │ │ │ -0006be50: 2020 3620 2020 2020 2020 2020 2020 2020 6 │ │ │ │ -0006be60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006be70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006bd50: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006bd60: 2030 2031 2032 2020 3320 2034 2020 2020 0 1 2 3 4 │ │ │ │ +0006bd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006bd80: 2020 2020 2020 2020 7c0a 7c6f 3130 203d |.|o10 = │ │ │ │ +0006bd90: 2074 6f74 616c 3a20 3120 3320 3620 3130 total: 1 3 6 10 │ │ │ │ +0006bda0: 2031 3520 2020 2020 2020 2020 2020 2020 15 │ │ │ │ +0006bdb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006bdc0: 0a7c 2020 2020 2020 2020 2020 303a 2031 .| 0: 1 │ │ │ │ +0006bdd0: 2033 2033 2020 3120 202e 2020 2020 2020 3 3 1 . │ │ │ │ +0006bde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006bdf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006be00: 2020 2031 3a20 2e20 2e20 3320 2039 2020 1: . . 3 9 │ │ │ │ +0006be10: 3920 2020 2020 2020 2020 2020 2020 2020 9 │ │ │ │ +0006be20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006be30: 2020 2020 2020 2020 2020 323a 202e 202e 2: . . │ │ │ │ +0006be40: 202e 2020 2e20 2036 2020 2020 2020 2020 . . 6 │ │ │ │ +0006be50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006be60: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006be70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006be80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006be90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bea0: 2020 2020 2020 7c0a 7c6f 3130 203a 2042 |.|o10 : B │ │ │ │ -0006beb0: 6574 7469 5461 6c6c 7920 2020 2020 2020 ettiTally │ │ │ │ +0006be90: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0006bea0: 3020 3a20 4265 7474 6954 616c 6c79 2020 0 : BettiTally │ │ │ │ +0006beb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bed0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0006bed0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0006bee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006bf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006bf10: 2d2d 2d2d 2b0a 7c69 3131 203a 2062 6574 ----+.|i11 : bet │ │ │ │ -0006bf20: 7469 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│ │ │ │ -0006bfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bff0: 7c0a 7c20 2020 2020 2020 2020 2030 3a20 |.| 0: │ │ │ │ -0006c000: 3120 3320 3320 2031 2020 2e20 2020 2020 1 3 3 1 . │ │ │ │ -0006c010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c020: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0006c030: 2020 2020 313a 202e 202e 2033 2020 3920 1: . . 3 9 │ │ │ │ -0006c040: 2039 2020 2020 2020 2020 2020 2020 2020 9 │ │ │ │ -0006c050: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006c060: 7c20 2020 2020 2020 2020 2032 3a20 2e20 | 2: . │ │ │ │ -0006c070: 2e20 2e20 202e 2020 3620 2020 2020 2020 . . . 6 │ │ │ │ -0006c080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c090: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006bf70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006bf80: 2020 2020 2020 2030 2031 2032 2020 3320 0 1 2 3 │ │ │ │ +0006bf90: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +0006bfa0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006bfb0: 7c6f 3131 203d 2074 6f74 616c 3a20 3120 |o11 = total: 1 │ │ │ │ +0006bfc0: 3320 3620 3130 2031 3520 2020 2020 2020 3 6 10 15 │ │ │ │ +0006bfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006bfe0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006bff0: 2020 303a 2031 2033 2033 2020 3120 202e 0: 1 3 3 1 . │ │ │ │ +0006c000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006c010: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006c020: 2020 2020 2020 2020 2031 3a20 2e20 2e20 1: . . │ │ │ │ +0006c030: 3320 2039 2020 3920 2020 2020 2020 2020 3 9 9 │ │ │ │ +0006c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006c050: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006c060: 323a 202e 202e 202e 2020 2e20 2036 2020 2: . . . . 6 │ │ │ │ +0006c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006c080: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0006c0d0: 3131 203a 2042 6574 7469 5461 6c6c 7920 11 : BettiTally │ │ │ │ +0006c0c0: 207c 0a7c 6f31 3120 3a20 4265 7474 6954 |.|o11 : BettiT │ │ │ │ +0006c0d0: 616c 6c79 2020 2020 2020 2020 2020 2020 ally │ │ │ │ 0006c0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c100: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0006c0f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0006c100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006c120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006c130: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 ----------+.|i12 │ │ │ │ -0006c140: 203a 2072 696e 6720 4747 2020 2020 2020 : ring GG │ │ │ │ +0006c120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0006c130: 0a7c 6931 3220 3a20 7269 6e67 2047 4720 .|i12 : ring GG │ │ │ │ +0006c140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c170: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006c160: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006c170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c1a0: 2020 2020 2020 2020 7c0a 7c6f 3132 203d |.|o12 = │ │ │ │ -0006c1b0: 2052 3120 2020 2020 2020 2020 2020 2020 R1 │ │ │ │ +0006c190: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006c1a0: 6f31 3220 3d20 5231 2020 2020 2020 2020 o12 = R1 │ │ │ │ +0006c1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c1c0: 2020 2020 2020 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2020 2020 2020 2020 2020 │ │ │ │ -0006c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c390: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0006c380: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0006c390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006c3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0006c3d0: 0a43 6176 6561 740a 3d3d 3d3d 3d3d 0a0a .Caveat.======.. │ │ │ │ -0006c3e0: 4620 6973 2061 7373 756d 6564 2074 6f20 F is assumed to │ │ │ │ -0006c3f0: 6265 2061 2068 6f6d 6f6c 6f67 6963 616c be a homological │ │ │ │ -0006c400: 2063 6f6d 706c 6578 2073 7461 7274 696e complex startin │ │ │ │ -0006c410: 6720 6672 6f6d 2046 5f30 2e20 5468 6520 g from F_0. The │ │ │ │ -0006c420: 6d61 7472 6978 2066 6620 6d75 7374 0a62 matrix ff must.b │ │ │ │ -0006c430: 6520 3178 312e 0a0a 5365 6520 616c 736f e 1x1...See also │ │ │ │ -0006c440: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ -0006c450: 6e6f 7465 2045 6973 656e 6275 6453 6861 note EisenbudSha │ │ │ │ -0006c460: 6d61 7368 3a20 4569 7365 6e62 7564 5368 mash: EisenbudSh │ │ │ │ -0006c470: 616d 6173 682c 202d 2d20 436f 6d70 7574 amash, -- Comput │ │ │ │ -0006c480: 6573 2074 6865 2045 6973 656e 6275 642d es the Eisenbud- │ │ │ │ -0006c490: 5368 616d 6173 680a 2020 2020 436f 6d70 Shamash. Comp │ │ │ │ -0006c4a0: 6c65 780a 2020 2a20 2a6e 6f74 6520 6d61 lex. * *note ma │ │ │ │ -0006c4b0: 6b65 486f 6d6f 746f 7069 6573 3a20 6d61 keHomotopies: ma │ │ │ │ -0006c4c0: 6b65 486f 6d6f 746f 7069 6573 2c20 2d2d keHomotopies, -- │ │ │ │ -0006c4d0: 2072 6574 7572 6e73 2061 2073 7973 7465 returns a syste │ │ │ │ -0006c4e0: 6d20 6f66 2068 6967 6865 720a 2020 2020 m of higher. │ │ │ │ -0006c4f0: 686f 6d6f 746f 7069 6573 0a0a 5761 7973 homotopies..Ways │ │ │ │ -0006c500: 2074 6f20 7573 6520 5368 616d 6173 683a to use Shamash: │ │ │ │ -0006c510: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0006c520: 3d3d 3d3d 3d0a 0a20 202a 2022 5368 616d =====.. * "Sham │ │ │ │ -0006c530: 6173 6828 4d61 7472 6978 2c43 6f6d 706c ash(Matrix,Compl │ │ │ │ -0006c540: 6578 2c5a 5a29 220a 2020 2a20 2253 6861 ex,ZZ)". * "Sha │ │ │ │ -0006c550: 6d61 7368 2852 696e 672c 436f 6d70 6c65 mash(Ring,Comple │ │ │ │ -0006c560: 782c 5a5a 2922 0a0a 466f 7220 7468 6520 x,ZZ)"..For the │ │ │ │ -0006c570: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -0006c580: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -0006c590: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -0006c5a0: 5368 616d 6173 683a 2053 6861 6d61 7368 Shamash: Shamash │ │ │ │ -0006c5b0: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -0006c5c0: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod function:.(M │ │ │ │ -0006c5d0: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -0006c5e0: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d odFunction,...-- │ │ │ │ +0006c3c0: 2d2d 2d2b 0a0a 4361 7665 6174 0a3d 3d3d ---+..Caveat.=== │ │ │ │ +0006c3d0: 3d3d 3d0a 0a46 2069 7320 6173 7375 6d65 ===..F is assume │ │ │ │ +0006c3e0: 6420 746f 2062 6520 6120 686f 6d6f 6c6f d to be a homolo │ │ │ │ +0006c3f0: 6769 6361 6c20 636f 6d70 6c65 7820 7374 gical complex st │ │ │ │ +0006c400: 6172 7469 6e67 2066 726f 6d20 465f 302e arting from F_0. │ │ │ │ +0006c410: 2054 6865 206d 6174 7269 7820 6666 206d The matrix ff m │ │ │ │ +0006c420: 7573 740a 6265 2031 7831 2e0a 0a53 6565 ust.be 1x1...See │ │ │ │ +0006c430: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ +0006c440: 2020 2a20 2a6e 6f74 6520 4569 7365 6e62 * *note Eisenb │ │ │ │ +0006c450: 7564 5368 616d 6173 683a 2045 6973 656e udShamash: Eisen │ │ │ │ +0006c460: 6275 6453 6861 6d61 7368 2c20 2d2d 2043 budShamash, -- C │ │ │ │ +0006c470: 6f6d 7075 7465 7320 7468 6520 4569 7365 omputes the Eise │ │ │ │ +0006c480: 6e62 7564 2d53 6861 6d61 7368 0a20 2020 nbud-Shamash. │ │ │ │ +0006c490: 2043 6f6d 706c 6578 0a20 202a 202a 6e6f Complex. * *no │ │ │ │ +0006c4a0: 7465 206d 616b 6548 6f6d 6f74 6f70 6965 te makeHomotopie │ │ │ │ +0006c4b0: 733a 206d 616b 6548 6f6d 6f74 6f70 6965 s: makeHomotopie │ │ │ │ +0006c4c0: 732c 202d 2d20 7265 7475 726e 7320 6120 s, -- returns a │ │ │ │ +0006c4d0: 7379 7374 656d 206f 6620 6869 6768 6572 system of higher │ │ │ │ +0006c4e0: 0a20 2020 2068 6f6d 6f74 6f70 6965 730a . homotopies. │ │ │ │ +0006c4f0: 0a57 6179 7320 746f 2075 7365 2053 6861 .Ways to use Sha │ │ │ │ +0006c500: 6d61 7368 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d mash:.========== │ │ │ │ +0006c510: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +0006c520: 2253 6861 6d61 7368 284d 6174 7269 782c "Shamash(Matrix, │ │ │ │ +0006c530: 436f 6d70 6c65 782c 5a5a 2922 0a20 202a Complex,ZZ)". * │ │ │ │ +0006c540: 2022 5368 616d 6173 6828 5269 6e67 2c43 "Shamash(Ring,C │ │ │ │ +0006c550: 6f6d 706c 6578 2c5a 5a29 220a 0a46 6f72 omplex,ZZ)"..For │ │ │ │ +0006c560: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +0006c570: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006c580: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +0006c590: 6e6f 7465 2053 6861 6d61 7368 3a20 5368 note Shamash: Sh │ │ │ │ +0006c5a0: 616d 6173 682c 2069 7320 6120 2a6e 6f74 amash, is a *not │ │ │ │ +0006c5b0: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +0006c5c0: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63 n:.(Macaulay2Doc │ │ │ │ +0006c5d0: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ +0006c5e0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 0006c5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006c630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -0006c640: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -0006c650: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -0006c660: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -0006c670: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -0006c680: 6179 322d 312e 3236 2e30 352b 6473 2f4d ay2-1.26.05+ds/M │ │ │ │ -0006c690: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -0006c6a0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ -0006c6b0: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -0006c6c0: 7469 6f6e 732e 6d32 3a34 3736 303a 302e tions.m2:4760:0. │ │ │ │ -0006c6d0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ -0006c6e0: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -0006c6f0: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ -0006c700: 6f64 653a 2073 706c 6974 7469 6e67 732c ode: splittings, │ │ │ │ -0006c710: 204e 6578 743a 2073 7461 626c 6548 6f6d Next: stableHom │ │ │ │ -0006c720: 2c20 5072 6576 3a20 5368 616d 6173 682c , Prev: Shamash, │ │ │ │ -0006c730: 2055 703a 2054 6f70 0a0a 7370 6c69 7474 Up: Top..splitt │ │ │ │ -0006c740: 696e 6773 202d 2d20 636f 6d70 7574 6520 ings -- compute │ │ │ │ -0006c750: 7468 6520 7370 6c69 7474 696e 6773 206f the splittings o │ │ │ │ -0006c760: 6620 6120 7370 6c69 7420 7269 6768 7420 f a split right │ │ │ │ -0006c770: 6578 6163 7420 7365 7175 656e 6365 0a2a exact sequence.* │ │ │ │ +0006c630: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0006c640: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0006c650: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0006c660: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0006c670: 6163 6175 6c61 7932 2d31 2e32 362e 3035 acaulay2-1.26.05 │ │ │ │ +0006c680: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +0006c690: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ +0006c6a0: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +0006c6b0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3437 esolutions.m2:47 │ │ │ │ +0006c6c0: 3630 3a30 2e0a 1f0a 4669 6c65 3a20 436f 60:0....File: Co │ │ │ │ +0006c6d0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +0006c6e0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ +0006c6f0: 666f 2c20 4e6f 6465 3a20 7370 6c69 7474 fo, Node: splitt │ │ │ │ +0006c700: 696e 6773 2c20 4e65 7874 3a20 7374 6162 ings, Next: stab │ │ │ │ +0006c710: 6c65 486f 6d2c 2050 7265 763a 2053 6861 leHom, Prev: Sha │ │ │ │ +0006c720: 6d61 7368 2c20 5570 3a20 546f 700a 0a73 mash, Up: Top..s │ │ │ │ +0006c730: 706c 6974 7469 6e67 7320 2d2d 2063 6f6d plittings -- com │ │ │ │ +0006c740: 7075 7465 2074 6865 2073 706c 6974 7469 pute the splitti │ │ │ │ +0006c750: 6e67 7320 6f66 2061 2073 706c 6974 2072 ngs of a split r │ │ │ │ +0006c760: 6967 6874 2065 7861 6374 2073 6571 7565 ight exact seque │ │ │ │ +0006c770: 6e63 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a nce.************ │ │ │ │ 0006c780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006c790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006c7a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006c7b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006c7c0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -0006c7d0: 0a20 2020 2020 2020 2078 203d 2073 706c . x = spl │ │ │ │ -0006c7e0: 6974 7469 6e67 7328 612c 6229 0a20 202a ittings(a,b). * │ │ │ │ -0006c7f0: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -0006c800: 2061 2c20 6120 2a6e 6f74 6520 6d61 7472 a, a *note matr │ │ │ │ -0006c810: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ -0006c820: 6329 4d61 7472 6978 2c2c 206d 6170 7320 c)Matrix,, maps │ │ │ │ -0006c830: 696e 746f 2074 6865 206b 6572 6e65 6c20 into the kernel │ │ │ │ -0006c840: 6f66 2062 0a20 2020 2020 202a 2062 2c20 of b. * b, │ │ │ │ -0006c850: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ -0006c860: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ -0006c870: 7472 6978 2c2c 2072 6570 7265 7365 6e74 trix,, represent │ │ │ │ -0006c880: 696e 6720 6120 7375 726a 6563 7469 6f6e ing a surjection │ │ │ │ -0006c890: 0a20 2020 2020 2020 2066 726f 6d20 7461 . from ta │ │ │ │ -0006c8a0: 7267 6574 2061 2074 6f20 6120 6672 6565 rget a to a free │ │ │ │ -0006c8b0: 206d 6f64 756c 650a 2020 2a20 4f75 7470 module. * Outp │ │ │ │ -0006c8c0: 7574 733a 0a20 2020 2020 202a 204c 2c20 uts:. * L, │ │ │ │ -0006c8d0: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d a *note list: (M │ │ │ │ -0006c8e0: 6163 6175 6c61 7932 446f 6329 4c69 7374 acaulay2Doc)List │ │ │ │ -0006c8f0: 2c2c 204c 203d 205c 7b73 6967 6d61 2c74 ,, L = \{sigma,t │ │ │ │ -0006c900: 6175 5c7d 2c20 7370 6c69 7474 696e 6773 au\}, splittings │ │ │ │ -0006c910: 206f 660a 2020 2020 2020 2020 612c 6220 of. a,b │ │ │ │ -0006c920: 7265 7370 6563 7469 7665 6c79 0a0a 4465 respectively..De │ │ │ │ -0006c930: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ -0006c940: 3d3d 3d3d 3d0a 0a41 7373 756d 696e 6720 =====..Assuming │ │ │ │ -0006c950: 7468 6174 2028 612c 6229 2061 7265 2074 that (a,b) are t │ │ │ │ -0006c960: 6865 206d 6170 7320 6f66 2061 2072 6967 he maps of a rig │ │ │ │ -0006c970: 6874 2065 7861 6374 2073 6571 7565 6e63 ht exact sequenc │ │ │ │ -0006c980: 650a 0a24 305c 746f 2041 5c74 6f20 425c e..$0\to A\to B\ │ │ │ │ -0006c990: 746f 2043 205c 746f 2030 240a 0a77 6974 to C \to 0$..wit │ │ │ │ -0006c9a0: 6820 422c 2043 2066 7265 652c 2074 6865 h B, C free, the │ │ │ │ -0006c9b0: 2073 6372 6970 7420 7072 6f64 7563 6573 script produces │ │ │ │ -0006c9c0: 2061 2070 6169 7220 6f66 206d 6170 7320 a pair of maps │ │ │ │ -0006c9d0: 7369 676d 612c 2074 6175 2077 6974 6820 sigma, tau with │ │ │ │ -0006c9e0: 2474 6175 3a20 4320 5c74 6f0a 4224 2061 $tau: C \to.B$ a │ │ │ │ -0006c9f0: 2073 706c 6974 7469 6e67 206f 6620 6220 splitting of b │ │ │ │ -0006ca00: 616e 6420 2473 6967 6d61 3a20 4220 5c74 and $sigma: B \t │ │ │ │ -0006ca10: 6f20 4124 2061 2073 706c 6974 7469 6e67 o A$ a splitting │ │ │ │ -0006ca20: 206f 6620 613b 2074 6861 7420 6973 2c0a of a; that is,. │ │ │ │ -0006ca30: 0a24 612a 7369 676d 612b 7461 752a 6220 .$a*sigma+tau*b │ │ │ │ -0006ca40: 3d20 315f 4224 0a0a 2473 6967 6d61 2a61 = 1_B$..$sigma*a │ │ │ │ -0006ca50: 203d 2031 5f41 240a 0a24 622a 7461 7520 = 1_A$..$b*tau │ │ │ │ -0006ca60: 3d20 315f 4324 0a0a 2b2d 2d2d 2d2d 2d2d = 1_C$..+------- │ │ │ │ +0006c7b0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ +0006c7c0: 6167 653a 200a 2020 2020 2020 2020 7820 age: . x │ │ │ │ +0006c7d0: 3d20 7370 6c69 7474 696e 6773 2861 2c62 = splittings(a,b │ │ │ │ +0006c7e0: 290a 2020 2a20 496e 7075 7473 3a0a 2020 ). * Inputs:. │ │ │ │ +0006c7f0: 2020 2020 2a20 612c 2061 202a 6e6f 7465 * a, a *note │ │ │ │ +0006c800: 206d 6174 7269 783a 2028 4d61 6361 756c matrix: (Macaul │ │ │ │ +0006c810: 6179 3244 6f63 294d 6174 7269 782c 2c20 ay2Doc)Matrix,, │ │ │ │ +0006c820: 6d61 7073 2069 6e74 6f20 7468 6520 6b65 maps into the ke │ │ │ │ +0006c830: 726e 656c 206f 6620 620a 2020 2020 2020 rnel of b. │ │ │ │ +0006c840: 2a20 622c 2061 202a 6e6f 7465 206d 6174 * b, a *note mat │ │ │ │ +0006c850: 7269 783a 2028 4d61 6361 756c 6179 3244 rix: (Macaulay2D │ │ │ │ +0006c860: 6f63 294d 6174 7269 782c 2c20 7265 7072 oc)Matrix,, repr │ │ │ │ +0006c870: 6573 656e 7469 6e67 2061 2073 7572 6a65 esenting a surje │ │ │ │ +0006c880: 6374 696f 6e0a 2020 2020 2020 2020 6672 ction. fr │ │ │ │ +0006c890: 6f6d 2074 6172 6765 7420 6120 746f 2061 om target a to a │ │ │ │ +0006c8a0: 2066 7265 6520 6d6f 6475 6c65 0a20 202a free module. * │ │ │ │ +0006c8b0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ +0006c8c0: 2a20 4c2c 2061 202a 6e6f 7465 206c 6973 * L, a *note lis │ │ │ │ +0006c8d0: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ +0006c8e0: 294c 6973 742c 2c20 4c20 3d20 5c7b 7369 )List,, L = \{si │ │ │ │ +0006c8f0: 676d 612c 7461 755c 7d2c 2073 706c 6974 gma,tau\}, split │ │ │ │ +0006c900: 7469 6e67 7320 6f66 0a20 2020 2020 2020 tings of. │ │ │ │ +0006c910: 2061 2c62 2072 6573 7065 6374 6976 656c a,b respectivel │ │ │ │ +0006c920: 790a 0a44 6573 6372 6970 7469 6f6e 0a3d y..Description.= │ │ │ │ +0006c930: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4173 7375 ==========..Assu │ │ │ │ +0006c940: 6d69 6e67 2074 6861 7420 2861 2c62 2920 ming that (a,b) │ │ │ │ +0006c950: 6172 6520 7468 6520 6d61 7073 206f 6620 are the maps of │ │ │ │ +0006c960: 6120 7269 6768 7420 6578 6163 7420 7365 a right exact se │ │ │ │ +0006c970: 7175 656e 6365 0a0a 2430 5c74 6f20 415c quence..$0\to A\ │ │ │ │ +0006c980: 746f 2042 5c74 6f20 4320 5c74 6f20 3024 to B\to C \to 0$ │ │ │ │ +0006c990: 0a0a 7769 7468 2042 2c20 4320 6672 6565 ..with B, C free │ │ │ │ +0006c9a0: 2c20 7468 6520 7363 7269 7074 2070 726f , the script pro │ │ │ │ +0006c9b0: 6475 6365 7320 6120 7061 6972 206f 6620 duces a pair of │ │ │ │ +0006c9c0: 6d61 7073 2073 6967 6d61 2c20 7461 7520 maps sigma, tau │ │ │ │ +0006c9d0: 7769 7468 2024 7461 753a 2043 205c 746f with $tau: C \to │ │ │ │ +0006c9e0: 0a42 2420 6120 7370 6c69 7474 696e 6720 .B$ a splitting │ │ │ │ +0006c9f0: 6f66 2062 2061 6e64 2024 7369 676d 613a of b and $sigma: │ │ │ │ +0006ca00: 2042 205c 746f 2041 2420 6120 7370 6c69 B \to A$ a spli │ │ │ │ +0006ca10: 7474 696e 6720 6f66 2061 3b20 7468 6174 tting of a; that │ │ │ │ +0006ca20: 2069 732c 0a0a 2461 2a73 6967 6d61 2b74 is,..$a*sigma+t │ │ │ │ +0006ca30: 6175 2a62 203d 2031 5f42 240a 0a24 7369 au*b = 1_B$..$si │ │ │ │ +0006ca40: 676d 612a 6120 3d20 315f 4124 0a0a 2462 gma*a = 1_A$..$b │ │ │ │ +0006ca50: 2a74 6175 203d 2031 5f43 240a 0a2b 2d2d *tau = 1_C$..+-- │ │ │ │ +0006ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006caa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006cab0: 2d2d 2d2b 0a7c 6931 203a 206b 6b3d 205a ---+.|i1 : kk= Z │ │ │ │ -0006cac0: 5a2f 3130 3120 2020 2020 2020 2020 2020 Z/101 │ │ │ │ +0006caa0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +0006cab0: 6b6b 3d20 5a5a 2f31 3031 2020 2020 2020 kk= ZZ/101 │ │ │ │ +0006cac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006caf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cb00: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006caf0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006cb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cb40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0006cb50: 6f31 203d 206b 6b20 2020 2020 2020 2020 o1 = kk │ │ │ │ +0006cb40: 2020 7c0a 7c6f 3120 3d20 6b6b 2020 2020 |.|o1 = kk │ │ │ │ +0006cb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cb90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006cb80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006cb90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0006cba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cbe0: 2020 2020 2020 207c 0a7c 6f31 203a 2051 |.|o1 : Q │ │ │ │ -0006cbf0: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +0006cbd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0006cbe0: 3120 3a20 5175 6f74 6965 6e74 5269 6e67 1 : QuotientRing │ │ │ │ +0006cbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cc30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0006cc20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0006cc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006cc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006cc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006cc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006cc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006cc80: 2d2b 0a7c 6932 203a 2053 203d 206b 6b5b -+.|i2 : S = kk[ │ │ │ │ -0006cc90: 782c 792c 7a5d 2020 2020 2020 2020 2020 x,y,z] │ │ │ │ +0006cc70: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 ------+.|i2 : S │ │ │ │ +0006cc80: 3d20 6b6b 5b78 2c79 2c7a 5d20 2020 2020 = kk[x,y,z] │ │ │ │ +0006cc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ccb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ccc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006ccd0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0006ccc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006ccd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ccf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cd10: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006cd20: 203d 2053 2020 2020 2020 2020 2020 2020 = S │ │ │ │ +0006cd10: 7c0a 7c6f 3220 3d20 5320 2020 2020 2020 |.|o2 = S │ │ │ │ +0006cd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cd60: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006cd50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006cd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cdb0: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c |.|o2 : Pol │ │ │ │ -0006cdc0: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ +0006cda0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0006cdb0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +0006cdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ce00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0006cdf0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0006ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ce20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ce40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0006ce50: 0a7c 6933 203a 2073 6574 5261 6e64 6f6d .|i3 : setRandom │ │ │ │ -0006ce60: 5365 6564 2030 2020 2020 2020 2020 2020 Seed 0 │ │ │ │ +0006ce40: 2d2d 2d2d 2b0a 7c69 3320 3a20 7365 7452 ----+.|i3 : setR │ │ │ │ +0006ce50: 616e 646f 6d53 6565 6420 3020 2020 2020 andomSeed 0 │ │ │ │ +0006ce60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ce70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ce80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ce90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0006cea0: 2d2d 2073 6574 7469 6e67 2072 616e 646f -- setting rando │ │ │ │ -0006ceb0: 6d20 7365 6564 2074 6f20 3020 2020 2020 m seed to 0 │ │ │ │ +0006ce90: 207c 0a7c 202d 2d20 7365 7474 696e 6720 |.| -- setting │ │ │ │ +0006cea0: 7261 6e64 6f6d 2073 6565 6420 746f 2030 random seed to 0 │ │ │ │ +0006ceb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ced0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cee0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006ced0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006cee0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006cef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cf30: 2020 2020 2020 7c0a 7c6f 3320 3d20 3020 |.|o3 = 0 │ │ │ │ +0006cf20: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +0006cf30: 203d 2030 2020 2020 2020 2020 2020 2020 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+0006cfc0: 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 203d -----+.|i4 : t = │ │ │ │ +0006cfd0: 2072 616e 646f 6d28 535e 7b32 3a2d 312c random(S^{2:-1, │ │ │ │ +0006cfe0: 323a 2d32 7d2c 2053 5e7b 333a 2d31 2c34 2:-2}, S^{3:-1,4 │ │ │ │ +0006cff0: 3a2d 327d 2920 2020 2020 2020 2020 2020 :-2}) │ │ │ │ 0006d000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d010: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006d010: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006d020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d060: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ -0006d070: 3d20 7b31 7d20 7c20 3234 2020 2d33 3620 = {1} | 24 -36 │ │ │ │ -0006d080: 2d33 3020 3339 782d 3433 792b 3435 7a20 -30 39x-43y+45z │ │ │ │ -0006d090: 2032 3178 2d31 3579 2d33 347a 2033 3478 21x-15y-34z 34x │ │ │ │ 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│ │ │ │ -0006d320: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006d310: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006d320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d360: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0006d370: 3520 3d20 7b7b 317d 207c 2030 2030 2031 5 = {{1} | 0 0 1 │ │ │ │ -0006d380: 2030 2030 2030 2020 2030 2020 7c2c 207b 0 0 0 0 |, { │ │ │ │ -0006d390: 317d 207c 202d 3237 2032 2020 3133 782d 1} | -27 2 13x- │ │ │ │ -0006d3a0: 3130 792b 3433 7a20 3530 782d 3334 792d 10y+43z 50x-34y- │ │ │ │ -0006d3b0: 3530 7a20 7c7d 2020 207c 0a7c 2020 2020 50z |} |.| │ │ │ │ -0006d3c0: 2020 7b32 7d20 7c20 3020 3020 3020 3020 {2} | 0 0 0 0 │ │ │ │ -0006d3d0: 3020 2d33 3120 2d36 207c 2020 7b31 7d20 0 -31 -6 | {1} │ │ │ │ -0006d3e0: 7c20 2d34 2020 3335 2032 3278 2b33 3279 | 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2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d830: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0006d840: 2020 2020 2020 2020 2032 3a20 3220 2e20 2: 2 . │ │ │ │ -0006d850: 2020 2020 2032 3a20 3420 2e20 2020 2020 2: 4 . │ │ │ │ +0006d830: 207c 0a7c 2020 2020 2020 2020 2020 323a |.| 2: │ │ │ │ +0006d840: 2032 202e 2020 2020 2020 323a 2034 202e 2 . 2: 4 . │ │ │ │ +0006d850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d880: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006d870: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006d880: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006d890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d8d0: 2020 2020 2020 7c0a 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0a57 6179 7320 746f 2075 7365 2073 +..Ways to use s │ │ │ │ -0006d980: 706c 6974 7469 6e67 733a 0a3d 3d3d 3d3d plittings:.===== │ │ │ │ -0006d990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0006d9a0: 3d3d 0a0a 2020 2a20 2273 706c 6974 7469 ==.. * "splitti │ │ │ │ -0006d9b0: 6e67 7328 4d61 7472 6978 2c4d 6174 7269 ngs(Matrix,Matri │ │ │ │ -0006d9c0: 7829 220a 0a46 6f72 2074 6865 2070 726f x)"..For the pro │ │ │ │ -0006d9d0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ -0006d9e0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -0006d9f0: 6f62 6a65 6374 202a 6e6f 7465 2073 706c object *note spl │ │ │ │ -0006da00: 6974 7469 6e67 733a 2073 706c 6974 7469 ittings: splitti │ │ │ │ -0006da10: 6e67 732c 2069 7320 6120 2a6e 6f74 6520 ngs, is a *note │ │ │ │ -0006da20: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ -0006da30: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ -0006da40: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ -0006da50: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +0006d960: 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 6f20 -----+..Ways to │ │ │ │ +0006d970: 7573 6520 7370 6c69 7474 696e 6773 3a0a use splittings:. │ │ │ │ +0006d980: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006d990: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7370 =======.. * "sp │ │ │ │ +0006d9a0: 6c69 7474 696e 6773 284d 6174 7269 782c littings(Matrix, │ │ │ │ +0006d9b0: 4d61 7472 6978 2922 0a0a 466f 7220 7468 Matrix)"..For th │ │ │ │ +0006d9c0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +0006d9d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0006d9e0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +0006d9f0: 6520 7370 6c69 7474 696e 6773 3a20 7370 e splittings: sp │ │ │ │ +0006da00: 6c69 7474 696e 6773 2c20 6973 2061 202a littings, is a * │ │ │ │ +0006da10: 6e6f 7465 206d 6574 686f 6420 6675 6e63 note method func │ │ │ │ +0006da20: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932 tion:.(Macaulay2 │ │ │ │ +0006da30: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ +0006da40: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ +0006da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006da60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006daa0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -0006dab0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -0006dac0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -0006dad0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -0006dae0: 6175 6c61 7932 2d31 2e32 362e 3035 2b64 aulay2-1.26.05+d │ │ │ │ -0006daf0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -0006db00: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ -0006db10: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -0006db20: 6f6c 7574 696f 6e73 2e6d 323a 3339 3235 olutions.m2:3925 │ │ │ │ -0006db30: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ -0006db40: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -0006db50: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ -0006db60: 2c20 4e6f 6465 3a20 7374 6162 6c65 486f , Node: stableHo │ │ │ │ -0006db70: 6d2c 204e 6578 743a 2073 756d 5477 6f4d m, Next: sumTwoM │ │ │ │ -0006db80: 6f6e 6f6d 6961 6c73 2c20 5072 6576 3a20 onomials, Prev: │ │ │ │ -0006db90: 7370 6c69 7474 696e 6773 2c20 5570 3a20 splittings, Up: │ │ │ │ -0006dba0: 546f 700a 0a73 7461 626c 6548 6f6d 202d Top..stableHom - │ │ │ │ -0006dbb0: 2d20 6d61 7020 6672 6f6d 2048 6f6d 284d - map from Hom(M │ │ │ │ -0006dbc0: 2c4e 2920 746f 2074 6865 2073 7461 626c ,N) to the stabl │ │ │ │ -0006dbd0: 6520 486f 6d20 6d6f 6475 6c65 0a2a 2a2a e Hom module.*** │ │ │ │ +0006da90: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +0006daa0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +0006dab0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +0006dac0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +0006dad0: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +0006dae0: 2e30 352b 6473 2f4d 322f 4d61 6361 756c .05+ds/M2/Macaul │ │ │ │ +0006daf0: 6179 322f 7061 636b 6167 6573 2f0a 436f ay2/packages/.Co │ │ │ │ +0006db00: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +0006db10: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32 onResolutions.m2 │ │ │ │ +0006db20: 3a33 3932 353a 302e 0a1f 0a46 696c 653a :3925:0....File: │ │ │ │ +0006db30: 2043 6f6d 706c 6574 6549 6e74 6572 7365 CompleteInterse │ │ │ │ +0006db40: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ +0006db50: 2e69 6e66 6f2c 204e 6f64 653a 2073 7461 .info, Node: sta │ │ │ │ +0006db60: 626c 6548 6f6d 2c20 4e65 7874 3a20 7375 bleHom, Next: su │ │ │ │ +0006db70: 6d54 776f 4d6f 6e6f 6d69 616c 732c 2050 mTwoMonomials, P │ │ │ │ +0006db80: 7265 763a 2073 706c 6974 7469 6e67 732c rev: splittings, │ │ │ │ +0006db90: 2055 703a 2054 6f70 0a0a 7374 6162 6c65 Up: Top..stable │ │ │ │ +0006dba0: 486f 6d20 2d2d 206d 6170 2066 726f 6d20 Hom -- map from │ │ │ │ +0006dbb0: 486f 6d28 4d2c 4e29 2074 6f20 7468 6520 Hom(M,N) to the │ │ │ │ +0006dbc0: 7374 6162 6c65 2048 6f6d 206d 6f64 756c stable Hom modul │ │ │ │ +0006dbd0: 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a e.************** │ │ │ │ 0006dbe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006dbf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006dc00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006dc10: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -0006dc20: 200a 2020 2020 2020 2020 7020 3d20 7374 . p = st │ │ │ │ -0006dc30: 6162 6c65 486f 6d28 4d2c 4e29 0a20 202a ableHom(M,N). * │ │ │ │ -0006dc40: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -0006dc50: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -0006dc60: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -0006dc70: 6329 4d6f 6475 6c65 2c2c 200a 2020 2020 c)Module,, . │ │ │ │ -0006dc80: 2020 2a20 4e2c 2061 202a 6e6f 7465 206d * N, a *note m │ │ │ │ -0006dc90: 6f64 756c 653a 2028 4d61 6361 756c 6179 odule: (Macaulay │ │ │ │ -0006dca0: 3244 6f63 294d 6f64 756c 652c 2c20 0a20 2Doc)Module,, . │ │ │ │ -0006dcb0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ -0006dcc0: 2020 2a20 702c 2061 202a 6e6f 7465 206d * p, a *note m │ │ │ │ -0006dcd0: 6174 7269 783a 2028 4d61 6361 756c 6179 atrix: (Macaulay │ │ │ │ -0006dce0: 3244 6f63 294d 6174 7269 782c 2c20 7072 2Doc)Matrix,, pr │ │ │ │ -0006dcf0: 6f6a 6563 7469 6f6e 2066 726f 6d20 486f ojection from Ho │ │ │ │ -0006dd00: 6d28 4d2c 4e29 2074 6f0a 2020 2020 2020 m(M,N) to. │ │ │ │ -0006dd10: 2020 7468 6520 7374 6162 6c65 2048 6f6d the stable Hom │ │ │ │ -0006dd20: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -0006dd30: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 2073 =========..The s │ │ │ │ -0006dd40: 7461 626c 6520 486f 6d20 6973 2048 6f6d table Hom is Hom │ │ │ │ -0006dd50: 284d 2c4e 292f 5420 7768 6572 6520 5420 (M,N)/T where T │ │ │ │ -0006dd60: 6973 2074 6865 2073 7562 6d6f 6475 6c65 is the submodule │ │ │ │ -0006dd70: 206f 6620 686f 6d6f 6d6f 7270 6869 736d of homomorphism │ │ │ │ -0006dd80: 7320 7468 6174 0a66 6163 746f 7220 7468 s that.factor th │ │ │ │ -0006dd90: 726f 7567 6820 6120 6672 6565 2063 6f76 rough a free cov │ │ │ │ -0006dda0: 6572 206f 6620 4e20 286f 722c 2065 7175 er of N (or, equ │ │ │ │ -0006ddb0: 6976 616c 656e 746c 792c 2074 6872 6f75 ivalently, throu │ │ │ │ -0006ddc0: 6768 2061 6e79 2070 726f 6a65 6374 6976 gh any projectiv │ │ │ │ -0006ddd0: 6529 0a0a 5365 6520 616c 736f 0a3d 3d3d e)..See also.=== │ │ │ │ -0006dde0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -0006ddf0: 2069 7353 7461 626c 7954 7269 7669 616c isStablyTrivial │ │ │ │ -0006de00: 3a20 6973 5374 6162 6c79 5472 6976 6961 : isStablyTrivia │ │ │ │ -0006de10: 6c2c 202d 2d20 7265 7475 726e 7320 7472 l, -- returns tr │ │ │ │ -0006de20: 7565 2069 6620 7468 6520 6d61 7020 676f ue if the map go │ │ │ │ -0006de30: 6573 2074 6f0a 2020 2020 3020 756e 6465 es to. 0 unde │ │ │ │ -0006de40: 7220 7374 6162 6c65 486f 6d0a 0a57 6179 r stableHom..Way │ │ │ │ -0006de50: 7320 746f 2075 7365 2073 7461 626c 6548 s to use stableH │ │ │ │ -0006de60: 6f6d 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d om:.============ │ │ │ │ -0006de70: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -0006de80: 2273 7461 626c 6548 6f6d 284d 6f64 756c "stableHom(Modul │ │ │ │ -0006de90: 652c 4d6f 6475 6c65 2922 0a0a 466f 7220 e,Module)"..For │ │ │ │ -0006dea0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -0006deb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0006dec0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -0006ded0: 6f74 6520 7374 6162 6c65 486f 6d3a 2073 ote stableHom: s │ │ │ │ -0006dee0: 7461 626c 6548 6f6d 2c20 6973 2061 202a tableHom, is a * │ │ │ │ -0006def0: 6e6f 7465 206d 6574 686f 6420 6675 6e63 note method func │ │ │ │ -0006df00: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932 tion:.(Macaulay2 │ │ │ │ -0006df10: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ -0006df20: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ +0006dc00: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +0006dc10: 7361 6765 3a20 0a20 2020 2020 2020 2070 sage: . p │ │ │ │ +0006dc20: 203d 2073 7461 626c 6548 6f6d 284d 2c4e = stableHom(M,N │ │ │ │ +0006dc30: 290a 2020 2a20 496e 7075 7473 3a0a 2020 ). * Inputs:. │ │ │ │ +0006dc40: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +0006dc50: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +0006dc60: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +0006dc70: 0a20 2020 2020 202a 204e 2c20 6120 2a6e . * N, a *n │ │ │ │ +0006dc80: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163 ote module: (Mac │ │ │ │ +0006dc90: 6175 6c61 7932 446f 6329 4d6f 6475 6c65 aulay2Doc)Module │ │ │ │ +0006dca0: 2c2c 200a 2020 2a20 4f75 7470 7574 733a ,, . * Outputs: │ │ │ │ +0006dcb0: 0a20 2020 2020 202a 2070 2c20 6120 2a6e . * p, a *n │ │ │ │ +0006dcc0: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ +0006dcd0: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ +0006dce0: 2c2c 2070 726f 6a65 6374 696f 6e20 6672 ,, projection fr │ │ │ │ +0006dcf0: 6f6d 2048 6f6d 284d 2c4e 2920 746f 0a20 om Hom(M,N) to. │ │ │ │ +0006dd00: 2020 2020 2020 2074 6865 2073 7461 626c the stabl │ │ │ │ +0006dd10: 6520 486f 6d0a 0a44 6573 6372 6970 7469 e Hom..Descripti │ │ │ │ +0006dd20: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +0006dd30: 5468 6520 7374 6162 6c65 2048 6f6d 2069 The stable Hom i │ │ │ │ +0006dd40: 7320 486f 6d28 4d2c 4e29 2f54 2077 6865 s Hom(M,N)/T whe │ │ │ │ +0006dd50: 7265 2054 2069 7320 7468 6520 7375 626d re T is the subm │ │ │ │ +0006dd60: 6f64 756c 6520 6f66 2068 6f6d 6f6d 6f72 odule of homomor │ │ │ │ +0006dd70: 7068 6973 6d73 2074 6861 740a 6661 6374 phisms that.fact │ │ │ │ +0006dd80: 6f72 2074 6872 6f75 6768 2061 2066 7265 or through a fre │ │ │ │ +0006dd90: 6520 636f 7665 7220 6f66 204e 2028 6f72 e cover of N (or │ │ │ │ +0006dda0: 2c20 6571 7569 7661 6c65 6e74 6c79 2c20 , equivalently, │ │ │ │ +0006ddb0: 7468 726f 7567 6820 616e 7920 7072 6f6a through any proj │ │ │ │ +0006ddc0: 6563 7469 7665 290a 0a53 6565 2061 6c73 ective)..See als │ │ │ │ +0006ddd0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +0006dde0: 2a6e 6f74 6520 6973 5374 6162 6c79 5472 *note isStablyTr │ │ │ │ +0006ddf0: 6976 6961 6c3a 2069 7353 7461 626c 7954 ivial: isStablyT │ │ │ │ +0006de00: 7269 7669 616c 2c20 2d2d 2072 6574 7572 rivial, -- retur │ │ │ │ +0006de10: 6e73 2074 7275 6520 6966 2074 6865 206d ns true if the m │ │ │ │ +0006de20: 6170 2067 6f65 7320 746f 0a20 2020 2030 ap goes to. 0 │ │ │ │ +0006de30: 2075 6e64 6572 2073 7461 626c 6548 6f6d under stableHom │ │ │ │ +0006de40: 0a0a 5761 7973 2074 6f20 7573 6520 7374 ..Ways to use st │ │ │ │ +0006de50: 6162 6c65 486f 6d3a 0a3d 3d3d 3d3d 3d3d ableHom:.======= │ │ │ │ +0006de60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0006de70: 0a20 202a 2022 7374 6162 6c65 486f 6d28 . * "stableHom( │ │ │ │ +0006de80: 4d6f 6475 6c65 2c4d 6f64 756c 6529 220a Module,Module)". │ │ │ │ +0006de90: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0006dea0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0006deb0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0006dec0: 6374 202a 6e6f 7465 2073 7461 626c 6548 ct *note stableH │ │ │ │ +0006ded0: 6f6d 3a20 7374 6162 6c65 486f 6d2c 2069 om: stableHom, i │ │ │ │ +0006dee0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ +0006def0: 2066 756e 6374 696f 6e3a 0a28 4d61 6361 function:.(Maca │ │ │ │ +0006df00: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ +0006df10: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ +0006df20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006df30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006df40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006df50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006df60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006df70: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -0006df80: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -0006df90: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -0006dfa0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -0006dfb0: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -0006dfc0: 2e30 352b 6473 2f4d 322f 4d61 6361 756c .05+ds/M2/Macaul │ │ │ │ -0006dfd0: 6179 322f 7061 636b 6167 6573 2f0a 436f ay2/packages/.Co │ │ │ │ -0006dfe0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -0006dff0: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32 onResolutions.m2 │ │ │ │ -0006e000: 3a34 3634 393a 302e 0a1f 0a46 696c 653a :4649:0....File: │ │ │ │ -0006e010: 2043 6f6d 706c 6574 6549 6e74 6572 7365 CompleteInterse │ │ │ │ -0006e020: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ -0006e030: 2e69 6e66 6f2c 204e 6f64 653a 2073 756d .info, Node: sum │ │ │ │ -0006e040: 5477 6f4d 6f6e 6f6d 6961 6c73 2c20 4e65 TwoMonomials, Ne │ │ │ │ -0006e050: 7874 3a20 5461 7465 5265 736f 6c75 7469 xt: TateResoluti │ │ │ │ -0006e060: 6f6e 2c20 5072 6576 3a20 7374 6162 6c65 on, Prev: stable │ │ │ │ -0006e070: 486f 6d2c 2055 703a 2054 6f70 0a0a 7375 Hom, Up: Top..su │ │ │ │ -0006e080: 6d54 776f 4d6f 6e6f 6d69 616c 7320 2d2d mTwoMonomials -- │ │ │ │ -0006e090: 2074 616c 6c79 2074 6865 2073 6571 7565 tally the seque │ │ │ │ -0006e0a0: 6e63 6573 206f 6620 4252 616e 6b73 2066 nces of BRanks f │ │ │ │ -0006e0b0: 6f72 2063 6572 7461 696e 2065 7861 6d70 or certain examp │ │ │ │ -0006e0c0: 6c65 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a les.************ │ │ │ │ +0006df60: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +0006df70: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +0006df80: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +0006df90: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +0006dfa0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +0006dfb0: 2d31 2e32 362e 3035 2b64 732f 4d32 2f4d -1.26.05+ds/M2/M │ │ │ │ +0006dfc0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +0006dfd0: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ +0006dfe0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +0006dff0: 6e73 2e6d 323a 3436 3439 3a30 2e0a 1f0a ns.m2:4649:0.... │ │ │ │ +0006e000: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ +0006e010: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ +0006e020: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ +0006e030: 3a20 7375 6d54 776f 4d6f 6e6f 6d69 616c : sumTwoMonomial │ │ │ │ +0006e040: 732c 204e 6578 743a 2054 6174 6552 6573 s, Next: TateRes │ │ │ │ +0006e050: 6f6c 7574 696f 6e2c 2050 7265 763a 2073 olution, Prev: s │ │ │ │ +0006e060: 7461 626c 6548 6f6d 2c20 5570 3a20 546f tableHom, Up: To │ │ │ │ +0006e070: 700a 0a73 756d 5477 6f4d 6f6e 6f6d 6961 p..sumTwoMonomia │ │ │ │ +0006e080: 6c73 202d 2d20 7461 6c6c 7920 7468 6520 ls -- tally the │ │ │ │ +0006e090: 7365 7175 656e 6365 7320 6f66 2042 5261 sequences of BRa │ │ │ │ +0006e0a0: 6e6b 7320 666f 7220 6365 7274 6169 6e20 nks for certain │ │ │ │ +0006e0b0: 6578 616d 706c 6573 0a2a 2a2a 2a2a 2a2a examples.******* │ │ │ │ +0006e0c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006e0d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006e0e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006e0f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006e100: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -0006e110: 7361 6765 3a20 0a20 2020 2020 2020 2073 sage: . s │ │ │ │ -0006e120: 756d 5477 6f4d 6f6e 6f6d 6961 6c73 2863 umTwoMonomials(c │ │ │ │ -0006e130: 2c64 290a 2020 2a20 496e 7075 7473 3a0a ,d). * Inputs:. │ │ │ │ -0006e140: 2020 2020 2020 2a20 632c 2061 6e20 2a6e * c, an *n │ │ │ │ -0006e150: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ -0006e160: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ -0006e170: 636f 6469 6d65 6e73 696f 6e20 696e 2077 codimension in w │ │ │ │ -0006e180: 6869 6368 2074 6f20 776f 726b 0a20 2020 hich to work. │ │ │ │ -0006e190: 2020 202a 2064 2c20 616e 202a 6e6f 7465 * d, an *note │ │ │ │ -0006e1a0: 2069 6e74 6567 6572 3a20 284d 6163 6175 integer: (Macau │ │ │ │ -0006e1b0: 6c61 7932 446f 6329 5a5a 2c2c 2064 6567 lay2Doc)ZZ,, deg │ │ │ │ -0006e1c0: 7265 6520 6f66 2074 6865 206d 6f6e 6f6d ree of the monom │ │ │ │ -0006e1d0: 6961 6c73 2074 6f20 7461 6b65 0a20 202a ials to take. * │ │ │ │ -0006e1e0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -0006e1f0: 2a20 542c 2061 202a 6e6f 7465 2074 616c * T, a *note tal │ │ │ │ -0006e200: 6c79 3a20 284d 6163 6175 6c61 7932 446f ly: (Macaulay2Do │ │ │ │ -0006e210: 6329 5461 6c6c 792c 2c20 0a0a 4465 7363 c)Tally,, ..Desc │ │ │ │ -0006e220: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -0006e230: 3d3d 3d0a 0a74 616c 6c69 6573 2074 6865 ===..tallies the │ │ │ │ -0006e240: 2073 6571 7565 6e63 6573 206f 6620 422d sequences of B- │ │ │ │ -0006e250: 7261 6e6b 7320 7468 6174 206f 6363 7572 ranks that occur │ │ │ │ -0006e260: 2066 6f72 2073 756d 7320 6f66 2070 6169 for sums of pai │ │ │ │ -0006e270: 7273 206f 6620 6d6f 6e6f 6d69 616c 7320 rs of monomials │ │ │ │ -0006e280: 696e 2052 0a3d 2053 2f28 642d 7468 2070 in R.= S/(d-th p │ │ │ │ -0006e290: 6f77 6572 7320 6f66 2074 6865 2076 6172 owers of the var │ │ │ │ -0006e2a0: 6961 626c 6573 292c 2077 6974 6820 6675 iables), with fu │ │ │ │ -0006e2b0: 6c6c 2063 6f6d 706c 6578 6974 7920 283d ll complexity (= │ │ │ │ -0006e2c0: 6329 3b20 7468 6174 2069 732c 2066 6f72 c); that is, for │ │ │ │ -0006e2d0: 2061 6e0a 6170 7072 6f70 7269 6174 6520 an.appropriate │ │ │ │ -0006e2e0: 7379 7a79 6779 204d 206f 6620 4d30 203d syzygy M of M0 = │ │ │ │ -0006e2f0: 2052 2f28 6d31 2b6d 3229 2077 6865 7265 R/(m1+m2) where │ │ │ │ -0006e300: 206d 3120 616e 6420 6d32 2061 7265 206d m1 and m2 are m │ │ │ │ -0006e310: 6f6e 6f6d 6961 6c73 206f 6620 7468 650a onomials of the. │ │ │ │ -0006e320: 7361 6d65 2064 6567 7265 652e 0a0a 2b2d same degree...+- │ │ │ │ +0006e0f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +0006e100: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +0006e110: 2020 2020 7375 6d54 776f 4d6f 6e6f 6d69 sumTwoMonomi │ │ │ │ +0006e120: 616c 7328 632c 6429 0a20 202a 2049 6e70 als(c,d). * Inp │ │ │ │ +0006e130: 7574 733a 0a20 2020 2020 202a 2063 2c20 uts:. * c, │ │ │ │ +0006e140: 616e 202a 6e6f 7465 2069 6e74 6567 6572 an *note integer │ │ │ │ +0006e150: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0006e160: 5a5a 2c2c 2063 6f64 696d 656e 7369 6f6e ZZ,, codimension │ │ │ │ +0006e170: 2069 6e20 7768 6963 6820 746f 2077 6f72 in which to wor │ │ │ │ +0006e180: 6b0a 2020 2020 2020 2a20 642c 2061 6e20 k. * d, an │ │ │ │ +0006e190: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ +0006e1a0: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ +0006e1b0: 2c20 6465 6772 6565 206f 6620 7468 6520 , degree of the │ │ │ │ +0006e1c0: 6d6f 6e6f 6d69 616c 7320 746f 2074 616b monomials to tak │ │ │ │ +0006e1d0: 650a 2020 2a20 4f75 7470 7574 733a 0a20 e. * Outputs:. │ │ │ │ +0006e1e0: 2020 2020 202a 2054 2c20 6120 2a6e 6f74 * T, a *not │ │ │ │ +0006e1f0: 6520 7461 6c6c 793a 2028 4d61 6361 756c e tally: (Macaul │ │ │ │ +0006e200: 6179 3244 6f63 2954 616c 6c79 2c2c 200a ay2Doc)Tally,, . │ │ │ │ +0006e210: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ +0006e220: 3d3d 3d3d 3d3d 3d3d 0a0a 7461 6c6c 6965 ========..tallie │ │ │ │ +0006e230: 7320 7468 6520 7365 7175 656e 6365 7320 s the sequences │ │ │ │ +0006e240: 6f66 2042 2d72 616e 6b73 2074 6861 7420 of B-ranks that │ │ │ │ +0006e250: 6f63 6375 7220 666f 7220 7375 6d73 206f occur for sums o │ │ │ │ +0006e260: 6620 7061 6972 7320 6f66 206d 6f6e 6f6d f pairs of monom │ │ │ │ +0006e270: 6961 6c73 2069 6e20 520a 3d20 532f 2864 ials in R.= S/(d │ │ │ │ +0006e280: 2d74 6820 706f 7765 7273 206f 6620 7468 -th powers of th │ │ │ │ +0006e290: 6520 7661 7269 6162 6c65 7329 2c20 7769 e variables), wi │ │ │ │ +0006e2a0: 7468 2066 756c 6c20 636f 6d70 6c65 7869 th full complexi │ │ │ │ +0006e2b0: 7479 2028 3d63 293b 2074 6861 7420 6973 ty (=c); that is │ │ │ │ +0006e2c0: 2c20 666f 7220 616e 0a61 7070 726f 7072 , for an.appropr │ │ │ │ +0006e2d0: 6961 7465 2073 797a 7967 7920 4d20 6f66 iate syzygy M of │ │ │ │ +0006e2e0: 204d 3020 3d20 522f 286d 312b 6d32 2920 M0 = R/(m1+m2) │ │ │ │ +0006e2f0: 7768 6572 6520 6d31 2061 6e64 206d 3220 where m1 and m2 │ │ │ │ +0006e300: 6172 6520 6d6f 6e6f 6d69 616c 7320 6f66 are monomials of │ │ │ │ +0006e310: 2074 6865 0a73 616d 6520 6465 6772 6565 the.same degree │ │ │ │ +0006e320: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ 0006e330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e360: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7365 ------+.|i1 : se │ │ │ │ -0006e370: 7452 616e 646f 6d53 6565 6420 3020 2020 tRandomSeed 0 │ │ │ │ +0006e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0006e360: 203a 2073 6574 5261 6e64 6f6d 5365 6564 : setRandomSeed │ │ │ │ +0006e370: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ 0006e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e3a0: 7c0a 7c20 2d2d 2073 6574 7469 6e67 2072 |.| -- setting r │ │ │ │ -0006e3b0: 616e 646f 6d20 7365 6564 2074 6f20 3020 andom seed to 0 │ │ │ │ -0006e3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e3d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006e390: 2020 2020 207c 0a7c 202d 2d20 7365 7474 |.| -- sett │ │ │ │ +0006e3a0: 696e 6720 7261 6e64 6f6d 2073 6565 6420 ing random seed │ │ │ │ +0006e3b0: 746f 2030 2020 2020 2020 2020 2020 2020 to 0 │ │ │ │ +0006e3c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006e3d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0006e3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e410: 2020 2020 7c0a 7c6f 3120 3d20 3020 2020 |.|o1 = 0 │ │ │ │ +0006e400: 2020 2020 2020 2020 207c 0a7c 6f31 203d |.|o1 = │ │ │ │ +0006e410: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ 0006e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e440: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006e450: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0006e440: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0006e450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e480: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -0006e490: 7375 6d54 776f 4d6f 6e6f 6d69 616c 7328 sumTwoMonomials( │ │ │ │ -0006e4a0: 322c 3329 2020 2020 2020 2020 2020 2020 2,3) │ │ │ │ -0006e4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e4c0: 2020 7c0a 7c20 2d2d 2075 7365 6420 312e |.| -- used 1. │ │ │ │ -0006e4d0: 3230 3331 3573 2028 6370 7529 3b20 302e 20315s (cpu); 0. │ │ │ │ -0006e4e0: 3536 3232 3136 7320 2874 6872 6561 6429 562216s (thread) │ │ │ │ -0006e4f0: 3b20 3073 2028 6763 2920 2020 7c0a 7c32 ; 0s (gc) |.|2 │ │ │ │ +0006e470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0006e480: 6932 203a 2073 756d 5477 6f4d 6f6e 6f6d i2 : sumTwoMonom │ │ │ │ +0006e490: 6961 6c73 2832 2c33 2920 2020 2020 2020 ials(2,3) │ │ │ │ +0006e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006e4b0: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ +0006e4c0: 6564 2031 2e32 3734 3734 7320 2863 7075 ed 1.27474s (cpu │ │ │ │ +0006e4d0: 293b 2030 2e35 3539 3832 3273 2028 7468 ); 0.559822s (th │ │ │ │ +0006e4e0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ +0006e4f0: 207c 0a7c 3220 2020 2020 2020 2020 2020 |.|2 │ │ │ │ 0006e500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e530: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7b |.|Tally{{ │ │ │ │ -0006e540: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20 {2, 2}, {1, 2}} │ │ │ │ -0006e550: 3d3e 2033 7d20 2020 2020 2020 2020 2020 => 3} │ │ │ │ -0006e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e570: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006e520: 2020 2020 2020 2020 2020 207c 0a7c 5461 |.|Ta │ │ │ │ +0006e530: 6c6c 797b 7b7b 322c 2032 7d2c 207b 312c lly{{{2, 2}, {1, │ │ │ │ +0006e540: 2032 7d7d 203d 3e20 337d 2020 2020 2020 2}} => 3} │ │ │ │ +0006e550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006e560: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006e570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e5a0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -0006e5b0: 2075 7365 6420 302e 3430 3938 3973 2028 used 0.40989s ( │ │ │ │ -0006e5c0: 6370 7529 3b20 302e 3137 3631 3435 7320 cpu); 0.176145s │ │ │ │ -0006e5d0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -0006e5e0: 2920 2020 7c0a 7c33 2020 2020 2020 2020 ) |.|3 │ │ │ │ +0006e590: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006e5a0: 0a7c 202d 2d20 7573 6564 2030 2e31 3938 .| -- used 0.198 │ │ │ │ +0006e5b0: 3438 3773 2028 6370 7529 3b20 302e 3133 487s (cpu); 0.13 │ │ │ │ +0006e5c0: 3930 3634 7320 2874 6872 6561 6429 3b20 9064s (thread); │ │ │ │ +0006e5d0: 3073 2028 6763 2920 207c 0a7c 3320 2020 0s (gc) |.|3 │ │ │ │ +0006e5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e610: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006e620: 7c54 616c 6c79 7b7b 7b32 2c20 327d 2c20 |Tally{{{2, 2}, │ │ │ │ -0006e630: 7b31 2c20 327d 7d20 3d3e 2031 7d20 2020 {1, 2}} => 1} │ │ │ │ -0006e640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e650: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006e610: 2020 207c 0a7c 5461 6c6c 797b 7b7b 322c |.|Tally{{{2, │ │ │ │ +0006e620: 2032 7d2c 207b 312c 2032 7d7d 203d 3e20 2}, {1, 2}} => │ │ │ │ +0006e630: 317d 2020 2020 2020 2020 2020 2020 2020 1} │ │ │ │ +0006e640: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e690: 2020 7c0a 7c20 2d2d 2075 7365 6420 342e |.| -- used 4. │ │ │ │ -0006e6a0: 3036 3865 2d30 3673 2028 6370 7529 3b20 068e-06s (cpu); │ │ │ │ -0006e6b0: 332e 3430 3665 2d30 3673 2028 7468 7265 3.406e-06s (thre │ │ │ │ -0006e6c0: 6164 293b 2030 7320 2867 6329 7c0a 7c34 ad); 0s (gc)|.|4 │ │ │ │ +0006e680: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ +0006e690: 6564 2033 2e38 3334 652d 3036 7320 2863 ed 3.834e-06s (c │ │ │ │ +0006e6a0: 7075 293b 2033 2e32 3536 652d 3036 7320 pu); 3.256e-06s │ │ │ │ +0006e6b0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +0006e6c0: 297c 0a7c 3420 2020 2020 2020 2020 2020 )|.|4 │ │ │ │ 0006e6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e700: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7d |.|Tally{} │ │ │ │ +0006e6f0: 2020 2020 2020 2020 2020 207c 0a7c 5461 |.|Ta │ │ │ │ +0006e700: 6c6c 797b 7d20 2020 2020 2020 2020 2020 lly{} │ │ │ │ 0006e710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e740: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0006e730: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0006e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 ----------+..See │ │ │ │ -0006e780: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -0006e790: 2020 2a20 2a6e 6f74 6520 7477 6f4d 6f6e * *note twoMon │ │ │ │ -0006e7a0: 6f6d 6961 6c73 3a20 7477 6f4d 6f6e 6f6d omials: twoMonom │ │ │ │ -0006e7b0: 6961 6c73 2c20 2d2d 2074 616c 6c79 2074 ials, -- tally t │ │ │ │ -0006e7c0: 6865 2073 6571 7565 6e63 6573 206f 6620 he sequences of │ │ │ │ -0006e7d0: 4252 616e 6b73 2066 6f72 0a20 2020 2063 BRanks for. c │ │ │ │ -0006e7e0: 6572 7461 696e 2065 7861 6d70 6c65 730a ertain examples. │ │ │ │ -0006e7f0: 0a57 6179 7320 746f 2075 7365 2073 756d .Ways to use sum │ │ │ │ -0006e800: 5477 6f4d 6f6e 6f6d 6961 6c73 3a0a 3d3d TwoMonomials:.== │ │ │ │ -0006e810: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0006e820: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -0006e830: 2273 756d 5477 6f4d 6f6e 6f6d 6961 6c73 "sumTwoMonomials │ │ │ │ -0006e840: 285a 5a2c 5a5a 2922 0a0a 466f 7220 7468 (ZZ,ZZ)"..For th │ │ │ │ -0006e850: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -0006e860: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -0006e870: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -0006e880: 6520 7375 6d54 776f 4d6f 6e6f 6d69 616c e sumTwoMonomial │ │ │ │ -0006e890: 733a 2073 756d 5477 6f4d 6f6e 6f6d 6961 s: sumTwoMonomia │ │ │ │ -0006e8a0: 6c73 2c20 6973 2061 202a 6e6f 7465 206d ls, is a *note m │ │ │ │ -0006e8b0: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a ethod function:. │ │ │ │ -0006e8c0: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -0006e8d0: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +0006e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0006e770: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ +0006e780: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2074 ===.. * *note t │ │ │ │ +0006e790: 776f 4d6f 6e6f 6d69 616c 733a 2074 776f woMonomials: two │ │ │ │ +0006e7a0: 4d6f 6e6f 6d69 616c 732c 202d 2d20 7461 Monomials, -- ta │ │ │ │ +0006e7b0: 6c6c 7920 7468 6520 7365 7175 656e 6365 lly the sequence │ │ │ │ +0006e7c0: 7320 6f66 2042 5261 6e6b 7320 666f 720a s of BRanks for. │ │ │ │ +0006e7d0: 2020 2020 6365 7274 6169 6e20 6578 616d certain exam │ │ │ │ +0006e7e0: 706c 6573 0a0a 5761 7973 2074 6f20 7573 ples..Ways to us │ │ │ │ +0006e7f0: 6520 7375 6d54 776f 4d6f 6e6f 6d69 616c e sumTwoMonomial │ │ │ │ +0006e800: 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d s:.============= │ │ │ │ +0006e810: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0006e820: 0a20 202a 2022 7375 6d54 776f 4d6f 6e6f . * "sumTwoMono │ │ │ │ +0006e830: 6d69 616c 7328 5a5a 2c5a 5a29 220a 0a46 mials(ZZ,ZZ)"..F │ │ │ │ +0006e840: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +0006e850: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +0006e860: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +0006e870: 202a 6e6f 7465 2073 756d 5477 6f4d 6f6e *note sumTwoMon │ │ │ │ +0006e880: 6f6d 6961 6c73 3a20 7375 6d54 776f 4d6f omials: sumTwoMo │ │ │ │ +0006e890: 6e6f 6d69 616c 732c 2069 7320 6120 2a6e nomials, is a *n │ │ │ │ +0006e8a0: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ +0006e8b0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ +0006e8c0: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ +0006e8d0: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ 0006e8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -0006e930: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -0006e940: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -0006e950: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -0006e960: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -0006e970: 756c 6179 322d 312e 3236 2e30 352b 6473 ulay2-1.26.05+ds │ │ │ │ -0006e980: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -0006e990: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ -0006e9a0: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ -0006e9b0: 6c75 7469 6f6e 732e 6d32 3a34 3531 323a lutions.m2:4512: │ │ │ │ -0006e9c0: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ -0006e9d0: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -0006e9e0: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ -0006e9f0: 204e 6f64 653a 2054 6174 6552 6573 6f6c Node: TateResol │ │ │ │ -0006ea00: 7574 696f 6e2c 204e 6578 743a 2074 656e ution, Next: ten │ │ │ │ -0006ea10: 736f 7257 6974 6843 6f6d 706f 6e65 6e74 sorWithComponent │ │ │ │ -0006ea20: 732c 2050 7265 763a 2073 756d 5477 6f4d s, Prev: sumTwoM │ │ │ │ -0006ea30: 6f6e 6f6d 6961 6c73 2c20 5570 3a20 546f onomials, Up: To │ │ │ │ -0006ea40: 700a 0a54 6174 6552 6573 6f6c 7574 696f p..TateResolutio │ │ │ │ -0006ea50: 6e20 2d2d 2054 6174 6552 6573 6f6c 7574 n -- TateResolut │ │ │ │ -0006ea60: 696f 6e20 6f66 2061 206d 6f64 756c 6520 ion of a module │ │ │ │ -0006ea70: 6f76 6572 2061 6e20 6578 7465 7269 6f72 over an exterior │ │ │ │ -0006ea80: 2061 6c67 6562 7261 0a2a 2a2a 2a2a 2a2a algebra.******* │ │ │ │ +0006e920: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +0006e930: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +0006e940: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +0006e950: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +0006e960: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ +0006e970: 3035 2b64 732f 4d32 2f4d 6163 6175 6c61 05+ds/M2/Macaula │ │ │ │ +0006e980: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d y2/packages/.Com │ │ │ │ +0006e990: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ +0006e9a0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a nResolutions.m2: │ │ │ │ +0006e9b0: 3435 3132 3a30 2e0a 1f0a 4669 6c65 3a20 4512:0....File: │ │ │ │ +0006e9c0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +0006e9d0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +0006e9e0: 696e 666f 2c20 4e6f 6465 3a20 5461 7465 info, Node: Tate │ │ │ │ +0006e9f0: 5265 736f 6c75 7469 6f6e 2c20 4e65 7874 Resolution, Next │ │ │ │ +0006ea00: 3a20 7465 6e73 6f72 5769 7468 436f 6d70 : tensorWithComp │ │ │ │ +0006ea10: 6f6e 656e 7473 2c20 5072 6576 3a20 7375 onents, Prev: su │ │ │ │ +0006ea20: 6d54 776f 4d6f 6e6f 6d69 616c 732c 2055 mTwoMonomials, U │ │ │ │ +0006ea30: 703a 2054 6f70 0a0a 5461 7465 5265 736f p: Top..TateReso │ │ │ │ +0006ea40: 6c75 7469 6f6e 202d 2d20 5461 7465 5265 lution -- TateRe │ │ │ │ +0006ea50: 736f 6c75 7469 6f6e 206f 6620 6120 6d6f solution of a mo │ │ │ │ +0006ea60: 6475 6c65 206f 7665 7220 616e 2065 7874 dule over an ext │ │ │ │ +0006ea70: 6572 696f 7220 616c 6765 6272 610a 2a2a erior algebra.** │ │ │ │ +0006ea80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006ea90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006eaa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006eab0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006eac0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -0006ead0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -0006eae0: 2020 2020 4620 3d20 5461 7465 5265 736f F = TateReso │ │ │ │ -0006eaf0: 6c75 7469 6f6e 284d 2c6c 6f77 6572 2c75 lution(M,lower,u │ │ │ │ -0006eb00: 7070 6572 290a 2020 2a20 496e 7075 7473 pper). * Inputs │ │ │ │ -0006eb10: 3a0a 2020 2020 2020 2a20 4d2c 2061 202a :. * M, a * │ │ │ │ -0006eb20: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -0006eb30: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -0006eb40: 652c 2c20 0a20 2020 2020 202a 206c 6f77 e,, . * low │ │ │ │ -0006eb50: 6572 2c20 616e 202a 6e6f 7465 2069 6e74 er, an *note int │ │ │ │ -0006eb60: 6567 6572 3a20 284d 6163 6175 6c61 7932 eger: (Macaulay2 │ │ │ │ -0006eb70: 446f 6329 5a5a 2c2c 200a 2020 2020 2020 Doc)ZZ,, . │ │ │ │ -0006eb80: 2a20 7570 7065 722c 2061 6e20 2a6e 6f74 * upper, an *not │ │ │ │ -0006eb90: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ -0006eba0: 756c 6179 3244 6f63 295a 5a2c 2c20 6c6f ulay2Doc)ZZ,, lo │ │ │ │ -0006ebb0: 7765 7220 616e 6420 7570 7065 7220 626f wer and upper bo │ │ │ │ -0006ebc0: 756e 6473 2066 6f72 0a20 2020 2020 2020 unds for. │ │ │ │ -0006ebd0: 2074 6865 2072 6573 6f6c 7574 696f 6e0a the resolution. │ │ │ │ -0006ebe0: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -0006ebf0: 2020 202a 2046 2c20 6120 2a6e 6f74 6520 * F, a *note │ │ │ │ -0006ec00: 636f 6d70 6c65 783a 2028 436f 6d70 6c65 complex: (Comple │ │ │ │ -0006ec10: 7865 7329 436f 6d70 6c65 782c 2c20 0a0a xes)Complex,, .. │ │ │ │ -0006ec20: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -0006ec30: 3d3d 3d3d 3d3d 3d0a 0a46 6f72 6d73 2061 =======..Forms a │ │ │ │ -0006ec40: 6e20 696e 7465 7276 616c 2c20 6c6f 7765 n interval, lowe │ │ │ │ -0006ec50: 722e 2e75 7070 6572 2c20 6f66 2061 2064 r..upper, of a d │ │ │ │ -0006ec60: 6f75 626c 7920 696e 6669 6e69 7465 2066 oubly infinite f │ │ │ │ -0006ec70: 7265 6520 7265 736f 6c75 7469 6f6e 206f ree resolution o │ │ │ │ -0006ec80: 6620 6120 610a 436f 6865 6e2d 4d61 6361 f a a.Cohen-Maca │ │ │ │ -0006ec90: 756c 6179 206d 6f64 756c 6520 6f76 6572 ulay module over │ │ │ │ -0006eca0: 2061 2047 6f72 656e 7374 6569 6e20 7269 a Gorenstein ri │ │ │ │ -0006ecb0: 6e67 2c20 7375 6368 2061 7320 616e 7920 ng, such as any │ │ │ │ -0006ecc0: 6d6f 6475 6c65 206f 7665 7220 616e 0a65 module over an.e │ │ │ │ -0006ecd0: 7874 6572 696f 7220 616c 6765 6272 6120 xterior algebra │ │ │ │ -0006ece0: 2861 6374 7561 6c6c 792c 2061 6e79 206d (actually, any m │ │ │ │ -0006ecf0: 6f64 756c 6520 6f76 6572 2061 6e79 2072 odule over any r │ │ │ │ -0006ed00: 696e 672e 290a 0a2b 2d2d 2d2d 2d2d 2d2d ing.)..+-------- │ │ │ │ +0006eac0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ +0006ead0: 0a20 2020 2020 2020 2046 203d 2054 6174 . F = Tat │ │ │ │ +0006eae0: 6552 6573 6f6c 7574 696f 6e28 4d2c 6c6f eResolution(M,lo │ │ │ │ +0006eaf0: 7765 722c 7570 7065 7229 0a20 202a 2049 wer,upper). * I │ │ │ │ +0006eb00: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ +0006eb10: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +0006eb20: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0006eb30: 4d6f 6475 6c65 2c2c 200a 2020 2020 2020 Module,, . │ │ │ │ +0006eb40: 2a20 6c6f 7765 722c 2061 6e20 2a6e 6f74 * lower, an *not │ │ │ │ +0006eb50: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ +0006eb60: 756c 6179 3244 6f63 295a 5a2c 2c20 0a20 ulay2Doc)ZZ,, . │ │ │ │ +0006eb70: 2020 2020 202a 2075 7070 6572 2c20 616e * upper, an │ │ │ │ +0006eb80: 202a 6e6f 7465 2069 6e74 6567 6572 3a20 *note integer: │ │ │ │ +0006eb90: 284d 6163 6175 6c61 7932 446f 6329 5a5a (Macaulay2Doc)ZZ │ │ │ │ +0006eba0: 2c2c 206c 6f77 6572 2061 6e64 2075 7070 ,, lower and upp │ │ │ │ +0006ebb0: 6572 2062 6f75 6e64 7320 666f 720a 2020 er bounds for. │ │ │ │ +0006ebc0: 2020 2020 2020 7468 6520 7265 736f 6c75 the resolu │ │ │ │ +0006ebd0: 7469 6f6e 0a20 202a 204f 7574 7075 7473 tion. * Outputs │ │ │ │ +0006ebe0: 3a0a 2020 2020 2020 2a20 462c 2061 202a :. * F, a * │ │ │ │ +0006ebf0: 6e6f 7465 2063 6f6d 706c 6578 3a20 2843 note complex: (C │ │ │ │ +0006ec00: 6f6d 706c 6578 6573 2943 6f6d 706c 6578 omplexes)Complex │ │ │ │ +0006ec10: 2c2c 200a 0a44 6573 6372 6970 7469 6f6e ,, ..Description │ │ │ │ +0006ec20: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 466f .===========..Fo │ │ │ │ +0006ec30: 726d 7320 616e 2069 6e74 6572 7661 6c2c rms an interval, │ │ │ │ +0006ec40: 206c 6f77 6572 2e2e 7570 7065 722c 206f lower..upper, o │ │ │ │ +0006ec50: 6620 6120 646f 7562 6c79 2069 6e66 696e f a doubly infin │ │ │ │ +0006ec60: 6974 6520 6672 6565 2072 6573 6f6c 7574 ite free resolut │ │ │ │ +0006ec70: 696f 6e20 6f66 2061 2061 0a43 6f68 656e ion of a a.Cohen │ │ │ │ +0006ec80: 2d4d 6163 6175 6c61 7920 6d6f 6475 6c65 -Macaulay module │ │ │ │ +0006ec90: 206f 7665 7220 6120 476f 7265 6e73 7465 over a Gorenste │ │ │ │ +0006eca0: 696e 2072 696e 672c 2073 7563 6820 6173 in ring, such as │ │ │ │ +0006ecb0: 2061 6e79 206d 6f64 756c 6520 6f76 6572 any module over │ │ │ │ +0006ecc0: 2061 6e0a 6578 7465 7269 6f72 2061 6c67 an.exterior alg │ │ │ │ +0006ecd0: 6562 7261 2028 6163 7475 616c 6c79 2c20 ebra (actually, │ │ │ │ +0006ece0: 616e 7920 6d6f 6475 6c65 206f 7665 7220 any module over │ │ │ │ +0006ecf0: 616e 7920 7269 6e67 2e29 0a0a 2b2d 2d2d any ring.)..+--- │ │ │ │ +0006ed00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ed10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ed20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ed30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ed40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ed50: 2d2d 2d2d 2d2b 0a7c 6931 203a 2045 203d -----+.|i1 : E = │ │ │ │ -0006ed60: 205a 5a2f 3130 315b 612c 622c 632c 2053 ZZ/101[a,b,c, S │ │ │ │ -0006ed70: 6b65 7743 6f6d 6d75 7461 7469 7665 3d3e kewCommutative=> │ │ │ │ -0006ed80: 7472 7565 5d20 2020 2020 2020 2020 2020 true] │ │ │ │ -0006ed90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006eda0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006ed40: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ +0006ed50: 3a20 4520 3d20 5a5a 2f31 3031 5b61 2c62 : E = ZZ/101[a,b │ │ │ │ +0006ed60: 2c63 2c20 536b 6577 436f 6d6d 7574 6174 ,c, SkewCommutat │ │ │ │ +0006ed70: 6976 653d 3e74 7275 655d 2020 2020 2020 ive=>true] │ │ │ │ +0006ed80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ed90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006eda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006edb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006edc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006edd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ede0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006edf0: 2020 2020 207c 0a7c 6f31 203d 2045 2020 |.|o1 = E │ │ │ │ +0006ede0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +0006edf0: 3d20 4520 2020 2020 2020 2020 2020 2020 = E │ │ │ │ 0006ee00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ee10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ee20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee40: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006ee30: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006ee40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ee50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ee60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ee70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee90: 2020 2020 207c 0a7c 6f31 203a 2050 6f6c |.|o1 : Pol │ │ │ │ -0006eea0: 796e 6f6d 6961 6c52 696e 672c 2033 2073 ynomialRing, 3 s │ │ │ │ -0006eeb0: 6b65 7720 636f 6d6d 7574 6174 6976 6520 kew commutative │ │ │ │ -0006eec0: 7661 7269 6162 6c65 2873 2920 2020 2020 variable(s) │ │ │ │ -0006eed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006eee0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0006ee80: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +0006ee90: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +0006eea0: 2c20 3320 736b 6577 2063 6f6d 6d75 7461 , 3 skew commuta │ │ │ │ +0006eeb0: 7469 7665 2076 6172 6961 626c 6528 7329 tive variable(s) │ │ │ │ +0006eec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006eed0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0006eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ef00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ef10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ef30: 2d2d 2d2d 2d2b 0a7c 6932 203a 204d 203d -----+.|i2 : M = │ │ │ │ -0006ef40: 2063 6f6b 6572 206d 6170 2845 5e32 2c20 coker map(E^2, │ │ │ │ -0006ef50: 455e 7b2d 317d 2c20 6d61 7472 6978 2261 E^{-1}, matrix"a │ │ │ │ -0006ef60: 623b 6263 2229 2020 2020 2020 2020 2020 b;bc") │ │ │ │ -0006ef70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ef80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ +0006ef30: 3a20 4d20 3d20 636f 6b65 7220 6d61 7028 : M = coker map( │ │ │ │ +0006ef40: 455e 322c 2045 5e7b 2d31 7d2c 206d 6174 E^2, E^{-1}, mat │ │ │ │ +0006ef50: 7269 7822 6162 3b62 6322 2920 2020 2020 rix"ab;bc") │ │ │ │ +0006ef60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ef70: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006ef80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006efa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006efb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006efc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006efd0: 2020 2020 207c 0a7c 6f32 203d 2063 6f6b |.|o2 = cok │ │ │ │ -0006efe0: 6572 6e65 6c20 7c20 6162 207c 2020 2020 ernel | ab | │ │ │ │ +0006efc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0006efd0: 3d20 636f 6b65 726e 656c 207c 2061 6220 = cokernel | ab │ │ │ │ +0006efe0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006eff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f020: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0006f030: 2020 2020 2020 7c20 6263 207c 2020 2020 | bc | │ │ │ │ +0006f010: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006f020: 2020 2020 2020 2020 2020 207c 2062 6320 | bc │ │ │ │ +0006f030: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006f040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f070: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006f060: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006f070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f0c0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0006f0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f0e0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +0006f0b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f0d0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +0006f0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f110: 2020 2020 207c 0a7c 6f32 203a 2045 2d6d |.|o2 : E-m │ │ │ │ -0006f120: 6f64 756c 652c 2071 756f 7469 656e 7420 odule, quotient │ │ │ │ -0006f130: 6f66 2045 2020 2020 2020 2020 2020 2020 of E │ │ │ │ +0006f100: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0006f110: 3a20 452d 6d6f 6475 6c65 2c20 7175 6f74 : E-module, quot │ │ │ │ +0006f120: 6965 6e74 206f 6620 4520 2020 2020 2020 ient of E │ │ │ │ +0006f130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f160: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0006f150: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0006f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006f1b0: 2d2d 2d2d 2d2b 0a7c 6933 203a 2070 7265 -----+.|i3 : pre │ │ │ │ -0006f1c0: 7365 6e74 6174 696f 6e20 4d20 2020 2020 sentation M │ │ │ │ +0006f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ +0006f1b0: 3a20 7072 6573 656e 7461 7469 6f6e 204d : presentation M │ │ │ │ +0006f1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f200: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006f1f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006f200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f250: 2020 2020 207c 0a7c 6f33 203d 207c 2061 |.|o3 = | a │ │ │ │ -0006f260: 6220 7c20 2020 2020 2020 2020 2020 2020 b | │ │ │ │ +0006f240: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +0006f250: 3d20 7c20 6162 207c 2020 2020 2020 2020 = | ab | │ │ │ │ +0006f260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f2a0: 2020 2020 207c 0a7c 2020 2020 207c 2062 |.| | b │ │ │ │ -0006f2b0: 6320 7c20 2020 2020 2020 2020 2020 2020 c | │ │ │ │ +0006f290: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006f2a0: 2020 7c20 6263 207c 2020 2020 2020 2020 | bc | │ │ │ │ +0006f2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f2f0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006f2e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006f2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f340: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0006f350: 2020 2020 2032 2020 2020 2020 3120 2020 2 1 │ │ │ │ +0006f330: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006f340: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0006f350: 2031 2020 2020 2020 2020 2020 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=======.. * "Ta │ │ │ │ +0006f900: 7465 5265 736f 6c75 7469 6f6e 284d 6f64 teResolution(Mod │ │ │ │ +0006f910: 756c 6529 220a 2020 2a20 2254 6174 6552 ule)". * "TateR │ │ │ │ +0006f920: 6573 6f6c 7574 696f 6e28 4d6f 6475 6c65 esolution(Module │ │ │ │ +0006f930: 2c5a 5a29 220a 2020 2a20 2254 6174 6552 ,ZZ)". * "TateR │ │ │ │ +0006f940: 6573 6f6c 7574 696f 6e28 4d6f 6475 6c65 esolution(Module │ │ │ │ +0006f950: 2c5a 5a2c 5a5a 2922 0a0a 466f 7220 7468 ,ZZ,ZZ)"..For th │ │ │ │ +0006f960: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +0006f970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0006f980: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +0006f990: 6520 5461 7465 5265 736f 6c75 7469 6f6e e TateResolution │ │ │ │ +0006f9a0: 3a20 5461 7465 5265 736f 6c75 7469 6f6e : TateResolution │ │ │ │ +0006f9b0: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ +0006f9c0: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod 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│ │ │ -0006faa0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -0006fab0: 0a43 6f6d 706c 6574 6549 6e74 6572 7365 .CompleteInterse │ │ │ │ -0006fac0: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ -0006fad0: 2e6d 323a 3334 3939 3a30 2e0a 1f0a 4669 .m2:3499:0....Fi │ │ │ │ -0006fae0: 6c65 3a20 436f 6d70 6c65 7465 496e 7465 le: CompleteInte │ │ │ │ -0006faf0: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -0006fb00: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20 ons.info, Node: │ │ │ │ -0006fb10: 7465 6e73 6f72 5769 7468 436f 6d70 6f6e tensorWithCompon │ │ │ │ -0006fb20: 656e 7473 2c20 4e65 7874 3a20 746f 4172 ents, Next: toAr │ │ │ │ -0006fb30: 7261 792c 2050 7265 763a 2054 6174 6552 ray, Prev: TateR │ │ │ │ -0006fb40: 6573 6f6c 7574 696f 6e2c 2055 703a 2054 esolution, Up: T │ │ │ │ -0006fb50: 6f70 0a0a 7465 6e73 6f72 5769 7468 436f op..tensorWithCo │ │ │ │ -0006fb60: 6d70 6f6e 656e 7473 202d 2d20 666f 726d mponents -- form │ │ │ │ -0006fb70: 7320 7468 6520 7465 6e73 6f72 2070 726f s the tensor pro │ │ │ │ -0006fb80: 6475 6374 2c20 7072 6573 6572 7669 6e67 duct, preserving │ │ │ │ -0006fb90: 2064 6972 6563 7420 7375 6d20 696e 666f direct sum info │ │ │ │ -0006fba0: 726d 6174 696f 6e0a 2a2a 2a2a 2a2a 2a2a rmation.******** │ │ │ │ +0006fa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ +0006fa40: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ +0006fa50: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ +0006fa60: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ +0006fa70: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ +0006fa80: 6179 322d 312e 3236 2e30 352b 6473 2f4d ay2-1.26.05+ds/M │ │ │ │ +0006fa90: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ +0006faa0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ +0006fab0: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ +0006fac0: 7469 6f6e 732e 6d32 3a33 3439 393a 302e tions.m2:3499:0. │ │ │ │ +0006fad0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ +0006fae0: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +0006faf0: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ +0006fb00: 6f64 653a 2074 656e 736f 7257 6974 6843 ode: tensorWithC │ │ │ │ +0006fb10: 6f6d 706f 6e65 6e74 732c 204e 6578 743a omponents, Next: │ │ │ │ +0006fb20: 2074 6f41 7272 6179 2c20 5072 6576 3a20 toArray, Prev: │ │ │ │ +0006fb30: 5461 7465 5265 736f 6c75 7469 6f6e 2c20 TateResolution, │ │ │ │ +0006fb40: 5570 3a20 546f 700a 0a74 656e 736f 7257 Up: Top..tensorW │ │ │ │ +0006fb50: 6974 6843 6f6d 706f 6e65 6e74 7320 2d2d ithComponents -- │ │ │ │ +0006fb60: 2066 6f72 6d73 2074 6865 2074 656e 736f forms the tenso │ │ │ │ +0006fb70: 7220 7072 6f64 7563 742c 2070 7265 7365 r product, prese │ │ │ │ +0006fb80: 7276 696e 6720 6469 7265 6374 2073 756d rving direct sum │ │ │ │ +0006fb90: 2069 6e66 6f72 6d61 7469 6f6e 0a2a 2a2a information.*** │ │ │ │ +0006fba0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006fbb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006fbc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006fbd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006fbe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006fbf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -0006fc00: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -0006fc10: 2054 203d 2074 656e 736f 7228 4d2c 4e29 T = tensor(M,N) │ │ │ │ -0006fc20: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ -0006fc30: 2020 202a 204d 2c20 6120 2a6e 6f74 6520 * M, a *note │ │ │ │ -0006fc40: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61 module: (Macaula │ │ │ │ -0006fc50: 7932 446f 6329 4d6f 6475 6c65 2c2c 200a y2Doc)Module,, . │ │ │ │ -0006fc60: 2020 2020 2020 2a20 4e2c 2061 202a 6e6f * N, a *no │ │ │ │ -0006fc70: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ -0006fc80: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ -0006fc90: 2c20 0a20 202a 204f 7574 7075 7473 3a0a , . * Outputs:. │ │ │ │ -0006fca0: 2020 2020 2020 2a20 542c 2061 202a 6e6f * T, a *no │ │ │ │ -0006fcb0: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ -0006fcc0: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ -0006fcd0: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a , ..Description. │ │ │ │ -0006fce0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620 ===========..If │ │ │ │ -0006fcf0: 4d20 616e 642f 6f72 204e 2061 7265 2064 M and/or N are d │ │ │ │ -0006fd00: 6972 6563 7420 7375 6d20 6d6f 6475 6c65 irect sum module │ │ │ │ -0006fd10: 7320 2869 7344 6972 6563 7453 756d 204d s (isDirectSum M │ │ │ │ -0006fd20: 203d 3d20 7472 7565 2920 7468 656e 2054 == true) then T │ │ │ │ -0006fd30: 2069 7320 7468 650a 6469 7265 6374 2073 is the.direct s │ │ │ │ -0006fd40: 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This SHOULD be │ │ │ │ -0006fd80: 2062 7569 6c74 0a69 6e74 6f20 4d2a 2a4e built.into M**N │ │ │ │ -0006fd90: 2c20 6275 7420 6973 6e27 7420 6173 206f , but isn't as o │ │ │ │ -0006fda0: 6620 4d32 2c20 762e 2031 2e37 0a0a 5365 f M2, v. 1.7..Se │ │ │ │ -0006fdb0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ -0006fdc0: 0a20 202a 202a 6e6f 7465 2048 6f6d 5769 . * *note HomWi │ │ │ │ -0006fdd0: 7468 436f 6d70 6f6e 656e 7473 3a20 486f thComponents: Ho │ │ │ │ -0006fde0: 6d57 6974 6843 6f6d 706f 6e65 6e74 732c mWithComponents, │ │ │ │ -0006fdf0: 202d 2d20 636f 6d70 7574 6573 2048 6f6d -- computes Hom │ │ │ │ -0006fe00: 2c20 7072 6573 6572 7669 6e67 0a20 2020 , preserving. │ │ │ │ -0006fe10: 2064 6972 6563 7420 7375 6d20 696e 666f direct sum info │ │ │ │ -0006fe20: 726d 6174 696f 6e0a 2020 2a20 2a6e 6f74 rmation. * *not │ │ │ │ -0006fe30: 6520 6475 616c 5769 7468 436f 6d70 6f6e e dualWithCompon │ │ │ │ -0006fe40: 656e 7473 3a20 6475 616c 5769 7468 436f ents: dualWithCo │ │ │ │ -0006fe50: 6d70 6f6e 656e 7473 2c20 2d2d 2064 7561 mponents, -- dua │ │ │ │ -0006fe60: 6c20 6d6f 6475 6c65 2070 7265 7365 7276 l module preserv │ │ │ │ -0006fe70: 696e 670a 2020 2020 6469 7265 6374 2073 ing. direct s │ │ │ │ -0006fe80: 756d 2069 6e66 6f72 6d61 7469 6f6e 0a0a um information.. │ │ │ │ -0006fe90: 5761 7973 2074 6f20 7573 6520 7465 6e73 Ways to use tens │ │ │ │ -0006fea0: 6f72 5769 7468 436f 6d70 6f6e 656e 7473 orWithComponents │ │ │ │ -0006feb0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -0006fec0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0006fed0: 3d3d 3d0a 0a20 202a 2022 7465 6e73 6f72 ===.. * "tensor │ │ │ │ -0006fee0: 5769 7468 436f 6d70 6f6e 656e 7473 284d WithComponents(M │ │ │ │ -0006fef0: 6f64 756c 652c 4d6f 6475 6c65 2922 0a0a odule,Module)".. │ │ │ │ -0006ff00: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -0006ff10: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -0006ff20: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -0006ff30: 7420 2a6e 6f74 6520 7465 6e73 6f72 5769 t *note tensorWi │ │ │ │ -0006ff40: 7468 436f 6d70 6f6e 656e 7473 3a20 7465 thComponents: te │ │ │ │ -0006ff50: 6e73 6f72 5769 7468 436f 6d70 6f6e 656e nsorWithComponen │ │ │ │ -0006ff60: 7473 2c20 6973 2061 202a 6e6f 7465 206d ts, is a *note m │ │ │ │ -0006ff70: 6574 686f 640a 6675 6e63 7469 6f6e 3a20 ethod.function: │ │ │ │ -0006ff80: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -0006ff90: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +0006fbf0: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +0006fc00: 2020 2020 2020 5420 3d20 7465 6e73 6f72 T = tensor │ │ │ │ +0006fc10: 284d 2c4e 290a 2020 2a20 496e 7075 7473 (M,N). * Inputs │ │ │ │ +0006fc20: 3a0a 2020 2020 2020 2a20 4d2c 2061 202a :. * M, a * │ │ │ │ +0006fc30: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ +0006fc40: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ +0006fc50: 652c 2c20 0a20 2020 2020 202a 204e 2c20 e,, . * N, │ │ │ │ +0006fc60: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20 a *note module: │ │ │ │ +0006fc70: 284d 6163 6175 6c61 7932 446f 6329 4d6f (Macaulay2Doc)Mo │ │ │ │ +0006fc80: 6475 6c65 2c2c 200a 2020 2a20 4f75 7470 dule,, . * Outp │ │ │ │ +0006fc90: 7574 733a 0a20 2020 2020 202a 2054 2c20 uts:. * T, │ │ │ │ +0006fca0: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20 a *note module: │ │ │ │ +0006fcb0: 284d 6163 6175 6c61 7932 446f 6329 4d6f (Macaulay2Doc)Mo │ │ │ │ +0006fcc0: 6475 6c65 2c2c 200a 0a44 6573 6372 6970 dule,, ..Descrip │ │ │ │ +0006fcd0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +0006fce0: 0a0a 4966 204d 2061 6e64 2f6f 7220 4e20 ..If M and/or N │ │ │ │ +0006fcf0: 6172 6520 6469 7265 6374 2073 756d 206d are direct sum m │ │ │ │ +0006fd00: 6f64 756c 6573 2028 6973 4469 7265 6374 odules (isDirect │ │ │ │ +0006fd10: 5375 6d20 4d20 3d3d 2074 7275 6529 2074 Sum M == true) t │ │ │ │ +0006fd20: 6865 6e20 5420 6973 2074 6865 0a64 6972 hen T is the.dir │ │ │ │ +0006fd30: 6563 7420 7375 6d20 6f66 2074 6865 2074 ect sum of the t │ │ │ │ +0006fd40: 656e 736f 7220 7072 6f64 7563 7473 2062 ensor products b │ │ │ │ +0006fd50: 6574 7765 656e 2074 6865 2063 6f6d 706f etween the compo │ │ │ │ +0006fd60: 6e65 6e74 732e 2054 6869 7320 5348 4f55 nents. This SHOU │ │ │ │ +0006fd70: 4c44 2062 6520 6275 696c 740a 696e 746f LD be built.into │ │ │ │ +0006fd80: 204d 2a2a 4e2c 2062 7574 2069 736e 2774 M**N, but isn't │ │ │ │ +0006fd90: 2061 7320 6f66 204d 322c 2076 2e20 312e as of M2, v. 1. │ │ │ │ +0006fda0: 370a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 7..See also.==== │ │ │ │ +0006fdb0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ +0006fdc0: 486f 6d57 6974 6843 6f6d 706f 6e65 6e74 HomWithComponent │ │ │ │ +0006fdd0: 733a 2048 6f6d 5769 7468 436f 6d70 6f6e s: HomWithCompon │ │ │ │ +0006fde0: 656e 7473 2c20 2d2d 2063 6f6d 7075 7465 ents, -- compute │ │ │ │ +0006fdf0: 7320 486f 6d2c 2070 7265 7365 7276 696e s Hom, preservin │ │ │ │ +0006fe00: 670a 2020 2020 6469 7265 6374 2073 756d g. direct sum │ │ │ │ +0006fe10: 2069 6e66 6f72 6d61 7469 6f6e 0a20 202a information. * │ │ │ │ +0006fe20: 202a 6e6f 7465 2064 7561 6c57 6974 6843 *note dualWithC │ │ │ │ +0006fe30: 6f6d 706f 6e65 6e74 733a 2064 7561 6c57 omponents: dualW │ │ │ │ +0006fe40: 6974 6843 6f6d 706f 6e65 6e74 732c 202d ithComponents, - │ │ │ │ +0006fe50: 2d20 6475 616c 206d 6f64 756c 6520 7072 - dual module pr │ │ │ │ +0006fe60: 6573 6572 7669 6e67 0a20 2020 2064 6972 eserving. dir │ │ │ │ +0006fe70: 6563 7420 7375 6d20 696e 666f 726d 6174 ect sum informat │ │ │ │ +0006fe80: 696f 6e0a 0a57 6179 7320 746f 2075 7365 ion..Ways to use │ │ │ │ +0006fe90: 2074 656e 736f 7257 6974 6843 6f6d 706f tensorWithCompo │ │ │ │ +0006fea0: 6e65 6e74 733a 0a3d 3d3d 3d3d 3d3d 3d3d nents:.========= │ │ │ │ +0006feb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006fec0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2274 ========.. * "t │ │ │ │ +0006fed0: 656e 736f 7257 6974 6843 6f6d 706f 6e65 ensorWithCompone │ │ │ │ +0006fee0: 6e74 7328 4d6f 6475 6c65 2c4d 6f64 756c nts(Module,Modul │ │ │ │ +0006fef0: 6529 220a 0a46 6f72 2074 6865 2070 726f e)"..For the pro │ │ │ │ +0006ff00: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +0006ff10: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +0006ff20: 6f62 6a65 6374 202a 6e6f 7465 2074 656e object *note ten │ │ │ │ +0006ff30: 736f 7257 6974 6843 6f6d 706f 6e65 6e74 sorWithComponent │ │ │ │ +0006ff40: 733a 2074 656e 736f 7257 6974 6843 6f6d s: tensorWithCom │ │ │ │ +0006ff50: 706f 6e65 6e74 732c 2069 7320 6120 2a6e ponents, is a *n │ │ │ │ +0006ff60: 6f74 6520 6d65 7468 6f64 0a66 756e 6374 ote method.funct │ │ │ │ +0006ff70: 696f 6e3a 2028 4d61 6361 756c 6179 3244 ion: (Macaulay2D │ │ │ │ +0006ff80: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ +0006ff90: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ 0006ffa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ffb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ffc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ffd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ffe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -0006fff0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -00070000: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -00070010: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -00070020: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -00070030: 756c 6179 322d 312e 3236 2e30 352b 6473 ulay2-1.26.05+ds │ │ │ │ -00070040: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -00070050: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ -00070060: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ -00070070: 6c75 7469 6f6e 732e 6d32 3a32 3636 373a lutions.m2:2667: │ │ │ │ -00070080: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ -00070090: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -000700a0: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ -000700b0: 204e 6f64 653a 2074 6f41 7272 6179 2c20 Node: toArray, │ │ │ │ -000700c0: 4e65 7874 3a20 7477 6f4d 6f6e 6f6d 6961 Next: twoMonomia │ │ │ │ -000700d0: 6c73 2c20 5072 6576 3a20 7465 6e73 6f72 ls, Prev: tensor │ │ │ │ -000700e0: 5769 7468 436f 6d70 6f6e 656e 7473 2c20 WithComponents, │ │ │ │ -000700f0: 5570 3a20 546f 700a 0a74 6f41 7272 6179 Up: Top..toArray │ │ │ │ -00070100: 202d 2d20 6d61 6b65 7320 616e 2061 7272 -- makes an arr │ │ │ │ -00070110: 6179 2066 726f 6d20 6120 4c69 7374 206f ay from a List o │ │ │ │ -00070120: 7220 6672 6f6d 2061 2073 696e 676c 6520 r from a single │ │ │ │ -00070130: 696e 7465 6765 720a 2a2a 2a2a 2a2a 2a2a integer.******** │ │ │ │ +0006ffe0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +0006fff0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +00070000: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +00070010: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +00070020: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ +00070030: 3035 2b64 732f 4d32 2f4d 6163 6175 6c61 05+ds/M2/Macaula │ │ │ │ +00070040: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d y2/packages/.Com │ │ │ │ +00070050: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ +00070060: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a nResolutions.m2: │ │ │ │ +00070070: 3236 3637 3a30 2e0a 1f0a 4669 6c65 3a20 2667:0....File: │ │ │ │ +00070080: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +00070090: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +000700a0: 696e 666f 2c20 4e6f 6465 3a20 746f 4172 info, Node: toAr │ │ │ │ +000700b0: 7261 792c 204e 6578 743a 2074 776f 4d6f ray, Next: twoMo │ │ │ │ +000700c0: 6e6f 6d69 616c 732c 2050 7265 763a 2074 nomials, Prev: t │ │ │ │ +000700d0: 656e 736f 7257 6974 6843 6f6d 706f 6e65 ensorWithCompone │ │ │ │ +000700e0: 6e74 732c 2055 703a 2054 6f70 0a0a 746f nts, Up: Top..to │ │ │ │ +000700f0: 4172 7261 7920 2d2d 206d 616b 6573 2061 Array -- makes a │ │ │ │ +00070100: 6e20 6172 7261 7920 6672 6f6d 2061 204c n array from a L │ │ │ │ +00070110: 6973 7420 6f72 2066 726f 6d20 6120 7369 ist or from a si │ │ │ │ +00070120: 6e67 6c65 2069 6e74 6567 6572 0a2a 2a2a ngle integer.*** │ │ │ │ +00070130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00070140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00070150: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00070160: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00070170: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -00070180: 653a 200a 2020 2020 2020 2020 6172 7220 e: . arr │ │ │ │ -00070190: 3d20 746f 4172 7261 7920 4c0a 2020 2020 = toArray L. │ │ │ │ -000701a0: 2020 2020 6172 7220 3d20 746f 4172 7261 arr = toArra │ │ │ │ -000701b0: 7920 6e0a 2020 2a20 496e 7075 7473 3a0a y n. * Inputs:. │ │ │ │ -000701c0: 2020 2020 2020 2a20 4c2c 2061 202a 6e6f * L, a *no │ │ │ │ -000701d0: 7465 206c 6973 743a 2028 4d61 6361 756c te list: (Macaul │ │ │ │ -000701e0: 6179 3244 6f63 294c 6973 742c 2c20 0a20 ay2Doc)List,, . │ │ │ │ -000701f0: 2020 2020 202a 206e 2c20 616e 202a 6e6f * n, an *no │ │ │ │ -00070200: 7465 2069 6e74 6567 6572 3a20 284d 6163 te integer: (Mac │ │ │ │ -00070210: 6175 6c61 7932 446f 6329 5a5a 2c2c 200a aulay2Doc)ZZ,, . │ │ │ │ -00070220: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -00070230: 2020 202a 2061 7272 2c20 616e 202a 6e6f * arr, an *no │ │ │ │ -00070240: 7465 2061 7272 6179 3a20 284d 6163 6175 te array: (Macau │ │ │ │ -00070250: 6c61 7932 446f 6329 4172 7261 792c 2c20 lay2Doc)Array,, │ │ │ │ -00070260: 0a0a 5761 7973 2074 6f20 7573 6520 746f ..Ways to use to │ │ │ │ -00070270: 4172 7261 793a 0a3d 3d3d 3d3d 3d3d 3d3d Array:.========= │ │ │ │ -00070280: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00070290: 2022 746f 4172 7261 7928 4c69 7374 2922 "toArray(List)" │ │ │ │ -000702a0: 0a20 202a 2022 746f 4172 7261 7928 5a5a . * "toArray(ZZ │ │ │ │ -000702b0: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ -000702c0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -000702d0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -000702e0: 626a 6563 7420 2a6e 6f74 6520 746f 4172 bject *note toAr │ │ │ │ -000702f0: 7261 793a 2074 6f41 7272 6179 2c20 6973 ray: toArray, is │ │ │ │ -00070300: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ -00070310: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ -00070320: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ -00070330: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +00070160: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00070170: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +00070180: 2061 7272 203d 2074 6f41 7272 6179 204c arr = toArray L │ │ │ │ +00070190: 0a20 2020 2020 2020 2061 7272 203d 2074 . arr = t │ │ │ │ +000701a0: 6f41 7272 6179 206e 0a20 202a 2049 6e70 oArray n. * Inp │ │ │ │ +000701b0: 7574 733a 0a20 2020 2020 202a 204c 2c20 uts:. * L, │ │ │ │ +000701c0: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d a *note list: (M │ │ │ │ +000701d0: 6163 6175 6c61 7932 446f 6329 4c69 7374 acaulay2Doc)List │ │ │ │ +000701e0: 2c2c 200a 2020 2020 2020 2a20 6e2c 2061 ,, . * n, a │ │ │ │ +000701f0: 6e20 2a6e 6f74 6520 696e 7465 6765 723a n *note integer: │ │ │ │ +00070200: 2028 4d61 6361 756c 6179 3244 6f63 295a (Macaulay2Doc)Z │ │ │ │ +00070210: 5a2c 2c20 0a20 202a 204f 7574 7075 7473 Z,, . * Outputs │ │ │ │ +00070220: 3a0a 2020 2020 2020 2a20 6172 722c 2061 :. * arr, a │ │ │ │ +00070230: 6e20 2a6e 6f74 6520 6172 7261 793a 2028 n *note array: ( │ │ │ │ +00070240: 4d61 6361 756c 6179 3244 6f63 2941 7272 Macaulay2Doc)Arr │ │ │ │ +00070250: 6179 2c2c 200a 0a57 6179 7320 746f 2075 ay,, ..Ways to u │ │ │ │ +00070260: 7365 2074 6f41 7272 6179 3a0a 3d3d 3d3d se toArray:.==== │ │ │ │ +00070270: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00070280: 0a0a 2020 2a20 2274 6f41 7272 6179 284c .. * "toArray(L │ │ │ │ +00070290: 6973 7429 220a 2020 2a20 2274 6f41 7272 ist)". * "toArr │ │ │ │ +000702a0: 6179 285a 5a29 220a 0a46 6f72 2074 6865 ay(ZZ)"..For the │ │ │ │ +000702b0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ +000702c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +000702d0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ +000702e0: 2074 6f41 7272 6179 3a20 746f 4172 7261 toArray: toArra │ │ │ │ +000702f0: 792c 2069 7320 6120 2a6e 6f74 6520 6d65 y, is a *note me │ │ │ │ +00070300: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ +00070310: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ +00070320: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +00070330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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s.m2:3943:0....F │ │ │ │ -00070420: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ -00070430: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ -00070440: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ -00070450: 2074 776f 4d6f 6e6f 6d69 616c 732c 2050 twoMonomials, P │ │ │ │ -00070460: 7265 763a 2074 6f41 7272 6179 2c20 5570 rev: toArray, Up │ │ │ │ -00070470: 3a20 546f 700a 0a74 776f 4d6f 6e6f 6d69 : Top..twoMonomi │ │ │ │ -00070480: 616c 7320 2d2d 2074 616c 6c79 2074 6865 als -- tally the │ │ │ │ -00070490: 2073 6571 7565 6e63 6573 206f 6620 4252 sequences of BR │ │ │ │ -000704a0: 616e 6b73 2066 6f72 2063 6572 7461 696e anks for certain │ │ │ │ -000704b0: 2065 7861 6d70 6c65 730a 2a2a 2a2a 2a2a examples.****** │ │ │ │ +00070370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +00070380: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +00070390: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +000703a0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +000703b0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +000703c0: 6c61 7932 2d31 2e32 362e 3035 2b64 732f lay2-1.26.05+ds/ │ │ │ │ +000703d0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +000703e0: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ +000703f0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +00070400: 7574 696f 6e73 2e6d 323a 3339 3433 3a30 utions.m2:3943:0 │ │ │ │ +00070410: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ +00070420: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ +00070430: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ +00070440: 4e6f 6465 3a20 7477 6f4d 6f6e 6f6d 6961 Node: twoMonomia │ │ │ │ +00070450: 6c73 2c20 5072 6576 3a20 746f 4172 7261 ls, Prev: toArra │ │ │ │ +00070460: 792c 2055 703a 2054 6f70 0a0a 7477 6f4d y, Up: Top..twoM │ │ │ │ +00070470: 6f6e 6f6d 6961 6c73 202d 2d20 7461 6c6c onomials -- tall │ │ │ │ +00070480: 7920 7468 6520 7365 7175 656e 6365 7320 y the sequences │ │ │ │ +00070490: 6f66 2042 5261 6e6b 7320 666f 7220 6365 of BRanks for ce │ │ │ │ +000704a0: 7274 6169 6e20 6578 616d 706c 6573 0a2a rtain examples.* │ │ │ │ +000704b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000704c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000704d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000704e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000704f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ -00070500: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -00070510: 2020 5420 3d20 5477 6f4d 6f6e 6f6d 6961 T = TwoMonomia │ │ │ │ -00070520: 6c73 2863 2c64 290a 2020 2a20 496e 7075 ls(c,d). * Inpu │ │ │ │ -00070530: 7473 3a0a 2020 2020 2020 2a20 632c 2061 ts:. * c, a │ │ │ │ -00070540: 6e20 2a6e 6f74 6520 696e 7465 6765 723a n *note integer: │ │ │ │ -00070550: 2028 4d61 6361 756c 6179 3244 6f63 295a (Macaulay2Doc)Z │ │ │ │ -00070560: 5a2c 2c20 636f 6469 6d65 6e73 696f 6e20 Z,, codimension │ │ │ │ -00070570: 696e 2077 6869 6368 2074 6f20 776f 726b in which to work │ │ │ │ -00070580: 0a20 2020 2020 202a 2064 2c20 616e 202a . * d, an * │ │ │ │ -00070590: 6e6f 7465 2069 6e74 6567 6572 3a20 284d note integer: (M │ │ │ │ -000705a0: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c acaulay2Doc)ZZ,, │ │ │ │ -000705b0: 2064 6567 7265 6520 6f66 2074 6865 206d degree of the m │ │ │ │ -000705c0: 6f6e 6f6d 6961 6c73 2074 6f20 7461 6b65 onomials to take │ │ │ │ -000705d0: 0a20 202a 202a 6e6f 7465 204f 7074 696f . * *note Optio │ │ │ │ -000705e0: 6e61 6c20 696e 7075 7473 3a20 284d 6163 nal inputs: (Mac │ │ │ │ -000705f0: 6175 6c61 7932 446f 6329 7573 696e 6720 aulay2Doc)using │ │ │ │ -00070600: 6675 6e63 7469 6f6e 7320 7769 7468 206f functions with o │ │ │ │ -00070610: 7074 696f 6e61 6c20 696e 7075 7473 2c3a ptional inputs,: │ │ │ │ -00070620: 0a20 2020 2020 202a 204f 7074 696d 6973 . * Optimis │ │ │ │ -00070630: 6d20 3d3e 202e 2e2e 2c20 6465 6661 756c m => ..., defaul │ │ │ │ -00070640: 7420 7661 6c75 6520 300a 2020 2a20 4f75 t value 0. * Ou │ │ │ │ -00070650: 7470 7574 733a 0a20 2020 2020 202a 2054 tputs:. * T │ │ │ │ -00070660: 2c20 6120 2a6e 6f74 6520 7461 6c6c 793a , a *note tally: │ │ │ │ -00070670: 2028 4d61 6361 756c 6179 3244 6f63 2954 (Macaulay2Doc)T │ │ │ │ -00070680: 616c 6c79 2c2c 200a 0a44 6573 6372 6970 ally,, ..Descrip │ │ │ │ -00070690: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -000706a0: 0a0a 7461 6c6c 6965 7320 7468 6520 7365 ..tallies the se │ │ │ │ -000706b0: 7175 656e 6365 7320 6f66 2042 2d72 616e quences of B-ran │ │ │ │ -000706c0: 6b73 2074 6861 7420 6f63 6375 7220 666f ks that occur fo │ │ │ │ -000706d0: 7220 6964 6561 6c73 2067 656e 6572 6174 r ideals generat │ │ │ │ -000706e0: 6564 2062 7920 7061 6972 7320 6f66 0a6d ed by pairs of.m │ │ │ │ -000706f0: 6f6e 6f6d 6961 6c73 2069 6e20 5220 3d20 onomials in R = │ │ │ │ -00070700: 532f 2864 2d74 6820 706f 7765 7273 206f S/(d-th powers o │ │ │ │ -00070710: 6620 7468 6520 7661 7269 6162 6c65 7329 f the variables) │ │ │ │ -00070720: 2c20 7769 7468 2066 756c 6c20 636f 6d70 , with full comp │ │ │ │ -00070730: 6c65 7869 7479 2028 3d63 293b 0a74 6861 lexity (=c);.tha │ │ │ │ -00070740: 7420 6973 2c20 666f 7220 616e 2061 7070 t is, for an app │ │ │ │ -00070750: 726f 7072 6961 7465 2073 797a 7967 7920 ropriate syzygy │ │ │ │ -00070760: 4d20 6f66 204d 3020 3d20 522f 286d 312c M of M0 = R/(m1, │ │ │ │ -00070770: 206d 3229 2077 6865 7265 206d 3120 616e m2) where m1 an │ │ │ │ -00070780: 6420 6d32 2061 7265 0a6d 6f6e 6f6d 6961 d m2 are.monomia │ │ │ │ -00070790: 6c73 206f 6620 7468 6520 7361 6d65 2064 ls of the same d │ │ │ │ -000707a0: 6567 7265 652e 0a0a 2b2d 2d2d 2d2d 2d2d egree...+------- │ │ │ │ +000704f0: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +00070500: 2020 2020 2020 2054 203d 2054 776f 4d6f T = TwoMo │ │ │ │ +00070510: 6e6f 6d69 616c 7328 632c 6429 0a20 202a nomials(c,d). * │ │ │ │ +00070520: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +00070530: 2063 2c20 616e 202a 6e6f 7465 2069 6e74 c, an *note int │ │ │ │ +00070540: 6567 6572 3a20 284d 6163 6175 6c61 7932 eger: (Macaulay2 │ │ │ │ +00070550: 446f 6329 5a5a 2c2c 2063 6f64 696d 656e Doc)ZZ,, codimen │ │ │ │ +00070560: 7369 6f6e 2069 6e20 7768 6963 6820 746f sion in which to │ │ │ │ +00070570: 2077 6f72 6b0a 2020 2020 2020 2a20 642c work. * d, │ │ │ │ +00070580: 2061 6e20 2a6e 6f74 6520 696e 7465 6765 an *note intege │ │ │ │ +00070590: 723a 2028 4d61 6361 756c 6179 3244 6f63 r: (Macaulay2Doc │ │ │ │ +000705a0: 295a 5a2c 2c20 6465 6772 6565 206f 6620 )ZZ,, degree of │ │ │ │ +000705b0: 7468 6520 6d6f 6e6f 6d69 616c 7320 746f the monomials to │ │ │ │ +000705c0: 2074 616b 650a 2020 2a20 2a6e 6f74 6520 take. * *note │ │ │ │ +000705d0: 4f70 7469 6f6e 616c 2069 6e70 7574 733a Optional inputs: │ │ │ │ +000705e0: 2028 4d61 6361 756c 6179 3244 6f63 2975 (Macaulay2Doc)u │ │ │ │ +000705f0: 7369 6e67 2066 756e 6374 696f 6e73 2077 sing functions w │ │ │ │ +00070600: 6974 6820 6f70 7469 6f6e 616c 2069 6e70 ith optional inp │ │ │ │ +00070610: 7574 732c 3a0a 2020 2020 2020 2a20 4f70 uts,:. * Op │ │ │ │ +00070620: 7469 6d69 736d 203d 3e20 2e2e 2e2c 2064 timism => ..., d │ │ │ │ +00070630: 6566 6175 6c74 2076 616c 7565 2030 0a20 efault value 0. │ │ │ │ +00070640: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +00070650: 2020 2a20 542c 2061 202a 6e6f 7465 2074 * T, a *note t │ │ │ │ +00070660: 616c 6c79 3a20 284d 6163 6175 6c61 7932 ally: (Macaulay2 │ │ │ │ +00070670: 446f 6329 5461 6c6c 792c 2c20 0a0a 4465 Doc)Tally,, ..De │ │ │ │ +00070680: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +00070690: 3d3d 3d3d 3d0a 0a74 616c 6c69 6573 2074 =====..tallies t │ │ │ │ +000706a0: 6865 2073 6571 7565 6e63 6573 206f 6620 he sequences of │ │ │ │ +000706b0: 422d 7261 6e6b 7320 7468 6174 206f 6363 B-ranks that occ │ │ │ │ +000706c0: 7572 2066 6f72 2069 6465 616c 7320 6765 ur for ideals ge │ │ │ │ +000706d0: 6e65 7261 7465 6420 6279 2070 6169 7273 nerated by pairs │ │ │ │ +000706e0: 206f 660a 6d6f 6e6f 6d69 616c 7320 696e of.monomials in │ │ │ │ +000706f0: 2052 203d 2053 2f28 642d 7468 2070 6f77 R = S/(d-th pow │ │ │ │ +00070700: 6572 7320 6f66 2074 6865 2076 6172 6961 ers of the varia │ │ │ │ +00070710: 626c 6573 292c 2077 6974 6820 6675 6c6c bles), with full │ │ │ │ +00070720: 2063 6f6d 706c 6578 6974 7920 283d 6329 complexity (=c) │ │ │ │ +00070730: 3b0a 7468 6174 2069 732c 2066 6f72 2061 ;.that is, for a │ │ │ │ +00070740: 6e20 6170 7072 6f70 7269 6174 6520 7379 n appropriate sy │ │ │ │ +00070750: 7a79 6779 204d 206f 6620 4d30 203d 2052 zygy M of M0 = R │ │ │ │ +00070760: 2f28 6d31 2c20 6d32 2920 7768 6572 6520 /(m1, m2) where │ │ │ │ +00070770: 6d31 2061 6e64 206d 3220 6172 650a 6d6f m1 and m2 are.mo │ │ │ │ +00070780: 6e6f 6d69 616c 7320 6f66 2074 6865 2073 nomials of the s │ │ │ │ +00070790: 616d 6520 6465 6772 6565 2e0a 0a2b 2d2d ame degree...+-- │ │ │ │ +000707a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000707b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000707c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000707d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000707e0: 2b0a 7c69 3120 3a20 7365 7452 616e 646f +.|i1 : setRando │ │ │ │ -000707f0: 6d53 6565 6420 3020 2020 2020 2020 2020 mSeed 0 │ │ │ │ -00070800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070810: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00070820: 2073 6574 7469 6e67 2072 616e 646f 6d20 setting random │ │ │ │ -00070830: 7365 6564 2074 6f20 3020 2020 2020 2020 seed to 0 │ │ │ │ -00070840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070850: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000707d0: 2d2d 2d2b 0a7c 6931 203a 2073 6574 5261 ---+.|i1 : setRa │ │ │ │ +000707e0: 6e64 6f6d 5365 6564 2030 2020 2020 2020 ndomSeed 0 │ │ │ │ +000707f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070800: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ +00070810: 2d20 7365 7474 696e 6720 7261 6e64 6f6d - setting random │ │ │ │ +00070820: 2073 6565 6420 746f 2030 2020 2020 2020 seed to 0 │ │ │ │ +00070830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070840: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00070850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070880: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00070890: 7c6f 3120 3d20 3020 2020 2020 2020 2020 |o1 = 0 │ │ │ │ +00070870: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +00070880: 203d 2030 2020 2020 2020 2020 2020 2020 = 0 │ │ │ │ +00070890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000708a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000708b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000708c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000708b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +000708c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000708d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000708e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000708f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070900: 2d2d 2b0a 7c69 3220 3a20 7477 6f4d 6f6e --+.|i2 : twoMon │ │ │ │ -00070910: 6f6d 6961 6c73 2832 2c33 2920 2020 2020 omials(2,3) │ │ │ │ -00070920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070930: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00070940: 2d2d 2075 7365 6420 302e 3731 3839 3134 -- used 0.718914 │ │ │ │ -00070950: 7320 2863 7075 293b 2030 2e35 3531 3831 s (cpu); 0.55181 │ │ │ │ -00070960: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ -00070970: 2867 6329 2020 7c0a 7c32 2020 2020 2020 (gc) |.|2 │ │ │ │ +000708e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +000708f0: 203a 2074 776f 4d6f 6e6f 6d69 616c 7328 : twoMonomials( │ │ │ │ +00070900: 322c 3329 2020 2020 2020 2020 2020 2020 2,3) │ │ │ │ +00070910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070920: 2020 207c 0a7c 202d 2d20 7573 6564 2031 |.| -- used 1 │ │ │ │ +00070930: 2e32 3436 3036 7320 2863 7075 293b 2030 .24606s (cpu); 0 │ │ │ │ +00070940: 2e37 3434 3537 3773 2028 7468 7265 6164 .744577s (thread │ │ │ │ +00070950: 293b 2030 7320 2867 6329 207c 0a7c 3220 ); 0s (gc) |.|2 │ │ │ │ +00070960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000709a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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-00070a20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00070a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070a50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00070a60: 7c20 2d2d 2075 7365 6420 302e 3530 3431 | -- used 0.5041 │ │ │ │ -00070a70: 3873 2028 6370 7529 3b20 302e 3337 3230 8s (cpu); 0.3720 │ │ │ │ -00070a80: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ -00070a90: 2867 6329 2020 2020 7c0a 7c33 2020 2020 (gc) |.|3 │ │ │ │ -00070aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070ad0: 2020 7c0a 7c54 616c 6c79 7b7b 7b32 2c20 |.|Tally{{{2, │ │ │ │ -00070ae0: 327d 2c20 7b31 2c20 327d 7d20 3d3e 2032 2}, {1, 2}} => 2 │ │ │ │ -00070af0: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ -00070b00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00070b10: 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│ +00070af0: 2033 7d2c 207b 322c 2033 7d7d 203d 3e20 3}, {2, 3}} => │ │ │ │ +00070b00: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00070b10: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00070b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070b40: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00070b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070b80: 7c0a 7c20 2d2d 2075 7365 6420 302e 3039 |.| -- used 0.09 │ │ │ │ -00070b90: 3832 3536 3673 2028 6370 7529 3b20 302e 82566s (cpu); 0. │ │ │ │ -00070ba0: 3039 3738 3832 3373 2028 7468 7265 6164 0978823s (thread │ │ │ │ -00070bb0: 293b 2030 7320 2867 6329 7c0a 7c34 2020 ); 0s (gc)|.|4 │ │ │ │ -00070bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070bf0: 2020 2020 7c0a 7c54 616c 6c79 7b7b 7b32 |.|Tally{{{2 │ │ │ │ -00070c00: 2c20 327d 2c20 7b31 2c20 327d 7d20 3d3e , 2}, {1, 2}} => │ │ │ │ -00070c10: 2031 7d20 2020 2020 2020 2020 2020 2020 1} │ │ │ │ -00070c20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00070c30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00070c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070c60: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -00070c70: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -00070c80: 2a20 2a6e 6f74 6520 7477 6f4d 6f6e 6f6d * *note twoMonom │ │ │ │ -00070c90: 6961 6c73 3a20 7477 6f4d 6f6e 6f6d 6961 ials: twoMonomia │ │ │ │ -00070ca0: 6c73 2c20 2d2d 2074 616c 6c79 2074 6865 ls, -- tally the │ │ │ │ -00070cb0: 2073 6571 7565 6e63 6573 206f 6620 4252 sequences of BR │ │ │ │ -00070cc0: 616e 6b73 2066 6f72 0a20 2020 2063 6572 anks for. cer │ │ │ │ -00070cd0: 7461 696e 2065 7861 6d70 6c65 730a 0a57 tain examples..W │ │ │ │ -00070ce0: 6179 7320 746f 2075 7365 2074 776f 4d6f ays to use twoMo │ │ │ │ -00070cf0: 6e6f 6d69 616c 733a 0a3d 3d3d 3d3d 3d3d nomials:.======= │ │ │ │ -00070d00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00070d10: 3d3d 0a0a 2020 2a20 2274 776f 4d6f 6e6f ==.. * "twoMono │ │ │ │ -00070d20: 6d69 616c 7328 5a5a 2c5a 5a29 220a 0a46 mials(ZZ,ZZ)"..F │ │ │ │ -00070d30: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00070d40: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00070d50: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00070d60: 202a 6e6f 7465 2074 776f 4d6f 6e6f 6d69 *note twoMonomi │ │ │ │ -00070d70: 616c 733a 2074 776f 4d6f 6e6f 6d69 616c als: twoMonomial │ │ │ │ -00070d80: 732c 2069 7320 6120 2a6e 6f74 6520 6d65 s, is a *note me │ │ │ │ -00070d90: 7468 6f64 2066 756e 6374 696f 6e20 7769 thod function wi │ │ │ │ -00070da0: 7468 0a6f 7074 696f 6e73 3a20 284d 6163 th.options: (Mac │ │ │ │ -00070db0: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -00070dc0: 4675 6e63 7469 6f6e 5769 7468 4f70 7469 FunctionWithOpti │ │ │ │ -00070dd0: 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ons,...--------- │ │ │ │ +00070b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070b50: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ +00070b60: 2e31 3332 3634 3273 2028 6370 7529 3b20 .132642s (cpu); │ │ │ │ +00070b70: 302e 3132 3035 3535 7320 2874 6872 6561 0.120555s (threa │ │ │ │ +00070b80: 6429 3b20 3073 2028 6763 297c 0a7c 3420 d); 0s (gc)|.|4 │ │ │ │ +00070b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070bc0: 2020 207c 0a7c 5461 6c6c 797b 7b7b 322c |.|Tally{{{2, │ │ │ │ +00070bd0: 2032 7d2c 207b 312c 2032 7d7d 203d 3e20 2}, {1, 2}} => │ │ │ │ +00070be0: 317d 2020 2020 2020 2020 2020 2020 2020 1} │ │ │ │ +00070bf0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00070c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00070c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00070c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00070c30: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ +00070c40: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +00070c50: 7465 2074 776f 4d6f 6e6f 6d69 616c 733a te twoMonomials: │ │ │ │ +00070c60: 2074 776f 4d6f 6e6f 6d69 616c 732c 202d twoMonomials, - │ │ │ │ +00070c70: 2d20 7461 6c6c 7920 7468 6520 7365 7175 - tally the sequ │ │ │ │ +00070c80: 656e 6365 7320 6f66 2042 5261 6e6b 7320 ences of BRanks │ │ │ │ +00070c90: 666f 720a 2020 2020 6365 7274 6169 6e20 for. certain │ │ │ │ +00070ca0: 6578 616d 706c 6573 0a0a 5761 7973 2074 examples..Ways t │ │ │ │ +00070cb0: 6f20 7573 6520 7477 6f4d 6f6e 6f6d 6961 o use twoMonomia │ │ │ │ +00070cc0: 6c73 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ls:.============ │ │ │ │ +00070cd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +00070ce0: 202a 2022 7477 6f4d 6f6e 6f6d 6961 6c73 * "twoMonomials │ │ │ │ +00070cf0: 285a 5a2c 5a5a 2922 0a0a 466f 7220 7468 (ZZ,ZZ)"..For th │ │ │ │ +00070d00: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +00070d10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +00070d20: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +00070d30: 6520 7477 6f4d 6f6e 6f6d 6961 6c73 3a20 e twoMonomials: │ │ │ │ +00070d40: 7477 6f4d 6f6e 6f6d 6961 6c73 2c20 6973 twoMonomials, is │ │ │ │ +00070d50: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ +00070d60: 6675 6e63 7469 6f6e 2077 6974 680a 6f70 function with.op │ │ │ │ +00070d70: 7469 6f6e 733a 2028 4d61 6361 756c 6179 tions: (Macaulay │ │ │ │ +00070d80: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ +00070d90: 696f 6e57 6974 684f 7074 696f 6e73 2c2e ionWithOptions,. │ │ │ │ +00070da0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +00070db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00070dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00070dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070e20: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -00070e30: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -00070e40: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00070e50: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00070e60: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00070e70: 362e 3035 2b64 732f 4d32 2f4d 6163 6175 6.05+ds/M2/Macau │ │ │ │ -00070e80: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ -00070e90: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -00070ea0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d ionResolutions.m │ │ │ │ -00070eb0: 323a 3435 3434 3a30 2e0a 1f0a 5461 6720 2:4544:0....Tag │ │ │ │ -00070ec0: 5461 626c 653a 0a4e 6f64 653a 2054 6f70 Table:.Node: Top │ │ │ │ -00070ed0: 7f33 3136 0a4e 6f64 653a 2041 5261 6e6b .316.Node: ARank │ │ │ │ -00070ee0: 737f 3336 3338 320a 4e6f 6465 3a20 4175 s.36382.Node: Au │ │ │ │ -00070ef0: 676d 656e 7461 7469 6f6e 7f33 3739 3438 gmentation.37948 │ │ │ │ -00070f00: 0a4e 6f64 653a 2042 4747 4c7f 3339 3136 .Node: BGGL.3916 │ │ │ │ -00070f10: 310a 4e6f 6465 3a20 624d 6170 737f 3432 1.Node: bMaps.42 │ │ │ │ -00070f20: 3539 380a 4e6f 6465 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│ 000096f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00009700: 7c69 3920 3a20 656c 6170 7365 6454 696d |i9 : elapsedTim │ │ │ │ 00009710: 6520 4120 3d20 636f 6e6e 6563 7469 6f6e e A = connection │ │ │ │ 00009720: 4d61 7472 6963 6573 2049 3b20 2020 2020 Matrices I; │ │ │ │ 00009730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009740: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00009750: 7c20 2d2d 2033 2e30 3131 3335 7320 656c | -- 3.01135s el │ │ │ │ +00009750: 7c20 2d2d 2032 2e35 3131 3238 7320 656c | -- 2.51128s el │ │ │ │ 00009760: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00009770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009790: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000097a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000097b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000097c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000097d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000097e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000097f0: 7c69 3130 203a 2065 6c61 7073 6564 5469 |i10 : elapsedTi │ │ │ │ 00009800: 6d65 2061 7373 6572 7420 6973 496e 7465 me assert isInte │ │ │ │ 00009810: 6772 6162 6c65 2041 2020 2020 2020 2020 grable A │ │ │ │ 00009820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009830: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00009840: 7c20 2d2d 2035 2e35 3733 3539 7320 656c | -- 5.57359s el │ │ │ │ +00009840: 7c20 2d2d 2034 2e31 3034 3734 7320 656c | -- 4.10474s el │ │ │ │ 00009850: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00009860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009880: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00009890: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000098a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000098b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -4559,15 +4559,15 @@ │ │ │ │ 00011ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011d00: 2b0a 7c69 3134 203a 2065 6c61 7073 6564 +.|i14 : elapsed │ │ │ │ 00011d10: 5469 6d65 2067 203d 2067 6175 6765 4d61 Time g = gaugeMa │ │ │ │ 00011d20: 7472 6978 2849 2c20 4229 3b20 2020 2020 trix(I, B); │ │ │ │ 00011d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011d50: 7c0a 7c20 2d2d 202e 3636 3638 3131 7320 |.| -- .666811s │ │ │ │ +00011d50: 7c0a 7c20 2d2d 202e 3531 3236 3832 7320 |.| -- .512682s │ │ │ │ 00011d60: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 00011d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011da0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00011db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4589,30 +4589,30 @@ │ │ │ │ 00011ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011ee0: 2b0a 7c69 3135 203a 2065 6c61 7073 6564 +.|i15 : elapsed │ │ │ │ 00011ef0: 5469 6d65 2041 3120 3d20 6761 7567 6554 Time A1 = gaugeT │ │ │ │ 00011f00: 7261 6e73 666f 726d 2867 2c20 4129 3b20 ransform(g, A); │ │ │ │ 00011f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011f30: 7c0a 7c20 2d2d 2031 2e36 3138 3831 7320 |.| -- 1.61881s │ │ │ │ -00011f40: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ +00011f30: 7c0a 7c20 2d2d 2031 2e30 3939 3873 2065 |.| -- 1.0998s e │ │ │ │ +00011f40: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 00011f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f80: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00011f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011fd0: 2b0a 7c69 3136 203a 2065 6c61 7073 6564 +.|i16 : elapsed │ │ │ │ 00011fe0: 5469 6d65 2061 7373 6572 7420 6973 496e Time assert isIn │ │ │ │ 00011ff0: 7465 6772 6162 6c65 2041 3120 2020 2020 tegrable A1 │ │ │ │ 00012000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012020: 7c0a 7c20 2d2d 2031 2e30 3437 3035 7320 |.| -- 1.04705s │ │ │ │ +00012020: 7c0a 7c20 2d2d 202e 3836 3239 3136 7320 |.| -- .862916s │ │ │ │ 00012030: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 00012040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012070: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00012080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5030,31 +5030,31 @@ │ │ │ │ 00013a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013a70: 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20 656c -----+.|i19 : el │ │ │ │ 00013a80: 6170 7365 6454 696d 6520 4132 203d 2067 apsedTime A2 = g │ │ │ │ 00013a90: 6175 6765 5472 616e 7366 6f72 6d28 6368 augeTransform(ch │ │ │ │ 00013aa0: 616e 6765 4570 732c 2041 3129 3b20 2020 angeEps, A1); │ │ │ │ 00013ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013ac0: 2020 2020 207c 0a7c 202d 2d20 2e35 3436 |.| -- .546 │ │ │ │ -00013ad0: 3232 3373 2065 6c61 7073 6564 2020 2020 223s elapsed │ │ │ │ +00013ac0: 2020 2020 207c 0a7c 202d 2d20 2e34 3136 |.| -- .416 │ │ │ │ +00013ad0: 3036 3473 2065 6c61 7073 6564 2020 2020 064s elapsed │ │ │ │ 00013ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013b10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00013b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b60: 2d2d 2d2d 2d2b 0a7c 6932 3020 3a20 656c -----+.|i20 : el │ │ │ │ 00013b70: 6170 7365 6454 696d 6520 6173 7365 7274 apsedTime assert │ │ │ │ 00013b80: 2069 7349 6e74 6567 7261 626c 6520 4132 isIntegrable A2 │ │ │ │ 00013b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013bb0: 2020 2020 207c 0a7c 202d 2d20 2e37 3234 |.| -- .724 │ │ │ │ -00013bc0: 3233 3773 2065 6c61 7073 6564 2020 2020 237s elapsed │ │ │ │ +00013bb0: 2020 2020 207c 0a7c 202d 2d20 2e36 3632 |.| -- .662 │ │ │ │ +00013bc0: 3038 3873 2065 6c61 7073 6564 2020 2020 088s elapsed │ │ │ │ 00013bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013c00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00013c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5440,30 +5440,30 @@ │ │ │ │ 000153f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00015410: 6937 203a 2065 6c61 7073 6564 5469 6d65 i7 : elapsedTime │ │ │ │ 00015420: 2041 203d 2063 6f6e 6e65 6374 696f 6e4d A = connectionM │ │ │ │ 00015430: 6174 7269 6365 7320 493b 2020 2020 2020 atrices I; │ │ │ │ 00015440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015450: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00015460: 202d 2d20 2e32 3137 3434 3973 2065 6c61 -- .217449s ela │ │ │ │ +00015460: 202d 2d20 2e31 3932 3636 3373 2065 6c61 -- .192663s ela │ │ │ │ 00015470: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 00015480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000154a0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 000154b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000154c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000154d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000154e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000154f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00015500: 6938 203a 2065 6c61 7073 6564 5469 6d65 i8 : elapsedTime │ │ │ │ 00015510: 2061 7373 6572 7420 6973 496e 7465 6772 assert isIntegr │ │ │ │ 00015520: 6162 6c65 2041 2020 2020 2020 2020 2020 able A │ │ │ │ 00015530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015540: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00015550: 202d 2d20 2e32 3831 3531 3273 2065 6c61 -- .281512s ela │ │ │ │ +00015550: 202d 2d20 2e32 3133 3432 3873 2065 6c61 -- .213428s ela │ │ │ │ 00015560: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 00015570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015590: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 000155a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000155b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000155c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ ├── ./usr/share/info/Cremona.info.gz │ │ │ ├── Cremona.info │ │ │ │ @@ -147,16 +147,16 @@ │ │ │ │ 00000920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000930: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2074 -------+.|i2 : t │ │ │ │ 00000940: 696d 6520 7068 6920 3d20 746f 4d61 7020 ime phi = toMap │ │ │ │ 00000950: 6d69 6e6f 7273 2833 2c6d 6174 7269 787b minors(3,matrix{ │ │ │ │ 00000960: 7b74 5f30 2e2e 745f 347d 2c7b 745f 312e {t_0..t_4},{t_1. │ │ │ │ 00000970: 2e74 5f35 7d2c 7b74 5f32 2e2e 745f 367d .t_5},{t_2..t_6} │ │ │ │ 00000980: 7d29 2020 2020 207c 0a7c 202d 2d20 7573 }) |.| -- us │ │ │ │ -00000990: 6564 2030 2e30 3033 3738 3633 3973 2028 ed 0.00378639s ( │ │ │ │ -000009a0: 6370 7529 3b20 302e 3030 3337 3834 3333 cpu); 0.00378433 │ │ │ │ +00000990: 6564 2030 2e30 3035 3134 3238 3573 2028 ed 0.00514285s ( │ │ │ │ +000009a0: 6370 7529 3b20 302e 3030 3531 3433 3039 cpu); 0.00514309 │ │ │ │ 000009b0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 000009c0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 000009d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000009e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000009f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -322,17 +322,17 @@ │ │ │ │ 00001410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001420: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 -------+.|i3 : t │ │ │ │ 00001430: 696d 6520 4a20 3d20 6b65 726e 656c 2870 ime J = kernel(p │ │ │ │ 00001440: 6869 2c32 2920 2020 2020 2020 2020 2020 hi,2) │ │ │ │ 00001450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001470: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001480: 6564 2030 2e31 3034 3339 3873 2028 6370 ed 0.104398s (cp │ │ │ │ -00001490: 7529 3b20 302e 3035 3736 3737 3573 2028 u); 0.0576775s ( │ │ │ │ -000014a0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00001480: 6564 2030 2e32 3031 3932 3773 2028 6370 ed 0.201927s (cp │ │ │ │ +00001490: 7529 3b20 302e 3130 3538 3973 2028 7468 u); 0.10589s (th │ │ │ │ +000014a0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 000014b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000014c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000014d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000014e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000014f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001510: 2020 2020 2020 207c 0a7c 6f33 203d 2069 |.|o3 = i │ │ │ │ @@ -387,18 +387,18 @@ │ │ │ │ 00001820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001830: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 -------+.|i4 : t │ │ │ │ 00001840: 696d 6520 6465 6772 6565 4d61 7020 7068 ime degreeMap ph │ │ │ │ 00001850: 6920 2020 2020 2020 2020 2020 2020 2020 i │ │ │ │ 00001860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001880: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001890: 6564 2030 2e30 3238 3434 3938 7320 2863 ed 0.0284498s (c │ │ │ │ -000018a0: 7075 293b 2030 2e30 3238 3435 3536 7320 pu); 0.0284556s │ │ │ │ -000018b0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -000018c0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00001890: 6564 2030 2e30 3434 3633 3773 2028 6370 ed 0.044637s (cp │ │ │ │ +000018a0: 7529 3b20 302e 3034 3436 3338 3673 2028 u); 0.0446386s ( │ │ │ │ +000018b0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +000018c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000018d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000018e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000018f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001920: 2020 2020 2020 207c 0a7c 6f34 203d 2031 |.|o4 = 1 │ │ │ │ 00001930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -412,17 +412,17 @@ │ │ │ │ 000019b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000019c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 -------+.|i5 : t │ │ │ │ 000019d0: 696d 6520 7072 6f6a 6563 7469 7665 4465 ime projectiveDe │ │ │ │ 000019e0: 6772 6565 7320 7068 6920 2020 2020 2020 grees phi │ │ │ │ 000019f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a10: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001a20: 6564 2030 2e38 3139 3534 3873 2028 6370 ed 0.819548s (cp │ │ │ │ -00001a30: 7529 3b20 302e 3439 3839 3434 7320 2874 u); 0.498944s (t │ │ │ │ -00001a40: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00001a20: 6564 2030 2e38 3537 3338 7320 2863 7075 ed 0.85738s (cpu │ │ │ │ +00001a30: 293b 2030 2e35 3832 3430 3773 2028 7468 ); 0.582407s (th │ │ │ │ +00001a40: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00001a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00001a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ab0: 2020 2020 2020 207c 0a7c 6f35 203d 207b |.|o5 = { │ │ │ │ @@ -447,17 +447,17 @@ │ │ │ │ 00001be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001bf0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2074 -------+.|i6 : t │ │ │ │ 00001c00: 696d 6520 7072 6f6a 6563 7469 7665 4465 ime projectiveDe │ │ │ │ 00001c10: 6772 6565 7328 7068 692c 4e75 6d44 6567 grees(phi,NumDeg │ │ │ │ 00001c20: 7265 6573 3d3e 3029 2020 2020 2020 2020 rees=>0) │ │ │ │ 00001c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001c40: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001c50: 6564 2030 2e31 3831 3138 3473 2028 6370 ed 0.181184s (cp │ │ │ │ -00001c60: 7529 3b20 302e 3130 3438 3337 7320 2874 u); 0.104837s (t │ │ │ │ -00001c70: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00001c50: 6564 2030 2e31 3738 3938 3773 2028 6370 ed 0.178987s (cp │ │ │ │ +00001c60: 7529 3b20 302e 3039 3632 3534 3673 2028 u); 0.0962546s ( │ │ │ │ +00001c70: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00001c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001c90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00001ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ce0: 2020 2020 2020 207c 0a7c 6f36 203d 207b |.|o6 = { │ │ │ │ @@ -482,15 +482,15 @@ │ │ │ │ 00001e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001e20: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2074 -------+.|i7 : t │ │ │ │ 00001e30: 696d 6520 7068 6920 3d20 746f 4d61 7028 ime phi = toMap( │ │ │ │ 00001e40: 7068 6920 2020 2020 2020 2020 2020 2020 phi │ │ │ │ 00001e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001e70: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001e80: 6564 2030 2e30 3032 3235 3431 3473 2028 ed 0.00225414s ( │ │ │ │ +00001e80: 6564 2030 2e30 3032 3632 3135 3773 2028 ed 0.00262157s ( │ │ │ │ 00001e90: 6370 7520 2020 2020 2020 2020 2020 2020 cpu │ │ │ │ 00001ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00001ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -567,15 +567,15 @@ │ │ │ │ 00002360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002370: 2d2d 2d2d 2d2d 2d7c 0a7c 2c44 6f6d 696e -------|.|,Domin │ │ │ │ 00002380: 616e 743d 3e4a 2920 2020 2020 2020 2020 ant=>J) │ │ │ │ 00002390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023c0: 2020 2020 2020 207c 0a7c 293b 2030 2e30 |.|); 0.0 │ │ │ │ -000023d0: 3032 3235 3833 3273 2028 7468 7265 6164 0225832s (thread │ │ │ │ +000023d0: 3032 3632 3535 3873 2028 7468 7265 6164 0262558s (thread │ │ │ │ 000023e0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 000023f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002410: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00002420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -832,16 +832,16 @@ │ │ │ │ 000033f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003400: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2074 -------+.|i8 : t │ │ │ │ 00003410: 696d 6520 7073 6920 3d20 696e 7665 7273 ime psi = invers │ │ │ │ 00003420: 654d 6170 2070 6869 2020 2020 2020 2020 eMap phi │ │ │ │ 00003430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003450: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00003460: 6564 2030 2e34 3733 3237 3973 2028 6370 ed 0.473279s (cp │ │ │ │ -00003470: 7529 3b20 302e 3338 3036 3735 7320 2874 u); 0.380675s (t │ │ │ │ +00003460: 6564 2030 2e34 3531 3331 3773 2028 6370 ed 0.451317s (cp │ │ │ │ +00003470: 7529 3b20 302e 3435 3133 3139 7320 2874 u); 0.451319s (t │ │ │ │ 00003480: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00003490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000034a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000034b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000034c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000034d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000034e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1117,18 +1117,18 @@ │ │ │ │ 000045c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000045d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2074 -------+.|i9 : t │ │ │ │ 000045e0: 696d 6520 6973 496e 7665 7273 654d 6170 ime isInverseMap │ │ │ │ 000045f0: 2870 6869 2c70 7369 2920 2020 2020 2020 (phi,psi) │ │ │ │ 00004600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004620: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00004630: 6564 2030 2e30 3039 3634 3037 3873 2028 ed 0.00964078s ( │ │ │ │ -00004640: 6370 7529 3b20 302e 3030 3936 3432 3939 cpu); 0.00964299 │ │ │ │ -00004650: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -00004660: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ +00004630: 6564 2030 2e30 3137 3337 3839 7320 2863 ed 0.0173789s (c │ │ │ │ +00004640: 7075 293b 2030 2e30 3137 3338 3032 7320 pu); 0.0173802s │ │ │ │ +00004650: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +00004660: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00004670: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00004680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000046a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000046b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000046c0: 2020 2020 2020 207c 0a7c 6f39 203d 2074 |.|o9 = t │ │ │ │ 000046d0: 7275 6520 2020 2020 2020 2020 2020 2020 rue │ │ │ │ @@ -1142,17 +1142,17 @@ │ │ │ │ 00004750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004760: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 -------+.|i10 : │ │ │ │ 00004770: 7469 6d65 2064 6567 7265 654d 6170 2070 time degreeMap p │ │ │ │ 00004780: 7369 2020 2020 2020 2020 2020 2020 2020 si │ │ │ │ 00004790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000047a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000047b0: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -000047c0: 6564 2030 2e33 3435 3739 3773 2028 6370 ed 0.345797s (cp │ │ │ │ -000047d0: 7529 3b20 302e 3139 3833 3932 7320 2874 u); 0.198392s (t │ │ │ │ -000047e0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +000047c0: 6564 2030 2e35 3532 3536 3873 2028 6370 ed 0.552568s (cp │ │ │ │ +000047d0: 7529 3b20 302e 3238 3037 7320 2874 6872 u); 0.2807s (thr │ │ │ │ +000047e0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 000047f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004800: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00004810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004850: 2020 2020 2020 207c 0a7c 6f31 3020 3d20 |.|o10 = │ │ │ │ @@ -1167,17 +1167,17 @@ │ │ │ │ 000048e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000048f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ 00004900: 7469 6d65 2070 726f 6a65 6374 6976 6544 time projectiveD │ │ │ │ 00004910: 6567 7265 6573 2070 7369 2020 2020 2020 egrees psi │ │ │ │ 00004920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004940: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00004950: 6564 2035 2e33 3831 3536 7320 2863 7075 ed 5.38156s (cpu │ │ │ │ -00004960: 293b 2034 2e34 3433 3432 7320 2874 6872 ); 4.44342s (thr │ │ │ │ -00004970: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +00004950: 6564 2036 2e37 3632 3873 2028 6370 7529 ed 6.7628s (cpu) │ │ │ │ +00004960: 3b20 362e 3333 3732 3873 2028 7468 7265 ; 6.33728s (thre │ │ │ │ +00004970: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00004980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004990: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000049a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049e0: 2020 2020 2020 207c 0a7c 6f31 3120 3d20 |.|o11 = │ │ │ │ @@ -1214,16 +1214,16 @@ │ │ │ │ 00004bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00004be0: 0a7c 6931 3220 3a20 7469 6d65 2070 6869 .|i12 : time phi │ │ │ │ 00004bf0: 203d 2072 6174 696f 6e61 6c4d 6170 206d = rationalMap m │ │ │ │ 00004c00: 696e 6f72 7328 332c 6d61 7472 6978 7b7b inors(3,matrix{{ │ │ │ │ 00004c10: 745f 302e 2e74 5f34 7d2c 7b74 5f31 2e2e t_0..t_4},{t_1.. │ │ │ │ 00004c20: 745f 357d 2c7b 745f 322e 2e74 5f36 207c t_5},{t_2..t_6 | │ │ │ │ 00004c30: 0a7c 202d 2d20 7573 6564 2030 2e30 3032 .| -- used 0.002 │ │ │ │ -00004c40: 3130 3431 3973 2028 6370 7529 3b20 302e 10419s (cpu); 0. │ │ │ │ -00004c50: 3030 3231 3035 3231 7320 2874 6872 6561 00210521s (threa │ │ │ │ +00004c40: 3630 3730 3573 2028 6370 7529 3b20 302e 60705s (cpu); 0. │ │ │ │ +00004c50: 3030 3236 3130 3836 7320 2874 6872 6561 00261086s (threa │ │ │ │ 00004c60: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00004c70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00004c80: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00004c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -1493,17 +1493,17 @@ │ │ │ │ 00005d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00005d60: 0a7c 6931 3320 3a20 7469 6d65 2070 6869 .|i13 : time phi │ │ │ │ 00005d70: 203d 2072 6174 696f 6e61 6c4d 6170 2870 = rationalMap(p │ │ │ │ 00005d80: 6869 2c44 6f6d 696e 616e 743d 3e32 2920 hi,Dominant=>2) │ │ │ │ 00005d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005da0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00005db0: 0a7c 202d 2d20 7573 6564 2030 2e30 3532 .| -- used 0.052 │ │ │ │ -00005dc0: 3132 3232 7320 2863 7075 293b 2030 2e30 1222s (cpu); 0.0 │ │ │ │ -00005dd0: 3532 3133 3135 7320 2874 6872 6561 6429 521315s (thread) │ │ │ │ +00005db0: 0a7c 202d 2d20 7573 6564 2030 2e30 3635 .| -- used 0.065 │ │ │ │ +00005dc0: 3137 3334 7320 2863 7075 293b 2030 2e30 1734s (cpu); 0.0 │ │ │ │ +00005dd0: 3635 3137 3636 7320 2874 6872 6561 6429 651766s (thread) │ │ │ │ 00005de0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00005df0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00005e00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00005e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005e40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -2153,17 +2153,17 @@ │ │ │ │ 00008680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 000086a0: 0a7c 6931 3420 3a20 7469 6d65 2070 6869 .|i14 : time phi │ │ │ │ 000086b0: 5e28 2d31 2920 2020 2020 2020 2020 2020 ^(-1) │ │ │ │ 000086c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000086d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000086e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000086f0: 0a7c 202d 2d20 7573 6564 2030 2e34 3935 .| -- used 0.495 │ │ │ │ -00008700: 3637 3573 2028 6370 7529 3b20 302e 3432 675s (cpu); 0.42 │ │ │ │ -00008710: 3735 3238 7320 2874 6872 6561 6429 3b20 7528s (thread); │ │ │ │ +000086f0: 0a7c 202d 2d20 7573 6564 2030 2e36 3937 .| -- used 0.697 │ │ │ │ +00008700: 3533 3273 2028 6370 7529 3b20 302e 3539 532s (cpu); 0.59 │ │ │ │ +00008710: 3932 3032 7320 2874 6872 6561 6429 3b20 9202s (thread); │ │ │ │ 00008720: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00008730: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00008740: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00008750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008780: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -2708,17 +2708,17 @@ │ │ │ │ 0000a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000a950: 0a7c 6931 3520 3a20 7469 6d65 2064 6567 .|i15 : time deg │ │ │ │ 0000a960: 7265 6573 2070 6869 5e28 2d31 2920 2020 rees phi^(-1) │ │ │ │ 0000a970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a990: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e34 3730 .| -- used 0.470 │ │ │ │ -0000a9b0: 3031 3773 2028 6370 7529 3b20 302e 3331 017s (cpu); 0.31 │ │ │ │ -0000a9c0: 3036 3534 7320 2874 6872 6561 6429 3b20 0654s (thread); │ │ │ │ +0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e33 3237 .| -- used 0.327 │ │ │ │ +0000a9b0: 3837 3173 2028 6370 7529 3b20 302e 3331 871s (cpu); 0.31 │ │ │ │ +0000a9c0: 3533 3132 7320 2874 6872 6561 6429 3b20 5312s (thread); │ │ │ │ 0000a9d0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 0000a9e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000a9f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000aa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -2743,17 +2743,17 @@ │ │ │ │ 0000ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000ab80: 0a7c 6931 3620 3a20 7469 6d65 2064 6567 .|i16 : time deg │ │ │ │ 0000ab90: 7265 6573 2070 6869 2020 2020 2020 2020 rees phi │ │ │ │ 0000aba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000abb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000abc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3137 .| -- used 0.017 │ │ │ │ -0000abe0: 3932 3737 7320 2863 7075 293b 2030 2e30 9277s (cpu); 0.0 │ │ │ │ -0000abf0: 3137 3438 3433 7320 2874 6872 6561 6429 174843s (thread) │ │ │ │ +0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3834 .| -- used 0.084 │ │ │ │ +0000abe0: 3931 3339 7320 2863 7075 293b 2030 2e30 9139s (cpu); 0.0 │ │ │ │ +0000abf0: 3238 3139 3034 7320 2874 6872 6561 6429 281904s (thread) │ │ │ │ 0000ac00: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0000ac10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000ac20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000ac30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -2778,18 +2778,18 @@ │ │ │ │ 0000ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000adb0: 0a7c 6931 3720 3a20 7469 6d65 2064 6573 .|i17 : time des │ │ │ │ 0000adc0: 6372 6962 6520 7068 6920 2020 2020 2020 cribe phi │ │ │ │ 0000add0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ade0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000adf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000ae00: 0a7c 202d 2d20 7573 6564 2030 2e30 3032 .| -- used 0.002 │ │ │ │ -0000ae10: 3936 3331 3873 2028 6370 7529 3b20 302e 96318s (cpu); 0. │ │ │ │ -0000ae20: 3030 3239 3634 3237 7320 2874 6872 6561 00296427s (threa │ │ │ │ -0000ae30: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +0000ae00: 0a7c 202d 2d20 7573 6564 2030 2e30 3034 .| -- used 0.004 │ │ │ │ +0000ae10: 3132 3331 7320 2863 7075 293b 2030 2e30 1231s (cpu); 0.0 │ │ │ │ +0000ae20: 3034 3132 3739 3973 2028 7468 7265 6164 0412799s (thread │ │ │ │ +0000ae30: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 0000ae40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000ae50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000ae60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000aea0: 0a7c 6f31 3720 3d20 7261 7469 6f6e 616c .|o17 = rational │ │ │ │ @@ -2843,17 +2843,17 @@ │ │ │ │ 0000b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000b1c0: 0a7c 6931 3820 3a20 7469 6d65 2064 6573 .|i18 : time des │ │ │ │ 0000b1d0: 6372 6962 6520 7068 695e 282d 3129 2020 cribe phi^(-1) │ │ │ │ 0000b1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b200: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3039 .| -- used 0.009 │ │ │ │ -0000b220: 3632 3273 2028 6370 7529 3b20 302e 3030 622s (cpu); 0.00 │ │ │ │ -0000b230: 3936 3233 3435 7320 2874 6872 6561 6429 962345s (thread) │ │ │ │ +0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3131 .| -- used 0.011 │ │ │ │ +0000b220: 3736 3134 7320 2863 7075 293b 2030 2e30 7614s (cpu); 0.0 │ │ │ │ +0000b230: 3131 3736 3733 7320 2874 6872 6561 6429 117673s (thread) │ │ │ │ 0000b240: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0000b250: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b260: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000b270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b2a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -2923,18 +2923,18 @@ │ │ │ │ 0000b6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000b6c0: 0a7c 6931 3920 3a20 7469 6d65 2028 662c .|i19 : time (f, │ │ │ │ 0000b6d0: 6729 203d 2067 7261 7068 2070 6869 5e2d g) = graph phi^- │ │ │ │ 0000b6e0: 313b 2066 3b20 2020 2020 2020 2020 2020 1; f; │ │ │ │ 0000b6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b700: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3039 .| -- used 0.009 │ │ │ │ -0000b720: 3030 3139 7320 2863 7075 293b 2030 2e30 0019s (cpu); 0.0 │ │ │ │ -0000b730: 3039 3030 3330 3773 2028 7468 7265 6164 0900307s (thread │ │ │ │ -0000b740: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ +0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3131 .| -- used 0.011 │ │ │ │ +0000b720: 3235 3636 7320 2863 7075 293b 2030 2e30 2566s (cpu); 0.0 │ │ │ │ +0000b730: 3131 3236 3138 7320 2874 6872 6561 6429 112618s (thread) │ │ │ │ +0000b740: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0000b750: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b760: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000b770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b7a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b7b0: 0a7c 6f32 3020 3a20 4d75 6c74 6968 6f6d .|o20 : Multihom │ │ │ │ @@ -2958,18 +2958,18 @@ │ │ │ │ 0000b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000b8f0: 0a7c 6932 3120 3a20 7469 6d65 2064 6567 .|i21 : time deg │ │ │ │ 0000b900: 7265 6573 2066 2020 2020 2020 2020 2020 rees f │ │ │ │ 0000b910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b930: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000b940: 0a7c 202d 2d20 7573 6564 2031 2e33 3532 .| -- used 1.352 │ │ │ │ -0000b950: 3837 7320 2863 7075 293b 2030 2e39 3137 87s (cpu); 0.917 │ │ │ │ -0000b960: 3339 3273 2028 7468 7265 6164 293b 2030 392s (thread); 0 │ │ │ │ -0000b970: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ +0000b940: 0a7c 202d 2d20 7573 6564 2031 2e33 3739 .| -- used 1.379 │ │ │ │ +0000b950: 3134 7320 2863 7075 293b 2031 2e30 3338 14s (cpu); 1.038 │ │ │ │ +0000b960: 3533 7320 2874 6872 6561 6429 3b20 3073 53s (thread); 0s │ │ │ │ +0000b970: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0000b980: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b990: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b9d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b9e0: 0a7c 6f32 3120 3d20 7b39 3034 2c20 3530 .|o21 = {904, 50 │ │ │ │ @@ -2993,17 +2993,17 @@ │ │ │ │ 0000bb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000bb20: 0a7c 6932 3220 3a20 7469 6d65 2064 6567 .|i22 : time deg │ │ │ │ 0000bb30: 7265 6520 6620 2020 2020 2020 2020 2020 ree f │ │ │ │ 0000bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bb60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000bb70: 0a7c 202d 2d20 7573 6564 2031 2e36 3835 .| -- used 1.685 │ │ │ │ -0000bb80: 3265 2d30 3573 2028 6370 7529 3b20 312e 2e-05s (cpu); 1. │ │ │ │ -0000bb90: 3635 3331 652d 3035 7320 2874 6872 6561 6531e-05s (threa │ │ │ │ +0000bb70: 0a7c 202d 2d20 7573 6564 2032 2e31 3833 .| -- used 2.183 │ │ │ │ +0000bb80: 3865 2d30 3573 2028 6370 7529 3b20 322e 8e-05s (cpu); 2. │ │ │ │ +0000bb90: 3039 3935 652d 3035 7320 2874 6872 6561 0995e-05s (threa │ │ │ │ 0000bba0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0000bbb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000bbc0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000bbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bc00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -3019,17 +3019,17 @@ │ │ │ │ 0000bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000bcb0: 0a7c 6932 3320 3a20 7469 6d65 2064 6573 .|i23 : time des │ │ │ │ 0000bcc0: 6372 6962 6520 6620 2020 2020 2020 2020 cribe f │ │ │ │ 0000bcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bcf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000bd00: 0a7c 202d 2d20 7573 6564 2030 2e30 3031 .| -- used 0.001 │ │ │ │ -0000bd10: 3530 3573 2028 6370 7529 3b20 302e 3030 505s (cpu); 0.00 │ │ │ │ -0000bd20: 3135 3036 3273 2028 7468 7265 6164 293b 15062s (thread); │ │ │ │ -0000bd30: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ +0000bd10: 3936 3637 3873 2028 6370 7529 3b20 302e 96678s (cpu); 0. │ │ │ │ +0000bd20: 3030 3139 3732 3634 7320 2874 6872 6561 00197264s (threa │ │ │ │ +0000bd30: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0000bd40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000bd50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000bd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bd90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000bda0: 0a7c 6f32 3320 3d20 7261 7469 6f6e 616c .|o23 = rational │ │ │ │ @@ -4676,16 +4676,16 @@ │ │ │ │ 00012430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012440: 2b0a 7c69 3420 3a20 7469 6d65 2070 7369 +.|i4 : time psi │ │ │ │ 00012450: 203d 2061 6273 7472 6163 7452 6174 696f = abstractRatio │ │ │ │ 00012460: 6e61 6c4d 6170 2850 342c 5035 2c66 2920 nalMap(P4,P5,f) │ │ │ │ 00012470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012490: 7c0a 7c20 2d2d 2075 7365 6420 302e 3030 |.| -- used 0.00 │ │ │ │ -000124a0: 3034 3430 3636 3673 2028 6370 7529 3b20 0440666s (cpu); │ │ │ │ -000124b0: 302e 3030 3034 3335 3230 3673 2028 7468 0.000435206s (th │ │ │ │ +000124a0: 3035 3634 3530 3573 2028 6370 7529 3b20 0564505s (cpu); │ │ │ │ +000124b0: 302e 3030 3035 3631 3730 3373 2028 7468 0.000561703s (th │ │ │ │ 000124c0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 000124d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000124e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000124f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4746,18 +4746,18 @@ │ │ │ │ 00012890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000128a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000128b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000128c0: 6935 203a 2074 696d 6520 7072 6f6a 6563 i5 : time projec │ │ │ │ 000128d0: 7469 7665 4465 6772 6565 7328 7073 692c tiveDegrees(psi, │ │ │ │ 000128e0: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ 000128f0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00012900: 2d20 7573 6564 2030 2e31 3134 3533 3973 - used 0.114539s │ │ │ │ -00012910: 2028 6370 7529 3b20 302e 3131 3435 7320 (cpu); 0.1145s │ │ │ │ -00012920: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00012930: 2920 2020 2020 2020 207c 0a7c 2020 2020 ) |.| │ │ │ │ +00012900: 2d20 7573 6564 2030 2e31 3438 3938 3273 - used 0.148982s │ │ │ │ +00012910: 2028 6370 7529 3b20 302e 3134 3839 3973 (cpu); 0.14899s │ │ │ │ +00012920: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +00012930: 6329 2020 2020 2020 207c 0a7c 2020 2020 c) |.| │ │ │ │ 00012940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012970: 2020 2020 2020 207c 0a7c 6f35 203d 2032 |.|o5 = 2 │ │ │ │ 00012980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000129a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4766,17 +4766,17 @@ │ │ │ │ 000129d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000129e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000129f0: 2d2d 2d2b 0a7c 6936 203a 2074 696d 6520 ---+.|i6 : time │ │ │ │ 00012a00: 7261 7469 6f6e 616c 4d61 7020 7073 6920 rationalMap psi │ │ │ │ 00012a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012a30: 207c 0a7c 202d 2d20 7573 6564 2030 2e35 |.| -- used 0.5 │ │ │ │ -00012a40: 3633 3936 3273 2028 6370 7529 3b20 302e 63962s (cpu); 0. │ │ │ │ -00012a50: 3339 3730 3837 7320 2874 6872 6561 6429 397087s (thread) │ │ │ │ -00012a60: 3b20 3073 2028 6763 2920 2020 2020 207c ; 0s (gc) | │ │ │ │ +00012a40: 3133 3138 7320 2863 7075 293b 2030 2e34 1318s (cpu); 0.4 │ │ │ │ +00012a50: 3039 3330 3173 2028 7468 7265 6164 293b 09301s (thread); │ │ │ │ +00012a60: 2030 7320 2867 6329 2020 2020 2020 207c 0s (gc) | │ │ │ │ 00012a70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00012a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012aa0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00012ab0: 6f36 203d 202d 2d20 7261 7469 6f6e 616c o6 = -- rational │ │ │ │ 00012ac0: 206d 6170 202d 2d20 2020 2020 2020 2020 map -- │ │ │ │ 00012ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5189,17 +5189,17 @@ │ │ │ │ 00014440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014450: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a --------+.|i14 : │ │ │ │ 00014460: 2074 696d 6520 5420 3d20 6162 7374 7261 time T = abstra │ │ │ │ 00014470: 6374 5261 7469 6f6e 616c 4d61 7028 492c ctRationalMap(I, │ │ │ │ 00014480: 224f 4144 5022 2920 2020 2020 2020 2020 "OADP") │ │ │ │ 00014490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144a0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -000144b0: 6420 302e 3033 3731 3834 3373 2028 6370 d 0.0371843s (cp │ │ │ │ -000144c0: 7529 3b20 302e 3033 3731 3833 3773 2028 u); 0.0371837s ( │ │ │ │ -000144d0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +000144b0: 6420 302e 3139 3735 3934 7320 2863 7075 d 0.197594s (cpu │ │ │ │ +000144c0: 293b 2030 2e31 3137 3637 3273 2028 7468 ); 0.117672s (th │ │ │ │ +000144d0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 000144e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00014500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014540: 2020 7c0a 7c6f 3134 203d 202d 2d20 7261 |.|o14 = -- ra │ │ │ │ @@ -5265,16 +5265,16 @@ │ │ │ │ 00014900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014930: 2d2d 2b0a 7c69 3135 203a 2074 696d 6520 --+.|i15 : time │ │ │ │ 00014940: 7072 6f6a 6563 7469 7665 4465 6772 6565 projectiveDegree │ │ │ │ 00014950: 7328 542c 3229 2020 2020 2020 2020 2020 s(T,2) │ │ │ │ 00014960: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00014970: 7365 6420 322e 3038 3431 3773 2028 6370 sed 2.08417s (cp │ │ │ │ -00014980: 7529 3b20 312e 3433 3139 3273 2028 7468 u); 1.43192s (th │ │ │ │ +00014970: 7365 6420 322e 3533 3839 3573 2028 6370 sed 2.53895s (cp │ │ │ │ +00014980: 7529 3b20 312e 3932 3936 3973 2028 7468 u); 1.92969s (th │ │ │ │ 00014990: 7265 6164 293b 2030 7320 2867 6329 7c0a read); 0s (gc)|. │ │ │ │ 000149a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000149b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149d0: 2020 2020 7c0a 7c6f 3135 203d 2033 2020 |.|o15 = 3 │ │ │ │ 000149e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5291,18 +5291,18 @@ │ │ │ │ 00014aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ 00014ad0: 3620 3a20 7469 6d65 2054 3220 3d20 5420 6 : time T2 = T │ │ │ │ 00014ae0: 2a20 5420 2020 2020 2020 2020 2020 2020 * T │ │ │ │ 00014af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b00: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00014b10: 7365 6420 342e 3737 3765 2d30 3573 2028 sed 4.777e-05s ( │ │ │ │ -00014b20: 6370 7529 3b20 342e 3735 3839 652d 3035 cpu); 4.7589e-05 │ │ │ │ -00014b30: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -00014b40: 6763 2920 207c 0a7c 2020 2020 2020 2020 gc) |.| │ │ │ │ +00014b10: 7365 6420 342e 3738 3231 652d 3035 7320 sed 4.7821e-05s │ │ │ │ +00014b20: 2863 7075 293b 2034 2e36 3339 3365 2d30 (cpu); 4.6393e-0 │ │ │ │ +00014b30: 3573 2028 7468 7265 6164 293b 2030 7320 5s (thread); 0s │ │ │ │ +00014b40: 2867 6329 207c 0a7c 2020 2020 2020 2020 (gc) |.| │ │ │ │ 00014b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b80: 2020 7c0a 7c6f 3136 203d 202d 2d20 7261 |.|o16 = -- ra │ │ │ │ 00014b90: 7469 6f6e 616c 206d 6170 202d 2d20 2020 tional map -- │ │ │ │ 00014ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014bb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -5344,17 +5344,17 @@ │ │ │ │ 00014df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014e20: 2d2b 0a7c 6931 3720 3a20 7469 6d65 2070 -+.|i17 : time p │ │ │ │ 00014e30: 726f 6a65 6374 6976 6544 6567 7265 6573 rojectiveDegrees │ │ │ │ 00014e40: 2854 322c 3229 2020 2020 2020 2020 2020 (T2,2) │ │ │ │ 00014e50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014e60: 7c20 2d2d 2075 7365 6420 332e 3336 3433 | -- used 3.3643 │ │ │ │ -00014e70: 3673 2028 6370 7529 3b20 322e 3335 3239 6s (cpu); 2.3529 │ │ │ │ -00014e80: 3373 2028 7468 7265 6164 293b 2030 7320 3s (thread); 0s │ │ │ │ +00014e60: 7c20 2d2d 2075 7365 6420 342e 3039 3235 | -- used 4.0925 │ │ │ │ +00014e70: 3373 2028 6370 7529 3b20 332e 3036 3737 3s (cpu); 3.0677 │ │ │ │ +00014e80: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ 00014e90: 2867 6329 2020 2020 2020 207c 0a7c 2020 (gc) |.| │ │ │ │ 00014ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ed0: 2020 2020 2020 2020 7c0a 7c6f 3137 203d |.|o17 = │ │ │ │ 00014ee0: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 00014ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5430,17 +5430,17 @@ │ │ │ │ 00015350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ 00015380: 3120 3a20 7469 6d65 2066 203d 2072 6174 1 : time f = rat │ │ │ │ 00015390: 696f 6e61 6c4d 6170 2054 2020 2020 2020 ionalMap T │ │ │ │ 000153a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000153b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000153c0: 0a7c 202d 2d20 7573 6564 2033 2e30 3031 .| -- used 3.001 │ │ │ │ -000153d0: 3035 7320 2863 7075 293b 2032 2e31 3437 05s (cpu); 2.147 │ │ │ │ -000153e0: 3334 7320 2874 6872 6561 6429 3b20 3073 34s (thread); 0s │ │ │ │ +000153c0: 0a7c 202d 2d20 7573 6564 2033 2e33 3939 .| -- used 3.399 │ │ │ │ +000153d0: 3637 7320 2863 7075 293b 2032 2e36 3034 67s (cpu); 2.604 │ │ │ │ +000153e0: 3937 7320 2874 6872 6561 6429 3b20 3073 97s (thread); 0s │ │ │ │ 000153f0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00015400: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00015410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015440: 2020 2020 2020 207c 0a7c 6f32 3120 3d20 |.|o21 = │ │ │ │ 00015450: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020 -- rational map │ │ │ │ @@ -6678,18 +6678,18 @@ │ │ │ │ 0001a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a160: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 0001a170: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ 0001a180: 6572 7365 4d61 703a 2073 7465 7020 3130 erseMap: step 10 │ │ │ │ 0001a190: 206f 6620 3130 2020 2020 2020 2020 2020 of 10 │ │ │ │ 0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a1b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001a1c0: 2d2d 2075 7365 6420 302e 3335 3234 3234 -- used 0.352424 │ │ │ │ -0001a1d0: 7320 2863 7075 293b 2030 2e32 3332 3233 s (cpu); 0.23223 │ │ │ │ -0001a1e0: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ -0001a1f0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0001a1c0: 2d2d 2075 7365 6420 302e 3434 3433 3137 -- used 0.444317 │ │ │ │ +0001a1d0: 7320 2863 7075 293b 2030 2e32 3836 3538 s (cpu); 0.28658 │ │ │ │ +0001a1e0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +0001a1f0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 0001a200: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001a210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a250: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 0001a260: 3320 3d20 2d2d 2072 6174 696f 6e61 6c20 3 = -- rational │ │ │ │ @@ -8043,17 +8043,17 @@ │ │ │ │ 0001f6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 0001f6c0: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ 0001f6d0: 6572 7365 4d61 703a 2073 7465 7020 3320 erseMap: step 3 │ │ │ │ 0001f6e0: 6f66 2033 2020 2020 2020 2020 2020 2020 of 3 │ │ │ │ 0001f6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f700: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001f710: 2d2d 2075 7365 6420 302e 3332 3436 3539 -- used 0.324659 │ │ │ │ -0001f720: 7320 2863 7075 293b 2030 2e31 3831 3935 s (cpu); 0.18195 │ │ │ │ -0001f730: 3573 2028 7468 7265 6164 293b 2030 7320 5s (thread); 0s │ │ │ │ +0001f710: 2d2d 2075 7365 6420 302e 3431 3937 3339 -- used 0.419739 │ │ │ │ +0001f720: 7320 2863 7075 293b 2030 2e32 3631 3838 s (cpu); 0.26188 │ │ │ │ +0001f730: 3373 2028 7468 7265 6164 293b 2030 7320 3s (thread); 0s │ │ │ │ 0001f740: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0001f750: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f7a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ @@ -10405,16 +10405,16 @@ │ │ │ │ 00028a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a50: 2020 2020 207c 0a7c 2d2d 2061 7070 726f |.|-- appro │ │ │ │ 00028a60: 7869 6d61 7465 496e 7665 7273 654d 6170 ximateInverseMap │ │ │ │ 00028a70: 3a20 7374 6570 2033 206f 6620 3320 2020 : step 3 of 3 │ │ │ │ 00028a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028aa0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00028ab0: 2032 2e35 3236 3437 7320 2863 7075 293b 2.52647s (cpu); │ │ │ │ -00028ac0: 2031 2e38 3136 3735 7320 2874 6872 6561 1.81675s (threa │ │ │ │ +00028ab0: 2032 2e33 3731 3335 7320 2863 7075 293b 2.37135s (cpu); │ │ │ │ +00028ac0: 2032 2e30 3334 3531 7320 2874 6872 6561 2.03451s (threa │ │ │ │ 00028ad0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00028ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028af0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00028b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -11710,17 +11710,17 @@ │ │ │ │ 0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dbe0: 2020 2020 207c 0a7c 4365 7274 6966 793a |.|Certify: │ │ │ │ 0002dbf0: 206f 7574 7075 7420 6365 7274 6966 6965 output certifie │ │ │ │ 0002dc00: 6421 2020 2020 2020 2020 2020 2020 2020 d! │ │ │ │ 0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc30: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0002dc40: 2033 2e36 3531 3439 7320 2863 7075 293b 3.65149s (cpu); │ │ │ │ -0002dc50: 2032 2e36 3737 3173 2028 7468 7265 6164 2.6771s (thread │ │ │ │ -0002dc60: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ +0002dc40: 2033 2e34 3533 3739 7320 2863 7075 293b 3.45379s (cpu); │ │ │ │ +0002dc50: 2032 2e39 3734 3936 7320 2874 6872 6561 2.97496s (threa │ │ │ │ +0002dc60: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0002dc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dcd0: 2020 2020 207c 0a7c 6f31 3020 3d20 2d2d |.|o10 = -- │ │ │ │ @@ -13366,17 +13366,17 @@ │ │ │ │ 00034350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00034360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00034370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00034380: 2b0a 7c69 3320 3a20 7469 6d65 2043 6865 +.|i3 : time Che │ │ │ │ 00034390: 726e 5363 6877 6172 747a 4d61 6350 6865 rnSchwartzMacPhe │ │ │ │ 000343a0: 7273 6f6e 2043 2020 2020 2020 2020 2020 rson C │ │ │ │ 000343b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000343c0: 7573 6564 2031 2e32 3335 3231 7320 2863 used 1.23521s (c │ │ │ │ -000343d0: 7075 293b 2030 2e39 3131 3537 3673 2028 pu); 0.911576s ( │ │ │ │ -000343e0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +000343c0: 7573 6564 2031 2e33 3033 3233 7320 2863 used 1.30323s (c │ │ │ │ +000343d0: 7075 293b 2031 2e30 3433 3439 7320 2874 pu); 1.04349s (t │ │ │ │ +000343e0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 000343f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00034400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034420: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00034430: 2020 2020 2034 2020 2020 2033 2020 2020 4 3 │ │ │ │ 00034440: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00034450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -13409,17 +13409,17 @@ │ │ │ │ 00034600: 4368 6572 6e53 6368 7761 7274 7a4d 6163 ChernSchwartzMac │ │ │ │ 00034610: 5068 6572 736f 6e28 432c 4365 7274 6966 Pherson(C,Certif │ │ │ │ 00034620: 793d 3e74 7275 6529 2020 2020 7c0a 7c43 y=>true) |.|C │ │ │ │ 00034630: 6572 7469 6679 3a20 6f75 7470 7574 2063 ertify: output c │ │ │ │ 00034640: 6572 7469 6669 6564 2120 2020 2020 2020 ertified! │ │ │ │ 00034650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034660: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00034670: 2031 2e32 3133 3437 7320 2863 7075 293b 1.21347s (cpu); │ │ │ │ -00034680: 2030 2e38 3331 3434 3573 2028 7468 7265 0.831445s (thre │ │ │ │ -00034690: 6164 293b 2030 7320 2867 6329 2020 7c0a ad); 0s (gc) |. │ │ │ │ +00034670: 2031 2e35 3236 3037 7320 2863 7075 293b 1.52607s (cpu); │ │ │ │ +00034680: 2031 2e30 3032 3032 7320 2874 6872 6561 1.00202s (threa │ │ │ │ +00034690: 6429 3b20 3073 2028 6763 2920 2020 7c0a d); 0s (gc) |. │ │ │ │ 000346a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000346b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000346c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000346d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000346e0: 2034 2020 2020 2033 2020 2020 2032 2020 4 3 2 │ │ │ │ 000346f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -13619,18 +13619,18 @@ │ │ │ │ 00035320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00035330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 -----------+.|i9 │ │ │ │ 00035340: 203a 2074 696d 6520 4368 6572 6e43 6c61 : time ChernCla │ │ │ │ 00035350: 7373 2047 2020 2020 2020 2020 2020 2020 ss G │ │ │ │ 00035360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035380: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00035390: 2d20 7573 6564 2030 2e32 3339 3036 3873 - used 0.239068s │ │ │ │ -000353a0: 2028 6370 7529 3b20 302e 3135 3034 3873 (cpu); 0.15048s │ │ │ │ -000353b0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -000353c0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ +00035390: 2d20 7573 6564 2030 2e31 3432 3536 3773 - used 0.142567s │ │ │ │ +000353a0: 2028 6370 7529 3b20 302e 3134 3235 3637 (cpu); 0.142567 │ │ │ │ +000353b0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +000353c0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 000353d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000353e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000353f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035420: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00035430: 2020 2020 2020 3920 2020 2020 2038 2020 9 8 │ │ │ │ @@ -13679,17 +13679,17 @@ │ │ │ │ 000356e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000356f0: 2020 2020 2020 2020 2020 207c 0a7c 4365 |.|Ce │ │ │ │ 00035700: 7274 6966 793a 206f 7574 7075 7420 6365 rtify: output ce │ │ │ │ 00035710: 7274 6966 6965 6421 2020 2020 2020 2020 rtified! │ │ │ │ 00035720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035740: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00035750: 2d20 7573 6564 2030 2e30 3130 3831 3333 - used 0.0108133 │ │ │ │ -00035760: 7320 2863 7075 293b 2030 2e30 3130 3338 s (cpu); 0.01038 │ │ │ │ -00035770: 3932 7320 2874 6872 6561 6429 3b20 3073 92s (thread); 0s │ │ │ │ +00035750: 2d20 7573 6564 2030 2e30 3236 3338 3738 - used 0.0263878 │ │ │ │ +00035760: 7320 2863 7075 293b 2030 2e30 3134 3132 s (cpu); 0.01412 │ │ │ │ +00035770: 3839 7320 2874 6872 6561 6429 3b20 3073 89s (thread); 0s │ │ │ │ 00035780: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00035790: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000357a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000357b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000357c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000357d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000357e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ @@ -16336,17 +16336,17 @@ │ │ │ │ 0003fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003fd00: 2d2b 0a7c 6935 203a 2074 696d 6520 6465 -+.|i5 : time de │ │ │ │ 0003fd10: 6772 6565 4d61 7020 7068 6920 2020 2020 greeMap phi │ │ │ │ 0003fd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fd50: 207c 0a7c 202d 2d20 7573 6564 2030 2e30 |.| -- used 0.0 │ │ │ │ -0003fd60: 3436 3430 3131 7320 2863 7075 293b 2030 464011s (cpu); 0 │ │ │ │ -0003fd70: 2e30 3436 3430 3439 7320 2874 6872 6561 .0464049s (threa │ │ │ │ -0003fd80: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +0003fd60: 3537 3630 3035 7320 2863 7075 293b 2030 576005s (cpu); 0 │ │ │ │ +0003fd70: 2e30 3537 3630 3473 2028 7468 7265 6164 .057604s (thread │ │ │ │ +0003fd80: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 0003fd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fda0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003fdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fdf0: 207c 0a7c 6f35 203d 2031 2020 2020 2020 |.|o5 = 1 │ │ │ │ @@ -17510,17 +17510,17 @@ │ │ │ │ 00044650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044670: 2d2b 0a7c 6937 203a 2074 696d 6520 6465 -+.|i7 : time de │ │ │ │ 00044680: 6772 6565 4d61 7020 7068 6927 2020 2020 greeMap phi' │ │ │ │ 00044690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000446a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000446b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000446c0: 207c 0a7c 202d 2d20 7573 6564 2031 2e33 |.| -- used 1.3 │ │ │ │ -000446d0: 3637 3237 7320 2863 7075 293b 2030 2e37 6727s (cpu); 0.7 │ │ │ │ -000446e0: 3433 3038 3873 2028 7468 7265 6164 293b 43088s (thread); │ │ │ │ +000446c0: 207c 0a7c 202d 2d20 7573 6564 2031 2e36 |.| -- used 1.6 │ │ │ │ +000446d0: 3637 3033 7320 2863 7075 293b 2030 2e38 6703s (cpu); 0.8 │ │ │ │ +000446e0: 3733 3239 3673 2028 7468 7265 6164 293b 73296s (thread); │ │ │ │ 000446f0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00044700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044710: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00044720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -18325,16 +18325,16 @@ │ │ │ │ 00047940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00047950: 2d2d 2b0a 7c69 3220 3a20 7469 6d65 2045 --+.|i2 : time E │ │ │ │ 00047960: 756c 6572 4368 6172 6163 7465 7269 7374 ulerCharacterist │ │ │ │ 00047970: 6963 2049 2020 2020 2020 2020 2020 2020 ic I │ │ │ │ 00047980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000479a0: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -000479b0: 3432 3739 3435 7320 2863 7075 293b 2030 427945s (cpu); 0 │ │ │ │ -000479c0: 2e31 3933 3936 3173 2028 7468 7265 6164 .193961s (thread │ │ │ │ +000479b0: 3534 3137 3337 7320 2863 7075 293b 2030 541737s (cpu); 0 │ │ │ │ +000479c0: 2e32 3638 3935 3573 2028 7468 7265 6164 .268955s (thread │ │ │ │ 000479d0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 000479e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000479f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00047a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -18355,16 +18355,16 @@ │ │ │ │ 00047b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047b30: 2020 7c0a 7c43 6572 7469 6679 3a20 6f75 |.|Certify: ou │ │ │ │ 00047b40: 7470 7574 2063 6572 7469 6669 6564 2120 tput certified! │ │ │ │ 00047b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047b80: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00047b90: 3031 3239 3233 3273 2028 6370 7529 3b20 0129232s (cpu); │ │ │ │ -00047ba0: 302e 3031 3232 3932 3473 2028 7468 7265 0.0122924s (thre │ │ │ │ +00047b90: 3035 3637 3232 3173 2028 6370 7529 3b20 0567221s (cpu); │ │ │ │ +00047ba0: 302e 3031 3931 3534 3773 2028 7468 7265 0.0191547s (thre │ │ │ │ 00047bb0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00047bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047bd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00047be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -19033,17 +19033,17 @@ │ │ │ │ 0004a580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0004a5a0: 3420 3a20 7469 6d65 2066 6f72 6365 496d 4 : time forceIm │ │ │ │ 0004a5b0: 6167 6528 5068 692c 6964 6561 6c20 305f age(Phi,ideal 0_ │ │ │ │ 0004a5c0: 2874 6172 6765 7420 5068 6929 2920 2020 (target Phi)) │ │ │ │ 0004a5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a5e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0004a5f0: 2d2d 2075 7365 6420 302e 3030 3036 3339 -- used 0.000639 │ │ │ │ -0004a600: 3333 3973 2028 6370 7529 3b20 302e 3030 339s (cpu); 0.00 │ │ │ │ -0004a610: 3036 3332 3032 3573 2028 7468 7265 6164 0632025s (thread │ │ │ │ +0004a5f0: 2d2d 2075 7365 6420 302e 3030 3038 3531 -- used 0.000851 │ │ │ │ +0004a600: 3539 3973 2028 6370 7529 3b20 302e 3030 599s (cpu); 0.00 │ │ │ │ +0004a610: 3038 3437 3133 3373 2028 7468 7265 6164 0847133s (thread │ │ │ │ 0004a620: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 0004a630: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0004a640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ @@ -19645,16 +19645,16 @@ │ │ │ │ 0004cbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004cbd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 -------+.|i3 : t │ │ │ │ 0004cbe0: 696d 6520 2870 312c 7032 2920 3d20 6772 ime (p1,p2) = gr │ │ │ │ 0004cbf0: 6170 6820 7068 693b 2020 2020 2020 2020 aph phi; │ │ │ │ 0004cc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004cc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004cc20: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -0004cc30: 6564 2030 2e30 3135 3037 3336 7320 2863 ed 0.0150736s (c │ │ │ │ -0004cc40: 7075 293b 2030 2e30 3134 3738 3235 7320 pu); 0.0147825s │ │ │ │ +0004cc30: 6564 2030 2e30 3330 3730 3733 7320 2863 ed 0.0307073s (c │ │ │ │ +0004cc40: 7075 293b 2030 2e30 3139 3433 3733 7320 pu); 0.0194373s │ │ │ │ 0004cc50: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 0004cc60: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0004cc70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 0004cc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004cc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004cca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004ccb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -20942,16 +20942,16 @@ │ │ │ │ 00051cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00051ce0: 2d2d 2d2d 2b0a 7c69 3920 3a20 7469 6d65 ----+.|i9 : time │ │ │ │ 00051cf0: 2067 203d 2067 7261 7068 2070 323b 2020 g = graph p2; │ │ │ │ 00051d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00051d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00051d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00051d30: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00051d40: 302e 3032 3939 3233 3173 2028 6370 7529 0.0299231s (cpu) │ │ │ │ -00051d50: 3b20 302e 3032 3936 3138 3173 2028 7468 ; 0.0296181s (th │ │ │ │ +00051d40: 302e 3034 3737 3331 3373 2028 6370 7529 0.0477313s (cpu) │ │ │ │ +00051d50: 3b20 302e 3033 3630 3036 3973 2028 7468 ; 0.0360069s (th │ │ │ │ 00051d60: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00051d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00051d80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 00051d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00051da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00051db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00051dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -21662,18 +21662,18 @@ │ │ │ │ 000549d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000549e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ 000549f0: 7469 6d65 2069 6465 616c 2070 6869 2020 time ideal phi │ │ │ │ 00054a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054a30: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00054a40: 7365 6420 302e 3030 3331 3236 3573 2028 sed 0.0031265s ( │ │ │ │ -00054a50: 6370 7529 3b20 302e 3030 3331 3233 3836 cpu); 0.00312386 │ │ │ │ -00054a60: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -00054a70: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ +00054a40: 7365 6420 302e 3030 3433 3034 3538 7320 sed 0.00430458s │ │ │ │ +00054a50: 2863 7075 293b 2030 2e30 3034 3330 3431 (cpu); 0.0043041 │ │ │ │ +00054a60: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ +00054a70: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00054a80: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00054a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054ad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00054ae0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ @@ -22297,18 +22297,18 @@ │ │ │ │ 00057180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00057190: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ 000571a0: 7469 6d65 2069 6465 616c 2070 6869 2720 time ideal phi' │ │ │ │ 000571b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000571c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000571d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000571e0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000571f0: 7365 6420 302e 3039 3231 3934 3173 2028 sed 0.0921941s ( │ │ │ │ -00057200: 6370 7529 3b20 302e 3039 3231 3937 3173 cpu); 0.0921971s │ │ │ │ -00057210: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -00057220: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ +000571f0: 7365 6420 302e 3132 3532 3037 7320 2863 sed 0.125207s (c │ │ │ │ +00057200: 7075 293b 2030 2e31 3235 3230 3973 2028 pu); 0.125209s ( │ │ │ │ +00057210: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00057220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057230: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00057240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057280: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ 00057290: 6964 6561 6c20 3120 2020 2020 2020 2020 ideal 1 │ │ │ │ @@ -24856,16 +24856,16 @@ │ │ │ │ 00061170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061180: 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 696d -----+.|i3 : tim │ │ │ │ 00061190: 6520 696e 7665 7273 6520 7068 6920 2020 e inverse phi │ │ │ │ 000611a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000611b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000611c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000611d0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000611e0: 2030 2e30 3535 3036 3938 7320 2863 7075 0.0550698s (cpu │ │ │ │ -000611f0: 293b 2030 2e30 3535 3036 3936 7320 2874 ); 0.0550696s (t │ │ │ │ +000611e0: 2030 2e30 3730 3338 3732 7320 2863 7075 0.0703872s (cpu │ │ │ │ +000611f0: 293b 2030 2e30 3730 3338 3731 7320 2874 ); 0.0703871s (t │ │ │ │ 00061200: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00061210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061220: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00061230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -27855,18 +27855,18 @@ │ │ │ │ 0006cce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ccf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ 0006cd00: 2074 696d 6520 7073 6920 3d20 696e 7665 time psi = inve │ │ │ │ 0006cd10: 7273 654d 6170 2070 6869 2020 2020 2020 rseMap phi │ │ │ │ 0006cd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd40: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -0006cd50: 7573 6564 2030 2e30 3738 3573 2028 6370 used 0.0785s (cp │ │ │ │ -0006cd60: 7529 3b20 302e 3037 3835 3031 3273 2028 u); 0.0785012s ( │ │ │ │ -0006cd70: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ -0006cd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006cd50: 7573 6564 2030 2e30 3935 3735 3036 7320 used 0.0957506s │ │ │ │ +0006cd60: 2863 7075 293b 2030 2e30 3935 3735 3038 (cpu); 0.0957508 │ │ │ │ +0006cd70: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +0006cd80: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 0006cd90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0006cda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cde0: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ 0006cdf0: 202d 2d20 7261 7469 6f6e 616c 206d 6170 -- rational map │ │ │ │ @@ -28540,16 +28540,16 @@ │ │ │ │ 0006f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006f7c0: 2d2b 0a7c 6935 203a 2074 696d 6520 7073 -+.|i5 : time ps │ │ │ │ 0006f7d0: 6920 3d20 696e 7665 7273 654d 6170 2070 i = inverseMap p │ │ │ │ 0006f7e0: 6869 2020 2020 2020 2020 2020 2020 2020 hi │ │ │ │ 0006f7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f810: 207c 0a7c 202d 2d20 7573 6564 2030 2e32 |.| -- used 0.2 │ │ │ │ -0006f820: 3838 3236 3273 2028 6370 7529 3b20 302e 88262s (cpu); 0. │ │ │ │ -0006f830: 3138 3935 3035 7320 2874 6872 6561 6429 189505s (thread) │ │ │ │ +0006f820: 3935 3636 3873 2028 6370 7529 3b20 302e 95668s (cpu); 0. │ │ │ │ +0006f830: 3230 3033 3731 7320 2874 6872 6561 6429 200371s (thread) │ │ │ │ 0006f840: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0006f850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f860: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006f870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -29536,16 +29536,16 @@ │ │ │ │ 000735f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00073600: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ 00073610: 7469 6d65 2069 7342 6972 6174 696f 6e61 time isBirationa │ │ │ │ 00073620: 6c20 7068 6920 2020 2020 2020 2020 2020 l phi │ │ │ │ 00073630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00073640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00073650: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00073660: 7365 6420 302e 3031 3939 3536 3873 2028 sed 0.0199568s ( │ │ │ │ -00073670: 6370 7529 3b20 302e 3031 3939 3534 3873 cpu); 0.0199548s │ │ │ │ +00073660: 7365 6420 302e 3032 3533 3835 3873 2028 sed 0.0253858s ( │ │ │ │ +00073670: 6370 7529 3b20 302e 3032 3533 3835 3773 cpu); 0.0253857s │ │ │ │ 00073680: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 00073690: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 000736a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000736b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000736c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000736d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000736e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -29566,18 +29566,18 @@ │ │ │ │ 000737d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000737e0: 2020 2020 2020 2020 7c0a 7c43 6572 7469 |.|Certi │ │ │ │ 000737f0: 6679 3a20 6f75 7470 7574 2063 6572 7469 fy: output certi │ │ │ │ 00073800: 6669 6564 2120 2020 2020 2020 2020 2020 fied! │ │ │ │ 00073810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00073820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00073830: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00073840: 7365 6420 302e 3031 3630 3532 3373 2028 sed 0.0160523s ( │ │ │ │ -00073850: 6370 7529 3b20 302e 3031 3534 3630 3573 cpu); 0.0154605s │ │ │ │ -00073860: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -00073870: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ +00073840: 7365 6420 302e 3033 3736 3933 7320 2863 sed 0.037693s (c │ │ │ │ +00073850: 7075 293b 2030 2e30 3137 3732 3138 7320 pu); 0.0177218s │ │ │ │ +00073860: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +00073870: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00073880: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00073890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000738a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000738b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000738c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000738d0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ 000738e0: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ @@ -29739,17 +29739,17 @@ │ │ │ │ 000742a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000742b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 000742c0: 0a7c 4365 7274 6966 793a 206f 7574 7075 .|Certify: outpu │ │ │ │ 000742d0: 7420 6365 7274 6966 6965 6421 2020 2020 t certified! │ │ │ │ 000742e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000742f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00074300: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00074310: 0a7c 202d 2d20 7573 6564 2032 2e38 3631 .| -- used 2.861 │ │ │ │ -00074320: 3839 7320 2863 7075 293b 2032 2e30 3835 89s (cpu); 2.085 │ │ │ │ -00074330: 3637 7320 2874 6872 6561 6429 3b20 3073 67s (thread); 0s │ │ │ │ +00074310: 0a7c 202d 2d20 7573 6564 2032 2e36 3938 .| -- used 2.698 │ │ │ │ +00074320: 3035 7320 2863 7075 293b 2032 2e33 3232 05s (cpu); 2.322 │ │ │ │ +00074330: 3335 7320 2874 6872 6561 6429 3b20 3073 35s (thread); 0s │ │ │ │ 00074340: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00074350: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00074360: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00074370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00074380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00074390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000743a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -30174,17 +30174,17 @@ │ │ │ │ 00075dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075de0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00075df0: 0a7c 4365 7274 6966 793a 206f 7574 7075 .|Certify: outpu │ │ │ │ 00075e00: 7420 6365 7274 6966 6965 6421 2020 2020 t certified! │ │ │ │ 00075e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075e30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00075e40: 0a7c 202d 2d20 7573 6564 2033 2e36 3337 .| -- used 3.637 │ │ │ │ -00075e50: 3631 7320 2863 7075 293b 2032 2e34 3039 61s (cpu); 2.409 │ │ │ │ -00075e60: 3634 7320 2874 6872 6561 6429 3b20 3073 64s (thread); 0s │ │ │ │ +00075e40: 0a7c 202d 2d20 7573 6564 2034 2e34 3438 .| -- used 4.448 │ │ │ │ +00075e50: 3939 7320 2863 7075 293b 2033 2e30 3838 99s (cpu); 3.088 │ │ │ │ +00075e60: 3333 7320 2874 6872 6561 6429 3b20 3073 33s (thread); 0s │ │ │ │ 00075e70: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00075e80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00075e90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00075ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075ed0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -31483,17 +31483,17 @@ │ │ │ │ 0007afa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0007afb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0007afc0: 7c69 3220 3a20 7469 6d65 206b 6572 6e65 |i2 : time kerne │ │ │ │ 0007afd0: 6c28 7068 692c 3129 2020 2020 2020 2020 l(phi,1) │ │ │ │ 0007afe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b000: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0007b010: 7c20 2d2d 2075 7365 6420 302e 3031 3737 | -- used 0.0177 │ │ │ │ -0007b020: 3131 3673 2028 6370 7529 3b20 302e 3031 116s (cpu); 0.01 │ │ │ │ -0007b030: 3737 3036 3673 2028 7468 7265 6164 293b 77066s (thread); │ │ │ │ +0007b010: 7c20 2d2d 2075 7365 6420 302e 3032 3233 | -- used 0.0223 │ │ │ │ +0007b020: 3932 3973 2028 6370 7529 3b20 302e 3032 929s (cpu); 0.02 │ │ │ │ +0007b030: 3233 3931 3273 2028 7468 7265 6164 293b 23912s (thread); │ │ │ │ 0007b040: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 0007b050: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0007b060: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0007b070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b0a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ @@ -31523,17 +31523,17 @@ │ │ │ │ 0007b220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0007b230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0007b240: 7c69 3320 3a20 7469 6d65 206b 6572 6e65 |i3 : time kerne │ │ │ │ 0007b250: 6c28 7068 692c 3229 2020 2020 2020 2020 l(phi,2) │ │ │ │ 0007b260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b280: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0007b290: 7c20 2d2d 2075 7365 6420 312e 3032 3039 | -- used 1.0209 │ │ │ │ -0007b2a0: 3373 2028 6370 7529 3b20 302e 3439 3133 3s (cpu); 0.4913 │ │ │ │ -0007b2b0: 3636 7320 2874 6872 6561 6429 3b20 3073 66s (thread); 0s │ │ │ │ +0007b290: 7c20 2d2d 2075 7365 6420 312e 3133 3535 | -- used 1.1355 │ │ │ │ +0007b2a0: 3173 2028 6370 7529 3b20 302e 3534 3535 1s (cpu); 0.5455 │ │ │ │ +0007b2b0: 3831 7320 2874 6872 6561 6429 3b20 3073 81s (thread); 0s │ │ │ │ 0007b2c0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0007b2d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0007b2e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0007b2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b320: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ @@ -32424,18 +32424,18 @@ │ │ │ │ 0007ea70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0007ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0007ea90: 7c69 3320 3a20 7469 6d65 2070 6172 616d |i3 : time param │ │ │ │ 0007eaa0: 6574 7269 7a65 204c 2020 2020 2020 2020 etrize L │ │ │ │ 0007eab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007eac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007ead0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0007eae0: 7c20 2d2d 2075 7365 6420 302e 3030 3433 | -- used 0.0043 │ │ │ │ -0007eaf0: 3437 3335 7320 2863 7075 293b 2030 2e30 4735s (cpu); 0.0 │ │ │ │ -0007eb00: 3034 3334 3230 3673 2028 7468 7265 6164 0434206s (thread │ │ │ │ -0007eb10: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ +0007eae0: 7c20 2d2d 2075 7365 6420 302e 3030 3730 | -- used 0.0070 │ │ │ │ +0007eaf0: 3732 3673 2028 6370 7529 3b20 302e 3030 726s (cpu); 0.00 │ │ │ │ +0007eb00: 3730 3733 3334 7320 2874 6872 6561 6429 707334s (thread) │ │ │ │ +0007eb10: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0007eb20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0007eb30: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0007eb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007eb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007eb70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0007eb80: 7c6f 3320 3d20 2d2d 2072 6174 696f 6e61 |o3 = -- rationa │ │ │ │ @@ -32934,17 +32934,17 @@ │ │ │ │ 00080a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00080a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00080a70: 7c69 3520 3a20 7469 6d65 2070 6172 616d |i5 : time param │ │ │ │ 00080a80: 6574 7269 7a65 2051 2020 2020 2020 2020 etrize Q │ │ │ │ 00080a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080ab0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00080ac0: 7c20 2d2d 2075 7365 6420 302e 3734 3930 | -- used 0.7490 │ │ │ │ -00080ad0: 3834 7320 2863 7075 293b 2030 2e34 3336 84s (cpu); 0.436 │ │ │ │ -00080ae0: 3336 3473 2028 7468 7265 6164 293b 2030 364s (thread); 0 │ │ │ │ +00080ac0: 7c20 2d2d 2075 7365 6420 302e 3537 3133 | -- used 0.5713 │ │ │ │ +00080ad0: 3736 7320 2863 7075 293b 2030 2e34 3337 76s (cpu); 0.437 │ │ │ │ +00080ae0: 3538 3173 2028 7468 7265 6164 293b 2030 581s (thread); 0 │ │ │ │ 00080af0: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 00080b00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00080b10: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00080b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080b50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ @@ -34395,17 +34395,17 @@ │ │ │ │ 000865a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000865b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000865c0: 6932 203a 2074 696d 6520 7020 3d20 706f i2 : time p = po │ │ │ │ 000865d0: 696e 7420 736f 7572 6365 2066 2020 2020 int source f │ │ │ │ 000865e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000865f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086600: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00086610: 202d 2d20 7573 6564 2030 2e32 3339 3037 -- used 0.23907 │ │ │ │ -00086620: 3873 2028 6370 7529 3b20 302e 3135 3134 8s (cpu); 0.1514 │ │ │ │ -00086630: 3332 7320 2874 6872 6561 6429 3b20 3073 32s (thread); 0s │ │ │ │ +00086610: 202d 2d20 7573 6564 2030 2e32 3939 3938 -- used 0.29998 │ │ │ │ +00086620: 3873 2028 6370 7529 3b20 302e 3230 3530 8s (cpu); 0.2050 │ │ │ │ +00086630: 3039 7320 2874 6872 6561 6429 3b20 3073 09s (thread); 0s │ │ │ │ 00086640: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00086650: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00086660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000866a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -34510,18 +34510,18 @@ │ │ │ │ 00086cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00086ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00086cf0: 6933 203a 2074 696d 6520 7020 3d3d 2066 i3 : time p == f │ │ │ │ 00086d00: 5e2a 2066 2070 2020 2020 2020 2020 2020 ^* f p │ │ │ │ 00086d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086d30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00086d40: 202d 2d20 7573 6564 2030 2e30 3933 3534 -- used 0.09354 │ │ │ │ -00086d50: 3235 7320 2863 7075 293b 2030 2e30 3933 25s (cpu); 0.093 │ │ │ │ -00086d60: 3535 3039 7320 2874 6872 6561 6429 3b20 5509s (thread); │ │ │ │ -00086d70: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +00086d40: 202d 2d20 7573 6564 2030 2e31 3337 3636 -- used 0.13766 │ │ │ │ +00086d50: 3173 2028 6370 7529 3b20 302e 3133 3736 1s (cpu); 0.1376 │ │ │ │ +00086d60: 3635 7320 2874 6872 6561 6429 3b20 3073 65s (thread); 0s │ │ │ │ +00086d70: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00086d80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00086d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086dd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00086de0: 6f33 203d 2074 7275 6520 2020 2020 2020 o3 = true │ │ │ │ @@ -34842,16 +34842,16 @@ │ │ │ │ 00088190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000881a0: 2020 207c 0a7c 4365 7274 6966 793a 206f |.|Certify: o │ │ │ │ 000881b0: 7574 7075 7420 6365 7274 6966 6965 6421 utput certified! │ │ │ │ 000881c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000881d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000881e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000881f0: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -00088200: 2e31 3230 3138 3673 2028 6370 7529 3b20 .120186s (cpu); │ │ │ │ -00088210: 302e 3034 3034 3038 3973 2028 7468 7265 0.0404089s (thre │ │ │ │ +00088200: 2e32 3132 3031 3273 2028 6370 7529 3b20 .212012s (cpu); │ │ │ │ +00088210: 302e 3035 3935 3736 3973 2028 7468 7265 0.0595769s (thre │ │ │ │ 00088220: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00088230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088240: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00088250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -34972,16 +34972,16 @@ │ │ │ │ 000889b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000889c0: 2020 207c 0a7c 4365 7274 6966 793a 206f |.|Certify: o │ │ │ │ 000889d0: 7574 7075 7420 6365 7274 6966 6965 6421 utput certified! │ │ │ │ 000889e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000889f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088a10: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -00088a20: 2e31 3232 3536 3773 2028 6370 7529 3b20 .122567s (cpu); │ │ │ │ -00088a30: 302e 3035 3337 3530 3773 2028 7468 7265 0.0537507s (thre │ │ │ │ +00088a20: 2e30 3331 3333 3773 2028 6370 7529 3b20 .031337s (cpu); │ │ │ │ +00088a30: 302e 3031 3432 3537 3273 2028 7468 7265 0.0142572s (thre │ │ │ │ 00088a40: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00088a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088a60: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00088a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -35296,17 +35296,17 @@ │ │ │ │ 00089df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00089e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00089e10: 2d2d 2d2b 0a7c 6937 203a 2074 696d 6520 ---+.|i7 : time │ │ │ │ 00089e20: 7072 6f6a 6563 7469 7665 4465 6772 6565 projectiveDegree │ │ │ │ 00089e30: 7320 7068 6920 2020 2020 2020 2020 2020 s phi │ │ │ │ 00089e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00089e60: 2020 207c 0a7c 202d 2d20 7573 6564 2034 |.| -- used 4 │ │ │ │ -00089e70: 2e36 3731 3765 2d30 3573 2028 6370 7529 .6717e-05s (cpu) │ │ │ │ -00089e80: 3b20 342e 3038 3436 652d 3035 7320 2874 ; 4.0846e-05s (t │ │ │ │ +00089e60: 2020 207c 0a7c 202d 2d20 7573 6564 2035 |.| -- used 5 │ │ │ │ +00089e70: 2e31 3738 3665 2d30 3573 2028 6370 7529 .1786e-05s (cpu) │ │ │ │ +00089e80: 3b20 342e 3538 3534 652d 3035 7320 2874 ; 4.5854e-05s (t │ │ │ │ 00089e90: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00089ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089eb0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00089ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -35332,17 +35332,17 @@ │ │ │ │ 0008a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0008a040: 2d2d 2d2b 0a7c 6938 203a 2074 696d 6520 ---+.|i8 : time │ │ │ │ 0008a050: 7072 6f6a 6563 7469 7665 4465 6772 6565 projectiveDegree │ │ │ │ 0008a060: 7328 7068 692c 4e75 6d44 6567 7265 6573 s(phi,NumDegrees │ │ │ │ 0008a070: 3d3e 3129 2020 2020 2020 2020 2020 2020 =>1) │ │ │ │ 0008a080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a090: 2020 207c 0a7c 202d 2d20 7573 6564 2032 |.| -- used 2 │ │ │ │ -0008a0a0: 2e31 3535 652d 3035 7320 2863 7075 293b .155e-05s (cpu); │ │ │ │ -0008a0b0: 2032 2e31 3338 652d 3035 7320 2874 6872 2.138e-05s (thr │ │ │ │ -0008a0c0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +0008a0a0: 2e35 3635 3465 2d30 3573 2028 6370 7529 .5654e-05s (cpu) │ │ │ │ +0008a0b0: 3b20 322e 3533 3838 652d 3035 7320 2874 ; 2.5388e-05s (t │ │ │ │ +0008a0c0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 0008a0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a0e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0008a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a130: 2020 207c 0a7c 6f38 203d 207b 342c 2031 |.|o8 = {4, 1 │ │ │ │ @@ -37824,17 +37824,17 @@ │ │ │ │ 00093bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00093c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00093c10: 2b0a 7c69 3420 3a20 7469 6d65 2070 6869 +.|i4 : time phi │ │ │ │ 00093c20: 2120 3b20 2020 2020 2020 2020 2020 2020 ! ; │ │ │ │ 00093c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00093c60: 7c0a 7c20 2d2d 2075 7365 6420 302e 3035 |.| -- used 0.05 │ │ │ │ -00093c70: 3337 3934 3673 2028 6370 7529 3b20 302e 37946s (cpu); 0. │ │ │ │ -00093c80: 3035 3333 3635 3573 2028 7468 7265 6164 0533655s (thread │ │ │ │ +00093c60: 7c0a 7c20 2d2d 2075 7365 6420 302e 3038 |.| -- used 0.08 │ │ │ │ +00093c70: 3239 3339 3173 2028 6370 7529 3b20 302e 29391s (cpu); 0. │ │ │ │ +00093c80: 3037 3036 3737 3573 2028 7468 7265 6164 0706775s (thread │ │ │ │ 00093c90: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 00093ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093cb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00093cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -37999,17 +37999,17 @@ │ │ │ │ 000946e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000946f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00094700: 2b0a 7c69 3920 3a20 7469 6d65 2070 6869 +.|i9 : time phi │ │ │ │ 00094710: 2120 3b20 2020 2020 2020 2020 2020 2020 ! ; │ │ │ │ 00094720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00094730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00094740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00094750: 7c0a 7c20 2d2d 2075 7365 6420 302e 3230 |.| -- used 0.20 │ │ │ │ -00094760: 3234 3739 7320 2863 7075 293b 2030 2e31 2479s (cpu); 0.1 │ │ │ │ -00094770: 3138 3939 3973 2028 7468 7265 6164 293b 18999s (thread); │ │ │ │ +00094750: 7c0a 7c20 2d2d 2075 7365 6420 302e 3330 |.| -- used 0.30 │ │ │ │ +00094760: 3534 3432 7320 2863 7075 293b 2030 2e31 5442s (cpu); 0.1 │ │ │ │ +00094770: 3334 3332 3973 2028 7468 7265 6164 293b 34329s (thread); │ │ │ │ 00094780: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00094790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000947a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000947b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000947c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000947d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000947e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -40043,17 +40043,17 @@ │ │ │ │ 0009c6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0009c6c0: 7c69 3620 3a20 7469 6d65 2070 6869 5e2a |i6 : time phi^* │ │ │ │ 0009c6d0: 2a20 7120 2020 2020 2020 2020 2020 2020 * q │ │ │ │ 0009c6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c700: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0009c710: 7c20 2d2d 2075 7365 6420 302e 3237 3038 | -- used 0.2708 │ │ │ │ -0009c720: 3437 7320 2863 7075 293b 2030 2e31 3934 47s (cpu); 0.194 │ │ │ │ -0009c730: 3731 3273 2028 7468 7265 6164 293b 2030 712s (thread); 0 │ │ │ │ +0009c710: 7c20 2d2d 2075 7365 6420 302e 3330 3634 | -- used 0.3064 │ │ │ │ +0009c720: 3636 7320 2863 7075 293b 2030 2e32 3135 66s (cpu); 0.215 │ │ │ │ +0009c730: 3430 3973 2028 7468 7265 6164 293b 2030 409s (thread); 0 │ │ │ │ 0009c740: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0009c750: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0009c760: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0009c770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c7a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ @@ -42163,18 +42163,18 @@ │ │ │ │ 000a4b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000a4b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000a4b40: 2d2b 0a7c 6933 203a 2074 696d 6520 7068 -+.|i3 : time ph │ │ │ │ 000a4b50: 6920 3d20 7261 7469 6f6e 616c 4d61 7028 i = rationalMap( │ │ │ │ 000a4b60: 562c 332c 3229 2020 2020 2020 2020 2020 V,3,2) │ │ │ │ 000a4b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000a4b90: 207c 0a7c 202d 2d20 7573 6564 2030 2e30 |.| -- used 0.0 │ │ │ │ -000a4ba0: 3938 3233 3636 7320 2863 7075 293b 2030 982366s (cpu); 0 │ │ │ │ -000a4bb0: 2e30 3938 3233 3936 7320 2874 6872 6561 .0982396s (threa │ │ │ │ -000a4bc0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +000a4b90: 207c 0a7c 202d 2d20 7573 6564 2030 2e31 |.| -- used 0.1 │ │ │ │ +000a4ba0: 3135 3833 3173 2028 6370 7529 3b20 302e 15831s (cpu); 0. │ │ │ │ +000a4bb0: 3131 3538 3239 7320 2874 6872 6561 6429 115829s (thread) │ │ │ │ +000a4bc0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 000a4bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4be0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000a4bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4c30: 207c 0a7c 6f33 203d 202d 2d20 7261 7469 |.|o3 = -- rati │ │ │ │ @@ -43696,18 +43696,18 @@ │ │ │ │ 000aaaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000aab00: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ 000aab10: 7469 6d65 2070 6869 203d 2072 6174 696f time phi = ratio │ │ │ │ 000aab20: 6e61 6c4d 6170 2044 2020 2020 2020 2020 nalMap D │ │ │ │ 000aab30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aab40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aab50: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000aab60: 7365 6420 302e 3033 3033 3836 3573 2028 sed 0.0303865s ( │ │ │ │ -000aab70: 6370 7529 3b20 302e 3033 3033 3834 3473 cpu); 0.0303844s │ │ │ │ -000aab80: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -000aab90: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ +000aab60: 7365 6420 302e 3033 3531 3834 7320 2863 sed 0.035184s (c │ │ │ │ +000aab70: 7075 293b 2030 2e30 3335 3138 3331 7320 pu); 0.0351831s │ │ │ │ +000aab80: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +000aab90: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 000aaba0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000aabb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aabc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aabd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aabe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aabf0: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ 000aac00: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020 -- rational map │ │ │ │ @@ -44706,16 +44706,16 @@ │ │ │ │ 000aea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000aea20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 --------+.|i7 : │ │ │ │ 000aea30: 7469 6d65 203f 2069 6d61 6765 2870 6869 time ? image(phi │ │ │ │ 000aea40: 2c22 4634 2229 2020 2020 2020 2020 2020 ,"F4") │ │ │ │ 000aea50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aea60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aea70: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000aea80: 7365 6420 312e 3231 3834 3573 2028 6370 sed 1.21845s (cp │ │ │ │ -000aea90: 7529 3b20 302e 3736 3831 3939 7320 2874 u); 0.768199s (t │ │ │ │ +000aea80: 7365 6420 312e 3535 3736 3673 2028 6370 sed 1.55766s (cp │ │ │ │ +000aea90: 7529 3b20 302e 3731 3735 3131 7320 2874 u); 0.717511s (t │ │ │ │ 000aeaa0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 000aeab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeac0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000aead0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46244,16 +46244,16 @@ │ │ │ │ 000b4a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b4a40: 2d2d 2d2d 2b0a 7c69 3420 3a20 7469 6d65 ----+.|i4 : time │ │ │ │ 000b4a50: 2053 6567 7265 436c 6173 7320 5820 2020 SegreClass X │ │ │ │ 000b4a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4a90: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -000b4aa0: 302e 3939 3735 3539 7320 2863 7075 293b 0.997559s (cpu); │ │ │ │ -000b4ab0: 2030 2e36 3336 3938 3773 2028 7468 7265 0.636987s (thre │ │ │ │ +000b4aa0: 302e 3833 3639 3733 7320 2863 7075 293b 0.836973s (cpu); │ │ │ │ +000b4ab0: 2030 2e35 3636 3932 3773 2028 7468 7265 0.566927s (thre │ │ │ │ 000b4ac0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 000b4ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4ae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000b4af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46299,16 +46299,16 @@ │ │ │ │ 000b4da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b4db0: 2d2d 2d2d 2b0a 7c69 3520 3a20 7469 6d65 ----+.|i5 : time │ │ │ │ 000b4dc0: 2053 6567 7265 436c 6173 7320 6c69 6674 SegreClass lift │ │ │ │ 000b4dd0: 2858 2c50 3729 2020 2020 2020 2020 2020 (X,P7) │ │ │ │ 000b4de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4e00: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -000b4e10: 302e 3539 3235 3032 7320 2863 7075 293b 0.592502s (cpu); │ │ │ │ -000b4e20: 2030 2e33 3234 3637 3573 2028 7468 7265 0.324675s (thre │ │ │ │ +000b4e10: 302e 3639 3536 3032 7320 2863 7075 293b 0.695602s (cpu); │ │ │ │ +000b4e20: 2030 2e34 3031 3330 3173 2028 7468 7265 0.401301s (thre │ │ │ │ 000b4e30: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 000b4e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4e50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000b4e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46359,16 +46359,16 @@ │ │ │ │ 000b5160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5170: 2020 2020 7c0a 7c43 6572 7469 6679 3a20 |.|Certify: │ │ │ │ 000b5180: 6f75 7470 7574 2063 6572 7469 6669 6564 output certified │ │ │ │ 000b5190: 2120 2020 2020 2020 2020 2020 2020 2020 ! │ │ │ │ 000b51a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b51b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b51c0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -000b51d0: 302e 3032 3137 3530 3273 2028 6370 7529 0.0217502s (cpu) │ │ │ │ -000b51e0: 3b20 302e 3032 3131 3839 3173 2028 7468 ; 0.0211891s (th │ │ │ │ +000b51d0: 302e 3033 3639 3639 3473 2028 6370 7529 0.0369694s (cpu) │ │ │ │ +000b51e0: 3b20 302e 3032 3531 3130 3873 2028 7468 ; 0.0251108s (th │ │ │ │ 000b51f0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 000b5200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5210: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000b5220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46419,17 +46419,17 @@ │ │ │ │ 000b5520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5530: 2020 2020 7c0a 7c43 6572 7469 6679 3a20 |.|Certify: │ │ │ │ 000b5540: 6f75 7470 7574 2063 6572 7469 6669 6564 output certified │ │ │ │ 000b5550: 2120 2020 2020 2020 2020 2020 2020 2020 ! │ │ │ │ 000b5560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5580: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -000b5590: 302e 3039 3539 3539 3473 2028 6370 7529 0.0959594s (cpu) │ │ │ │ -000b55a0: 3b20 302e 3039 3535 3935 3973 2028 7468 ; 0.0955959s (th │ │ │ │ -000b55b0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ +000b5590: 302e 3333 3836 3439 7320 2863 7075 293b 0.338649s (cpu); │ │ │ │ +000b55a0: 2030 2e31 3730 3635 3373 2028 7468 7265 0.170653s (thre │ │ │ │ +000b55b0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 000b55c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b55d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000b55e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b55f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5620: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ @@ -46535,25 +46535,25 @@ │ │ │ │ 000b5c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5c70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b5c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5cc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000b5cd0: 6f39 2020 2020 205a 5a20 2020 2020 2020 o9 ZZ │ │ │ │ +000b5cd0: 6f39 203d 2020 205a 5a20 2020 2020 2020 o9 = ZZ │ │ │ │ 000b5ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000b5d20: 203d 202d 2d2d 2d2d 2d5b 7820 2e2e 7820 = ------[x ..x │ │ │ │ -000b5d30: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +000b5d20: 202d 2d2d 2d2d 2d5b 7820 2e2e 7820 5d20 ------[x ..x ] │ │ │ │ +000b5d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000b5d70: 2020 2031 3030 3030 3320 2030 2020 2036 100003 0 6 │ │ │ │ +000b5d70: 2031 3030 3030 3320 2030 2020 2036 2020 100003 0 6 │ │ │ │ 000b5d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5db0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b5dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46570,18 +46570,18 @@ │ │ │ │ 000b5e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b5ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000b5eb0: 6931 3020 3a20 7469 6d65 2070 6869 203d i10 : time phi = │ │ │ │ 000b5ec0: 2069 6e76 6572 7365 4d61 7020 746f 4d61 inverseMap toMa │ │ │ │ 000b5ed0: 7028 6d69 6e6f 7273 2832 2c6d 6174 7269 p(minors(2,matri │ │ │ │ 000b5ee0: 787b 7b78 5f30 2c78 5f31 2c78 5f33 2c78 x{{x_0,x_1,x_3,x │ │ │ │ 000b5ef0: 5f34 2c78 5f35 7d2c 7b78 5f31 2c7c 0a7c _4,x_5},{x_1,|.| │ │ │ │ -000b5f00: 202d 2d20 7573 6564 2030 2e32 3236 3939 -- used 0.22699 │ │ │ │ -000b5f10: 3673 2028 6370 7529 3b20 302e 3130 3234 6s (cpu); 0.1024 │ │ │ │ -000b5f20: 3535 7320 2874 6872 6561 6429 3b20 3073 55s (thread); 0s │ │ │ │ -000b5f30: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +000b5f00: 202d 2d20 7573 6564 2030 2e30 3730 3031 -- used 0.07001 │ │ │ │ +000b5f10: 3934 7320 2863 7075 293b 2030 2e30 3730 94s (cpu); 0.070 │ │ │ │ +000b5f20: 3032 3034 7320 2874 6872 6561 6429 3b20 0204s (thread); │ │ │ │ +000b5f30: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 000b5f40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b5f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5f90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b5fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46775,17 +46775,17 @@ │ │ │ │ 000b6b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b6b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000b6b80: 6931 3120 3a20 7469 6d65 2053 6567 7265 i11 : time Segre │ │ │ │ 000b6b90: 436c 6173 7320 7068 6920 2020 2020 2020 Class phi │ │ │ │ 000b6ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000b6bd0: 202d 2d20 7573 6564 2030 2e31 3734 3130 -- used 0.17410 │ │ │ │ -000b6be0: 3573 2028 6370 7529 3b20 302e 3137 3431 5s (cpu); 0.1741 │ │ │ │ -000b6bf0: 3136 7320 2874 6872 6561 6429 3b20 3073 16s (thread); 0s │ │ │ │ +000b6bd0: 202d 2d20 7573 6564 2030 2e34 3434 3335 -- used 0.44435 │ │ │ │ +000b6be0: 3473 2028 6370 7529 3b20 302e 3330 3032 4s (cpu); 0.3002 │ │ │ │ +000b6bf0: 3337 7320 2874 6872 6561 6429 3b20 3073 37s (thread); 0s │ │ │ │ 000b6c00: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000b6c10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b6c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6c60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -46935,17 +46935,17 @@ │ │ │ │ 000b7560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7570: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b7580: 2020 2020 2020 7469 6d65 2053 6567 7265 time Segre │ │ │ │ 000b7590: 436c 6173 7320 4220 2020 2020 2020 2020 Class B │ │ │ │ 000b75a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b75b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b75c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000b75d0: 202d 2d20 7573 6564 2030 2e35 3134 3337 -- used 0.51437 │ │ │ │ -000b75e0: 3373 2028 6370 7529 3b20 302e 3239 3732 3s (cpu); 0.2972 │ │ │ │ -000b75f0: 3937 7320 2874 6872 6561 6429 3b20 3073 97s (thread); 0s │ │ │ │ +000b75d0: 202d 2d20 7573 6564 2030 2e35 3038 3637 -- used 0.50867 │ │ │ │ +000b75e0: 3373 2028 6370 7529 3b20 302e 3334 3034 3s (cpu); 0.3404 │ │ │ │ +000b75f0: 3636 7320 2874 6872 6561 6429 3b20 3073 66s (thread); 0s │ │ │ │ 000b7600: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000b7610: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b7620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7660: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -46995,16 +46995,16 @@ │ │ │ │ 000b7920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7930: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b7940: 2020 2020 2020 7469 6d65 2053 6567 7265 time Segre │ │ │ │ 000b7950: 436c 6173 7320 6c69 6674 2842 2c61 6d62 Class lift(B,amb │ │ │ │ 000b7960: 6965 6e74 2072 696e 6720 4229 2020 2020 ient ring B) │ │ │ │ 000b7970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7980: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000b7990: 202d 2d20 7573 6564 2031 2e35 3534 3634 -- used 1.55464 │ │ │ │ -000b79a0: 7320 2863 7075 293b 2030 2e39 3335 3431 s (cpu); 0.93541 │ │ │ │ +000b7990: 202d 2d20 7573 6564 2031 2e37 3738 3935 -- used 1.77895 │ │ │ │ +000b79a0: 7320 2863 7075 293b 2031 2e31 3030 3234 s (cpu); 1.10024 │ │ │ │ 000b79b0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 000b79c0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 000b79d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b79e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b79f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -47245,16 +47245,16 @@ │ │ │ │ 000b88c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b88d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000b88e0: 7c69 3120 3a20 7469 6d65 2061 7070 6c79 |i1 : time apply │ │ │ │ 000b88f0: 2831 2e2e 3132 2c69 202d 3e20 6465 7363 (1..12,i -> desc │ │ │ │ 000b8900: 7269 6265 2073 7065 6369 616c 4372 656d ribe specialCrem │ │ │ │ 000b8910: 6f6e 6154 7261 6e73 666f 726d 6174 696f onaTransformatio │ │ │ │ 000b8920: 6e28 692c 5a5a 2f33 3333 3129 2920 7c0a n(i,ZZ/3331)) |. │ │ │ │ -000b8930: 7c20 2d2d 2075 7365 6420 312e 3238 3830 | -- used 1.2880 │ │ │ │ -000b8940: 3473 2028 6370 7529 3b20 312e 3030 3739 4s (cpu); 1.0079 │ │ │ │ +000b8930: 7c20 2d2d 2075 7365 6420 312e 3233 3335 | -- used 1.2335 │ │ │ │ +000b8940: 3273 2028 6370 7529 3b20 312e 3132 3637 2s (cpu); 1.1267 │ │ │ │ 000b8950: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ 000b8960: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000b8970: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000b8980: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000b8990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b89a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b89b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -48068,16 +48068,16 @@ │ │ │ │ 000bbc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000bbc40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ 000bbc50: 2074 696d 6520 7370 6563 6961 6c43 7562 time specialCub │ │ │ │ 000bbc60: 6963 5472 616e 7366 6f72 6d61 7469 6f6e icTransformation │ │ │ │ 000bbc70: 2039 2020 2020 2020 2020 2020 2020 2020 9 │ │ │ │ 000bbc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbc90: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000bbca0: 7573 6564 2030 2e30 3932 3930 3032 7320 used 0.0929002s │ │ │ │ -000bbcb0: 2863 7075 293b 2030 2e30 3932 3930 3133 (cpu); 0.0929013 │ │ │ │ +000bbca0: 7573 6564 2030 2e30 3936 3633 3537 7320 used 0.0966357s │ │ │ │ +000bbcb0: 2863 7075 293b 2030 2e30 3936 3633 3331 (cpu); 0.0966331 │ │ │ │ 000bbcc0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 000bbcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbce0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 000bbcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -50518,18 +50518,18 @@ │ │ │ │ 000c5550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000c5560: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ 000c5570: 2074 696d 6520 6465 7363 7269 6265 206f time describe o │ │ │ │ 000c5580: 6f20 2020 2020 2020 2020 2020 2020 2020 o │ │ │ │ 000c5590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c55a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c55b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000c55c0: 7573 6564 2030 2e30 3139 3236 3536 7320 used 0.0192656s │ │ │ │ -000c55d0: 2863 7075 293b 2030 2e30 3139 3236 3437 (cpu); 0.0192647 │ │ │ │ -000c55e0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -000c55f0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ +000c55c0: 7573 6564 2030 2e30 3230 3634 3237 7320 used 0.0206427s │ │ │ │ +000c55d0: 2863 7075 293b 2030 2e30 3230 3634 3373 (cpu); 0.020643s │ │ │ │ +000c55e0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +000c55f0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 000c5600: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 000c5610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c5620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c5630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c5640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c5650: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ 000c5660: 2072 6174 696f 6e61 6c20 6d61 7020 6465 rational map de │ │ │ │ @@ -50732,17 +50732,17 @@ │ │ │ │ 000c62b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000c62c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ 000c62d0: 203a 2074 696d 6520 7370 6563 6961 6c51 : time specialQ │ │ │ │ 000c62e0: 7561 6472 6174 6963 5472 616e 7366 6f72 uadraticTransfor │ │ │ │ 000c62f0: 6d61 7469 6f6e 2034 2020 2020 2020 2020 mation 4 │ │ │ │ 000c6300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c6310: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -000c6320: 2d20 7573 6564 2030 2e30 3737 3338 3737 - used 0.0773877 │ │ │ │ -000c6330: 7320 2863 7075 293b 2030 2e30 3737 3338 s (cpu); 0.07738 │ │ │ │ -000c6340: 3834 7320 2874 6872 6561 6429 3b20 3073 84s (thread); 0s │ │ │ │ +000c6320: 2d20 7573 6564 2030 2e30 3831 3135 3738 - used 0.0811578 │ │ │ │ +000c6330: 7320 2863 7075 293b 2030 2e30 3831 3135 s (cpu); 0.08115 │ │ │ │ +000c6340: 3638 7320 2874 6872 6561 6429 3b20 3073 68s (thread); 0s │ │ │ │ 000c6350: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000c6360: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000c6370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c6380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c6390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c63a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c63b0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ @@ -51287,18 +51287,18 @@ │ │ │ │ 000c8560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000c8570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ 000c8580: 203a 2074 696d 6520 6465 7363 7269 6265 : time describe │ │ │ │ 000c8590: 206f 6f20 2020 2020 2020 2020 2020 2020 oo │ │ │ │ 000c85a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c85b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c85c0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -000c85d0: 2d20 7573 6564 2030 2e30 3036 3738 3637 - used 0.0067867 │ │ │ │ -000c85e0: 3873 2028 6370 7529 3b20 302e 3030 3637 8s (cpu); 0.0067 │ │ │ │ -000c85f0: 3834 3133 7320 2874 6872 6561 6429 3b20 8413s (thread); │ │ │ │ -000c8600: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +000c85d0: 2d20 7573 6564 2030 2e30 3131 3432 3632 - used 0.0114262 │ │ │ │ +000c85e0: 7320 2863 7075 293b 2030 2e30 3131 3432 s (cpu); 0.01142 │ │ │ │ +000c85f0: 3735 7320 2874 6872 6561 6429 3b20 3073 75s (thread); 0s │ │ │ │ +000c8600: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000c8610: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000c8620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c8630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c8640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c8650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c8660: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ 000c8670: 203d 2072 6174 696f 6e61 6c20 6d61 7020 = rational map │ │ │ │ @@ -52398,17 +52398,17 @@ │ │ │ │ 000ccad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000ccae0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ 000ccaf0: 7469 6d65 2070 6869 2720 3d20 7661 6c75 time phi' = valu │ │ │ │ 000ccb00: 6520 7374 723b 2020 2020 2020 2020 2020 e str; │ │ │ │ 000ccb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccb30: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -000ccb40: 302e 3032 3232 3033 3273 2028 6370 7529 0.0222032s (cpu) │ │ │ │ -000ccb50: 3b20 302e 3032 3232 3033 3173 2028 7468 ; 0.0222031s (th │ │ │ │ -000ccb60: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ +000ccb40: 302e 3032 3639 3938 3873 2028 6370 7529 0.0269988s (cpu) │ │ │ │ +000ccb50: 3b20 302e 3032 3639 3938 7320 2874 6872 ; 0.026998s (thr │ │ │ │ +000ccb60: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 000ccb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccb80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000ccb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccbc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 000ccbd0: 3420 3a20 5261 7469 6f6e 616c 4d61 7020 4 : RationalMap │ │ │ │ @@ -52422,16 +52422,16 @@ │ │ │ │ 000ccc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000ccc60: 2d2d 2d2d 2b0a 7c69 3520 3a20 7469 6d65 ----+.|i5 : time │ │ │ │ 000ccc70: 2064 6573 6372 6962 6520 7068 6927 2020 describe phi' │ │ │ │ 000ccc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cccb0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3030 |.| -- used 0.00 │ │ │ │ -000cccc0: 3533 3737 3436 7320 2863 7075 293b 2030 537746s (cpu); 0 │ │ │ │ -000cccd0: 2e30 3035 3337 3833 3273 2028 7468 7265 .00537832s (thre │ │ │ │ +000cccc0: 3637 3132 3831 7320 2863 7075 293b 2030 671281s (cpu); 0 │ │ │ │ +000cccd0: 2e30 3036 3731 3830 3173 2028 7468 7265 .00671801s (thre │ │ │ │ 000ccce0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 000cccf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000ccd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccd40: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ @@ -52488,17 +52488,17 @@ │ │ │ │ 000cd070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000cd080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 000cd090: 3620 3a20 7469 6d65 2064 6573 6372 6962 6 : time describ │ │ │ │ 000cd0a0: 6520 696e 7665 7273 6520 7068 6927 2020 e inverse phi' │ │ │ │ 000cd0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd0d0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000cd0e0: 7365 6420 302e 3030 3432 3535 3938 7320 sed 0.00425598s │ │ │ │ -000cd0f0: 2863 7075 293b 2030 2e30 3034 3235 3730 (cpu); 0.0042570 │ │ │ │ -000cd100: 3273 2028 7468 7265 6164 293b 2030 7320 2s (thread); 0s │ │ │ │ +000cd0e0: 7365 6420 302e 3030 3539 3939 3337 7320 sed 0.00599937s │ │ │ │ +000cd0f0: 2863 7075 293b 2030 2e30 3036 3030 3634 (cpu); 0.0060064 │ │ │ │ +000cd100: 3973 2028 7468 7265 6164 293b 2030 7320 9s (thread); 0s │ │ │ │ 000cd110: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000cd120: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000cd130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd170: 7c0a 7c6f 3620 3d20 7261 7469 6f6e 616c |.|o6 = rational │ │ ├── ./usr/share/info/DGAlgebras.info.gz │ │ │ ├── DGAlgebras.info │ │ │ │ @@ -3619,16 +3619,16 @@ │ │ │ │ 0000e220: 4220 3d20 4848 2042 2020 2020 2020 2020 B = HH B │ │ │ │ 0000e230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e260: 2020 2020 2020 207c 0a7c 4669 6e64 696e |.|Findin │ │ │ │ 0000e270: 6720 6561 7379 2072 656c 6174 696f 6e73 g easy relations │ │ │ │ 0000e280: 2020 2020 2020 2020 2020 203a 2020 2d2d : -- │ │ │ │ -0000e290: 2075 7365 6420 302e 3032 3037 3530 3573 used 0.0207505s │ │ │ │ -0000e2a0: 2028 6370 7529 3b20 302e 3031 3936 3937 (cpu); 0.019697 │ │ │ │ +0000e290: 2075 7365 6420 302e 3033 3531 3633 3873 used 0.0351638s │ │ │ │ +0000e2a0: 2028 6370 7529 3b20 302e 3032 3239 3638 (cpu); 0.022968 │ │ │ │ 0000e2b0: 3473 2020 2020 207c 0a7c 2020 2020 2020 4s |.| │ │ │ │ 0000e2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e300: 2020 2020 2020 207c 0a7c 6f35 203d 2048 |.|o5 = H │ │ │ │ 0000e310: 4220 2020 2020 2020 2020 2020 2020 2020 B │ │ │ │ @@ -3883,16 +3883,16 @@ │ │ │ │ 0000f2a0: 6c67 6562 7261 2843 2c47 656e 4465 6772 lgebra(C,GenDegr │ │ │ │ 0000f2b0: 6565 4c69 6d69 743d 3e34 2c52 656c 4465 eeLimit=>4,RelDe │ │ │ │ 0000f2c0: 6772 6565 4c69 6d69 743d 3e34 2920 2020 greeLimit=>4) │ │ │ │ 0000f2d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0000f2e0: 7c46 696e 6469 6e67 2065 6173 7920 7265 |Finding easy re │ │ │ │ 0000f2f0: 6c61 7469 6f6e 7320 2020 2020 2020 2020 lations │ │ │ │ 0000f300: 2020 3a20 202d 2d20 7573 6564 2030 2e30 : -- used 0.0 │ │ │ │ -0000f310: 3139 3430 3236 7320 2863 7075 293b 2030 194026s (cpu); 0 │ │ │ │ -0000f320: 2e30 3137 3831 3833 7320 2020 2020 7c0a .0178183s |. │ │ │ │ +0000f310: 3530 3934 3534 7320 2863 7075 293b 2030 509454s (cpu); 0 │ │ │ │ +0000f320: 2e30 3236 3631 3973 2020 2020 2020 7c0a .026619s |. │ │ │ │ 0000f330: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0000f340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f370: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0000f380: 7c20 2020 2020 2020 5a5a 2020 2020 2020 | ZZ │ │ │ │ 0000f390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5287,16 +5287,16 @@ │ │ │ │ 00014a60: 4b52 2020 2020 2020 2020 2020 2020 2020 KR │ │ │ │ 00014a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014a90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00014aa0: 0a7c 4669 6e64 696e 6720 6561 7379 2072 .|Finding easy r │ │ │ │ 00014ab0: 656c 6174 696f 6e73 2020 2020 2020 2020 elations │ │ │ │ 00014ac0: 2020 203a 2020 2d2d 2075 7365 6420 302e : -- used 0. │ │ │ │ -00014ad0: 3032 3039 3239 3273 2028 6370 7529 3b20 0209292s (cpu); │ │ │ │ -00014ae0: 302e 3031 3938 3330 3573 2020 2020 207c 0.0198305s | │ │ │ │ +00014ad0: 3033 3339 3736 3273 2028 6370 7529 3b20 0339762s (cpu); │ │ │ │ +00014ae0: 302e 3032 3131 3131 7320 2020 2020 207c 0.021111s | │ │ │ │ 00014af0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00014b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00014b40: 0a7c 6f37 203d 2048 4b52 2020 2020 2020 .|o7 = HKR │ │ │ │ 00014b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5402,16 +5402,16 @@ │ │ │ │ 00015190: 4820 6b6f 737a 756c 436f 6d70 6c65 7844 H koszulComplexD │ │ │ │ 000151a0: 4741 2052 2720 2020 2020 2020 2020 2020 GA R' │ │ │ │ 000151b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000151c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 000151d0: 0a7c 4669 6e64 696e 6720 6561 7379 2072 .|Finding easy r │ │ │ │ 000151e0: 656c 6174 696f 6e73 2020 2020 2020 2020 elations │ │ │ │ 000151f0: 2020 203a 2020 2d2d 2075 7365 6420 302e : -- used 0. │ │ │ │ -00015200: 3733 3831 3335 7320 2863 7075 293b 2030 738135s (cpu); 0 │ │ │ │ -00015210: 2e35 3739 3438 3673 2020 2020 2020 207c .579486s | │ │ │ │ +00015200: 3837 3133 3338 7320 2863 7075 293b 2030 871338s (cpu); 0 │ │ │ │ +00015210: 2e37 3437 3232 3173 2020 2020 2020 207c .747221s | │ │ │ │ 00015220: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00015230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015260: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00015270: 0a7c 6f31 3020 3d20 484b 5227 2020 2020 .|o10 = HKR' │ │ │ │ 00015280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7271,16 +7271,16 @@ │ │ │ │ 0001c660: 2048 4820 6720 2020 2020 2020 2020 2020 HH g │ │ │ │ 0001c670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c6a0: 2020 7c0a 7c46 696e 6469 6e67 2065 6173 |.|Finding eas │ │ │ │ 0001c6b0: 7920 7265 6c61 7469 6f6e 7320 2020 2020 y relations │ │ │ │ 0001c6c0: 2020 2020 2020 3a20 202d 2d20 7573 6564 : -- used │ │ │ │ -0001c6d0: 2030 2e30 3136 3333 3331 7320 2863 7075 0.0163331s (cpu │ │ │ │ -0001c6e0: 293b 2030 2e30 3135 3032 3637 7320 2020 ); 0.0150267s │ │ │ │ +0001c6d0: 2030 2e30 3338 3436 3338 7320 2863 7075 0.0384638s (cpu │ │ │ │ +0001c6e0: 293b 2030 2e30 3233 3935 3435 7320 2020 ); 0.0239545s │ │ │ │ 0001c6f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001c700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c740: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001c750: 2020 2020 2020 2020 2020 2020 2020 205a Z │ │ │ │ @@ -31565,17 +31565,17 @@ │ │ │ │ 0007b4c0: 484d 203d 2068 6f6d 6f6c 6f67 7920 4d20 HM = homology M │ │ │ │ 0007b4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b500: 2020 2020 2020 2020 7c0a 7c46 696e 6469 |.|Findi │ │ │ │ 0007b510: 6e67 2065 6173 7920 7265 6c61 7469 6f6e ng easy relation │ │ │ │ 0007b520: 7320 2020 2020 2020 2020 2020 3a20 202d s : - │ │ │ │ -0007b530: 2d20 7573 6564 2030 2e30 3131 3935 3038 - used 0.0119508 │ │ │ │ -0007b540: 7320 2863 7075 293b 2030 2e30 3130 3736 s (cpu); 0.01076 │ │ │ │ -0007b550: 3037 7320 2020 2020 7c0a 7c20 2020 2020 07s |.| │ │ │ │ +0007b530: 2d20 7573 6564 2030 2e30 3536 3839 3638 - used 0.0568968 │ │ │ │ +0007b540: 7320 2863 7075 293b 2030 2e30 3231 3436 s (cpu); 0.02146 │ │ │ │ +0007b550: 3533 7320 2020 2020 7c0a 7c20 2020 2020 53s |.| │ │ │ │ 0007b560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b5a0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ 0007b5b0: 636f 6b65 726e 656c 207b 302c 2030 7d20 cokernel {0, 0} │ │ │ │ 0007b5c0: 7c20 585f 3220 585f 3120 3020 2020 3020 | X_2 X_1 0 0 │ │ │ │ @@ -68914,17 +68914,17 @@ │ │ │ │ 0010d310: 3a20 4841 203d 2068 6f6d 6f6c 6f67 7941 : HA = homologyA │ │ │ │ 0010d320: 6c67 6562 7261 2041 2020 2020 2020 2020 lgebra A │ │ │ │ 0010d330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0010d340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0010d350: 2020 2020 2020 2020 2020 7c0a 7c46 696e |.|Fin │ │ │ │ 0010d360: 6469 6e67 2065 6173 7920 7265 6c61 7469 ding easy relati │ │ │ │ 0010d370: 6f6e 7320 2020 2020 2020 2020 2020 3a20 ons : │ │ │ │ -0010d380: 202d 2d20 7573 6564 2030 2e30 3139 3237 -- used 0.01927 │ │ │ │ -0010d390: 3031 7320 2863 7075 293b 2030 2e30 3137 01s (cpu); 0.017 │ │ │ │ -0010d3a0: 3034 3731 7320 2020 2020 7c0a 7c20 2020 0471s |.| │ │ │ │ +0010d380: 202d 2d20 7573 6564 2030 2e30 3836 3433 -- used 0.08643 │ │ │ │ +0010d390: 3939 7320 2863 7075 293b 2030 2e30 3238 99s (cpu); 0.028 │ │ │ │ +0010d3a0: 3836 3931 7320 2020 2020 7c0a 7c20 2020 8691s |.| │ │ │ │ 0010d3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0010d3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0010d3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0010d3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0010d3f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ 0010d400: 3d20 4841 2020 2020 2020 2020 2020 2020 = HA │ │ │ │ 0010d410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -79727,16 +79727,16 @@ │ │ │ │ 001376e0: 523b 2020 2020 2020 2020 2020 2020 2020 R; │ │ │ │ 001376f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00137700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00137710: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00137720: 0a7c 4669 6e64 696e 6720 6561 7379 2072 .|Finding easy r │ │ │ │ 00137730: 656c 6174 696f 6e73 2020 2020 2020 2020 elations │ │ │ │ 00137740: 2020 203a 2020 2d2d 2075 7365 6420 302e : -- used 0. │ │ │ │ -00137750: 3031 3638 3335 3873 2028 6370 7529 3b20 0168358s (cpu); │ │ │ │ -00137760: 302e 3031 3538 3639 3373 2020 2020 207c 0.0158693s | │ │ │ │ +00137750: 3230 3732 3336 7320 2863 7075 293b 2030 207236s (cpu); 0 │ │ │ │ +00137760: 2e30 3432 3636 3338 7320 2020 2020 207c .0426638s | │ │ │ │ 00137770: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ 00137780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00137790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001377a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001377b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ 001377c0: 0a7c 2874 6872 6561 6429 3b20 3073 2028 .|(thread); 0s ( │ │ │ │ 001377d0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ @@ -81099,16 +81099,16 @@ │ │ │ │ 0013cca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013ccb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013ccc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013ccd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0013cce0: 0a7c 4669 6e64 696e 6720 6561 7379 2072 .|Finding easy r │ │ │ │ 0013ccf0: 656c 6174 696f 6e73 2020 2020 2020 2020 elations │ │ │ │ 0013cd00: 2020 203a 2020 2d2d 2075 7365 6420 302e : -- used 0. │ │ │ │ -0013cd10: 3033 3036 3131 3573 2028 6370 7529 3b20 0306115s (cpu); │ │ │ │ -0013cd20: 302e 3032 3835 3836 3573 2020 2020 207c 0.0285865s | │ │ │ │ +0013cd10: 3331 3834 3538 7320 2863 7075 293b 2030 318458s (cpu); 0 │ │ │ │ +0013cd20: 2e30 3736 3733 3733 7320 2020 2020 207c .0767373s | │ │ │ │ 0013cd30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0013cd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013cd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013cd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013cd70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0013cd80: 0a7c 6f33 203d 2048 4120 2020 2020 2020 .|o3 = HA │ │ │ │ 0013cd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -81475,16 +81475,16 @@ │ │ │ │ 0013e420: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ 0013e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013e440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013e450: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0013e460: 0a7c 4669 6e64 696e 6720 6561 7379 2072 .|Finding easy r │ │ │ │ 0013e470: 656c 6174 696f 6e73 2020 2020 2020 2020 elations │ │ │ │ 0013e480: 2020 203a 2020 2d2d 2075 7365 6420 302e : -- used 0. │ │ │ │ -0013e490: 3031 3432 3430 3173 2028 6370 7529 3b20 0142401s (cpu); │ │ │ │ -0013e4a0: 302e 3031 3334 3739 3373 2020 2020 207c 0.0134793s | │ │ │ │ +0013e490: 3033 3031 3637 3873 2028 6370 7529 3b20 0301678s (cpu); │ │ │ │ +0013e4a0: 302e 3031 3736 3839 3573 2020 2020 207c 0.0176895s | │ │ │ │ 0013e4b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0013e4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013e4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013e4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013e4f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0013e500: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0013e510: 2020 2020 2020 2020 2020 205a 5a20 2020 ZZ │ │ │ │ @@ -81946,16 +81946,16 @@ │ │ │ │ 00140190: 6c6f 6779 2069 644d 2020 2020 2020 2020 logy idM │ │ │ │ 001401a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001401b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001401c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001401d0: 7c0a 7c46 696e 6469 6e67 2065 6173 7920 |.|Finding easy │ │ │ │ 001401e0: 7265 6c61 7469 6f6e 7320 2020 2020 2020 relations │ │ │ │ 001401f0: 2020 2020 3a20 202d 2d20 7573 6564 2030 : -- used 0 │ │ │ │ -00140200: 2e30 3139 3639 3733 7320 2863 7075 293b .0196973s (cpu); │ │ │ │ -00140210: 2030 2e30 3138 3237 3831 7320 2020 2020 0.0182781s │ │ │ │ +00140200: 2e30 3237 3034 3834 7320 2863 7075 293b .0270484s (cpu); │ │ │ │ +00140210: 2030 2e30 3133 3334 3635 7320 2020 2020 0.0133465s │ │ │ │ 00140220: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00140230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00140240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00140250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00140260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00140270: 7c0a 7c6f 3520 3d20 7b30 2c20 307d 207c |.|o5 = {0, 0} | │ │ │ │ 00140280: 2031 2030 2030 2030 207c 2020 2020 2020 1 0 0 0 | │ │ │ │ @@ -82390,16 +82390,16 @@ │ │ │ │ 00141d50: 6c6f 6779 416c 6765 6272 6128 4129 2020 logyAlgebra(A) │ │ │ │ 00141d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00141d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00141d80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00141d90: 0a7c 4669 6e64 696e 6720 6561 7379 2072 .|Finding easy r │ │ │ │ 00141da0: 656c 6174 696f 6e73 2020 2020 2020 2020 elations │ │ │ │ 00141db0: 2020 203a 2020 2d2d 2075 7365 6420 302e : -- used 0. │ │ │ │ -00141dc0: 3031 3830 3537 3973 2028 6370 7529 3b20 0180579s (cpu); │ │ │ │ -00141dd0: 302e 3031 3730 3537 3973 2020 2020 207c 0.0170579s | │ │ │ │ +00141dc0: 3231 3431 3535 7320 2863 7075 293b 2030 214155s (cpu); 0 │ │ │ │ +00141dd0: 2e30 3337 3034 3131 7320 2020 2020 207c .0370411s | │ │ │ │ 00141de0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00141df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00141e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00141e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00141e20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00141e30: 0a7c 6f34 203d 2048 4120 2020 2020 2020 .|o4 = HA │ │ │ │ 00141e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -82552,16 +82552,16 @@ │ │ │ │ 00142770: 3d20 686f 6d6f 6c6f 6779 416c 6765 6272 = homologyAlgebr │ │ │ │ 00142780: 6128 4129 2020 2020 2020 2020 2020 2020 a(A) │ │ │ │ 00142790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001427a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001427b0: 2020 2020 207c 0a7c 4669 6e64 696e 6720 |.|Finding │ │ │ │ 001427c0: 6561 7379 2072 656c 6174 696f 6e73 2020 easy relations │ │ │ │ 001427d0: 2020 2020 2020 2020 203a 2020 2d2d 2075 : -- u │ │ │ │ -001427e0: 7365 6420 302e 3039 3230 3536 3773 2028 sed 0.0920567s ( │ │ │ │ -001427f0: 6370 7529 3b20 302e 3038 3839 3739 3173 cpu); 0.0889791s │ │ │ │ +001427e0: 7365 6420 302e 3133 3330 3834 7320 2863 sed 0.133084s (c │ │ │ │ +001427f0: 7075 293b 2030 2e31 3131 3230 3973 2020 pu); 0.111209s │ │ │ │ 00142800: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00142810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00142820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00142830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00142840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00142850: 2020 2020 207c 0a7c 6f38 203d 2048 4120 |.|o8 = HA │ │ │ │ 00142860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -82957,16 +82957,16 @@ │ │ │ │ 001440c0: 203d 2068 6f6d 6f6c 6f67 7941 6c67 6562 = homologyAlgeb │ │ │ │ 001440d0: 7261 2841 2920 2020 2020 2020 2020 2020 ra(A) │ │ │ │ 001440e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001440f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00144100: 2020 2020 207c 0a7c 4669 6e64 696e 6720 |.|Finding │ │ │ │ 00144110: 6561 7379 2072 656c 6174 696f 6e73 2020 easy relations │ │ │ │ 00144120: 2020 2020 2020 2020 203a 2020 2d2d 2075 : -- u │ │ │ │ -00144130: 7365 6420 302e 3139 3735 3233 7320 2863 sed 0.197523s (c │ │ │ │ -00144140: 7075 293b 2030 2e30 3937 3432 3736 7320 pu); 0.0974276s │ │ │ │ +00144130: 7365 6420 302e 3335 3137 3739 7320 2863 sed 0.351779s (c │ │ │ │ +00144140: 7075 293b 2030 2e31 3134 3834 3773 2020 pu); 0.114847s │ │ │ │ 00144150: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00144160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00144170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00144180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00144190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001441a0: 2020 2020 207c 0a7c 6f31 3620 3d20 4841 |.|o16 = HA │ │ │ │ 001441b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -83169,16 +83169,16 @@ │ │ │ │ 00144e00: 6f6d 6f6c 6f67 7941 6c67 6562 7261 2842 omologyAlgebra(B │ │ │ │ 00144e10: 2c47 656e 4465 6772 6565 4c69 6d69 743d ,GenDegreeLimit= │ │ │ │ 00144e20: 3e37 2c52 656c 4465 6772 6565 4c69 6d69 >7,RelDegreeLimi │ │ │ │ 00144e30: 743d 3e31 3429 2020 2020 2020 2020 2020 t=>14) │ │ │ │ 00144e40: 207c 0a7c 4669 6e64 696e 6720 6561 7379 |.|Finding easy │ │ │ │ 00144e50: 2072 656c 6174 696f 6e73 2020 2020 2020 relations │ │ │ │ 00144e60: 2020 2020 203a 2020 2d2d 2075 7365 6420 : -- used │ │ │ │ -00144e70: 302e 3132 3131 3639 7320 2863 7075 293b 0.121169s (cpu); │ │ │ │ -00144e80: 2030 2e30 3439 3237 3738 7320 2020 2020 0.0492778s │ │ │ │ +00144e70: 302e 3334 3336 3435 7320 2863 7075 293b 0.343645s (cpu); │ │ │ │ +00144e80: 2030 2e30 3831 3230 3436 7320 2020 2020 0.0812046s │ │ │ │ 00144e90: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00144ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00144eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00144ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00144ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00144ee0: 207c 0a7c 6f32 3120 3d20 4842 2020 2020 |.|o21 = HB │ │ │ │ 00144ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -84014,17 +84014,17 @@ │ │ │ │ 001482d0: 3720 3a20 4820 3d20 4848 284b 5229 2020 7 : H = HH(KR) │ │ │ │ 001482e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001482f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00148300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00148310: 2020 2020 2020 2020 2020 2020 7c0a 7c46 |.|F │ │ │ │ 00148320: 696e 6469 6e67 2065 6173 7920 7265 6c61 inding easy rela │ │ │ │ 00148330: 7469 6f6e 7320 2020 2020 2020 2020 2020 tions │ │ │ │ -00148340: 3a20 202d 2d20 7573 6564 2030 2e30 3134 : -- used 0.014 │ │ │ │ -00148350: 3737 3531 7320 2863 7075 293b 2030 2e30 7751s (cpu); 0.0 │ │ │ │ -00148360: 3133 3539 7320 2020 2020 2020 7c0a 7c20 1359s |.| │ │ │ │ +00148340: 3a20 202d 2d20 7573 6564 2030 2e31 3836 : -- used 0.186 │ │ │ │ +00148350: 3836 3573 2028 6370 7529 3b20 302e 3034 865s (cpu); 0.04 │ │ │ │ +00148360: 3031 3630 3573 2020 2020 2020 7c0a 7c20 01605s |.| │ │ │ │ 00148370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00148380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00148390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001483a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001483b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 001483c0: 3720 3d20 4820 2020 2020 2020 2020 2020 7 = H │ │ │ │ 001483d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -84390,16 +84390,16 @@ │ │ │ │ 00149a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00149a60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ 00149a70: 484b 5220 3d20 4848 284b 5229 2020 2020 HKR = HH(KR) │ │ │ │ 00149a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00149a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00149aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00149ab0: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -00149ac0: 2e31 3034 3032 3873 2028 6370 7529 3b20 .104028s (cpu); │ │ │ │ -00149ad0: 302e 3130 3136 3135 7320 2874 6872 6561 0.101615s (threa │ │ │ │ +00149ac0: 2e31 3436 3739 3273 2028 6370 7529 3b20 .146792s (cpu); │ │ │ │ +00149ad0: 302e 3132 3431 3533 7320 2874 6872 6561 0.124153s (threa │ │ │ │ 00149ae0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00149af0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00149b00: 7c46 696e 6469 6e67 2065 6173 7920 7265 |Finding easy re │ │ │ │ 00149b10: 6c61 7469 6f6e 7320 2020 2020 2020 2020 lations │ │ │ │ 00149b20: 2020 3a20 2020 2020 2020 2020 2020 2020 : │ │ │ │ 00149b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00149b40: 2020 2020 2020 2020 207c 0a7c 6f36 203d |.|o6 = │ │ │ │ @@ -91683,17 +91683,17 @@ │ │ │ │ 00166220: 207a 3132 3320 3d20 6d61 7373 6579 5472 z123 = masseyTr │ │ │ │ 00166230: 6970 6c65 5072 6f64 7563 7428 4b52 2c7a ipleProduct(KR,z │ │ │ │ 00166240: 312c 7a32 2c7a 3329 2020 2020 2020 2020 1,z2,z3) │ │ │ │ 00166250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00166260: 2020 2020 2020 2020 7c0a 7c46 696e 6469 |.|Findi │ │ │ │ 00166270: 6e67 2065 6173 7920 7265 6c61 7469 6f6e ng easy relation │ │ │ │ 00166280: 7320 2020 2020 2020 2020 2020 3a20 202d s : - │ │ │ │ -00166290: 2d20 7573 6564 2030 2e36 3039 3837 3873 - used 0.609878s │ │ │ │ -001662a0: 2028 6370 7529 3b20 302e 3530 3536 3037 (cpu); 0.505607 │ │ │ │ -001662b0: 7320 2020 2020 2020 7c0a 7c20 2020 2020 s |.| │ │ │ │ +00166290: 2d20 7573 6564 2030 2e36 3636 3437 7320 - used 0.66647s │ │ │ │ +001662a0: 2863 7075 293b 2030 2e36 3532 3538 3373 (cpu); 0.652583s │ │ │ │ +001662b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 001662c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001662d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001662e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001662f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00166300: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00166310: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ 00166320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -92187,17 +92187,17 @@ │ │ │ │ 001681a0: 3a20 4820 3d20 4848 284b 5229 2020 2020 : H = HH(KR) │ │ │ │ 001681b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001681c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001681d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001681e0: 2020 2020 2020 2020 2020 7c0a 7c46 696e |.|Fin │ │ │ │ 001681f0: 6469 6e67 2065 6173 7920 7265 6c61 7469 ding easy relati │ │ │ │ 00168200: 6f6e 7320 2020 2020 2020 2020 2020 3a20 ons : │ │ │ │ -00168210: 202d 2d20 7573 6564 2030 2e31 3631 3638 -- used 0.16168 │ │ │ │ -00168220: 3173 2028 6370 7529 3b20 302e 3135 3932 1s (cpu); 0.1592 │ │ │ │ -00168230: 3735 7320 2020 2020 2020 7c0a 7c20 2020 75s |.| │ │ │ │ +00168210: 202d 2d20 7573 6564 2030 2e32 3034 3438 -- used 0.20448 │ │ │ │ +00168220: 3173 2028 6370 7529 3b20 302e 3139 3133 1s (cpu); 0.1913 │ │ │ │ +00168230: 3339 7320 2020 2020 2020 7c0a 7c20 2020 39s |.| │ │ │ │ 00168240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00168250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00168260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00168270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00168280: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ 00168290: 3d20 4820 2020 2020 2020 2020 2020 2020 = H │ │ │ │ 001682a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -98850,16 +98850,16 @@ │ │ │ │ 00182210: 4842 203d 2074 6f72 416c 6765 6272 6128 HB = torAlgebra( │ │ │ │ 00182220: 522c 532c 4765 6e44 6567 7265 654c 696d R,S,GenDegreeLim │ │ │ │ 00182230: 6974 3d3e 342c 5265 6c44 6567 7265 654c it=>4,RelDegreeL │ │ │ │ 00182240: 696d 6974 3d3e 3829 2020 2020 2020 2020 imit=>8) │ │ │ │ 00182250: 2020 2020 2020 2020 7c0a 7c46 696e 6469 |.|Findi │ │ │ │ 00182260: 6e67 2065 6173 7920 7265 6c61 7469 6f6e ng easy relation │ │ │ │ 00182270: 7320 2020 2020 2020 2020 2020 3a20 202d s : - │ │ │ │ -00182280: 2d20 7573 6564 2030 2e35 3037 3937 3873 - used 0.507978s │ │ │ │ -00182290: 2028 6370 7529 3b20 302e 3431 3133 3331 (cpu); 0.411331 │ │ │ │ +00182280: 2d20 7573 6564 2030 2e37 3436 3035 3773 - used 0.746057s │ │ │ │ +00182290: 2028 6370 7529 3b20 302e 3538 3633 3232 (cpu); 0.586322 │ │ │ │ 001822a0: 7320 2020 2020 2020 7c0a 7c20 2020 2020 s |.| │ │ │ │ 001822b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001822c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001822d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001822e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001822f0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ 00182300: 4842 2020 2020 2020 2020 2020 2020 2020 HB │ │ ├── ./usr/share/info/EdgeIdeals.info.gz │ │ │ ├── EdgeIdeals.info │ │ │ │ @@ -7842,16 +7842,16 @@ │ │ │ │ 0001ea10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea20: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001ea30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea60: 2020 7c0a 7c6f 3420 3d20 4879 7065 7247 |.|o4 = HyperG │ │ │ │ 0001ea70: 7261 7068 7b22 6564 6765 7322 203d 3e20 raph{"edges" => │ │ │ │ -0001ea80: 7b7b 632c 2064 7d2c 207b 622c 2063 7d2c {{c, d}, {b, c}, │ │ │ │ -0001ea90: 207b 612c 2065 7d7d 7d20 2020 2020 2020 {a, e}}} │ │ │ │ +0001ea80: 7b7b 622c 2063 7d2c 207b 612c 2065 7d2c {{b, c}, {a, e}, │ │ │ │ +0001ea90: 207b 632c 2064 7d7d 7d20 2020 2020 2020 {c, d}}} │ │ │ │ 0001eaa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0001eab0: 2020 2020 2020 2022 7269 6e67 2220 3d3e "ring" => │ │ │ │ 0001eac0: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 0001ead0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eae0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0001eaf0: 2020 2020 2020 2020 2022 7665 7274 6963 "vertic │ │ │ │ 0001eb00: 6573 2220 3d3e 207b 612c 2062 2c20 632c es" => {a, b, c, │ │ │ │ @@ -21467,17 +21467,17 @@ │ │ │ │ 00053da0: 2020 207c 0a7c 6f34 203d 2048 7970 6572 |.|o4 = Hyper │ │ │ │ 00053db0: 4772 6170 687b 2265 6467 6573 2220 3d3e Graph{"edges" => │ │ │ │ 00053dc0: 207b 7b78 202c 2078 202c 2078 207d 2c20 {{x , x , x }, │ │ │ │ 00053dd0: 7b78 202c 2078 207d 2c20 7b78 202c 2078 {x , x }, {x , x │ │ │ │ 00053de0: 202c 2078 202c 2078 207d 7d7d 2020 2020 , x , x }}} │ │ │ │ 00053df0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00053e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053e10: 2020 2020 3120 2020 3320 2020 3420 2020 1 3 4 │ │ │ │ -00053e20: 2020 3320 2020 3520 2020 2020 3120 2020 3 5 1 │ │ │ │ -00053e30: 3220 2020 3420 2020 3520 2020 2020 2020 2 4 5 │ │ │ │ +00053e10: 2020 2020 3320 2020 3420 2020 3520 2020 3 4 5 │ │ │ │ +00053e20: 2020 3220 2020 3420 2020 2020 3120 2020 2 4 1 │ │ │ │ +00053e30: 3220 2020 3320 2020 3520 2020 2020 2020 2 3 5 │ │ │ │ 00053e40: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00053e50: 2020 2020 2020 2272 696e 6722 203d 3e20 "ring" => │ │ │ │ 00053e60: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 00053e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053e90: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00053ea0: 2020 2020 2020 2276 6572 7469 6365 7322 "vertices" │ │ │ │ @@ -22868,17 +22868,17 @@ │ │ │ │ 00059530: 2020 207c 0a7c 6f33 203d 2047 7261 7068 |.|o3 = Graph │ │ │ │ 00059540: 7b22 6564 6765 7322 203d 3e20 7b7b 7820 {"edges" => {{x │ │ │ │ 00059550: 2c20 7820 7d2c 207b 7820 2c20 7820 7d2c , x }, {x , x }, │ │ │ │ 00059560: 207b 7820 2c20 7820 7d2c 207b 7820 2c20 {x , x }, {x , │ │ │ │ 00059570: 7820 7d2c 207b 7820 2c20 7820 7d7d 7d20 x }, {x , x }}} │ │ │ │ 00059580: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00059590: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ -000595a0: 2020 2032 2020 2020 2032 2020 2033 2020 2 2 3 │ │ │ │ -000595b0: 2020 2033 2020 2034 2020 2020 2034 2020 3 4 4 │ │ │ │ -000595c0: 2035 2020 2020 2035 2020 2036 2020 2020 5 5 6 │ │ │ │ +000595a0: 2020 2032 2020 2020 2033 2020 2034 2020 2 3 4 │ │ │ │ +000595b0: 2020 2034 2020 2035 2020 2020 2031 2020 4 5 1 │ │ │ │ +000595c0: 2036 2020 2020 2035 2020 2036 2020 2020 6 5 6 │ │ │ │ 000595d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000595e0: 2022 7269 6e67 2220 3d3e 2052 2020 2020 "ring" => R │ │ │ │ 000595f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059620: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00059630: 2022 7665 7274 6963 6573 2220 3d3e 207b "vertices" => { │ │ │ │ @@ -22978,16 +22978,16 @@ │ │ │ │ 00059c10: 2020 207c 0a7c 6f37 203d 2047 7261 7068 |.|o7 = Graph │ │ │ │ 00059c20: 7b22 6564 6765 7322 203d 3e20 7b7b 7820 {"edges" => {{x │ │ │ │ 00059c30: 2c20 7820 7d2c 207b 7820 2c20 7820 7d2c , x }, {x , x }, │ │ │ │ 00059c40: 207b 7820 2c20 7820 7d2c 207b 7820 2c20 {x , x }, {x , │ │ │ │ 00059c50: 7820 7d7d 7d20 2020 2020 2020 2020 2020 x }}} │ │ │ │ 00059c60: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00059c70: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ -00059c80: 2020 2032 2020 2020 2031 2020 2033 2020 2 1 3 │ │ │ │ -00059c90: 2020 2034 2020 2035 2020 2020 2034 2020 4 5 4 │ │ │ │ +00059c80: 2020 2033 2020 2020 2032 2020 2033 2020 3 2 3 │ │ │ │ +00059c90: 2020 2034 2020 2036 2020 2020 2035 2020 4 6 5 │ │ │ │ 00059ca0: 2036 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ 00059cb0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00059cc0: 2022 7269 6e67 2220 3d3e 2052 2020 2020 "ring" => R │ │ │ │ 00059cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059d00: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ ├── ./usr/share/info/EigenSolver.info.gz │ │ │ ├── EigenSolver.info │ │ │ │ @@ -171,15 +171,15 @@ │ │ │ │ 00000aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000ac0: 2b0a 7c69 3320 3a20 656c 6170 7365 6454 +.|i3 : elapsedT │ │ │ │ 00000ad0: 696d 6520 736f 6c73 203d 207a 6572 6f44 ime sols = zeroD │ │ │ │ 00000ae0: 696d 536f 6c76 6520 493b 2020 2020 2020 imSolve I; │ │ │ │ 00000af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00000b10: 7c0a 7c20 2d2d 202e 3236 3431 3239 7320 |.| -- .264129s │ │ │ │ +00000b10: 7c0a 7c20 2d2d 202e 3235 3730 3736 7320 |.| -- .257076s │ │ │ │ 00000b20: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 00000b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000b60: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00000b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ ├── ./usr/share/info/Elimination.info.gz │ │ │ ├── Elimination.info │ │ │ │ @@ -336,16 +336,16 @@ │ │ │ │ 000014f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00001520: 6934 203a 2074 696d 6520 656c 696d 696e i4 : time elimin │ │ │ │ 00001530: 6174 6528 782c 6964 6561 6c28 662c 6729 ate(x,ideal(f,g) │ │ │ │ 00001540: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00001550: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00001560: 7573 6564 2030 2e30 3032 3432 3735 7320 used 0.0024275s │ │ │ │ -00001570: 2863 7075 293b 2030 2e30 3032 3432 3432 (cpu); 0.0024242 │ │ │ │ +00001560: 7573 6564 2030 2e30 3032 3738 3931 7320 used 0.0027891s │ │ │ │ +00001570: 2863 7075 293b 2030 2e30 3032 3738 3830 (cpu); 0.0027880 │ │ │ │ 00001580: 3273 2028 7468 7265 6164 293b 2030 7320 2s (thread); 0s │ │ │ │ 00001590: 2867 6329 207c 0a7c 2020 2020 2020 2020 (gc) |.| │ │ │ │ 000015a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000015b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000015c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000015d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000015e0: 2020 2020 2020 2020 2020 3220 2020 2032 2 2 │ │ │ │ @@ -366,17 +366,17 @@ │ │ │ │ 000016d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000016e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000016f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00001700: 6935 203a 2074 696d 6520 6964 6561 6c20 i5 : time ideal │ │ │ │ 00001710: 7265 7375 6c74 616e 7428 662c 672c 7829 resultant(f,g,x) │ │ │ │ 00001720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001730: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00001740: 7573 6564 2030 2e30 3031 3439 3535 3973 used 0.00149559s │ │ │ │ -00001750: 2028 6370 7529 3b20 302e 3030 3134 3936 (cpu); 0.001496 │ │ │ │ -00001760: 3333 7320 2874 6872 6561 6429 3b20 3073 33s (thread); 0s │ │ │ │ +00001740: 7573 6564 2030 2e30 3031 3638 3832 3573 used 0.00168825s │ │ │ │ +00001750: 2028 6370 7529 3b20 302e 3030 3136 3839 (cpu); 0.001689 │ │ │ │ +00001760: 3833 7320 2874 6872 6561 6429 3b20 3073 83s (thread); 0s │ │ │ │ 00001770: 2028 6763 297c 0a7c 2020 2020 2020 2020 (gc)|.| │ │ │ │ 00001780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000017a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000017b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000017c0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ 000017d0: 2032 2020 2020 2020 2020 2020 2020 2032 2 2 │ │ │ │ @@ -620,17 +620,17 @@ │ │ │ │ 000026b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000026c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000026d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000026e0: 2b0a 7c69 3420 3a20 7469 6d65 2065 6c69 +.|i4 : time eli │ │ │ │ 000026f0: 6d69 6e61 7465 2878 2c69 6465 616c 2866 minate(x,ideal(f │ │ │ │ 00002700: 2c67 2929 2020 2020 2020 2020 2020 2020 ,g)) │ │ │ │ 00002710: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00002720: 2d2d 2075 7365 6420 302e 3030 3235 3030 -- used 0.002500 │ │ │ │ -00002730: 3834 7320 2863 7075 293b 2030 2e30 3032 84s (cpu); 0.002 │ │ │ │ -00002740: 3439 3735 3573 2028 7468 7265 6164 293b 49755s (thread); │ │ │ │ +00002720: 2d2d 2075 7365 6420 302e 3030 3332 3230 -- used 0.003220 │ │ │ │ +00002730: 3534 7320 2863 7075 293b 2030 2e30 3033 54s (cpu); 0.003 │ │ │ │ +00002740: 3231 3739 3773 2028 7468 7265 6164 293b 21797s (thread); │ │ │ │ 00002750: 2030 7320 2867 6329 7c0a 7c20 2020 2020 0s (gc)|.| │ │ │ │ 00002760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002790: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000027a0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ 000027b0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ @@ -650,17 +650,17 @@ │ │ │ │ 00002890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000028a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000028b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000028c0: 2b0a 7c69 3520 3a20 7469 6d65 2069 6465 +.|i5 : time ide │ │ │ │ 000028d0: 616c 2072 6573 756c 7461 6e74 2866 2c67 al resultant(f,g │ │ │ │ 000028e0: 2c78 2920 2020 2020 2020 2020 2020 2020 ,x) │ │ │ │ 000028f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00002900: 2d2d 2075 7365 6420 302e 3030 3135 3137 -- used 0.001517 │ │ │ │ -00002910: 3637 7320 2863 7075 293b 2030 2e30 3031 67s (cpu); 0.001 │ │ │ │ -00002920: 3531 3833 3873 2028 7468 7265 6164 293b 51838s (thread); │ │ │ │ +00002900: 2d2d 2075 7365 6420 302e 3030 3138 3832 -- used 0.001882 │ │ │ │ +00002910: 3138 7320 2863 7075 293b 2030 2e30 3031 18s (cpu); 0.001 │ │ │ │ +00002920: 3838 3434 3773 2028 7468 7265 6164 293b 88447s (thread); │ │ │ │ 00002930: 2030 7320 2867 6329 7c0a 7c20 2020 2020 0s (gc)|.| │ │ │ │ 00002940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002970: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00002980: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ 00002990: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ @@ -995,16 +995,16 @@ │ │ │ │ 00003e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003e30: 2d2d 2b0a 7c69 3420 3a20 7469 6d65 2065 --+.|i4 : time e │ │ │ │ 00003e40: 6c69 6d69 6e61 7465 2869 6465 616c 2866 liminate(ideal(f │ │ │ │ 00003e50: 2c67 292c 7829 2020 2020 2020 2020 2020 ,g),x) │ │ │ │ 00003e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003e80: 2020 7c0a 7c20 2d2d 2075 7365 6420 312e |.| -- used 1. │ │ │ │ -00003e90: 3732 3031 3673 2028 6370 7529 3b20 312e 72016s (cpu); 1. │ │ │ │ -00003ea0: 3338 3833 3273 2028 7468 7265 6164 293b 38832s (thread); │ │ │ │ +00003e90: 3636 3133 3473 2028 6370 7529 3b20 312e 66134s (cpu); 1. │ │ │ │ +00003ea0: 3338 3539 3773 2028 7468 7265 6164 293b 38597s (thread); │ │ │ │ 00003eb0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00003ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003ed0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00003ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1275,16 +1275,16 @@ │ │ │ │ 00004fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004fb0: 2d2d 2b0a 7c69 3520 3a20 7469 6d65 2069 --+.|i5 : time i │ │ │ │ 00004fc0: 6465 616c 2072 6573 756c 7461 6e74 2866 deal resultant(f │ │ │ │ 00004fd0: 2c67 2c78 2920 2020 2020 2020 2020 2020 ,g,x) │ │ │ │ 00004fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005000: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00005010: 3031 3733 3535 3873 2028 6370 7529 3b20 0173558s (cpu); │ │ │ │ -00005020: 302e 3031 3733 3537 3373 2028 7468 7265 0.0173573s (thre │ │ │ │ +00005010: 3031 3732 3535 3273 2028 6370 7529 3b20 0172552s (cpu); │ │ │ │ +00005020: 302e 3031 3732 3536 3773 2028 7468 7265 0.0172567s (thre │ │ │ │ 00005030: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00005040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005050: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00005060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1917,16 +1917,16 @@ │ │ │ │ 000077c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000077d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ 000077e0: 3a20 7469 6d65 2065 6c69 6d69 6e61 7465 : time eliminate │ │ │ │ 000077f0: 2869 6465 616c 2866 2c67 292c 7829 2020 (ideal(f,g),x) │ │ │ │ 00007800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007820: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00007830: 2075 7365 6420 312e 3639 3234 3673 2028 used 1.69246s ( │ │ │ │ -00007840: 6370 7529 3b20 312e 3430 3839 3173 2028 cpu); 1.40891s ( │ │ │ │ +00007830: 2075 7365 6420 312e 3635 3933 3873 2028 used 1.65938s ( │ │ │ │ +00007840: 6370 7529 3b20 312e 3435 3836 3773 2028 cpu); 1.45867s ( │ │ │ │ 00007850: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00007860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007870: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00007880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000078a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000078b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2197,18 +2197,18 @@ │ │ │ │ 00008940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008950: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ 00008960: 3a20 7469 6d65 2069 6465 616c 2072 6573 : time ideal res │ │ │ │ 00008970: 756c 7461 6e74 2866 2c67 2c78 2920 2020 ultant(f,g,x) │ │ │ │ 00008980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000089a0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -000089b0: 2075 7365 6420 302e 3031 3734 3430 3173 used 0.0174401s │ │ │ │ -000089c0: 2028 6370 7529 3b20 302e 3031 3734 3433 (cpu); 0.017443 │ │ │ │ -000089d0: 3373 2028 7468 7265 6164 293b 2030 7320 3s (thread); 0s │ │ │ │ -000089e0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +000089b0: 2075 7365 6420 302e 3134 3638 3638 7320 used 0.146868s │ │ │ │ +000089c0: 2863 7075 293b 2030 2e30 3438 3038 3835 (cpu); 0.0480885 │ │ │ │ +000089d0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +000089e0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 000089f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00008a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008a40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00008a50: 2020 2020 2020 2020 2020 2037 2020 2020 7 │ │ ├── ./usr/share/info/EnumerationCurves.info.gz │ │ │ ├── EnumerationCurves.info │ │ │ │ @@ -256,16 +256,16 @@ │ │ │ │ 00000ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001000: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7469 ------+.|i1 : ti │ │ │ │ 00001010: 6d65 2066 6f72 206e 2066 726f 6d20 3220 me for n from 2 │ │ │ │ 00001020: 746f 2031 3020 6c69 7374 206c 696e 6573 to 10 list lines │ │ │ │ 00001030: 4879 7065 7273 7572 6661 6365 286e 2920 Hypersurface(n) │ │ │ │ 00001040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001050: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00001060: 6420 302e 3032 3839 3534 3173 2028 6370 d 0.0289541s (cp │ │ │ │ -00001070: 7529 3b20 302e 3032 3839 3534 3373 2028 u); 0.0289543s ( │ │ │ │ +00001060: 6420 302e 3032 3932 3434 3573 2028 6370 d 0.0292445s (cp │ │ │ │ +00001070: 7529 3b20 302e 3032 3932 3435 3273 2028 u); 0.0292452s ( │ │ │ │ 00001080: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00001090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000010a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000010b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000010c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000010d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000010e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -649,17 +649,17 @@ │ │ │ │ 00002880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000028a0: 2d2d 2b0a 7c69 3720 3a20 7469 6d65 2066 --+.|i7 : time f │ │ │ │ 000028b0: 6f72 2044 2069 6e20 5420 6c69 7374 2072 or D in T list r │ │ │ │ 000028c0: 6174 696f 6e61 6c43 7572 7665 2832 2c44 ationalCurve(2,D │ │ │ │ 000028d0: 2920 2d20 7261 7469 6f6e 616c 4375 7276 ) - rationalCurv │ │ │ │ 000028e0: 6528 312c 4429 2f38 7c0a 7c20 2d2d 2075 e(1,D)/8|.| -- u │ │ │ │ -000028f0: 7365 6420 302e 3334 3635 3638 7320 2863 sed 0.346568s (c │ │ │ │ -00002900: 7075 293b 2030 2e32 3830 3831 3173 2028 pu); 0.280811s ( │ │ │ │ -00002910: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +000028f0: 7365 6420 302e 3335 3733 3673 2028 6370 sed 0.35736s (cp │ │ │ │ +00002900: 7529 3b20 302e 3239 3931 3436 7320 2874 u); 0.299146s (t │ │ │ │ +00002910: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00002920: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00002930: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00002940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002970: 2020 2020 7c0a 7c6f 3720 3d20 7b36 3039 |.|o7 = {609 │ │ │ │ 00002980: 3235 302c 2039 3232 3838 2c20 3532 3831 250, 92288, 5281 │ │ │ │ @@ -685,16 +685,16 @@ │ │ │ │ 00002ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00002af0: 3820 3a20 7469 6d65 2072 6174 696f 6e61 8 : time rationa │ │ │ │ 00002b00: 6c43 7572 7665 2833 2920 2020 2020 2020 lCurve(3) │ │ │ │ 00002b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002b20: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00002b30: 6564 2030 2e31 3335 3734 3473 2028 6370 ed 0.135744s (cp │ │ │ │ -00002b40: 7529 3b20 302e 3133 3537 3438 7320 2874 u); 0.135748s (t │ │ │ │ +00002b30: 6564 2030 2e31 3339 3436 3373 2028 6370 ed 0.139463s (cp │ │ │ │ +00002b40: 7529 3b20 302e 3133 3932 3431 7320 2874 u); 0.139241s (t │ │ │ │ 00002b50: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00002b60: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00002b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002b90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00002ba0: 2020 2020 2038 3536 3435 3735 3030 3020 8564575000 │ │ │ │ 00002bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -718,16 +718,16 @@ │ │ │ │ 00002cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00002d00: 0a7c 6939 203a 2074 696d 6520 666f 7220 .|i9 : time for │ │ │ │ 00002d10: 4420 696e 2054 206c 6973 7420 7261 7469 D in T list rati │ │ │ │ 00002d20: 6f6e 616c 4375 7276 6528 332c 4429 2020 onalCurve(3,D) │ │ │ │ 00002d30: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00002d40: 2075 7365 6420 352e 3633 3737 3773 2028 used 5.63777s ( │ │ │ │ -00002d50: 6370 7529 3b20 342e 3636 3937 3973 2028 cpu); 4.66979s ( │ │ │ │ +00002d40: 2075 7365 6420 352e 3139 3832 3773 2028 used 5.19827s ( │ │ │ │ +00002d50: 6370 7529 3b20 342e 3631 3837 3573 2028 cpu); 4.61875s ( │ │ │ │ 00002d60: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00002d70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00002d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002db0: 7c0a 7c20 2020 2020 2038 3536 3435 3735 |.| 8564575 │ │ │ │ 00002dc0: 3030 3020 2034 3232 3639 3038 3136 2020 000 422690816 │ │ │ │ @@ -757,274 +757,275 @@ │ │ │ │ 00002f40: 6e20 6120 6765 6e65 7261 6c20 7175 696e n a general quin │ │ │ │ 00002f50: 7469 6320 7468 7265 6566 6f6c 6420 6361 tic threefold ca │ │ │ │ 00002f60: 6e20 6265 0a63 6f6d 7075 7465 6420 6173 n be.computed as │ │ │ │ 00002f70: 2066 6f6c 6c6f 7773 3a0a 0a0a 0a2b 2d2d follows:....+-- │ │ │ │ 00002f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00002fb0: 2d2d 2b0a 7c69 3130 203a 2074 696d 6520 --+.|i10 : time │ │ │ │ -00002fc0: 7261 7469 6f6e 616c 4375 7276 6528 3329 rationalCurve(3) │ │ │ │ -00002fd0: 202d 2072 6174 696f 6e61 6c43 7572 7665 - rationalCurve │ │ │ │ -00002fe0: 2831 292f 3237 2020 207c 0a7c 202d 2d20 (1)/27 |.| -- │ │ │ │ -00002ff0: 7573 6564 2030 2e31 3335 3335 3773 2028 used 0.135357s ( │ │ │ │ -00003000: 6370 7529 3b20 302e 3133 3533 3673 2028 cpu); 0.13536s ( │ │ │ │ -00003010: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ -00003020: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00002fb0: 2d2d 2d2b 0a7c 6931 3020 3a20 7469 6d65 ---+.|i10 : time │ │ │ │ +00002fc0: 2072 6174 696f 6e61 6c43 7572 7665 2833 rationalCurve(3 │ │ │ │ +00002fd0: 2920 2d20 7261 7469 6f6e 616c 4375 7276 ) - rationalCurv │ │ │ │ +00002fe0: 6528 3129 2f32 3720 2020 207c 0a7c 202d e(1)/27 |.| - │ │ │ │ +00002ff0: 2d20 7573 6564 2030 2e31 3435 3236 3273 - used 0.145262s │ │ │ │ +00003000: 2028 6370 7529 3b20 302e 3134 3532 3639 (cpu); 0.145269 │ │ │ │ +00003010: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +00003020: 6763 297c 0a7c 2020 2020 2020 2020 2020 gc)|.| │ │ │ │ 00003030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003050: 2020 2020 2020 207c 0a7c 6f31 3020 3d20 |.|o10 = │ │ │ │ -00003060: 3331 3732 3036 3337 3520 2020 2020 2020 317206375 │ │ │ │ +00003050: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +00003060: 3020 3d20 3331 3732 3036 3337 3520 2020 0 = 317206375 │ │ │ │ 00003070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003080: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00003090: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00003080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003090: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000030a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000030b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000030c0: 2020 2020 207c 0a7c 6f31 3020 3a20 5151 |.|o10 : QQ │ │ │ │ -000030d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000030c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +000030d0: 3020 3a20 5151 2020 2020 2020 2020 2020 0 : QQ │ │ │ │ 000030e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000030f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00003100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000030f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003100: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00003110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003130: 2d2d 2d2b 0a0a 5468 6520 6e75 6d62 6572 ---+..The number │ │ │ │ -00003140: 7320 6f66 2072 6174 696f 6e61 6c20 6375 s of rational cu │ │ │ │ -00003150: 7276 6573 206f 6620 6465 6772 6565 2033 rves of degree 3 │ │ │ │ -00003160: 206f 6e20 6765 6e65 7261 6c20 636f 6d70 on general comp │ │ │ │ -00003170: 6c65 7465 2069 6e74 6572 7365 6374 696f lete intersectio │ │ │ │ -00003180: 6e0a 4361 6c61 6269 2d59 6175 2074 6872 n.Calabi-Yau thr │ │ │ │ -00003190: 6565 666f 6c64 7320 6361 6e20 6265 2063 eefolds can be c │ │ │ │ -000031a0: 6f6d 7075 7465 6420 6173 2066 6f6c 6c6f omputed as follo │ │ │ │ -000031b0: 7773 3a0a 0a0a 0a2b 2d2d 2d2d 2d2d 2d2d ws:....+-------- │ │ │ │ +00003130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468 -----------+..Th │ │ │ │ +00003140: 6520 6e75 6d62 6572 7320 6f66 2072 6174 e numbers of rat │ │ │ │ +00003150: 696f 6e61 6c20 6375 7276 6573 206f 6620 ional curves of │ │ │ │ +00003160: 6465 6772 6565 2033 206f 6e20 6765 6e65 degree 3 on gene │ │ │ │ +00003170: 7261 6c20 636f 6d70 6c65 7465 2069 6e74 ral complete int │ │ │ │ +00003180: 6572 7365 6374 696f 6e0a 4361 6c61 6269 ersection.Calabi │ │ │ │ +00003190: 2d59 6175 2074 6872 6565 666f 6c64 7320 -Yau threefolds │ │ │ │ +000031a0: 6361 6e20 6265 2063 6f6d 7075 7465 6420 can be computed │ │ │ │ +000031b0: 6173 2066 6f6c 6c6f 7773 3a0a 0a0a 0a2b as follows:....+ │ │ │ │ 000031c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000031d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000031e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000031f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00003200: 6931 3120 3a20 7469 6d65 2066 6f72 2044 i11 : time for D │ │ │ │ -00003210: 2069 6e20 5420 6c69 7374 2072 6174 696f in T list ratio │ │ │ │ -00003220: 6e61 6c43 7572 7665 2833 2c44 2920 2d20 nalCurve(3,D) - │ │ │ │ -00003230: 7261 7469 6f6e 616c 4375 7276 6528 312c rationalCurve(1, │ │ │ │ -00003240: 4429 2f32 377c 0a7c 202d 2d20 7573 6564 D)/27|.| -- used │ │ │ │ -00003250: 2035 2e34 3638 3732 7320 2863 7075 293b 5.46872s (cpu); │ │ │ │ -00003260: 2034 2e35 3938 3633 7320 2874 6872 6561 4.59863s (threa │ │ │ │ -00003270: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ -00003280: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00003290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000031f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003200: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 7469 -----+.|i11 : ti │ │ │ │ +00003210: 6d65 2066 6f72 2044 2069 6e20 5420 6c69 me for D in T li │ │ │ │ +00003220: 7374 2072 6174 696f 6e61 6c43 7572 7665 st rationalCurve │ │ │ │ +00003230: 2833 2c44 2920 2d20 7261 7469 6f6e 616c (3,D) - rational │ │ │ │ +00003240: 4375 7276 6528 312c 4429 2f32 377c 0a7c Curve(1,D)/27|.| │ │ │ │ +00003250: 202d 2d20 7573 6564 2035 2e33 3434 3538 -- used 5.34458 │ │ │ │ +00003260: 7320 2863 7075 293b 2034 2e36 3639 3431 s (cpu); 4.66941 │ │ │ │ +00003270: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +00003280: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ +00003290: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000032a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000032b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000032c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000032d0: 2020 2020 207c 0a7c 6f31 3120 3d20 7b33 |.|o11 = {3 │ │ │ │ -000032e0: 3137 3230 3633 3735 2c20 3135 3635 3531 17206375, 156551 │ │ │ │ -000032f0: 3638 2c20 3634 3234 3332 362c 2031 3631 68, 6424326, 161 │ │ │ │ -00003300: 3135 3034 2c20 3431 3632 3536 7d20 2020 1504, 416256} │ │ │ │ -00003310: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00003320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000032d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000032e0: 6f31 3120 3d20 7b33 3137 3230 3633 3735 o11 = {317206375 │ │ │ │ +000032f0: 2c20 3135 3635 3531 3638 2c20 3634 3234 , 15655168, 6424 │ │ │ │ +00003300: 3332 362c 2031 3631 3135 3034 2c20 3431 326, 1611504, 41 │ │ │ │ +00003310: 3632 3536 7d20 2020 2020 2020 2020 2020 6256} │ │ │ │ +00003320: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00003330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003360: 2020 2020 207c 0a7c 6f31 3120 3a20 4c69 |.|o11 : Li │ │ │ │ -00003370: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ +00003360: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00003370: 6f31 3120 3a20 4c69 7374 2020 2020 2020 o11 : List │ │ │ │ 00003380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000033a0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -000033b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000033a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000033b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 000033c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000033d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000033e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000033f0: 2d2d 2d2d 2d2b 0a0a 466f 7220 7261 7469 -----+..For rati │ │ │ │ -00003400: 6f6e 616c 2063 7572 7665 7320 6f66 2064 onal curves of d │ │ │ │ -00003410: 6567 7265 6520 343a 0a0a 0a0a 2b2d 2d2d egree 4:....+--- │ │ │ │ -00003420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000033f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +00003400: 466f 7220 7261 7469 6f6e 616c 2063 7572 For rational cur │ │ │ │ +00003410: 7665 7320 6f66 2064 6567 7265 6520 343a ves of degree 4: │ │ │ │ +00003420: 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d ....+----------- │ │ │ │ 00003430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003450: 2b0a 7c69 3132 203a 2074 696d 6520 7261 +.|i12 : time ra │ │ │ │ -00003460: 7469 6f6e 616c 4375 7276 6528 3429 2020 tionalCurve(4) │ │ │ │ -00003470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003480: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00003490: 6420 312e 3731 3939 3673 2028 6370 7529 d 1.71996s (cpu) │ │ │ │ -000034a0: 3b20 312e 3434 3437 3373 2028 7468 7265 ; 1.44473s (thre │ │ │ │ -000034b0: 6164 293b 2030 7320 2867 6329 7c0a 7c20 ad); 0s (gc)|.| │ │ │ │ -000034c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003450: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a --------+.|i12 : │ │ │ │ +00003460: 2074 696d 6520 7261 7469 6f6e 616c 4375 time rationalCu │ │ │ │ +00003470: 7276 6528 3429 2020 2020 2020 2020 2020 rve(4) │ │ │ │ +00003480: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00003490: 7c20 2d2d 2075 7365 6420 312e 3633 3136 | -- used 1.6316 │ │ │ │ +000034a0: 3273 2028 6370 7529 3b20 312e 3434 3030 2s (cpu); 1.4400 │ │ │ │ +000034b0: 3173 2028 7468 7265 6164 293b 2030 7320 1s (thread); 0s │ │ │ │ +000034c0: 2867 6329 7c0a 7c20 2020 2020 2020 2020 (gc)|.| │ │ │ │ 000034d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000034e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000034f0: 2020 7c0a 7c20 2020 2020 2031 3535 3137 |.| 15517 │ │ │ │ -00003500: 3932 3637 3936 3837 3520 2020 2020 2020 926796875 │ │ │ │ -00003510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003520: 2020 2020 2020 2020 7c0a 7c6f 3132 203d |.|o12 = │ │ │ │ -00003530: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d20 -------------- │ │ │ │ -00003540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003550: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00003560: 7c20 2020 2020 2020 2020 2020 2036 3420 | 64 │ │ │ │ -00003570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000034f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00003500: 2020 2031 3535 3137 3932 3637 3936 3837 1551792679687 │ │ │ │ +00003510: 3520 2020 2020 2020 2020 2020 2020 2020 5 │ │ │ │ +00003520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003530: 7c0a 7c6f 3132 203d 202d 2d2d 2d2d 2d2d |.|o12 = ------- │ │ │ │ +00003540: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020 ------- │ │ │ │ +00003550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003560: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00003570: 2020 2020 2036 3420 2020 2020 2020 2020 64 │ │ │ │ 00003580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003590: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00003590: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000035a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000035b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000035c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3132 |.|o12 │ │ │ │ -000035d0: 203a 2051 5120 2020 2020 2020 2020 2020 : QQ │ │ │ │ +000035c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000035d0: 2020 7c0a 7c6f 3132 203a 2051 5120 2020 |.|o12 : QQ │ │ │ │ 000035e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000035f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003600: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00003600: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 00003610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003630: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2074 ------+.|i13 : t │ │ │ │ -00003640: 696d 6520 7261 7469 6f6e 616c 4375 7276 ime rationalCurv │ │ │ │ -00003650: 6528 342c 7b34 2c32 7d29 2020 2020 2020 e(4,{4,2}) │ │ │ │ -00003660: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00003670: 2d2d 2075 7365 6420 372e 3838 3834 3773 -- used 7.88847s │ │ │ │ -00003680: 2028 6370 7529 3b20 352e 3832 3236 3473 (cpu); 5.82264s │ │ │ │ -00003690: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -000036a0: 6329 7c0a 7c20 2020 2020 2020 2020 2020 c)|.| │ │ │ │ +00003630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00003640: 7c69 3133 203a 2074 696d 6520 7261 7469 |i13 : time rati │ │ │ │ +00003650: 6f6e 616c 4375 7276 6528 342c 7b34 2c32 onalCurve(4,{4,2 │ │ │ │ +00003660: 7d29 2020 2020 2020 2020 2020 2020 2020 }) │ │ │ │ +00003670: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ +00003680: 372e 3034 3737 3173 2028 6370 7529 3b20 7.04771s (cpu); │ │ │ │ +00003690: 352e 3739 3536 3473 2028 7468 7265 6164 5.79564s (thread │ │ │ │ +000036a0: 293b 2030 7320 2867 6329 7c0a 7c20 2020 ); 0s (gc)|.| │ │ │ │ 000036b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000036c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000036d0: 2020 2020 2020 2020 7c0a 7c6f 3133 203d |.|o13 = │ │ │ │ -000036e0: 2033 3838 3339 3134 3038 3420 2020 2020 3883914084 │ │ │ │ -000036f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003700: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00003710: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000036d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000036e0: 7c0a 7c6f 3133 203d 2033 3838 3339 3134 |.|o13 = 3883914 │ │ │ │ +000036f0: 3038 3420 2020 2020 2020 2020 2020 2020 084 │ │ │ │ +00003700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003710: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00003720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003740: 2020 2020 7c0a 7c6f 3133 203a 2051 5120 |.|o13 : QQ │ │ │ │ -00003750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003740: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00003750: 3133 203a 2051 5120 2020 2020 2020 2020 13 : QQ │ │ │ │ 00003760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003770: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00003780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003780: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00003790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000037a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000037b0: 2b0a 0a54 6865 206e 756d 6265 7220 6f66 +..The number of │ │ │ │ -000037c0: 2072 6174 696f 6e61 6c20 6375 7276 6573 rational curves │ │ │ │ -000037d0: 206f 6620 6465 6772 6565 2034 206f 6e20 of degree 4 on │ │ │ │ -000037e0: 6120 6765 6e65 7261 6c20 7175 696e 7469 a general quinti │ │ │ │ -000037f0: 6320 7468 7265 6566 6f6c 6420 6361 6e20 c threefold can │ │ │ │ -00003800: 6265 0a63 6f6d 7075 7465 6420 6173 2066 be.computed as f │ │ │ │ -00003810: 6f6c 6c6f 7773 3a0a 0a0a 0a2b 2d2d 2d2d ollows:....+---- │ │ │ │ -00003820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000037b0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6865 206e --------+..The n │ │ │ │ +000037c0: 756d 6265 7220 6f66 2072 6174 696f 6e61 umber of rationa │ │ │ │ +000037d0: 6c20 6375 7276 6573 206f 6620 6465 6772 l curves of degr │ │ │ │ +000037e0: 6565 2034 206f 6e20 6120 6765 6e65 7261 ee 4 on a genera │ │ │ │ +000037f0: 6c20 7175 696e 7469 6320 7468 7265 6566 l quintic threef │ │ │ │ +00003800: 6f6c 6420 6361 6e20 6265 0a63 6f6d 7075 old can be.compu │ │ │ │ +00003810: 7465 6420 6173 2066 6f6c 6c6f 7773 3a0a ted as follows:. │ │ │ │ +00003820: 0a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ 00003830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00003850: 0a7c 6931 3420 3a20 7469 6d65 2072 6174 .|i14 : time rat │ │ │ │ -00003860: 696f 6e61 6c43 7572 7665 2834 2920 2d20 ionalCurve(4) - │ │ │ │ -00003870: 7261 7469 6f6e 616c 4375 7276 6528 3229 rationalCurve(2) │ │ │ │ -00003880: 2f38 2020 207c 0a7c 202d 2d20 7573 6564 /8 |.| -- used │ │ │ │ -00003890: 2031 2e39 3531 3632 7320 2863 7075 293b 1.95162s (cpu); │ │ │ │ -000038a0: 2031 2e36 3033 3232 7320 2874 6872 6561 1.60322s (threa │ │ │ │ -000038b0: 6429 3b20 3073 2028 6763 297c 0a7c 2020 d); 0s (gc)|.| │ │ │ │ -000038c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003850: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 -------+.|i14 : │ │ │ │ +00003860: 7469 6d65 2072 6174 696f 6e61 6c43 7572 time rationalCur │ │ │ │ +00003870: 7665 2834 2920 2d20 7261 7469 6f6e 616c ve(4) - rational │ │ │ │ +00003880: 4375 7276 6528 3229 2f38 2020 207c 0a7c Curve(2)/8 |.| │ │ │ │ +00003890: 202d 2d20 7573 6564 2031 2e36 3231 3739 -- used 1.62179 │ │ │ │ +000038a0: 7320 2863 7075 293b 2031 2e34 3235 3637 s (cpu); 1.42567 │ │ │ │ +000038b0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +000038c0: 6763 297c 0a7c 2020 2020 2020 2020 2020 gc)|.| │ │ │ │ 000038d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000038e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000038f0: 207c 0a7c 6f31 3420 3d20 3234 3234 3637 |.|o14 = 242467 │ │ │ │ -00003900: 3533 3030 3030 2020 2020 2020 2020 2020 530000 │ │ │ │ +000038f0: 2020 2020 2020 2020 207c 0a7c 6f31 3420 |.|o14 │ │ │ │ +00003900: 3d20 3234 3234 3637 3533 3030 3030 2020 = 242467530000 │ │ │ │ 00003910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003920: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00003930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003920: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00003930: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00003940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003950: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00003960: 6f31 3420 3a20 5151 2020 2020 2020 2020 o14 : QQ │ │ │ │ +00003950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003960: 2020 2020 207c 0a7c 6f31 3420 3a20 5151 |.|o14 : QQ │ │ │ │ 00003970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003990: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00003990: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 000039a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000039b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000039c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468 6520 ---------+..The │ │ │ │ -000039d0: 6e75 6d62 6572 7320 6f66 2072 6174 696f numbers of ratio │ │ │ │ -000039e0: 6e61 6c20 6375 7276 6573 206f 6620 6465 nal curves of de │ │ │ │ -000039f0: 6772 6565 2034 206f 6e20 6765 6e65 7261 gree 4 on genera │ │ │ │ -00003a00: 6c20 636f 6d70 6c65 7465 2069 6e74 6572 l complete inter │ │ │ │ -00003a10: 7365 6374 696f 6e73 206f 660a 7479 7065 sections of.type │ │ │ │ -00003a20: 7320 2834 2c32 2920 616e 6420 2833 2c33 s (4,2) and (3,3 │ │ │ │ -00003a30: 2920 696e 205c 6d61 7468 6262 2050 5e35 ) in \mathbb P^5 │ │ │ │ -00003a40: 2063 616e 2062 6520 636f 6d70 7574 6564 can be computed │ │ │ │ -00003a50: 2061 7320 666f 6c6c 6f77 733a 0a0a 0a0a as follows:.... │ │ │ │ -00003a60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +000039c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000039d0: 2d2b 0a0a 5468 6520 6e75 6d62 6572 7320 -+..The numbers │ │ │ │ +000039e0: 6f66 2072 6174 696f 6e61 6c20 6375 7276 of rational curv │ │ │ │ +000039f0: 6573 206f 6620 6465 6772 6565 2034 206f es of degree 4 o │ │ │ │ +00003a00: 6e20 6765 6e65 7261 6c20 636f 6d70 6c65 n general comple │ │ │ │ +00003a10: 7465 2069 6e74 6572 7365 6374 696f 6e73 te intersections │ │ │ │ +00003a20: 206f 660a 7479 7065 7320 2834 2c32 2920 of.types (4,2) │ │ │ │ +00003a30: 616e 6420 2833 2c33 2920 696e 205c 6d61 and (3,3) in \ma │ │ │ │ +00003a40: 7468 6262 2050 5e35 2063 616e 2062 6520 thbb P^5 can be │ │ │ │ +00003a50: 636f 6d70 7574 6564 2061 7320 666f 6c6c computed as foll │ │ │ │ +00003a60: 6f77 733a 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d ows:....+------- │ │ │ │ 00003a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00003aa0: 6931 3520 3a20 7469 6d65 2072 6174 696f i15 : time ratio │ │ │ │ -00003ab0: 6e61 6c43 7572 7665 2834 2c7b 342c 327d nalCurve(4,{4,2} │ │ │ │ -00003ac0: 2920 2d20 7261 7469 6f6e 616c 4375 7276 ) - rationalCurv │ │ │ │ -00003ad0: 6528 322c 7b34 2c32 7d29 2f38 7c0a 7c20 e(2,{4,2})/8|.| │ │ │ │ -00003ae0: 2d2d 2075 7365 6420 382e 3437 3533 3873 -- used 8.47538s │ │ │ │ -00003af0: 2028 6370 7529 3b20 362e 3233 3338 3473 (cpu); 6.23384s │ │ │ │ -00003b00: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -00003b10: 6329 2020 2020 2020 2020 207c 0a7c 2020 c) |.| │ │ │ │ -00003b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003aa0: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 7469 -----+.|i15 : ti │ │ │ │ +00003ab0: 6d65 2072 6174 696f 6e61 6c43 7572 7665 me rationalCurve │ │ │ │ +00003ac0: 2834 2c7b 342c 327d 2920 2d20 7261 7469 (4,{4,2}) - rati │ │ │ │ +00003ad0: 6f6e 616c 4375 7276 6528 322c 7b34 2c32 onalCurve(2,{4,2 │ │ │ │ +00003ae0: 7d29 2f38 7c0a 7c20 2d2d 2075 7365 6420 })/8|.| -- used │ │ │ │ +00003af0: 362e 3937 3937 3973 2028 6370 7529 3b20 6.97979s (cpu); │ │ │ │ +00003b00: 352e 3831 3238 3373 2028 7468 7265 6164 5.81283s (thread │ │ │ │ +00003b10: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ +00003b20: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00003b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003b50: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ -00003b60: 203d 2033 3838 3339 3032 3532 3820 2020 = 3883902528 │ │ │ │ -00003b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003b60: 2020 7c0a 7c6f 3135 203d 2033 3838 3339 |.|o15 = 38839 │ │ │ │ +00003b70: 3032 3532 3820 2020 2020 2020 2020 2020 02528 │ │ │ │ 00003b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003b90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00003ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003ba0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00003bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003bd0: 2020 2020 2020 2020 7c0a 7c6f 3135 203a |.|o15 : │ │ │ │ -00003be0: 2051 5120 2020 2020 2020 2020 2020 2020 QQ │ │ │ │ +00003bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003be0: 7c0a 7c6f 3135 203a 2051 5120 2020 2020 |.|o15 : QQ │ │ │ │ 00003bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003c10: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00003c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003c10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00003c20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00003c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003c50: 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a 2074 ------+.|i16 : t │ │ │ │ -00003c60: 696d 6520 7261 7469 6f6e 616c 4375 7276 ime rationalCurv │ │ │ │ -00003c70: 6528 342c 7b33 2c33 7d29 202d 2072 6174 e(4,{3,3}) - rat │ │ │ │ -00003c80: 696f 6e61 6c43 7572 7665 2832 2c7b 332c ionalCurve(2,{3, │ │ │ │ -00003c90: 337d 292f 387c 0a7c 202d 2d20 7573 6564 3})/8|.| -- used │ │ │ │ -00003ca0: 2037 2e37 3935 3273 2028 6370 7529 3b20 7.7952s (cpu); │ │ │ │ -00003cb0: 352e 3931 3437 7320 2874 6872 6561 6429 5.9147s (thread) │ │ │ │ -00003cc0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ -00003cd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00003c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00003c60: 7c69 3136 203a 2074 696d 6520 7261 7469 |i16 : time rati │ │ │ │ +00003c70: 6f6e 616c 4375 7276 6528 342c 7b33 2c33 onalCurve(4,{3,3 │ │ │ │ +00003c80: 7d29 202d 2072 6174 696f 6e61 6c43 7572 }) - rationalCur │ │ │ │ +00003c90: 7665 2832 2c7b 332c 337d 292f 387c 0a7c ve(2,{3,3})/8|.| │ │ │ │ +00003ca0: 202d 2d20 7573 6564 2037 2e31 3031 3332 -- used 7.10132 │ │ │ │ +00003cb0: 7320 2863 7075 293b 2035 2e37 3835 3131 s (cpu); 5.78511 │ │ │ │ +00003cc0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +00003cd0: 6763 2920 2020 2020 2020 2020 7c0a 7c20 gc) |.| │ │ │ │ 00003ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003d10: 2020 207c 0a7c 6f31 3620 3d20 3131 3339 |.|o16 = 1139 │ │ │ │ -00003d20: 3434 3833 3834 2020 2020 2020 2020 2020 448384 │ │ │ │ +00003d10: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +00003d20: 3620 3d20 3131 3339 3434 3833 3834 2020 6 = 1139448384 │ │ │ │ 00003d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003d50: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00003d50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00003d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003d90: 207c 0a7c 6f31 3620 3a20 5151 2020 2020 |.|o16 : QQ │ │ │ │ -00003da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003d90: 2020 2020 2020 2020 207c 0a7c 6f31 3620 |.|o16 │ │ │ │ +00003da0: 3a20 5151 2020 2020 2020 2020 2020 2020 : QQ │ │ │ │ 00003db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003dd0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00003dd0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 00003de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00003e10: 0a0a 5761 7973 2074 6f20 7573 6520 7261 ..Ways to use ra │ │ │ │ -00003e20: 7469 6f6e 616c 4375 7276 653a 0a3d 3d3d tionalCurve:.=== │ │ │ │ -00003e30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00003e40: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7261 =======.. * "ra │ │ │ │ -00003e50: 7469 6f6e 616c 4375 7276 6528 5a5a 2922 tionalCurve(ZZ)" │ │ │ │ -00003e60: 0a20 202a 2022 7261 7469 6f6e 616c 4375 . * "rationalCu │ │ │ │ -00003e70: 7276 6528 5a5a 2c4c 6973 7429 220a 0a46 rve(ZZ,List)"..F │ │ │ │ -00003e80: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00003e90: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00003ea0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00003eb0: 202a 6e6f 7465 2072 6174 696f 6e61 6c43 *note rationalC │ │ │ │ -00003ec0: 7572 7665 3a20 7261 7469 6f6e 616c 4375 urve: rationalCu │ │ │ │ -00003ed0: 7276 652c 2069 7320 6120 2a6e 6f74 6520 rve, is a *note │ │ │ │ -00003ee0: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ -00003ef0: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ -00003f00: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ -00003f10: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +00003e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003e10: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 -------+..Ways t │ │ │ │ +00003e20: 6f20 7573 6520 7261 7469 6f6e 616c 4375 o use rationalCu │ │ │ │ +00003e30: 7276 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d rve:.=========== │ │ │ │ +00003e40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +00003e50: 0a20 202a 2022 7261 7469 6f6e 616c 4375 . * "rationalCu │ │ │ │ +00003e60: 7276 6528 5a5a 2922 0a20 202a 2022 7261 rve(ZZ)". * "ra │ │ │ │ +00003e70: 7469 6f6e 616c 4375 7276 6528 5a5a 2c4c tionalCurve(ZZ,L │ │ │ │ +00003e80: 6973 7429 220a 0a46 6f72 2074 6865 2070 ist)"..For the p │ │ │ │ +00003e90: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +00003ea0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +00003eb0: 6520 6f62 6a65 6374 202a 6e6f 7465 2072 e object *note r │ │ │ │ +00003ec0: 6174 696f 6e61 6c43 7572 7665 3a20 7261 ationalCurve: ra │ │ │ │ +00003ed0: 7469 6f6e 616c 4375 7276 652c 2069 7320 tionalCurve, is │ │ │ │ +00003ee0: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ +00003ef0: 756e 6374 696f 6e3a 0a28 4d61 6361 756c unction:.(Macaul │ │ │ │ +00003f00: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +00003f10: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ 00003f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003f60: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -00003f70: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -00003f80: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -00003f90: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -00003fa0: 6175 6c61 7932 2d31 2e32 362e 3035 2b64 aulay2-1.26.05+d │ │ │ │ -00003fb0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -00003fc0: 6163 6b61 6765 732f 0a45 6e75 6d65 7261 ackages/.Enumera │ │ │ │ -00003fd0: 7469 6f6e 4375 7276 6573 2e6d 323a 3938 tionCurves.m2:98 │ │ │ │ -00003fe0: 313a 302e 0a1f 0a54 6167 2054 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2075 7365 6420 2e30 3033 3436 -- used .00346 │ │ │ │ -000079d0: 3431 3720 7365 636f 6e64 7320 2020 2020 417 seconds │ │ │ │ +000079c0: 2020 2d2d 2075 7365 6420 2e30 3033 3734 -- used .00374 │ │ │ │ +000079d0: 3036 3720 7365 636f 6e64 7320 2020 2020 067 seconds │ │ │ │ 000079e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000079f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007a00: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007a10: 2020 2d2d 2075 7365 6420 2e30 3034 3639 -- used .00469 │ │ │ │ -00007a20: 3035 3820 7365 636f 6e64 7320 2020 2020 058 seconds │ │ │ │ +00007a10: 2020 2d2d 2075 7365 6420 2e30 3035 3332 -- used .00532 │ │ │ │ +00007a20: 3932 3720 7365 636f 6e64 7320 2020 2020 927 seconds │ │ │ │ 00007a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007a50: 2020 2020 2020 2020 2020 7c0a 7c28 3136 |.|(16 │ │ │ │ 00007a60: 2c20 3236 2920 2020 2020 2020 2020 2020 , 26) │ │ │ │ 00007a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1967,61 +1967,61 @@ │ │ │ │ 00007ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007af0: 2020 2020 2020 2020 2020 7c0a 7c35 2020 |.|5 │ │ │ │ 00007b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007b40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007b50: 2020 2d2d 2075 7365 6420 2e30 3038 3235 -- used .00825 │ │ │ │ -00007b60: 3539 3120 7365 636f 6e64 7320 2020 2020 591 seconds │ │ │ │ +00007b50: 2020 2d2d 2075 7365 6420 2e30 3038 3333 -- used .00833 │ │ │ │ +00007b60: 3035 3920 7365 636f 6e64 7320 2020 2020 059 seconds │ │ │ │ 00007b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007b90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007ba0: 2020 2d2d 2075 7365 6420 2e30 3234 3239 -- used .02429 │ │ │ │ -00007bb0: 3236 2073 6563 6f6e 6473 2020 2020 2020 26 seconds │ │ │ │ +00007ba0: 2020 2d2d 2075 7365 6420 2e30 3433 3539 -- used .04359 │ │ │ │ +00007bb0: 3533 2073 6563 6f6e 6473 2020 2020 2020 53 seconds │ │ │ │ 00007bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007be0: 2020 2020 2020 2020 2020 7c0a 7c28 3235 |.|(25 │ │ │ │ 00007bf0: 2c20 3630 2920 2020 2020 2020 2020 2020 , 60) │ │ │ │ 00007c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007c30: 2020 2020 2020 2020 2020 7c0a 7c36 2020 |.|6 │ │ │ │ 00007c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007c80: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007c90: 2020 2d2d 2075 7365 6420 2e30 3137 3539 -- used .01759 │ │ │ │ -00007ca0: 3936 2073 6563 6f6e 6473 2020 2020 2020 96 seconds │ │ │ │ +00007c90: 2020 2d2d 2075 7365 6420 2e30 3139 3136 -- used .01916 │ │ │ │ +00007ca0: 3036 2073 6563 6f6e 6473 2020 2020 2020 06 seconds │ │ │ │ 00007cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007cd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007ce0: 2020 2d2d 2075 7365 6420 2e32 3837 3030 -- used .28700 │ │ │ │ -00007cf0: 3220 7365 636f 6e64 7320 2020 2020 2020 2 seconds │ │ │ │ +00007ce0: 2020 2d2d 2075 7365 6420 2e33 3431 3538 -- used .34158 │ │ │ │ +00007cf0: 3920 7365 636f 6e64 7320 2020 2020 2020 9 seconds │ │ │ │ 00007d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d20: 2020 2020 2020 2020 2020 7c0a 7c28 3336 |.|(36 │ │ │ │ 00007d30: 2c20 3132 3029 2020 2020 2020 2020 2020 , 120) │ │ │ │ 00007d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d70: 2020 2020 2020 2020 2020 7c0a 7c37 2020 |.|7 │ │ │ │ 00007d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007dc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007dd0: 2020 2d2d 2075 7365 6420 2e30 3338 3132 -- used .03812 │ │ │ │ +00007dd0: 2020 2d2d 2075 7365 6420 2e30 3432 3139 -- used .04219 │ │ │ │ 00007de0: 3531 2073 6563 6f6e 6473 2020 2020 2020 51 seconds │ │ │ │ 00007df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007e20: 2020 2d2d 2075 7365 6420 2e39 3330 3633 -- used .93063 │ │ │ │ -00007e30: 3920 7365 636f 6e64 7320 2020 2020 2020 9 seconds │ │ │ │ +00007e20: 2020 2d2d 2075 7365 6420 2e39 3630 3633 -- used .96063 │ │ │ │ +00007e30: 3420 7365 636f 6e64 7320 2020 2020 2020 4 seconds │ │ │ │ 00007e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e60: 2020 2020 2020 2020 2020 7c0a 7c28 3439 |.|(49 │ │ │ │ 00007e70: 2c20 3231 3729 2020 2020 2020 2020 2020 , 217) │ │ │ │ 00007e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/FastMinors.info.gz │ │ │ ├── FastMinors.info │ │ │ │ @@ -4166,18 +4166,18 @@ │ │ │ │ 00010450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00010470: 3238 203a 2074 696d 6520 6469 6d20 284a 28 : time dim (J │ │ │ │ 00010480: 202b 2063 686f 6f73 6547 6f6f 644d 696e + chooseGoodMin │ │ │ │ 00010490: 6f72 7328 382c 2036 2c20 4d2c 204a 2c20 ors(8, 6, M, J, │ │ │ │ 000104a0: 5374 7261 7465 6779 3d3e 5261 6e64 6f6d Strategy=>Random │ │ │ │ 000104b0: 2929 2020 2020 2020 2020 2020 7c0a 7c20 )) |.| │ │ │ │ -000104c0: 2d2d 2075 7365 6420 302e 3130 3931 3239 -- used 0.109129 │ │ │ │ -000104d0: 7320 2863 7075 293b 2030 2e30 3732 3933 s (cpu); 0.07293 │ │ │ │ -000104e0: 3973 2028 7468 7265 6164 293b 2030 7320 9s (thread); 0s │ │ │ │ -000104f0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +000104c0: 2d2d 2075 7365 6420 302e 3036 3837 3933 -- used 0.068793 │ │ │ │ +000104d0: 3873 2028 6370 7529 3b20 302e 3036 3931 8s (cpu); 0.0691 │ │ │ │ +000104e0: 3031 3873 2028 7468 7265 6164 293b 2030 018s (thread); 0 │ │ │ │ +000104f0: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 00010500: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00010510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010550: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 00010560: 3238 203d 2032 2020 2020 2020 2020 2020 28 = 2 │ │ │ │ @@ -4191,17 +4191,17 @@ │ │ │ │ 000105e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000105f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00010600: 3239 203a 2074 696d 6520 6469 6d20 284a 29 : time dim (J │ │ │ │ 00010610: 202b 2063 686f 6f73 6547 6f6f 644d 696e + chooseGoodMin │ │ │ │ 00010620: 6f72 7328 382c 2036 2c20 4d2c 204a 2c20 ors(8, 6, M, J, │ │ │ │ 00010630: 5374 7261 7465 6779 3d3e 4c65 7853 6d61 Strategy=>LexSma │ │ │ │ 00010640: 6c6c 6573 7429 2920 2020 2020 7c0a 7c20 llest)) |.| │ │ │ │ -00010650: 2d2d 2075 7365 6420 302e 3333 3133 3235 -- 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│ 00010790: 3330 203a 2074 696d 6520 6469 6d20 284a 30 : time dim (J │ │ │ │ 000107a0: 202b 2063 686f 6f73 6547 6f6f 644d 696e + chooseGoodMin │ │ │ │ 000107b0: 6f72 7328 382c 2036 2c20 4d2c 204a 2c20 ors(8, 6, M, J, │ │ │ │ 000107c0: 5374 7261 7465 6779 3d3e 4c65 7853 6d61 Strategy=>LexSma │ │ │ │ 000107d0: 6c6c 6573 7454 6572 6d29 2920 7c0a 7c20 llestTerm)) |.| │ │ │ │ -000107e0: 2d2d 2075 7365 6420 302e 3533 3639 3537 -- used 0.536957 │ │ │ │ -000107f0: 7320 2863 7075 293b 2030 2e33 3431 3635 s (cpu); 0.34165 │ │ │ │ -00010800: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ +000107e0: 2d2d 2075 7365 6420 302e 3632 3439 3035 -- used 0.624905 │ │ │ │ +000107f0: 7320 2863 7075 293b 2030 2e33 3839 3531 s (cpu); 0.38951 │ │ │ │ +00010800: 3173 2028 7468 7265 6164 293b 2030 7320 1s (thread); 0s │ │ │ │ 00010810: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00010820: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00010830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010870: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ @@ -4241,18 +4241,18 @@ │ │ │ │ 00010900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00010920: 3331 203a 2074 696d 6520 6469 6d20 284a 31 : time dim (J │ │ │ │ 00010930: 202b 2063 686f 6f73 6547 6f6f 644d 696e + chooseGoodMin │ │ │ │ 00010940: 6f72 7328 382c 2036 2c20 4d2c 204a 2c20 ors(8, 6, M, J, │ │ │ │ 00010950: 5374 7261 7465 6779 3d3e 4c65 784c 6172 Strategy=>LexLar │ │ │ │ 00010960: 6765 7374 2929 2020 2020 2020 7c0a 7c20 gest)) |.| │ │ │ │ -00010970: 2d2d 2075 7365 6420 302e 3434 3130 3539 -- used 0.441059 │ │ │ │ -00010980: 7320 2863 7075 293b 2030 2e32 3533 3030 s (cpu); 0.25300 │ │ │ │ -00010990: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ -000109a0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00010970: 2d2d 2075 7365 6420 302e 3437 3635 3373 -- used 0.47653s │ │ │ │ +00010980: 2028 6370 7529 3b20 302e 3237 3238 3937 (cpu); 0.272897 │ │ │ │ +00010990: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +000109a0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 000109b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000109c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000109d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000109e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000109f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010a00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 00010a10: 3331 203d 2032 2020 2020 2020 2020 2020 31 = 2 │ │ │ │ @@ -4266,17 +4266,17 @@ │ │ │ │ 00010a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00010ab0: 3332 203a 2074 696d 6520 6469 6d20 284a 32 : time dim (J │ │ │ │ 00010ac0: 202b 2063 686f 6f73 6547 6f6f 644d 696e + chooseGoodMin │ │ │ │ 00010ad0: 6f72 7328 382c 2036 2c20 4d2c 204a 2c20 ors(8, 6, M, J, │ │ │ │ 00010ae0: 5374 7261 7465 6779 3d3e 4752 6576 4c65 Strategy=>GRevLe │ │ │ │ 00010af0: 7853 6d61 6c6c 6573 7429 2920 7c0a 7c20 xSmallest)) |.| │ │ │ │ -00010b00: 2d2d 2075 7365 6420 302e 3433 3233 3735 -- used 0.432375 │ │ │ │ -00010b10: 7320 2863 7075 293b 2030 2e32 3336 3932 s (cpu); 0.23692 │ │ │ │ -00010b20: 3273 2028 7468 7265 6164 293b 2030 7320 2s (thread); 0s │ │ │ │ +00010b00: 2d2d 2075 7365 6420 302e 3434 3537 3834 -- used 0.445784 │ │ │ │ +00010b10: 7320 2863 7075 293b 2030 2e32 3334 3539 s (cpu); 0.23459 │ │ │ │ +00010b20: 3973 2028 7468 7265 6164 293b 2030 7320 9s (thread); 0s │ │ │ │ 00010b30: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00010b40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00010b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010b90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ @@ -4291,16 +4291,16 @@ │ │ │ │ 00010c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00010c40: 3333 203a 2074 696d 6520 6469 6d20 284a 33 : time dim (J │ │ │ │ 00010c50: 202b 2063 686f 6f73 6547 6f6f 644d 696e + chooseGoodMin │ │ │ │ 00010c60: 6f72 7328 382c 2036 2c20 4d2c 204a 2c20 ors(8, 6, M, J, │ │ │ │ 00010c70: 5374 7261 7465 6779 3d3e 2020 2020 2020 Strategy=> │ │ │ │ 00010c80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00010c90: 2d2d 2075 7365 6420 302e 3436 3235 3373 -- used 0.46253s │ │ │ │ -00010ca0: 2028 6370 7529 3b20 302e 3235 3032 3734 (cpu); 0.250274 │ │ │ │ +00010c90: 2d2d 2075 7365 6420 302e 3638 3337 3473 -- used 0.68374s │ │ │ │ +00010ca0: 2028 6370 7529 3b20 302e 3332 3039 3331 (cpu); 0.320931 │ │ │ │ 00010cb0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 00010cc0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 00010cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00010ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4326,17 +4326,17 @@ │ │ │ │ 00010e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00010e70: 3334 203a 2074 696d 6520 6469 6d20 284a 34 : time dim (J │ │ │ │ 00010e80: 202b 2063 686f 6f73 6547 6f6f 644d 696e + chooseGoodMin │ │ │ │ 00010e90: 6f72 7328 382c 2036 2c20 4d2c 204a 2c20 ors(8, 6, M, J, │ │ │ │ 00010ea0: 5374 7261 7465 6779 3d3e 4752 6576 4c65 Strategy=>GRevLe │ │ │ │ 00010eb0: 784c 6172 6765 7374 2929 2020 7c0a 7c20 xLargest)) |.| │ │ │ │ -00010ec0: 2d2d 2075 7365 6420 302e 3432 3132 3738 -- used 0.421278 │ │ │ │ -00010ed0: 7320 2863 7075 293b 2030 2e32 3134 3838 s (cpu); 0.21488 │ │ │ │ -00010ee0: 3873 2028 7468 7265 6164 293b 2030 7320 8s (thread); 0s │ │ │ │ +00010ec0: 2d2d 2075 7365 6420 302e 3435 3739 3836 -- used 0.457986 │ │ │ │ +00010ed0: 7320 2863 7075 293b 2030 2e32 3030 3832 s (cpu); 0.20082 │ │ │ │ +00010ee0: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ 00010ef0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00010f00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00010f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010f50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ @@ -4351,18 +4351,18 @@ │ │ │ │ 00010fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00011000: 3335 203a 2074 696d 6520 6469 6d20 284a 35 : time dim (J │ │ │ │ 00011010: 202b 2063 686f 6f73 6547 6f6f 644d 696e + chooseGoodMin │ │ │ │ 00011020: 6f72 7328 382c 2036 2c20 4d2c 204a 2c20 ors(8, 6, M, J, │ │ │ │ 00011030: 5374 7261 7465 6779 3d3e 506f 696e 7473 Strategy=>Points │ │ │ │ 00011040: 2929 2020 2020 2020 2020 2020 7c0a 7c20 )) |.| │ │ │ │ -00011050: 2d2d 2075 7365 6420 3135 2e38 3236 7320 -- used 15.826s │ │ │ │ -00011060: 2863 7075 293b 2039 2e34 3735 3734 7320 (cpu); 9.47574s │ │ │ │ -00011070: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00011080: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00011050: 2d2d 2075 7365 6420 3139 2e36 3939 3973 -- used 19.6999s │ │ │ │ +00011060: 2028 6370 7529 3b20 3131 2e37 3237 3173 (cpu); 11.7271s │ │ │ │ +00011070: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +00011080: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 00011090: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000110a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000110b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000110c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000110d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000110e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 000110f0: 3335 203d 2031 2020 2020 2020 2020 2020 35 = 1 │ │ │ │ @@ -4475,17 +4475,17 @@ │ │ │ │ 000117a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000117b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000117c0: 7c69 3337 203a 2074 696d 6520 6368 6f6f |i37 : time choo │ │ │ │ 000117d0: 7365 476f 6f64 4d69 6e6f 7273 2832 302c seGoodMinors(20, │ │ │ │ 000117e0: 2036 2c20 4d2c 204a 2c20 5374 7261 7465 6, M, J, Strate │ │ │ │ 000117f0: 6779 3d3e 5374 7261 7465 6779 4465 6661 gy=>StrategyDefa │ │ │ │ 00011800: 756c 742c 2020 2020 2020 2020 2020 7c0a ult, |. │ │ │ │ -00011810: 7c20 2d2d 2075 7365 6420 302e 3430 3136 | -- used 0.4016 │ │ │ │ -00011820: 3138 7320 2863 7075 293b 2030 2e33 3334 18s (cpu); 0.334 │ │ │ │ -00011830: 3432 3573 2028 7468 7265 6164 293b 2030 425s (thread); 0 │ │ │ │ +00011810: 7c20 2d2d 2075 7365 6420 302e 3435 3031 | -- used 0.4501 │ │ │ │ +00011820: 3933 7320 2863 7075 293b 2030 2e33 3735 93s (cpu); 0.375 │ │ │ │ +00011830: 3433 3673 2028 7468 7265 6164 293b 2030 436s (thread); 0 │ │ │ │ 00011840: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 00011850: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00011860: 7c69 6e74 6572 6e61 6c43 686f 6f73 654d |internalChooseM │ │ │ │ 00011870: 696e 6f72 3a20 4368 6f6f 7369 6e67 2052 inor: Choosing R │ │ │ │ 00011880: 616e 646f 6d20 2020 2020 2020 2020 2020 andom │ │ │ │ 00011890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000118a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ @@ -4886,17 +4886,17 @@ │ │ │ │ 00013150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013160: 2d2d 2d2d 2d2b 0a7c 6934 3320 3a20 7469 -----+.|i43 : ti │ │ │ │ 00013170: 6d65 2064 696d 2028 4a20 2b20 6368 6f6f me dim (J + choo │ │ │ │ 00013180: 7365 476f 6f64 4d69 6e6f 7273 2831 2c20 seGoodMinors(1, │ │ │ │ 00013190: 362c 204d 2c20 4a2c 2053 7472 6174 6567 6, M, J, Strateg │ │ │ │ 000131a0: 793d 3e50 6f69 6e74 732c 2020 2020 2020 y=>Points, │ │ │ │ 000131b0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000131c0: 2030 2e36 3739 3632 3273 2028 6370 7529 0.679622s (cpu) │ │ │ │ -000131d0: 3b20 302e 3533 3834 3573 2028 7468 7265 ; 0.53845s (thre │ │ │ │ -000131e0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ +000131c0: 2030 2e38 3432 3834 3773 2028 6370 7529 0.842847s (cpu) │ │ │ │ +000131d0: 3b20 302e 3638 3434 3732 7320 2874 6872 ; 0.684472s (thr │ │ │ │ +000131e0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 000131f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013200: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00013210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013250: 2020 2020 207c 0a7c 6f34 3320 3d20 3220 |.|o43 = 2 │ │ │ │ @@ -5021,17 +5021,17 @@ │ │ │ │ 000139c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000139d0: 2d2d 2d2d 2d2b 0a7c 6934 3620 3a20 7469 -----+.|i46 : ti │ │ │ │ 000139e0: 6d65 2064 696d 2028 4a20 2b20 6368 6f6f me dim (J + choo │ │ │ │ 000139f0: 7365 476f 6f64 4d69 6e6f 7273 2831 2c20 seGoodMinors(1, │ │ │ │ 00013a00: 362c 204d 2c20 4a2c 2053 7472 6174 6567 6, M, J, Strateg │ │ │ │ 00013a10: 793d 3e50 6f69 6e74 732c 2020 2020 2020 y=>Points, │ │ │ │ 00013a20: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00013a30: 2030 2e35 3933 3833 3173 2028 6370 7529 0.593831s (cpu) │ │ │ │ -00013a40: 3b20 302e 3433 3939 3273 2028 7468 7265 ; 0.43992s (thre │ │ │ │ -00013a50: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ +00013a30: 2030 2e37 3330 3530 3673 2028 6370 7529 0.730506s (cpu) │ │ │ │ +00013a40: 3b20 302e 3534 3634 3431 7320 2874 6872 ; 0.546441s (thr │ │ │ │ +00013a50: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00013a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013a70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00013a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013ac0: 2020 2020 207c 0a7c 6f34 3620 3d20 3220 |.|o46 = 2 │ │ │ │ @@ -5113,16 +5113,16 @@ │ │ │ │ 00013f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ 00013fa0: 3720 3a20 7469 6d65 2072 6567 756c 6172 7 : time regular │ │ │ │ 00013fb0: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 00013fc0: 2053 2f4a 2c20 4d61 784d 696e 6f72 7320 S/J, MaxMinors │ │ │ │ 00013fd0: 3d3e 2031 3030 2c20 2020 2020 2020 2020 => 100, │ │ │ │ 00013fe0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00013ff0: 2d20 7573 6564 2032 2e30 3233 3637 7320 - used 2.02367s │ │ │ │ -00014000: 2863 7075 293b 2031 2e37 3231 3938 7320 (cpu); 1.72198s │ │ │ │ +00013ff0: 2d20 7573 6564 2032 2e34 3331 3037 7320 - used 2.43107s │ │ │ │ +00014000: 2863 7075 293b 2032 2e31 3738 3734 7320 (cpu); 2.17874s │ │ │ │ 00014010: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00014020: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00014030: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00014040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5148,18 +5148,18 @@ │ │ │ │ 000141b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000141c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ 000141d0: 3820 3a20 7469 6d65 2072 6567 756c 6172 8 : time regular │ │ │ │ 000141e0: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 000141f0: 2053 2f4a 2c20 4d61 784d 696e 6f72 7320 S/J, MaxMinors │ │ │ │ 00014200: 3d3e 2031 3030 2c20 2020 2020 2020 2020 => 100, │ │ │ │ 00014210: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00014220: 2d20 7573 6564 2031 2e30 3839 3332 7320 - used 1.08932s │ │ │ │ -00014230: 2863 7075 293b 2030 2e39 3633 3630 3173 (cpu); 0.963601s │ │ │ │ -00014240: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -00014250: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ +00014220: 2d20 7573 6564 2031 2e35 3232 3834 7320 - used 1.52284s │ │ │ │ +00014230: 2863 7075 293b 2031 2e32 3737 3839 7320 (cpu); 1.27789s │ │ │ │ +00014240: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +00014250: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00014260: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00014270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000142a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000142b0: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ 000142c0: 3820 3d20 7472 7565 2020 2020 2020 2020 8 = true │ │ │ │ @@ -5183,31 +5183,31 @@ │ │ │ │ 000143e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000143f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ 00014400: 3920 3a20 7469 6d65 2072 6567 756c 6172 9 : time regular │ │ │ │ 00014410: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 00014420: 2053 2f4a 2c20 4d61 784d 696e 6f72 7320 S/J, MaxMinors │ │ │ │ 00014430: 3d3e 2031 3030 2c20 5374 7261 7465 6779 => 100, Strategy │ │ │ │ 00014440: 3d3e 5261 6e64 6f6d 2920 207c 0a7c 202d =>Random) |.| - │ │ │ │ -00014450: 2d20 7573 6564 2032 2e38 3939 3338 7320 - used 2.89938s │ │ │ │ -00014460: 2863 7075 293b 2032 2e35 3632 3139 7320 (cpu); 2.56219s │ │ │ │ +00014450: 2d20 7573 6564 2033 2e34 3732 3534 7320 - used 3.47254s │ │ │ │ +00014460: 2863 7075 293b 2033 2e32 3338 3135 7320 (cpu); 3.23815s │ │ │ │ 00014470: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00014480: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00014490: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 000144a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000144b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000144c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000144d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000144e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ 000144f0: 3020 3a20 7469 6d65 2072 6567 756c 6172 0 : time regular │ │ │ │ 00014500: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 00014510: 2053 2f4a 2c20 4d61 784d 696e 6f72 7320 S/J, MaxMinors │ │ │ │ 00014520: 3d3e 2031 3030 2c20 2020 2020 2020 2020 => 100, │ │ │ │ 00014530: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00014540: 2d20 7573 6564 2032 2e33 3638 3737 7320 - used 2.36877s │ │ │ │ -00014550: 2863 7075 293b 2031 2e39 3036 3731 7320 (cpu); 1.90671s │ │ │ │ +00014540: 2d20 7573 6564 2032 2e38 3833 3739 7320 - used 2.88379s │ │ │ │ +00014550: 2863 7075 293b 2032 2e32 3938 3038 7320 (cpu); 2.29808s │ │ │ │ 00014560: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00014570: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00014580: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ 00014590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000145a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000145b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000145c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5223,16 +5223,16 @@ │ │ │ │ 00014660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ 00014680: 3120 3a20 7469 6d65 2072 6567 756c 6172 1 : time regular │ │ │ │ 00014690: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 000146a0: 2053 2f4a 2c20 4d61 784d 696e 6f72 7320 S/J, MaxMinors │ │ │ │ 000146b0: 3d3e 2031 3030 2c20 2020 2020 2020 2020 => 100, │ │ │ │ 000146c0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -000146d0: 2d20 7573 6564 2030 2e33 3737 3530 3973 - used 0.377509s │ │ │ │ -000146e0: 2028 6370 7529 3b20 302e 3330 3934 3733 (cpu); 0.309473 │ │ │ │ +000146d0: 2d20 7573 6564 2030 2e33 3237 3033 3773 - used 0.327037s │ │ │ │ +000146e0: 2028 6370 7529 3b20 302e 3332 3930 3337 (cpu); 0.329037 │ │ │ │ 000146f0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 00014700: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 00014710: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00014720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5258,16 +5258,16 @@ │ │ │ │ 00014890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000148a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ 000148b0: 3220 3a20 7469 6d65 2072 6567 756c 6172 2 : time regular │ │ │ │ 000148c0: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 000148d0: 2053 2f4a 2c20 4d61 784d 696e 6f72 7320 S/J, MaxMinors │ │ │ │ 000148e0: 3d3e 2031 3030 2c20 2020 2020 2020 2020 => 100, │ │ │ │ 000148f0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00014900: 2d20 7573 6564 2032 2e38 3236 3431 7320 - used 2.82641s │ │ │ │ -00014910: 2863 7075 293b 2032 2e32 3335 3239 7320 (cpu); 2.23529s │ │ │ │ +00014900: 2d20 7573 6564 2033 2e34 3638 3038 7320 - used 3.46808s │ │ │ │ +00014910: 2863 7075 293b 2032 2e37 3336 3137 7320 (cpu); 2.73617s │ │ │ │ 00014920: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00014930: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00014940: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ 00014950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5283,16 +5283,16 @@ │ │ │ │ 00014a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ 00014a40: 3320 3a20 7469 6d65 2072 6567 756c 6172 3 : time regular │ │ │ │ 00014a50: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 00014a60: 2053 2f4a 2c20 4d61 784d 696e 6f72 7320 S/J, MaxMinors │ │ │ │ 00014a70: 3d3e 2031 3030 2c20 2020 2020 2020 2020 => 100, │ │ │ │ 00014a80: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00014a90: 2d20 7573 6564 2033 2e33 3837 3031 7320 - used 3.38701s │ │ │ │ -00014aa0: 2863 7075 293b 2032 2e37 3335 3238 7320 (cpu); 2.73528s │ │ │ │ +00014a90: 2d20 7573 6564 2034 2e30 3531 3535 7320 - used 4.05155s │ │ │ │ +00014aa0: 2863 7075 293b 2033 2e33 3434 3834 7320 (cpu); 3.34484s │ │ │ │ 00014ab0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00014ac0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00014ad0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ 00014ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5308,16 +5308,16 @@ │ │ │ │ 00014bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ 00014bd0: 3420 3a20 7469 6d65 2072 6567 756c 6172 4 : time regular │ │ │ │ 00014be0: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 00014bf0: 2053 2f4a 2c20 4d61 784d 696e 6f72 7320 S/J, MaxMinors │ │ │ │ 00014c00: 3d3e 2031 3030 2c20 5374 7261 7465 6779 => 100, Strategy │ │ │ │ 00014c10: 3d3e 506f 696e 7473 2920 207c 0a7c 202d =>Points) |.| - │ │ │ │ -00014c20: 2d20 7573 6564 2035 382e 3931 3337 7320 - used 58.9137s │ │ │ │ -00014c30: 2863 7075 293b 2034 372e 3735 3435 7320 (cpu); 47.7545s │ │ │ │ +00014c20: 2d20 7573 6564 2036 342e 3535 3731 7320 - used 64.5571s │ │ │ │ +00014c30: 2863 7075 293b 2035 352e 3531 3234 7320 (cpu); 55.5124s │ │ │ │ 00014c40: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00014c50: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00014c60: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00014c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5333,16 +5333,16 @@ │ │ │ │ 00014d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ 00014d60: 3520 3a20 7469 6d65 2072 6567 756c 6172 5 : time regular │ │ │ │ 00014d70: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 00014d80: 2053 2f4a 2c20 4d61 784d 696e 6f72 7320 S/J, MaxMinors │ │ │ │ 00014d90: 3d3e 2031 3030 2c20 2020 2020 2020 2020 => 100, │ │ │ │ 00014da0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00014db0: 2d20 7573 6564 2032 2e39 3933 3937 7320 - used 2.99397s │ │ │ │ -00014dc0: 2863 7075 293b 2032 2e34 3132 3638 7320 (cpu); 2.41268s │ │ │ │ +00014db0: 2d20 7573 6564 2033 2e39 3930 3735 7320 - used 3.99075s │ │ │ │ +00014dc0: 2863 7075 293b 2032 2e36 3537 3932 7320 (cpu); 2.65792s │ │ │ │ 00014dd0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00014de0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00014df0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00014e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5998,17 +5998,17 @@ │ │ │ │ 000176d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000176e0: 2d2d 2d2b 0a7c 6937 203a 2074 696d 6520 ---+.|i7 : time │ │ │ │ 000176f0: 6973 436f 6469 6d41 744c 6561 7374 2833 isCodimAtLeast(3 │ │ │ │ 00017700: 2c20 4a29 2020 2020 2020 2020 2020 2020 , J) │ │ │ │ 00017710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017730: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -00017740: 2e30 3032 3134 3530 3973 2028 6370 7529 .00214509s (cpu) │ │ │ │ -00017750: 3b20 302e 3030 3330 3435 3437 7320 2874 ; 0.00304547s (t │ │ │ │ -00017760: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00017740: 2e30 3030 3137 3137 3934 7320 2863 7075 .000171794s (cpu │ │ │ │ +00017750: 293b 2030 2e30 3033 3030 3934 3673 2028 ); 0.00300946s ( │ │ │ │ +00017760: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00017770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017780: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00017790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000177a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000177b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000177c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000177d0: 2020 207c 0a7c 6f37 203d 2074 7275 6520 |.|o7 = true │ │ │ │ @@ -6208,18 +6208,18 @@ │ │ │ │ 000183f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018400: 2d2d 2d2d 2b0a 7c69 3920 3a20 7469 6d65 ----+.|i9 : time │ │ │ │ 00018410: 2069 7343 6f64 696d 4174 4c65 6173 7428 isCodimAtLeast( │ │ │ │ 00018420: 352c 2049 2c20 5061 6972 4c69 6d69 7420 5, I, PairLimit │ │ │ │ 00018430: 3d3e 2035 2c20 5665 7262 6f73 653d 3e74 => 5, Verbose=>t │ │ │ │ 00018440: 7275 6529 2020 2020 2020 2020 2020 2020 rue) │ │ │ │ 00018450: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00018460: 302e 3030 3039 3133 3139 3273 2028 6370 0.000913192s (cp │ │ │ │ -00018470: 7529 3b20 302e 3030 3237 3536 3234 7320 u); 0.00275624s │ │ │ │ -00018480: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00018490: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00018460: 302e 3030 3430 3331 3432 7320 2863 7075 0.00403142s (cpu │ │ │ │ +00018470: 293b 2030 2e30 3033 3231 3938 3773 2028 ); 0.00321987s ( │ │ │ │ +00018480: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00018490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000184a0: 2020 2020 7c0a 7c69 7343 6f64 696d 4174 |.|isCodimAt │ │ │ │ 000184b0: 4c65 6173 743a 2043 6f6d 7075 7469 6e67 Least: Computing │ │ │ │ 000184c0: 2063 6f64 696d 206f 6620 6d6f 6e6f 6d69 codim of monomi │ │ │ │ 000184d0: 616c 7320 6261 7365 6420 6f6e 2069 6465 als based on ide │ │ │ │ 000184e0: 616c 2067 656e 6572 6174 6f72 732e 2020 al generators. │ │ │ │ 000184f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00018500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -6238,16 +6238,16 @@ │ │ │ │ 000185d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000185e0: 2d2d 2d2d 2b0a 7c69 3130 203a 2074 696d ----+.|i10 : tim │ │ │ │ 000185f0: 6520 6973 436f 6469 6d41 744c 6561 7374 e isCodimAtLeast │ │ │ │ 00018600: 2835 2c20 492c 2050 6169 724c 696d 6974 (5, I, PairLimit │ │ │ │ 00018610: 203d 3e20 3230 302c 2056 6572 626f 7365 => 200, Verbose │ │ │ │ 00018620: 3d3e 6661 6c73 6529 2020 2020 2020 2020 =>false) │ │ │ │ 00018630: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00018640: 302e 3030 3031 3334 3131 3273 2028 6370 0.000134112s (cp │ │ │ │ -00018650: 7529 3b20 302e 3030 3234 3135 3738 7320 u); 0.00241578s │ │ │ │ +00018640: 302e 3030 3031 3438 3231 3673 2028 6370 0.000148216s (cp │ │ │ │ +00018650: 7529 3b20 302e 3030 3238 3133 3638 7320 u); 0.00281368s │ │ │ │ 00018660: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00018670: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00018680: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00018690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000186a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000186b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000186c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7475,16 +7475,16 @@ │ │ │ │ 0001d320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d330: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ 0001d340: 2074 696d 6520 7072 6f6a 4469 6d28 6d6f time projDim(mo │ │ │ │ 0001d350: 6475 6c65 2049 2c20 5374 7261 7465 6779 dule I, Strategy │ │ │ │ 0001d360: 3d3e 5374 7261 7465 6779 5261 6e64 6f6d =>StrategyRandom │ │ │ │ 0001d370: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0001d380: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -0001d390: 302e 3237 3939 3538 7320 2863 7075 293b 0.279958s (cpu); │ │ │ │ -0001d3a0: 2030 2e31 3437 3436 3573 2028 7468 7265 0.147465s (thre │ │ │ │ +0001d390: 302e 3331 3037 3437 7320 2863 7075 293b 0.310747s (cpu); │ │ │ │ +0001d3a0: 2030 2e31 3534 3638 3473 2028 7468 7265 0.154684s (thre │ │ │ │ 0001d3b0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 0001d3c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0001d3d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001d3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d410: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ @@ -7498,17 +7498,17 @@ │ │ │ │ 0001d490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d4b0: 2b0a 7c69 3520 3a20 7469 6d65 2070 726f +.|i5 : time pro │ │ │ │ 0001d4c0: 6a44 696d 286d 6f64 756c 6520 492c 2053 jDim(module I, S │ │ │ │ 0001d4d0: 7472 6174 6567 793d 3e53 7472 6174 6567 trategy=>Strateg │ │ │ │ 0001d4e0: 7952 616e 646f 6d2c 204d 696e 4469 6d65 yRandom, MinDime │ │ │ │ 0001d4f0: 6e73 696f 6e20 3d3e 2031 297c 0a7c 202d nsion => 1)|.| - │ │ │ │ -0001d500: 2d20 7573 6564 2030 2e30 3130 3033 3439 - used 0.0100349 │ │ │ │ -0001d510: 7320 2863 7075 293b 2030 2e30 3132 3139 s (cpu); 0.01219 │ │ │ │ -0001d520: 3434 7320 2874 6872 6561 6429 3b20 3073 44s (thread); 0s │ │ │ │ +0001d500: 2d20 7573 6564 2030 2e30 3132 3530 3133 - used 0.0125013 │ │ │ │ +0001d510: 7320 2863 7075 293b 2030 2e30 3134 3534 s (cpu); 0.01454 │ │ │ │ +0001d520: 3539 7320 2874 6872 6561 6429 3b20 3073 59s (thread); 0s │ │ │ │ 0001d530: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0001d540: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0001d550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d590: 207c 0a7c 6f35 203d 2031 2020 2020 2020 |.|o5 = 1 │ │ │ │ @@ -7716,17 +7716,17 @@ │ │ │ │ 0001e230: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0001e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e260: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 -------+.|i3 : t │ │ │ │ 0001e270: 696d 6520 4932 203d 2072 6563 7572 7369 ime I2 = recursi │ │ │ │ 0001e280: 7665 4d69 6e6f 7273 2834 2c20 4d2c 2054 veMinors(4, M, T │ │ │ │ 0001e290: 6872 6561 6473 3d3e 3029 3b20 2020 207c hreads=>0); | │ │ │ │ -0001e2a0: 0a7c 202d 2d20 7573 6564 2030 2e35 3232 .| -- used 0.522 │ │ │ │ -0001e2b0: 3434 3773 2028 6370 7529 3b20 302e 3436 447s (cpu); 0.46 │ │ │ │ -0001e2c0: 3633 3633 7320 2874 6872 6561 6429 3b20 6363s (thread); │ │ │ │ +0001e2a0: 0a7c 202d 2d20 7573 6564 2030 2e35 3636 .| -- used 0.566 │ │ │ │ +0001e2b0: 3639 3773 2028 6370 7529 3b20 302e 3530 697s (cpu); 0.50 │ │ │ │ +0001e2c0: 3938 3733 7320 2874 6872 6561 6429 3b20 9873s (thread); │ │ │ │ 0001e2d0: 3073 2028 6763 297c 0a7c 2020 2020 2020 0s (gc)|.| │ │ │ │ 0001e2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e300: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0001e310: 0a7c 6f33 203a 2049 6465 616c 206f 6620 .|o3 : Ideal of │ │ │ │ 0001e320: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 0001e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7734,17 +7734,17 @@ │ │ │ │ 0001e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0001e380: 0a7c 6934 203a 2074 696d 6520 4931 203d .|i4 : time I1 = │ │ │ │ 0001e390: 206d 696e 6f72 7328 342c 204d 2c20 5374 minors(4, M, St │ │ │ │ 0001e3a0: 7261 7465 6779 3d3e 436f 6661 6374 6f72 rategy=>Cofactor │ │ │ │ 0001e3b0: 293b 2020 2020 207c 0a7c 202d 2d20 7573 ); |.| -- us │ │ │ │ -0001e3c0: 6564 2031 2e35 3139 3632 7320 2863 7075 ed 1.51962s (cpu │ │ │ │ -0001e3d0: 293b 2031 2e32 3838 3732 7320 2874 6872 ); 1.28872s (thr │ │ │ │ -0001e3e0: 6561 6429 3b20 3073 2028 6763 2920 207c ead); 0s (gc) | │ │ │ │ +0001e3c0: 6564 2031 2e34 3935 3036 7320 2863 7075 ed 1.49506s (cpu │ │ │ │ +0001e3d0: 293b 2031 2e33 3630 3873 2028 7468 7265 ); 1.3608s (thre │ │ │ │ +0001e3e0: 6164 293b 2030 7320 2867 6329 2020 207c ad); 0s (gc) | │ │ │ │ 0001e3f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001e400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e420: 2020 2020 2020 207c 0a7c 6f34 203a 2049 |.|o4 : I │ │ │ │ 0001e430: 6465 616c 206f 6620 5220 2020 2020 2020 deal of R │ │ │ │ 0001e440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e450: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -8161,18 +8161,18 @@ │ │ │ │ 0001fe00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fe10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fe20: 2d2b 0a7c 6938 203a 2074 696d 6520 7265 -+.|i8 : time re │ │ │ │ 0001fe30: 6775 6c61 7249 6e43 6f64 696d 656e 7369 gularInCodimensi │ │ │ │ 0001fe40: 6f6e 2831 2c20 5329 2020 2020 2020 2020 on(1, S) │ │ │ │ 0001fe50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fe60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fe70: 207c 0a7c 202d 2d20 7573 6564 2030 2e39 |.| -- used 0.9 │ │ │ │ -0001fe80: 3030 3630 3473 2028 6370 7529 3b20 302e 00604s (cpu); 0. │ │ │ │ -0001fe90: 3538 3937 3636 7320 2874 6872 6561 6429 589766s (thread) │ │ │ │ -0001fea0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +0001fe70: 207c 0a7c 202d 2d20 7573 6564 2031 2e30 |.| -- used 1.0 │ │ │ │ +0001fe80: 3136 3234 7320 2863 7075 293b 2030 2e36 1624s (cpu); 0.6 │ │ │ │ +0001fe90: 3837 3835 3873 2028 7468 7265 6164 293b 87858s (thread); │ │ │ │ +0001fea0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 0001feb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fec0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001fed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ff00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ff10: 207c 0a7c 6f38 203d 2074 7275 6520 2020 |.|o8 = true │ │ │ │ @@ -8186,17 +8186,17 @@ │ │ │ │ 0001ff90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ffa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ffb0: 2d2b 0a7c 6939 203a 2074 696d 6520 7265 -+.|i9 : time re │ │ │ │ 0001ffc0: 6775 6c61 7249 6e43 6f64 696d 656e 7369 gularInCodimensi │ │ │ │ 0001ffd0: 6f6e 2832 2c20 5329 2020 2020 2020 2020 on(2, S) │ │ │ │ 0001ffe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020000: 207c 0a7c 202d 2d20 7573 6564 2037 2e38 |.| -- used 7.8 │ │ │ │ -00020010: 3438 3438 7320 2863 7075 293b 2035 2e31 4848s (cpu); 5.1 │ │ │ │ -00020020: 3136 3738 7320 2874 6872 6561 6429 3b20 1678s (thread); │ │ │ │ +00020000: 207c 0a7c 202d 2d20 7573 6564 2039 2e31 |.| -- used 9.1 │ │ │ │ +00020010: 3534 3131 7320 2863 7075 293b 2036 2e32 5411s (cpu); 6.2 │ │ │ │ +00020020: 3033 3637 7320 2874 6872 6561 6429 3b20 0367s (thread); │ │ │ │ 00020030: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00020040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020050: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00020060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -8316,18 +8316,18 @@ │ │ │ │ 000207b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000207c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 ----------+.|i12 │ │ │ │ 000207d0: 203a 2074 696d 6520 2864 696d 2073 696e : time (dim sin │ │ │ │ 000207e0: 6775 6c61 724c 6f63 7573 2028 5229 2920 gularLocus (R)) │ │ │ │ 000207f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020810: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00020820: 2075 7365 6420 302e 3031 3939 3635 3773 used 0.0199657s │ │ │ │ -00020830: 2028 6370 7529 3b20 302e 3031 3939 3539 (cpu); 0.019959 │ │ │ │ -00020840: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ -00020850: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00020820: 2075 7365 6420 302e 3032 3030 3031 7320 used 0.020001s │ │ │ │ +00020830: 2863 7075 293b 2030 2e30 3230 3836 3637 (cpu); 0.0208667 │ │ │ │ +00020840: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +00020850: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 00020860: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00020870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000208a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000208b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3132 |.|o12 │ │ │ │ 000208c0: 203d 202d 3120 2020 2020 2020 2020 2020 = -1 │ │ │ │ @@ -8341,16 +8341,16 @@ │ │ │ │ 00020940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020950: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 ----------+.|i13 │ │ │ │ 00020960: 203a 2074 696d 6520 7265 6775 6c61 7249 : time regularI │ │ │ │ 00020970: 6e43 6f64 696d 656e 7369 6f6e 2832 2c20 nCodimension(2, │ │ │ │ 00020980: 5229 2020 2020 2020 2020 2020 2020 2020 R) │ │ │ │ 00020990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000209a0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -000209b0: 2075 7365 6420 302e 3436 3637 3638 7320 used 0.466768s │ │ │ │ -000209c0: 2863 7075 293b 2030 2e32 3833 3432 3873 (cpu); 0.283428s │ │ │ │ +000209b0: 2075 7365 6420 302e 3439 3431 3336 7320 used 0.494136s │ │ │ │ +000209c0: 2863 7075 293b 2030 2e32 3937 3434 3873 (cpu); 0.297448s │ │ │ │ 000209d0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 000209e0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 000209f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00020a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8366,18 +8366,18 @@ │ │ │ │ 00020ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 ----------+.|i14 │ │ │ │ 00020af0: 203a 2074 696d 6520 7265 6775 6c61 7249 : time regularI │ │ │ │ 00020b00: 6e43 6f64 696d 656e 7369 6f6e 2832 2c20 nCodimension(2, │ │ │ │ 00020b10: 5229 2020 2020 2020 2020 2020 2020 2020 R) │ │ │ │ 00020b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020b30: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00020b40: 2075 7365 6420 302e 3431 3132 3936 7320 used 0.411296s │ │ │ │ -00020b50: 2863 7075 293b 2030 2e32 3338 3333 3673 (cpu); 0.238336s │ │ │ │ -00020b60: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -00020b70: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ +00020b40: 2075 7365 6420 302e 3436 3139 7320 2863 used 0.4619s (c │ │ │ │ +00020b50: 7075 293b 2030 2e32 3338 3839 3773 2028 pu); 0.238897s ( │ │ │ │ +00020b60: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00020b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020b80: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00020b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020bd0: 2020 2020 2020 2020 2020 7c0a 7c6f 3134 |.|o14 │ │ │ │ 00020be0: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ @@ -8391,16 +8391,16 @@ │ │ │ │ 00020c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 ----------+.|i15 │ │ │ │ 00020c80: 203a 2074 696d 6520 7265 6775 6c61 7249 : time regularI │ │ │ │ 00020c90: 6e43 6f64 696d 656e 7369 6f6e 2832 2c20 nCodimension(2, │ │ │ │ 00020ca0: 5229 2020 2020 2020 2020 2020 2020 2020 R) │ │ │ │ 00020cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020cc0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00020cd0: 2075 7365 6420 302e 3330 3139 3137 7320 used 0.301917s │ │ │ │ -00020ce0: 2863 7075 293b 2030 2e31 3735 3831 3273 (cpu); 0.175812s │ │ │ │ +00020cd0: 2075 7365 6420 302e 3333 3337 3839 7320 used 0.333789s │ │ │ │ +00020ce0: 2863 7075 293b 2030 2e31 3839 3830 3873 (cpu); 0.189808s │ │ │ │ 00020cf0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 00020d00: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 00020d10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00020d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -10118,17 +10118,17 @@ │ │ │ │ 00027850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027860: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ 00027870: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ 00027880: 6720 5261 6e64 6f6d 2020 2020 2020 2020 g Random │ │ │ │ 00027890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000278a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000278b0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000278c0: 7365 4d69 202d 2d20 7573 6564 2038 2e32 seMi -- used 8.2 │ │ │ │ -000278d0: 3132 3337 7320 2863 7075 293b 2035 2e34 1237s (cpu); 5.4 │ │ │ │ -000278e0: 3633 3331 7320 2874 6872 6561 6429 3b20 6331s (thread); │ │ │ │ +000278c0: 7365 4d69 202d 2d20 7573 6564 2038 2e39 seMi -- used 8.9 │ │ │ │ +000278d0: 3530 3736 7320 2863 7075 293b 2036 2e31 5076s (cpu); 6.1 │ │ │ │ +000278e0: 3630 3834 7320 2874 6872 6561 6429 3b20 6084s (thread); │ │ │ │ 000278f0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00027900: 207c 0a7c 6e6f 723a 2043 686f 6f73 696e |.|nor: Choosin │ │ │ │ 00027910: 6720 4c65 7853 6d61 6c6c 6573 7454 6572 g LexSmallestTer │ │ │ │ 00027920: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ 00027930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027950: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ @@ -12262,16 +12262,16 @@ │ │ │ │ 0002fe50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fe60: 2d2d 2d2b 0a7c 6931 3720 3a20 7469 6d65 ---+.|i17 : time │ │ │ │ 0002fe70: 2072 6567 756c 6172 496e 436f 6469 6d65 regularInCodime │ │ │ │ 0002fe80: 6e73 696f 6e28 322c 2053 2c20 5665 7262 nsion(2, S, Verb │ │ │ │ 0002fe90: 6f73 653d 3e74 7275 652c 204d 6178 4d69 ose=>true, MaxMi │ │ │ │ 0002fea0: 6e6f 7273 3d3e 3330 2920 2020 2020 2020 nors=>30) │ │ │ │ 0002feb0: 2020 207c 0a7c 202d 2d20 7573 6564 2031 |.| -- used 1 │ │ │ │ -0002fec0: 2e35 3736 3132 7320 2863 7075 293b 2031 .57612s (cpu); 1 │ │ │ │ -0002fed0: 2e30 3635 3233 7320 2874 6872 6561 6429 .06523s (thread) │ │ │ │ +0002fec0: 2e38 3232 3338 7320 2863 7075 293b 2031 .82238s (cpu); 1 │ │ │ │ +0002fed0: 2e33 3339 3537 7320 2874 6872 6561 6429 .33957s (thread) │ │ │ │ 0002fee0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0002fef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ff00: 2020 207c 0a7c 7265 6775 6c61 7249 6e43 |.|regularInC │ │ │ │ 0002ff10: 6f64 696d 656e 7369 6f6e 3a20 7269 6e67 odimension: ring │ │ │ │ 0002ff20: 2064 696d 656e 7369 6f6e 203d 332c 2074 dimension =3, t │ │ │ │ 0002ff30: 6865 7265 2061 7265 2031 3733 3235 2070 here are 17325 p │ │ │ │ 0002ff40: 6f73 7369 626c 6520 3420 6279 2034 206d ossible 4 by 4 m │ │ │ │ @@ -12874,16 +12874,16 @@ │ │ │ │ 00032490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000324a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000324b0: 2d2d 2d2d 2b0a 7c69 3231 203a 2074 696d ----+.|i21 : tim │ │ │ │ 000324c0: 6520 7265 6775 6c61 7249 6e43 6f64 696d e regularInCodim │ │ │ │ 000324d0: 656e 7369 6f6e 2832 2c20 522c 2053 7472 ension(2, R, Str │ │ │ │ 000324e0: 6174 6567 793d 3e53 7472 6174 6567 7943 ategy=>StrategyC │ │ │ │ 000324f0: 7572 7265 6e74 297c 0a7c 202d 2d20 7573 urrent)|.| -- us │ │ │ │ -00032500: 6564 2030 2e34 3235 3634 3173 2028 6370 ed 0.425641s (cp │ │ │ │ -00032510: 7529 3b20 302e 3233 3036 3533 7320 2874 u); 0.230653s (t │ │ │ │ +00032500: 6564 2030 2e34 3430 3338 3773 2028 6370 ed 0.440387s (cp │ │ │ │ +00032510: 7529 3b20 302e 3236 3137 3232 7320 2874 u); 0.261722s (t │ │ │ │ 00032520: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00032530: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00032540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032570: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00032580: 6f32 3120 3d20 7472 7565 2020 2020 2020 o21 = true │ │ │ │ @@ -12895,16 +12895,16 @@ │ │ │ │ 000325e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000325f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032600: 2d2d 2d2b 0a7c 6932 3220 3a20 7469 6d65 ---+.|i22 : time │ │ │ │ 00032610: 2072 6567 756c 6172 496e 436f 6469 6d65 regularInCodime │ │ │ │ 00032620: 6e73 696f 6e28 322c 2052 2c20 5374 7261 nsion(2, R, Stra │ │ │ │ 00032630: 7465 6779 3d3e 5374 7261 7465 6779 4375 tegy=>StrategyCu │ │ │ │ 00032640: 7272 656e 7429 7c0a 7c20 2d2d 2075 7365 rrent)|.| -- use │ │ │ │ -00032650: 6420 302e 3134 3534 3131 7320 2863 7075 d 0.145411s (cpu │ │ │ │ -00032660: 293b 2030 2e30 3830 3432 3937 7320 2874 ); 0.0804297s (t │ │ │ │ +00032650: 6420 302e 3135 3330 3233 7320 2863 7075 d 0.153023s (cpu │ │ │ │ +00032660: 293b 2030 2e30 3838 3036 3832 7320 2874 ); 0.0880682s (t │ │ │ │ 00032670: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00032680: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00032690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000326a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000326b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000326c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 000326d0: 3232 203d 2074 7275 6520 2020 2020 2020 22 = true │ │ │ │ @@ -12916,17 +12916,17 @@ │ │ │ │ 00032730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032750: 2d2d 2b0a 7c69 3233 203a 2074 696d 6520 --+.|i23 : time │ │ │ │ 00032760: 7265 6775 6c61 7249 6e43 6f64 696d 656e regularInCodimen │ │ │ │ 00032770: 7369 6f6e 2831 2c20 532c 2053 7472 6174 sion(1, S, Strat │ │ │ │ 00032780: 6567 793d 3e53 7472 6174 6567 7943 7572 egy=>StrategyCur │ │ │ │ 00032790: 7265 6e74 297c 0a7c 202d 2d20 7573 6564 rent)|.| -- used │ │ │ │ -000327a0: 2031 2e33 3233 3038 7320 2863 7075 293b 1.32308s (cpu); │ │ │ │ -000327b0: 2030 2e38 3338 3432 3973 2028 7468 7265 0.838429s (thre │ │ │ │ -000327c0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ +000327a0: 2031 2e35 3331 3032 7320 2863 7075 293b 1.53102s (cpu); │ │ │ │ +000327b0: 2031 2e30 3234 3332 7320 2874 6872 6561 1.02432s (threa │ │ │ │ +000327c0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 000327d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000327e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000327f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032810: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ 00032820: 3320 3d20 7472 7565 2020 2020 2020 2020 3 = true │ │ │ │ 00032830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -12937,16 +12937,16 @@ │ │ │ │ 00032880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000328a0: 2d2b 0a7c 6932 3420 3a20 7469 6d65 2072 -+.|i24 : time r │ │ │ │ 000328b0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ 000328c0: 696f 6e28 312c 2053 2c20 5374 7261 7465 ion(1, S, Strate │ │ │ │ 000328d0: 6779 3d3e 5374 7261 7465 6779 4375 7272 gy=>StrategyCurr │ │ │ │ 000328e0: 656e 7429 7c0a 7c20 2d2d 2075 7365 6420 ent)|.| -- used │ │ │ │ -000328f0: 302e 3738 3030 3235 7320 2863 7075 293b 0.780025s (cpu); │ │ │ │ -00032900: 2030 2e34 3639 3238 3773 2028 7468 7265 0.469287s (thre │ │ │ │ +000328f0: 302e 3837 3632 3336 7320 2863 7075 293b 0.876236s (cpu); │ │ │ │ +00032900: 2030 2e35 3436 3733 3573 2028 7468 7265 0.546735s (thre │ │ │ │ 00032910: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00032920: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00032930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032960: 2020 2020 2020 2020 2020 7c0a 7c6f 3234 |.|o24 │ │ │ │ 00032970: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ @@ -12974,42 +12974,42 @@ │ │ │ │ 00032ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00032b00: 3237 203a 2074 696d 6520 7265 6775 6c61 27 : time regula │ │ │ │ 00032b10: 7249 6e43 6f64 696d 656e 7369 6f6e 2832 rInCodimension(2 │ │ │ │ 00032b20: 2c20 522c 2053 7472 6174 6567 793d 3e53 , R, Strategy=>S │ │ │ │ 00032b30: 7472 6174 6567 7943 7572 7265 6e74 297c trategyCurrent)| │ │ │ │ -00032b40: 0a7c 202d 2d20 7573 6564 2032 2e39 3336 .| -- used 2.936 │ │ │ │ -00032b50: 3939 7320 2863 7075 293b 2031 2e37 3734 99s (cpu); 1.774 │ │ │ │ -00032b60: 3336 7320 2874 6872 6561 6429 3b20 3073 36s (thread); 0s │ │ │ │ +00032b40: 0a7c 202d 2d20 7573 6564 2033 2e33 3231 .| -- used 3.321 │ │ │ │ +00032b50: 3833 7320 2863 7075 293b 2032 2e31 3338 83s (cpu); 2.138 │ │ │ │ +00032b60: 3339 7320 2874 6872 6561 6429 3b20 3073 39s (thread); 0s │ │ │ │ 00032b70: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00032b80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00032b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032bc0: 2d2d 2d2d 2d2b 0a7c 6932 3820 3a20 7469 -----+.|i28 : ti │ │ │ │ 00032bd0: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 00032be0: 6d65 6e73 696f 6e28 322c 2052 2c20 5374 mension(2, R, St │ │ │ │ 00032bf0: 7261 7465 6779 3d3e 5374 7261 7465 6779 rategy=>Strategy │ │ │ │ 00032c00: 4375 7272 656e 7429 7c0a 7c20 2d2d 2075 Current)|.| -- u │ │ │ │ -00032c10: 7365 6420 322e 3833 3335 3673 2028 6370 sed 2.83356s (cp │ │ │ │ -00032c20: 7529 3b20 312e 3733 3632 3173 2028 7468 u); 1.73621s (th │ │ │ │ +00032c10: 7365 6420 332e 3134 3837 3373 2028 6370 sed 3.14873s (cp │ │ │ │ +00032c20: 7529 3b20 322e 3034 3238 3973 2028 7468 u); 2.04289s (th │ │ │ │ 00032c30: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00032c40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00032c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00032c90: 7c69 3239 203a 2074 696d 6520 7265 6775 |i29 : time regu │ │ │ │ 00032ca0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ 00032cb0: 2831 2c20 532c 2053 7472 6174 6567 793d (1, S, Strategy= │ │ │ │ 00032cc0: 3e53 7472 6174 6567 7943 7572 7265 6e74 >StrategyCurrent │ │ │ │ 00032cd0: 297c 0a7c 202d 2d20 7573 6564 2030 2e36 )|.| -- used 0.6 │ │ │ │ -00032ce0: 3331 3235 3273 2028 6370 7529 3b20 302e 31252s (cpu); 0. │ │ │ │ -00032cf0: 3433 3035 3633 7320 2874 6872 6561 6429 430563s (thread) │ │ │ │ +00032ce0: 3135 3835 3273 2028 6370 7529 3b20 302e 15852s (cpu); 0. │ │ │ │ +00032cf0: 3433 3031 3232 7320 2874 6872 6561 6429 430122s (thread) │ │ │ │ 00032d00: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00032d10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00032d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032d50: 2020 2020 2020 207c 0a7c 6f32 3920 3d20 |.|o29 = │ │ │ │ 00032d60: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ @@ -13020,18 +13020,18 @@ │ │ │ │ 00032db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00032de0: 6933 3020 3a20 7469 6d65 2072 6567 756c i30 : time regul │ │ │ │ 00032df0: 6172 496e 436f 6469 6d65 6e73 696f 6e28 arInCodimension( │ │ │ │ 00032e00: 312c 2053 2c20 5374 7261 7465 6779 3d3e 1, S, Strategy=> │ │ │ │ 00032e10: 5374 7261 7465 6779 4375 7272 656e 7429 StrategyCurrent) │ │ │ │ -00032e20: 7c0a 7c20 2d2d 2075 7365 6420 302e 3331 |.| -- used 0.31 │ │ │ │ -00032e30: 3431 3735 7320 2863 7075 293b 2030 2e31 4175s (cpu); 0.1 │ │ │ │ -00032e40: 3838 3337 3173 2028 7468 7265 6164 293b 88371s (thread); │ │ │ │ -00032e50: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ +00032e20: 7c0a 7c20 2d2d 2075 7365 6420 302e 3337 |.| -- used 0.37 │ │ │ │ +00032e30: 3632 3834 7320 2863 7075 293b 2030 2e32 6284s (cpu); 0.2 │ │ │ │ +00032e40: 3336 3838 7320 2874 6872 6561 6429 3b20 3688s (thread); │ │ │ │ +00032e50: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00032e60: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00032e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032ea0: 2020 2020 2020 7c0a 7c6f 3330 203d 2074 |.|o30 = t │ │ │ │ 00032eb0: 7275 6520 2020 2020 2020 2020 2020 2020 rue │ │ │ │ 00032ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -13041,18 +13041,18 @@ │ │ │ │ 00032f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00032f30: 3331 203a 2074 696d 6520 7265 6775 6c61 31 : time regula │ │ │ │ 00032f40: 7249 6e43 6f64 696d 656e 7369 6f6e 2831 rInCodimension(1 │ │ │ │ 00032f50: 2c20 532c 2053 7472 6174 6567 793d 3e53 , S, Strategy=>S │ │ │ │ 00032f60: 7472 6174 6567 7952 616e 646f 6d29 207c trategyRandom) | │ │ │ │ -00032f70: 0a7c 202d 2d20 7573 6564 2030 2e35 3733 .| -- used 0.573 │ │ │ │ -00032f80: 3231 3273 2028 6370 7529 3b20 302e 3434 212s (cpu); 0.44 │ │ │ │ -00032f90: 3436 3134 7320 2874 6872 6561 6429 3b20 4614s (thread); │ │ │ │ -00032fa0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +00032f70: 0a7c 202d 2d20 7573 6564 2030 2e37 3332 .| -- used 0.732 │ │ │ │ +00032f80: 3338 7320 2863 7075 293b 2030 2e35 3936 38s (cpu); 0.596 │ │ │ │ +00032f90: 3738 3573 2028 7468 7265 6164 293b 2030 785s (thread); 0 │ │ │ │ +00032fa0: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 00032fb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00032fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032ff0: 2020 2020 207c 0a7c 6f33 3120 3d20 7472 |.|o31 = tr │ │ │ │ 00033000: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ 00033010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -13062,17 +13062,17 @@ │ │ │ │ 00033050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ 00033080: 3220 3a20 7469 6d65 2072 6567 756c 6172 2 : time regular │ │ │ │ 00033090: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 000330a0: 2053 2c20 5374 7261 7465 6779 3d3e 5374 S, Strategy=>St │ │ │ │ 000330b0: 7261 7465 6779 5261 6e64 6f6d 2920 7c0a rategyRandom) |. │ │ │ │ -000330c0: 7c20 2d2d 2075 7365 6420 312e 3436 3139 | -- used 1.4619 │ │ │ │ -000330d0: 3673 2028 6370 7529 3b20 312e 3034 3038 6s (cpu); 1.0408 │ │ │ │ -000330e0: 3873 2028 7468 7265 6164 293b 2030 7320 8s (thread); 0s │ │ │ │ +000330c0: 7c20 2d2d 2075 7365 6420 312e 3538 3830 | -- used 1.5880 │ │ │ │ +000330d0: 3473 2028 6370 7529 3b20 312e 3134 3437 4s (cpu); 1.1447 │ │ │ │ +000330e0: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ 000330f0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00033100: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00033110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033140: 2020 2020 7c0a 7c6f 3332 203d 2074 7275 |.|o32 = tru │ │ │ │ 00033150: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ @@ -13333,35 +13333,35 @@ │ │ │ │ 00034140: 2033 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 3...+---------- │ │ │ │ 00034150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00034160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00034170: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ 00034180: 2074 696d 6520 7265 6775 6c61 7249 6e43 time regularInC │ │ │ │ 00034190: 6f64 696d 656e 7369 6f6e 2831 2c20 532f odimension(1, S/ │ │ │ │ 000341a0: 4a29 2020 2020 2020 2020 2020 2020 207c J) | │ │ │ │ -000341b0: 0a7c 202d 2d20 7573 6564 2033 2e30 3731 .| -- used 3.071 │ │ │ │ -000341c0: 3838 7320 2863 7075 293b 2031 2e38 3730 88s (cpu); 1.870 │ │ │ │ -000341d0: 3739 7320 2874 6872 6561 6429 3b20 3073 79s (thread); 0s │ │ │ │ +000341b0: 0a7c 202d 2d20 7573 6564 2033 2e35 3339 .| -- used 3.539 │ │ │ │ +000341c0: 3536 7320 2863 7075 293b 2032 2e31 3439 56s (cpu); 2.149 │ │ │ │ +000341d0: 3239 7320 2874 6872 6561 6429 3b20 3073 29s (thread); 0s │ │ │ │ 000341e0: 2028 6763 297c 0a7c 2020 2020 2020 2020 (gc)|.| │ │ │ │ 000341f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034210: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ 00034220: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ 00034230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034250: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00034260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00034270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00034280: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 -------+.|i5 : t │ │ │ │ 00034290: 696d 6520 7265 6775 6c61 7249 6e43 6f64 ime regularInCod │ │ │ │ 000342a0: 696d 656e 7369 6f6e 2832 2c20 532f 4a29 imension(2, S/J) │ │ │ │ 000342b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000342c0: 202d 2d20 7573 6564 2031 322e 3633 3673 -- used 12.636s │ │ │ │ -000342d0: 2028 6370 7529 3b20 382e 3032 3636 3373 (cpu); 8.02663s │ │ │ │ -000342e0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -000342f0: 6329 207c 0a7c 2020 2020 2020 2020 2020 c) |.| │ │ │ │ +000342c0: 202d 2d20 7573 6564 2031 342e 3336 3136 -- used 14.3616 │ │ │ │ +000342d0: 7320 2863 7075 293b 2039 2e33 3037 3837 s (cpu); 9.30787 │ │ │ │ +000342e0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +000342f0: 6763 297c 0a7c 2020 2020 2020 2020 2020 gc)|.| │ │ │ │ 00034300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034320: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ 00034330: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ 00034340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034350: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00034360: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ @@ -13784,17 +13784,17 @@ │ │ │ │ 00035d70: 6e73 696f 6e20 636f 6d70 7574 6564 2c20 nsion computed, │ │ │ │ 00035d80: 3d20 3320 2020 2020 2020 207c 0a7c 696e = 3 |.|in │ │ │ │ 00035d90: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f ternalChooseMino │ │ │ │ 00035da0: 723a 2043 686f 6f73 696e 6720 4752 6576 r: Choosing GRev │ │ │ │ 00035db0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020 LexSmallest │ │ │ │ 00035dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035dd0: 2020 2020 2020 2020 2020 207c 0a7c 696e |.|in │ │ │ │ -00035de0: 7465 726e 616c 202d 2d20 7573 6564 2034 ternal -- used 4 │ │ │ │ -00035df0: 2e32 3130 3039 7320 2863 7075 293b 2032 .21009s (cpu); 2 │ │ │ │ -00035e00: 2e37 3830 3939 7320 2874 6872 6561 6429 .78099s (thread) │ │ │ │ +00035de0: 7465 726e 616c 202d 2d20 7573 6564 2033 ternal -- used 3 │ │ │ │ +00035df0: 2e38 3635 3435 7320 2863 7075 293b 2032 .86545s (cpu); 2 │ │ │ │ +00035e00: 2e33 3736 3138 7320 2874 6872 6561 6429 .37618s (thread) │ │ │ │ 00035e10: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00035e20: 2020 2020 2020 2020 2020 207c 0a7c 4368 |.|Ch │ │ │ │ 00035e30: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73 ooseMinor: Choos │ │ │ │ 00035e40: 696e 6720 4752 6576 4c65 7853 6d61 6c6c ing GRevLexSmall │ │ │ │ 00035e50: 6573 7420 2020 2020 2020 2020 2020 2020 est │ │ │ │ 00035e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035e70: 2020 2020 2020 2020 2020 207c 0a7c 696e |.|in │ │ │ │ @@ -14898,17 +14898,17 @@ │ │ │ │ 0003a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0003a330: 0a7c 6937 203a 2074 696d 6520 7265 6775 .|i7 : time regu │ │ │ │ 0003a340: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ 0003a350: 2831 2c20 532f 4a2c 204d 6178 4d69 6e6f (1, S/J, MaxMino │ │ │ │ 0003a360: 7273 3d3e 3130 2c20 5665 7262 6f73 653d rs=>10, Verbose= │ │ │ │ 0003a370: 3e74 7275 6529 2020 2020 2020 2020 207c >true) | │ │ │ │ -0003a380: 0a7c 202d 2d20 7573 6564 2030 2e32 3832 .| -- used 0.282 │ │ │ │ -0003a390: 3330 3373 2028 6370 7529 3b20 302e 3135 303s (cpu); 0.15 │ │ │ │ -0003a3a0: 3332 3273 2028 7468 7265 6164 293b 2030 322s (thread); 0 │ │ │ │ +0003a380: 0a7c 202d 2d20 7573 6564 2030 2e33 3538 .| -- used 0.358 │ │ │ │ +0003a390: 3137 7320 2863 7075 293b 2030 2e32 3232 17s (cpu); 0.222 │ │ │ │ +0003a3a0: 3639 3473 2028 7468 7265 6164 293b 2030 694s (thread); 0 │ │ │ │ 0003a3b0: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0003a3c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0003a3d0: 0a7c 7265 6775 6c61 7249 6e43 6f64 696d .|regularInCodim │ │ │ │ 0003a3e0: 656e 7369 6f6e 3a20 7269 6e67 2064 696d ension: ring dim │ │ │ │ 0003a3f0: 656e 7369 6f6e 203d 342c 2074 6865 7265 ension =4, there │ │ │ │ 0003a400: 2061 7265 2031 3436 3531 3238 2070 6f73 are 1465128 pos │ │ │ │ 0003a410: 7369 626c 6520 3520 6279 2035 206d 697c sible 5 by 5 mi| │ │ │ │ @@ -15136,17 +15136,17 @@ │ │ │ │ 0003b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0003b210: 3820 3a20 7469 6d65 2072 6567 756c 6172 8 : time regular │ │ │ │ 0003b220: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 0003b230: 2053 2f4a 2c20 4d61 784d 696e 6f72 733d S/J, MaxMinors= │ │ │ │ 0003b240: 3e31 302c 2053 7472 6174 6567 793d 3e53 >10, Strategy=>S │ │ │ │ 0003b250: 7472 6174 6567 7952 616e 646f 7c0a 7c20 trategyRando|.| │ │ │ │ -0003b260: 2d2d 2075 7365 6420 302e 3231 3636 3537 -- used 0.216657 │ │ │ │ -0003b270: 7320 2863 7075 293b 2030 2e31 3537 3233 s (cpu); 0.15723 │ │ │ │ -0003b280: 3273 2028 7468 7265 6164 293b 2030 7320 2s (thread); 0s │ │ │ │ +0003b260: 2d2d 2075 7365 6420 302e 3231 3137 3638 -- used 0.211768 │ │ │ │ +0003b270: 7320 2863 7075 293b 2030 2e31 3436 3535 s (cpu); 0.14655 │ │ │ │ +0003b280: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ 0003b290: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0003b2a0: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ 0003b2b0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ 0003b2c0: 696f 6e3a 2072 696e 6720 6469 6d65 6e73 ion: ring dimens │ │ │ │ 0003b2d0: 696f 6e20 3d34 2c20 7468 6572 6520 6172 ion =4, there ar │ │ │ │ 0003b2e0: 6520 3134 3635 3132 3820 706f 7373 6962 e 1465128 possib │ │ │ │ 0003b2f0: 6c65 2035 2062 7920 3520 6d69 7c0a 7c72 le 5 by 5 mi|.|r │ │ │ │ @@ -15413,16 +15413,16 @@ │ │ │ │ 0003c340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003c350: 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 7469 ------+.|i9 : ti │ │ │ │ 0003c360: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 0003c370: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20 mension(1, S/J, │ │ │ │ 0003c380: 4d61 784d 696e 6f72 733d 3e31 302c 204d MaxMinors=>10, M │ │ │ │ 0003c390: 696e 4d69 6e6f 7273 4675 6e63 7469 6f6e inMinorsFunction │ │ │ │ 0003c3a0: 203d 3e20 742d 7c0a 7c20 2d2d 2075 7365 => t-|.| -- use │ │ │ │ -0003c3b0: 6420 302e 3436 3931 3931 7320 2863 7075 d 0.469191s (cpu │ │ │ │ -0003c3c0: 293b 2030 2e32 3737 3537 3673 2028 7468 ); 0.277576s (th │ │ │ │ +0003c3b0: 6420 302e 3436 3630 3931 7320 2863 7075 d 0.466091s (cpu │ │ │ │ +0003c3c0: 293b 2030 2e32 3836 3233 3173 2028 7468 ); 0.286231s (th │ │ │ │ 0003c3d0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 0003c3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003c3f0: 2020 2020 2020 7c0a 7c72 6567 756c 6172 |.|regular │ │ │ │ 0003c400: 496e 436f 6469 6d65 6e73 696f 6e3a 2072 InCodimension: r │ │ │ │ 0003c410: 696e 6720 6469 6d65 6e73 696f 6e20 3d34 ing dimension =4 │ │ │ │ 0003c420: 2c20 7468 6572 6520 6172 6520 3134 3635 , there are 1465 │ │ │ │ 0003c430: 3132 3820 706f 7373 6962 6c65 2035 2062 128 possible 5 b │ │ │ │ @@ -15752,17 +15752,17 @@ │ │ │ │ 0003d870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d880: 2d2d 2d2b 0a7c 6931 3020 3a20 7469 6d65 ---+.|i10 : time │ │ │ │ 0003d890: 2072 6567 756c 6172 496e 436f 6469 6d65 regularInCodime │ │ │ │ 0003d8a0: 6e73 696f 6e28 312c 2053 2f4a 2c20 4d61 nsion(1, S/J, Ma │ │ │ │ 0003d8b0: 784d 696e 6f72 733d 3e32 352c 2043 6f64 xMinors=>25, Cod │ │ │ │ 0003d8c0: 696d 4368 6563 6b46 756e 6374 696f 6e20 imCheckFunction │ │ │ │ 0003d8d0: 3d3e 207c 0a7c 202d 2d20 7573 6564 2031 => |.| -- used 1 │ │ │ │ -0003d8e0: 2e32 3833 3639 7320 2863 7075 293b 2030 .28369s (cpu); 0 │ │ │ │ -0003d8f0: 2e38 3339 3238 7320 2874 6872 6561 6429 .83928s (thread) │ │ │ │ -0003d900: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +0003d8e0: 2e33 3239 3834 7320 2863 7075 293b 2030 .32984s (cpu); 0 │ │ │ │ +0003d8f0: 2e38 3931 3831 3673 2028 7468 7265 6164 .891816s (thread │ │ │ │ +0003d900: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 0003d910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d920: 2020 207c 0a7c 7265 6775 6c61 7249 6e43 |.|regularInC │ │ │ │ 0003d930: 6f64 696d 656e 7369 6f6e 3a20 7269 6e67 odimension: ring │ │ │ │ 0003d940: 2064 696d 656e 7369 6f6e 203d 342c 2074 dimension =4, t │ │ │ │ 0003d950: 6865 7265 2061 7265 2031 3436 3531 3238 here are 1465128 │ │ │ │ 0003d960: 2070 6f73 7369 626c 6520 3520 6279 2035 possible 5 by 5 │ │ │ │ 0003d970: 206d 697c 0a7c 7265 6775 6c61 7249 6e43 mi|.|regularInC │ │ │ │ @@ -16296,17 +16296,17 @@ │ │ │ │ 0003fa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003fa80: 2d2d 2d2d 2b0a 7c69 3131 203a 2074 696d ----+.|i11 : tim │ │ │ │ 0003fa90: 6520 7265 6775 6c61 7249 6e43 6f64 696d e regularInCodim │ │ │ │ 0003faa0: 656e 7369 6f6e 2831 2c20 532f 4a2c 204d ension(1, S/J, M │ │ │ │ 0003fab0: 6178 4d69 6e6f 7273 3d3e 3235 2c20 5573 axMinors=>25, Us │ │ │ │ 0003fac0: 654f 6e6c 7946 6173 7443 6f64 696d 203d eOnlyFastCodim = │ │ │ │ 0003fad0: 3e20 7472 7c0a 7c20 2d2d 2075 7365 6420 > tr|.| -- used │ │ │ │ -0003fae0: 302e 3939 3631 3334 7320 2863 7075 293b 0.996134s (cpu); │ │ │ │ -0003faf0: 2030 2e36 3135 3336 3873 2028 7468 7265 0.615368s (thre │ │ │ │ -0003fb00: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ +0003fae0: 312e 3130 3236 3573 2028 6370 7529 3b20 1.10265s (cpu); │ │ │ │ +0003faf0: 302e 3638 3838 3638 7320 2874 6872 6561 0.688868s (threa │ │ │ │ +0003fb00: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0003fb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fb20: 2020 2020 7c0a 7c72 6567 756c 6172 496e |.|regularIn │ │ │ │ 0003fb30: 436f 6469 6d65 6e73 696f 6e3a 2072 696e Codimension: rin │ │ │ │ 0003fb40: 6720 6469 6d65 6e73 696f 6e20 3d34 2c20 g dimension =4, │ │ │ │ 0003fb50: 7468 6572 6520 6172 6520 3134 3635 3132 there are 146512 │ │ │ │ 0003fb60: 3820 706f 7373 6962 6c65 2035 2062 7920 8 possible 5 by │ │ │ │ 0003fb70: 3520 6d69 7c0a 7c72 6567 756c 6172 496e 5 mi|.|regularIn │ │ │ │ @@ -17187,16 +17187,16 @@ │ │ │ │ 00043220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00043230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00043240: 2d2d 2d2d 2d2b 0a7c 6932 203a 2065 6c61 -----+.|i2 : ela │ │ │ │ 00043250: 7073 6564 5469 6d65 2072 6567 756c 6172 psedTime regular │ │ │ │ 00043260: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 00043270: 2054 2c20 5374 7261 7465 6779 3d3e 5374 T, Strategy=>St │ │ │ │ 00043280: 7261 7465 6779 4465 6661 756c 7429 2020 rategyDefault) │ │ │ │ -00043290: 2020 2020 207c 0a7c 202d 2d20 332e 3833 |.| -- 3.83 │ │ │ │ -000432a0: 3839 3173 2065 6c61 7073 6564 2020 2020 891s elapsed │ │ │ │ +00043290: 2020 2020 207c 0a7c 202d 2d20 332e 3230 |.| -- 3.20 │ │ │ │ +000432a0: 3234 3673 2065 6c61 7073 6564 2020 2020 246s elapsed │ │ │ │ 000432b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000432c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000432d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000432e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000432f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -17542,15 +17542,15 @@ │ │ │ │ 00044850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044870: 2d2d 2b0a 7c69 3420 3a20 656c 6170 7365 --+.|i4 : elapse │ │ │ │ 00044880: 6454 696d 6520 7265 6775 6c61 7249 6e43 dTime regularInC │ │ │ │ 00044890: 6f64 696d 656e 7369 6f6e 2831 2c20 542c odimension(1, T, │ │ │ │ 000448a0: 2053 7472 6174 6567 793d 3e4c 6578 536d Strategy=>LexSm │ │ │ │ 000448b0: 616c 6c65 7374 5465 726d 297c 0a7c 202d allestTerm)|.| - │ │ │ │ -000448c0: 2d20 312e 3731 3534 3473 2065 6c61 7073 - 1.71544s elaps │ │ │ │ +000448c0: 2d20 312e 3337 3731 3273 2065 6c61 7073 - 1.37712s elaps │ │ │ │ 000448d0: 6564 2020 2020 2020 2020 2020 2020 2020 ed │ │ │ │ 000448e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000448f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044900: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00044910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/FiniteFittingIdeals.info.gz │ │ │ ├── FiniteFittingIdeals.info │ │ │ │ @@ -1017,17 +1017,17 @@ │ │ │ │ 00003f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00003fa0: 0a7c 6931 3520 3a20 7469 6d65 2049 3d63 .|i15 : time I=c │ │ │ │ 00003fb0: 6f31 4669 7474 696e 6728 4b33 2920 2020 o1Fitting(K3) │ │ │ │ 00003fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003fe0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00003ff0: 2075 7365 6420 302e 3030 3234 3436 3536 used 0.00244656 │ │ │ │ -00004000: 7320 2863 7075 293b 2030 2e30 3032 3434 s (cpu); 0.00244 │ │ │ │ -00004010: 3335 3273 2028 7468 7265 6164 293b 2030 352s (thread); 0 │ │ │ │ +00003ff0: 2075 7365 6420 302e 3030 3238 3034 3238 used 0.00280428 │ │ │ │ +00004000: 7320 2863 7075 293b 2030 2e30 3032 3830 s (cpu); 0.00280 │ │ │ │ +00004010: 3031 3673 2028 7468 7265 6164 293b 2030 016s (thread); 0 │ │ │ │ 00004020: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 00004030: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00004040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004080: 7c0a 7c6f 3135 203d 2069 6465 616c 2028 |.|o15 = ideal ( │ │ │ │ @@ -1055,17 +1055,17 @@ │ │ │ │ 000041e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000041f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 -------+.|i16 : │ │ │ │ 00004200: 7469 6d65 204a 3d66 6974 7469 6e67 4964 time J=fittingId │ │ │ │ 00004210: 6561 6c28 322d 312c 636f 6b65 7220 4b33 eal(2-1,coker K3 │ │ │ │ 00004220: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ 00004230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004240: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00004250: 3030 3634 3835 3373 2028 6370 7529 3b20 0064853s (cpu); │ │ │ │ -00004260: 302e 3030 3634 3835 3539 7320 2874 6872 0.00648559s (thr │ │ │ │ -00004270: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +00004250: 3030 3637 3139 3336 7320 2863 7075 293b 00671936s (cpu); │ │ │ │ +00004260: 2030 2e30 3036 3732 3231 3873 2028 7468 0.00672218s (th │ │ │ │ +00004270: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00004280: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00004290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000042a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000042b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000042c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000042d0: 2020 2020 2020 2020 7c0a 7c6f 3136 203a |.|o16 : │ │ │ │ 000042e0: 2049 6465 616c 206f 6620 5220 2020 2020 Ideal of R │ │ ├── ./usr/share/info/ForeignFunctions.info.gz │ │ │ ├── ForeignFunctions.info │ │ │ │ @@ -3506,15 +3506,15 @@ │ │ │ │ 0000db10: 7820 3d20 6d61 6c6c 6f63 2038 2020 2020 x = malloc 8 │ │ │ │ 0000db20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000db30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000db40: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000db50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000db60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000db70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000db80: 0a7c 6f31 3720 3d20 3078 3766 3731 6630 .|o17 = 0x7f71f0 │ │ │ │ +0000db80: 0a7c 6f31 3720 3d20 3078 3766 6262 6434 .|o17 = 0x7fbbd4 │ │ │ │ 0000db90: 3036 6137 3030 2020 2020 2020 2020 2020 06a700 │ │ │ │ 0000dba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000dbb0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000dbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000dbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000dbf0: 2020 2020 2020 207c 0a7c 6f31 3720 3a20 |.|o17 : │ │ │ │ @@ -3640,15 +3640,15 @@ │ │ │ │ 0000e370: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2070 -------+.|i2 : p │ │ │ │ 0000e380: 6565 6b20 6d70 6672 2020 2020 2020 2020 eek mpfr │ │ │ │ 0000e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e3a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000e3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e3c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0000e3d0: 6f32 203d 2053 6861 7265 644c 6962 7261 o2 = SharedLibra │ │ │ │ -0000e3e0: 7279 7b30 7837 6665 3466 3733 3839 6163 ry{0x7fe4f7389ac │ │ │ │ +0000e3e0: 7279 7b30 7837 6661 3332 3264 6235 6163 ry{0x7fa322db5ac │ │ │ │ 0000e3f0: 302c 206d 7066 727d 7c0a 2b2d 2d2d 2d2d 0, mpfr}|.+----- │ │ │ │ 0000e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e420: 2d2d 2d2b 0a2a 204d 656e 753a 0a0a 2a20 ---+.* Menu:..* │ │ │ │ 0000e430: 6f70 656e 5368 6172 6564 4c69 6272 6172 openSharedLibrar │ │ │ │ 0000e440: 793a 3a20 2020 2020 2020 2020 2020 6f70 y:: op │ │ │ │ 0000e450: 656e 2061 2073 6861 7265 6420 6c69 6272 en a shared libr │ │ │ │ @@ -5395,29 +5395,29 @@ │ │ │ │ 00015120: 6520 706f 696e 7465 722e 0a0a 2b2d 2d2d e pointer...+--- │ │ │ │ 00015130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00015150: 6931 203a 2070 7472 203d 2061 6464 7265 i1 : ptr = addre │ │ │ │ 00015160: 7373 2069 6e74 2030 2020 2020 2020 2020 ss int 0 │ │ │ │ 00015170: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00015180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015190: 2020 207c 0a7c 6f31 203d 2030 7837 6665 |.|o1 = 0x7fe │ │ │ │ -000151a0: 3465 3738 3839 6635 3020 2020 2020 2020 4e7889f50 │ │ │ │ +00015190: 2020 207c 0a7c 6f31 203d 2030 7837 6661 |.|o1 = 0x7fa │ │ │ │ +000151a0: 3330 6533 3262 6631 3020 2020 2020 2020 30e32bf10 │ │ │ │ 000151b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000151c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000151d0: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ 000151e0: 2050 6f69 6e74 6572 2020 2020 2020 2020 Pointer │ │ │ │ 000151f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00015200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00015220: 0a7c 6932 203a 2076 6f69 6473 7461 7220 .|i2 : voidstar │ │ │ │ 00015230: 7074 7220 2020 2020 2020 2020 2020 2020 ptr │ │ │ │ 00015240: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00015250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015260: 2020 2020 207c 0a7c 6f32 203d 2030 7837 |.|o2 = 0x7 │ │ │ │ -00015270: 6665 3465 3738 3839 6635 3020 2020 2020 fe4e7889f50 │ │ │ │ +00015270: 6661 3330 6533 3262 6631 3020 2020 2020 fa30e32bf10 │ │ │ │ 00015280: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00015290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000152a0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ 000152b0: 203a 2046 6f72 6569 676e 4f62 6a65 6374 : ForeignObject │ │ │ │ 000152c0: 206f 6620 7479 7065 2076 6f69 642a 7c0a of type void*|. │ │ │ │ 000152d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000152e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5498,16 +5498,16 @@ │ │ │ │ 00015790: 7970 6520 696e 7433 327c 0a2b 2d2d 2d2d ype int32|.+---- │ │ │ │ 000157a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000157b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 000157c0: 3220 3a20 7074 7220 3d20 6164 6472 6573 2 : ptr = addres │ │ │ │ 000157d0: 7320 7820 2020 2020 2020 2020 2020 207c s x | │ │ │ │ 000157e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000157f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015800: 2020 7c0a 7c6f 3220 3d20 3078 3766 6534 |.|o2 = 0x7fe4 │ │ │ │ -00015810: 6537 3635 3164 6230 2020 2020 2020 2020 e7651db0 │ │ │ │ +00015800: 2020 7c0a 7c6f 3220 3d20 3078 3766 6133 |.|o2 = 0x7fa3 │ │ │ │ +00015810: 3064 6164 3264 3730 2020 2020 2020 2020 0dad2d70 │ │ │ │ 00015820: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00015830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015840: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ 00015850: 506f 696e 7465 7220 2020 2020 2020 2020 Pointer │ │ │ │ 00015860: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00015870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ @@ -5648,16 +5648,16 @@ │ │ │ │ 000160f0: 696e 7465 7273 2e0a 0a2b 2d2d 2d2d 2d2d inters...+------ │ │ │ │ 00016100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ 00016120: 203a 2070 7472 203d 2076 6f69 6473 7461 : ptr = voidsta │ │ │ │ 00016130: 7220 6164 6472 6573 7320 696e 7420 357c r address int 5| │ │ │ │ 00016140: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00016150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016160: 2020 207c 0a7c 6f31 203d 2030 7837 6665 |.|o1 = 0x7fe │ │ │ │ -00016170: 3465 3736 3531 6232 3020 2020 2020 2020 4e7651b20 │ │ │ │ +00016160: 2020 207c 0a7c 6f31 203d 2030 7837 6661 |.|o1 = 0x7fa │ │ │ │ +00016170: 3330 6461 6432 6233 3020 2020 2020 2020 30dad2b30 │ │ │ │ 00016180: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00016190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000161a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ 000161b0: 203a 2046 6f72 6569 676e 4f62 6a65 6374 : ForeignObject │ │ │ │ 000161c0: 206f 6620 7479 7065 2076 6f69 642a 207c of type void* | │ │ │ │ 000161d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 000161e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5746,15 +5746,15 @@ │ │ │ │ 00016710: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ 00016720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016730: 2d2d 2d2d 2b0a 7c69 3120 3a20 7074 7220 ----+.|i1 : ptr │ │ │ │ 00016740: 3d20 6765 744d 656d 6f72 7920 3820 2020 = getMemory 8 │ │ │ │ 00016750: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00016760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016770: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ -00016780: 3d20 3078 3766 6534 6632 6261 6339 6230 = 0x7fe4f2bac9b0 │ │ │ │ +00016780: 3d20 3078 3766 6133 3065 6330 6538 3630 = 0x7fa30ec0e860 │ │ │ │ 00016790: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000167a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000167b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000167c0: 7c0a 7c6f 3120 3a20 466f 7265 6967 6e4f |.|o1 : ForeignO │ │ │ │ 000167d0: 626a 6563 7420 6f66 2074 7970 6520 766f bject of type vo │ │ │ │ 000167e0: 6964 2a7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d id*|.+---------- │ │ │ │ 000167f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5769,16 +5769,16 @@ │ │ │ │ 00016880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000168a0: 7c69 3220 3a20 7074 7220 3d20 6765 744d |i2 : ptr = getM │ │ │ │ 000168b0: 656d 6f72 7928 382c 2041 746f 6d69 6320 emory(8, Atomic │ │ │ │ 000168c0: 3d3e 2074 7275 6529 7c0a 7c20 2020 2020 => true)|.| │ │ │ │ 000168d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000168e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000168f0: 2020 7c0a 7c6f 3220 3d20 3078 3766 6534 |.|o2 = 0x7fe4 │ │ │ │ -00016900: 6537 3635 3162 6530 2020 2020 2020 2020 e7651be0 │ │ │ │ +000168f0: 2020 7c0a 7c6f 3220 3d20 3078 3766 6133 |.|o2 = 0x7fa3 │ │ │ │ +00016900: 3064 6164 3262 6130 2020 2020 2020 2020 0dad2ba0 │ │ │ │ 00016910: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00016920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016940: 2020 2020 2020 7c0a 7c6f 3220 3a20 466f |.|o2 : Fo │ │ │ │ 00016950: 7265 6967 6e4f 626a 6563 7420 6f66 2074 reignObject of t │ │ │ │ 00016960: 7970 6520 766f 6964 2a20 2020 2020 2020 ype void* │ │ │ │ 00016970: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ @@ -5797,16 +5797,16 @@ │ │ │ │ 00016a40: 6361 6c6c 792e 0a0a 2b2d 2d2d 2d2d 2d2d cally...+------- │ │ │ │ 00016a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016a60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ 00016a70: 2070 7472 203d 2067 6574 4d65 6d6f 7279 ptr = getMemory │ │ │ │ 00016a80: 2069 6e74 2020 2020 2020 2020 7c0a 7c20 int |.| │ │ │ │ 00016a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016aa0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016ab0: 0a7c 6f33 203d 2030 7837 6665 3465 3736 .|o3 = 0x7fe4e76 │ │ │ │ -00016ac0: 3531 6166 3020 2020 2020 2020 2020 2020 51af0 │ │ │ │ +00016ab0: 0a7c 6f33 203d 2030 7837 6661 3330 6461 .|o3 = 0x7fa30da │ │ │ │ +00016ac0: 6432 6139 3020 2020 2020 2020 2020 2020 d2a90 │ │ │ │ 00016ad0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00016ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016af0: 2020 2020 207c 0a7c 6f33 203a 2046 6f72 |.|o3 : For │ │ │ │ 00016b00: 6569 676e 4f62 6a65 6374 206f 6620 7479 eignObject of ty │ │ │ │ 00016b10: 7065 2076 6f69 642a 7c0a 2b2d 2d2d 2d2d pe void*|.+----- │ │ │ │ 00016b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 -----------+..Se │ │ │ │ @@ -5967,18 +5967,18 @@ │ │ │ │ 000174e0: 7320 696e 7420 312c 2061 6464 7265 7373 s int 1, address │ │ │ │ 000174f0: 2069 6e74 2032 7d20 2020 2020 2020 2020 int 2} │ │ │ │ 00017500: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00017510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017540: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017550: 0a7c 6f33 203d 207b 3078 3766 6534 6537 .|o3 = {0x7fe4e7 │ │ │ │ -00017560: 3637 3561 3130 2c20 3078 3766 6534 6537 675a10, 0x7fe4e7 │ │ │ │ -00017570: 3637 3561 3030 2c20 3078 3766 6534 6537 675a00, 0x7fe4e7 │ │ │ │ -00017580: 3637 3539 6630 7d20 2020 2020 2020 2020 6759f0} │ │ │ │ +00017550: 0a7c 6f33 203d 207b 3078 3766 6133 3064 .|o3 = {0x7fa30d │ │ │ │ +00017560: 6165 6261 3430 2c20 3078 3766 6133 3064 aeba40, 0x7fa30d │ │ │ │ +00017570: 6165 6261 3330 2c20 3078 3766 6133 3064 aeba30, 0x7fa30d │ │ │ │ +00017580: 6165 6261 3230 7d20 2020 2020 2020 2020 aeba20} │ │ │ │ 00017590: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000175a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000175b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000175c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000175d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000175e0: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ 000175f0: 2046 6f72 6569 676e 4f62 6a65 6374 206f ForeignObject o │ │ │ │ @@ -6433,17 +6433,17 @@ │ │ │ │ 00019200: 7373 2069 6e74 2030 2c20 6164 6472 6573 ss int 0, addres │ │ │ │ 00019210: 7320 696e 7420 312c 2061 6464 7265 7373 s int 1, address │ │ │ │ 00019220: 2069 6e74 2032 7d7c 0a7c 2020 2020 2020 int 2}|.| │ │ │ │ 00019230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019260: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ -00019270: 207b 3078 3766 6534 6537 3637 3538 3130 {0x7fe4e7675810 │ │ │ │ -00019280: 2c20 3078 3766 6534 6537 3637 3538 3030 , 0x7fe4e7675800 │ │ │ │ -00019290: 2c20 3078 3766 6534 6537 3637 3537 6630 , 0x7fe4e76757f0 │ │ │ │ +00019270: 207b 3078 3766 6133 3064 6165 6238 3830 {0x7fa30daeb880 │ │ │ │ +00019280: 2c20 3078 3766 6133 3064 6165 6238 3730 , 0x7fa30daeb870 │ │ │ │ +00019290: 2c20 3078 3766 6133 3064 6165 6238 3630 , 0x7fa30daeb860 │ │ │ │ 000192a0: 7d20 2020 2020 2020 2020 207c 0a7c 2020 } |.| │ │ │ │ 000192b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000192c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000192d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000192e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000192f0: 6f32 203a 2046 6f72 6569 676e 4f62 6a65 o2 : ForeignObje │ │ │ │ 00019300: 6374 206f 6620 7479 7065 2076 6f69 642a ct of type void* │ │ │ │ @@ -7911,15 +7911,15 @@ │ │ │ │ 0001ee60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ee70: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001ee80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ee90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eeb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0001eec0: 7c6f 3220 3d20 4861 7368 5461 626c 657b |o2 = HashTable{ │ │ │ │ -0001eed0: 2262 6172 2220 3d3e 2036 2e39 3437 3431 "bar" => 6.94741 │ │ │ │ +0001eed0: 2262 6172 2220 3d3e 2036 2e39 3333 3632 "bar" => 6.93362 │ │ │ │ 0001eee0: 652d 3331 307d 2020 2020 2020 2020 2020 e-310} │ │ │ │ 0001eef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ef00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001ef10: 2020 2020 2020 2020 2020 2020 2020 2266 "f │ │ │ │ 0001ef20: 6f6f 2220 3d3e 2032 3720 2020 2020 2020 oo" => 27 │ │ │ │ 0001ef30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ef40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8162,26 +8162,26 @@ │ │ │ │ 0001fe10: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 7065 ------+.|i2 : pe │ │ │ │ 0001fe20: 656b 2078 2020 2020 2020 2020 2020 2020 ek x │ │ │ │ 0001fe30: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0001fe40: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0001fe50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fe60: 2020 2020 2020 7c0a 7c6f 3220 3d20 696e |.|o2 = in │ │ │ │ 0001fe70: 7433 327b 4164 6472 6573 7320 3d3e 2030 t32{Address => 0 │ │ │ │ -0001fe80: 7837 6665 3465 3736 3333 3237 307d 7c0a x7fe4e7633270}|. │ │ │ │ +0001fe80: 7837 6661 3330 6461 6163 3331 307d 7c0a x7fa30daac310}|. │ │ │ │ 0001fe90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0001fea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001feb0: 2d2d 2d2d 2d2d 2b0a 0a54 6f20 6765 7420 ------+..To get │ │ │ │ 0001fec0: 7468 6973 2c20 7573 6520 2a6e 6f74 6520 this, use *note │ │ │ │ 0001fed0: 6164 6472 6573 733a 2061 6464 7265 7373 address: address │ │ │ │ 0001fee0: 2c2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...+----------- │ │ │ │ 0001fef0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ 0001ff00: 6164 6472 6573 7320 7820 2020 2020 7c0a address x |. │ │ │ │ 0001ff10: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0001ff20: 2020 2020 7c0a 7c6f 3320 3d20 3078 3766 |.|o3 = 0x7f │ │ │ │ -0001ff30: 6534 6537 3633 3332 3730 7c0a 7c20 2020 e4e7633270|.| │ │ │ │ +0001ff30: 6133 3064 6161 6333 3130 7c0a 7c20 2020 a30daac310|.| │ │ │ │ 0001ff40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ff50: 7c0a 7c6f 3320 3a20 506f 696e 7465 7220 |.|o3 : Pointer │ │ │ │ 0001ff60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 0001ff70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a55 ------------+..U │ │ │ │ 0001ff80: 7365 202a 6e6f 7465 2063 6c61 7373 3a20 se *note class: │ │ │ │ 0001ff90: 284d 6163 6175 6c61 7932 446f 6329 636c (Macaulay2Doc)cl │ │ │ │ 0001ffa0: 6173 732c 2074 6f20 6465 7465 726d 696e ass, to determin │ │ │ │ @@ -8883,29 +8883,29 @@ │ │ │ │ 00022b20: 626a 6563 7473 2e0a 0a2b 2d2d 2d2d 2d2d bjects...+------ │ │ │ │ 00022b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ 00022b50: 3a20 7820 3d20 766f 6964 7374 6172 2061 : x = voidstar a │ │ │ │ 00022b60: 6464 7265 7373 2069 6e74 2035 207c 0a7c ddress int 5 |.| │ │ │ │ 00022b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022b90: 7c0a 7c6f 3520 3d20 3078 3766 6534 6537 |.|o5 = 0x7fe4e7 │ │ │ │ -00022ba0: 3635 3166 6130 2020 2020 2020 2020 2020 651fa0 │ │ │ │ +00022b90: 7c0a 7c6f 3520 3d20 3078 3766 6133 3064 |.|o5 = 0x7fa30d │ │ │ │ +00022ba0: 6164 3230 3130 2020 2020 2020 2020 2020 ad2010 │ │ │ │ 00022bb0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00022bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022bd0: 2020 2020 2020 7c0a 7c6f 3520 3a20 466f |.|o5 : Fo │ │ │ │ 00022be0: 7265 6967 6e4f 626a 6563 7420 6f66 2074 reignObject of t │ │ │ │ 00022bf0: 7970 6520 766f 6964 2a7c 0a2b 2d2d 2d2d ype void*|.+---- │ │ │ │ 00022c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00022c20: 3620 3a20 7661 6c75 6520 7820 2020 2020 6 : value x │ │ │ │ 00022c30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00022c40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00022c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c60: 2020 7c0a 7c6f 3620 3d20 3078 3766 6534 |.|o6 = 0x7fe4 │ │ │ │ -00022c70: 6537 3635 3166 6130 2020 2020 2020 2020 e7651fa0 │ │ │ │ +00022c60: 2020 7c0a 7c6f 3620 3d20 3078 3766 6133 |.|o6 = 0x7fa3 │ │ │ │ +00022c70: 3064 6164 3230 3130 2020 2020 2020 2020 0dad2010 │ │ │ │ 00022c80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00022c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ca0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20 |.|o6 : │ │ │ │ 00022cb0: 506f 696e 7465 7220 2020 2020 2020 2020 Pointer │ │ │ │ 00022cc0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00022cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ @@ -9432,50 +9432,50 @@ │ │ │ │ 00024d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024d80: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2063 -------+.|i5 : c │ │ │ │ 00024d90: 6f6c 6c65 6374 4761 7262 6167 6528 2920 ollectGarbage() │ │ │ │ 00024da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024dc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00024dd0: 6672 6565 696e 6720 6d65 6d6f 7279 2061 freeing memory a │ │ │ │ -00024de0: 7420 3078 3766 6534 6463 3037 6663 6630 t 0x7fe4dc07fcf0 │ │ │ │ +00024de0: 7420 3078 3766 6132 6638 3037 6664 3530 t 0x7fa2f807fd50 │ │ │ │ 00024df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024e10: 2020 207c 0a7c 6672 6565 696e 6720 6d65 |.|freeing me │ │ │ │ -00024e20: 6d6f 7279 2061 7420 3078 3766 6534 6463 mory at 0x7fe4dc │ │ │ │ -00024e30: 3037 6636 3530 2020 2020 2020 2020 2020 07f650 │ │ │ │ +00024e20: 6d6f 7279 2061 7420 3078 3766 6132 6638 mory at 0x7fa2f8 │ │ │ │ +00024e30: 3037 6664 3730 2020 2020 2020 2020 2020 07fd70 │ │ │ │ 00024e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024e50: 2020 2020 2020 2020 207c 0a7c 6672 6565 |.|free │ │ │ │ 00024e60: 696e 6720 6d65 6d6f 7279 2061 7420 3078 ing memory at 0x │ │ │ │ -00024e70: 3766 6534 6463 3037 6664 3330 2020 2020 7fe4dc07fd30 │ │ │ │ +00024e70: 3766 6132 6638 3037 6664 3930 2020 2020 7fa2f807fd90 │ │ │ │ 00024e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024e90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00024ea0: 0a7c 6672 6565 696e 6720 6d65 6d6f 7279 .|freeing memory │ │ │ │ -00024eb0: 2061 7420 3078 3766 6534 6463 3037 6664 at 0x7fe4dc07fd │ │ │ │ -00024ec0: 3130 2020 2020 2020 2020 2020 2020 2020 10 │ │ │ │ +00024eb0: 2061 7420 3078 3766 6132 6638 3037 6664 at 0x7fa2f807fd │ │ │ │ +00024ec0: 6230 2020 2020 2020 2020 2020 2020 2020 b0 │ │ │ │ 00024ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024ee0: 2020 2020 207c 0a7c 6672 6565 696e 6720 |.|freeing │ │ │ │ -00024ef0: 6d65 6d6f 7279 2061 7420 3078 3766 6534 memory at 0x7fe4 │ │ │ │ -00024f00: 6463 3037 6664 6230 2020 2020 2020 2020 dc07fdb0 │ │ │ │ +00024ef0: 6d65 6d6f 7279 2061 7420 3078 3766 6132 memory at 0x7fa2 │ │ │ │ +00024f00: 6638 3037 6636 3330 2020 2020 2020 2020 f807f630 │ │ │ │ 00024f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f20: 2020 2020 2020 2020 2020 207c 0a7c 6672 |.|fr │ │ │ │ 00024f30: 6565 696e 6720 6d65 6d6f 7279 2061 7420 eeing memory at │ │ │ │ -00024f40: 3078 3766 6534 6463 3037 6664 3930 2020 0x7fe4dc07fd90 │ │ │ │ +00024f40: 3078 3766 6132 6638 3037 6663 6630 2020 0x7fa2f807fcf0 │ │ │ │ 00024f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f70: 207c 0a7c 6672 6565 696e 6720 6d65 6d6f |.|freeing memo │ │ │ │ -00024f80: 7279 2061 7420 3078 3766 6534 6463 3037 ry at 0x7fe4dc07 │ │ │ │ -00024f90: 6664 3730 2020 2020 2020 2020 2020 2020 fd70 │ │ │ │ +00024f80: 7279 2061 7420 3078 3766 6132 6638 3037 ry at 0x7fa2f807 │ │ │ │ +00024f90: 6664 3130 2020 2020 2020 2020 2020 2020 fd10 │ │ │ │ 00024fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024fb0: 2020 2020 2020 207c 0a7c 6672 6565 696e |.|freein │ │ │ │ 00024fc0: 6720 6d65 6d6f 7279 2061 7420 3078 3766 g memory at 0x7f │ │ │ │ -00024fd0: 6534 6463 3037 6664 3530 2020 2020 2020 e4dc07fd50 │ │ │ │ +00024fd0: 6132 6638 3037 6636 3530 2020 2020 2020 a2f807f650 │ │ │ │ 00024fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024ff0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00025000: 6672 6565 696e 6720 6d65 6d6f 7279 2061 freeing memory a │ │ │ │ -00025010: 7420 3078 3766 6534 6463 3037 6636 3330 t 0x7fe4dc07f630 │ │ │ │ +00025010: 7420 3078 3766 6132 6638 3037 6664 3330 t 0x7fa2f807fd30 │ │ │ │ 00025020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025040: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00025050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025080: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 ---------+..See │ │ │ │ @@ -9549,49 +9549,49 @@ │ │ │ │ 000254c0: 2d2d 2d2b 0a7c 6932 203a 2070 6565 6b20 ---+.|i2 : peek │ │ │ │ 000254d0: 7820 2020 2020 2020 2020 2020 2020 2020 x │ │ │ │ 000254e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000254f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025510: 2020 207c 0a7c 6f32 203d 2069 6e74 3332 |.|o2 = int32 │ │ │ │ 00025520: 7b41 6464 7265 7373 203d 3e20 3078 3766 {Address => 0x7f │ │ │ │ -00025530: 6534 6537 3633 3332 6430 7d7c 0a2b 2d2d e4e76332d0}|.+-- │ │ │ │ +00025530: 6133 3064 6161 6333 3330 7d7c 0a2b 2d2d a30daac330}|.+-- │ │ │ │ 00025540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025560: 2d2d 2d2b 0a0a 5468 6573 6520 706f 696e ---+..These poin │ │ │ │ 00025570: 7465 7273 2063 616e 2062 6520 6163 6365 ters can be acce │ │ │ │ 00025580: 7373 6564 2075 7369 6e67 202a 6e6f 7465 ssed using *note │ │ │ │ 00025590: 2061 6464 7265 7373 3a20 6164 6472 6573 address: addres │ │ │ │ 000255a0: 732c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d s,...+---------- │ │ │ │ 000255b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ 000255c0: 3a20 7074 7220 3d20 6164 6472 6573 7320 : ptr = address │ │ │ │ 000255d0: 787c 0a7c 2020 2020 2020 2020 2020 2020 x|.| │ │ │ │ 000255e0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -000255f0: 3078 3766 6534 6537 3633 3332 6430 207c 0x7fe4e76332d0 | │ │ │ │ +000255f0: 3078 3766 6133 3064 6161 6333 3330 207c 0x7fa30daac330 | │ │ │ │ 00025600: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00025610: 2020 2020 2020 7c0a 7c6f 3320 3a20 506f |.|o3 : Po │ │ │ │ 00025620: 696e 7465 7220 2020 2020 2020 207c 0a2b inter |.+ │ │ │ │ 00025630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025640: 2d2d 2d2d 2b0a 0a53 696d 706c 6520 6172 ----+..Simple ar │ │ │ │ 00025650: 6974 686d 6574 6963 2063 616e 2062 6520 ithmetic can be │ │ │ │ 00025660: 7065 7266 6f72 6d65 6420 6f6e 2070 6f69 performed on poi │ │ │ │ 00025670: 6e74 6572 732e 0a0a 2b2d 2d2d 2d2d 2d2d nters...+------- │ │ │ │ 00025680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00025690: 3420 3a20 7074 7220 2b20 3520 2020 2020 4 : ptr + 5 │ │ │ │ 000256a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000256b0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ -000256c0: 3078 3766 6534 6537 3633 3332 6435 7c0a 0x7fe4e76332d5|. │ │ │ │ +000256c0: 3078 3766 6133 3064 6161 6333 3335 7c0a 0x7fa30daac335|. │ │ │ │ 000256d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000256e0: 2020 2020 7c0a 7c6f 3420 3a20 506f 696e |.|o4 : Poin │ │ │ │ 000256f0: 7465 7220 2020 2020 2020 7c0a 2b2d 2d2d ter |.+--- │ │ │ │ 00025700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025710: 2b0a 7c69 3520 3a20 7074 7220 2d20 3320 +.|i5 : ptr - 3 │ │ │ │ 00025720: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00025730: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00025740: 3520 3d20 3078 3766 6534 6537 3633 3332 5 = 0x7fe4e76332 │ │ │ │ -00025750: 6364 7c0a 7c20 2020 2020 2020 2020 2020 cd|.| │ │ │ │ +00025740: 3520 3d20 3078 3766 6133 3064 6161 6333 5 = 0x7fa30daac3 │ │ │ │ +00025750: 3264 7c0a 7c20 2020 2020 2020 2020 2020 2d|.| │ │ │ │ 00025760: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20 |.|o5 : │ │ │ │ 00025770: 506f 696e 7465 7220 2020 2020 2020 7c0a Pointer |. │ │ │ │ 00025780: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00025790: 2d2d 2d2d 2b0a 2a20 4d65 6e75 3a0a 0a2a ----+.* Menu:..* │ │ │ │ 000257a0: 206e 756c 6c50 6f69 6e74 6572 3a3a 2020 nullPointer:: │ │ │ │ 000257b0: 2020 2020 2020 2020 2020 2020 2020 2074 t │ │ │ │ 000257c0: 6865 206e 756c 6c20 706f 696e 7465 720a he null pointer. │ │ │ │ @@ -9760,15 +9760,15 @@ │ │ │ │ 000261f0: 740a 7573 6564 2062 7920 6c69 6266 6669 t.used by libffi │ │ │ │ 00026200: 2074 6f20 6964 656e 7469 6679 2074 6865 to identify the │ │ │ │ 00026210: 2074 7970 652e 0a0a 2b2d 2d2d 2d2d 2d2d type...+------- │ │ │ │ 00026220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00026230: 3120 3a20 6164 6472 6573 7320 696e 7420 1 : address int │ │ │ │ 00026240: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00026250: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -00026260: 3078 3535 3566 6337 6632 6234 3030 7c0a 0x555fc7f2b400|. │ │ │ │ +00026260: 3078 3536 3334 3061 3938 3934 3030 7c0a 0x56340a989400|. │ │ │ │ 00026270: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00026280: 2020 2020 7c0a 7c6f 3120 3a20 506f 696e |.|o1 : Poin │ │ │ │ 00026290: 7465 7220 2020 2020 2020 7c0a 2b2d 2d2d ter |.+--- │ │ │ │ 000262a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000262b0: 2b0a 0a49 6620 7820 6973 2061 2066 6f72 +..If x is a for │ │ │ │ 000262c0: 6569 676e 206f 626a 6563 742c 2074 6865 eign object, the │ │ │ │ 000262d0: 6e20 7468 6973 2072 6574 7572 6e73 2074 n this returns t │ │ │ │ @@ -9777,16 +9777,16 @@ │ │ │ │ 00026300: 6861 7665 7320 6c69 6b65 2074 6865 2026 haves like the & │ │ │ │ 00026310: 2022 6164 6472 6573 732d 6f66 2220 6f70 "address-of" op │ │ │ │ 00026320: 6572 6174 6f72 2069 6e20 432e 0a0a 2b2d erator in C...+- │ │ │ │ 00026330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026340: 2d2d 2b0a 7c69 3220 3a20 6164 6472 6573 --+.|i2 : addres │ │ │ │ 00026350: 7320 696e 7420 3520 7c0a 7c20 2020 2020 s int 5 |.| │ │ │ │ 00026360: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00026370: 7c6f 3220 3d20 3078 3766 6534 6537 3635 |o2 = 0x7fe4e765 │ │ │ │ -00026380: 3164 3030 7c0a 7c20 2020 2020 2020 2020 1d00|.| │ │ │ │ +00026370: 7c6f 3220 3d20 3078 3766 6133 3064 6164 |o2 = 0x7fa30dad │ │ │ │ +00026380: 3264 3630 7c0a 7c20 2020 2020 2020 2020 2d60|.| │ │ │ │ 00026390: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ 000263a0: 3a20 506f 696e 7465 7220 2020 2020 2020 : Pointer │ │ │ │ 000263b0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 000263c0: 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320 746f ------+..Ways to │ │ │ │ 000263d0: 2075 7365 2061 6464 7265 7373 3a0a 3d3d use address:.== │ │ │ │ 000263e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 000263f0: 3d3d 0a0a 2020 2a20 2261 6464 7265 7373 ==.. * "address │ │ │ │ @@ -9868,16 +9868,16 @@ │ │ │ │ 000268b0: 7970 6520 696e 7433 327c 0a2b 2d2d 2d2d ype int32|.+---- │ │ │ │ 000268c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000268d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 000268e0: 3220 3a20 7074 7220 3d20 6164 6472 6573 2 : ptr = addres │ │ │ │ 000268f0: 7320 7820 2020 2020 2020 2020 2020 207c s x | │ │ │ │ 00026900: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00026910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026920: 2020 7c0a 7c6f 3220 3d20 3078 3766 6534 |.|o2 = 0x7fe4 │ │ │ │ -00026930: 6537 3635 3164 6230 2020 2020 2020 2020 e7651db0 │ │ │ │ +00026920: 2020 7c0a 7c6f 3220 3d20 3078 3766 6133 |.|o2 = 0x7fa3 │ │ │ │ +00026930: 3064 6164 3264 3430 2020 2020 2020 2020 0dad2d40 │ │ │ │ 00026940: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00026950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026960: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ 00026970: 506f 696e 7465 7220 2020 2020 2020 2020 Pointer │ │ │ │ 00026980: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00026990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000269a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ ├── ./usr/share/info/FourTiTwo.info.gz │ │ │ ├── FourTiTwo.info │ │ │ │ @@ -838,25 +838,25 @@ │ │ │ │ 00003450: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00003460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003470: 2d2d 2d2d 2b0a 7c69 3220 3a20 7320 3d20 ----+.|i2 : s = │ │ │ │ 00003480: 7465 6d70 6f72 6172 7946 696c 654e 616d temporaryFileNam │ │ │ │ 00003490: 6528 2920 2020 2020 207c 0a7c 2020 2020 e() |.| │ │ │ │ 000034a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000034b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000034c0: 7c6f 3220 3d20 2f74 6d70 2f4d 322d 3135 |o2 = /tmp/M2-15 │ │ │ │ -000034d0: 3135 302d 302f 3020 2020 2020 2020 2020 150-0/0 │ │ │ │ +000034c0: 7c6f 3220 3d20 2f74 6d70 2f4d 322d 3138 |o2 = /tmp/M2-18 │ │ │ │ +000034d0: 3938 302d 302f 3020 2020 2020 2020 2020 980-0/0 │ │ │ │ 000034e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 000034f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003500: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ 00003510: 4620 3d20 6f70 656e 4f75 7428 7329 2020 F = openOut(s) │ │ │ │ 00003520: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00003530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003550: 2020 7c0a 7c6f 3320 3d20 2f74 6d70 2f4d |.|o3 = /tmp/M │ │ │ │ -00003560: 322d 3135 3135 302d 302f 3020 2020 2020 2-15150-0/0 │ │ │ │ +00003560: 322d 3138 3938 302d 302f 3020 2020 2020 2-18980-0/0 │ │ │ │ 00003570: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00003580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003590: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 000035a0: 3320 3a20 4669 6c65 2020 2020 2020 2020 3 : File │ │ │ │ 000035b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000035c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 000035d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -866,15 +866,15 @@ │ │ │ │ 00003610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003630: 2b0a 7c69 3520 3a20 636c 6f73 6528 4629 +.|i5 : close(F) │ │ │ │ 00003640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003650: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00003660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003670: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ -00003680: 3d20 2f74 6d70 2f4d 322d 3135 3135 302d = /tmp/M2-15150- │ │ │ │ +00003680: 3d20 2f74 6d70 2f4d 322d 3138 3938 302d = /tmp/M2-18980- │ │ │ │ 00003690: 302f 3020 2020 2020 2020 2020 2020 207c 0/0 | │ │ │ │ 000036a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000036b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000036c0: 2020 2020 7c0a 7c6f 3520 3a20 4669 6c65 |.|o5 : File │ │ │ │ 000036d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000036e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 000036f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ ├── ./usr/share/info/FrobeniusThresholds.info.gz │ │ │ ├── FrobeniusThresholds.info │ │ │ │ @@ -2692,16 +2692,16 @@ │ │ │ │ 0000a830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a850: 2d2d 2d2d 2d2b 0a7c 6932 3720 3a20 7469 -----+.|i27 : ti │ │ │ │ 0000a860: 6d65 206e 756d 6572 6963 2066 7074 2866 me numeric fpt(f │ │ │ │ 0000a870: 2c20 4465 7074 684f 6653 6561 7263 6820 , DepthOfSearch │ │ │ │ 0000a880: 3d3e 2033 2c20 4669 6e61 6c41 7474 656d => 3, FinalAttem │ │ │ │ 0000a890: 7074 203d 3e20 7472 7565 297c 0a7c 202d pt => true)|.| - │ │ │ │ -0000a8a0: 2d20 7573 6564 2033 2e36 3934 3433 7320 - used 3.69443s │ │ │ │ -0000a8b0: 2863 7075 293b 2031 2e35 3834 3939 7320 (cpu); 1.58499s │ │ │ │ +0000a8a0: 2d20 7573 6564 2034 2e32 3333 3939 7320 - used 4.23399s │ │ │ │ +0000a8b0: 2863 7075 293b 2031 2e38 3337 3934 7320 (cpu); 1.83794s │ │ │ │ 0000a8c0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 0000a8d0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0000a8e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a920: 2020 2020 2020 207c 0a7c 6f32 3720 3d20 |.|o27 = │ │ │ │ @@ -2723,17 +2723,17 @@ │ │ │ │ 0000aa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000aa40: 0a7c 6932 3820 3a20 7469 6d65 2066 7074 .|i28 : time fpt │ │ │ │ 0000aa50: 2866 2c20 4465 7074 684f 6653 6561 7263 (f, DepthOfSearc │ │ │ │ 0000aa60: 6820 3d3e 2033 2c20 4174 7465 6d70 7473 h => 3, Attempts │ │ │ │ 0000aa70: 203d 3e20 3729 2020 2020 2020 2020 2020 => 7) │ │ │ │ 0000aa80: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0000aa90: 2032 2e30 3434 3935 7320 2863 7075 293b 2.04495s (cpu); │ │ │ │ -0000aaa0: 2030 2e39 3837 3234 3573 2028 7468 7265 0.987245s (thre │ │ │ │ -0000aab0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ +0000aa90: 2032 2e33 3839 3873 2028 6370 7529 3b20 2.3898s (cpu); │ │ │ │ +0000aaa0: 312e 3035 3336 3473 2028 7468 7265 6164 1.05364s (thread │ │ │ │ +0000aab0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 0000aac0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000aad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab10: 207c 0a7c 2020 2020 2020 3120 2020 2020 |.| 1 │ │ │ │ 0000ab20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2762,18 +2762,18 @@ │ │ │ │ 0000ac90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000acb0: 2d2d 2d2d 2d2b 0a7c 6932 3920 3a20 7469 -----+.|i29 : ti │ │ │ │ 0000acc0: 6d65 2066 7074 2866 2c20 4465 7074 684f me fpt(f, DepthO │ │ │ │ 0000acd0: 6653 6561 7263 6820 3d3e 2034 2920 2020 fSearch => 4) │ │ │ │ 0000ace0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000acf0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -0000ad00: 2d20 7573 6564 2031 2e37 3639 3573 2028 - used 1.7695s ( │ │ │ │ -0000ad10: 6370 7529 3b20 302e 3737 3832 3439 7320 cpu); 0.778249s │ │ │ │ -0000ad20: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -0000ad30: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +0000ad00: 2d20 7573 6564 2032 2e30 3235 3033 7320 - used 2.02503s │ │ │ │ +0000ad10: 2863 7075 293b 2030 2e38 3639 3836 3273 (cpu); 0.869862s │ │ │ │ +0000ad20: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +0000ad30: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 0000ad40: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000ad50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ad60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ad70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ad80: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0000ad90: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 0000ada0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -3670,17 +3670,17 @@ │ │ │ │ 0000e550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e580: 2b0a 7c69 3135 203a 2074 696d 6520 6672 +.|i15 : time fr │ │ │ │ 0000e590: 6f62 656e 6975 734e 7528 332c 2066 2920 obeniusNu(3, f) │ │ │ │ 0000e5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e5b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0000e5c0: 7c20 2d2d 2075 7365 6420 302e 3030 3432 | -- used 0.0042 │ │ │ │ -0000e5d0: 3037 3632 7320 2863 7075 293b 2030 2e30 0762s (cpu); 0.0 │ │ │ │ -0000e5e0: 3034 3230 3438 3373 2028 7468 7265 6164 0420483s (thread │ │ │ │ +0000e5c0: 7c20 2d2d 2075 7365 6420 302e 3030 3533 | -- used 0.0053 │ │ │ │ +0000e5d0: 3734 3931 7320 2863 7075 293b 2030 2e30 7491s (cpu); 0.0 │ │ │ │ +0000e5e0: 3035 3337 3432 3973 2028 7468 7265 6164 0537429s (thread │ │ │ │ 0000e5f0: 293b 2030 7320 2867 6329 2020 7c0a 7c20 ); 0s (gc) |.| │ │ │ │ 0000e600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e630: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ 0000e640: 203d 2033 3735 3620 2020 2020 2020 2020 = 3756 │ │ │ │ 0000e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -3690,16 +3690,16 @@ │ │ │ │ 0000e690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e6b0: 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a 2074 ------+.|i16 : t │ │ │ │ 0000e6c0: 696d 6520 6672 6f62 656e 6975 734e 7528 ime frobeniusNu( │ │ │ │ 0000e6d0: 332c 2066 2c20 5573 6553 7065 6369 616c 3, f, UseSpecial │ │ │ │ 0000e6e0: 416c 676f 7269 7468 6d73 203d 3e20 6661 Algorithms => fa │ │ │ │ 0000e6f0: 6c73 6529 7c0a 7c20 2d2d 2075 7365 6420 lse)|.| -- used │ │ │ │ -0000e700: 302e 3539 3434 3732 7320 2863 7075 293b 0.594472s (cpu); │ │ │ │ -0000e710: 2030 2e33 3131 3631 3773 2028 7468 7265 0.311617s (thre │ │ │ │ +0000e700: 302e 3635 3836 3632 7320 2863 7075 293b 0.658662s (cpu); │ │ │ │ +0000e710: 2030 2e33 3532 3731 3673 2028 7468 7265 0.352716s (thre │ │ │ │ 0000e720: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 0000e730: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e770: 7c0a 7c6f 3136 203d 2033 3735 3620 2020 |.|o16 = 3756 │ │ │ │ 0000e780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -3805,17 +3805,17 @@ │ │ │ │ 0000edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000edd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ede0: 2b0a 7c69 3139 203a 2074 696d 6520 6672 +.|i19 : time fr │ │ │ │ 0000edf0: 6f62 656e 6975 734e 7528 342c 2066 2920 obeniusNu(4, f) │ │ │ │ 0000ee00: 2d2d 2043 6f6e 7461 696e 6d65 6e74 5465 -- ContainmentTe │ │ │ │ 0000ee10: 7374 2069 7320 7365 7420 746f 2046 726f st is set to Fro │ │ │ │ 0000ee20: 6265 6e69 7573 526f 6f74 2c20 6279 2020 beniusRoot, by │ │ │ │ -0000ee30: 7c0a 7c20 2d2d 2075 7365 6420 302e 3532 |.| -- used 0.52 │ │ │ │ -0000ee40: 3234 3739 7320 2863 7075 293b 2030 2e32 2479s (cpu); 0.2 │ │ │ │ -0000ee50: 3432 3134 3873 2028 7468 7265 6164 293b 42148s (thread); │ │ │ │ +0000ee30: 7c0a 7c20 2d2d 2075 7365 6420 302e 3538 |.| -- used 0.58 │ │ │ │ +0000ee40: 3533 3933 7320 2863 7075 293b 2030 2e32 5393s (cpu); 0.2 │ │ │ │ +0000ee50: 3732 3630 3773 2028 7468 7265 6164 293b 72607s (thread); │ │ │ │ 0000ee60: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 0000ee70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ee80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000ee90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -3840,17 +3840,17 @@ │ │ │ │ 0000eff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f010: 2b0a 7c69 3230 203a 2074 696d 6520 6672 +.|i20 : time fr │ │ │ │ 0000f020: 6f62 656e 6975 734e 7528 342c 2066 2c20 obeniusNu(4, f, │ │ │ │ 0000f030: 436f 6e74 6169 6e6d 656e 7454 6573 7420 ContainmentTest │ │ │ │ 0000f040: 3d3e 2053 7461 6e64 6172 6450 6f77 6572 => StandardPower │ │ │ │ 0000f050: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0000f060: 7c0a 7c20 2d2d 2075 7365 6420 312e 3633 |.| -- used 1.63 │ │ │ │ -0000f070: 3135 3973 2028 6370 7529 3b20 312e 3137 159s (cpu); 1.17 │ │ │ │ -0000f080: 3139 3273 2028 7468 7265 6164 293b 2030 192s (thread); 0 │ │ │ │ +0000f060: 7c0a 7c20 2d2d 2075 7365 6420 312e 3535 |.| -- used 1.55 │ │ │ │ +0000f070: 3633 3273 2028 6370 7529 3b20 312e 3330 632s (cpu); 1.30 │ │ │ │ +0000f080: 3735 3373 2028 7468 7265 6164 293b 2030 753s (thread); 0 │ │ │ │ 0000f090: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0000f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4015,18 +4015,18 @@ │ │ │ │ 0000fae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000faf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000fb00: 0a7c 6932 3720 3a20 7469 6d65 2066 726f .|i27 : time fro │ │ │ │ 0000fb10: 6265 6e69 7573 4e75 2835 2c20 6629 202d beniusNu(5, f) - │ │ │ │ 0000fb20: 2d20 7573 6573 2062 696e 6172 7920 7365 - uses binary se │ │ │ │ 0000fb30: 6172 6368 2028 6465 6661 756c 7429 2020 arch (default) │ │ │ │ 0000fb40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000fb50: 0a7c 202d 2d20 7573 6564 2031 2e34 3634 .| -- used 1.464 │ │ │ │ -0000fb60: 3773 2028 6370 7529 3b20 302e 3735 3931 7s (cpu); 0.7591 │ │ │ │ -0000fb70: 3539 7320 2874 6872 6561 6429 3b20 3073 59s (thread); 0s │ │ │ │ -0000fb80: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0000fb50: 0a7c 202d 2d20 7573 6564 2031 2e36 3437 .| -- used 1.647 │ │ │ │ +0000fb60: 3031 7320 2863 7075 293b 2030 2e38 3437 01s (cpu); 0.847 │ │ │ │ +0000fb70: 3730 3173 2028 7468 7265 6164 293b 2030 701s (thread); 0 │ │ │ │ +0000fb80: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0000fb90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000fba0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000fbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fbe0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000fbf0: 0a7c 6f32 3720 3d20 3131 3234 2020 2020 .|o27 = 1124 │ │ │ │ @@ -4040,17 +4040,17 @@ │ │ │ │ 0000fc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000fc90: 0a7c 6932 3820 3a20 7469 6d65 2066 726f .|i28 : time fro │ │ │ │ 0000fca0: 6265 6e69 7573 4e75 2835 2c20 662c 2053 beniusNu(5, f, S │ │ │ │ 0000fcb0: 6561 7263 6820 3d3e 204c 696e 6561 7229 earch => Linear) │ │ │ │ 0000fcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fcd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000fce0: 0a7c 202d 2d20 7573 6564 2032 2e31 3332 .| -- used 2.132 │ │ │ │ -0000fcf0: 3537 7320 2863 7075 293b 2031 2e30 3332 57s (cpu); 1.032 │ │ │ │ -0000fd00: 3034 7320 2874 6872 6561 6429 3b20 3073 04s (thread); 0s │ │ │ │ +0000fce0: 0a7c 202d 2d20 7573 6564 2032 2e33 3730 .| -- used 2.370 │ │ │ │ +0000fcf0: 3131 7320 2863 7075 293b 2031 2e32 3037 11s (cpu); 1.207 │ │ │ │ +0000fd00: 3139 7320 2874 6872 6561 6429 3b20 3073 19s (thread); 0s │ │ │ │ 0000fd10: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0000fd20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000fd30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000fd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fd70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -4085,18 +4085,18 @@ │ │ │ │ 0000ff40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ff50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000ff60: 0a7c 6933 3020 3a20 7469 6d65 2066 726f .|i30 : time fro │ │ │ │ 0000ff70: 6265 6e69 7573 4e75 2832 2c20 4d2c 204d beniusNu(2, M, M │ │ │ │ 0000ff80: 5e32 2920 2d2d 2075 7365 7320 6269 6e61 ^2) -- uses bina │ │ │ │ 0000ff90: 7279 2073 6561 7263 6820 2864 6566 6175 ry search (defau │ │ │ │ 0000ffa0: 6c74 2920 2020 2020 2020 2020 2020 207c lt) | │ │ │ │ -0000ffb0: 0a7c 202d 2d20 7573 6564 2031 2e39 3430 .| -- used 1.940 │ │ │ │ -0000ffc0: 3637 7320 2863 7075 293b 2031 2e34 3334 67s (cpu); 1.434 │ │ │ │ -0000ffd0: 3973 2028 7468 7265 6164 293b 2030 7320 9s (thread); 0s │ │ │ │ -0000ffe0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0000ffb0: 0a7c 202d 2d20 7573 6564 2031 2e38 3435 .| -- used 1.845 │ │ │ │ +0000ffc0: 3738 7320 2863 7075 293b 2031 2e35 3037 78s (cpu); 1.507 │ │ │ │ +0000ffd0: 3739 7320 2874 6872 6561 6429 3b20 3073 79s (thread); 0s │ │ │ │ +0000ffe0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0000fff0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00010000: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00010010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010040: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00010050: 0a7c 6f33 3020 3d20 3937 2020 2020 2020 .|o30 = 97 │ │ │ │ @@ -4110,17 +4110,17 @@ │ │ │ │ 000100d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000100e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 000100f0: 0a7c 6933 3120 3a20 7469 6d65 2066 726f .|i31 : time fro │ │ │ │ 00010100: 6265 6e69 7573 4e75 2832 2c20 4d2c 204d beniusNu(2, M, M │ │ │ │ 00010110: 5e32 2c20 5365 6172 6368 203d 3e20 4c69 ^2, Search => Li │ │ │ │ 00010120: 6e65 6172 2920 2d2d 2062 7574 206c 696e near) -- but lin │ │ │ │ 00010130: 6561 7220 7365 6172 6368 2067 6574 737c ear search gets| │ │ │ │ -00010140: 0a7c 202d 2d20 7573 6564 2030 2e36 3430 .| -- used 0.640 │ │ │ │ -00010150: 3331 3573 2028 6370 7529 3b20 302e 3439 315s (cpu); 0.49 │ │ │ │ -00010160: 3338 3933 7320 2874 6872 6561 6429 3b20 3893s (thread); │ │ │ │ +00010140: 0a7c 202d 2d20 7573 6564 2030 2e35 3732 .| -- used 0.572 │ │ │ │ +00010150: 3134 3173 2028 6370 7529 3b20 302e 3439 141s (cpu); 0.49 │ │ │ │ +00010160: 3431 3634 7320 2874 6872 6561 6429 3b20 4164s (thread); │ │ │ │ 00010170: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00010180: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00010190: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000101a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000101b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000101c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000101d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ ├── ./usr/share/info/GKMVarieties.info.gz │ │ │ ├── GKMVarieties.info │ │ │ │ @@ -17563,18 +17563,18 @@ │ │ │ │ 000449a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000449b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000449c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3720 ---------+.|i27 │ │ │ │ 000449d0: 3a20 7469 6d65 2043 203d 206f 7262 6974 : time C = orbit │ │ │ │ 000449e0: 436c 6f73 7572 6528 582c 4d61 7429 2020 Closure(X,Mat) │ │ │ │ 000449f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044a00: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00044a10: 7573 6564 2030 2e36 3038 3339 3773 2028 used 0.608397s ( │ │ │ │ -00044a20: 6370 7529 3b20 302e 3335 3836 3039 7320 cpu); 0.358609s │ │ │ │ -00044a30: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00044a40: 2920 2020 2020 2020 207c 0a7c 2020 2020 ) |.| │ │ │ │ +00044a10: 7573 6564 2032 2e30 3735 3432 7320 2863 used 2.07542s (c │ │ │ │ +00044a20: 7075 293b 2030 2e34 3936 3634 7320 2874 pu); 0.49664s (t │ │ │ │ +00044a30: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00044a40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00044a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044a80: 2020 2020 2020 2020 207c 0a7c 6f32 3720 |.|o27 │ │ │ │ 00044a90: 3d20 616e 2022 6571 7569 7661 7269 616e = an "equivarian │ │ │ │ 00044aa0: 7420 4b2d 636c 6173 7322 206f 6e20 6120 t K-class" on a │ │ │ │ 00044ab0: 474b 4d20 7661 7269 6574 7920 2020 2020 GKM variety │ │ │ │ @@ -17591,17 +17591,17 @@ │ │ │ │ 00044b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044b80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3820 ---------+.|i28 │ │ │ │ 00044b90: 3a20 7469 6d65 2043 203d 206f 7262 6974 : time C = orbit │ │ │ │ 00044ba0: 436c 6f73 7572 6528 582c 4d61 742c 2052 Closure(X,Mat, R │ │ │ │ 00044bb0: 5245 464d 6574 686f 6420 3d3e 2074 7275 REFMethod => tru │ │ │ │ 00044bc0: 6529 2020 2020 2020 207c 0a7c 202d 2d20 e) |.| -- │ │ │ │ -00044bd0: 7573 6564 2031 2e38 3933 3138 7320 2863 used 1.89318s (c │ │ │ │ -00044be0: 7075 293b 2030 2e39 3837 3534 3273 2028 pu); 0.987542s ( │ │ │ │ -00044bf0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00044bd0: 7573 6564 2033 2e31 3038 3938 7320 2863 used 3.10898s (c │ │ │ │ +00044be0: 7075 293b 2031 2e30 3337 3631 7320 2874 pu); 1.03761s (t │ │ │ │ +00044bf0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00044c00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00044c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044c40: 2020 2020 2020 2020 207c 0a7c 6f32 3820 |.|o28 │ │ │ │ 00044c50: 3d20 616e 2022 6571 7569 7661 7269 616e = an "equivarian │ │ │ │ 00044c60: 7420 4b2d 636c 6173 7322 206f 6e20 6120 t K-class" on a │ │ ├── ./usr/share/info/Graphs.info.gz │ │ │ ├── Graphs.info │ │ │ │ @@ -18078,16 +18078,16 @@ │ │ │ │ 000469d0: 7973 2048 2020 2020 2020 2020 2020 2020 ys H │ │ │ │ 000469e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000469f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00046a00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00046a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046a30: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -00046a40: 7b6e 6577 4469 6772 6170 682c 206d 6170 {newDigraph, map │ │ │ │ -00046a50: 2c20 6469 6772 6170 687d 2020 2020 2020 , digraph} │ │ │ │ +00046a40: 7b64 6967 7261 7068 2c20 6d61 702c 206e {digraph, map, n │ │ │ │ +00046a50: 6577 4469 6772 6170 687d 2020 2020 2020 ewDigraph} │ │ │ │ 00046a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046a70: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00046a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046aa0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ 00046ab0: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ 00046ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/GroebnerStrata.info.gz │ │ │ ├── GroebnerStrata.info │ │ │ │ @@ -2491,25 +2491,25 @@ │ │ │ │ 00009ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009bb0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00009bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009c00: 2020 2020 207c 0a7c 6f31 3220 3d20 7c20 |.|o12 = | │ │ │ │ -00009c10: 2d32 3620 2d32 3420 3620 3339 2031 3720 -26 -24 6 39 17 │ │ │ │ -00009c20: 3238 202d 3436 202d 3132 2031 3120 2d32 28 -46 -12 11 -2 │ │ │ │ -00009c30: 3920 2d31 3020 2d34 3820 2d33 3620 2d33 9 -10 -48 -36 -3 │ │ │ │ -00009c40: 3020 3339 202d 3239 202d 3820 2d32 3220 0 39 -29 -8 -22 │ │ │ │ -00009c50: 2d31 3020 207c 0a7c 2020 2020 2020 2d2d -10 |.| -- │ │ │ │ +00009c10: 2d31 3420 3438 2034 3120 2d33 3220 2d33 -14 48 41 -32 -3 │ │ │ │ +00009c20: 3920 3330 2034 3820 3336 2032 202d 3239 9 30 48 36 2 -29 │ │ │ │ +00009c30: 202d 3330 202d 3233 2031 3920 3139 202d -30 -23 19 19 - │ │ │ │ +00009c40: 3130 202d 3239 202d 3820 2d32 3220 3234 10 -29 -8 -22 24 │ │ │ │ +00009c50: 202d 3133 207c 0a7c 2020 2020 2020 2d2d -13 |.| -- │ │ │ │ 00009c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009ca0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3131 -----|.| 11 │ │ │ │ -00009cb0: 2031 3920 3139 2032 3420 2d32 3920 7c20 19 19 24 -29 | │ │ │ │ +00009ca0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2d33 -----|.| -3 │ │ │ │ +00009cb0: 3620 2d33 3020 2d32 3920 2d31 3020 7c20 6 -30 -29 -10 | │ │ │ │ 00009cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009cf0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00009d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2536,25 +2536,25 @@ │ │ │ │ 00009e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009e80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00009e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ed0: 2020 2020 207c 0a7c 6f31 3320 3d20 7c20 |.|o13 = | │ │ │ │ -00009ee0: 2d33 3020 2d34 3120 2d33 3920 3330 202d -30 -41 -39 30 - │ │ │ │ -00009ef0: 3133 2033 3220 3137 202d 3132 2032 3920 13 32 17 -12 29 │ │ │ │ -00009f00: 3235 202d 3439 2031 3120 3433 2032 3120 25 -49 11 43 21 │ │ │ │ -00009f10: 2d34 3220 2d34 2031 3920 2d34 3720 3339 -42 -4 19 -47 39 │ │ │ │ -00009f20: 202d 3338 207c 0a7c 2020 2020 2020 2d2d -38 |.| -- │ │ │ │ +00009ee0: 2d33 3920 2d32 3220 3131 2032 3620 2d32 -39 -22 11 26 -2 │ │ │ │ +00009ef0: 3820 3331 2033 3820 3234 202d 3331 202d 8 31 38 24 -31 - │ │ │ │ +00009f00: 3136 202d 3136 202d 3720 2d34 3120 2d32 16 -16 -7 -41 -2 │ │ │ │ +00009f10: 3420 3138 202d 3137 2031 3920 3334 202d 4 18 -17 19 34 - │ │ │ │ +00009f20: 3338 2033 397c 0a7c 2020 2020 2020 2d2d 38 39|.| -- │ │ │ │ 00009f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009f70: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2d31 -----|.| -1 │ │ │ │ -00009f80: 3620 3020 2d32 3420 3334 207c 2020 2020 6 0 -24 34 | │ │ │ │ +00009f80: 3620 3020 2d34 3720 3231 207c 2020 2020 6 0 -47 21 | │ │ │ │ 00009f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009fc0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00009fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2582,52 +2582,52 @@ │ │ │ │ 0000a150: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000a160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a1a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000a1b0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -0000a1c0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +0000a1c0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ 0000a1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a1e0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +0000a1e0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0000a1f0: 2020 2020 207c 0a7c 6f31 3420 3d20 6964 |.|o14 = id │ │ │ │ -0000a200: 6561 6c20 2861 2020 2b20 3131 622a 6320 eal (a + 11b*c │ │ │ │ -0000a210: 2b20 3339 6320 202d 2032 3961 2a64 202b + 39c - 29a*d + │ │ │ │ -0000a220: 2031 3762 2a64 202d 2032 3463 2a64 202d 17b*d - 24c*d - │ │ │ │ -0000a230: 2032 3664 202c 2061 2a62 202b 2033 3962 26d , a*b + 39b │ │ │ │ -0000a240: 2a63 202d 207c 0a7c 2020 2020 2020 2d2d *c - |.| -- │ │ │ │ +0000a200: 6561 6c20 2861 2020 2b20 3262 2a63 202d eal (a + 2b*c - │ │ │ │ +0000a210: 2033 3263 2020 2d20 3239 612a 6420 2d20 32c - 29a*d - │ │ │ │ +0000a220: 3339 622a 6420 2b20 3438 632a 6420 2d20 39b*d + 48c*d - │ │ │ │ +0000a230: 3134 6420 2c20 612a 6220 2d20 3130 622a 14d , a*b - 10b* │ │ │ │ +0000a240: 6320 2d20 207c 0a7c 2020 2020 2020 2d2d c - |.| -- │ │ │ │ 0000a250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a290: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ 0000a2a0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0000a2b0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -0000a2c0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0000a2d0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0000a2e0: 2020 2020 207c 0a7c 2020 2020 2020 3130 |.| 10 │ │ │ │ -0000a2f0: 6320 202d 2032 3961 2a64 202d 2034 3862 c - 29a*d - 48b │ │ │ │ -0000a300: 2a64 202b 2032 3863 2a64 202b 2036 6420 *d + 28c*d + 6d │ │ │ │ -0000a310: 2c20 6220 202b 2031 3962 2a63 202d 2038 , b + 19b*c - 8 │ │ │ │ -0000a320: 6320 202b 2031 3961 2a64 202d 2032 3262 c + 19a*d - 22b │ │ │ │ -0000a330: 2a64 202d 207c 0a7c 2020 2020 2020 2d2d *d - |.| -- │ │ │ │ +0000a2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000a2c0: 3220 2020 3220 2020 2020 2020 2020 2020 2 2 │ │ │ │ +0000a2d0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0000a2e0: 2020 2020 207c 0a7c 2020 2020 2020 3330 |.| 30 │ │ │ │ +0000a2f0: 6320 202d 2032 3961 2a64 202d 2032 3362 c - 29a*d - 23b │ │ │ │ +0000a300: 2a64 202b 2033 3063 2a64 202b 2034 3164 *d + 30c*d + 41d │ │ │ │ +0000a310: 202c 2062 2020 2d20 3336 622a 6320 2d20 , b - 36b*c - │ │ │ │ +0000a320: 3863 2020 2d20 3330 612a 6420 2d20 3232 8c - 30a*d - 22 │ │ │ │ +0000a330: 622a 6420 2b7c 0a7c 2020 2020 2020 2d2d b*d +|.| -- │ │ │ │ 0000a340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a380: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ 0000a390: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ 0000a3a0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ 0000a3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a3c0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -0000a3d0: 2020 2020 207c 0a7c 2020 2020 2020 3336 |.| 36 │ │ │ │ -0000a3e0: 632a 6420 2d20 3436 6420 2c20 612a 6320 c*d - 46d , a*c │ │ │ │ -0000a3f0: 2b20 3234 622a 6320 2d20 3130 6320 202d + 24b*c - 10c - │ │ │ │ -0000a400: 2032 3961 2a64 202b 2031 3162 2a64 202d 29a*d + 11b*d - │ │ │ │ -0000a410: 2033 3063 2a64 202d 2031 3264 2029 2020 30c*d - 12d ) │ │ │ │ +0000a3d0: 2020 2020 207c 0a7c 2020 2020 2020 3139 |.| 19 │ │ │ │ +0000a3e0: 632a 6420 2b20 3438 6420 2c20 612a 6320 c*d + 48d , a*c │ │ │ │ +0000a3f0: 2d20 3239 622a 6320 2b20 3234 6320 202d - 29b*c + 24c - │ │ │ │ +0000a400: 2031 3061 2a64 202d 2031 3362 2a64 202b 10a*d - 13b*d + │ │ │ │ +0000a410: 2031 3963 2a64 202b 2033 3664 2029 2020 19c*d + 36d ) │ │ │ │ 0000a420: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000a430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a470: 2020 2020 207c 0a7c 6f31 3420 3a20 4964 |.|o14 : Id │ │ │ │ 0000a480: 6561 6c20 6f66 2053 2020 2020 2020 2020 eal of S │ │ │ │ @@ -2651,48 +2651,48 @@ │ │ │ │ 0000a5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a5b0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000a5c0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ 0000a5d0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ 0000a5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a5f0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ 0000a600: 2020 2020 207c 0a7c 6f31 3520 3d20 6964 |.|o15 = id │ │ │ │ -0000a610: 6561 6c20 2861 2020 2b20 3239 622a 6320 eal (a + 29b*c │ │ │ │ -0000a620: 2b20 3330 6320 202b 2032 3561 2a64 202d + 30c + 25a*d - │ │ │ │ -0000a630: 2031 3362 2a64 202d 2034 3163 2a64 202d 13b*d - 41c*d - │ │ │ │ -0000a640: 2033 3064 202c 2061 2a62 202d 2034 3262 30d , a*b - 42b │ │ │ │ +0000a610: 6561 6c20 2861 2020 2d20 3331 622a 6320 eal (a - 31b*c │ │ │ │ +0000a620: 2b20 3236 6320 202d 2031 3661 2a64 202d + 26c - 16a*d - │ │ │ │ +0000a630: 2032 3862 2a64 202d 2032 3263 2a64 202d 28b*d - 22c*d - │ │ │ │ +0000a640: 2033 3964 202c 2061 2a62 202b 2031 3862 39d , a*b + 18b │ │ │ │ 0000a650: 2a63 202d 207c 0a7c 2020 2020 2020 2d2d *c - |.| -- │ │ │ │ 0000a660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a6a0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ 0000a6b0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0000a6c0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ 0000a6d0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0000a6e0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0000a6f0: 2020 2020 207c 0a7c 2020 2020 2020 3439 |.| 49 │ │ │ │ -0000a700: 6320 202d 2034 612a 6420 2b20 3131 622a c - 4a*d + 11b* │ │ │ │ -0000a710: 6420 2b20 3332 632a 6420 2d20 3339 6420 d + 32c*d - 39d │ │ │ │ +0000a6f0: 2020 2020 207c 0a7c 2020 2020 2020 3136 |.| 16 │ │ │ │ +0000a700: 6320 202d 2031 3761 2a64 202d 2037 622a c - 17a*d - 7b* │ │ │ │ +0000a710: 6420 2b20 3331 632a 6420 2b20 3131 6420 d + 31c*d + 11d │ │ │ │ 0000a720: 2c20 6220 202d 2031 3662 2a63 202b 2031 , b - 16b*c + 1 │ │ │ │ -0000a730: 3963 2020 2d20 3437 622a 6420 2b20 3433 9c - 47b*d + 43 │ │ │ │ +0000a730: 3963 2020 2b20 3334 622a 6420 2d20 3431 9c + 34b*d - 41 │ │ │ │ 0000a740: 632a 6420 2b7c 0a7c 2020 2020 2020 2d2d c*d +|.| -- │ │ │ │ 0000a750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a790: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ 0000a7a0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0000a7b0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ 0000a7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a7d0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -0000a7e0: 2020 2020 207c 0a7c 2020 2020 2020 3137 |.| 17 │ │ │ │ -0000a7f0: 6420 2c20 612a 6320 2d20 3234 622a 6320 d , a*c - 24b*c │ │ │ │ -0000a800: 2b20 3339 6320 202b 2033 3461 2a64 202d + 39c + 34a*d - │ │ │ │ -0000a810: 2033 3862 2a64 202b 2032 3163 2a64 202d 38b*d + 21c*d - │ │ │ │ -0000a820: 2031 3264 2029 2020 2020 2020 2020 2020 12d ) │ │ │ │ +0000a7e0: 2020 2020 207c 0a7c 2020 2020 2020 3338 |.| 38 │ │ │ │ +0000a7f0: 6420 2c20 612a 6320 2d20 3437 622a 6320 d , a*c - 47b*c │ │ │ │ +0000a800: 2d20 3338 6320 202b 2032 3161 2a64 202b - 38c + 21a*d + │ │ │ │ +0000a810: 2033 3962 2a64 202d 2032 3463 2a64 202b 39b*d - 24c*d + │ │ │ │ +0000a820: 2032 3464 2029 2020 2020 2020 2020 2020 24d ) │ │ │ │ 0000a830: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000a840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a880: 2020 2020 207c 0a7c 6f31 3520 3a20 4964 |.|o15 : Id │ │ │ │ 0000a890: 6561 6c20 6f66 2053 2020 2020 2020 2020 eal of S │ │ │ │ @@ -2716,26 +2716,26 @@ │ │ │ │ 0000a9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a9c0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000a9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a9e0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ 0000a9f0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ 0000aa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa10: 2032 2020 207c 0a7c 6f31 3620 3d20 7b69 2 |.|o16 = {i │ │ │ │ -0000aa20: 6465 616c 2028 6120 2b20 3234 6220 2d20 deal (a + 24b - │ │ │ │ -0000aa30: 3130 6320 2d20 3137 642c 2062 2020 2b20 10c - 17d, b + │ │ │ │ -0000aa40: 3139 622a 6320 2d20 3863 2020 2b20 3237 19b*c - 8c + 27 │ │ │ │ -0000aa50: 622a 6420 2d20 3438 632a 6420 2d20 3236 b*d - 48c*d - 26 │ │ │ │ +0000aa20: 6465 616c 2028 6120 2d20 3239 6220 2b20 deal (a - 29b + │ │ │ │ +0000aa30: 3234 6320 2d20 3434 642c 2062 2020 2d20 24c - 44d, b - │ │ │ │ +0000aa40: 3336 622a 6320 2d20 3863 2020 2b20 3137 36b*c - 8c + 17 │ │ │ │ +0000aa50: 622a 6420 2b20 3332 632a 6420 2b20 3431 b*d + 32c*d + 41 │ │ │ │ 0000aa60: 6420 292c 207c 0a7c 2020 2020 2020 2d2d d ), |.| -- │ │ │ │ 0000aa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aa90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aab0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 6964 -----|.| id │ │ │ │ -0000aac0: 6561 6c20 2863 202d 2032 3964 2c20 6220 eal (c - 29d, b │ │ │ │ -0000aad0: 2b20 3233 642c 2061 202d 2031 3464 297d + 23d, a - 14d)} │ │ │ │ +0000aac0: 6561 6c20 2863 202d 2031 3064 2c20 6220 eal (c - 10d, b │ │ │ │ +0000aad0: 2b20 3333 642c 2061 202b 2031 3064 297d + 33d, a + 10d)} │ │ │ │ 0000aae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000ab10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2756,25 +2756,25 @@ │ │ │ │ 0000ac30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac40: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000ac50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac90: 2020 2020 207c 0a7c 6f31 3720 3d20 7b69 |.|o17 = {i │ │ │ │ -0000aca0: 6465 616c 2028 6220 2d20 3432 6320 2d20 deal (b - 42c - │ │ │ │ -0000acb0: 3138 642c 2061 202b 2034 3163 202b 2033 18d, a + 41c + 3 │ │ │ │ -0000acc0: 3364 292c 2069 6465 616c 2028 6220 2b20 3d), ideal (b + │ │ │ │ -0000acd0: 3236 6320 2d20 3239 642c 2061 202d 2034 26c - 29d, a - 4 │ │ │ │ -0000ace0: 3463 202d 207c 0a7c 2020 2020 2020 2d2d 4c - |.| -- │ │ │ │ +0000aca0: 6465 616c 2028 6220 2d20 3432 6320 2b20 deal (b - 42c + │ │ │ │ +0000acb0: 3130 642c 2061 202b 2038 6320 2d20 3364 10d, a + 8c - 3d │ │ │ │ +0000acc0: 292c 2069 6465 616c 2028 6220 2b20 3236 ), ideal (b + 26 │ │ │ │ +0000acd0: 6320 2b20 3234 642c 2061 202d 2032 3863 c + 24d, a - 28c │ │ │ │ +0000ace0: 202d 2020 207c 0a7c 2020 2020 2020 2d2d - |.| -- │ │ │ │ 0000acf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ad00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ad10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ad20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ad30: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3964 -----|.| 9d │ │ │ │ -0000ad40: 297d 2020 2020 2020 2020 2020 2020 2020 )} │ │ │ │ +0000ad30: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3239 -----|.| 29 │ │ │ │ +0000ad40: 6429 7d20 2020 2020 2020 2020 2020 2020 d)} │ │ │ │ 0000ad50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ad60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ad70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ad80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000ad90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ada0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000adb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8735,28 +8735,28 @@ │ │ │ │ 000221e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000221f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00022200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022240: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -00022250: 3420 3d20 7c20 2d32 3220 3431 202d 3131 4 = | -22 41 -11 │ │ │ │ -00022260: 2031 3920 2d34 3320 2d37 2031 3520 3420 19 -43 -7 15 4 │ │ │ │ -00022270: 3231 202d 3336 2033 3120 3330 2033 3720 21 -36 31 30 37 │ │ │ │ -00022280: 3139 202d 3920 2d34 3420 3330 2031 3920 19 -9 -44 30 19 │ │ │ │ -00022290: 2d33 3820 3120 3437 2032 347c 0a7c 2020 -38 1 47 24|.| │ │ │ │ +00022250: 3420 3d20 7c20 3433 2033 3520 2d34 3320 4 = | 43 35 -43 │ │ │ │ +00022260: 3720 3338 2033 3120 3437 2034 3820 3436 7 38 31 47 48 46 │ │ │ │ +00022270: 2032 3120 3820 3130 2036 202d 3330 202d 21 8 10 6 -30 - │ │ │ │ +00022280: 3430 2031 3020 2d32 3720 2d31 3020 2d35 40 10 -27 -10 -5 │ │ │ │ +00022290: 3020 3330 202d 3231 2020 207c 0a7c 2020 0 30 -21 |.| │ │ │ │ 000222a0: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 000222b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000222c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000222d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000222e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -000222f0: 2020 2020 2d32 3920 2d31 3620 3136 202d -29 -16 16 - │ │ │ │ -00022300: 3239 202d 3330 2032 3120 2d31 3020 2d32 29 -30 21 -10 -2 │ │ │ │ -00022310: 3220 3339 202d 3234 202d 3239 202d 3820 2 39 -24 -29 -8 │ │ │ │ -00022320: 2d33 3620 2d33 3820 7c20 2020 2020 2020 -36 -38 | │ │ │ │ +000222f0: 2020 2020 2d33 3820 2d31 3620 2d32 3920 -38 -16 -29 │ │ │ │ +00022300: 3331 202d 3336 2033 3920 2d32 3920 3139 31 -36 39 -29 19 │ │ │ │ +00022310: 2032 3420 2d32 3420 2d38 2031 3920 2d32 24 -24 -8 19 -2 │ │ │ │ +00022320: 3920 3231 202d 3232 207c 2020 2020 2020 9 21 -22 | │ │ │ │ 00022330: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00022340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022380: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00022390: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ @@ -8780,28 +8780,28 @@ │ │ │ │ 000224b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000224c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000224d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000224e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000224f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022510: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -00022520: 3520 3d20 7c20 2d34 3820 2d34 3620 3136 5 = | -48 -46 16 │ │ │ │ -00022530: 2031 3720 2d31 202d 3433 2031 3520 2d31 17 -1 -43 15 -1 │ │ │ │ -00022540: 2031 3220 2d31 3820 2d36 202d 3238 2031 12 -18 -6 -28 1 │ │ │ │ -00022550: 3420 2d32 3820 2d39 2033 3220 2d32 3220 4 -28 -9 32 -22 │ │ │ │ -00022560: 2d33 3920 3620 2d34 3720 207c 0a7c 2020 -39 6 -47 |.| │ │ │ │ +00022520: 3520 3d20 7c20 3138 2031 3320 2d34 3820 5 = | 18 13 -48 │ │ │ │ +00022530: 3130 2032 3720 2d33 3320 3133 2034 2033 10 27 -33 13 4 3 │ │ │ │ +00022540: 3720 3333 202d 3135 2034 3620 3432 202d 7 33 -15 46 42 - │ │ │ │ +00022550: 3437 202d 3335 2032 3320 3435 202d 3133 47 -35 23 45 -13 │ │ │ │ +00022560: 2033 3320 2d34 3320 3120 377c 0a7c 2020 33 -43 1 7|.| │ │ │ │ 00022570: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00022580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000225a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000225b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -000225c0: 2020 2020 3238 202d 3337 202d 3437 2033 28 -37 -47 3 │ │ │ │ -000225d0: 3820 2d31 3620 2d31 3520 3334 2032 3720 8 -16 -15 34 27 │ │ │ │ -000225e0: 2d31 3320 2d34 3320 3232 2031 3620 3020 -13 -43 22 16 0 │ │ │ │ -000225f0: 2d31 3820 3139 2032 207c 2020 2020 2020 -18 19 2 | │ │ │ │ +000225c0: 2020 2020 3220 2d34 3720 3436 2031 3920 2 -47 46 19 │ │ │ │ +000225d0: 3136 2031 3420 2d31 3820 3334 2033 3820 16 14 -18 34 38 │ │ │ │ +000225e0: 2d31 3520 3020 2d33 3920 3232 202d 3238 -15 0 -39 22 -28 │ │ │ │ +000225f0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00022600: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00022610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022650: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00022660: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ @@ -8826,81 +8826,81 @@ │ │ │ │ 00022790: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000227a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000227b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000227c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000227d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000227e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000227f0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -00022800: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00022800: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ 00022810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022820: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +00022820: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ 00022830: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ 00022840: 3620 3d20 6964 6561 6c20 2861 2020 2d20 6 = ideal (a - │ │ │ │ -00022850: 3962 2a63 202d 2033 3663 2020 2b20 3330 9b*c - 36c + 30 │ │ │ │ -00022860: 612a 6420 2d20 3762 2a64 202d 2031 3163 a*d - 7b*d - 11c │ │ │ │ -00022870: 2a64 202d 2032 3264 202c 2061 2a62 202b *d - 22d , a*b + │ │ │ │ -00022880: 2031 3662 2a63 202d 2020 207c 0a7c 2020 16b*c - |.| │ │ │ │ +00022850: 3430 622a 6320 2b20 3231 6320 202b 2031 40b*c + 21c + 1 │ │ │ │ +00022860: 3061 2a64 202b 2033 3162 2a64 202d 2034 0a*d + 31b*d - 4 │ │ │ │ +00022870: 3363 2a64 202b 2034 3364 202c 2061 2a62 3c*d + 43d , a*b │ │ │ │ +00022880: 202b 2033 3162 2a63 202d 207c 0a7c 2020 + 31b*c - |.| │ │ │ │ 00022890: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 000228a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000228b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000228c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000228d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ 000228e0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ 000228f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022900: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00022910: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00022900: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00022910: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ 00022920: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00022930: 2020 2020 3338 6320 202b 2034 3761 2a64 38c + 47a*d │ │ │ │ -00022940: 202b 2033 3762 2a64 202b 2032 3163 2a64 + 37b*d + 21c*d │ │ │ │ -00022950: 202b 2034 3164 202c 2061 2a63 202d 2032 + 41d , a*c - 2 │ │ │ │ -00022960: 3462 2a63 202d 2032 3963 2020 2b20 3231 4b*c - 29c + 21 │ │ │ │ -00022970: 612a 6420 2b20 622a 6420 2b7c 0a7c 2020 a*d + b*d +|.| │ │ │ │ +00022930: 2020 2020 3530 6320 202d 2032 3161 2a64 50c - 21a*d │ │ │ │ +00022940: 202b 2036 622a 6420 2b20 3436 632a 6420 + 6b*d + 46c*d │ │ │ │ +00022950: 2b20 3335 6420 2c20 612a 6320 2d20 3862 + 35d , a*c - 8b │ │ │ │ +00022960: 2a63 202d 2033 3663 2020 2d20 3239 612a *c - 36c - 29a* │ │ │ │ +00022970: 6420 2b20 3330 622a 6420 2d7c 0a7c 2020 d + 30b*d -|.| │ │ │ │ 00022980: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00022990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000229a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000229b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000229c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ 000229d0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ 000229e0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 000229f0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00022a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a10: 2032 2020 2020 2032 2020 207c 0a7c 2020 2 2 |.| │ │ │ │ -00022a20: 2020 2020 3139 632a 6420 2d20 3433 6420 19c*d - 43d │ │ │ │ -00022a30: 2c20 6220 202d 2033 3662 2a63 202d 2031 , b - 36b*c - 1 │ │ │ │ -00022a40: 3063 2020 2d20 3239 612a 6420 2b20 3234 0c - 29a*d + 24 │ │ │ │ -00022a50: 622a 6420 2d20 3434 632a 6420 2b20 3135 b*d - 44c*d + 15 │ │ │ │ -00022a60: 6420 2c20 622a 6320 202d 207c 0a7c 2020 d , b*c - |.| │ │ │ │ +00022a20: 2020 2020 3330 632a 6420 2b20 3338 6420 30c*d + 38d │ │ │ │ +00022a30: 2c20 6220 202b 2032 3162 2a63 202b 2031 , b + 21b*c + 1 │ │ │ │ +00022a40: 3963 2020 2b20 3139 612a 6420 2d20 3338 9c + 19a*d - 38 │ │ │ │ +00022a50: 622a 6420 2b20 3130 632a 6420 2b20 3437 b*d + 10c*d + 47 │ │ │ │ +00022a60: 6420 2c20 622a 6320 202b 207c 0a7c 2020 d , b*c + |.| │ │ │ │ 00022a70: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00022a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ 00022ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ad0: 2032 2020 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ -00022ae0: 2020 2020 3220 2020 2020 2020 2032 2020 2 2 │ │ │ │ -00022af0: 2020 2020 3320 2020 3320 2020 2020 2020 3 3 │ │ │ │ -00022b00: 2020 2020 2020 2020 2032 207c 0a7c 2020 2 |.| │ │ │ │ -00022b10: 2020 2020 3232 622a 632a 6420 2d20 3239 22b*c*d - 29 │ │ │ │ -00022b20: 6320 6420 2d20 3330 612a 6420 202b 2033 c d - 30a*d + 3 │ │ │ │ -00022b30: 3062 2a64 2020 2b20 3331 632a 6420 202b 0b*d + 31c*d + │ │ │ │ -00022b40: 2031 3964 202c 2063 2020 2d20 3338 622a 19d , c - 38b* │ │ │ │ -00022b50: 632a 6420 2b20 3339 6320 647c 0a7c 2020 c*d + 39c d|.| │ │ │ │ +00022ae0: 2020 2020 3220 2020 2020 2020 3220 2020 2 2 │ │ │ │ +00022af0: 2020 3320 2020 3320 2020 2020 2020 2020 3 3 │ │ │ │ +00022b00: 2020 2020 2020 2032 2020 207c 0a7c 2020 2 |.| │ │ │ │ +00022b10: 2020 2020 3234 622a 632a 6420 2d20 3136 24b*c*d - 16 │ │ │ │ +00022b20: 6320 6420 2b20 3339 612a 6420 202d 2032 c d + 39a*d - 2 │ │ │ │ +00022b30: 3762 2a64 2020 2b20 3863 2a64 2020 2b20 7b*d + 8c*d + │ │ │ │ +00022b40: 3764 202c 2063 2020 2d20 3232 622a 632a 7d , c - 22b*c* │ │ │ │ +00022b50: 6420 2d20 3234 6320 6420 2d7c 0a7c 2020 d - 24c d -|.| │ │ │ │ 00022b60: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00022b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -00022bb0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -00022bc0: 2020 2032 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ +00022bb0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +00022bc0: 2020 3220 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ 00022bd0: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00022be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022bf0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00022c00: 2020 2020 2d20 3861 2a64 2020 2d20 3136 - 8a*d - 16 │ │ │ │ -00022c10: 622a 6420 202b 2031 3963 2a64 2020 2b20 b*d + 19c*d + │ │ │ │ -00022c20: 3464 2029 2020 2020 2020 2020 2020 2020 4d ) │ │ │ │ +00022c00: 2020 2020 3239 612a 6420 202d 2032 3962 29a*d - 29b │ │ │ │ +00022c10: 2a64 2020 2d20 3130 632a 6420 202b 2034 *d - 10c*d + 4 │ │ │ │ +00022c20: 3864 2029 2020 2020 2020 2020 2020 2020 8d ) │ │ │ │ 00022c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c40: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00022c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c90: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ @@ -8971,80 +8971,80 @@ │ │ │ │ 000230a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000230b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000230c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000230d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000230e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000230f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00023100: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -00023110: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00023110: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ 00023120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023130: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +00023130: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ 00023140: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ 00023150: 3820 3d20 6964 6561 6c20 2861 2020 2d20 8 = ideal (a - │ │ │ │ -00023160: 3962 2a63 202d 2031 3863 2020 2d20 3238 9b*c - 18c - 28 │ │ │ │ -00023170: 612a 6420 2d20 3433 622a 6420 2b20 3136 a*d - 43b*d + 16 │ │ │ │ -00023180: 632a 6420 2d20 3438 6420 2c20 612a 6220 c*d - 48d , a*b │ │ │ │ -00023190: 2d20 3136 622a 6320 2b20 207c 0a7c 2020 - 16b*c + |.| │ │ │ │ +00023160: 3335 622a 6320 2b20 3333 6320 202b 2034 35b*c + 33c + 4 │ │ │ │ +00023170: 3661 2a64 202d 2033 3362 2a64 202d 2034 6a*d - 33b*d - 4 │ │ │ │ +00023180: 3863 2a64 202b 2031 3864 202c 2061 2a62 8c*d + 18d , a*b │ │ │ │ +00023190: 202b 2034 3662 2a63 202b 207c 0a7c 2020 + 46b*c + |.| │ │ │ │ 000231a0: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 000231b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000231c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000231d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000231e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -000231f0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +000231f0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ 00023200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023210: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -00023220: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +00023210: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00023220: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ 00023230: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00023240: 2020 2020 3663 2020 2b20 3238 612a 6420 6c + 28a*d │ │ │ │ -00023250: 2b20 3134 622a 6420 2b20 3132 632a 6420 + 14b*d + 12c*d │ │ │ │ -00023260: 2d20 3436 6420 2c20 612a 6320 2b20 3136 - 46d , a*c + 16 │ │ │ │ -00023270: 622a 6320 2d20 3135 6320 202b 2032 3761 b*c - 15c + 27a │ │ │ │ -00023280: 2a64 202d 2034 3762 2a64 207c 0a7c 2020 *d - 47b*d |.| │ │ │ │ +00023240: 2020 2020 3333 6320 202b 2061 2a64 202b 33c + a*d + │ │ │ │ +00023250: 2034 3262 2a64 202b 2033 3763 2a64 202b 42b*d + 37c*d + │ │ │ │ +00023260: 2031 3364 202c 2061 2a63 202d 2031 3562 13d , a*c - 15b │ │ │ │ +00023270: 2a63 202b 2031 3963 2020 2b20 3134 612a *c + 19c + 14a* │ │ │ │ +00023280: 6420 2d20 3433 622a 6420 2d7c 0a7c 2020 d - 43b*d -|.| │ │ │ │ 00023290: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 000232a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000232b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000232c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000232d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ 000232e0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ 000232f0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00023300: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00023310: 2020 2020 2020 2020 2032 2020 2020 2032 2 2 │ │ │ │ +00023310: 2020 2020 2020 2020 3220 2020 2020 3220 2 2 │ │ │ │ 00023320: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00023330: 2020 2020 2d20 3238 632a 6420 2d20 6420 - 28c*d - d │ │ │ │ -00023340: 2c20 6220 202b 2031 3962 2a63 202d 2031 , b + 19b*c - 1 │ │ │ │ -00023350: 3363 2020 2d20 3337 622a 6420 2b20 3332 3c - 37b*d + 32 │ │ │ │ -00023360: 632a 6420 2b20 3135 6420 2c20 622a 6320 c*d + 15d , b*c │ │ │ │ -00023370: 202d 2034 3362 2a63 2a64 207c 0a7c 2020 - 43b*c*d |.| │ │ │ │ +00023330: 2020 2020 3437 632a 6420 2b20 3237 6420 47c*d + 27d │ │ │ │ +00023340: 2c20 6220 202b 2032 3262 2a63 202d 2031 , b + 22b*c - 1 │ │ │ │ +00023350: 3863 2020 2b20 3762 2a64 202b 2032 3363 8c + 7b*d + 23c │ │ │ │ +00023360: 2a64 202b 2031 3364 202c 2062 2a63 2020 *d + 13d , b*c │ │ │ │ +00023370: 2b20 3334 622a 632a 6420 2b7c 0a7c 2020 + 34b*c*d +|.| │ │ │ │ 00023380: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00023390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000233a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000233b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000233c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -000233d0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -000233e0: 2020 2032 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ -000233f0: 2020 2020 3220 2020 2020 2033 2020 2033 2 3 3 │ │ │ │ -00023400: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -00023410: 2020 2020 2020 2020 2032 207c 0a7c 2020 2 |.| │ │ │ │ -00023420: 2020 2020 2d20 3437 6320 6420 2b20 3334 - 47c d + 34 │ │ │ │ -00023430: 612a 6420 202d 2032 3262 2a64 2020 2d20 a*d - 22b*d - │ │ │ │ -00023440: 3663 2a64 2020 2b20 3137 6420 2c20 6320 6c*d + 17d , c │ │ │ │ -00023450: 202b 2032 622a 632a 6420 2b20 3232 6320 + 2b*c*d + 22c │ │ │ │ -00023460: 6420 2d20 3138 612a 6420 207c 0a7c 2020 d - 18a*d |.| │ │ │ │ +000233d0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +000233e0: 3220 2020 2020 2020 2032 2020 2020 2020 2 2 │ │ │ │ +000233f0: 2020 3220 2020 2020 2033 2020 2033 2020 2 3 3 │ │ │ │ +00023400: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +00023410: 2020 2020 2020 2020 3220 207c 0a7c 2020 2 |.| │ │ │ │ +00023420: 2020 2020 3263 2064 202b 2031 3661 2a64 2c d + 16a*d │ │ │ │ +00023430: 2020 2b20 3435 622a 6420 202d 2031 3563 + 45b*d - 15c │ │ │ │ +00023440: 2a64 2020 2b20 3130 6420 2c20 6320 202d *d + 10d , c - │ │ │ │ +00023450: 2032 3862 2a63 2a64 202b 2033 3863 2064 28b*c*d + 38c d │ │ │ │ +00023460: 202d 2033 3961 2a64 2020 2d7c 0a7c 2020 - 39a*d -|.| │ │ │ │ 00023470: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00023480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000234a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000234b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -000234c0: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -000234d0: 2020 2020 3220 2020 2033 2020 2020 2020 2 3 │ │ │ │ +000234c0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +000234d0: 2020 3220 2020 2020 3320 2020 2020 2020 2 3 │ │ │ │ 000234e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000234f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023500: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00023510: 2020 2020 2b20 3338 622a 6420 202d 2033 + 38b*d - 3 │ │ │ │ -00023520: 3963 2a64 2020 2d20 6420 2920 2020 2020 9c*d - d ) │ │ │ │ +00023510: 2020 2020 3437 622a 6420 202d 2031 3363 47b*d - 13c │ │ │ │ +00023520: 2a64 2020 2b20 3464 2029 2020 2020 2020 *d + 4d ) │ │ │ │ 00023530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023550: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00023560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9128,111 +9128,111 @@ │ │ │ │ 00023a70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00023a80: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ 00023a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c6f ------------|.|o │ │ │ │ 00023ad0: 3230 203d 207c 6964 6561 6c20 2863 202d 20 = |ideal (c - │ │ │ │ -00023ae0: 2032 3364 2c20 6220 2b20 3764 2c20 6120 23d, b + 7d, a │ │ │ │ -00023af0: 2b20 3238 6429 2020 2020 2020 2020 2020 + 28d) │ │ │ │ +00023ae0: 2031 3664 2c20 6220 2b20 3331 642c 2061 16d, b + 31d, a │ │ │ │ +00023af0: 202b 2031 3264 2920 2020 2020 2020 2020 + 12d) │ │ │ │ 00023b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00023b20: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ 00023b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ -00023b70: 2020 2020 207c 6964 6561 6c20 2863 202b |ideal (c + │ │ │ │ -00023b80: 2032 3164 2c20 6220 2b20 3336 642c 2061 21d, b + 36d, a │ │ │ │ -00023b90: 202d 2031 3264 2920 2020 2020 2020 2020 - 12d) │ │ │ │ +00023b70: 2020 2020 207c 6964 6561 6c20 2863 202d |ideal (c - │ │ │ │ +00023b80: 2032 3964 2c20 6220 2b20 3239 642c 2061 29d, b + 29d, a │ │ │ │ +00023b90: 202d 2032 3764 2920 2020 2020 2020 2020 - 27d) │ │ │ │ 00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023bb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00023bc0: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ 00023bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ 00023c10: 2020 2020 207c 6964 6561 6c20 2863 202b |ideal (c + │ │ │ │ -00023c20: 2031 3364 2c20 6220 2d20 3232 642c 2061 13d, b - 22d, a │ │ │ │ -00023c30: 202d 2032 3864 2920 2020 2020 2020 2020 - 28d) │ │ │ │ +00023c20: 2034 3164 2c20 6220 2b20 3335 642c 2061 41d, b + 35d, a │ │ │ │ +00023c30: 202d 2032 3564 2920 2020 2020 2020 2020 - 25d) │ │ │ │ 00023c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00023c60: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ 00023c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ 00023cb0: 2020 2020 207c 2020 2020 2020 2020 2020 | │ │ │ │ 00023cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023cd0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00023ce0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00023cd0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00023ce0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ 00023cf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00023d00: 2020 2020 207c 6964 6561 6c20 2861 202d |ideal (a - │ │ │ │ -00023d10: 2032 3462 202d 2032 3963 202b 2032 3264 24b - 29c + 22d │ │ │ │ -00023d20: 2c20 6320 202b 2033 3362 2a64 202b 2034 , c + 33b*d + 4 │ │ │ │ -00023d30: 3363 2a64 202b 2034 3164 202c 2062 2a63 3c*d + 41d , b*c │ │ │ │ -00023d40: 202d 2031 3662 2a64 202b 2034 7c0a 7c20 - 16b*d + 4|.| │ │ │ │ +00023d10: 2038 6220 2d20 3336 6320 2b20 3337 642c 8b - 36c + 37d, │ │ │ │ +00023d20: 2063 2020 2d20 3562 2a64 202b 2034 3663 c - 5b*d + 46c │ │ │ │ +00023d30: 2a64 202b 2034 3164 202c 2062 2a63 202b *d + 41d , b*c + │ │ │ │ +00023d40: 2033 3062 2a64 202d 2032 3463 7c0a 7c20 30b*d - 24c|.| │ │ │ │ 00023d50: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ 00023d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2d ------------|.|- │ │ │ │ 00023da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2d ------------|.|- │ │ │ │ 00023df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023e10: 2d2d 2b20 2020 2020 2020 2020 2020 2020 --+ │ │ │ │ +00023e10: 2d2d 2d2b 2020 2020 2020 2020 2020 2020 ---+ │ │ │ │ 00023e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00023e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e60: 2020 7c20 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00023e60: 2020 207c 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00023e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e80: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 00023e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023eb0: 2d2d 2b20 2020 2020 2020 2020 2020 2020 --+ │ │ │ │ +00023eb0: 2d2d 2d2b 2020 2020 2020 2020 2020 2020 ---+ │ │ │ │ 00023ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ed0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00023ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f00: 2020 7c20 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00023f00: 2020 207c 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00023f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023f20: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 00023f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023f50: 2d2d 2b20 2020 2020 2020 2020 2020 2020 --+ │ │ │ │ +00023f50: 2d2d 2d2b 2020 2020 2020 2020 2020 2020 ---+ │ │ │ │ 00023f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023f70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00023f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023fa0: 2020 7c20 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00023fa0: 2020 207c 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00023fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023fc0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 00023fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ff0: 2d2d 2b20 2020 2020 2020 2020 2020 2020 --+ │ │ │ │ +00023ff0: 2d2d 2d2b 2020 2020 2020 2020 2020 2020 ---+ │ │ │ │ 00024000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024010: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00024020: 2020 2020 2020 2032 2020 2032 2020 2020 2 2 │ │ │ │ +00024020: 2020 2020 2020 3220 2020 3220 2020 2020 2 2 │ │ │ │ 00024030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024040: 3220 7c20 2020 2020 2020 2020 2020 2020 2 | │ │ │ │ +00024040: 2032 207c 2020 2020 2020 2020 2020 2020 2 | │ │ │ │ 00024050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024060: 2020 2020 2020 2020 2020 2020 7c0a 7c63 |.|c │ │ │ │ -00024070: 2a64 202d 2032 6420 2c20 6220 202d 2039 *d - 2d , b - 9 │ │ │ │ -00024080: 622a 6420 2d20 3863 2a64 202d 2031 3964 b*d - 8c*d - 19d │ │ │ │ -00024090: 2029 7c20 2020 2020 2020 2020 2020 2020 )| │ │ │ │ +00024060: 2020 2020 2020 2020 2020 2020 7c0a 7c2a |.|* │ │ │ │ +00024070: 6420 2d20 3964 202c 2062 2020 2d20 3137 d - 9d , b - 17 │ │ │ │ +00024080: 622a 6420 2b20 3231 632a 6420 2d20 3334 b*d + 21c*d - 34 │ │ │ │ +00024090: 6420 297c 2020 2020 2020 2020 2020 2020 d )| │ │ │ │ 000240a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000240b0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 000240c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000240d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000240e0: 2d2d 2b20 2020 2020 2020 2020 2020 2020 --+ │ │ │ │ +000240e0: 2d2d 2d2b 2020 2020 2020 2020 2020 2020 ---+ │ │ │ │ 000240f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024100: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00024110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ @@ -9245,3475 +9245,3535 @@ │ │ │ │ 000241c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000241d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000241e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000241f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00024200: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ 00024210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 2020 -------------+ │ │ │ │ -00024240: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00024250: 3231 203d 207c 6964 6561 6c20 2863 202d 21 = |ideal (c - │ │ │ │ -00024260: 2033 3264 2c20 6220 2d20 3564 2c20 6120 32d, b - 5d, a │ │ │ │ -00024270: 2d20 3239 6429 2020 2020 2020 2020 2020 - 29d) │ │ │ │ -00024280: 2020 2020 2020 2020 2020 2020 207c 2020 | │ │ │ │ -00024290: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000242a0: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ -000242b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000242c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000242d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 2020 -------------+ │ │ │ │ -000242e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000242f0: 2020 2020 207c 6964 6561 6c20 2863 202b |ideal (c + │ │ │ │ -00024300: 2034 3364 2c20 6220 2d20 3437 642c 2061 43d, b - 47d, a │ │ │ │ -00024310: 202d 2032 3764 2920 2020 2020 2020 2020 - 27d) │ │ │ │ -00024320: 2020 2020 2020 2020 2020 2020 207c 2020 | │ │ │ │ -00024330: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00024340: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ +00024230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00024250: 2020 2020 207c 2020 2020 2020 2020 2020 | │ │ │ │ +00024260: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +00024270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024280: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00024290: 2032 2020 2020 2020 2020 2020 7c0a 7c6f 2 |.|o │ │ │ │ +000242a0: 3231 203d 207c 6964 6561 6c20 2861 2a63 21 = |ideal (a*c │ │ │ │ +000242b0: 202d 2031 3562 2a63 202b 2031 3963 2020 - 15b*c + 19c │ │ │ │ +000242c0: 2b20 3134 612a 6420 2d20 3433 622a 6420 + 14a*d - 43b*d │ │ │ │ +000242d0: 2d20 3437 632a 6420 2b20 3237 6420 2c20 - 47c*d + 27d , │ │ │ │ +000242e0: 6220 202b 2032 3262 2a63 202d 7c0a 7c20 b + 22b*c -|.| │ │ │ │ +000242f0: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ +00024300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2d ------------|.|- │ │ │ │ +00024340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 2020 -------------+ │ │ │ │ -00024380: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00024390: 2020 2020 207c 6964 6561 6c20 2863 202b |ideal (c + │ │ │ │ -000243a0: 2032 3464 2c20 6220 2d20 3439 642c 2061 24d, b - 49d, a │ │ │ │ -000243b0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -000243c0: 2020 2020 2020 2020 2020 2020 207c 2020 | │ │ │ │ -000243d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000243e0: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ -000243f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 2020 -------------+ │ │ │ │ -00024420: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00024430: 2020 2020 207c 6964 6561 6c20 2863 202b |ideal (c + │ │ │ │ -00024440: 2031 3464 2c20 6220 2b20 3331 642c 2061 14d, b + 31d, a │ │ │ │ -00024450: 202d 2031 3664 2920 2020 2020 2020 2020 - 16d) │ │ │ │ -00024460: 2020 2020 2020 2020 2020 2020 207c 2020 | │ │ │ │ -00024470: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00024480: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ +00024370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2d ------------|.|- │ │ │ │ +00024390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000243a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000243b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000243c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000243d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +000243e0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000243f0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +00024400: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00024410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024420: 2020 2020 2020 2020 2020 3220 7c0a 7c20 2 |.| │ │ │ │ +00024430: 3138 6320 202b 2037 622a 6420 2b20 3233 18c + 7b*d + 23 │ │ │ │ +00024440: 632a 6420 2b20 3133 6420 2c20 612a 6220 c*d + 13d , a*b │ │ │ │ +00024450: 2b20 3436 622a 6320 2b20 3333 6320 202b + 46b*c + 33c + │ │ │ │ +00024460: 2061 2a64 202b 2034 3262 2a64 202b 2033 a*d + 42b*d + 3 │ │ │ │ +00024470: 3763 2a64 202b 2031 3364 202c 7c0a 7c2d 7c*d + 13d ,|.|- │ │ │ │ +00024480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000244a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000244b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 2020 -------------+ │ │ │ │ -000244c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000244d0: 2020 2020 207c 2020 2020 2020 2020 2020 | │ │ │ │ -000244e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000244f0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -00024500: 2020 2020 2020 2020 2020 2032 207c 2020 2 | │ │ │ │ -00024510: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00024520: 2020 2020 207c 6964 6561 6c20 2862 202b |ideal (b + │ │ │ │ -00024530: 2031 3163 202b 2032 3264 2c20 6120 2b20 11c + 22d, a + │ │ │ │ -00024540: 3131 6320 2b20 3432 642c 2063 2020 2d20 11c + 42d, c - │ │ │ │ -00024550: 3433 632a 6420 2b20 3331 6420 297c 2020 43c*d + 31d )| │ │ │ │ -00024560: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00024570: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ -00024580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000245a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 2020 -------------+ │ │ │ │ -000245b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -000245c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000245d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000245e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000245f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 ------------+..W │ │ │ │ -00024610: 6520 6361 6e20 6465 7465 726d 696e 6520 e can determine │ │ │ │ -00024620: 7768 6174 2074 6865 7365 2072 6570 7265 what these repre │ │ │ │ -00024630: 7365 6e74 2e20 204f 6e65 2073 686f 756c sent. One shoul │ │ │ │ -00024640: 6420 6265 2061 2073 6574 206f 6620 3620 d be a set of 6 │ │ │ │ -00024650: 706f 696e 7473 2c20 7768 6572 650a 3520 points, where.5 │ │ │ │ -00024660: 6c69 6520 6f6e 2061 2070 6c61 6e65 2e20 lie on a plane. │ │ │ │ -00024670: 2054 6865 206f 7468 6572 2073 686f 756c The other shoul │ │ │ │ -00024680: 6420 6265 2036 2070 6f69 6e74 7320 7769 d be 6 points wi │ │ │ │ -00024690: 7468 2033 2070 6f69 6e74 7320 6f6e 206f th 3 points on o │ │ │ │ -000246a0: 6e65 206c 696e 652c 2061 6e64 0a74 6865 ne line, and.the │ │ │ │ -000246b0: 206f 7468 6572 2033 2070 6f69 6e74 7320 other 3 points │ │ │ │ -000246c0: 6f6e 2061 2073 6b65 7720 6c69 6e65 2e0a on a skew line.. │ │ │ │ -000246d0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ -000246e0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7261 ==.. * *note ra │ │ │ │ -000246f0: 6e64 6f6d 506f 696e 744f 6e52 6174 696f ndomPointOnRatio │ │ │ │ -00024700: 6e61 6c56 6172 6965 7479 3a0a 2020 2020 nalVariety:. │ │ │ │ -00024710: 7261 6e64 6f6d 506f 696e 744f 6e52 6174 randomPointOnRat │ │ │ │ -00024720: 696f 6e61 6c56 6172 6965 7479 5f6c 7049 ionalVariety_lpI │ │ │ │ -00024730: 6465 616c 5f72 702c 202d 2d20 6669 6e64 deal_rp, -- find │ │ │ │ -00024740: 2061 2072 616e 646f 6d20 706f 696e 7420 a random point │ │ │ │ -00024750: 6f6e 2061 0a20 2020 2076 6172 6965 7479 on a. variety │ │ │ │ -00024760: 2074 6861 7420 6361 6e20 6265 2064 6574 that can be det │ │ │ │ -00024770: 6563 7465 6420 746f 2062 6520 7261 7469 ected to be rati │ │ │ │ -00024780: 6f6e 616c 0a0a 5761 7973 2074 6f20 7573 onal..Ways to us │ │ │ │ -00024790: 6520 6e6f 6e6d 696e 696d 616c 4d61 7073 e nonminimalMaps │ │ │ │ -000247a0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -000247b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -000247c0: 202a 2022 6e6f 6e6d 696e 696d 616c 4d61 * "nonminimalMa │ │ │ │ -000247d0: 7073 2849 6465 616c 2922 0a0a 466f 7220 ps(Ideal)"..For │ │ │ │ -000247e0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -000247f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00024800: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -00024810: 6f74 6520 6e6f 6e6d 696e 696d 616c 4d61 ote nonminimalMa │ │ │ │ -00024820: 7073 3a20 6e6f 6e6d 696e 696d 616c 4d61 ps: nonminimalMa │ │ │ │ -00024830: 7073 2c20 6973 2061 202a 6e6f 7465 206d ps, is a *note m │ │ │ │ -00024840: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a ethod function:. │ │ │ │ -00024850: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -00024860: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ -00024870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000248a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000248b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -000248c0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -000248d0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -000248e0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -000248f0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -00024900: 756c 6179 322d 312e 3236 2e30 352b 6473 ulay2-1.26.05+ds │ │ │ │ -00024910: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -00024920: 636b 6167 6573 2f0a 4772 6f65 626e 6572 ckages/.Groebner │ │ │ │ -00024930: 5374 7261 7461 2e6d 323a 3130 3133 3a30 Strata.m2:1013:0 │ │ │ │ -00024940: 2e0a 1f0a 4669 6c65 3a20 4772 6f65 626e ....File: Groebn │ │ │ │ -00024950: 6572 5374 7261 7461 2e69 6e66 6f2c 204e erStrata.info, N │ │ │ │ -00024960: 6f64 653a 2072 616e 646f 6d50 6f69 6e74 ode: randomPoint │ │ │ │ -00024970: 4f6e 5261 7469 6f6e 616c 5661 7269 6574 OnRationalVariet │ │ │ │ -00024980: 795f 6c70 4964 6561 6c5f 7270 2c20 4e65 y_lpIdeal_rp, Ne │ │ │ │ -00024990: 7874 3a20 7261 6e64 6f6d 506f 696e 7473 xt: randomPoints │ │ │ │ -000249a0: 4f6e 5261 7469 6f6e 616c 5661 7269 6574 OnRationalVariet │ │ │ │ -000249b0: 795f 6c70 4964 6561 6c5f 636d 5a5a 5f72 y_lpIdeal_cmZZ_r │ │ │ │ -000249c0: 702c 2050 7265 763a 206e 6f6e 6d69 6e69 p, Prev: nonmini │ │ │ │ -000249d0: 6d61 6c4d 6170 732c 2055 703a 2054 6f70 malMaps, Up: Top │ │ │ │ -000249e0: 0a0a 7261 6e64 6f6d 506f 696e 744f 6e52 ..randomPointOnR │ │ │ │ -000249f0: 6174 696f 6e61 6c56 6172 6965 7479 2849 ationalVariety(I │ │ │ │ -00024a00: 6465 616c 2920 2d2d 2066 696e 6420 6120 deal) -- find a │ │ │ │ -00024a10: 7261 6e64 6f6d 2070 6f69 6e74 206f 6e20 random point on │ │ │ │ -00024a20: 6120 7661 7269 6574 7920 7468 6174 2063 a variety that c │ │ │ │ -00024a30: 616e 2062 6520 6465 7465 6374 6564 2074 an be detected t │ │ │ │ -00024a40: 6f20 6265 2072 6174 696f 6e61 6c0a 2a2a o be rational.** │ │ │ │ -00024a50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024a60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024a70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024a80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024a90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024aa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024ab0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 *********.. * F │ │ │ │ -00024ac0: 756e 6374 696f 6e3a 202a 6e6f 7465 2072 unction: *note r │ │ │ │ -00024ad0: 616e 646f 6d50 6f69 6e74 4f6e 5261 7469 andomPointOnRati │ │ │ │ -00024ae0: 6f6e 616c 5661 7269 6574 793a 0a20 2020 onalVariety:. │ │ │ │ -00024af0: 2072 616e 646f 6d50 6f69 6e74 4f6e 5261 randomPointOnRa │ │ │ │ -00024b00: 7469 6f6e 616c 5661 7269 6574 795f 6c70 tionalVariety_lp │ │ │ │ -00024b10: 4964 6561 6c5f 7270 2c0a 2020 2a20 5573 Ideal_rp,. * Us │ │ │ │ -00024b20: 6167 653a 200a 2020 2020 2020 2020 7261 age: . ra │ │ │ │ -00024b30: 6e64 6f6d 506f 696e 744f 6e52 6174 696f ndomPointOnRatio │ │ │ │ -00024b40: 6e61 6c56 6172 6965 7479 2049 0a20 2020 nalVariety I. │ │ │ │ -00024b50: 2020 2020 2072 616e 646f 6d50 6f69 6e74 randomPoint │ │ │ │ -00024b60: 4f6e 5261 7469 6f6e 616c 5661 7269 6574 OnRationalVariet │ │ │ │ -00024b70: 790a 2020 2a20 496e 7075 7473 3a0a 2020 y. * Inputs:. │ │ │ │ -00024b80: 2020 2020 2a20 492c 2061 6e20 2a6e 6f74 * I, an *not │ │ │ │ -00024b90: 6520 6964 6561 6c3a 2028 4d61 6361 756c e ideal: (Macaul │ │ │ │ -00024ba0: 6179 3244 6f63 2949 6465 616c 2c2c 2041 ay2Doc)Ideal,, A │ │ │ │ -00024bb0: 6e20 6964 6561 6c20 696e 2061 2070 6f6c n ideal in a pol │ │ │ │ -00024bc0: 796e 6f6d 6961 6c20 7269 6e67 0a20 2020 ynomial ring. │ │ │ │ -00024bd0: 2020 2020 2024 5324 206f 7665 7220 6120 $S$ over a │ │ │ │ -00024be0: 6669 656c 642c 2077 6869 6368 2064 6566 field, which def │ │ │ │ -00024bf0: 696e 6573 2061 2070 7269 6d65 2069 6465 ines a prime ide │ │ │ │ -00024c00: 616c 0a20 202a 204f 7574 7075 7473 3a0a al. * Outputs:. │ │ │ │ -00024c10: 2020 2020 2020 2a20 6120 2a6e 6f74 6520 * a *note │ │ │ │ -00024c20: 6d61 7472 6978 3a20 284d 6163 6175 6c61 matrix: (Macaula │ │ │ │ -00024c30: 7932 446f 6329 4d61 7472 6978 2c2c 2041 y2Doc)Matrix,, A │ │ │ │ -00024c40: 206f 6e65 2072 6f77 206d 6174 7269 7820 one row matrix │ │ │ │ -00024c50: 6f76 6572 2074 6865 2062 6173 650a 2020 over the base. │ │ │ │ -00024c60: 2020 2020 2020 6669 656c 6420 6f66 2024 field of $ │ │ │ │ -00024c70: 5324 2c20 7265 7072 6573 656e 7469 6e67 S$, representing │ │ │ │ -00024c80: 2061 2072 616e 646f 6d6c 7920 6368 6f73 a randomly chos │ │ │ │ -00024c90: 656e 2070 6f69 6e74 206f 6e20 7468 6520 en point on the │ │ │ │ -00024ca0: 7a65 726f 206c 6f63 7573 206f 660a 2020 zero locus of. │ │ │ │ -00024cb0: 2020 2020 2020 2449 242e 2020 6e75 6c6c $I$. null │ │ │ │ -00024cc0: 2069 7320 7265 7475 726e 6564 2069 6e20 is returned in │ │ │ │ -00024cd0: 7468 6520 6361 7365 2077 6865 6e20 7468 the case when th │ │ │ │ -00024ce0: 6520 726f 7574 696e 6520 6361 6e6e 6f74 e routine cannot │ │ │ │ -00024cf0: 2064 6574 6572 6d69 6e65 2069 660a 2020 determine if. │ │ │ │ -00024d00: 2020 2020 2020 7468 6520 7661 7269 6574 the variet │ │ │ │ -00024d10: 7920 6973 2072 6174 696f 6e61 6c20 616e y is rational an │ │ │ │ -00024d20: 6420 6972 7265 6475 6369 626c 652e 0a0a d irreducible... │ │ │ │ -00024d30: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -00024d40: 3d3d 3d3d 3d3d 3d0a 0a41 7320 6120 6669 =======..As a fi │ │ │ │ -00024d50: 7273 7420 6578 616d 706c 652c 2077 6520 rst example, we │ │ │ │ -00024d60: 6669 6e64 2061 2072 616e 646f 6d20 706f find a random po │ │ │ │ -00024d70: 696e 7420 6f6e 2074 6865 2056 6572 6f6e int on the Veron │ │ │ │ -00024d80: 6573 6520 7375 7266 6163 6520 696e 2024 ese surface in $ │ │ │ │ -00024d90: 5c50 505e 3524 2e0a 0a2b 2d2d 2d2d 2d2d \PP^5$...+------ │ │ │ │ -00024da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024de0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b -------+.|i1 : k │ │ │ │ -00024df0: 6b20 3d20 5a5a 2f31 3031 3b20 2020 2020 k = ZZ/101; │ │ │ │ -00024e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024e30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00024e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024e80: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 -------+.|i2 : S │ │ │ │ -00024e90: 203d 206b 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+00024ad0: 7261 6e64 6f6d 506f 696e 744f 6e52 6174 randomPointOnRat │ │ │ │ +00024ae0: 696f 6e61 6c56 6172 6965 7479 5f6c 7049 ionalVariety_lpI │ │ │ │ +00024af0: 6465 616c 5f72 702c 202d 2d20 6669 6e64 deal_rp, -- find │ │ │ │ +00024b00: 2061 2072 616e 646f 6d20 706f 696e 7420 a random point │ │ │ │ +00024b10: 6f6e 2061 0a20 2020 2076 6172 6965 7479 on a. variety │ │ │ │ +00024b20: 2074 6861 7420 6361 6e20 6265 2064 6574 that can be det │ │ │ │ +00024b30: 6563 7465 6420 746f 2062 6520 7261 7469 ected to be rati │ │ │ │ +00024b40: 6f6e 616c 0a0a 5761 7973 2074 6f20 7573 onal..Ways to us │ │ │ │ +00024b50: 6520 6e6f 6e6d 696e 696d 616c 4d61 7073 e nonminimalMaps │ │ │ │ +00024b60: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +00024b70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +00024b80: 202a 2022 6e6f 6e6d 696e 696d 616c 4d61 * "nonminimalMa │ │ │ │ +00024b90: 7073 2849 6465 616c 2922 0a0a 466f 7220 ps(Ideal)"..For │ │ │ │ +00024ba0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +00024bb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00024bc0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +00024bd0: 6f74 6520 6e6f 6e6d 696e 696d 616c 4d61 ote nonminimalMa │ │ │ │ +00024be0: 7073 3a20 6e6f 6e6d 696e 696d 616c 4d61 ps: nonminimalMa │ │ │ │ +00024bf0: 7073 2c20 6973 2061 202a 6e6f 7465 206d ps, is a *note m │ │ │ │ +00024c00: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a ethod function:. │ │ │ │ +00024c10: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +00024c20: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +00024c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +00024c80: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +00024c90: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +00024ca0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +00024cb0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +00024cc0: 756c 6179 322d 312e 3236 2e30 352b 6473 ulay2-1.26.05+ds │ │ │ │ +00024cd0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +00024ce0: 636b 6167 6573 2f0a 4772 6f65 626e 6572 ckages/.Groebner │ │ │ │ +00024cf0: 5374 7261 7461 2e6d 323a 3130 3133 3a30 Strata.m2:1013:0 │ │ │ │ +00024d00: 2e0a 1f0a 4669 6c65 3a20 4772 6f65 626e ....File: Groebn │ │ │ │ +00024d10: 6572 5374 7261 7461 2e69 6e66 6f2c 204e erStrata.info, N │ │ │ │ +00024d20: 6f64 653a 2072 616e 646f 6d50 6f69 6e74 ode: randomPoint │ │ │ │ +00024d30: 4f6e 5261 7469 6f6e 616c 5661 7269 6574 OnRationalVariet │ │ │ │ +00024d40: 795f 6c70 4964 6561 6c5f 7270 2c20 4e65 y_lpIdeal_rp, Ne │ │ │ │ +00024d50: 7874 3a20 7261 6e64 6f6d 506f 696e 7473 xt: randomPoints │ │ │ │ +00024d60: 4f6e 5261 7469 6f6e 616c 5661 7269 6574 OnRationalVariet │ │ │ │ +00024d70: 795f 6c70 4964 6561 6c5f 636d 5a5a 5f72 y_lpIdeal_cmZZ_r │ │ │ │ +00024d80: 702c 2050 7265 763a 206e 6f6e 6d69 6e69 p, Prev: nonmini │ │ │ │ +00024d90: 6d61 6c4d 6170 732c 2055 703a 2054 6f70 malMaps, Up: Top │ │ │ │ +00024da0: 0a0a 7261 6e64 6f6d 506f 696e 744f 6e52 ..randomPointOnR │ │ │ │ +00024db0: 6174 696f 6e61 6c56 6172 6965 7479 2849 ationalVariety(I │ │ │ │ +00024dc0: 6465 616c 2920 2d2d 2066 696e 6420 6120 deal) -- find a │ │ │ │ +00024dd0: 7261 6e64 6f6d 2070 6f69 6e74 206f 6e20 random point on │ │ │ │ +00024de0: 6120 7661 7269 6574 7920 7468 6174 2063 a variety that c │ │ │ │ +00024df0: 616e 2062 6520 6465 7465 6374 6564 2074 an be detected t │ │ │ │ +00024e00: 6f20 6265 2072 6174 696f 6e61 6c0a 2a2a o be rational.** │ │ │ │ +00024e10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024e20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024e30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024e40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024e50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024e60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024e70: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 *********.. * F │ │ │ │ +00024e80: 756e 6374 696f 6e3a 202a 6e6f 7465 2072 unction: *note r │ │ │ │ +00024e90: 616e 646f 6d50 6f69 6e74 4f6e 5261 7469 andomPointOnRati │ │ │ │ +00024ea0: 6f6e 616c 5661 7269 6574 793a 0a20 2020 onalVariety:. │ │ │ │ +00024eb0: 2072 616e 646f 6d50 6f69 6e74 4f6e 5261 randomPointOnRa │ │ │ │ +00024ec0: 7469 6f6e 616c 5661 7269 6574 795f 6c70 tionalVariety_lp │ │ │ │ +00024ed0: 4964 6561 6c5f 7270 2c0a 2020 2a20 5573 Ideal_rp,. * Us │ │ │ │ +00024ee0: 6167 653a 200a 2020 2020 2020 2020 7261 age: . ra │ │ │ │ +00024ef0: 6e64 6f6d 506f 696e 744f 6e52 6174 696f ndomPointOnRatio │ │ │ │ +00024f00: 6e61 6c56 6172 6965 7479 2049 0a20 2020 nalVariety I. │ │ │ │ +00024f10: 2020 2020 2072 616e 646f 6d50 6f69 6e74 randomPoint │ │ │ │ +00024f20: 4f6e 5261 7469 6f6e 616c 5661 7269 6574 OnRationalVariet │ │ │ │ +00024f30: 790a 2020 2a20 496e 7075 7473 3a0a 2020 y. * Inputs:. │ │ │ │ +00024f40: 2020 2020 2a20 492c 2061 6e20 2a6e 6f74 * I, an *not │ │ │ │ +00024f50: 6520 6964 6561 6c3a 2028 4d61 6361 756c e ideal: (Macaul │ │ │ │ +00024f60: 6179 3244 6f63 2949 6465 616c 2c2c 2041 ay2Doc)Ideal,, A │ │ │ │ +00024f70: 6e20 6964 6561 6c20 696e 2061 2070 6f6c n ideal in a pol │ │ │ │ +00024f80: 796e 6f6d 6961 6c20 7269 6e67 0a20 2020 ynomial ring. │ │ │ │ +00024f90: 2020 2020 2024 5324 206f 7665 7220 6120 $S$ over a │ │ │ │ +00024fa0: 6669 656c 642c 2077 6869 6368 2064 6566 field, which def │ │ │ │ +00024fb0: 696e 6573 2061 2070 7269 6d65 2069 6465 ines a prime ide │ │ │ │ +00024fc0: 616c 0a20 202a 204f 7574 7075 7473 3a0a al. * Outputs:. │ │ │ │ +00024fd0: 2020 2020 2020 2a20 6120 2a6e 6f74 6520 * a *note │ │ │ │ +00024fe0: 6d61 7472 6978 3a20 284d 6163 6175 6c61 matrix: (Macaula │ │ │ │ +00024ff0: 7932 446f 6329 4d61 7472 6978 2c2c 2041 y2Doc)Matrix,, A │ │ │ │ +00025000: 206f 6e65 2072 6f77 206d 6174 7269 7820 one row matrix │ │ │ │ +00025010: 6f76 6572 2074 6865 2062 6173 650a 2020 over the base. │ │ │ │ +00025020: 2020 2020 2020 6669 656c 6420 6f66 2024 field of $ │ │ │ │ +00025030: 5324 2c20 7265 7072 6573 656e 7469 6e67 S$, representing │ │ │ │ +00025040: 2061 2072 616e 646f 6d6c 7920 6368 6f73 a randomly chos │ │ │ │ +00025050: 656e 2070 6f69 6e74 206f 6e20 7468 6520 en point on the │ │ │ │ +00025060: 7a65 726f 206c 6f63 7573 206f 660a 2020 zero locus of. │ │ │ │ +00025070: 2020 2020 2020 2449 242e 2020 6e75 6c6c $I$. null │ │ │ │ +00025080: 2069 7320 7265 7475 726e 6564 2069 6e20 is returned in │ │ │ │ +00025090: 7468 6520 6361 7365 2077 6865 6e20 7468 the case when th │ │ │ │ +000250a0: 6520 726f 7574 696e 6520 6361 6e6e 6f74 e routine cannot │ │ │ │ +000250b0: 2064 6574 6572 6d69 6e65 2069 660a 2020 determine if. │ │ │ │ +000250c0: 2020 2020 2020 7468 6520 7661 7269 6574 the variet │ │ │ │ +000250d0: 7920 6973 2072 6174 696f 6e61 6c20 616e y is rational an │ │ │ │ +000250e0: 6420 6972 7265 6475 6369 626c 652e 0a0a d irreducible... │ │ │ │ +000250f0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00025100: 3d3d 3d3d 3d3d 3d0a 0a41 7320 6120 6669 =======..As a fi │ │ │ │ +00025110: 7273 7420 6578 616d 706c 652c 2077 6520 rst example, we │ │ │ │ +00025120: 6669 6e64 2061 2072 616e 646f 6d20 706f find a random po │ │ │ │ +00025130: 696e 7420 6f6e 2074 6865 2056 6572 6f6e int on the Veron │ │ │ │ +00025140: 6573 6520 7375 7266 6163 6520 696e 2024 ese surface in $ │ │ │ │ +00025150: 5c50 505e 3524 2e0a 0a2b 2d2d 2d2d 2d2d \PP^5$...+------ │ │ │ │ +00025160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000251a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b -------+.|i1 : k │ │ │ │ +000251b0: 6b20 3d20 5a5a 2f31 3031 3b20 2020 2020 k = ZZ/101; │ │ │ │ 000251c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000251d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000251e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000251f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00025200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025240: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2070 -------+.|i4 : p │ │ │ │ -00025250: 7420 3d20 7261 6e64 6f6d 506f 696e 744f t = randomPointO │ │ │ │ -00025260: 6e52 6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ -00025270: 2049 2020 2020 2020 2020 2020 2020 2020 I │ │ │ │ +00025240: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 -------+.|i2 : S │ │ │ │ +00025250: 203d 206b 6b5b 612e 2e66 5d3b 2020 2020 = kk[a..f]; │ │ │ │ +00025260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025290: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000252a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000252b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000252c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000252d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000252e0: 2020 2020 2020 207c 0a7c 6f34 203d 207c |.|o4 = | │ │ │ │ -000252f0: 202d 3235 2032 3020 2d33 3020 2d31 3620 -25 20 -30 -16 │ │ │ │ -00025300: 3234 202d 3336 207c 2020 2020 2020 2020 24 -36 | │ │ │ │ -00025310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025290: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000252a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000252b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000252c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000252d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000252e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2049 -------+.|i3 : I │ │ │ │ +000252f0: 203d 206d 696e 6f72 7328 322c 2067 656e = minors(2, gen │ │ │ │ +00025300: 6572 6963 5379 6d6d 6574 7269 634d 6174 ericSymmetricMat │ │ │ │ +00025310: 7269 7828 532c 2033 2929 2020 2020 2020 rix(S, 3)) │ │ │ │ 00025320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025330: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00025340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025380: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00025390: 2020 2020 2020 2020 3120 2020 2020 2020 1 │ │ │ │ -000253a0: 3620 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ +00025390: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +000253a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000253b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000253c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000253d0: 2020 2020 2020 207c 0a7c 6f34 203a 204d |.|o4 : M │ │ │ │ -000253e0: 6174 7269 7820 6b6b 2020 3c2d 2d20 6b6b atrix kk <-- kk │ │ │ │ -000253f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025420: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000253c0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +000253d0: 2020 2020 2020 207c 0a7c 6f33 203d 2069 |.|o3 = i │ │ │ │ +000253e0: 6465 616c 2028 2d20 6220 202b 2061 2a64 deal (- b + a*d │ │ │ │ +000253f0: 2c20 2d20 622a 6320 2b20 612a 652c 202d , - b*c + a*e, - │ │ │ │ +00025400: 2063 2a64 202b 2062 2a65 2c20 2d20 622a c*d + b*e, - b* │ │ │ │ +00025410: 6320 2b20 612a 652c 202d 2063 2020 2b20 c + a*e, - c + │ │ │ │ +00025420: 612a 662c 202d 207c 0a7c 2020 2020 202d a*f, - |.| - │ │ │ │ 00025430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025470: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2073 -------+.|i5 : s │ │ │ │ -00025480: 7562 2849 2c20 7074 2920 3d3d 2030 2020 ub(I, pt) == 0 │ │ │ │ +00025470: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 -------|.| │ │ │ │ +00025480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000254a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000254a0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ 000254b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000254c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000254d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000254e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000254f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000254c0: 2020 2020 2020 207c 0a7c 2020 2020 2063 |.| c │ │ │ │ +000254d0: 2a65 202b 2062 2a66 2c20 2d20 632a 6420 *e + b*f, - c*d │ │ │ │ +000254e0: 2b20 622a 652c 202d 2063 2a65 202b 2062 + b*e, - c*e + b │ │ │ │ +000254f0: 2a66 2c20 2d20 6520 202b 2064 2a66 2920 *f, - e + d*f) │ │ │ │ 00025500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025510: 2020 2020 2020 207c 0a7c 6f35 203d 2074 |.|o5 = t │ │ │ │ -00025520: 7275 6520 2020 2020 2020 2020 2020 2020 rue │ │ │ │ +00025510: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00025520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025560: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00025570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000255a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000255b0: 2d2d 2d2d 2d2d 2d2b 0a2b 2d2d 2d2d 2d2d -------+.+------ │ │ │ │ +00025560: 2020 2020 2020 207c 0a7c 6f33 203a 2049 |.|o3 : I │ │ │ │ +00025570: 6465 616c 206f 6620 5320 2020 2020 2020 deal of S │ │ │ │ +00025580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000255a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000255b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 000255c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000255d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000255e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000255f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025600: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2053 -------+.|i6 : S │ │ │ │ -00025610: 203d 206b 6b5b 612e 2e64 5d3b 2020 2020 = kk[a..d]; │ │ │ │ -00025620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025600: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2070 -------+.|i4 : p │ │ │ │ +00025610: 7420 3d20 7261 6e64 6f6d 506f 696e 744f t = randomPointO │ │ │ │ +00025620: 6e52 6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ +00025630: 2049 2020 2020 2020 2020 2020 2020 2020 I │ │ │ │ 00025640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025650: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00025660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000256a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2046 -------+.|i7 : F │ │ │ │ -000256b0: 203d 2067 726f 6562 6e65 7246 616d 696c = groebnerFamil │ │ │ │ -000256c0: 7920 6964 6561 6c22 6132 2c61 622c 6163 y ideal"a2,ab,ac │ │ │ │ -000256d0: 2c62 3222 2020 2020 2020 2020 2020 2020 ,b2" │ │ │ │ +00025650: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00025660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000256a0: 2020 2020 2020 207c 0a7c 6f34 203d 207c |.|o4 = | │ │ │ │ +000256b0: 2031 2034 3920 3234 202d 3233 202d 3336 1 49 24 -23 -36 │ │ │ │ +000256c0: 202d 3330 207c 2020 2020 2020 2020 2020 -30 | │ │ │ │ +000256d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000256e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000256f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00025700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025740: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00025750: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -00025760: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +00025750: 2020 2020 2020 2020 3120 2020 2020 2020 1 │ │ │ │ +00025760: 3620 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ 00025770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025780: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -00025790: 2020 2020 2020 207c 0a7c 6f37 203d 2069 |.|o7 = i │ │ │ │ -000257a0: 6465 616c 2028 6120 202b 2074 2062 2a63 deal (a + t b*c │ │ │ │ -000257b0: 202b 2074 2061 2a64 202b 2074 2063 2020 + t a*d + t c │ │ │ │ -000257c0: 2b20 7420 622a 6420 2b20 7420 632a 6420 + t b*d + t c*d │ │ │ │ -000257d0: 2b20 7420 6420 2c20 612a 6220 2b20 7420 + t d , a*b + t │ │ │ │ -000257e0: 622a 6320 2b20 207c 0a7c 2020 2020 2020 b*c + |.| │ │ │ │ -000257f0: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ -00025800: 2020 2020 3320 2020 2020 2020 3220 2020 3 2 │ │ │ │ -00025810: 2020 2034 2020 2020 2020 2035 2020 2020 4 5 │ │ │ │ -00025820: 2020 2036 2020 2020 2020 2020 2020 2037 6 7 │ │ │ │ -00025830: 2020 2020 2020 207c 0a7c 2020 2020 202d |.| - │ │ │ │ -00025840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025880: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 -------|.| │ │ │ │ -00025890: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00025780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025790: 2020 2020 2020 207c 0a7c 6f34 203a 204d |.|o4 : M │ │ │ │ +000257a0: 6174 7269 7820 6b6b 2020 3c2d 2d20 6b6b atrix kk <-- kk │ │ │ │ +000257b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000257c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000257d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000257e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000257f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025830: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2073 -------+.|i5 : s │ │ │ │ +00025840: 7562 2849 2c20 7074 2920 3d3d 2030 2020 ub(I, pt) == 0 │ │ │ │ +00025850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025880: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00025890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000258a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000258b0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +000258b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000258c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000258d0: 2020 2032 2020 207c 0a7c 2020 2020 2074 2 |.| t │ │ │ │ -000258e0: 2061 2a64 202b 2074 2063 2020 2b20 7420 a*d + t c + t │ │ │ │ -000258f0: 2062 2a64 202b 2074 2020 632a 6420 2b20 b*d + t c*d + │ │ │ │ -00025900: 7420 2064 202c 2061 2a63 202b 2074 2020 t d , a*c + t │ │ │ │ -00025910: 622a 6320 2b20 7420 2061 2a64 202b 2074 b*c + t a*d + t │ │ │ │ -00025920: 2020 6320 202b 207c 0a7c 2020 2020 2020 c + |.| │ │ │ │ -00025930: 3920 2020 2020 2020 3820 2020 2020 2031 9 8 1 │ │ │ │ -00025940: 3020 2020 2020 2020 3131 2020 2020 2020 0 11 │ │ │ │ -00025950: 2031 3220 2020 2020 2020 2020 2020 3133 12 13 │ │ │ │ -00025960: 2020 2020 2020 2031 3520 2020 2020 2020 15 │ │ │ │ -00025970: 3134 2020 2020 207c 0a7c 2020 2020 202d 14 |.| - │ │ │ │ +000258d0: 2020 2020 2020 207c 0a7c 6f35 203d 2074 |.|o5 = t │ │ │ │ +000258e0: 7275 6520 2020 2020 2020 2020 2020 2020 rue │ │ │ │ +000258f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025920: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00025930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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0a7c 2020 2020 2074 |.| t │ │ │ │ -00025a20: 2020 622a 6420 2b20 7420 2063 2a64 202b b*d + t c*d + │ │ │ │ -00025a30: 2074 2020 6420 2c20 6220 202b 2074 2020 t d , b + t │ │ │ │ -00025a40: 622a 6320 2b20 7420 2061 2a64 202b 2074 b*c + t a*d + t │ │ │ │ -00025a50: 2020 6320 202b 2074 2020 622a 6420 2b20 c + t b*d + │ │ │ │ -00025a60: 7420 2063 2a64 207c 0a7c 2020 2020 2020 t c*d |.| │ │ │ │ -00025a70: 3136 2020 2020 2020 2031 3720 2020 2020 16 17 │ │ │ │ -00025a80: 2020 3138 2020 2020 2020 2020 2020 3139 18 19 │ │ │ │ -00025a90: 2020 2020 2020 2032 3120 2020 2020 2020 21 │ │ │ │ -00025aa0: 3230 2020 2020 2020 3232 2020 2020 2020 20 22 │ │ │ │ -00025ab0: 2032 3320 2020 207c 0a7c 2020 2020 202d 23 |.| - │ │ │ │ -00025ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025b00: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 -------|.| │ │ │ │ -00025b10: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -00025b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025a10: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00025a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025a60: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2046 -------+.|i7 : F │ │ │ │ +00025a70: 203d 2067 726f 6562 6e65 7246 616d 696c = groebnerFamil │ │ │ │ +00025a80: 7920 6964 6561 6c22 6132 2c61 622c 6163 y ideal"a2,ab,ac │ │ │ │ +00025a90: 2c62 3222 2020 2020 2020 2020 2020 2020 ,b2" │ │ │ │ +00025aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 2020 2020 2020 24 │ │ │ │ -00025bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025bf0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00025c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c40: 2020 2020 2020 207c 0a7c 6f37 203a 2049 |.|o7 : I │ │ │ │ -00025c50: 6465 616c 206f 6620 6b6b 5b74 202c 2074 deal of kk[t , t │ │ │ │ -00025c60: 202c 2074 2020 2c20 7420 2c20 7420 2c20 , t , t , t , │ │ │ │ -00025c70: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -00025c80: 202c 2074 202c 2074 202c 2074 2020 2c20 , t , t , t , │ │ │ │ -00025c90: 7420 202c 2074 207c 0a7c 2020 2020 2020 t , t |.| │ │ │ │ -00025ca0: 2020 2020 2020 2020 2020 2020 3620 2020 6 │ │ │ │ -00025cb0: 3520 2020 3132 2020 2032 2020 2034 2020 5 12 2 4 │ │ │ │ -00025cc0: 2031 3120 2020 3138 2020 2032 3420 2020 11 18 24 │ │ │ │ -00025cd0: 3120 2020 3320 2020 3820 2020 3130 2020 1 3 8 10 │ │ │ │ -00025ce0: 2031 3720 2020 327c 0a7c 2d2d 2d2d 2d2d 17 2|.|------ │ │ │ │ -00025cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025d30: 2d2d 2d2d 2d2d 2d7c 0a7c 202c 2074 202c -------|.| , t , │ │ │ │ -00025d40: 2074 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -00025d50: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ -00025d60: 202c 2074 2020 2c20 7420 205d 5b61 2e2e , t , t ][a.. │ │ │ │ -00025d70: 645d 2020 2020 2020 2020 2020 2020 2020 d] │ │ │ │ -00025d80: 2020 2020 2020 207c 0a7c 3320 2020 3720 |.|3 7 │ │ │ │ -00025d90: 2020 3920 2020 3134 2020 2031 3620 2020 9 14 16 │ │ │ │ -00025da0: 3230 2020 2032 3220 2020 3133 2020 2031 20 22 13 1 │ │ │ │ -00025db0: 3520 2020 3139 2020 2032 3120 2020 2020 5 19 21 │ │ │ │ -00025dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025dd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00025de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e20: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 204a -------+.|i8 : J │ │ │ │ -00025e30: 203d 2067 726f 6562 6e65 7253 7472 6174 = groebnerStrat │ │ │ │ -00025e40: 756d 2046 2020 2020 2020 2020 2020 2020 um F │ │ │ │ -00025e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025e70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00025e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00025ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b40: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00025b50: 2020 2020 2020 207c 0a7c 6f37 203d 2069 |.|o7 = i │ │ │ │ +00025b60: 6465 616c 2028 6120 202b 2074 2062 2a63 deal (a + t b*c │ │ │ │ +00025b70: 202b 2074 2061 2a64 202b 2074 2063 2020 + t a*d + t c │ │ │ │ +00025b80: 2b20 7420 622a 6420 2b20 7420 632a 6420 + t b*d + t c*d │ │ │ │ +00025b90: 2b20 7420 6420 2c20 612a 6220 2b20 7420 + t d , a*b + t │ │ │ │ +00025ba0: 622a 6320 2b20 207c 0a7c 2020 2020 2020 b*c + |.| │ │ │ │ +00025bb0: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ +00025bc0: 2020 2020 3320 2020 2020 2020 3220 2020 3 2 │ │ │ │ +00025bd0: 2020 2034 2020 2020 2020 2035 2020 2020 4 5 │ │ │ │ +00025be0: 2020 2036 2020 2020 2020 2020 2020 2037 6 7 │ │ │ │ +00025bf0: 2020 2020 2020 207c 0a7c 2020 2020 202d |.| - │ │ │ │ +00025c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025c40: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 -------|.| │ │ │ │ +00025c50: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00025c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025c70: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00025c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025c90: 2020 2032 2020 207c 0a7c 2020 2020 2074 2 |.| t │ │ │ │ +00025ca0: 2061 2a64 202b 2074 2063 2020 2b20 7420 a*d + t c + t │ │ │ │ +00025cb0: 2062 2a64 202b 2074 2020 632a 6420 2b20 b*d + t c*d + │ │ │ │ +00025cc0: 7420 2064 202c 2061 2a63 202b 2074 2020 t d , a*c + t │ │ │ │ +00025cd0: 622a 6320 2b20 7420 2061 2a64 202b 2074 b*c + t a*d + t │ │ │ │ +00025ce0: 2020 6320 202b 207c 0a7c 2020 2020 2020 c + |.| │ │ │ │ +00025cf0: 3920 2020 2020 2020 3820 2020 2020 2031 9 8 1 │ │ │ │ +00025d00: 3020 2020 2020 2020 3131 2020 2020 2020 0 11 │ │ │ │ +00025d10: 2031 3220 2020 2020 2020 2020 2020 3133 12 13 │ │ │ │ +00025d20: 2020 2020 2020 2031 3520 2020 2020 2020 15 │ │ │ │ +00025d30: 3134 2020 2020 207c 0a7c 2020 2020 202d 14 |.| - │ │ │ │ +00025d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025d80: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 -------|.| │ │ │ │ +00025d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025da0: 2020 2020 2032 2020 2032 2020 2020 2020 2 2 │ │ │ │ +00025db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025dc0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00025dd0: 2020 2020 2020 207c 0a7c 2020 2020 2074 |.| t │ │ │ │ +00025de0: 2020 622a 6420 2b20 7420 2063 2a64 202b b*d + t c*d + │ │ │ │ +00025df0: 2074 2020 6420 2c20 6220 202b 2074 2020 t d , b + t │ │ │ │ +00025e00: 622a 6320 2b20 7420 2061 2a64 202b 2074 b*c + t a*d + t │ │ │ │ +00025e10: 2020 6320 202b 2074 2020 622a 6420 2b20 c + t b*d + │ │ │ │ +00025e20: 7420 2063 2a64 207c 0a7c 2020 2020 2020 t c*d |.| │ │ │ │ +00025e30: 3136 2020 2020 2020 2031 3720 2020 2020 16 17 │ │ │ │ +00025e40: 2020 3138 2020 2020 2020 2020 2020 3139 18 19 │ │ │ │ +00025e50: 2020 2020 2020 2032 3120 2020 2020 2020 21 │ │ │ │ +00025e60: 3230 2020 2020 2020 3232 2020 2020 2020 20 22 │ │ │ │ +00025e70: 2032 3320 2020 207c 0a7c 2020 2020 202d 23 |.| - │ │ │ │ +00025e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025ec0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 -------|.| │ │ │ │ +00025ed0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ 00025ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025f10: 2020 2020 2020 207c 0a7c 6f38 203d 2069 |.|o8 = i │ │ │ │ -00025f20: 6465 616c 2028 2d20 7420 202b 2074 2020 deal (- t + t │ │ │ │ -00025f30: 202d 2074 2020 7420 202c 202d 2074 2020 - t t , - t │ │ │ │ -00025f40: 2d20 7420 2074 2020 2c20 2d20 7420 2020 - t t , - t │ │ │ │ -00025f50: 2b20 7420 2020 2b20 7420 7420 2020 2d20 + t + t t - │ │ │ │ -00025f60: 7420 2074 2020 207c 0a7c 2020 2020 2020 t t |.| │ │ │ │ -00025f70: 2020 2020 2020 2020 2037 2020 2020 3134 7 14 │ │ │ │ -00025f80: 2020 2020 3133 2031 3920 2020 2020 3820 13 19 8 │ │ │ │ -00025f90: 2020 2032 3020 3133 2020 2020 2031 3020 20 13 10 │ │ │ │ -00025fa0: 2020 2031 3720 2020 2039 2031 3320 2020 17 9 13 │ │ │ │ -00025fb0: 2032 3220 3133 207c 0a7c 2020 2020 202d 22 13 |.| - │ │ │ │ -00025fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026000: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 -------|.| │ │ │ │ -00026010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026020: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00026030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026050: 2020 2020 2020 207c 0a7c 2020 2020 202d |.| - │ │ │ │ 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---------------- │ │ │ │ +000286a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000286b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000286c0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 -------|.| │ │ │ │ +000286d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000286e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000286f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028700: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00028710: 2020 2020 2020 207c 0a7c 2020 2020 2074 |.| t │ │ │ │ +00028720: 2074 2020 7420 2020 2d20 7420 7420 2074 t t - t t t │ │ │ │ +00028730: 2020 202d 2074 2074 2020 7420 2074 2020 - t t t t │ │ │ │ +00028740: 2c20 2d20 7420 202b 2074 2074 2020 202d , - t + t t - │ │ │ │ +00028750: 2074 2020 202d 2074 2020 7420 202d 2074 t - t t - t │ │ │ │ +00028760: 2074 2020 2d20 207c 0a7c 2020 2020 2020 t - |.| │ │ │ │ +00028770: 3820 3134 2031 3520 2020 2031 2031 3420 8 14 15 1 14 │ │ │ │ +00028780: 3231 2020 2020 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+ t t + │ │ │ │ +00028820: 7420 7420 2074 2020 202b 2074 2020 7420 t t t + t t │ │ │ │ +00028830: 7420 2020 2b20 7420 7420 2074 2020 202d t + t t t - │ │ │ │ +00028840: 2074 2074 2020 7420 2020 2d20 2020 2020 t t t - │ │ │ │ 00028850: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00028860: 2020 3138 2031 2032 3120 2020 2031 3820 18 1 21 18 │ │ │ │ -00028870: 3720 3133 2032 3120 2020 2020 2020 2020 7 13 21 │ │ │ │ -00028880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000288a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000288b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000288c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000288d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000288e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000288f0: 2020 2020 2020 207c 0a7c 6f38 203a 2049 |.|o8 : I │ │ │ │ -00028900: 6465 616c 206f 6620 6b6b 5b74 202c 2074 deal of kk[t , t │ │ │ │ -00028910: 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2020 2020 202d 9 |.| - │ │ │ │ 000289a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000289b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000289c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000289d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000289e0: 2d2d 2d2d 2d2d 2d7c 0a7c 202c 2074 202c -------|.| , t , │ │ │ │ -000289f0: 2074 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -00028a00: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ -00028a10: 202c 2074 2020 2c20 7420 205d 2020 2020 , t , t ] │ │ │ │ -00028a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a30: 2020 2020 2020 207c 0a7c 3320 2020 3720 |.|3 7 │ │ │ │ -00028a40: 2020 3920 2020 3134 2020 2031 3620 2020 9 14 16 │ │ │ │ -00028a50: 3230 2020 2032 3220 2020 3133 2020 2031 20 22 13 1 │ │ │ │ -00028a60: 3520 2020 3139 2020 2032 3120 2020 2020 5 19 21 │ │ │ │ -00028a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000289e0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2074 -------|.| t │ │ │ │ +000289f0: 2020 7420 7420 2020 2b20 7420 2074 2074 t t + t t t │ │ │ │ +00028a00: 2020 202b 2074 2074 2020 7420 2020 2b20 + t t t + │ │ │ │ +00028a10: 7420 2074 2074 2020 202d 2074 2074 2020 t t t - t t │ │ │ │ +00028a20: 7420 2020 2d20 7420 2074 2074 2020 7420 t - t t t t │ │ │ │ +00028a30: 202c 202d 2020 207c 0a7c 2020 2020 2020 , - |.| │ │ │ │ +00028a40: 3233 2037 2031 3620 2020 2032 3420 3720 23 7 16 24 7 │ │ │ │ +00028a50: 3133 2020 2020 3820 3137 2031 3520 2020 13 8 17 15 │ │ │ │ +00028a60: 2031 3120 3720 3135 2020 2020 3120 3137 11 7 15 1 17 │ │ │ │ +00028a70: 2032 3120 2020 2031 3720 3720 3133 2032 21 17 7 13 2 │ │ │ │ +00028a80: 3120 2020 2020 207c 0a7c 2020 2020 202d 1 |.| - │ │ │ │ 00028a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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│ │ │ │ -00028b90: 7472 696d 3b20 2020 2020 2020 2020 2020 trim; │ │ │ │ -00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028bc0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00028bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028c10: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ -00028c20: 2363 6f6d 7073 4a20 3d3d 2032 2020 2020 #compsJ == 2 │ │ │ │ -00028c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028ad0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2074 -------|.| t │ │ │ │ +00028ae0: 2020 7420 2020 2b20 7420 7420 2020 2b20 t + t t + │ │ │ │ +00028af0: 7420 2074 2020 2d20 7420 2074 2020 202d t t - t t - │ │ │ │ +00028b00: 2074 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+00028be0: 2074 2020 7420 2029 2020 2020 2020 2020 t t ) │ │ │ │ +00028bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028c10: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00028c20: 2020 3138 2031 2032 3120 2020 2031 3820 18 1 21 18 │ │ │ │ +00028c30: 3720 3133 2032 3120 2020 2020 2020 2020 7 13 21 │ │ │ │ 00028c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00028c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028cb0: 2020 2020 2020 207c 0a7c 6f31 3120 3d20 |.|o11 = │ │ │ │ -00028cc0: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ -00028cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d00: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00028d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028d50: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 -------+.|i12 : │ │ │ │ -00028d60: 636f 6d70 734a 2f64 696d 2020 2020 2020 compsJ/dim │ │ │ │ -00028d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028da0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00028db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028cb0: 2020 2020 2020 207c 0a7c 6f38 203a 2049 |.|o8 : I │ │ │ │ +00028cc0: 6465 616c 206f 6620 6b6b 5b74 202c 2074 deal of kk[t , t │ │ │ │ +00028cd0: 202c 2074 2020 2c20 7420 2c20 7420 2c20 , t , t , t , │ │ │ │ +00028ce0: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ +00028cf0: 202c 2074 202c 2074 202c 2074 2020 2c20 , t , t , t , │ │ │ │ +00028d00: 7420 202c 2074 207c 0a7c 2020 2020 2020 t , t |.| │ │ │ │ +00028d10: 2020 2020 2020 2020 2020 2020 3620 2020 6 │ │ │ │ +00028d20: 3520 2020 3132 2020 2032 2020 2034 2020 5 12 2 4 │ │ │ │ +00028d30: 2031 3120 2020 3138 2020 2032 3420 2020 11 18 24 │ │ │ │ +00028d40: 3120 2020 3320 2020 3820 2020 3130 2020 1 3 8 10 │ │ │ │ +00028d50: 2031 3720 2020 327c 0a7c 2d2d 2d2d 2d2d 17 2|.|------ │ │ │ │ +00028d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028da0: 2d2d 2d2d 2d2d 2d7c 0a7c 202c 2074 202c -------|.| , t , │ │ │ │ +00028db0: 2074 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ +00028dc0: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ +00028dd0: 202c 2074 2020 2c20 7420 205d 2020 2020 , t , t ] │ │ │ │ 00028de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028df0: 2020 2020 2020 207c 0a7c 6f31 3220 3d20 |.|o12 = │ │ │ │ -00028e00: 7b31 312c 2038 7d20 2020 2020 2020 2020 {11, 8} │ │ │ │ -00028e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028df0: 2020 2020 2020 207c 0a7c 3320 2020 3720 |.|3 7 │ │ │ │ +00028e00: 2020 3920 2020 3134 2020 2031 3620 2020 9 14 16 │ │ │ │ +00028e10: 3230 2020 2032 3220 2020 3133 2020 2031 20 22 13 1 │ │ │ │ +00028e20: 3520 2020 3139 2020 2032 3120 2020 2020 5 19 21 │ │ │ │ 00028e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e40: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00028e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e90: 2020 2020 2020 207c 0a7c 6f31 3220 3a20 |.|o12 : │ │ │ │ -00028ea0: 4c69 7374 2020 2020 2020 2020 2020 2020 List │ │ │ │ -00028eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028e40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00028e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028e90: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2063 -------+.|i9 : c │ │ │ │ +00028ea0: 6f6d 7073 4a20 3d20 6465 636f 6d70 6f73 ompsJ = decompos │ │ │ │ +00028eb0: 6520 4a3b 2020 2020 2020 2020 2020 2020 e J; │ │ │ │ 00028ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ee0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00028ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028f30: 2d2d 2d2d 2d2d 2d2b 0a0a 5468 6572 6520 -------+..There │ │ │ │ -00028f40: 6172 6520 3220 636f 6d70 6f6e 656e 7473 are 2 components │ │ │ │ -00028f50: 2e20 2057 6520 6174 7465 6d70 7420 746f . We attempt to │ │ │ │ -00028f60: 2066 696e 6420 6120 706f 696e 7420 6f6e find a point on │ │ │ │ -00028f70: 2074 6865 2066 6972 7374 2063 6f6d 706f the first compo │ │ │ │ -00028f80: 6e65 6e74 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d nent..+--------- │ │ │ │ +00028f30: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 -------+.|i10 : │ │ │ │ +00028f40: 636f 6d70 734a 203d 2063 6f6d 7073 4a2f compsJ = compsJ/ │ │ │ │ +00028f50: 7472 696d 3b20 2020 2020 2020 2020 2020 trim; │ │ │ │ +00028f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028f80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00028f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028fd0: 2d2d 2d2d 2b0a 7c69 3133 203a 2070 7431 ----+.|i13 : pt1 │ │ │ │ -00028fe0: 203d 2072 616e 646f 6d50 6f69 6e74 4f6e = randomPointOn │ │ │ │ -00028ff0: 5261 7469 6f6e 616c 5661 7269 6574 7920 RationalVariety │ │ │ │ -00029000: 636f 6d70 734a 5f30 2020 2020 2020 2020 compsJ_0 │ │ │ │ +00028fd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ +00028fe0: 2363 6f6d 7073 4a20 3d3d 2032 2020 2020 #compsJ == 2 │ │ │ │ +00028ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029020: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00029020: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00029030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029070: 2020 2020 7c0a 7c6f 3133 203d 207c 202d |.|o13 = | - │ │ │ │ -00029080: 3232 2033 3020 2d33 3620 2d31 3720 2d33 22 30 -36 -17 -3 │ │ │ │ -00029090: 3620 3339 2032 3920 2d34 3920 2d34 3520 6 39 29 -49 -45 │ │ │ │ -000290a0: 3139 2031 3120 3231 202d 3239 202d 3820 19 11 21 -29 -8 │ │ │ │ -000290b0: 3137 202d 3338 202d 3239 2033 3220 2d32 17 -38 -29 32 -2 │ │ │ │ -000290c0: 3420 2020 7c0a 7c20 2020 2020 202d 2d2d 4 |.| --- │ │ │ │ +00029070: 2020 2020 2020 207c 0a7c 6f31 3120 3d20 |.|o11 = │ │ │ │ +00029080: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ +00029090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000290a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000290b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000290c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 000290d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000290e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000290f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029110: 2d2d 2d2d 7c0a 7c20 2020 2020 202d 3130 ----|.| -10 │ │ │ │ -00029120: 202d 3239 202d 3232 2031 3920 2d31 3620 -29 -22 19 -16 │ │ │ │ -00029130: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00029110: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 -------+.|i12 : │ │ │ │ +00029120: 636f 6d70 734a 2f64 696d 2020 2020 2020 compsJ/dim │ │ │ │ +00029130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029160: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00029160: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00029170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000291a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000291b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000291c0: 2020 2020 2020 3120 2020 2020 2020 3234 1 24 │ │ │ │ +000291b0: 2020 2020 2020 207c 0a7c 6f31 3220 3d20 |.|o12 = │ │ │ │ +000291c0: 7b31 312c 2038 7d20 2020 2020 2020 2020 {11, 8} │ │ │ │ 000291d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000291e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000291f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029200: 2020 2020 7c0a 7c6f 3133 203a 204d 6174 |.|o13 : Mat │ │ │ │ -00029210: 7269 7820 6b6b 2020 3c2d 2d20 6b6b 2020 rix kk <-- kk │ │ │ │ +00029200: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00029210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029250: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00029260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000292a0: 2d2d 2d2d 2b0a 7c69 3134 203a 2046 3120 ----+.|i14 : F1 │ │ │ │ -000292b0: 3d20 7375 6228 462c 2028 7661 7273 2053 = sub(F, (vars S │ │ │ │ -000292c0: 297c 7074 3129 2020 2020 2020 2020 2020 )|pt1) │ │ │ │ -000292d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000292e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000292f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00029300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029340: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00029350: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -00029360: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -00029370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029380: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00029390: 2020 2020 7c0a 7c6f 3134 203d 2069 6465 |.|o14 = ide │ │ │ │ -000293a0: 616c 2028 6120 202d 2034 3562 2a63 202d al (a - 45b*c - │ │ │ │ -000293b0: 2031 3763 2020 2b20 3139 612a 6420 2d20 17c + 19a*d - │ │ │ │ -000293c0: 3336 622a 6420 2b20 3330 632a 6420 2d20 36b*d + 30c*d - │ │ │ │ -000293d0: 3232 6420 2c20 612a 6220 2b20 3137 622a 22d , a*b + 17b* │ │ │ │ -000293e0: 6320 2b20 7c0a 7c20 2020 2020 202d 2d2d c + |.| --- │ │ │ │ -000293f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029430: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ -00029440: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00029450: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -00029460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029470: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00029480: 2020 2020 7c0a 7c20 2020 2020 2031 3163 |.| 11c │ │ │ │ -00029490: 2020 2d20 3338 612a 6420 2b20 3231 622a - 38a*d + 21b* │ │ │ │ -000294a0: 6420 2b20 3339 632a 6420 2d20 3336 6420 d + 39c*d - 36d │ │ │ │ -000294b0: 2c20 612a 6320 2d20 3239 622a 6320 2d20 , a*c - 29b*c - │ │ │ │ -000294c0: 3239 6320 202d 2032 3261 2a64 202b 2033 29c - 22a*d + 3 │ │ │ │ -000294d0: 3262 2a64 7c0a 7c20 2020 2020 202d 2d2d 2b*d|.| --- │ │ │ │ -000294e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000294f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029520: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ -00029530: 2020 2020 2020 2020 2020 3220 2020 3220 2 2 │ │ │ │ -00029540: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00029250: 2020 2020 2020 207c 0a7c 6f31 3220 3a20 |.|o12 : │ │ │ │ +00029260: 4c69 7374 2020 2020 2020 2020 2020 2020 List │ │ │ │ +00029270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000292a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000292b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000292c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000292d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000292e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000292f0: 2d2d 2d2d 2d2d 2d2b 0a0a 5468 6572 6520 -------+..There │ │ │ │ +00029300: 6172 6520 3220 636f 6d70 6f6e 656e 7473 are 2 components │ │ │ │ +00029310: 2e20 2057 6520 6174 7465 6d70 7420 746f . We attempt to │ │ │ │ +00029320: 2066 696e 6420 6120 706f 696e 7420 6f6e find a point on │ │ │ │ +00029330: 2074 6865 2066 6972 7374 2063 6f6d 706f the first compo │ │ │ │ +00029340: 6e65 6e74 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d nent..+--------- │ │ │ │ +00029350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029390: 2d2d 2d2d 2b0a 7c69 3133 203a 2070 7431 ----+.|i13 : pt1 │ │ │ │ +000293a0: 203d 2072 616e 646f 6d50 6f69 6e74 4f6e = randomPointOn │ │ │ │ +000293b0: 5261 7469 6f6e 616c 5661 7269 6574 7920 RationalVariety │ │ │ │ +000293c0: 636f 6d70 734a 5f30 2020 2020 2020 2020 compsJ_0 │ │ │ │ +000293d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000293e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000293f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029430: 2020 2020 7c0a 7c6f 3133 203d 207c 2032 |.|o13 = | 2 │ │ │ │ +00029440: 202d 3138 2036 2034 2032 3020 2d31 3120 -18 6 4 20 -11 │ │ │ │ +00029450: 3434 202d 3133 2030 2031 3920 3131 202d 44 -13 0 19 11 - │ │ │ │ +00029460: 3437 202d 3239 202d 3820 2d31 3920 2d32 47 -29 -8 -19 -2 │ │ │ │ +00029470: 3220 3139 202d 3230 202d 3239 202d 3338 2 19 -20 -29 -38 │ │ │ │ +00029480: 202d 3234 7c0a 7c20 2020 2020 202d 2d2d -24|.| --- │ │ │ │ +00029490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000294a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000294b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000294c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000294d0: 2d2d 2d2d 7c0a 7c20 2020 2020 202d 3136 ----|.| -16 │ │ │ │ +000294e0: 202d 3130 202d 3239 207c 2020 2020 2020 -10 -29 | │ │ │ │ +000294f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029520: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00029530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029560: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -00029570: 2020 2020 7c0a 7c20 2020 2020 202d 2032 |.| - 2 │ │ │ │ -00029580: 3963 2a64 202b 2032 3964 202c 2062 2020 9c*d + 29d , b │ │ │ │ -00029590: 2b20 3139 622a 6320 2d20 3234 6320 202d + 19b*c - 24c - │ │ │ │ -000295a0: 2031 3661 2a64 202d 2031 3062 2a64 202d 16a*d - 10b*d - │ │ │ │ -000295b0: 2038 632a 6420 2d20 3439 6420 2920 2020 8c*d - 49d ) │ │ │ │ -000295c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000295d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029570: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00029580: 2020 2020 2020 3120 2020 2020 2020 3234 1 24 │ │ │ │ +00029590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000295a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000295b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000295c0: 2020 2020 7c0a 7c6f 3133 203a 204d 6174 |.|o13 : Mat │ │ │ │ +000295d0: 7269 7820 6b6b 2020 3c2d 2d20 6b6b 2020 rix kk <-- kk │ │ │ │ 000295e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000295f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029610: 2020 2020 7c0a 7c6f 3134 203a 2049 6465 |.|o14 : Ide │ │ │ │ -00029620: 616c 206f 6620 5320 2020 2020 2020 2020 al of S │ │ │ │ -00029630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029660: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00029670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000296a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000296b0: 2d2d 2d2d 2b0a 7c69 3135 203a 2064 6563 ----+.|i15 : dec │ │ │ │ -000296c0: 6f6d 706f 7365 2046 3120 2020 2020 2020 ompose F1 │ │ │ │ +00029610: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00029620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029660: 2d2d 2d2d 2b0a 7c69 3134 203a 2046 3120 ----+.|i14 : F1 │ │ │ │ +00029670: 3d20 7375 6228 462c 2028 7661 7273 2053 = sub(F, (vars S │ │ │ │ +00029680: 297c 7074 3129 2020 2020 2020 2020 2020 )|pt1) │ │ │ │ +00029690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000296a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000296b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000296c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000296d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000296e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000296f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029700: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00029710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029710: 2020 2020 2032 2020 2020 2032 2020 2020 2 2 │ │ │ │ 00029720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029750: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00029760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029770: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -00029780: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -00029790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000297a0: 2032 2020 7c0a 7c6f 3135 203d 207b 6964 2 |.|o15 = {id │ │ │ │ -000297b0: 6561 6c20 2861 202d 2032 3962 202d 2032 eal (a - 29b - 2 │ │ │ │ -000297c0: 3963 202b 2034 3064 2c20 6220 202b 2031 9c + 40d, b + 1 │ │ │ │ -000297d0: 3962 2a63 202d 2032 3463 2020 2b20 3331 9b*c - 24c + 31 │ │ │ │ -000297e0: 622a 6420 2b20 3333 632a 6420 2d20 3135 b*d + 33c*d - 15 │ │ │ │ -000297f0: 6420 292c 7c0a 7c20 2020 2020 202d 2d2d d ),|.| --- │ │ │ │ -00029800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029840: 2d2d 2d2d 7c0a 7c20 2020 2020 2069 6465 ----|.| ide │ │ │ │ -00029850: 616c 2028 6320 2d20 3232 642c 2062 202b al (c - 22d, b + │ │ │ │ -00029860: 2032 3264 2c20 6120 2b20 3264 297d 2020 22d, a + 2d)} │ │ │ │ -00029870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029890: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000298a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000298b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000298c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000298d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000298e0: 2020 2020 7c0a 7c6f 3135 203a 204c 6973 |.|o15 : Lis │ │ │ │ -000298f0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -00029900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029730: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +00029740: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00029750: 2020 2020 7c0a 7c6f 3134 203d 2069 6465 |.|o14 = ide │ │ │ │ +00029760: 616c 2028 6120 202b 2034 6320 202b 2031 al (a + 4c + 1 │ │ │ │ +00029770: 3961 2a64 202b 2032 3062 2a64 202d 2031 9a*d + 20b*d - 1 │ │ │ │ +00029780: 3863 2a64 202b 2032 6420 2c20 612a 6220 8c*d + 2d , a*b │ │ │ │ +00029790: 2d20 3139 622a 6320 2b20 3131 6320 202d - 19b*c + 11c - │ │ │ │ +000297a0: 2020 2020 7c0a 7c20 2020 2020 202d 2d2d |.| --- │ │ │ │ +000297b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000297c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000297d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000297e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000297f0: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ +00029800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029810: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +00029820: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +00029830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029840: 2020 2020 7c0a 7c20 2020 2020 2032 3261 |.| 22a │ │ │ │ +00029850: 2a64 202d 2034 3762 2a64 202d 2031 3163 *d - 47b*d - 11c │ │ │ │ +00029860: 2a64 202b 2036 6420 2c20 612a 6320 2d20 *d + 6d , a*c - │ │ │ │ +00029870: 3234 622a 6320 2b20 3139 6320 202d 2031 24b*c + 19c - 1 │ │ │ │ +00029880: 3661 2a64 202d 2032 3062 2a64 202d 2032 6a*d - 20b*d - 2 │ │ │ │ +00029890: 3963 2a64 7c0a 7c20 2020 2020 202d 2d2d 9c*d|.| --- │ │ │ │ +000298a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000298b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000298c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000298d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000298e0: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ +000298f0: 2020 3220 2020 3220 2020 2020 2020 2020 2 2 │ │ │ │ +00029900: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ 00029910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029930: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00029940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029980: 2d2d 2d2d 2b0a 0a57 6520 6174 7465 6d70 ----+..We attemp │ │ │ │ -00029990: 7420 746f 2066 696e 6420 6120 706f 696e t to find a poin │ │ │ │ -000299a0: 7420 6f6e 2074 6865 2073 6563 6f6e 6420 t on the second │ │ │ │ -000299b0: 636f 6d70 6f6e 656e 7420 696e 2070 6172 component in par │ │ │ │ -000299c0: 616d 6574 6572 2073 7061 6365 2c20 616e ameter space, an │ │ │ │ -000299d0: 6420 6974 730a 636f 7272 6573 706f 6e64 d its.correspond │ │ │ │ -000299e0: 696e 6720 6964 6561 6c2e 0a0a 2b2d 2d2d ing ideal...+--- │ │ │ │ -000299f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136 ----------+.|i16 │ │ │ │ -00029a40: 203a 2070 7432 203d 2072 616e 646f 6d50 : pt2 = randomP │ │ │ │ -00029a50: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ -00029a60: 7269 6574 7920 636f 6d70 734a 5f31 2020 riety compsJ_1 │ │ │ │ -00029a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029a80: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00029920: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00029930: 2020 2020 7c0a 7c20 2020 2020 202b 2034 |.| + 4 │ │ │ │ +00029940: 3464 202c 2062 2020 2d20 3130 622a 6320 4d , b - 10b*c │ │ │ │ +00029950: 2d20 3239 6320 202d 2032 3961 2a64 202d - 29c - 29a*d - │ │ │ │ +00029960: 2033 3862 2a64 202d 2038 632a 6420 2d20 38b*d - 8c*d - │ │ │ │ +00029970: 3133 6420 2920 2020 2020 2020 2020 2020 13d ) │ │ │ │ +00029980: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00029990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000299a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000299b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000299c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000299d0: 2020 2020 7c0a 7c6f 3134 203a 2049 6465 |.|o14 : Ide │ │ │ │ +000299e0: 616c 206f 6620 5320 2020 2020 2020 2020 al of S │ │ │ │ +000299f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029a20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00029a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029a70: 2d2d 2d2d 2b0a 7c69 3135 203a 2064 6563 ----+.|i15 : dec │ │ │ │ +00029a80: 6f6d 706f 7365 2046 3120 2020 2020 2020 ompose F1 │ │ │ │ 00029a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ad0: 2020 2020 2020 2020 2020 7c0a 7c6f 3136 |.|o16 │ │ │ │ -00029ae0: 203d 207c 202d 3134 2034 3020 2d35 2032 = | -14 40 -5 2 │ │ │ │ -00029af0: 3620 2d34 3820 2d32 3620 2d33 3520 3431 6 -48 -26 -35 41 │ │ │ │ -00029b00: 202d 3820 2d31 3520 2d33 3820 3331 202d -8 -15 -38 31 - │ │ │ │ -00029b10: 3133 2032 3920 3231 2031 3620 3339 2032 13 29 21 16 39 2 │ │ │ │ -00029b20: 3120 2d31 3820 3139 2020 7c0a 7c20 2020 1 -18 19 |.| │ │ │ │ -00029b30: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------- │ │ │ │ -00029b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b70: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ -00029b80: 2020 202d 3437 202d 3339 2033 3420 3020 -47 -39 34 0 │ │ │ │ -00029b90: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00029ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029bc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00029bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00029c20: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ -00029c30: 2020 2020 3234 2020 2020 2020 2020 2020 24 │ │ │ │ +00029ac0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00029ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029b10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00029b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029b30: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +00029b40: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00029b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029b60: 2032 2020 7c0a 7c6f 3135 203d 207b 6964 2 |.|o15 = {id │ │ │ │ +00029b70: 6561 6c20 2861 202d 2032 3462 202b 2031 eal (a - 24b + 1 │ │ │ │ +00029b80: 3963 202d 2032 3864 2c20 6220 202d 2031 9c - 28d, b - 1 │ │ │ │ +00029b90: 3062 2a63 202d 2032 3963 2020 2d20 3237 0b*c - 29c - 27 │ │ │ │ +00029ba0: 622a 6420 2b20 3338 632a 6420 2d20 3137 b*d + 38c*d - 17 │ │ │ │ +00029bb0: 6420 292c 7c0a 7c20 2020 2020 202d 2d2d d ),|.| --- │ │ │ │ +00029bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029c00: 2d2d 2d2d 7c0a 7c20 2020 2020 2069 6465 ----|.| ide │ │ │ │ +00029c10: 616c 2028 6320 2d20 3136 642c 2062 202d al (c - 16d, b - │ │ │ │ +00029c20: 2033 3364 2c20 6120 2d20 3332 6429 7d20 33d, a - 32d)} │ │ │ │ +00029c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c60: 2020 2020 2020 2020 2020 7c0a 7c6f 3136 |.|o16 │ │ │ │ -00029c70: 203a 204d 6174 7269 7820 6b6b 2020 3c2d : Matrix kk <- │ │ │ │ -00029c80: 2d20 6b6b 2020 2020 2020 2020 2020 2020 - kk │ │ │ │ +00029c50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00029c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029cb0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00029cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137 ----------+.|i17 │ │ │ │ -00029d10: 203a 2046 3220 3d20 7375 6228 462c 2028 : F2 = sub(F, ( │ │ │ │ -00029d20: 7661 7273 2053 297c 7074 3229 2020 2020 vars S)|pt2) │ │ │ │ -00029d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029d50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00029d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029da0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00029db0: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -00029dc0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -00029dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029de0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -00029df0: 2020 2020 2020 2020 2020 7c0a 7c6f 3137 |.|o17 │ │ │ │ -00029e00: 203d 2069 6465 616c 2028 6120 202d 2038 = ideal (a - 8 │ │ │ │ -00029e10: 622a 6320 2b20 3236 6320 202d 2031 3561 b*c + 26c - 15a │ │ │ │ -00029e20: 2a64 202d 2034 3862 2a64 202b 2034 3063 *d - 48b*d + 40c │ │ │ │ -00029e30: 2a64 202d 2031 3464 202c 2061 2a62 202b *d - 14d , a*b + │ │ │ │ -00029e40: 2032 3162 2a63 202d 2020 7c0a 7c20 2020 21b*c - |.| │ │ │ │ -00029e50: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------- │ │ │ │ -00029e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029e90: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ -00029ea0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00029eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ec0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -00029ed0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -00029ee0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00029ef0: 2020 2033 3863 2020 2b20 3136 612a 6420 38c + 16a*d │ │ │ │ -00029f00: 2b20 3331 622a 6420 2d20 3236 632a 6420 + 31b*d - 26c*d │ │ │ │ -00029f10: 2d20 3564 202c 2061 2a63 202d 2034 3762 - 5d , a*c - 47b │ │ │ │ -00029f20: 2a63 202b 2033 3963 2020 2d20 3339 612a *c + 39c - 39a* │ │ │ │ -00029f30: 6420 2b20 3231 622a 6420 7c0a 7c20 2020 d + 21b*d |.| │ │ │ │ -00029f40: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------- │ │ │ │ -00029f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029f80: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ +00029ca0: 2020 2020 7c0a 7c6f 3135 203a 204c 6973 |.|o15 : Lis │ │ │ │ +00029cb0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00029cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029cf0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00029d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029d40: 2d2d 2d2d 2b0a 0a57 6520 6174 7465 6d70 ----+..We attemp │ │ │ │ +00029d50: 7420 746f 2066 696e 6420 6120 706f 696e t to find a poin │ │ │ │ +00029d60: 7420 6f6e 2074 6865 2073 6563 6f6e 6420 t on the second │ │ │ │ +00029d70: 636f 6d70 6f6e 656e 7420 696e 2070 6172 component in par │ │ │ │ +00029d80: 616d 6574 6572 2073 7061 6365 2c20 616e ameter space, an │ │ │ │ +00029d90: 6420 6974 730a 636f 7272 6573 706f 6e64 d its.correspond │ │ │ │ +00029da0: 696e 6720 6964 6561 6c2e 0a0a 2b2d 2d2d ing ideal...+--- │ │ │ │ +00029db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136 ----------+.|i16 │ │ │ │ +00029e00: 203a 2070 7432 203d 2072 616e 646f 6d50 : pt2 = randomP │ │ │ │ +00029e10: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ +00029e20: 7269 6574 7920 636f 6d70 734a 5f31 2020 riety compsJ_1 │ │ │ │ +00029e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00029e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e90: 2020 2020 2020 2020 2020 7c0a 7c6f 3136 |.|o16 │ │ │ │ +00029ea0: 203d 207c 202d 3820 3238 2032 3520 3330 = | -8 28 25 30 │ │ │ │ +00029eb0: 202d 3132 202d 3432 202d 3230 202d 3335 -12 -42 -20 -35 │ │ │ │ +00029ec0: 202d 3720 2d33 3720 2d35 2034 3520 3139 -7 -37 -5 45 19 │ │ │ │ +00029ed0: 2032 3020 2d32 3420 3334 2033 3920 3231 20 -24 34 39 21 │ │ │ │ +00029ee0: 202d 3437 202d 3339 2020 7c0a 7c20 2020 -47 -39 |.| │ │ │ │ +00029ef0: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------- │ │ │ │ +00029f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029f30: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ +00029f40: 2020 202d 3133 202d 3138 2033 3420 3020 -13 -18 34 0 │ │ │ │ +00029f50: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00029f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f80: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00029f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029fa0: 3220 2020 3220 2020 2020 2020 2020 2020 2 2 │ │ │ │ -00029fb0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00029fc0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00029fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029fd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00029fe0: 2020 202d 2031 3363 2a64 202d 2033 3564 - 13c*d - 35d │ │ │ │ -00029ff0: 202c 2062 2020 2b20 3334 622a 6320 2d20 , b + 34b*c - │ │ │ │ -0002a000: 3138 6320 202b 2031 3962 2a64 202b 2032 18c + 19b*d + 2 │ │ │ │ -0002a010: 3963 2a64 202b 2034 3164 2029 2020 2020 9c*d + 41d ) │ │ │ │ -0002a020: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0002a030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029fe0: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ +00029ff0: 2020 2020 3234 2020 2020 2020 2020 2020 24 │ │ │ │ +0002a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a020: 2020 2020 2020 2020 2020 7c0a 7c6f 3136 |.|o16 │ │ │ │ +0002a030: 203a 204d 6174 7269 7820 6b6b 2020 3c2d : Matrix kk <- │ │ │ │ +0002a040: 2d20 6b6b 2020 2020 2020 2020 2020 2020 - kk │ │ │ │ 0002a050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a070: 2020 2020 2020 2020 2020 7c0a 7c6f 3137 |.|o17 │ │ │ │ -0002a080: 203a 2049 6465 616c 206f 6620 5320 2020 : Ideal of S │ │ │ │ -0002a090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a0c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -0002a0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a110: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 ----------+.|i18 │ │ │ │ -0002a120: 203a 2064 6563 6f6d 706f 7365 2046 3220 : decompose F2 │ │ │ │ +0002a070: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002a080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137 ----------+.|i17 │ │ │ │ +0002a0d0: 203a 2046 3220 3d20 7375 6228 462c 2028 : F2 = sub(F, ( │ │ │ │ +0002a0e0: 7661 7273 2053 297c 7074 3229 2020 2020 vars S)|pt2) │ │ │ │ +0002a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a110: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002a120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a160: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0002a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a170: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +0002a180: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ 0002a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a1b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3138 |.|o18 │ │ │ │ -0002a1c0: 203d 207b 6964 6561 6c20 2862 202b 2031 = {ideal (b + 1 │ │ │ │ -0002a1d0: 3963 202d 2031 3864 2c20 6120 2b20 3233 9c - 18d, a + 23 │ │ │ │ -0002a1e0: 6320 2b20 3433 6429 2c20 6964 6561 6c20 c + 43d), ideal │ │ │ │ -0002a1f0: 2862 202b 2031 3563 202b 2033 3764 2c20 (b + 15c + 37d, │ │ │ │ -0002a200: 6120 2b20 3337 6320 2b20 7c0a 7c20 2020 a + 37c + |.| │ │ │ │ +0002a1a0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0002a1b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3137 |.|o17 │ │ │ │ +0002a1c0: 203d 2069 6465 616c 2028 6120 202d 2037 = ideal (a - 7 │ │ │ │ +0002a1d0: 622a 6320 2b20 3330 6320 202d 2033 3761 b*c + 30c - 37a │ │ │ │ +0002a1e0: 2a64 202d 2031 3262 2a64 202b 2032 3863 *d - 12b*d + 28c │ │ │ │ +0002a1f0: 2a64 202d 2038 6420 2c20 612a 6220 2d20 *d - 8d , a*b - │ │ │ │ +0002a200: 3234 622a 6320 2d20 2020 7c0a 7c20 2020 24b*c - |.| │ │ │ │ 0002a210: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------- │ │ │ │ 0002a220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a250: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ -0002a260: 2020 2032 3664 297d 2020 2020 2020 2020 26d)} │ │ │ │ +0002a260: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ 0002a270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a280: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a290: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ 0002a2a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0002a2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a2f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3138 |.|o18 │ │ │ │ -0002a300: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ -0002a310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a340: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -0002a350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a390: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 7420 ----------+..It │ │ │ │ -0002a3a0: 7475 726e 7320 6f75 7420 7468 6174 2074 turns out that t │ │ │ │ -0002a3b0: 6869 7320 6973 2074 6865 2069 6465 616c his is the ideal │ │ │ │ -0002a3c0: 206f 6620 3220 736b 6577 206c 696e 6573 of 2 skew lines │ │ │ │ -0002a3d0: 2c20 6a75 7374 206e 6f74 2064 6566 696e , just not defin │ │ │ │ -0002a3e0: 6564 206f 7665 7220 7468 6973 0a66 6965 ed over this.fie │ │ │ │ -0002a3f0: 6c64 2e0a 0a43 6176 6561 740a 3d3d 3d3d ld...Caveat.==== │ │ │ │ -0002a400: 3d3d 0a0a 5468 6973 2072 6f75 7469 6e65 ==..This routine │ │ │ │ -0002a410: 2065 7870 6563 7473 2074 6865 2069 6e70 expects the inp │ │ │ │ -0002a420: 7574 2074 6f20 7265 7072 6573 656e 7420 ut to represent │ │ │ │ -0002a430: 616e 2069 7272 6564 7563 6962 6c65 2076 an irreducible v │ │ │ │ -0002a440: 6172 6965 7479 0a0a 5365 6520 616c 736f ariety..See also │ │ │ │ -0002a450: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ -0002a460: 6e6f 7465 2072 616e 646f 6d50 6f69 6e74 note randomPoint │ │ │ │ -0002a470: 734f 6e52 6174 696f 6e61 6c56 6172 6965 sOnRationalVarie │ │ │ │ -0002a480: 7479 3a0a 2020 2020 7261 6e64 6f6d 506f ty:. randomPo │ │ │ │ -0002a490: 696e 7473 4f6e 5261 7469 6f6e 616c 5661 intsOnRationalVa │ │ │ │ -0002a4a0: 7269 6574 795f 6c70 4964 6561 6c5f 636d riety_lpIdeal_cm │ │ │ │ -0002a4b0: 5a5a 5f72 702c 202d 2d20 6669 6e64 2072 ZZ_rp, -- find r │ │ │ │ -0002a4c0: 616e 646f 6d20 706f 696e 7473 206f 6e20 andom points on │ │ │ │ -0002a4d0: 610a 2020 2020 7661 7269 6574 7920 7468 a. variety th │ │ │ │ -0002a4e0: 6174 2063 616e 2062 6520 6465 7465 6374 at can be detect │ │ │ │ -0002a4f0: 6564 2074 6f20 6265 2072 6174 696f 6e61 ed to be rationa │ │ │ │ -0002a500: 6c0a 0a57 6179 7320 746f 2075 7365 2074 l..Ways to use t │ │ │ │ -0002a510: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d his method:.==== │ │ │ │ -0002a520: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002a530: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -0002a540: 7261 6e64 6f6d 506f 696e 744f 6e52 6174 randomPointOnRat │ │ │ │ -0002a550: 696f 6e61 6c56 6172 6965 7479 2849 6465 ionalVariety(Ide │ │ │ │ -0002a560: 616c 293a 0a20 2020 2072 616e 646f 6d50 al):. randomP │ │ │ │ -0002a570: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ -0002a580: 7269 6574 795f 6c70 4964 6561 6c5f 7270 riety_lpIdeal_rp │ │ │ │ -0002a590: 2c20 2d2d 2066 696e 6420 6120 7261 6e64 , -- find a rand │ │ │ │ -0002a5a0: 6f6d 2070 6f69 6e74 206f 6e20 610a 2020 om point on a. │ │ │ │ -0002a5b0: 2020 7661 7269 6574 7920 7468 6174 2063 variety that c │ │ │ │ -0002a5c0: 616e 2062 6520 6465 7465 6374 6564 2074 an be detected t │ │ │ │ -0002a5d0: 6f20 6265 2072 6174 696f 6e61 6c0a 2d2d o be rational.-- │ │ │ │ +0002a2b0: 2020 2035 6320 202b 2033 3461 2a64 202b 5c + 34a*d + │ │ │ │ +0002a2c0: 2034 3562 2a64 202d 2034 3263 2a64 202b 45b*d - 42c*d + │ │ │ │ +0002a2d0: 2032 3564 202c 2061 2a63 202d 2031 3362 25d , a*c - 13b │ │ │ │ +0002a2e0: 2a63 202b 2033 3963 2020 2d20 3138 612a *c + 39c - 18a* │ │ │ │ +0002a2f0: 6420 2b20 3231 622a 6420 7c0a 7c20 2020 d + 21b*d |.| │ │ │ │ +0002a300: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------- │ │ │ │ +0002a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a340: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ +0002a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a360: 3220 2020 3220 2020 2020 2020 2020 2020 2 2 │ │ │ │ +0002a370: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a380: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0002a390: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002a3a0: 2020 202b 2031 3963 2a64 202d 2032 3064 + 19c*d - 20d │ │ │ │ +0002a3b0: 202c 2062 2020 2b20 3334 622a 6320 2d20 , b + 34b*c - │ │ │ │ +0002a3c0: 3437 6320 202d 2033 3962 2a64 202b 2032 47c - 39b*d + 2 │ │ │ │ +0002a3d0: 3063 2a64 202d 2033 3564 2029 2020 2020 0c*d - 35d ) │ │ │ │ +0002a3e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002a3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a430: 2020 2020 2020 2020 2020 7c0a 7c6f 3137 |.|o17 │ │ │ │ +0002a440: 203a 2049 6465 616c 206f 6620 5320 2020 : Ideal of S │ │ │ │ +0002a450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a480: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002a490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 ----------+.|i18 │ │ │ │ +0002a4e0: 203a 2064 6563 6f6d 706f 7365 2046 3220 : decompose F2 │ │ │ │ +0002a4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a520: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002a530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a570: 2020 2020 2020 2020 2020 7c0a 7c6f 3138 |.|o18 │ │ │ │ +0002a580: 203d 207b 6964 6561 6c20 2862 202d 2031 = {ideal (b - 1 │ │ │ │ +0002a590: 3263 202b 2034 3864 2c20 6120 2d20 3136 2c + 48d, a - 16 │ │ │ │ +0002a5a0: 6320 2b20 6429 2c20 6964 6561 6c20 2862 c + d), ideal (b │ │ │ │ +0002a5b0: 202b 2034 3663 202b 2031 3464 2c20 6120 + 46c + 14d, a │ │ │ │ +0002a5c0: 2b20 3331 6320 2d20 2020 7c0a 7c20 2020 + 31c - |.| │ │ │ │ +0002a5d0: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------- │ │ │ │ 0002a5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -0002a630: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -0002a640: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -0002a650: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -0002a660: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -0002a670: 6179 322d 312e 3236 2e30 352b 6473 2f4d ay2-1.26.05+ds/M │ │ │ │ -0002a680: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -0002a690: 6167 6573 2f0a 4772 6f65 626e 6572 5374 ages/.GroebnerSt │ │ │ │ -0002a6a0: 7261 7461 2e6d 323a 3934 303a 302e 0a1f rata.m2:940:0... │ │ │ │ -0002a6b0: 0a46 696c 653a 2047 726f 6562 6e65 7253 .File: GroebnerS │ │ │ │ -0002a6c0: 7472 6174 612e 696e 666f 2c20 4e6f 6465 trata.info, Node │ │ │ │ -0002a6d0: 3a20 7261 6e64 6f6d 506f 696e 7473 4f6e : randomPointsOn │ │ │ │ -0002a6e0: 5261 7469 6f6e 616c 5661 7269 6574 795f RationalVariety_ │ │ │ │ -0002a6f0: 6c70 4964 6561 6c5f 636d 5a5a 5f72 702c lpIdeal_cmZZ_rp, │ │ │ │ -0002a700: 204e 6578 743a 2073 6d61 6c6c 6572 4d6f Next: smallerMo │ │ │ │ -0002a710: 6e6f 6d69 616c 732c 2050 7265 763a 2072 nomials, Prev: r │ │ │ │ -0002a720: 616e 646f 6d50 6f69 6e74 4f6e 5261 7469 andomPointOnRati │ │ │ │ -0002a730: 6f6e 616c 5661 7269 6574 795f 6c70 4964 onalVariety_lpId │ │ │ │ -0002a740: 6561 6c5f 7270 2c20 5570 3a20 546f 700a eal_rp, Up: Top. │ │ │ │ -0002a750: 0a72 616e 646f 6d50 6f69 6e74 734f 6e52 .randomPointsOnR │ │ │ │ -0002a760: 6174 696f 6e61 6c56 6172 6965 7479 2849 ationalVariety(I │ │ │ │ -0002a770: 6465 616c 2c5a 5a29 202d 2d20 6669 6e64 deal,ZZ) -- find │ │ │ │ -0002a780: 2072 616e 646f 6d20 706f 696e 7473 206f random points o │ │ │ │ -0002a790: 6e20 6120 7661 7269 6574 7920 7468 6174 n a variety that │ │ │ │ -0002a7a0: 2063 616e 2062 6520 6465 7465 6374 6564 can be detected │ │ │ │ -0002a7b0: 2074 6f20 6265 2072 6174 696f 6e61 6c0a to be rational. │ │ │ │ -0002a7c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002a7d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002a7e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002a7f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002a800: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002a810: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002a820: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -0002a830: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e * Function: *n │ │ │ │ -0002a840: 6f74 6520 7261 6e64 6f6d 506f 696e 7473 ote randomPoints │ │ │ │ -0002a850: 4f6e 5261 7469 6f6e 616c 5661 7269 6574 OnRationalVariet │ │ │ │ -0002a860: 793a 0a20 2020 2072 616e 646f 6d50 6f69 y:. randomPoi │ │ │ │ -0002a870: 6e74 734f 6e52 6174 696f 6e61 6c56 6172 ntsOnRationalVar │ │ │ │ -0002a880: 6965 7479 5f6c 7049 6465 616c 5f63 6d5a iety_lpIdeal_cmZ │ │ │ │ -0002a890: 5a5f 7270 2c0a 2020 2a20 5573 6167 653a Z_rp,. * Usage: │ │ │ │ -0002a8a0: 200a 2020 2020 2020 2020 7261 6e64 6f6d . random │ │ │ │ -0002a8b0: 506f 696e 7473 4f6e 5261 7469 6f6e 616c PointsOnRational │ │ │ │ -0002a8c0: 5661 7269 6574 7928 492c 206e 290a 2020 Variety(I, n). │ │ │ │ -0002a8d0: 2020 2020 2020 7261 6e64 6f6d 506f 696e randomPoin │ │ │ │ -0002a8e0: 744f 6e52 6174 696f 6e61 6c56 6172 6965 tOnRationalVarie │ │ │ │ -0002a8f0: 7479 0a20 202a 2049 6e70 7574 733a 0a20 ty. * Inputs:. │ │ │ │ -0002a900: 2020 2020 202a 2049 2c20 616e 202a 6e6f * I, an *no │ │ │ │ -0002a910: 7465 2069 6465 616c 3a20 284d 6163 6175 te ideal: (Macau │ │ │ │ -0002a920: 6c61 7932 446f 6329 4964 6561 6c2c 2c20 lay2Doc)Ideal,, │ │ │ │ -0002a930: 416e 2069 6465 616c 2069 6e20 6120 706f An ideal in a po │ │ │ │ -0002a940: 6c79 6e6f 6d69 616c 2072 696e 670a 2020 lynomial ring. │ │ │ │ -0002a950: 2020 2020 2020 2453 2420 6f76 6572 2061 $S$ over a │ │ │ │ -0002a960: 2066 6965 6c64 2c20 7768 6963 6820 6465 field, which de │ │ │ │ -0002a970: 6669 6e65 7320 6120 7072 696d 6520 6964 fines a prime id │ │ │ │ -0002a980: 6561 6c0a 2020 2020 2020 2a20 6e2c 2061 eal. * n, a │ │ │ │ -0002a990: 6e20 2a6e 6f74 6520 696e 7465 6765 723a n *note integer: │ │ │ │ -0002a9a0: 2028 4d61 6361 756c 6179 3244 6f63 295a (Macaulay2Doc)Z │ │ │ │ -0002a9b0: 5a2c 2c20 5468 6520 6e75 6d62 6572 206f Z,, The number o │ │ │ │ -0002a9c0: 6620 706f 696e 7473 2074 6f0a 2020 2020 f points to. │ │ │ │ -0002a9d0: 2020 2020 6765 6e65 7261 7465 0a20 202a generate. * │ │ │ │ -0002a9e0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -0002a9f0: 2a20 6120 2a6e 6f74 6520 6c69 7374 3a20 * a *note list: │ │ │ │ -0002aa00: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ -0002aa10: 7374 2c2c 2041 206c 6973 7420 6f66 2024 st,, A list of $ │ │ │ │ -0002aa20: 6e24 206f 6e65 2072 6f77 206d 6174 7269 n$ one row matri │ │ │ │ -0002aa30: 6365 7320 6f76 6572 0a20 2020 2020 2020 ces over. │ │ │ │ -0002aa40: 2074 6865 2062 6173 6520 6669 656c 6420 the base field │ │ │ │ -0002aa50: 6f66 2024 5324 2c20 7468 6174 2061 7265 of $S$, that are │ │ │ │ -0002aa60: 2072 616e 646f 6d6c 7920 6368 6f73 656e randomly chosen │ │ │ │ -0002aa70: 2070 6f69 6e74 7320 6f6e 2024 4924 2e20 points on $I$. │ │ │ │ -0002aa80: 206e 756c 6c20 6973 0a20 2020 2020 2020 null is. │ │ │ │ -0002aa90: 2072 6574 7572 6e65 6420 696e 2074 6865 returned in the │ │ │ │ -0002aaa0: 2063 6173 6520 7768 656e 2074 6865 2072 case when the r │ │ │ │ -0002aab0: 6f75 7469 6e65 2063 616e 6e6f 7420 6465 outine cannot de │ │ │ │ -0002aac0: 7465 726d 696e 6520 6966 2074 6865 2076 termine if the v │ │ │ │ -0002aad0: 6172 6965 7479 0a20 2020 2020 2020 2069 ariety. i │ │ │ │ -0002aae0: 7320 7261 7469 6f6e 616c 2061 6e64 2069 s rational and i │ │ │ │ -0002aaf0: 7272 6564 7563 6962 6c65 2e0a 0a44 6573 rreducible...Des │ │ │ │ -0002ab00: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -0002ab10: 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ====..+--------- │ │ │ │ -0002ab20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ab30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ab40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ab60: 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b 203d ----+.|i1 : kk = │ │ │ │ -0002ab70: 205a 5a2f 3130 313b 2020 2020 2020 2020 ZZ/101; │ │ │ │ -0002ab80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ab90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002abb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002abd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002abe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002abf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ac00: 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 3d20 ----+.|i2 : S = │ │ │ │ -0002ac10: 6b6b 5b61 2e2e 665d 3b20 2020 2020 2020 kk[a..f]; │ │ │ │ -0002ac20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ac30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ac40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ac50: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002ac60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ac70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ac80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ac90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002aca0: 2d2d 2d2d 2b0a 7c69 3320 3a20 4920 3d20 ----+.|i3 : I = │ │ │ │ -0002acb0: 6d69 6e6f 7273 2832 2c20 6765 6e65 7269 minors(2, generi │ │ │ │ -0002acc0: 6353 796d 6d65 7472 6963 4d61 7472 6978 cSymmetricMatrix │ │ │ │ -0002acd0: 2853 2c20 3329 2920 2020 2020 2020 2020 (S, 3)) │ │ │ │ -0002ace0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002acf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002ad00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ad10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ad20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ad30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ad40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002ad50: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -0002ad60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ad70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ad80: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -0002ad90: 2020 2020 7c0a 7c6f 3320 3d20 6964 6561 |.|o3 = idea │ │ │ │ -0002ada0: 6c20 282d 2062 2020 2b20 612a 642c 202d l (- b + a*d, - │ │ │ │ -0002adb0: 2062 2a63 202b 2061 2a65 2c20 2d20 632a b*c + a*e, - c* │ │ │ │ -0002adc0: 6420 2b20 622a 652c 202d 2062 2a63 202b d + b*e, - b*c + │ │ │ │ -0002add0: 2061 2a65 2c20 2d20 6320 202b 2061 2a66 a*e, - c + a*f │ │ │ │ -0002ade0: 2c20 2d20 7c0a 7c20 2020 2020 2d2d 2d2d , - |.| ---- │ │ │ │ -0002adf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ae00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ae10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ae20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ae30: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ -0002ae40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ae50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ae60: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -0002ae70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ae80: 2020 2020 7c0a 7c20 2020 2020 632a 6520 |.| c*e │ │ │ │ -0002ae90: 2b20 622a 662c 202d 2063 2a64 202b 2062 + b*f, - c*d + b │ │ │ │ -0002aea0: 2a65 2c20 2d20 632a 6520 2b20 622a 662c *e, - c*e + b*f, │ │ │ │ -0002aeb0: 202d 2065 2020 2b20 642a 6629 2020 2020 - e + d*f) │ │ │ │ -0002aec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aed0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002aee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002af00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002af10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002af20: 2020 2020 7c0a 7c6f 3320 3a20 4964 6561 |.|o3 : Idea │ │ │ │ -0002af30: 6c20 6f66 2053 2020 2020 2020 2020 2020 l of S │ │ │ │ +0002a610: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ +0002a620: 2020 2035 6429 7d20 2020 2020 2020 2020 5d)} │ │ │ │ +0002a630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a660: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002a670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a6b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3138 |.|o18 │ │ │ │ +0002a6c0: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +0002a6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a700: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002a710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a750: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 7420 ----------+..It │ │ │ │ +0002a760: 7475 726e 7320 6f75 7420 7468 6174 2074 turns out that t │ │ │ │ +0002a770: 6869 7320 6973 2074 6865 2069 6465 616c his is the ideal │ │ │ │ +0002a780: 206f 6620 3220 736b 6577 206c 696e 6573 of 2 skew lines │ │ │ │ +0002a790: 2c20 6a75 7374 206e 6f74 2064 6566 696e , just not defin │ │ │ │ +0002a7a0: 6564 206f 7665 7220 7468 6973 0a66 6965 ed over this.fie │ │ │ │ +0002a7b0: 6c64 2e0a 0a43 6176 6561 740a 3d3d 3d3d ld...Caveat.==== │ │ │ │ +0002a7c0: 3d3d 0a0a 5468 6973 2072 6f75 7469 6e65 ==..This routine │ │ │ │ +0002a7d0: 2065 7870 6563 7473 2074 6865 2069 6e70 expects the inp │ │ │ │ +0002a7e0: 7574 2074 6f20 7265 7072 6573 656e 7420 ut to represent │ │ │ │ +0002a7f0: 616e 2069 7272 6564 7563 6962 6c65 2076 an irreducible v │ │ │ │ +0002a800: 6172 6965 7479 0a0a 5365 6520 616c 736f ariety..See also │ │ │ │ +0002a810: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +0002a820: 6e6f 7465 2072 616e 646f 6d50 6f69 6e74 note randomPoint │ │ │ │ +0002a830: 734f 6e52 6174 696f 6e61 6c56 6172 6965 sOnRationalVarie │ │ │ │ +0002a840: 7479 3a0a 2020 2020 7261 6e64 6f6d 506f ty:. randomPo │ │ │ │ +0002a850: 696e 7473 4f6e 5261 7469 6f6e 616c 5661 intsOnRationalVa │ │ │ │ +0002a860: 7269 6574 795f 6c70 4964 6561 6c5f 636d riety_lpIdeal_cm │ │ │ │ +0002a870: 5a5a 5f72 702c 202d 2d20 6669 6e64 2072 ZZ_rp, -- find r │ │ │ │ +0002a880: 616e 646f 6d20 706f 696e 7473 206f 6e20 andom points on │ │ │ │ +0002a890: 610a 2020 2020 7661 7269 6574 7920 7468 a. variety th │ │ │ │ +0002a8a0: 6174 2063 616e 2062 6520 6465 7465 6374 at can be detect │ │ │ │ +0002a8b0: 6564 2074 6f20 6265 2072 6174 696f 6e61 ed to be rationa │ │ │ │ +0002a8c0: 6c0a 0a57 6179 7320 746f 2075 7365 2074 l..Ways to use t │ │ │ │ +0002a8d0: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d his method:.==== │ │ │ │ +0002a8e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002a8f0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ +0002a900: 7261 6e64 6f6d 506f 696e 744f 6e52 6174 randomPointOnRat │ │ │ │ +0002a910: 696f 6e61 6c56 6172 6965 7479 2849 6465 ionalVariety(Ide │ │ │ │ +0002a920: 616c 293a 0a20 2020 2072 616e 646f 6d50 al):. randomP │ │ │ │ +0002a930: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ +0002a940: 7269 6574 795f 6c70 4964 6561 6c5f 7270 riety_lpIdeal_rp │ │ │ │ +0002a950: 2c20 2d2d 2066 696e 6420 6120 7261 6e64 , -- find a rand │ │ │ │ +0002a960: 6f6d 2070 6f69 6e74 206f 6e20 610a 2020 om point on a. │ │ │ │ +0002a970: 2020 7661 7269 6574 7920 7468 6174 2063 variety that c │ │ │ │ +0002a980: 616e 2062 6520 6465 7465 6374 6564 2074 an be detected t │ │ │ │ +0002a990: 6f20 6265 2072 6174 696f 6e61 6c0a 2d2d o be rational.-- │ │ │ │ +0002a9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ +0002a9f0: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ +0002aa00: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ +0002aa10: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ +0002aa20: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ +0002aa30: 6179 322d 312e 3236 2e30 352b 6473 2f4d ay2-1.26.05+ds/M │ │ │ │ +0002aa40: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ +0002aa50: 6167 6573 2f0a 4772 6f65 626e 6572 5374 ages/.GroebnerSt │ │ │ │ +0002aa60: 7261 7461 2e6d 323a 3934 303a 302e 0a1f rata.m2:940:0... │ │ │ │ +0002aa70: 0a46 696c 653a 2047 726f 6562 6e65 7253 .File: GroebnerS │ │ │ │ +0002aa80: 7472 6174 612e 696e 666f 2c20 4e6f 6465 trata.info, Node │ │ │ │ +0002aa90: 3a20 7261 6e64 6f6d 506f 696e 7473 4f6e : randomPointsOn │ │ │ │ +0002aaa0: 5261 7469 6f6e 616c 5661 7269 6574 795f RationalVariety_ │ │ │ │ +0002aab0: 6c70 4964 6561 6c5f 636d 5a5a 5f72 702c lpIdeal_cmZZ_rp, │ │ │ │ +0002aac0: 204e 6578 743a 2073 6d61 6c6c 6572 4d6f Next: smallerMo │ │ │ │ +0002aad0: 6e6f 6d69 616c 732c 2050 7265 763a 2072 nomials, Prev: r │ │ │ │ +0002aae0: 616e 646f 6d50 6f69 6e74 4f6e 5261 7469 andomPointOnRati │ │ │ │ +0002aaf0: 6f6e 616c 5661 7269 6574 795f 6c70 4964 onalVariety_lpId │ │ │ │ +0002ab00: 6561 6c5f 7270 2c20 5570 3a20 546f 700a eal_rp, Up: Top. │ │ │ │ +0002ab10: 0a72 616e 646f 6d50 6f69 6e74 734f 6e52 .randomPointsOnR │ │ │ │ +0002ab20: 6174 696f 6e61 6c56 6172 6965 7479 2849 ationalVariety(I │ │ │ │ +0002ab30: 6465 616c 2c5a 5a29 202d 2d20 6669 6e64 deal,ZZ) -- find │ │ │ │ +0002ab40: 2072 616e 646f 6d20 706f 696e 7473 206f random points o │ │ │ │ +0002ab50: 6e20 6120 7661 7269 6574 7920 7468 6174 n a variety that │ │ │ │ +0002ab60: 2063 616e 2062 6520 6465 7465 6374 6564 can be detected │ │ │ │ +0002ab70: 2074 6f20 6265 2072 6174 696f 6e61 6c0a to be rational. │ │ │ │ +0002ab80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002ab90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002aba0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002abb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002abc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002abd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002abe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +0002abf0: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e * Function: *n │ │ │ │ +0002ac00: 6f74 6520 7261 6e64 6f6d 506f 696e 7473 ote randomPoints │ │ │ │ +0002ac10: 4f6e 5261 7469 6f6e 616c 5661 7269 6574 OnRationalVariet │ │ │ │ +0002ac20: 793a 0a20 2020 2072 616e 646f 6d50 6f69 y:. randomPoi │ │ │ │ +0002ac30: 6e74 734f 6e52 6174 696f 6e61 6c56 6172 ntsOnRationalVar │ │ │ │ +0002ac40: 6965 7479 5f6c 7049 6465 616c 5f63 6d5a iety_lpIdeal_cmZ │ │ │ │ +0002ac50: 5a5f 7270 2c0a 2020 2a20 5573 6167 653a Z_rp,. * Usage: │ │ │ │ +0002ac60: 200a 2020 2020 2020 2020 7261 6e64 6f6d . random │ │ │ │ +0002ac70: 506f 696e 7473 4f6e 5261 7469 6f6e 616c PointsOnRational │ │ │ │ +0002ac80: 5661 7269 6574 7928 492c 206e 290a 2020 Variety(I, n). │ │ │ │ +0002ac90: 2020 2020 2020 7261 6e64 6f6d 506f 696e randomPoin │ │ │ │ +0002aca0: 744f 6e52 6174 696f 6e61 6c56 6172 6965 tOnRationalVarie │ │ │ │ +0002acb0: 7479 0a20 202a 2049 6e70 7574 733a 0a20 ty. * Inputs:. │ │ │ │ +0002acc0: 2020 2020 202a 2049 2c20 616e 202a 6e6f * I, an *no │ │ │ │ +0002acd0: 7465 2069 6465 616c 3a20 284d 6163 6175 te ideal: (Macau │ │ │ │ +0002ace0: 6c61 7932 446f 6329 4964 6561 6c2c 2c20 lay2Doc)Ideal,, │ │ │ │ +0002acf0: 416e 2069 6465 616c 2069 6e20 6120 706f An ideal in a po │ │ │ │ +0002ad00: 6c79 6e6f 6d69 616c 2072 696e 670a 2020 lynomial ring. │ │ │ │ +0002ad10: 2020 2020 2020 2453 2420 6f76 6572 2061 $S$ over a │ │ │ │ +0002ad20: 2066 6965 6c64 2c20 7768 6963 6820 6465 field, which de │ │ │ │ +0002ad30: 6669 6e65 7320 6120 7072 696d 6520 6964 fines a prime id │ │ │ │ +0002ad40: 6561 6c0a 2020 2020 2020 2a20 6e2c 2061 eal. * n, a │ │ │ │ +0002ad50: 6e20 2a6e 6f74 6520 696e 7465 6765 723a n *note integer: │ │ │ │ +0002ad60: 2028 4d61 6361 756c 6179 3244 6f63 295a (Macaulay2Doc)Z │ │ │ │ +0002ad70: 5a2c 2c20 5468 6520 6e75 6d62 6572 206f Z,, The number o │ │ │ │ +0002ad80: 6620 706f 696e 7473 2074 6f0a 2020 2020 f points to. │ │ │ │ +0002ad90: 2020 2020 6765 6e65 7261 7465 0a20 202a generate. * │ │ │ │ +0002ada0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ +0002adb0: 2a20 6120 2a6e 6f74 6520 6c69 7374 3a20 * a *note list: │ │ │ │ +0002adc0: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ +0002add0: 7374 2c2c 2041 206c 6973 7420 6f66 2024 st,, A list of $ │ │ │ │ +0002ade0: 6e24 206f 6e65 2072 6f77 206d 6174 7269 n$ one row matri │ │ │ │ +0002adf0: 6365 7320 6f76 6572 0a20 2020 2020 2020 ces over. │ │ │ │ +0002ae00: 2074 6865 2062 6173 6520 6669 656c 6420 the base field │ │ │ │ +0002ae10: 6f66 2024 5324 2c20 7468 6174 2061 7265 of $S$, that are │ │ │ │ +0002ae20: 2072 616e 646f 6d6c 7920 6368 6f73 656e randomly chosen │ │ │ │ +0002ae30: 2070 6f69 6e74 7320 6f6e 2024 4924 2e20 points on $I$. │ │ │ │ +0002ae40: 206e 756c 6c20 6973 0a20 2020 2020 2020 null is. │ │ │ │ +0002ae50: 2072 6574 7572 6e65 6420 696e 2074 6865 returned in the │ │ │ │ +0002ae60: 2063 6173 6520 7768 656e 2074 6865 2072 case when the r │ │ │ │ +0002ae70: 6f75 7469 6e65 2063 616e 6e6f 7420 6465 outine cannot de │ │ │ │ +0002ae80: 7465 726d 696e 6520 6966 2074 6865 2076 termine if the v │ │ │ │ +0002ae90: 6172 6965 7479 0a20 2020 2020 2020 2069 ariety. i │ │ │ │ +0002aea0: 7320 7261 7469 6f6e 616c 2061 6e64 2069 s rational and i │ │ │ │ +0002aeb0: 7272 6564 7563 6962 6c65 2e0a 0a44 6573 rreducible...Des │ │ │ │ +0002aec0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +0002aed0: 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ====..+--------- │ │ │ │ +0002aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002aef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002af00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002af10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002af20: 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b 203d ----+.|i1 : kk = │ │ │ │ +0002af30: 205a 5a2f 3130 313b 2020 2020 2020 2020 ZZ/101; │ │ │ │ 0002af40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002af50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002af60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002af70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0002af80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002afa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002afb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002afc0: 2d2d 2d2d 2b0a 7c69 3420 3a20 7074 7320 ----+.|i4 : pts │ │ │ │ -0002afd0: 3d20 7261 6e64 6f6d 506f 696e 7473 4f6e = randomPointsOn │ │ │ │ -0002afe0: 5261 7469 6f6e 616c 5661 7269 6574 7928 RationalVariety( │ │ │ │ -0002aff0: 492c 2034 2920 2020 2020 2020 2020 2020 I, 4) │ │ │ │ +0002afc0: 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 3d20 ----+.|i2 : S = │ │ │ │ +0002afd0: 6b6b 5b61 2e2e 665d 3b20 2020 2020 2020 kk[a..f]; │ │ │ │ +0002afe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b010: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002b020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b060: 2020 2020 7c0a 7c6f 3420 3d20 7b7c 202d |.|o4 = {| - │ │ │ │ -0002b070: 3235 2032 3020 2d33 3020 2d31 3620 3234 25 20 -30 -16 24 │ │ │ │ -0002b080: 202d 3336 207c 2c20 7c20 3139 202d 3239 -36 |, | 19 -29 │ │ │ │ -0002b090: 2031 3920 3233 202d 3239 2031 3920 7c2c 19 23 -29 19 |, │ │ │ │ -0002b0a0: 207c 202d 3434 2034 3620 2d38 2037 202d | -44 46 -8 7 - │ │ │ │ -0002b0b0: 3130 2020 7c0a 7c20 2020 2020 2d2d 2d2d 10 |.| ---- │ │ │ │ -0002b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b100: 2d2d 2d2d 7c0a 7c20 2020 2020 2d32 3920 ----|.| -29 │ │ │ │ -0002b110: 7c2c 207c 2038 2034 3120 2d32 3420 3436 |, | 8 41 -24 46 │ │ │ │ -0002b120: 202d 3232 202d 3239 207c 7d20 2020 2020 -22 -29 |} │ │ │ │ +0002b010: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002b020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b060: 2d2d 2d2d 2b0a 7c69 3320 3a20 4920 3d20 ----+.|i3 : I = │ │ │ │ +0002b070: 6d69 6e6f 7273 2832 2c20 6765 6e65 7269 minors(2, generi │ │ │ │ +0002b080: 6353 796d 6d65 7472 6963 4d61 7472 6978 cSymmetricMatrix │ │ │ │ +0002b090: 2853 2c20 3329 2920 2020 2020 2020 2020 (S, 3)) │ │ │ │ +0002b0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b0b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b100: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b110: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +0002b120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b150: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002b160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b1a0: 2020 2020 7c0a 7c6f 3420 3a20 4c69 7374 |.|o4 : List │ │ │ │ -0002b1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b1f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002b200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b240: 2d2d 2d2d 2b0a 7c69 3520 3a20 666f 7220 ----+.|i5 : for │ │ │ │ -0002b250: 7020 696e 2070 7473 206c 6973 7420 7375 p in pts list su │ │ │ │ -0002b260: 6228 492c 2070 2920 3d3d 2030 2020 2020 b(I, p) == 0 │ │ │ │ -0002b270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b140: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +0002b150: 2020 2020 7c0a 7c6f 3320 3d20 6964 6561 |.|o3 = idea │ │ │ │ +0002b160: 6c20 282d 2062 2020 2b20 612a 642c 202d l (- b + a*d, - │ │ │ │ +0002b170: 2062 2a63 202b 2061 2a65 2c20 2d20 632a b*c + a*e, - c* │ │ │ │ +0002b180: 6420 2b20 622a 652c 202d 2062 2a63 202b d + b*e, - b*c + │ │ │ │ +0002b190: 2061 2a65 2c20 2d20 6320 202b 2061 2a66 a*e, - c + a*f │ │ │ │ +0002b1a0: 2c20 2d20 7c0a 7c20 2020 2020 2d2d 2d2d , - |.| ---- │ │ │ │ +0002b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b1f0: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ +0002b200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b220: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +0002b230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b240: 2020 2020 7c0a 7c20 2020 2020 632a 6520 |.| c*e │ │ │ │ +0002b250: 2b20 622a 662c 202d 2063 2a64 202b 2062 + b*f, - c*d + b │ │ │ │ +0002b260: 2a65 2c20 2d20 632a 6520 2b20 622a 662c *e, - c*e + b*f, │ │ │ │ +0002b270: 202d 2065 2020 2b20 642a 6629 2020 2020 - e + d*f) │ │ │ │ 0002b280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b290: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0002b2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b2e0: 2020 2020 7c0a 7c6f 3520 3d20 7b74 7275 |.|o5 = {tru │ │ │ │ -0002b2f0: 652c 2074 7275 652c 2074 7275 652c 2074 e, true, true, t │ │ │ │ -0002b300: 7275 657d 2020 2020 2020 2020 2020 2020 rue} │ │ │ │ +0002b2e0: 2020 2020 7c0a 7c6f 3320 3a20 4964 6561 |.|o3 : Idea │ │ │ │ +0002b2f0: 6c20 6f66 2053 2020 2020 2020 2020 2020 l of S │ │ │ │ +0002b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b330: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002b340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b380: 2020 2020 7c0a 7c6f 3520 3a20 4c69 7374 |.|o5 : List │ │ │ │ -0002b390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b330: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b380: 2d2d 2d2d 2b0a 7c69 3420 3a20 7074 7320 ----+.|i4 : pts │ │ │ │ +0002b390: 3d20 7261 6e64 6f6d 506f 696e 7473 4f6e = randomPointsOn │ │ │ │ +0002b3a0: 5261 7469 6f6e 616c 5661 7269 6574 7928 RationalVariety( │ │ │ │ +0002b3b0: 492c 2034 2920 2020 2020 2020 2020 2020 I, 4) │ │ │ │ 0002b3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b3d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002b3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b420: 2d2d 2d2d 2b0a 2b2d 2d2d 2d2d 2d2d 2d2d ----+.+--------- │ │ │ │ -0002b430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b470: 2d2d 2d2d 2b0a 7c69 3620 3a20 5320 3d20 ----+.|i6 : S = │ │ │ │ -0002b480: 6b6b 5b61 2e2e 645d 3b20 2020 2020 2020 kk[a..d]; │ │ │ │ -0002b490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b4c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002b4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b510: 2d2d 2d2d 2b0a 7c69 3720 3a20 4620 3d20 ----+.|i7 : F = │ │ │ │ -0002b520: 6772 6f65 626e 6572 4661 6d69 6c79 2069 groebnerFamily i │ │ │ │ -0002b530: 6465 616c 2261 322c 6162 2c61 632c 6232 deal"a2,ab,ac,b2 │ │ │ │ -0002b540: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ +0002b3d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b420: 2020 2020 7c0a 7c6f 3420 3d20 7b7c 202d |.|o4 = {| - │ │ │ │ +0002b430: 3235 2032 3020 2d33 3020 2d31 3620 3234 25 20 -30 -16 24 │ │ │ │ +0002b440: 202d 3336 207c 2c20 7c20 3139 202d 3239 -36 |, | 19 -29 │ │ │ │ +0002b450: 2031 3920 3233 202d 3239 2031 3920 7c2c 19 23 -29 19 |, │ │ │ │ +0002b460: 207c 202d 3434 2034 3620 2d38 2037 202d | -44 46 -8 7 - │ │ │ │ +0002b470: 3130 2020 7c0a 7c20 2020 2020 2d2d 2d2d 10 |.| ---- │ │ │ │ +0002b480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b4c0: 2d2d 2d2d 7c0a 7c20 2020 2020 2d32 3920 ----|.| -29 │ │ │ │ +0002b4d0: 7c2c 207c 2038 2034 3120 2d32 3420 3436 |, | 8 41 -24 46 │ │ │ │ +0002b4e0: 202d 3232 202d 3239 207c 7d20 2020 2020 -22 -29 |} │ │ │ │ +0002b4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b510: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b560: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b560: 2020 2020 7c0a 7c6f 3420 3a20 4c69 7374 |.|o4 : List │ │ │ │ 0002b570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b5b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002b5c0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -0002b5d0: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -0002b5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b5f0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002b600: 2020 2020 7c0a 7c6f 3720 3d20 6964 6561 |.|o7 = idea │ │ │ │ -0002b610: 6c20 2861 2020 2b20 7420 622a 6320 2b20 l (a + t b*c + │ │ │ │ -0002b620: 7420 612a 6420 2b20 7420 6320 202b 2074 t a*d + t c + t │ │ │ │ -0002b630: 2062 2a64 202b 2074 2063 2a64 202b 2074 b*d + t c*d + t │ │ │ │ -0002b640: 2064 202c 2061 2a62 202b 2074 2062 2a63 d , a*b + t b*c │ │ │ │ -0002b650: 202b 2020 7c0a 7c20 2020 2020 2020 2020 + |.| │ │ │ │ -0002b660: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ -0002b670: 2033 2020 2020 2020 2032 2020 2020 2020 3 2 │ │ │ │ -0002b680: 3420 2020 2020 2020 3520 2020 2020 2020 4 5 │ │ │ │ -0002b690: 3620 2020 2020 2020 2020 2020 3720 2020 6 7 │ │ │ │ -0002b6a0: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d |.| ---- │ │ │ │ -0002b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b6f0: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ -0002b700: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0002b5b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002b5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b600: 2d2d 2d2d 2b0a 7c69 3520 3a20 666f 7220 ----+.|i5 : for │ │ │ │ +0002b610: 7020 696e 2070 7473 206c 6973 7420 7375 p in pts list su │ │ │ │ +0002b620: 6228 492c 2070 2920 3d3d 2030 2020 2020 b(I, p) == 0 │ │ │ │ +0002b630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b650: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b6a0: 2020 2020 7c0a 7c6f 3520 3d20 7b74 7275 |.|o5 = {tru │ │ │ │ +0002b6b0: 652c 2074 7275 652c 2074 7275 652c 2074 e, true, true, t │ │ │ │ +0002b6c0: 7275 657d 2020 2020 2020 2020 2020 2020 rue} │ │ │ │ +0002b6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b6f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b720: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002b720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b740: 3220 2020 7c0a 7c20 2020 2020 7420 612a 2 |.| t a* │ │ │ │ -0002b750: 6420 2b20 7420 6320 202b 2074 2020 622a d + t c + t b* │ │ │ │ -0002b760: 6420 2b20 7420 2063 2a64 202b 2074 2020 d + t c*d + t │ │ │ │ -0002b770: 6420 2c20 612a 6320 2b20 7420 2062 2a63 d , a*c + t b*c │ │ │ │ -0002b780: 202b 2074 2020 612a 6420 2b20 7420 2063 + t a*d + t c │ │ │ │ -0002b790: 2020 2b20 7c0a 7c20 2020 2020 2039 2020 + |.| 9 │ │ │ │ -0002b7a0: 2020 2020 2038 2020 2020 2020 3130 2020 8 10 │ │ │ │ -0002b7b0: 2020 2020 2031 3120 2020 2020 2020 3132 11 12 │ │ │ │ -0002b7c0: 2020 2020 2020 2020 2020 2031 3320 2020 13 │ │ │ │ -0002b7d0: 2020 2020 3135 2020 2020 2020 2031 3420 15 14 │ │ │ │ -0002b7e0: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d |.| ---- │ │ │ │ +0002b740: 2020 2020 7c0a 7c6f 3520 3a20 4c69 7374 |.|o5 : List │ │ │ │ +0002b750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b790: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b7e0: 2d2d 2d2d 2b0a 2b2d 2d2d 2d2d 2d2d 2d2d ----+.+--------- │ │ │ │ 0002b7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b830: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ -0002b840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b850: 2020 3220 2020 3220 2020 2020 2020 2020 2 2 │ │ │ │ +0002b830: 2d2d 2d2d 2b0a 7c69 3620 3a20 5320 3d20 ----+.|i6 : S = │ │ │ │ +0002b840: 6b6b 5b61 2e2e 645d 3b20 2020 2020 2020 kk[a..d]; │ │ │ │ +0002b850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b870: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002b880: 2020 2020 7c0a 7c20 2020 2020 7420 2062 |.| t b │ │ │ │ -0002b890: 2a64 202b 2074 2020 632a 6420 2b20 7420 *d + t c*d + t │ │ │ │ -0002b8a0: 2064 202c 2062 2020 2b20 7420 2062 2a63 d , b + t b*c │ │ │ │ -0002b8b0: 202b 2074 2020 612a 6420 2b20 7420 2063 + t a*d + t c │ │ │ │ -0002b8c0: 2020 2b20 7420 2062 2a64 202b 2074 2020 + t b*d + t │ │ │ │ -0002b8d0: 632a 6420 7c0a 7c20 2020 2020 2031 3620 c*d |.| 16 │ │ │ │ -0002b8e0: 2020 2020 2020 3137 2020 2020 2020 2031 17 1 │ │ │ │ -0002b8f0: 3820 2020 2020 2020 2020 2031 3920 2020 8 19 │ │ │ │ -0002b900: 2020 2020 3231 2020 2020 2020 2032 3020 21 20 │ │ │ │ -0002b910: 2020 2020 2032 3220 2020 2020 2020 3233 22 23 │ │ │ │ -0002b920: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d |.| ---- │ │ │ │ -0002b930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b970: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ -0002b980: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002b990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b880: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002b890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b8d0: 2d2d 2d2d 2b0a 7c69 3720 3a20 4620 3d20 ----+.|i7 : F = │ │ │ │ +0002b8e0: 6772 6f65 626e 6572 4661 6d69 6c79 2069 groebnerFamily i │ │ │ │ +0002b8f0: 6465 616c 2261 322c 6162 2c61 632c 6232 deal"a2,ab,ac,b2 │ │ │ │ +0002b900: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ +0002b910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b920: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b970: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b980: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +0002b990: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ 0002b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b9c0: 2020 2020 7c0a 7c20 2020 2020 2b20 7420 |.| + t │ │ │ │ -0002b9d0: 2064 2029 2020 2020 2020 2020 2020 2020 d ) │ │ │ │ -0002b9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba10: 2020 2020 7c0a 7c20 2020 2020 2020 2032 |.| 2 │ │ │ │ -0002ba20: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -0002ba30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba60: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002ba70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002baa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bab0: 2020 2020 7c0a 7c6f 3720 3a20 4964 6561 |.|o7 : Idea │ │ │ │ -0002bac0: 6c20 6f66 206b 6b5b 7420 2c20 7420 2c20 l of kk[t , t , │ │ │ │ -0002bad0: 7420 202c 2074 202c 2074 202c 2074 2020 t , t , t , t │ │ │ │ -0002bae0: 2c20 7420 202c 2074 2020 2c20 7420 2c20 , t , t , t , │ │ │ │ -0002baf0: 7420 2c20 7420 2c20 7420 202c 2074 2020 t , t , t , t │ │ │ │ -0002bb00: 2c20 7420 7c0a 7c20 2020 2020 2020 2020 , t |.| │ │ │ │ -0002bb10: 2020 2020 2020 2020 2036 2020 2035 2020 6 5 │ │ │ │ -0002bb20: 2031 3220 2020 3220 2020 3420 2020 3131 12 2 4 11 │ │ │ │ -0002bb30: 2020 2031 3820 2020 3234 2020 2031 2020 18 24 1 │ │ │ │ -0002bb40: 2033 2020 2038 2020 2031 3020 2020 3137 3 8 10 17 │ │ │ │ -0002bb50: 2020 2032 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2|.|--------- │ │ │ │ -0002bb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bba0: 2d2d 2d2d 7c0a 7c20 2c20 7420 2c20 7420 ----|.| , t , t │ │ │ │ -0002bbb0: 2c20 7420 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ -0002bbc0: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ -0002bbd0: 7420 202c 2074 2020 5d5b 612e 2e64 5d20 t , t ][a..d] │ │ │ │ -0002bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bbf0: 2020 2020 7c0a 7c33 2020 2037 2020 2039 |.|3 7 9 │ │ │ │ -0002bc00: 2020 2031 3420 2020 3136 2020 2032 3020 14 16 20 │ │ │ │ -0002bc10: 2020 3232 2020 2031 3320 2020 3135 2020 22 13 15 │ │ │ │ -0002bc20: 2031 3920 2020 3231 2020 2020 2020 2020 19 21 │ │ │ │ -0002bc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002bc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bc90: 2d2d 2d2d 2b0a 7c69 3820 3a20 4a20 3d20 ----+.|i8 : J = │ │ │ │ -0002bca0: 6772 6f65 626e 6572 5374 7261 7475 6d20 groebnerStratum │ │ │ │ -0002bcb0: 463b 2020 2020 2020 2020 2020 2020 2020 F; │ │ │ │ -0002bcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bce0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002bcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bd30: 2020 2020 7c0a 7c6f 3820 3a20 4964 6561 |.|o8 : Idea │ │ │ │ -0002bd40: 6c20 6f66 206b 6b5b 7420 2c20 7420 2c20 l of kk[t , t , │ │ │ │ -0002bd50: 7420 202c 2074 202c 2074 202c 2074 2020 t , t , t , t │ │ │ │ -0002bd60: 2c20 7420 202c 2074 2020 2c20 7420 2c20 , t , t , t , │ │ │ │ -0002bd70: 7420 2c20 7420 2c20 7420 202c 2074 2020 t , t , t , t │ │ │ │ -0002bd80: 2c20 2020 7c0a 7c20 2020 2020 2020 2020 , |.| │ │ │ │ -0002bd90: 2020 2020 2020 2020 2036 2020 2035 2020 6 5 │ │ │ │ -0002bda0: 2031 3220 2020 3220 2020 3420 2020 3131 12 2 4 11 │ │ │ │ -0002bdb0: 2020 2031 3820 2020 3234 2020 2031 2020 18 24 1 │ │ │ │ -0002bdc0: 2033 2020 2038 2020 2031 3020 2020 3137 3 8 10 17 │ │ │ │ -0002bdd0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d |.|--------- │ │ │ │ -0002bde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002be00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002be10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002be20: 2d2d 2d2d 7c0a 7c74 2020 2c20 7420 2c20 ----|.|t , t , │ │ │ │ -0002be30: 7420 2c20 7420 202c 2074 2020 2c20 7420 t , t , t , t │ │ │ │ -0002be40: 202c 2074 2020 2c20 7420 202c 2074 2020 , t , t , t │ │ │ │ -0002be50: 2c20 7420 202c 2074 2020 5d20 2020 2020 , t , t ] │ │ │ │ +0002b9b0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002b9c0: 2020 2020 7c0a 7c6f 3720 3d20 6964 6561 |.|o7 = idea │ │ │ │ +0002b9d0: 6c20 2861 2020 2b20 7420 622a 6320 2b20 l (a + t b*c + │ │ │ │ +0002b9e0: 7420 612a 6420 2b20 7420 6320 202b 2074 t a*d + t c + t │ │ │ │ +0002b9f0: 2062 2a64 202b 2074 2063 2a64 202b 2074 b*d + t c*d + t │ │ │ │ +0002ba00: 2064 202c 2061 2a62 202b 2074 2062 2a63 d , a*b + t b*c │ │ │ │ +0002ba10: 202b 2020 7c0a 7c20 2020 2020 2020 2020 + |.| │ │ │ │ +0002ba20: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ +0002ba30: 2033 2020 2020 2020 2032 2020 2020 2020 3 2 │ │ │ │ +0002ba40: 3420 2020 2020 2020 3520 2020 2020 2020 4 5 │ │ │ │ +0002ba50: 3620 2020 2020 2020 2020 2020 3720 2020 6 7 │ │ │ │ +0002ba60: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d |.| ---- │ │ │ │ +0002ba70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ba80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ba90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002baa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bab0: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ +0002bac0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0002bad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bae0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002baf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb00: 3220 2020 7c0a 7c20 2020 2020 7420 612a 2 |.| t a* │ │ │ │ +0002bb10: 6420 2b20 7420 6320 202b 2074 2020 622a d + t c + t b* │ │ │ │ +0002bb20: 6420 2b20 7420 2063 2a64 202b 2074 2020 d + t c*d + t │ │ │ │ +0002bb30: 6420 2c20 612a 6320 2b20 7420 2062 2a63 d , a*c + t b*c │ │ │ │ +0002bb40: 202b 2074 2020 612a 6420 2b20 7420 2063 + t a*d + t c │ │ │ │ +0002bb50: 2020 2b20 7c0a 7c20 2020 2020 2039 2020 + |.| 9 │ │ │ │ +0002bb60: 2020 2020 2038 2020 2020 2020 3130 2020 8 10 │ │ │ │ +0002bb70: 2020 2020 2031 3120 2020 2020 2020 3132 11 12 │ │ │ │ +0002bb80: 2020 2020 2020 2020 2020 2031 3320 2020 13 │ │ │ │ +0002bb90: 2020 2020 3135 2020 2020 2020 2031 3420 15 14 │ │ │ │ +0002bba0: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d |.| ---- │ │ │ │ +0002bbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bbf0: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ +0002bc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bc10: 2020 3220 2020 3220 2020 2020 2020 2020 2 2 │ │ │ │ +0002bc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bc30: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002bc40: 2020 2020 7c0a 7c20 2020 2020 7420 2062 |.| t b │ │ │ │ +0002bc50: 2a64 202b 2074 2020 632a 6420 2b20 7420 *d + t c*d + t │ │ │ │ +0002bc60: 2064 202c 2062 2020 2b20 7420 2062 2a63 d , b + t b*c │ │ │ │ +0002bc70: 202b 2074 2020 612a 6420 2b20 7420 2063 + t a*d + t c │ │ │ │ +0002bc80: 2020 2b20 7420 2062 2a64 202b 2074 2020 + t b*d + t │ │ │ │ +0002bc90: 632a 6420 7c0a 7c20 2020 2020 2031 3620 c*d |.| 16 │ │ │ │ +0002bca0: 2020 2020 2020 3137 2020 2020 2020 2031 17 1 │ │ │ │ +0002bcb0: 3820 2020 2020 2020 2020 2031 3920 2020 8 19 │ │ │ │ +0002bcc0: 2020 2020 3231 2020 2020 2020 2032 3020 21 20 │ │ │ │ +0002bcd0: 2020 2020 2032 3220 2020 2020 2020 3233 22 23 │ │ │ │ +0002bce0: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d |.| ---- │ │ │ │ +0002bcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bd30: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ +0002bd40: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002bd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bd80: 2020 2020 7c0a 7c20 2020 2020 2b20 7420 |.| + t │ │ │ │ +0002bd90: 2064 2029 2020 2020 2020 2020 2020 2020 d ) │ │ │ │ +0002bda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bdd0: 2020 2020 7c0a 7c20 2020 2020 2020 2032 |.| 2 │ │ │ │ +0002bde0: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +0002bdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002be30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002be60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002be70: 2020 2020 7c0a 7c20 3233 2020 2037 2020 |.| 23 7 │ │ │ │ -0002be80: 2039 2020 2031 3420 2020 3136 2020 2032 9 14 16 2 │ │ │ │ -0002be90: 3020 2020 3232 2020 2031 3320 2020 3135 0 22 13 15 │ │ │ │ -0002bea0: 2020 2031 3920 2020 3231 2020 2020 2020 19 21 │ │ │ │ -0002beb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bec0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002bed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bf10: 2d2d 2d2d 2b0a 7c69 3920 3a20 636f 6d70 ----+.|i9 : comp │ │ │ │ -0002bf20: 734a 203d 2064 6563 6f6d 706f 7365 204a sJ = decompose J │ │ │ │ -0002bf30: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ -0002bf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002bf70: 2d2d 2d2d 2d2d 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2020 2020 2020 2036 2020 2035 2020 6 5 │ │ │ │ +0002bee0: 2031 3220 2020 3220 2020 3420 2020 3131 12 2 4 11 │ │ │ │ +0002bef0: 2020 2031 3820 2020 3234 2020 2031 2020 18 24 1 │ │ │ │ +0002bf00: 2033 2020 2038 2020 2031 3020 2020 3137 3 8 10 17 │ │ │ │ +0002bf10: 2020 2032 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2|.|--------- │ │ │ │ +0002bf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bf30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bf50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bf60: 2d2d 2d2d 7c0a 7c20 2c20 7420 2c20 7420 ----|.| , t , t │ │ │ │ +0002bf70: 2c20 7420 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ +0002bf80: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ +0002bf90: 7420 202c 2074 2020 5d5b 612e 2e64 5d20 t , t ][a..d] │ │ │ │ +0002bfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bfb0: 2020 2020 7c0a 7c33 2020 2037 2020 2039 |.|3 7 9 │ │ │ │ +0002bfc0: 2020 2031 3420 2020 3136 2020 2032 3020 14 16 20 │ │ │ │ +0002bfd0: 2020 3232 2020 2031 3320 2020 3135 2020 22 13 15 │ │ │ │ +0002bfe0: 2031 3920 2020 3231 2020 2020 2020 2020 19 21 │ │ │ │ 0002bff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c000: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0002c010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c050: 2d2d 2d2d 2b0a 7c69 3131 203a 2023 636f ----+.|i11 : #co │ │ │ │ -0002c060: 6d70 734a 203d 3d20 3220 2020 2020 2020 mpsJ == 2 │ │ │ │ -0002c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c050: 2d2d 2d2d 2b0a 7c69 3820 3a20 4a20 3d20 ----+.|i8 : J = │ │ │ │ +0002c060: 6772 6f65 626e 6572 5374 7261 7475 6d20 groebnerStratum │ │ │ │ +0002c070: 463b 2020 2020 2020 2020 2020 2020 2020 F; │ │ │ │ 0002c080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c0a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0002c0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c0f0: 2020 2020 7c0a 7c6f 3131 203d 2074 7275 |.|o11 = tru │ │ │ │ -0002c100: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ -0002c110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c140: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002c150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c190: 2d2d 2d2d 2b0a 7c69 3132 203a 2063 6f6d ----+.|i12 : com │ │ │ │ -0002c1a0: 7073 4a2f 6469 6d20 2020 2020 2020 2020 psJ/dim │ │ │ │ -0002c1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c1e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002c1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c0f0: 2020 2020 7c0a 7c6f 3820 3a20 4964 6561 |.|o8 : Idea │ │ │ │ +0002c100: 6c20 6f66 206b 6b5b 7420 2c20 7420 2c20 l of kk[t , t , │ │ │ │ +0002c110: 7420 202c 2074 202c 2074 202c 2074 2020 t , t , t , t │ │ │ │ +0002c120: 2c20 7420 202c 2074 2020 2c20 7420 2c20 , t , t , t , │ │ │ │ +0002c130: 7420 2c20 7420 2c20 7420 202c 2074 2020 t , t , t , t │ │ │ │ +0002c140: 2c20 2020 7c0a 7c20 2020 2020 2020 2020 , |.| │ │ │ │ +0002c150: 2020 2020 2020 2020 2036 2020 2035 2020 6 5 │ │ │ │ +0002c160: 2031 3220 2020 3220 2020 3420 2020 3131 12 2 4 11 │ │ │ │ +0002c170: 2020 2031 3820 2020 3234 2020 2031 2020 18 24 1 │ │ │ │ +0002c180: 2033 2020 2038 2020 2031 3020 2020 3137 3 8 10 17 │ │ │ │ +0002c190: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d |.|--------- │ │ │ │ +0002c1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c1e0: 2d2d 2d2d 7c0a 7c74 2020 2c20 7420 2c20 ----|.|t , t , │ │ │ │ +0002c1f0: 7420 2c20 7420 202c 2074 2020 2c20 7420 t , t , t , t │ │ │ │ +0002c200: 202c 2074 2020 2c20 7420 202c 2074 2020 , t , t , t │ │ │ │ +0002c210: 2c20 7420 202c 2074 2020 5d20 2020 2020 , t , t ] │ │ │ │ 0002c220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c230: 2020 2020 7c0a 7c6f 3132 203d 207b 3131 |.|o12 = {11 │ │ │ │ -0002c240: 2c20 387d 2020 2020 2020 2020 2020 2020 , 8} │ │ │ │ -0002c250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c230: 2020 2020 7c0a 7c20 3233 2020 2037 2020 |.| 23 7 │ │ │ │ +0002c240: 2039 2020 2031 3420 2020 3136 2020 2032 9 14 16 2 │ │ │ │ +0002c250: 3020 2020 3232 2020 2031 3320 2020 3135 0 22 13 15 │ │ │ │ +0002c260: 2020 2031 3920 2020 3231 2020 2020 2020 19 21 │ │ │ │ 0002c270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c280: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002c290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2d0: 2020 2020 7c0a 7c6f 3132 203a 204c 6973 |.|o12 : Lis │ │ │ │ -0002c2e0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -0002c2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c280: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002c290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c2d0: 2d2d 2d2d 2b0a 7c69 3920 3a20 636f 6d70 ----+.|i9 : comp │ │ │ │ +0002c2e0: 734a 203d 2064 6563 6f6d 706f 7365 204a sJ = decompose J │ │ │ │ +0002c2f0: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ 0002c300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c320: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0002c330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c370: 2d2d 2d2d 2b0a 0a54 6865 7265 2061 7265 ----+..There are │ │ │ │ -0002c380: 2032 2063 6f6d 706f 6e65 6e74 732e 2020 2 components. │ │ │ │ -0002c390: 5765 2061 7474 656d 7074 2074 6f20 6669 We attempt to fi │ │ │ │ -0002c3a0: 6e64 2070 6f69 6e74 7320 6f6e 2065 6163 nd points on eac │ │ │ │ -0002c3b0: 6820 6f66 2074 6865 7365 2074 776f 0a63 h of these two.c │ │ │ │ -0002c3c0: 6f6d 706f 6e65 6e74 732e 2057 6520 6172 omponents. We ar │ │ │ │ -0002c3d0: 6520 7375 6363 6573 7366 756c 2e20 2054 e successful. T │ │ │ │ -0002c3e0: 6869 7320 696e 6469 6361 7465 7320 7468 his indicates th │ │ │ │ -0002c3f0: 6174 2074 6865 2063 6f72 7265 7370 6f6e at the correspon │ │ │ │ -0002c400: 6469 6e67 2076 6172 6965 7469 6573 0a61 ding varieties.a │ │ │ │ -0002c410: 7265 2062 6f74 6820 7261 7469 6f6e 616c re both rational │ │ │ │ -0002c420: 2e20 416c 736f 2c20 6966 2077 6520 6361 . Also, if we ca │ │ │ │ -0002c430: 6e20 6669 6e64 206f 6e65 2070 6f69 6e74 n find one point │ │ │ │ -0002c440: 2c20 7765 2063 616e 2066 696e 6420 6173 , we can find as │ │ │ │ -0002c450: 206d 616e 7920 6173 2077 650a 7761 6e74 many as we.want │ │ │ │ -0002c460: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ -0002c470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c4b0: 2d2b 0a7c 6931 3320 3a20 6e65 744c 6973 -+.|i13 : netLis │ │ │ │ -0002c4c0: 7420 7261 6e64 6f6d 506f 696e 7473 4f6e t randomPointsOn │ │ │ │ -0002c4d0: 5261 7469 6f6e 616c 5661 7269 6574 7928 RationalVariety( │ │ │ │ -0002c4e0: 636f 6d70 734a 5f30 2c20 3130 2920 2020 compsJ_0, 10) │ │ │ │ +0002c370: 2d2d 2d2d 2b0a 7c69 3130 203a 2063 6f6d ----+.|i10 : com │ │ │ │ +0002c380: 7073 4a20 3d20 636f 6d70 734a 2f74 7269 psJ = compsJ/tri │ │ │ │ +0002c390: 6d3b 2020 2020 2020 2020 2020 2020 2020 m; │ │ │ │ +0002c3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c3c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002c3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c410: 2d2d 2d2d 2b0a 7c69 3131 203a 2023 636f ----+.|i11 : #co │ │ │ │ +0002c420: 6d70 734a 203d 3d20 3220 2020 2020 2020 mpsJ == 2 │ │ │ │ +0002c430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c460: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002c470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c4b0: 2020 2020 7c0a 7c6f 3131 203d 2074 7275 |.|o11 = tru │ │ │ │ +0002c4c0: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +0002c4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c500: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002c510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c550: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -0002c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c5a0: 2d7c 0a7c 6f31 3320 3d20 7c7c 2034 3220 -|.|o13 = || 42 │ │ │ │ -0002c5b0: 2d35 3020 2d35 3020 3338 202d 3339 2036 -50 -50 38 -39 6 │ │ │ │ -0002c5c0: 202d 3120 3437 2034 3920 2d31 3820 3138 -1 47 49 -18 18 │ │ │ │ -0002c5d0: 202d 3238 202d 3437 2031 3920 3438 2033 -28 -47 19 48 3 │ │ │ │ -0002c5e0: 3420 2d31 3320 3131 202d 3136 202d 3338 4 -13 11 -16 -38 │ │ │ │ -0002c5f0: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -0002c600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c640: 2d7c 0a7c 2020 2020 2020 7c7c 202d 3434 -|.| || -44 │ │ │ │ -0002c650: 202d 3331 202d 3920 2d32 3120 2d34 3220 -31 -9 -21 -42 │ │ │ │ -0002c660: 2d33 3620 2d34 3720 2d32 3020 3338 202d -36 -47 -20 38 - │ │ │ │ -0002c670: 3334 202d 3320 2d34 3320 3232 2031 3620 34 -3 -43 22 16 │ │ │ │ -0002c680: 2d33 3520 3220 2d34 3820 3332 202d 3238 -35 2 -48 32 -28 │ │ │ │ -0002c690: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -0002c6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c6e0: 2d7c 0a7c 2020 2020 2020 7c7c 2034 3520 -|.| || 45 │ │ │ │ -0002c6f0: 2d32 3920 2d31 202d 3420 3432 202d 3335 -29 -1 -4 42 -35 │ │ │ │ -0002c700: 2034 202d 3133 2031 3820 2d31 3720 3120 4 -13 18 -17 1 │ │ │ │ -0002c710: 3231 2033 3920 2d32 3320 3530 2031 3520 21 39 -23 50 15 │ │ │ │ -0002c720: 2d31 3120 2d31 3120 3139 2034 3720 2d31 -11 -11 19 47 -1 │ │ │ │ -0002c730: 367c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d 6|.| +----- │ │ │ │ -0002c740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c780: 2d7c 0a7c 2020 2020 2020 7c7c 2033 3520 -|.| || 35 │ │ │ │ -0002c790: 2d34 3420 2d33 3320 2d38 2032 3120 2d32 -44 -33 -8 21 -2 │ │ │ │ -0002c7a0: 202d 3434 202d 3230 2031 3920 2d32 3820 -44 -20 19 -28 │ │ │ │ -0002c7b0: 3139 2032 3720 3131 2034 3020 3334 2033 19 27 11 40 34 3 │ │ │ │ -0002c7c0: 3320 3120 2d31 3420 3335 2033 3620 3131 3 1 -14 35 36 11 │ │ │ │ -0002c7d0: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -0002c7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c820: 2d7c 0a7c 2020 2020 2020 7c7c 2033 3320 -|.| || 33 │ │ │ │ -0002c830: 3437 2034 3620 3136 202d 3232 202d 3235 47 46 16 -22 -25 │ │ │ │ -0002c840: 202d 3434 202d 3336 202d 3330 202d 3337 -44 -36 -30 -37 │ │ │ │ -0002c850: 2033 3020 2d32 3520 2d34 3720 3239 202d 30 -25 -47 29 - │ │ │ │ -0002c860: 3431 2032 202d 3133 202d 3431 202d 3437 41 2 -13 -41 -47 │ │ │ │ -0002c870: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -0002c880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c8c0: 2d7c 0a7c 2020 2020 2020 7c7c 2034 3320 -|.| || 43 │ │ │ │ -0002c8d0: 3120 2d34 3120 3233 202d 3432 202d 3134 1 -41 23 -42 -14 │ │ │ │ -0002c8e0: 2033 3720 2d35 3020 2d33 3220 2d32 3020 37 -50 -32 -20 │ │ │ │ -0002c8f0: 2d35 202d 3439 202d 3920 3332 202d 3138 -5 -49 -9 32 -18 │ │ │ │ -0002c900: 202d 3232 2032 3420 3433 202d 3138 2033 -22 24 43 -18 3 │ │ │ │ -0002c910: 307c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d 0|.| +----- │ │ │ │ +0002c500: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002c510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c550: 2d2d 2d2d 2b0a 7c69 3132 203a 2063 6f6d ----+.|i12 : com │ │ │ │ +0002c560: 7073 4a2f 6469 6d20 2020 2020 2020 2020 psJ/dim │ │ │ │ +0002c570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c5a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002c5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c5f0: 2020 2020 7c0a 7c6f 3132 203d 207b 3131 |.|o12 = {11 │ │ │ │ +0002c600: 2c20 387d 2020 2020 2020 2020 2020 2020 , 8} │ │ │ │ +0002c610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c640: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002c650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c690: 2020 2020 7c0a 7c6f 3132 203a 204c 6973 |.|o12 : Lis │ │ │ │ +0002c6a0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +0002c6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c6e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002c6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c730: 2d2d 2d2d 2b0a 0a54 6865 7265 2061 7265 ----+..There are │ │ │ │ +0002c740: 2032 2063 6f6d 706f 6e65 6e74 732e 2020 2 components. │ │ │ │ +0002c750: 5765 2061 7474 656d 7074 2074 6f20 6669 We attempt to fi │ │ │ │ +0002c760: 6e64 2070 6f69 6e74 7320 6f6e 2065 6163 nd points on eac │ │ │ │ +0002c770: 6820 6f66 2074 6865 7365 2074 776f 0a63 h of these two.c │ │ │ │ +0002c780: 6f6d 706f 6e65 6e74 732e 2057 6520 6172 omponents. We ar │ │ │ │ +0002c790: 6520 7375 6363 6573 7366 756c 2e20 2054 e successful. T │ │ │ │ +0002c7a0: 6869 7320 696e 6469 6361 7465 7320 7468 his indicates th │ │ │ │ +0002c7b0: 6174 2074 6865 2063 6f72 7265 7370 6f6e at the correspon │ │ │ │ +0002c7c0: 6469 6e67 2076 6172 6965 7469 6573 0a61 ding varieties.a │ │ │ │ +0002c7d0: 7265 2062 6f74 6820 7261 7469 6f6e 616c re both rational │ │ │ │ +0002c7e0: 2e20 416c 736f 2c20 6966 2077 6520 6361 . Also, if we ca │ │ │ │ +0002c7f0: 6e20 6669 6e64 206f 6e65 2070 6f69 6e74 n find one point │ │ │ │ +0002c800: 2c20 7765 2063 616e 2066 696e 6420 6173 , we can find as │ │ │ │ +0002c810: 206d 616e 7920 6173 2077 650a 7761 6e74 many as we.want │ │ │ │ +0002c820: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ +0002c830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c870: 2d2b 0a7c 6931 3320 3a20 6e65 744c 6973 -+.|i13 : netLis │ │ │ │ +0002c880: 7420 7261 6e64 6f6d 506f 696e 7473 4f6e t randomPointsOn │ │ │ │ +0002c890: 5261 7469 6f6e 616c 5661 7269 6574 7928 RationalVariety( │ │ │ │ +0002c8a0: 636f 6d70 734a 5f30 2c20 3130 2920 2020 compsJ_0, 10) │ │ │ │ +0002c8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c8c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002c8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c910: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ 0002c920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c960: 2d7c 0a7c 2020 2020 2020 7c7c 2031 3220 -|.| || 12 │ │ │ │ -0002c970: 3820 3236 2031 3520 3232 2031 3220 3020 8 26 15 22 12 0 │ │ │ │ -0002c980: 2d35 2036 2031 3720 2d32 3120 2d31 3820 -5 6 17 -21 -18 │ │ │ │ -0002c990: 2d33 3320 2d34 3920 2d31 3920 3333 202d -33 -49 -19 33 - │ │ │ │ -0002c9a0: 3230 2030 202d 3135 202d 3438 2033 3920 20 0 -15 -48 39 │ │ │ │ -0002c9b0: 307c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d 0|.| +----- │ │ │ │ +0002c960: 2d7c 0a7c 6f31 3320 3d20 7c7c 2034 3220 -|.|o13 = || 42 │ │ │ │ +0002c970: 2d35 3020 2d35 3020 3338 202d 3339 2036 -50 -50 38 -39 6 │ │ │ │ +0002c980: 202d 3120 3437 2034 3920 2d31 3820 3138 -1 47 49 -18 18 │ │ │ │ +0002c990: 202d 3238 202d 3437 2031 3920 3438 2033 -28 -47 19 48 3 │ │ │ │ +0002c9a0: 3420 2d31 3320 3131 202d 3136 202d 3338 4 -13 11 -16 -38 │ │ │ │ +0002c9b0: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ 0002c9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ca00: 2d7c 0a7c 2020 2020 2020 7c7c 202d 3438 -|.| || -48 │ │ │ │ -0002ca10: 202d 3237 202d 3820 2d33 3320 2d33 3520 -27 -8 -33 -35 │ │ │ │ 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2020 2020 2020 2020 2020 │ │ │ │ 0002e0d0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ 0002e0e0: 2d2d 2d2d 2d2d 2b20 2020 2020 2020 2020 ------+ │ │ │ │ 0002e0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e120: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002e130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e170: 2d2b 0a0a 4361 7665 6174 0a3d 3d3d 3d3d -+..Caveat.===== │ │ │ │ -0002e180: 3d0a 0a54 6869 7320 726f 7574 696e 6520 =..This routine │ │ │ │ -0002e190: 6578 7065 6374 7320 7468 6520 696e 7075 expects the inpu │ │ │ │ -0002e1a0: 7420 746f 2072 6570 7265 7365 6e74 2061 t to represent a │ │ │ │ -0002e1b0: 6e20 6972 7265 6475 6369 626c 6520 7661 n irreducible va │ │ │ │ -0002e1c0: 7269 6574 790a 0a53 6565 2061 6c73 6f0a riety..See also. │ │ │ │ -0002e1d0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ -0002e1e0: 6f74 6520 7261 6e64 6f6d 506f 696e 744f ote randomPointO │ │ │ │ -0002e1f0: 6e52 6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ -0002e200: 2849 6465 616c 293a 0a20 2020 2072 616e (Ideal):. ran │ │ │ │ -0002e210: 646f 6d50 6f69 6e74 4f6e 5261 7469 6f6e domPointOnRation │ │ │ │ -0002e220: 616c 5661 7269 6574 795f 6c70 4964 6561 alVariety_lpIdea │ │ │ │ -0002e230: 6c5f 7270 2c20 2d2d 2066 696e 6420 6120 l_rp, -- find a │ │ │ │ -0002e240: 7261 6e64 6f6d 2070 6f69 6e74 206f 6e20 random point on │ │ │ │ -0002e250: 610a 2020 2020 7661 7269 6574 7920 7468 a. variety th │ │ │ │ -0002e260: 6174 2063 616e 2062 6520 6465 7465 6374 at can be detect │ │ │ │ -0002e270: 6564 2074 6f20 6265 2072 6174 696f 6e61 ed to be rationa │ │ │ │ -0002e280: 6c0a 0a57 6179 7320 746f 2075 7365 2074 l..Ways to use t │ │ │ │ -0002e290: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d his method:.==== │ │ │ │ -0002e2a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002e2b0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -0002e2c0: 7261 6e64 6f6d 506f 696e 7473 4f6e 5261 randomPointsOnRa │ │ │ │ -0002e2d0: 7469 6f6e 616c 5661 7269 6574 7928 4964 tionalVariety(Id │ │ │ │ -0002e2e0: 6561 6c2c 5a5a 293a 0a20 2020 2072 616e eal,ZZ):. ran │ │ │ │ -0002e2f0: 646f 6d50 6f69 6e74 734f 6e52 6174 696f domPointsOnRatio │ │ │ │ -0002e300: 6e61 6c56 6172 6965 7479 5f6c 7049 6465 nalVariety_lpIde │ │ │ │ -0002e310: 616c 5f63 6d5a 5a5f 7270 2c20 2d2d 2066 al_cmZZ_rp, -- f │ │ │ │ -0002e320: 696e 6420 7261 6e64 6f6d 2070 6f69 6e74 ind random point │ │ │ │ -0002e330: 7320 6f6e 2061 0a20 2020 2076 6172 6965 s on a. varie │ │ │ │ -0002e340: 7479 2074 6861 7420 6361 6e20 6265 2064 ty that can be d │ │ │ │ -0002e350: 6574 6563 7465 6420 746f 2062 6520 7261 etected to be ra │ │ │ │ -0002e360: 7469 6f6e 616c 0a2d 2d2d 2d2d 2d2d 2d2d tional.--------- │ │ │ │ -0002e370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e3b0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -0002e3c0: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -0002e3d0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -0002e3e0: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -0002e3f0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -0002e400: 362e 3035 2b64 732f 4d32 2f4d 6163 6175 6.05+ds/M2/Macau │ │ │ │ -0002e410: 6c61 7932 2f70 6163 6b61 6765 732f 0a47 lay2/packages/.G │ │ │ │ -0002e420: 726f 6562 6e65 7253 7472 6174 612e 6d32 roebnerStrata.m2 │ │ │ │ -0002e430: 3a38 3834 3a30 2e0a 1f0a 4669 6c65 3a20 :884:0....File: │ │ │ │ -0002e440: 4772 6f65 626e 6572 5374 7261 7461 2e69 GroebnerStrata.i │ │ │ │ -0002e450: 6e66 6f2c 204e 6f64 653a 2073 6d61 6c6c nfo, Node: small │ │ │ │ -0002e460: 6572 4d6f 6e6f 6d69 616c 732c 204e 6578 erMonomials, Nex │ │ │ │ -0002e470: 743a 2073 7461 6e64 6172 644d 6f6e 6f6d t: standardMonom │ │ │ │ -0002e480: 6961 6c73 2c20 5072 6576 3a20 7261 6e64 ials, Prev: rand │ │ │ │ -0002e490: 6f6d 506f 696e 7473 4f6e 5261 7469 6f6e omPointsOnRation │ │ │ │ -0002e4a0: 616c 5661 7269 6574 795f 6c70 4964 6561 alVariety_lpIdea │ │ │ │ -0002e4b0: 6c5f 636d 5a5a 5f72 702c 2055 703a 2054 l_cmZZ_rp, Up: T │ │ │ │ -0002e4c0: 6f70 0a0a 736d 616c 6c65 724d 6f6e 6f6d op..smallerMonom │ │ │ │ -0002e4d0: 6961 6c73 202d 2d20 7265 7475 726e 7320 ials -- returns │ │ │ │ -0002e4e0: 7468 6520 7374 616e 6461 7264 206d 6f6e the standard mon │ │ │ │ -0002e4f0: 6f6d 6961 6c73 2073 6d61 6c6c 6572 2062 omials smaller b │ │ │ │ -0002e500: 7574 206f 6620 7468 6520 7361 6d65 2064 ut of the same d │ │ │ │ -0002e510: 6567 7265 6520 6173 2067 6976 656e 206d egree as given m │ │ │ │ -0002e520: 6f6e 6f6d 6961 6c28 7329 0a2a 2a2a 2a2a onomial(s).***** │ │ │ │ -0002e530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002e540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002e550: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002e560: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002e570: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002e580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002e590: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ -0002e5a0: 2020 2020 2020 204c 203d 2073 6d61 6c6c L = small │ │ │ │ -0002e5b0: 6572 4d6f 6e6f 6d69 616c 7320 4d0a 2020 erMonomials M. │ │ │ │ -0002e5c0: 2020 2020 2020 4c20 3d20 736d 616c 6c65 L = smalle │ │ │ │ -0002e5d0: 724d 6f6e 6f6d 6961 6c73 284d 2c20 6d29 rMonomials(M, m) │ │ │ │ -0002e5e0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ -0002e5f0: 2020 202a 204d 2c20 616e 202a 6e6f 7465 * M, an *note │ │ │ │ -0002e600: 2069 6465 616c 3a20 284d 6163 6175 6c61 ideal: (Macaula │ │ │ │ -0002e610: 7932 446f 6329 4964 6561 6c2c 2c20 244d y2Doc)Ideal,, $M │ │ │ │ -0002e620: 2420 7368 6f75 6c64 2062 6520 6120 6d6f $ should be a mo │ │ │ │ -0002e630: 6e6f 6d69 616c 2069 6465 616c 0a20 2020 nomial ideal. │ │ │ │ -0002e640: 2020 2020 2028 616e 2069 6465 616c 2067 (an ideal g │ │ │ │ -0002e650: 656e 6572 6174 6564 2062 7920 6d6f 6e6f enerated by mono │ │ │ │ -0002e660: 6d69 616c 7329 0a20 2020 2020 202a 206d mials). * m │ │ │ │ -0002e670: 2c20 6120 2a6e 6f74 6520 7269 6e67 2065 , a *note ring e │ │ │ │ -0002e680: 6c65 6d65 6e74 3a20 284d 6163 6175 6c61 lement: (Macaula │ │ │ │ -0002e690: 7932 446f 6329 5269 6e67 456c 656d 656e y2Doc)RingElemen │ │ │ │ -0002e6a0: 742c 2c20 6f70 7469 6f6e 616c 2c0a 2020 t,, optional,. │ │ │ │ -0002e6b0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ -0002e6c0: 202a 204c 2c20 6120 2a6e 6f74 6520 6c69 * L, a *note li │ │ │ │ -0002e6d0: 7374 3a20 284d 6163 6175 6c61 7932 446f st: (Macaulay2Do │ │ │ │ -0002e6e0: 6329 4c69 7374 2c2c 2061 206c 6973 7420 c)List,, a list │ │ │ │ -0002e6f0: 6f66 206c 6973 7473 3a20 666f 7220 6561 of lists: for ea │ │ │ │ -0002e700: 6368 0a20 2020 2020 2020 2067 656e 6572 ch. gener │ │ │ │ -0002e710: 6174 6f72 2024 6d24 206f 6620 244d 242c ator $m$ of $M$, │ │ │ │ -0002e720: 2074 6865 206c 6973 7420 6f66 2061 6c6c the list of all │ │ │ │ -0002e730: 206d 6f6e 6f6d 6961 6c73 206f 6620 7468 monomials of th │ │ │ │ -0002e740: 6520 7361 6d65 2064 6567 7265 6520 6173 e same degree as │ │ │ │ -0002e750: 0a20 2020 2020 2020 2024 6d24 2c20 6e6f . $m$, no │ │ │ │ -0002e760: 7420 696e 2074 6865 206d 6f6e 6f6d 6961 t in the monomia │ │ │ │ -0002e770: 6c20 6964 6561 6c20 616e 6420 736d 616c l ideal and smal │ │ │ │ -0002e780: 6c65 7220 7468 616e 2074 6861 7420 6765 ler than that ge │ │ │ │ -0002e790: 6e65 7261 746f 7220 696e 2074 6865 0a20 nerator in the. │ │ │ │ -0002e7a0: 2020 2020 2020 2074 6572 6d20 6f72 6465 term orde │ │ │ │ -0002e7b0: 7220 6f66 2074 6865 2061 6d62 6965 6e74 r of the ambient │ │ │ │ -0002e7c0: 2072 696e 672e 2020 4966 2069 6e73 7465 ring. If inste │ │ │ │ -0002e7d0: 6164 2024 6d24 2069 7320 6769 7665 6e2c ad $m$ is given, │ │ │ │ -0002e7e0: 2074 6865 206c 6973 7420 6f66 0a20 2020 the list of. │ │ │ │ -0002e7f0: 2020 2020 2074 6865 2073 7461 6e64 6172 the standar │ │ │ │ -0002e800: 6420 6d6f 6e6f 6d69 616c 7320 6f66 2074 d monomials of t │ │ │ │ -0002e810: 6865 2073 616d 6520 6465 6772 6565 2c20 he same degree, │ │ │ │ -0002e820: 736d 616c 6c65 7220 7468 616e 2024 6d24 smaller than $m$ │ │ │ │ -0002e830: 2c20 6973 0a20 2020 2020 2020 2072 6574 , is. ret │ │ │ │ -0002e840: 7572 6e65 642e 0a0a 4465 7363 7269 7074 urned...Descript │ │ │ │ -0002e850: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -0002e860: 0a49 6e70 7574 7469 6e67 2061 6e20 6964 .Inputting an id │ │ │ │ -0002e870: 6561 6c20 244d 2420 7265 7475 726e 7320 eal $M$ returns │ │ │ │ -0002e880: 7468 6520 736d 616c 6c65 7220 6d6f 6e6f the smaller mono │ │ │ │ -0002e890: 6d69 616c 7320 6f66 2065 6163 6820 6f66 mials of each of │ │ │ │ -0002e8a0: 2074 6865 2067 6976 656e 0a67 656e 6572 the given.gener │ │ │ │ -0002e8b0: 6174 6f72 7320 6f66 2074 6865 2069 6465 ators of the ide │ │ │ │ -0002e8c0: 616c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d al...+---------- │ │ │ │ -0002e8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e910: 2d2d 2d2b 0a7c 6931 203a 2052 203d 205a ---+.|i1 : R = Z │ │ │ │ -0002e920: 5a2f 3332 3030 335b 612e 2e64 5d3b 2020 Z/32003[a..d]; │ │ │ │ -0002e930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e960: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0002e970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e9b0: 2d2d 2d2b 0a7c 6932 203a 204d 203d 2069 ---+.|i2 : M = i │ │ │ │ -0002e9c0: 6465 616c 2028 615e 322c 2062 5e32 2c20 deal (a^2, b^2, │ │ │ │ -0002e9d0: 612a 622a 6329 3b20 2020 2020 2020 2020 a*b*c); │ │ │ │ -0002e9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea00: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002ea10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea50: 2020 207c 0a7c 6f32 203a 2049 6465 616c |.|o2 : Ideal │ │ │ │ -0002ea60: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ -0002ea70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eaa0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0002eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ead0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eaf0: 2d2d 2d2b 0a7c 6933 203a 204c 3120 3d20 ---+.|i3 : L1 = │ │ │ │ -0002eb00: 736d 616c 6c65 724d 6f6e 6f6d 6961 6c73 smallerMonomials │ │ │ │ -0002eb10: 204d 2020 2020 2020 2020 2020 2020 2020 M │ │ │ │ -0002eb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb40: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002eb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb90: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002eba0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ -0002ebb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ebc0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002ebd0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002ebe0: 2020 207c 0a7c 6f33 203d 207b 7b61 2a62 |.|o3 = {{a*b │ │ │ │ -0002ebf0: 2c20 612a 632c 2062 2a63 2c20 6320 2c20 , a*c, b*c, c , │ │ │ │ -0002ec00: 612a 642c 2062 2a64 2c20 632a 642c 2064 a*d, b*d, c*d, d │ │ │ │ -0002ec10: 207d 2c20 7b61 2a63 2c20 622a 632c 2063 }, {a*c, b*c, c │ │ │ │ -0002ec20: 202c 2061 2a64 2c20 622a 642c 2063 2a64 , a*d, b*d, c*d │ │ │ │ -0002ec30: 2c20 207c 0a7c 2020 2020 202d 2d2d 2d2d , |.| ----- │ │ │ │ -0002ec40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ec50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ec60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ec70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ec80: 2d2d 2d7c 0a7c 2020 2020 2020 3220 2020 ---|.| 2 │ │ │ │ -0002ec90: 2020 2020 3220 2020 2020 3220 2020 3320 2 2 3 │ │ │ │ -0002eca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ecb0: 2020 2020 2020 2032 2020 2020 2020 3220 2 2 │ │ │ │ -0002ecc0: 2020 2020 3220 2020 2020 3220 2020 3320 2 2 3 │ │ │ │ -0002ecd0: 2020 207c 0a7c 2020 2020 2064 207d 2c20 |.| d }, │ │ │ │ -0002ece0: 7b61 2a63 202c 2062 2a63 202c 2063 202c {a*c , b*c , c , │ │ │ │ -0002ecf0: 2061 2a62 2a64 2c20 612a 632a 642c 2062 a*b*d, a*c*d, b │ │ │ │ -0002ed00: 2a63 2a64 2c20 6320 642c 2061 2a64 202c *c*d, c d, a*d , │ │ │ │ -0002ed10: 2062 2a64 202c 2063 2a64 202c 2064 207d b*d , c*d , d } │ │ │ │ -0002ed20: 7d20 207c 0a7c 2020 2020 2020 2020 2020 } |.| │ │ │ │ -0002ed30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed70: 2020 207c 0a7c 6f33 203a 204c 6973 7420 |.|o3 : List │ │ │ │ -0002ed80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e120: 207c 0a7c 202d 3436 2032 3320 3020 7c20 |.| -46 23 0 | │ │ │ │ +0002e130: 2020 2020 2020 7c20 2020 2020 2020 2020 | │ │ │ │ +0002e140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e170: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ +0002e180: 2d2d 2d2d 2d2d 2b20 2020 2020 2020 2020 ------+ │ │ │ │ +0002e190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e1c0: 207c 0a7c 3720 3020 7c20 2020 2020 2020 |.|7 0 | │ │ │ │ +0002e1d0: 2020 2020 2020 7c20 2020 2020 2020 2020 | │ │ │ │ +0002e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e210: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ +0002e220: 2d2d 2d2d 2d2d 2b20 2020 2020 2020 2020 ------+ │ │ │ │ +0002e230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e260: 207c 0a7c 3720 2d32 3320 3139 2030 207c |.|7 -23 19 0 | │ │ │ │ +0002e270: 2020 2020 2020 7c20 2020 2020 2020 2020 | │ │ │ │ +0002e280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e2b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ +0002e2c0: 2d2d 2d2d 2d2d 2b20 2020 2020 2020 2020 ------+ │ │ │ │ +0002e2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e300: 207c 0a7c 3820 3437 202d 3333 2030 207c |.|8 47 -33 0 | │ │ │ │ +0002e310: 2020 2020 2020 7c20 2020 2020 2020 2020 | │ │ │ │ +0002e320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e350: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ +0002e360: 2d2d 2d2d 2d2d 2b20 2020 2020 2020 2020 ------+ │ │ │ │ +0002e370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3a0: 207c 0a7c 2035 2034 3220 3020 7c20 2020 |.| 5 42 0 | │ │ │ │ +0002e3b0: 2020 2020 2020 7c20 2020 2020 2020 2020 | │ │ │ │ +0002e3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ +0002e400: 2d2d 2d2d 2d2d 2b20 2020 2020 2020 2020 ------+ │ │ │ │ +0002e410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e440: 207c 0a7c 3220 2d32 3920 3020 7c20 2020 |.|2 -29 0 | │ │ │ │ +0002e450: 2020 2020 2020 7c20 2020 2020 2020 2020 | │ │ │ │ +0002e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e490: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ +0002e4a0: 2d2d 2d2d 2d2d 2b20 2020 2020 2020 2020 ------+ │ │ │ │ +0002e4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e4e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e530: 2d2b 0a0a 4361 7665 6174 0a3d 3d3d 3d3d -+..Caveat.===== │ │ │ │ +0002e540: 3d0a 0a54 6869 7320 726f 7574 696e 6520 =..This routine │ │ │ │ +0002e550: 6578 7065 6374 7320 7468 6520 696e 7075 expects the inpu │ │ │ │ +0002e560: 7420 746f 2072 6570 7265 7365 6e74 2061 t to represent a │ │ │ │ +0002e570: 6e20 6972 7265 6475 6369 626c 6520 7661 n irreducible va │ │ │ │ +0002e580: 7269 6574 790a 0a53 6565 2061 6c73 6f0a riety..See also. │ │ │ │ +0002e590: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +0002e5a0: 6f74 6520 7261 6e64 6f6d 506f 696e 744f ote randomPointO │ │ │ │ +0002e5b0: 6e52 6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ +0002e5c0: 2849 6465 616c 293a 0a20 2020 2072 616e (Ideal):. ran │ │ │ │ +0002e5d0: 646f 6d50 6f69 6e74 4f6e 5261 7469 6f6e domPointOnRation │ │ │ │ +0002e5e0: 616c 5661 7269 6574 795f 6c70 4964 6561 alVariety_lpIdea │ │ │ │ +0002e5f0: 6c5f 7270 2c20 2d2d 2066 696e 6420 6120 l_rp, -- find a │ │ │ │ +0002e600: 7261 6e64 6f6d 2070 6f69 6e74 206f 6e20 random point on │ │ │ │ +0002e610: 610a 2020 2020 7661 7269 6574 7920 7468 a. variety th │ │ │ │ +0002e620: 6174 2063 616e 2062 6520 6465 7465 6374 at can be detect │ │ │ │ +0002e630: 6564 2074 6f20 6265 2072 6174 696f 6e61 ed to be rationa │ │ │ │ +0002e640: 6c0a 0a57 6179 7320 746f 2075 7365 2074 l..Ways to use t │ │ │ │ +0002e650: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d his method:.==== │ │ │ │ +0002e660: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002e670: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ +0002e680: 7261 6e64 6f6d 506f 696e 7473 4f6e 5261 randomPointsOnRa │ │ │ │ +0002e690: 7469 6f6e 616c 5661 7269 6574 7928 4964 tionalVariety(Id │ │ │ │ +0002e6a0: 6561 6c2c 5a5a 293a 0a20 2020 2072 616e eal,ZZ):. ran │ │ │ │ +0002e6b0: 646f 6d50 6f69 6e74 734f 6e52 6174 696f domPointsOnRatio │ │ │ │ +0002e6c0: 6e61 6c56 6172 6965 7479 5f6c 7049 6465 nalVariety_lpIde │ │ │ │ +0002e6d0: 616c 5f63 6d5a 5a5f 7270 2c20 2d2d 2066 al_cmZZ_rp, -- f │ │ │ │ +0002e6e0: 696e 6420 7261 6e64 6f6d 2070 6f69 6e74 ind random point │ │ │ │ +0002e6f0: 7320 6f6e 2061 0a20 2020 2076 6172 6965 s on a. varie │ │ │ │ +0002e700: 7479 2074 6861 7420 6361 6e20 6265 2064 ty that can be d │ │ │ │ +0002e710: 6574 6563 7465 6420 746f 2062 6520 7261 etected to be ra │ │ │ │ +0002e720: 7469 6f6e 616c 0a2d 2d2d 2d2d 2d2d 2d2d tional.--------- │ │ │ │ +0002e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e770: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +0002e780: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +0002e790: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +0002e7a0: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +0002e7b0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +0002e7c0: 362e 3035 2b64 732f 4d32 2f4d 6163 6175 6.05+ds/M2/Macau │ │ │ │ +0002e7d0: 6c61 7932 2f70 6163 6b61 6765 732f 0a47 lay2/packages/.G │ │ │ │ +0002e7e0: 726f 6562 6e65 7253 7472 6174 612e 6d32 roebnerStrata.m2 │ │ │ │ +0002e7f0: 3a38 3834 3a30 2e0a 1f0a 4669 6c65 3a20 :884:0....File: │ │ │ │ +0002e800: 4772 6f65 626e 6572 5374 7261 7461 2e69 GroebnerStrata.i │ │ │ │ +0002e810: 6e66 6f2c 204e 6f64 653a 2073 6d61 6c6c nfo, Node: small │ │ │ │ +0002e820: 6572 4d6f 6e6f 6d69 616c 732c 204e 6578 erMonomials, Nex │ │ │ │ +0002e830: 743a 2073 7461 6e64 6172 644d 6f6e 6f6d t: standardMonom │ │ │ │ +0002e840: 6961 6c73 2c20 5072 6576 3a20 7261 6e64 ials, Prev: rand │ │ │ │ +0002e850: 6f6d 506f 696e 7473 4f6e 5261 7469 6f6e omPointsOnRation │ │ │ │ +0002e860: 616c 5661 7269 6574 795f 6c70 4964 6561 alVariety_lpIdea │ │ │ │ +0002e870: 6c5f 636d 5a5a 5f72 702c 2055 703a 2054 l_cmZZ_rp, Up: T │ │ │ │ +0002e880: 6f70 0a0a 736d 616c 6c65 724d 6f6e 6f6d op..smallerMonom │ │ │ │ +0002e890: 6961 6c73 202d 2d20 7265 7475 726e 7320 ials -- returns │ │ │ │ +0002e8a0: 7468 6520 7374 616e 6461 7264 206d 6f6e the standard mon │ │ │ │ +0002e8b0: 6f6d 6961 6c73 2073 6d61 6c6c 6572 2062 omials smaller b │ │ │ │ +0002e8c0: 7574 206f 6620 7468 6520 7361 6d65 2064 ut of the same d │ │ │ │ +0002e8d0: 6567 7265 6520 6173 2067 6976 656e 206d egree as given m │ │ │ │ +0002e8e0: 6f6e 6f6d 6961 6c28 7329 0a2a 2a2a 2a2a onomial(s).***** │ │ │ │ +0002e8f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002e900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002e910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002e920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002e930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002e940: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002e950: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +0002e960: 2020 2020 2020 204c 203d 2073 6d61 6c6c L = small │ │ │ │ +0002e970: 6572 4d6f 6e6f 6d69 616c 7320 4d0a 2020 erMonomials M. │ │ │ │ +0002e980: 2020 2020 2020 4c20 3d20 736d 616c 6c65 L = smalle │ │ │ │ +0002e990: 724d 6f6e 6f6d 6961 6c73 284d 2c20 6d29 rMonomials(M, m) │ │ │ │ +0002e9a0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ +0002e9b0: 2020 202a 204d 2c20 616e 202a 6e6f 7465 * M, an *note │ │ │ │ +0002e9c0: 2069 6465 616c 3a20 284d 6163 6175 6c61 ideal: (Macaula │ │ │ │ +0002e9d0: 7932 446f 6329 4964 6561 6c2c 2c20 244d y2Doc)Ideal,, $M │ │ │ │ +0002e9e0: 2420 7368 6f75 6c64 2062 6520 6120 6d6f $ should be a mo │ │ │ │ +0002e9f0: 6e6f 6d69 616c 2069 6465 616c 0a20 2020 nomial ideal. │ │ │ │ +0002ea00: 2020 2020 2028 616e 2069 6465 616c 2067 (an ideal g │ │ │ │ +0002ea10: 656e 6572 6174 6564 2062 7920 6d6f 6e6f enerated by mono │ │ │ │ +0002ea20: 6d69 616c 7329 0a20 2020 2020 202a 206d mials). * m │ │ │ │ +0002ea30: 2c20 6120 2a6e 6f74 6520 7269 6e67 2065 , a *note ring e │ │ │ │ +0002ea40: 6c65 6d65 6e74 3a20 284d 6163 6175 6c61 lement: (Macaula │ │ │ │ +0002ea50: 7932 446f 6329 5269 6e67 456c 656d 656e y2Doc)RingElemen │ │ │ │ +0002ea60: 742c 2c20 6f70 7469 6f6e 616c 2c0a 2020 t,, optional,. │ │ │ │ +0002ea70: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +0002ea80: 202a 204c 2c20 6120 2a6e 6f74 6520 6c69 * L, a *note li │ │ │ │ +0002ea90: 7374 3a20 284d 6163 6175 6c61 7932 446f st: (Macaulay2Do │ │ │ │ +0002eaa0: 6329 4c69 7374 2c2c 2061 206c 6973 7420 c)List,, a list │ │ │ │ +0002eab0: 6f66 206c 6973 7473 3a20 666f 7220 6561 of lists: for ea │ │ │ │ +0002eac0: 6368 0a20 2020 2020 2020 2067 656e 6572 ch. gener │ │ │ │ +0002ead0: 6174 6f72 2024 6d24 206f 6620 244d 242c ator $m$ of $M$, │ │ │ │ +0002eae0: 2074 6865 206c 6973 7420 6f66 2061 6c6c the list of all │ │ │ │ +0002eaf0: 206d 6f6e 6f6d 6961 6c73 206f 6620 7468 monomials of th │ │ │ │ +0002eb00: 6520 7361 6d65 2064 6567 7265 6520 6173 e same degree as │ │ │ │ +0002eb10: 0a20 2020 2020 2020 2024 6d24 2c20 6e6f . $m$, no │ │ │ │ +0002eb20: 7420 696e 2074 6865 206d 6f6e 6f6d 6961 t in the monomia │ │ │ │ +0002eb30: 6c20 6964 6561 6c20 616e 6420 736d 616c l ideal and smal │ │ │ │ +0002eb40: 6c65 7220 7468 616e 2074 6861 7420 6765 ler than that ge │ │ │ │ +0002eb50: 6e65 7261 746f 7220 696e 2074 6865 0a20 nerator in the. │ │ │ │ +0002eb60: 2020 2020 2020 2074 6572 6d20 6f72 6465 term orde │ │ │ │ +0002eb70: 7220 6f66 2074 6865 2061 6d62 6965 6e74 r of the ambient │ │ │ │ +0002eb80: 2072 696e 672e 2020 4966 2069 6e73 7465 ring. If inste │ │ │ │ +0002eb90: 6164 2024 6d24 2069 7320 6769 7665 6e2c ad $m$ is given, │ │ │ │ +0002eba0: 2074 6865 206c 6973 7420 6f66 0a20 2020 the list of. │ │ │ │ +0002ebb0: 2020 2020 2074 6865 2073 7461 6e64 6172 the standar │ │ │ │ +0002ebc0: 6420 6d6f 6e6f 6d69 616c 7320 6f66 2074 d monomials of t │ │ │ │ +0002ebd0: 6865 2073 616d 6520 6465 6772 6565 2c20 he same degree, │ │ │ │ +0002ebe0: 736d 616c 6c65 7220 7468 616e 2024 6d24 smaller than $m$ │ │ │ │ +0002ebf0: 2c20 6973 0a20 2020 2020 2020 2072 6574 , is. ret │ │ │ │ +0002ec00: 7572 6e65 642e 0a0a 4465 7363 7269 7074 urned...Descript │ │ │ │ +0002ec10: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +0002ec20: 0a49 6e70 7574 7469 6e67 2061 6e20 6964 .Inputting an id │ │ │ │ +0002ec30: 6561 6c20 244d 2420 7265 7475 726e 7320 eal $M$ returns │ │ │ │ +0002ec40: 7468 6520 736d 616c 6c65 7220 6d6f 6e6f the smaller mono │ │ │ │ +0002ec50: 6d69 616c 7320 6f66 2065 6163 6820 6f66 mials of each of │ │ │ │ +0002ec60: 2074 6865 2067 6976 656e 0a67 656e 6572 the given.gener │ │ │ │ +0002ec70: 6174 6f72 7320 6f66 2074 6865 2069 6465 ators of the ide │ │ │ │ +0002ec80: 616c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d al...+---------- │ │ │ │ +0002ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ecc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ecd0: 2d2d 2d2b 0a7c 6931 203a 2052 203d 205a ---+.|i1 : R = Z │ │ │ │ +0002ece0: 5a2f 3332 3030 335b 612e 2e64 5d3b 2020 Z/32003[a..d]; │ │ │ │ +0002ecf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ed00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ed10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ed20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002ed30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ed40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ed50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ed60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ed70: 2d2d 2d2b 0a7c 6932 203a 204d 203d 2069 ---+.|i2 : M = i │ │ │ │ +0002ed80: 6465 616c 2028 615e 322c 2062 5e32 2c20 deal (a^2, b^2, │ │ │ │ +0002ed90: 612a 622a 6329 3b20 2020 2020 2020 2020 a*b*c); │ │ │ │ 0002eda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002edb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002edc0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0002edd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ede0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002edf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ee00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ee10: 2d2d 2d2b 0a7c 6934 203a 2073 6d61 6c6c ---+.|i4 : small │ │ │ │ -0002ee20: 6572 4d6f 6e6f 6d69 616c 7328 4d2c 2062 erMonomials(M, b │ │ │ │ -0002ee30: 5e32 2920 2020 2020 2020 2020 2020 2020 ^2) │ │ │ │ +0002edc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002edd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ede0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002edf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ee00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ee10: 2020 207c 0a7c 6f32 203a 2049 6465 616c |.|o2 : Ideal │ │ │ │ +0002ee20: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ +0002ee30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ee40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ee50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee60: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002ee70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eeb0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002eec0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -0002eed0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0002ee60: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ee90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002eeb0: 2d2d 2d2b 0a7c 6933 203a 204c 3120 3d20 ---+.|i3 : L1 = │ │ │ │ +0002eec0: 736d 616c 6c65 724d 6f6e 6f6d 6961 6c73 smallerMonomials │ │ │ │ +0002eed0: 204d 2020 2020 2020 2020 2020 2020 2020 M │ │ │ │ 0002eee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002eef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ef00: 2020 207c 0a7c 6f34 203d 207b 612a 632c |.|o4 = {a*c, │ │ │ │ -0002ef10: 2062 2a63 2c20 6320 2c20 612a 642c 2062 b*c, c , a*d, b │ │ │ │ -0002ef20: 2a64 2c20 632a 642c 2064 207d 2020 2020 *d, c*d, d } │ │ │ │ +0002ef00: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002ef10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ef20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ef30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ef40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ef50: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002ef60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ef60: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ 0002ef70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ef80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002efa0: 2020 207c 0a7c 6f34 203a 204c 6973 7420 |.|o4 : List │ │ │ │ -0002efb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002efc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002efd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002efe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eff0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002ef80: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002ef90: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002efa0: 2020 207c 0a7c 6f33 203d 207b 7b61 2a62 |.|o3 = {{a*b │ │ │ │ +0002efb0: 2c20 612a 632c 2062 2a63 2c20 6320 2c20 , a*c, b*c, c , │ │ │ │ +0002efc0: 612a 642c 2062 2a64 2c20 632a 642c 2064 a*d, b*d, c*d, d │ │ │ │ +0002efd0: 207d 2c20 7b61 2a63 2c20 622a 632c 2063 }, {a*c, b*c, c │ │ │ │ +0002efe0: 202c 2061 2a64 2c20 622a 642c 2063 2a64 , a*d, b*d, c*d │ │ │ │ +0002eff0: 2c20 207c 0a7c 2020 2020 202d 2d2d 2d2d , |.| ----- │ │ │ │ 0002f000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f040: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ -0002f050: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ -0002f060: 7465 2074 6169 6c4d 6f6e 6f6d 6961 6c73 te tailMonomials │ │ │ │ -0002f070: 3a20 7461 696c 4d6f 6e6f 6d69 616c 732c : tailMonomials, │ │ │ │ -0002f080: 202d 2d20 6669 6e64 2074 6169 6c20 6d6f -- find tail mo │ │ │ │ -0002f090: 6e6f 6d69 616c 730a 2020 2a20 2a6e 6f74 nomials. * *not │ │ │ │ -0002f0a0: 6520 7374 616e 6461 7264 4d6f 6e6f 6d69 e standardMonomi │ │ │ │ -0002f0b0: 616c 733a 2073 7461 6e64 6172 644d 6f6e als: standardMon │ │ │ │ -0002f0c0: 6f6d 6961 6c73 2c20 2d2d 2063 6f6d 7075 omials, -- compu │ │ │ │ -0002f0d0: 7465 7320 7374 616e 6461 7264 206d 6f6e tes standard mon │ │ │ │ -0002f0e0: 6f6d 6961 6c73 0a0a 5761 7973 2074 6f20 omials..Ways to │ │ │ │ -0002f0f0: 7573 6520 736d 616c 6c65 724d 6f6e 6f6d use smallerMonom │ │ │ │ -0002f100: 6961 6c73 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d ials:.========== │ │ │ │ -0002f110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002f120: 3d3d 3d0a 0a20 202a 2022 736d 616c 6c65 ===.. * "smalle │ │ │ │ -0002f130: 724d 6f6e 6f6d 6961 6c73 2849 6465 616c rMonomials(Ideal │ │ │ │ -0002f140: 2922 0a20 202a 2022 736d 616c 6c65 724d )". * "smallerM │ │ │ │ -0002f150: 6f6e 6f6d 6961 6c73 2849 6465 616c 2c52 onomials(Ideal,R │ │ │ │ -0002f160: 696e 6745 6c65 6d65 6e74 2922 0a0a 466f ingElement)"..Fo │ │ │ │ -0002f170: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -0002f180: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0002f190: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -0002f1a0: 2a6e 6f74 6520 736d 616c 6c65 724d 6f6e *note smallerMon │ │ │ │ -0002f1b0: 6f6d 6961 6c73 3a20 736d 616c 6c65 724d omials: smallerM │ │ │ │ -0002f1c0: 6f6e 6f6d 6961 6c73 2c20 6973 2061 202a onomials, is a * │ │ │ │ -0002f1d0: 6e6f 7465 206d 6574 686f 640a 6675 6e63 note method.func │ │ │ │ -0002f1e0: 7469 6f6e 3a20 284d 6163 6175 6c61 7932 tion: (Macaulay2 │ │ │ │ -0002f1f0: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ -0002f200: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ -0002f210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f250: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -0002f260: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -0002f270: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -0002f280: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -0002f290: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -0002f2a0: 2e30 352b 6473 2f4d 322f 4d61 6361 756c .05+ds/M2/Macaul │ │ │ │ -0002f2b0: 6179 322f 7061 636b 6167 6573 2f0a 4772 ay2/packages/.Gr │ │ │ │ -0002f2c0: 6f65 626e 6572 5374 7261 7461 2e6d 323a oebnerStrata.m2: │ │ │ │ -0002f2d0: 3532 353a 302e 0a1f 0a46 696c 653a 2047 525:0....File: G │ │ │ │ -0002f2e0: 726f 6562 6e65 7253 7472 6174 612e 696e roebnerStrata.in │ │ │ │ -0002f2f0: 666f 2c20 4e6f 6465 3a20 7374 616e 6461 fo, Node: standa │ │ │ │ -0002f300: 7264 4d6f 6e6f 6d69 616c 732c 204e 6578 rdMonomials, Nex │ │ │ │ -0002f310: 743a 2074 6169 6c4d 6f6e 6f6d 6961 6c73 t: tailMonomials │ │ │ │ -0002f320: 2c20 5072 6576 3a20 736d 616c 6c65 724d , Prev: smallerM │ │ │ │ -0002f330: 6f6e 6f6d 6961 6c73 2c20 5570 3a20 546f onomials, Up: To │ │ │ │ -0002f340: 700a 0a73 7461 6e64 6172 644d 6f6e 6f6d p..standardMonom │ │ │ │ -0002f350: 6961 6c73 202d 2d20 636f 6d70 7574 6573 ials -- computes │ │ │ │ -0002f360: 2073 7461 6e64 6172 6420 6d6f 6e6f 6d69 standard monomi │ │ │ │ -0002f370: 616c 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a als.************ │ │ │ │ -0002f380: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002f390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002f3a0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -0002f3b0: 200a 2020 2020 2020 2020 4c20 3d20 7374 . L = st │ │ │ │ -0002f3c0: 616e 6461 7264 4d6f 6e6f 6d69 616c 7320 andardMonomials │ │ │ │ -0002f3d0: 4d0a 2020 2020 2020 2020 4c20 3d20 7374 M. L = st │ │ │ │ -0002f3e0: 616e 6461 7264 4d6f 6e6f 6d69 616c 7328 andardMonomials( │ │ │ │ -0002f3f0: 642c 204d 290a 2020 2a20 496e 7075 7473 d, M). * Inputs │ │ │ │ -0002f400: 3a0a 2020 2020 2020 2a20 4d2c 2061 6e20 :. * M, an │ │ │ │ -0002f410: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61 *note ideal: (Ma │ │ │ │ -0002f420: 6361 756c 6179 3244 6f63 2949 6465 616c caulay2Doc)Ideal │ │ │ │ -0002f430: 2c2c 204d 2073 686f 756c 6420 6265 2061 ,, M should be a │ │ │ │ -0002f440: 206d 6f6e 6f6d 6961 6c20 6964 6561 6c0a monomial ideal. │ │ │ │ -0002f450: 2020 2020 2020 2a20 642c 2061 202a 6e6f * d, a *no │ │ │ │ -0002f460: 7465 206c 6973 743a 2028 4d61 6361 756c te list: (Macaul │ │ │ │ -0002f470: 6179 3244 6f63 294c 6973 742c 2c20 6120 ay2Doc)List,, a │ │ │ │ -0002f480: 6465 6772 6565 0a20 202a 204f 7574 7075 degree. * Outpu │ │ │ │ -0002f490: 7473 3a0a 2020 2020 2020 2a20 4c2c 2061 ts:. * L, a │ │ │ │ -0002f4a0: 202a 6e6f 7465 206c 6973 743a 2028 4d61 *note list: (Ma │ │ │ │ -0002f4b0: 6361 756c 6179 3244 6f63 294c 6973 742c caulay2Doc)List, │ │ │ │ -0002f4c0: 2c20 4c20 6973 2061 206c 6973 7420 6f66 , L is a list of │ │ │ │ -0002f4d0: 206c 6973 7473 206f 6620 7374 616e 6461 lists of standa │ │ │ │ -0002f4e0: 7264 0a20 2020 2020 2020 206d 6f6e 6f6d rd. monom │ │ │ │ -0002f4f0: 6961 6c73 2066 6f72 2074 6865 2069 6465 ials for the ide │ │ │ │ -0002f500: 616c 2024 4d24 2c20 6f6e 6520 666f 7220 al $M$, one for │ │ │ │ -0002f510: 6561 6368 2067 656e 6572 6174 6f72 206f each generator o │ │ │ │ -0002f520: 6620 244d 240a 0a44 6573 6372 6970 7469 f $M$..Descripti │ │ │ │ -0002f530: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -0002f540: 4120 6d6f 6e6f 6d69 616c 2024 6d24 2069 A monomial $m$ i │ │ │ │ -0002f550: 7320 7374 616e 6461 7264 2077 6974 6820 s standard with │ │ │ │ -0002f560: 7265 7370 6563 7420 746f 2061 206d 6f6e respect to a mon │ │ │ │ -0002f570: 6f6d 6961 6c20 6964 6561 6c20 244d 2420 omial ideal $M$ │ │ │ │ -0002f580: 616e 6420 6120 6765 6e65 7261 746f 720a and a generator. │ │ │ │ -0002f590: 2467 2420 6f66 2024 4d24 2069 6620 246d $g$ of $M$ if $m │ │ │ │ -0002f5a0: 2420 6973 206f 6620 7468 6520 7361 6d65 $ is of the same │ │ │ │ -0002f5b0: 2064 6567 7265 6520 6173 2024 6724 2062 degree as $g$ b │ │ │ │ -0002f5c0: 7574 2069 7320 6e6f 7420 616e 2065 6c65 ut is not an ele │ │ │ │ -0002f5d0: 6d65 6e74 206f 6620 244d 242e 0a0a 496e ment of $M$...In │ │ │ │ -0002f5e0: 7075 7474 696e 6720 616e 2069 6465 616c putting an ideal │ │ │ │ -0002f5f0: 2024 4d24 2072 6574 7572 6e73 2074 6865 $M$ returns the │ │ │ │ -0002f600: 2073 7461 6e64 6172 6420 6d6f 6e6f 6d69 standard monomi │ │ │ │ -0002f610: 616c 7320 6f66 2065 6163 6820 6f66 2074 als of each of t │ │ │ │ -0002f620: 6865 2067 6976 656e 0a67 656e 6572 6174 he given.generat │ │ │ │ -0002f630: 6f72 7320 6f66 2074 6865 2069 6465 616c ors of the ideal │ │ │ │ -0002f640: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ -0002f650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f690: 2d2b 0a7c 6931 203a 2052 203d 205a 5a2f -+.|i1 : R = ZZ/ │ │ │ │ -0002f6a0: 3332 3030 335b 612e 2e64 5d3b 2020 2020 32003[a..d]; │ │ │ │ -0002f6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f6e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f730: 2d2b 0a7c 6932 203a 204d 203d 2069 6465 -+.|i2 : M = ide │ │ │ │ -0002f740: 616c 2028 615e 322c 2061 2a62 2c20 625e al (a^2, a*b, b^ │ │ │ │ -0002f750: 332c 2061 2a63 293b 2020 2020 2020 2020 3, a*c); │ │ │ │ -0002f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f780: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f7d0: 207c 0a7c 6f32 203a 2049 6465 616c 206f |.|o2 : Ideal o │ │ │ │ -0002f7e0: 6620 5220 2020 2020 2020 2020 2020 2020 f R │ │ │ │ -0002f7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f820: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002f830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f870: 2d2b 0a7c 6933 203a 204c 3120 3d20 7374 -+.|i3 : L1 = st │ │ │ │ -0002f880: 616e 6461 7264 4d6f 6e6f 6d69 616c 7320 andardMonomials │ │ │ │ -0002f890: 4d20 2020 2020 2020 2020 2020 2020 2020 M │ │ │ │ -0002f8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f8c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002f8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f910: 207c 0a7c 2020 2020 2020 2020 3220 2020 |.| 2 │ │ │ │ -0002f920: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -0002f930: 2020 2020 2020 2020 3220 2020 2020 3220 2 2 │ │ │ │ -0002f940: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -0002f950: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -0002f960: 207c 0a7c 6f33 203d 207b 7b62 202c 2062 |.|o3 = {{b , b │ │ │ │ -0002f970: 2a63 2c20 6320 2c20 612a 642c 2062 2a64 *c, c , a*d, b*d │ │ │ │ -0002f980: 2c20 632a 642c 2064 207d 2c20 7b62 202c , c*d, d }, {b , │ │ │ │ -0002f990: 2062 2a63 2c20 6320 2c20 612a 642c 2062 b*c, c , a*d, b │ │ │ │ -0002f9a0: 2a64 2c20 632a 642c 2064 207d 2c20 2020 *d, c*d, d }, │ │ │ │ -0002f9b0: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -0002f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fa00: 2d7c 0a7c 2020 2020 2020 2032 2020 2020 -|.| 2 │ │ │ │ -0002fa10: 2020 3220 2020 3320 2020 3220 2020 2020 2 3 2 │ │ │ │ -0002fa20: 2020 2020 2020 3220 2020 2020 2032 2020 2 2 │ │ │ │ -0002fa30: 2020 2032 2020 2020 2032 2020 2033 2020 2 2 3 │ │ │ │ -0002fa40: 2020 2032 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ -0002fa50: 207c 0a7c 2020 2020 207b 6220 632c 2062 |.| {b c, b │ │ │ │ -0002fa60: 2a63 202c 2063 202c 2062 2064 2c20 622a *c , c , b d, b* │ │ │ │ -0002fa70: 632a 642c 2063 2064 2c20 612a 6420 2c20 c*d, c d, a*d , │ │ │ │ -0002fa80: 622a 6420 2c20 632a 6420 2c20 6420 7d2c b*d , c*d , d }, │ │ │ │ -0002fa90: 207b 6220 2c20 622a 632c 2063 202c 2020 {b , b*c, c , │ │ │ │ -0002faa0: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ +0002f040: 2d2d 2d7c 0a7c 2020 2020 2020 3220 2020 ---|.| 2 │ │ │ │ +0002f050: 2020 2020 3220 2020 2020 3220 2020 3320 2 2 3 │ │ │ │ +0002f060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f070: 2020 2020 2020 2032 2020 2020 2020 3220 2 2 │ │ │ │ +0002f080: 2020 2020 3220 2020 2020 3220 2020 3320 2 2 3 │ │ │ │ +0002f090: 2020 207c 0a7c 2020 2020 2064 207d 2c20 |.| d }, │ │ │ │ +0002f0a0: 7b61 2a63 202c 2062 2a63 202c 2063 202c {a*c , b*c , c , │ │ │ │ +0002f0b0: 2061 2a62 2a64 2c20 612a 632a 642c 2062 a*b*d, a*c*d, b │ │ │ │ +0002f0c0: 2a63 2a64 2c20 6320 642c 2061 2a64 202c *c*d, c d, a*d , │ │ │ │ +0002f0d0: 2062 2a64 202c 2063 2a64 202c 2064 207d b*d , c*d , d } │ │ │ │ +0002f0e0: 7d20 207c 0a7c 2020 2020 2020 2020 2020 } |.| │ │ │ │ +0002f0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f130: 2020 207c 0a7c 6f33 203a 204c 6973 7420 |.|o3 : List │ │ │ │ +0002f140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f180: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f1d0: 2d2d 2d2b 0a7c 6934 203a 2073 6d61 6c6c ---+.|i4 : small │ │ │ │ +0002f1e0: 6572 4d6f 6e6f 6d69 616c 7328 4d2c 2062 erMonomials(M, b │ │ │ │ +0002f1f0: 5e32 2920 2020 2020 2020 2020 2020 2020 ^2) │ │ │ │ +0002f200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f220: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f270: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f280: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0002f290: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0002f2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f2c0: 2020 207c 0a7c 6f34 203d 207b 612a 632c |.|o4 = {a*c, │ │ │ │ +0002f2d0: 2062 2a63 2c20 6320 2c20 612a 642c 2062 b*c, c , a*d, b │ │ │ │ +0002f2e0: 2a64 2c20 632a 642c 2064 207d 2020 2020 *d, c*d, d } │ │ │ │ +0002f2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f310: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f360: 2020 207c 0a7c 6f34 203a 204c 6973 7420 |.|o4 : List │ │ │ │ +0002f370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f3b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002f3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f400: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ +0002f410: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +0002f420: 7465 2074 6169 6c4d 6f6e 6f6d 6961 6c73 te tailMonomials │ │ │ │ +0002f430: 3a20 7461 696c 4d6f 6e6f 6d69 616c 732c : tailMonomials, │ │ │ │ +0002f440: 202d 2d20 6669 6e64 2074 6169 6c20 6d6f -- find tail mo │ │ │ │ +0002f450: 6e6f 6d69 616c 730a 2020 2a20 2a6e 6f74 nomials. * *not │ │ │ │ +0002f460: 6520 7374 616e 6461 7264 4d6f 6e6f 6d69 e standardMonomi │ │ │ │ +0002f470: 616c 733a 2073 7461 6e64 6172 644d 6f6e als: standardMon │ │ │ │ +0002f480: 6f6d 6961 6c73 2c20 2d2d 2063 6f6d 7075 omials, -- compu │ │ │ │ +0002f490: 7465 7320 7374 616e 6461 7264 206d 6f6e tes standard mon │ │ │ │ +0002f4a0: 6f6d 6961 6c73 0a0a 5761 7973 2074 6f20 omials..Ways to │ │ │ │ +0002f4b0: 7573 6520 736d 616c 6c65 724d 6f6e 6f6d use smallerMonom │ │ │ │ +0002f4c0: 6961 6c73 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d ials:.========== │ │ │ │ +0002f4d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002f4e0: 3d3d 3d0a 0a20 202a 2022 736d 616c 6c65 ===.. * "smalle │ │ │ │ +0002f4f0: 724d 6f6e 6f6d 6961 6c73 2849 6465 616c rMonomials(Ideal │ │ │ │ +0002f500: 2922 0a20 202a 2022 736d 616c 6c65 724d )". * "smallerM │ │ │ │ +0002f510: 6f6e 6f6d 6961 6c73 2849 6465 616c 2c52 onomials(Ideal,R │ │ │ │ +0002f520: 696e 6745 6c65 6d65 6e74 2922 0a0a 466f ingElement)"..Fo │ │ │ │ +0002f530: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +0002f540: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +0002f550: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +0002f560: 2a6e 6f74 6520 736d 616c 6c65 724d 6f6e *note smallerMon │ │ │ │ +0002f570: 6f6d 6961 6c73 3a20 736d 616c 6c65 724d omials: smallerM │ │ │ │ +0002f580: 6f6e 6f6d 6961 6c73 2c20 6973 2061 202a onomials, is a * │ │ │ │ +0002f590: 6e6f 7465 206d 6574 686f 640a 6675 6e63 note method.func │ │ │ │ +0002f5a0: 7469 6f6e 3a20 284d 6163 6175 6c61 7932 tion: (Macaulay2 │ │ │ │ +0002f5b0: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ +0002f5c0: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ +0002f5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f610: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +0002f620: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +0002f630: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +0002f640: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +0002f650: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +0002f660: 2e30 352b 6473 2f4d 322f 4d61 6361 756c .05+ds/M2/Macaul │ │ │ │ +0002f670: 6179 322f 7061 636b 6167 6573 2f0a 4772 ay2/packages/.Gr │ │ │ │ +0002f680: 6f65 626e 6572 5374 7261 7461 2e6d 323a oebnerStrata.m2: │ │ │ │ +0002f690: 3532 353a 302e 0a1f 0a46 696c 653a 2047 525:0....File: G │ │ │ │ +0002f6a0: 726f 6562 6e65 7253 7472 6174 612e 696e roebnerStrata.in │ │ │ │ +0002f6b0: 666f 2c20 4e6f 6465 3a20 7374 616e 6461 fo, Node: standa │ │ │ │ +0002f6c0: 7264 4d6f 6e6f 6d69 616c 732c 204e 6578 rdMonomials, Nex │ │ │ │ +0002f6d0: 743a 2074 6169 6c4d 6f6e 6f6d 6961 6c73 t: tailMonomials │ │ │ │ +0002f6e0: 2c20 5072 6576 3a20 736d 616c 6c65 724d , Prev: smallerM │ │ │ │ +0002f6f0: 6f6e 6f6d 6961 6c73 2c20 5570 3a20 546f onomials, Up: To │ │ │ │ +0002f700: 700a 0a73 7461 6e64 6172 644d 6f6e 6f6d p..standardMonom │ │ │ │ +0002f710: 6961 6c73 202d 2d20 636f 6d70 7574 6573 ials -- computes │ │ │ │ +0002f720: 2073 7461 6e64 6172 6420 6d6f 6e6f 6d69 standard monomi │ │ │ │ +0002f730: 616c 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a als.************ │ │ │ │ +0002f740: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002f750: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002f760: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ +0002f770: 200a 2020 2020 2020 2020 4c20 3d20 7374 . L = st │ │ │ │ +0002f780: 616e 6461 7264 4d6f 6e6f 6d69 616c 7320 andardMonomials │ │ │ │ +0002f790: 4d0a 2020 2020 2020 2020 4c20 3d20 7374 M. L = st │ │ │ │ +0002f7a0: 616e 6461 7264 4d6f 6e6f 6d69 616c 7328 andardMonomials( │ │ │ │ +0002f7b0: 642c 204d 290a 2020 2a20 496e 7075 7473 d, M). * Inputs │ │ │ │ +0002f7c0: 3a0a 2020 2020 2020 2a20 4d2c 2061 6e20 :. * M, an │ │ │ │ +0002f7d0: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61 *note ideal: (Ma │ │ │ │ +0002f7e0: 6361 756c 6179 3244 6f63 2949 6465 616c caulay2Doc)Ideal │ │ │ │ +0002f7f0: 2c2c 204d 2073 686f 756c 6420 6265 2061 ,, M should be a │ │ │ │ +0002f800: 206d 6f6e 6f6d 6961 6c20 6964 6561 6c0a monomial ideal. │ │ │ │ +0002f810: 2020 2020 2020 2a20 642c 2061 202a 6e6f * d, a *no │ │ │ │ +0002f820: 7465 206c 6973 743a 2028 4d61 6361 756c te list: (Macaul │ │ │ │ +0002f830: 6179 3244 6f63 294c 6973 742c 2c20 6120 ay2Doc)List,, a │ │ │ │ +0002f840: 6465 6772 6565 0a20 202a 204f 7574 7075 degree. * Outpu │ │ │ │ +0002f850: 7473 3a0a 2020 2020 2020 2a20 4c2c 2061 ts:. * L, a │ │ │ │ +0002f860: 202a 6e6f 7465 206c 6973 743a 2028 4d61 *note list: (Ma │ │ │ │ +0002f870: 6361 756c 6179 3244 6f63 294c 6973 742c caulay2Doc)List, │ │ │ │ +0002f880: 2c20 4c20 6973 2061 206c 6973 7420 6f66 , L is a list of │ │ │ │ +0002f890: 206c 6973 7473 206f 6620 7374 616e 6461 lists of standa │ │ │ │ +0002f8a0: 7264 0a20 2020 2020 2020 206d 6f6e 6f6d rd. monom │ │ │ │ +0002f8b0: 6961 6c73 2066 6f72 2074 6865 2069 6465 ials for the ide │ │ │ │ +0002f8c0: 616c 2024 4d24 2c20 6f6e 6520 666f 7220 al $M$, one for │ │ │ │ +0002f8d0: 6561 6368 2067 656e 6572 6174 6f72 206f each generator o │ │ │ │ +0002f8e0: 6620 244d 240a 0a44 6573 6372 6970 7469 f $M$..Descripti │ │ │ │ +0002f8f0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +0002f900: 4120 6d6f 6e6f 6d69 616c 2024 6d24 2069 A monomial $m$ i │ │ │ │ +0002f910: 7320 7374 616e 6461 7264 2077 6974 6820 s standard with │ │ │ │ +0002f920: 7265 7370 6563 7420 746f 2061 206d 6f6e respect to a mon │ │ │ │ +0002f930: 6f6d 6961 6c20 6964 6561 6c20 244d 2420 omial ideal $M$ │ │ │ │ +0002f940: 616e 6420 6120 6765 6e65 7261 746f 720a and a generator. │ │ │ │ +0002f950: 2467 2420 6f66 2024 4d24 2069 6620 246d $g$ of $M$ if $m │ │ │ │ +0002f960: 2420 6973 206f 6620 7468 6520 7361 6d65 $ is of the same │ │ │ │ +0002f970: 2064 6567 7265 6520 6173 2024 6724 2062 degree as $g$ b │ │ │ │ +0002f980: 7574 2069 7320 6e6f 7420 616e 2065 6c65 ut is not an ele │ │ │ │ +0002f990: 6d65 6e74 206f 6620 244d 242e 0a0a 496e ment of $M$...In │ │ │ │ +0002f9a0: 7075 7474 696e 6720 616e 2069 6465 616c putting an ideal │ │ │ │ +0002f9b0: 2024 4d24 2072 6574 7572 6e73 2074 6865 $M$ returns the │ │ │ │ +0002f9c0: 2073 7461 6e64 6172 6420 6d6f 6e6f 6d69 standard monomi │ │ │ │ +0002f9d0: 616c 7320 6f66 2065 6163 6820 6f66 2074 als of each of t │ │ │ │ +0002f9e0: 6865 2067 6976 656e 0a67 656e 6572 6174 he given.generat │ │ │ │ +0002f9f0: 6f72 7320 6f66 2074 6865 2069 6465 616c ors of the ideal │ │ │ │ +0002fa00: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ +0002fa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fa50: 2d2b 0a7c 6931 203a 2052 203d 205a 5a2f -+.|i1 : R = ZZ/ │ │ │ │ +0002fa60: 3332 3030 335b 612e 2e64 5d3b 2020 2020 32003[a..d]; │ │ │ │ +0002fa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fa90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002faa0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0002fab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002faf0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ -0002fb00: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -0002fb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002faf0: 2d2b 0a7c 6932 203a 204d 203d 2069 6465 -+.|i2 : M = ide │ │ │ │ +0002fb00: 616c 2028 615e 322c 2061 2a62 2c20 625e al (a^2, a*b, b^ │ │ │ │ +0002fb10: 332c 2061 2a63 293b 2020 2020 2020 2020 3, a*c); │ │ │ │ 0002fb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fb40: 207c 0a7c 2020 2020 2061 2a64 2c20 622a |.| a*d, b* │ │ │ │ -0002fb50: 642c 2063 2a64 2c20 6420 7d7d 2020 2020 d, c*d, d }} │ │ │ │ +0002fb40: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002fb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fb90: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002fba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fb90: 207c 0a7c 6f32 203a 2049 6465 616c 206f |.|o2 : Ideal o │ │ │ │ +0002fba0: 6620 5220 2020 2020 2020 2020 2020 2020 f R │ │ │ │ 0002fbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fbe0: 207c 0a7c 6f33 203a 204c 6973 7420 2020 |.|o3 : List │ │ │ │ -0002fbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fc30: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002fc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fc80: 2d2b 0a7c 6934 203a 2073 7461 6e64 6172 -+.|i4 : standar │ │ │ │ -0002fc90: 644d 6f6e 6f6d 6961 6c73 287b 337d 2c20 dMonomials({3}, │ │ │ │ -0002fca0: 4d29 2020 2020 2020 2020 2020 2020 2020 M) │ │ │ │ +0002fbe0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002fbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fc30: 2d2b 0a7c 6933 203a 204c 3120 3d20 7374 -+.|i3 : L1 = st │ │ │ │ +0002fc40: 616e 6461 7264 4d6f 6e6f 6d69 616c 7320 andardMonomials │ │ │ │ +0002fc50: 4d20 2020 2020 2020 2020 2020 2020 2020 M │ │ │ │ +0002fc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fc80: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002fc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fcd0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002fce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fd20: 207c 0a7c 2020 2020 2020 2032 2020 2020 |.| 2 │ │ │ │ -0002fd30: 2020 3220 2020 3320 2020 3220 2020 2020 2 3 2 │ │ │ │ -0002fd40: 2020 2020 2020 3220 2020 2020 2032 2020 2 2 │ │ │ │ -0002fd50: 2020 2032 2020 2020 2032 2020 2033 2020 2 2 3 │ │ │ │ -0002fd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fd70: 207c 0a7c 6f34 203d 207b 6220 632c 2062 |.|o4 = {b c, b │ │ │ │ -0002fd80: 2a63 202c 2063 202c 2062 2064 2c20 622a *c , c , b d, b* │ │ │ │ -0002fd90: 632a 642c 2063 2064 2c20 612a 6420 2c20 c*d, c d, a*d , │ │ │ │ -0002fda0: 622a 6420 2c20 632a 6420 2c20 6420 7d20 b*d , c*d , d } │ │ │ │ -0002fdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fdc0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002fdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fe00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fe10: 207c 0a7c 6f34 203a 204c 6973 7420 2020 |.|o4 : List │ │ │ │ -0002fe20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fe30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fe40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fe50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fe60: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002fcd0: 207c 0a7c 2020 2020 2020 2020 3220 2020 |.| 2 │ │ │ │ +0002fce0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +0002fcf0: 2020 2020 2020 2020 3220 2020 2020 3220 2 2 │ │ │ │ +0002fd00: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0002fd10: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0002fd20: 207c 0a7c 6f33 203d 207b 7b62 202c 2062 |.|o3 = {{b , b │ │ │ │ +0002fd30: 2a63 2c20 6320 2c20 612a 642c 2062 2a64 *c, c , a*d, b*d │ │ │ │ +0002fd40: 2c20 632a 642c 2064 207d 2c20 7b62 202c , c*d, d }, {b , │ │ │ │ +0002fd50: 2062 2a63 2c20 6320 2c20 612a 642c 2062 b*c, c , a*d, b │ │ │ │ +0002fd60: 2a64 2c20 632a 642c 2064 207d 2c20 2020 *d, c*d, d }, │ │ │ │ +0002fd70: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ +0002fd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fdc0: 2d7c 0a7c 2020 2020 2020 2032 2020 2020 -|.| 2 │ │ │ │ +0002fdd0: 2020 3220 2020 3320 2020 3220 2020 2020 2 3 2 │ │ │ │ +0002fde0: 2020 2020 2020 3220 2020 2020 2032 2020 2 2 │ │ │ │ +0002fdf0: 2020 2032 2020 2020 2032 2020 2033 2020 2 2 3 │ │ │ │ +0002fe00: 2020 2032 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ +0002fe10: 207c 0a7c 2020 2020 207b 6220 632c 2062 |.| {b c, b │ │ │ │ +0002fe20: 2a63 202c 2063 202c 2062 2064 2c20 622a *c , c , b d, b* │ │ │ │ +0002fe30: 632a 642c 2063 2064 2c20 612a 6420 2c20 c*d, c d, a*d , │ │ │ │ +0002fe40: 622a 6420 2c20 632a 6420 2c20 6420 7d2c b*d , c*d , d }, │ │ │ │ +0002fe50: 207b 6220 2c20 622a 632c 2063 202c 2020 {b , b*c, c , │ │ │ │ +0002fe60: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ 0002fe70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fe90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002feb0: 2d2b 0a0a 496e 7075 7474 696e 6720 616e -+..Inputting an │ │ │ │ -0002fec0: 2069 6e74 6567 6572 2024 6424 2028 6f72 integer $d$ (or │ │ │ │ -0002fed0: 2064 6567 7265 6520 2464 2429 2061 6e64 degree $d$) and │ │ │ │ -0002fee0: 2061 6e20 6964 6561 6c20 6769 7665 7320 an ideal gives │ │ │ │ -0002fef0: 7468 6520 7374 616e 6461 7264 0a6d 6f6e the standard.mon │ │ │ │ -0002ff00: 6f6d 6961 6c73 2066 6f72 2074 6865 2073 omials for the s │ │ │ │ -0002ff10: 7065 6369 6669 6564 2069 6465 616c 2069 pecified ideal i │ │ │ │ -0002ff20: 6e20 6465 6772 6565 2024 6424 2e0a 0a2b n degree $d$...+ │ │ │ │ -0002ff30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ff40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ff50: 2d2d 2d2d 2d2b 0a7c 6935 203a 2073 7461 -----+.|i5 : sta │ │ │ │ -0002ff60: 6e64 6172 644d 6f6e 6f6d 6961 6c73 2832 ndardMonomials(2 │ │ │ │ -0002ff70: 2c20 4d29 2020 2020 2020 2020 207c 0a7c , M) |.| │ │ │ │ +0002feb0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ +0002fec0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +0002fed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ff00: 207c 0a7c 2020 2020 2061 2a64 2c20 622a |.| a*d, b* │ │ │ │ +0002ff10: 642c 2063 2a64 2c20 6420 7d7d 2020 2020 d, c*d, d }} │ │ │ │ +0002ff20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ff30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ff40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ff50: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002ff60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ff70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ff80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ff90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ffa0: 2020 2020 207c 0a7c 2020 2020 2020 2032 |.| 2 │ │ │ │ -0002ffb0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -0002ffc0: 2020 2020 2020 2020 2020 2032 207c 0a7c 2 |.| │ │ │ │ -0002ffd0: 6f35 203d 207b 6220 2c20 622a 632c 2063 o5 = {b , b*c, c │ │ │ │ -0002ffe0: 202c 2061 2a64 2c20 622a 642c 2063 2a64 , a*d, b*d, c*d │ │ │ │ -0002fff0: 2c20 6420 7d7c 0a7c 2020 2020 2020 2020 , d }|.| │ │ │ │ -00030000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030010: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00030020: 6f35 203a 204c 6973 7420 2020 2020 2020 o5 : List │ │ │ │ -00030030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030040: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00030050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -00030070: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ -00030080: 3d0a 0a20 202a 202a 6e6f 7465 2074 6169 =.. * *note tai │ │ │ │ -00030090: 6c4d 6f6e 6f6d 6961 6c73 3a20 7461 696c lMonomials: tail │ │ │ │ -000300a0: 4d6f 6e6f 6d69 616c 732c 202d 2d20 6669 Monomials, -- fi │ │ │ │ -000300b0: 6e64 2074 6169 6c20 6d6f 6e6f 6d69 616c nd tail monomial │ │ │ │ -000300c0: 730a 2020 2a20 2a6e 6f74 6520 736d 616c s. * *note smal │ │ │ │ -000300d0: 6c65 724d 6f6e 6f6d 6961 6c73 3a20 736d lerMonomials: sm │ │ │ │ -000300e0: 616c 6c65 724d 6f6e 6f6d 6961 6c73 2c20 allerMonomials, │ │ │ │ -000300f0: 2d2d 2072 6574 7572 6e73 2074 6865 2073 -- returns the s │ │ │ │ -00030100: 7461 6e64 6172 6420 6d6f 6e6f 6d69 616c tandard monomial │ │ │ │ -00030110: 730a 2020 2020 736d 616c 6c65 7220 6275 s. smaller bu │ │ │ │ -00030120: 7420 6f66 2074 6865 2073 616d 6520 6465 t of the same de │ │ │ │ -00030130: 6772 6565 2061 7320 6769 7665 6e20 6d6f gree as given mo │ │ │ │ -00030140: 6e6f 6d69 616c 2873 290a 0a57 6179 7320 nomial(s)..Ways │ │ │ │ -00030150: 746f 2075 7365 2073 7461 6e64 6172 644d to use standardM │ │ │ │ -00030160: 6f6e 6f6d 6961 6c73 3a0a 3d3d 3d3d 3d3d onomials:.====== │ │ │ │ -00030170: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00030180: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2273 ========.. * "s │ │ │ │ -00030190: 7461 6e64 6172 644d 6f6e 6f6d 6961 6c73 tandardMonomials │ │ │ │ -000301a0: 2849 6465 616c 2922 0a20 202a 2022 7374 (Ideal)". * "st │ │ │ │ -000301b0: 616e 6461 7264 4d6f 6e6f 6d69 616c 7328 andardMonomials( │ │ │ │ -000301c0: 4c69 7374 2c49 6465 616c 2922 0a20 202a List,Ideal)". * │ │ │ │ -000301d0: 2022 7374 616e 6461 7264 4d6f 6e6f 6d69 "standardMonomi │ │ │ │ -000301e0: 616c 7328 5a5a 2c49 6465 616c 2922 0a0a als(ZZ,Ideal)".. │ │ │ │ -000301f0: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -00030200: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -00030210: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -00030220: 7420 2a6e 6f74 6520 7374 616e 6461 7264 t *note standard │ │ │ │ -00030230: 4d6f 6e6f 6d69 616c 733a 2073 7461 6e64 Monomials: stand │ │ │ │ -00030240: 6172 644d 6f6e 6f6d 6961 6c73 2c20 6973 ardMonomials, is │ │ │ │ -00030250: 2061 202a 6e6f 7465 206d 6574 686f 640a a *note method. │ │ │ │ -00030260: 6675 6e63 7469 6f6e 3a20 284d 6163 6175 function: (Macau │ │ │ │ -00030270: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ -00030280: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ -00030290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000302a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000302b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000302c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000302d0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -000302e0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -000302f0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -00030300: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -00030310: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -00030320: 312e 3236 2e30 352b 6473 2f4d 322f 4d61 1.26.05+ds/M2/Ma │ │ │ │ -00030330: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -00030340: 2f0a 4772 6f65 626e 6572 5374 7261 7461 /.GroebnerStrata │ │ │ │ -00030350: 2e6d 323a 3438 393a 302e 0a1f 0a46 696c .m2:489:0....Fil │ │ │ │ -00030360: 653a 2047 726f 6562 6e65 7253 7472 6174 e: GroebnerStrat │ │ │ │ -00030370: 612e 696e 666f 2c20 4e6f 6465 3a20 7461 a.info, Node: ta │ │ │ │ -00030380: 696c 4d6f 6e6f 6d69 616c 732c 2050 7265 ilMonomials, Pre │ │ │ │ -00030390: 763a 2073 7461 6e64 6172 644d 6f6e 6f6d v: standardMonom │ │ │ │ -000303a0: 6961 6c73 2c20 5570 3a20 546f 700a 0a74 ials, Up: Top..t │ │ │ │ -000303b0: 6169 6c4d 6f6e 6f6d 6961 6c73 202d 2d20 ailMonomials -- │ │ │ │ -000303c0: 6669 6e64 2074 6169 6c20 6d6f 6e6f 6d69 find tail monomi │ │ │ │ -000303d0: 616c 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a als.************ │ │ │ │ -000303e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000303f0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ -00030400: 6167 653a 200a 2020 2020 2020 2020 4c20 age: . L │ │ │ │ -00030410: 3d20 7461 696c 4d6f 6e6f 6d69 616c 7320 = tailMonomials │ │ │ │ -00030420: 4d0a 2020 2020 2020 2020 4c20 3d20 7461 M. L = ta │ │ │ │ -00030430: 696c 4d6f 6e6f 6d69 616c 7328 4d2c 206d ilMonomials(M, m │ │ │ │ -00030440: 290a 2020 2a20 496e 7075 7473 3a0a 2020 ). * Inputs:. │ │ │ │ -00030450: 2020 2020 2a20 4d2c 2061 6e20 2a6e 6f74 * M, an *not │ │ │ │ -00030460: 6520 6964 6561 6c3a 2028 4d61 6361 756c e ideal: (Macaul │ │ │ │ -00030470: 6179 3244 6f63 2949 6465 616c 2c2c 2024 ay2Doc)Ideal,, $ │ │ │ │ -00030480: 4d24 2073 686f 756c 6420 6265 2061 206d M$ should be a m │ │ │ │ -00030490: 6f6e 6f6d 6961 6c20 6964 6561 6c0a 2020 onomial ideal. │ │ │ │ -000304a0: 2020 2020 2020 2861 6e20 6964 6561 6c20 (an ideal │ │ │ │ -000304b0: 6765 6e65 7261 7465 6420 6279 206d 6f6e generated by mon │ │ │ │ -000304c0: 6f6d 6961 6c73 290a 2020 2020 2020 2a20 omials). * │ │ │ │ -000304d0: 6d2c 2061 202a 6e6f 7465 2072 696e 6720 m, a *note ring │ │ │ │ -000304e0: 656c 656d 656e 743a 2028 4d61 6361 756c element: (Macaul │ │ │ │ -000304f0: 6179 3244 6f63 2952 696e 6745 6c65 6d65 ay2Doc)RingEleme │ │ │ │ -00030500: 6e74 2c2c 206f 7074 696f 6e61 6c2c 206f nt,, optional, o │ │ │ │ -00030510: 6e6c 790a 2020 2020 2020 2020 7265 7475 nly. retu │ │ │ │ -00030520: 726e 2061 2073 696e 676c 6520 6c69 7374 rn a single list │ │ │ │ -00030530: 206f 6620 7468 6520 7461 696c 206d 6f6e of the tail mon │ │ │ │ -00030540: 6f6d 6961 6c73 2066 6f72 2074 6869 7320 omials for this │ │ │ │ -00030550: 6d6f 6e6f 6d69 616c 0a20 202a 202a 6e6f monomial. * *no │ │ │ │ -00030560: 7465 204f 7074 696f 6e61 6c20 696e 7075 te Optional inpu │ │ │ │ -00030570: 7473 3a20 284d 6163 6175 6c61 7932 446f ts: (Macaulay2Do │ │ │ │ -00030580: 6329 7573 696e 6720 6675 6e63 7469 6f6e c)using function │ │ │ │ -00030590: 7320 7769 7468 206f 7074 696f 6e61 6c20 s with optional │ │ │ │ -000305a0: 696e 7075 7473 2c3a 0a20 2020 2020 202a inputs,:. * │ │ │ │ -000305b0: 2041 6c6c 5374 616e 6461 7264 203d 3e20 AllStandard => │ │ │ │ -000305c0: 6120 2a6e 6f74 6520 426f 6f6c 6561 6e20 a *note Boolean │ │ │ │ -000305d0: 7661 6c75 653a 2028 4d61 6361 756c 6179 value: (Macaulay │ │ │ │ -000305e0: 3244 6f63 2942 6f6f 6c65 616e 2c2c 2064 2Doc)Boolean,, d │ │ │ │ -000305f0: 6566 6175 6c74 0a20 2020 2020 2020 2076 efault. v │ │ │ │ -00030600: 616c 7565 2066 616c 7365 2c20 7768 6963 alue false, whic │ │ │ │ -00030610: 6820 6d6f 6e6f 6d69 616c 7320 7368 6f75 h monomials shou │ │ │ │ -00030620: 6c64 2062 6520 636f 6e73 6964 6572 6564 ld be considered │ │ │ │ -00030630: 2074 6169 6c20 6d6f 6e6f 6d69 616c 7320 tail monomials │ │ │ │ -00030640: 6f66 2061 0a20 2020 2020 2020 206d 6f6e of a. mon │ │ │ │ -00030650: 6f6d 6961 6c20 246d 243a 2065 6974 6865 omial $m$: eithe │ │ │ │ -00030660: 7220 616c 6c20 7374 616e 6461 7264 206d r all standard m │ │ │ │ -00030670: 6f6e 6f6d 6961 6c73 206f 6620 6120 6769 onomials of a gi │ │ │ │ -00030680: 7665 6e20 6465 6772 6565 2c20 6f72 2061 ven degree, or a │ │ │ │ -00030690: 6c6c 0a20 2020 2020 2020 206d 6f6e 6f6d ll. monom │ │ │ │ -000306a0: 6961 6c73 2073 6d61 6c6c 6572 2074 6861 ials smaller tha │ │ │ │ -000306b0: 6e20 246d 2420 696e 2074 6865 2067 6976 n $m$ in the giv │ │ │ │ -000306c0: 656e 2074 6572 6d20 6f72 6465 7220 2862 en term order (b │ │ │ │ -000306d0: 7574 2073 7469 6c6c 206f 6620 7468 650a ut still of the. │ │ │ │ -000306e0: 2020 2020 2020 2020 7361 6d65 2064 6567 same deg │ │ │ │ -000306f0: 7265 6529 0a20 202a 204f 7574 7075 7473 ree). * Outputs │ │ │ │ -00030700: 3a0a 2020 2020 2020 2a20 4c2c 2061 202a :. * L, a * │ │ │ │ -00030710: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ -00030720: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ -00030730: 6120 6c69 7374 206f 6620 6c69 7374 733a a list of lists: │ │ │ │ -00030740: 2066 6f72 2065 6163 680a 2020 2020 2020 for each. │ │ │ │ -00030750: 2020 6765 6e65 7261 746f 7220 246d 2420 generator $m$ │ │ │ │ -00030760: 6f66 2024 4d24 2c20 7468 6520 6c69 7374 of $M$, the list │ │ │ │ -00030770: 206f 6620 616c 6c20 7461 696c 206d 6f6e of all tail mon │ │ │ │ -00030780: 6f6d 6961 6c73 2049 6620 696e 7374 6561 omials If instea │ │ │ │ -00030790: 6420 246d 2420 6973 0a20 2020 2020 2020 d $m$ is. │ │ │ │ -000307a0: 2067 6976 656e 2c20 7468 6520 6c69 7374 given, the list │ │ │ │ -000307b0: 206f 6620 7468 6520 7461 696c 206d 6f6e of the tail mon │ │ │ │ -000307c0: 6f6d 6961 6c73 206f 6620 246d 2420 6973 omials of $m$ is │ │ │ │ -000307d0: 2072 6574 7572 6e65 640a 0a44 6573 6372 returned..Descr │ │ │ │ -000307e0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -000307f0: 3d3d 0a0a 496e 7075 7474 696e 6720 616e ==..Inputting an │ │ │ │ -00030800: 2069 6465 616c 2024 4d24 2067 656e 6572 ideal $M$ gener │ │ │ │ -00030810: 6174 6564 2062 7920 6d6f 6e6f 6d69 616c ated by monomial │ │ │ │ -00030820: 7320 7265 7475 726e 7320 6120 6c69 7374 s returns a list │ │ │ │ -00030830: 206f 6620 6c69 7374 7320 6f66 2074 6169 of lists of tai │ │ │ │ -00030840: 6c0a 6d6f 6e6f 6d69 616c 7320 666f 7220 l.monomials for │ │ │ │ -00030850: 6561 6368 2067 656e 6572 6174 6f72 206f each generator o │ │ │ │ -00030860: 6620 244d 2420 2869 6e20 7468 6520 7361 f $M$ (in the sa │ │ │ │ -00030870: 6d65 206f 7264 6572 292e 0a0a 2b2d 2d2d me order)...+--- │ │ │ │ -00030880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000308a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000308b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000308c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ -000308d0: 3a20 5220 3d20 5a5a 2f33 3230 3033 5b61 : R = ZZ/32003[a │ │ │ │ -000308e0: 2e2e 645d 3b20 2020 2020 2020 2020 2020 ..d]; │ │ │ │ -000308f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030910: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00030920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030960: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ -00030970: 3a20 4d20 3d20 6964 6561 6c20 2861 5e32 : M = ideal (a^2 │ │ │ │ -00030980: 2c20 625e 322c 2061 2a62 2a63 293b 2020 , b^2, a*b*c); │ │ │ │ -00030990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000309a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000309b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -000309c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000309d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000309e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000309f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a00: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ -00030a10: 3a20 4964 6561 6c20 6f66 2052 2020 2020 : Ideal of R │ │ │ │ -00030a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a50: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00030a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ -00030ab0: 3a20 7461 696c 4d6f 6e6f 6d69 616c 7320 : tailMonomials │ │ │ │ -00030ac0: 4d20 2020 2020 2020 2020 2020 2020 2020 M │ │ │ │ -00030ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030af0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00030b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00030b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b60: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -00030b70: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -00030b80: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -00030b90: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ -00030ba0: 3d20 7b7b 612a 622c 2061 2a63 2c20 622a = {{a*b, a*c, b* │ │ │ │ -00030bb0: 632c 2063 202c 2061 2a64 2c20 622a 642c c, c , a*d, b*d, │ │ │ │ -00030bc0: 2063 2a64 2c20 6420 7d2c 207b 612a 632c c*d, d }, {a*c, │ │ │ │ -00030bd0: 2062 2a63 2c20 6320 2c20 612a 642c 2062 b*c, c , a*d, b │ │ │ │ -00030be0: 2a64 2c20 632a 642c 2020 7c0a 7c20 2020 *d, c*d, |.| │ │ │ │ -00030bf0: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d -------------- │ │ │ │ -00030c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030c30: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ -00030c40: 2020 2032 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ -00030c50: 2032 2020 2033 2020 2020 2020 2020 2020 2 3 │ │ │ │ -00030c60: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -00030c70: 2020 2020 2032 2020 2020 2032 2020 2020 2 2 │ │ │ │ -00030c80: 2032 2020 2033 2020 2020 7c0a 7c20 2020 2 3 |.| │ │ │ │ -00030c90: 2020 6420 7d2c 207b 612a 6320 2c20 622a d }, {a*c , b* │ │ │ │ -00030ca0: 6320 2c20 6320 2c20 612a 622a 642c 2061 c , c , a*b*d, a │ │ │ │ -00030cb0: 2a63 2a64 2c20 622a 632a 642c 2063 2064 *c*d, b*c*d, c d │ │ │ │ -00030cc0: 2c20 612a 6420 2c20 622a 6420 2c20 632a , a*d , b*d , c* │ │ │ │ -00030cd0: 6420 2c20 6420 7d7d 2020 7c0a 7c20 2020 d , d }} |.| │ │ │ │ -00030ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030d20: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ -00030d30: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ -00030d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ffa0: 207c 0a7c 6f33 203a 204c 6973 7420 2020 |.|o3 : List │ │ │ │ +0002ffb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ffc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ffd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ffe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fff0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00030000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030030: 2d2d 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│ │ │ +00030130: 207c 0a7c 6f34 203d 207b 6220 632c 2062 |.|o4 = {b c, b │ │ │ │ +00030140: 2a63 202c 2063 202c 2062 2064 2c20 622a *c , c , b d, b* │ │ │ │ +00030150: 632a 642c 2063 2064 2c20 612a 6420 2c20 c*d, c d, a*d , │ │ │ │ +00030160: 622a 6420 2c20 632a 6420 2c20 6420 7d20 b*d , c*d , d } │ │ │ │ +00030170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030180: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00030190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000301a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000301b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000301c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000301d0: 207c 0a7c 6f34 203a 204c 6973 7420 2020 |.|o4 : List │ │ │ │ +000301e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000301f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030220: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00030230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030270: 2d2b 0a0a 496e 7075 7474 696e 6720 616e -+..Inputting an │ │ │ │ +00030280: 2069 6e74 6567 6572 2024 6424 2028 6f72 integer $d$ (or │ │ │ │ +00030290: 2064 6567 7265 6520 2464 2429 2061 6e64 degree $d$) and │ │ │ │ +000302a0: 2061 6e20 6964 6561 6c20 6769 7665 7320 an ideal gives │ │ │ │ +000302b0: 7468 6520 7374 616e 6461 7264 0a6d 6f6e the standard.mon │ │ │ │ +000302c0: 6f6d 6961 6c73 2066 6f72 2074 6865 2073 omials for the s │ │ │ │ +000302d0: 7065 6369 6669 6564 2069 6465 616c 2069 pecified ideal i │ │ │ │ +000302e0: 6e20 6465 6772 6565 2024 6424 2e0a 0a2b n degree $d$...+ │ │ │ │ +000302f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030310: 2d2d 2d2d 2d2b 0a7c 6935 203a 2073 7461 -----+.|i5 : sta │ │ │ │ +00030320: 6e64 6172 644d 6f6e 6f6d 6961 6c73 2832 ndardMonomials(2 │ │ │ │ +00030330: 2c20 4d29 2020 2020 2020 2020 207c 0a7c , M) |.| │ │ │ │ +00030340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030360: 2020 2020 207c 0a7c 2020 2020 2020 2032 |.| 2 │ │ │ │ +00030370: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +00030380: 2020 2020 2020 2020 2020 2032 207c 0a7c 2 |.| │ │ │ │ +00030390: 6f35 203d 207b 6220 2c20 622a 632c 2063 o5 = {b , b*c, c │ │ │ │ +000303a0: 202c 2061 2a64 2c20 622a 642c 2063 2a64 , a*d, b*d, c*d │ │ │ │ +000303b0: 2c20 6420 7d7c 0a7c 2020 2020 2020 2020 , d }|.| │ │ │ │ +000303c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000303d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000303e0: 6f35 203a 204c 6973 7420 2020 2020 2020 o5 : List │ │ │ │ +000303f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030400: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00030410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +00030430: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +00030440: 3d0a 0a20 202a 202a 6e6f 7465 2074 6169 =.. * *note tai │ │ │ │ +00030450: 6c4d 6f6e 6f6d 6961 6c73 3a20 7461 696c lMonomials: tail │ │ │ │ +00030460: 4d6f 6e6f 6d69 616c 732c 202d 2d20 6669 Monomials, -- fi │ │ │ │ +00030470: 6e64 2074 6169 6c20 6d6f 6e6f 6d69 616c nd tail monomial │ │ │ │ +00030480: 730a 2020 2a20 2a6e 6f74 6520 736d 616c s. * *note smal │ │ │ │ +00030490: 6c65 724d 6f6e 6f6d 6961 6c73 3a20 736d lerMonomials: sm │ │ │ │ +000304a0: 616c 6c65 724d 6f6e 6f6d 6961 6c73 2c20 allerMonomials, │ │ │ │ +000304b0: 2d2d 2072 6574 7572 6e73 2074 6865 2073 -- returns the s │ │ │ │ +000304c0: 7461 6e64 6172 6420 6d6f 6e6f 6d69 616c tandard monomial │ │ │ │ +000304d0: 730a 2020 2020 736d 616c 6c65 7220 6275 s. smaller bu │ │ │ │ +000304e0: 7420 6f66 2074 6865 2073 616d 6520 6465 t of the same de │ │ │ │ +000304f0: 6772 6565 2061 7320 6769 7665 6e20 6d6f gree as given mo │ │ │ │ +00030500: 6e6f 6d69 616c 2873 290a 0a57 6179 7320 nomial(s)..Ways │ │ │ │ +00030510: 746f 2075 7365 2073 7461 6e64 6172 644d to use standardM │ │ │ │ +00030520: 6f6e 6f6d 6961 6c73 3a0a 3d3d 3d3d 3d3d onomials:.====== │ │ │ │ +00030530: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00030540: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2273 ========.. * "s │ │ │ │ +00030550: 7461 6e64 6172 644d 6f6e 6f6d 6961 6c73 tandardMonomials │ │ │ │ +00030560: 2849 6465 616c 2922 0a20 202a 2022 7374 (Ideal)". * "st │ │ │ │ +00030570: 616e 6461 7264 4d6f 6e6f 6d69 616c 7328 andardMonomials( │ │ │ │ +00030580: 4c69 7374 2c49 6465 616c 2922 0a20 202a List,Ideal)". * │ │ │ │ +00030590: 2022 7374 616e 6461 7264 4d6f 6e6f 6d69 "standardMonomi │ │ │ │ +000305a0: 616c 7328 5a5a 2c49 6465 616c 2922 0a0a als(ZZ,Ideal)".. │ │ │ │ +000305b0: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ +000305c0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ +000305d0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ +000305e0: 7420 2a6e 6f74 6520 7374 616e 6461 7264 t *note standard │ │ │ │ +000305f0: 4d6f 6e6f 6d69 616c 733a 2073 7461 6e64 Monomials: stand │ │ │ │ +00030600: 6172 644d 6f6e 6f6d 6961 6c73 2c20 6973 ardMonomials, is │ │ │ │ +00030610: 2061 202a 6e6f 7465 206d 6574 686f 640a a *note method. │ │ │ │ +00030620: 6675 6e63 7469 6f6e 3a20 284d 6163 6175 function: (Macau │ │ │ │ +00030630: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ +00030640: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +00030650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030660: 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666f 2c20 4e6f 6465 3a20 7461 a.info, Node: ta │ │ │ │ +00030740: 696c 4d6f 6e6f 6d69 616c 732c 2050 7265 ilMonomials, Pre │ │ │ │ +00030750: 763a 2073 7461 6e64 6172 644d 6f6e 6f6d v: standardMonom │ │ │ │ +00030760: 6961 6c73 2c20 5570 3a20 546f 700a 0a74 ials, Up: Top..t │ │ │ │ +00030770: 6169 6c4d 6f6e 6f6d 6961 6c73 202d 2d20 ailMonomials -- │ │ │ │ +00030780: 6669 6e64 2074 6169 6c20 6d6f 6e6f 6d69 find tail monomi │ │ │ │ +00030790: 616c 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a als.************ │ │ │ │ +000307a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000307b0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ +000307c0: 6167 653a 200a 2020 2020 2020 2020 4c20 age: . 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L = ta │ │ │ │ +000307f0: 696c 4d6f 6e6f 6d69 616c 7328 4d2c 206d ilMonomials(M, m │ │ │ │ +00030800: 290a 2020 2a20 496e 7075 7473 3a0a 2020 ). * Inputs:. │ │ │ │ +00030810: 2020 2020 2a20 4d2c 2061 6e20 2a6e 6f74 * M, an *not │ │ │ │ +00030820: 6520 6964 6561 6c3a 2028 4d61 6361 756c e ideal: (Macaul │ │ │ │ +00030830: 6179 3244 6f63 2949 6465 616c 2c2c 2024 ay2Doc)Ideal,, $ │ │ │ │ +00030840: 4d24 2073 686f 756c 6420 6265 2061 206d M$ should be a m │ │ │ │ +00030850: 6f6e 6f6d 6961 6c20 6964 6561 6c0a 2020 onomial ideal. │ │ │ │ +00030860: 2020 2020 2020 2861 6e20 6964 6561 6c20 (an ideal │ │ │ │ +00030870: 6765 6e65 7261 7465 6420 6279 206d 6f6e generated by mon │ │ │ │ +00030880: 6f6d 6961 6c73 290a 2020 2020 2020 2a20 omials). * │ │ │ │ +00030890: 6d2c 2061 202a 6e6f 7465 2072 696e 6720 m, a *note ring │ │ │ │ +000308a0: 656c 656d 656e 743a 2028 4d61 6361 756c element: (Macaul │ │ │ │ +000308b0: 6179 3244 6f63 2952 696e 6745 6c65 6d65 ay2Doc)RingEleme │ │ │ │ +000308c0: 6e74 2c2c 206f 7074 696f 6e61 6c2c 206f nt,, optional, o │ │ │ │ +000308d0: 6e6c 790a 2020 2020 2020 2020 7265 7475 nly. retu │ │ │ │ +000308e0: 726e 2061 2073 696e 676c 6520 6c69 7374 rn a single list │ │ │ │ +000308f0: 206f 6620 7468 6520 7461 696c 206d 6f6e of the tail mon │ │ │ │ +00030900: 6f6d 6961 6c73 2066 6f72 2074 6869 7320 omials for this │ │ │ │ +00030910: 6d6f 6e6f 6d69 616c 0a20 202a 202a 6e6f monomial. * *no │ │ │ │ +00030920: 7465 204f 7074 696f 6e61 6c20 696e 7075 te Optional inpu │ │ │ │ +00030930: 7473 3a20 284d 6163 6175 6c61 7932 446f ts: (Macaulay2Do │ │ │ │ +00030940: 6329 7573 696e 6720 6675 6e63 7469 6f6e c)using function │ │ │ │ +00030950: 7320 7769 7468 206f 7074 696f 6e61 6c20 s with optional │ │ │ │ +00030960: 696e 7075 7473 2c3a 0a20 2020 2020 202a inputs,:. * │ │ │ │ +00030970: 2041 6c6c 5374 616e 6461 7264 203d 3e20 AllStandard => │ │ │ │ +00030980: 6120 2a6e 6f74 6520 426f 6f6c 6561 6e20 a *note Boolean │ │ │ │ +00030990: 7661 6c75 653a 2028 4d61 6361 756c 6179 value: (Macaulay │ │ │ │ +000309a0: 3244 6f63 2942 6f6f 6c65 616e 2c2c 2064 2Doc)Boolean,, d │ │ │ │ +000309b0: 6566 6175 6c74 0a20 2020 2020 2020 2076 efault. v │ │ │ │ +000309c0: 616c 7565 2066 616c 7365 2c20 7768 6963 alue false, whic │ │ │ │ +000309d0: 6820 6d6f 6e6f 6d69 616c 7320 7368 6f75 h monomials shou │ │ │ │ +000309e0: 6c64 2062 6520 636f 6e73 6964 6572 6564 ld be considered │ │ │ │ +000309f0: 2074 6169 6c20 6d6f 6e6f 6d69 616c 7320 tail monomials │ │ │ │ +00030a00: 6f66 2061 0a20 2020 2020 2020 206d 6f6e of a. mon │ │ │ │ +00030a10: 6f6d 6961 6c20 246d 243a 2065 6974 6865 omial $m$: eithe │ │ │ │ +00030a20: 7220 616c 6c20 7374 616e 6461 7264 206d r all standard m │ │ │ │ +00030a30: 6f6e 6f6d 6961 6c73 206f 6620 6120 6769 onomials of a gi │ │ │ │ +00030a40: 7665 6e20 6465 6772 6565 2c20 6f72 2061 ven degree, or a │ │ │ │ +00030a50: 6c6c 0a20 2020 2020 2020 206d 6f6e 6f6d ll. monom │ │ │ │ +00030a60: 6961 6c73 2073 6d61 6c6c 6572 2074 6861 ials smaller tha │ │ │ │ +00030a70: 6e20 246d 2420 696e 2074 6865 2067 6976 n $m$ in the giv │ │ │ │ +00030a80: 656e 2074 6572 6d20 6f72 6465 7220 2862 en term order (b │ │ │ │ +00030a90: 7574 2073 7469 6c6c 206f 6620 7468 650a ut still of the. │ │ │ │ +00030aa0: 2020 2020 2020 2020 7361 6d65 2064 6567 same deg │ │ │ │ +00030ab0: 7265 6529 0a20 202a 204f 7574 7075 7473 ree). * Outputs │ │ │ │ +00030ac0: 3a0a 2020 2020 2020 2a20 4c2c 2061 202a :. * L, a * │ │ │ │ +00030ad0: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ +00030ae0: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ +00030af0: 6120 6c69 7374 206f 6620 6c69 7374 733a a list of lists: │ │ │ │ +00030b00: 2066 6f72 2065 6163 680a 2020 2020 2020 for each. │ │ │ │ +00030b10: 2020 6765 6e65 7261 746f 7220 246d 2420 generator $m$ │ │ │ │ +00030b20: 6f66 2024 4d24 2c20 7468 6520 6c69 7374 of $M$, the list │ │ │ │ +00030b30: 206f 6620 616c 6c20 7461 696c 206d 6f6e of all tail mon │ │ │ │ +00030b40: 6f6d 6961 6c73 2049 6620 696e 7374 6561 omials If instea │ │ │ │ +00030b50: 6420 246d 2420 6973 0a20 2020 2020 2020 d $m$ is. │ │ │ │ +00030b60: 2067 6976 656e 2c20 7468 6520 6c69 7374 given, the list │ │ │ │ +00030b70: 206f 6620 7468 6520 7461 696c 206d 6f6e of the tail mon │ │ │ │ +00030b80: 6f6d 6961 6c73 206f 6620 246d 2420 6973 omials of $m$ is │ │ │ │ +00030b90: 2072 6574 7572 6e65 640a 0a44 6573 6372 returned..Descr │ │ │ │ +00030ba0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +00030bb0: 3d3d 0a0a 496e 7075 7474 696e 6720 616e ==..Inputting an │ │ │ │ +00030bc0: 2069 6465 616c 2024 4d24 2067 656e 6572 ideal $M$ gener │ │ │ │ +00030bd0: 6174 6564 2062 7920 6d6f 6e6f 6d69 616c ated by monomial │ │ │ │ +00030be0: 7320 7265 7475 726e 7320 6120 6c69 7374 s returns a list │ │ │ │ +00030bf0: 206f 6620 6c69 7374 7320 6f66 2074 6169 of lists of tai │ │ │ │ +00030c00: 6c0a 6d6f 6e6f 6d69 616c 7320 666f 7220 l.monomials for │ │ │ │ +00030c10: 6561 6368 2067 656e 6572 6174 6f72 206f each generator o │ │ │ │ +00030c20: 6620 244d 2420 2869 6e20 7468 6520 7361 f $M$ (in the sa │ │ │ │ +00030c30: 6d65 206f 7264 6572 292e 0a0a 2b2d 2d2d me order)...+--- │ │ │ │ +00030c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ +00030c90: 3a20 5220 3d20 5a5a 2f33 3230 3033 5b61 : R = ZZ/32003[a │ │ │ │ +00030ca0: 2e2e 645d 3b20 2020 2020 2020 2020 2020 ..d]; │ │ │ │ +00030cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030cd0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00030ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ +00030d30: 3a20 4d20 3d20 6964 6561 6c20 2861 5e32 : M = ideal (a^2 │ │ │ │ +00030d40: 2c20 625e 322c 2061 2a62 2a63 293b 2020 , b^2, a*b*c); │ │ │ │ 00030d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030d70: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00030d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ -00030dd0: 3a20 7461 696c 4d6f 6e6f 6d69 616c 7328 : tailMonomials( │ │ │ │ -00030de0: 4d2c 2041 6c6c 5374 616e 6461 7264 203d M, AllStandard = │ │ │ │ -00030df0: 3e20 7472 7565 2920 2020 2020 2020 2020 > true) │ │ │ │ +00030d70: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00030d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030dc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +00030dd0: 3a20 4964 6561 6c20 6f66 2052 2020 2020 : Ideal of R │ │ │ │ +00030de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00030e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e60: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00030e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e80: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -00030e90: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -00030ea0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -00030eb0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ -00030ec0: 3d20 7b7b 612a 622c 2061 2a63 2c20 622a = {{a*b, a*c, b* │ │ │ │ -00030ed0: 632c 2063 202c 2061 2a64 2c20 622a 642c c, c , a*d, b*d, │ │ │ │ -00030ee0: 2063 2a64 2c20 6420 7d2c 207b 612a 622c c*d, d }, {a*b, │ │ │ │ -00030ef0: 2061 2a63 2c20 622a 632c 2063 202c 2061 a*c, b*c, c , a │ │ │ │ -00030f00: 2a64 2c20 622a 642c 2020 7c0a 7c20 2020 *d, b*d, |.| │ │ │ │ -00030f10: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d -------------- │ │ │ │ -00030f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030f50: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ -00030f60: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -00030f70: 3220 2020 2020 3220 2020 3320 2020 2020 2 2 3 │ │ │ │ -00030f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030f90: 2020 2032 2020 2020 2020 3220 2020 2020 2 2 │ │ │ │ -00030fa0: 3220 2020 2020 3220 2020 7c0a 7c20 2020 2 2 |.| │ │ │ │ -00030fb0: 2020 632a 642c 2064 207d 2c20 7b61 2a63 c*d, d }, {a*c │ │ │ │ -00030fc0: 202c 2062 2a63 202c 2063 202c 2061 2a62 , b*c , c , a*b │ │ │ │ -00030fd0: 2a64 2c20 612a 632a 642c 2062 2a63 2a64 *d, a*c*d, b*c*d │ │ │ │ -00030fe0: 2c20 6320 642c 2061 2a64 202c 2062 2a64 , c d, a*d , b*d │ │ │ │ -00030ff0: 202c 2063 2a64 202c 2020 7c0a 7c20 2020 , c*d , |.| │ │ │ │ -00031000: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d -------------- │ │ │ │ -00031010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031040: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ -00031050: 2020 2033 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -00031060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031090: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -000310a0: 2020 6420 7d7d 2020 2020 2020 2020 2020 d }} │ │ │ │ +00030e10: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00030e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ +00030e70: 3a20 7461 696c 4d6f 6e6f 6d69 616c 7320 : tailMonomials │ │ │ │ +00030e80: 4d20 2020 2020 2020 2020 2020 2020 2020 M │ │ │ │ +00030e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030eb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00030ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030f00: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00030f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030f20: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00030f30: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +00030f40: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +00030f50: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +00030f60: 3d20 7b7b 612a 622c 2061 2a63 2c20 622a = {{a*b, a*c, b* │ │ │ │ +00030f70: 632c 2063 202c 2061 2a64 2c20 622a 642c c, c , a*d, b*d, │ │ │ │ +00030f80: 2063 2a64 2c20 6420 7d2c 207b 612a 632c c*d, d }, {a*c, │ │ │ │ +00030f90: 2062 2a63 2c20 6320 2c20 612a 642c 2062 b*c, c , a*d, b │ │ │ │ +00030fa0: 2a64 2c20 632a 642c 2020 7c0a 7c20 2020 *d, c*d, |.| │ │ │ │ +00030fb0: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d -------------- │ │ │ │ +00030fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ +00031000: 2020 2032 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ +00031010: 2032 2020 2033 2020 2020 2020 2020 2020 2 3 │ │ │ │ +00031020: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +00031030: 2020 2020 2032 2020 2020 2032 2020 2020 2 2 │ │ │ │ +00031040: 2032 2020 2033 2020 2020 7c0a 7c20 2020 2 3 |.| │ │ │ │ +00031050: 2020 6420 7d2c 207b 612a 6320 2c20 622a d }, {a*c , b* │ │ │ │ +00031060: 6320 2c20 6320 2c20 612a 622a 642c 2061 c , c , a*b*d, a │ │ │ │ +00031070: 2a63 2a64 2c20 622a 632a 642c 2063 2064 *c*d, b*c*d, c d │ │ │ │ +00031080: 2c20 612a 6420 2c20 622a 6420 2c20 632a , a*d , b*d , c* │ │ │ │ +00031090: 6420 2c20 6420 7d7d 2020 7c0a 7c20 2020 d , d }} |.| │ │ │ │ +000310a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000310b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000310c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000310d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000310e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -000310f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000310e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +000310f0: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ 00031100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031130: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ -00031140: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ -00031150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031180: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00031190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000311a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000311b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000311c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000311d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ -000311e0: 3a20 7461 696c 4d6f 6e6f 6d69 616c 7328 : tailMonomials( │ │ │ │ -000311f0: 4d2c 2062 5e32 2920 2020 2020 2020 2020 M, b^2) │ │ │ │ +00031130: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00031140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031180: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ +00031190: 3a20 7461 696c 4d6f 6e6f 6d69 616c 7328 : tailMonomials( │ │ │ │ +000311a0: 4d2c 2041 6c6c 5374 616e 6461 7264 203d M, AllStandard = │ │ │ │ +000311b0: 3e20 7472 7565 2920 2020 2020 2020 2020 > true) │ │ │ │ +000311c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000311d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000311e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000311f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031220: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00031230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031270: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00031280: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -00031290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000312a0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000312b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000312c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ -000312d0: 3d20 7b61 2a63 2c20 622a 632c 2063 202c = {a*c, b*c, c , │ │ │ │ -000312e0: 2061 2a64 2c20 622a 642c 2063 2a64 2c20 a*d, b*d, c*d, │ │ │ │ -000312f0: 6420 7d20 2020 2020 2020 2020 2020 2020 d } │ │ │ │ -00031300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031310: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00031320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031240: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00031250: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +00031260: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +00031270: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ +00031280: 3d20 7b7b 612a 622c 2061 2a63 2c20 622a = {{a*b, a*c, b* │ │ │ │ +00031290: 632c 2063 202c 2061 2a64 2c20 622a 642c c, c , a*d, b*d, │ │ │ │ +000312a0: 2063 2a64 2c20 6420 7d2c 207b 612a 622c c*d, d }, {a*b, │ │ │ │ +000312b0: 2061 2a63 2c20 622a 632c 2063 202c 2061 a*c, b*c, c , a │ │ │ │ +000312c0: 2a64 2c20 622a 642c 2020 7c0a 7c20 2020 *d, b*d, |.| │ │ │ │ +000312d0: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d -------------- │ │ │ │ +000312e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000312f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031310: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ +00031320: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +00031330: 3220 2020 2020 3220 2020 3320 2020 2020 2 2 3 │ │ │ │ 00031340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031360: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ -00031370: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ -00031380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000313a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000313b0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -000313c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031350: 2020 2032 2020 2020 2020 3220 2020 2020 2 2 │ │ │ │ +00031360: 3220 2020 2020 3220 2020 7c0a 7c20 2020 2 2 |.| │ │ │ │ +00031370: 2020 632a 642c 2064 207d 2c20 7b61 2a63 c*d, d }, {a*c │ │ │ │ +00031380: 202c 2062 2a63 202c 2063 202c 2061 2a62 , b*c , c , a*b │ │ │ │ +00031390: 2a64 2c20 612a 632a 642c 2062 2a63 2a64 *d, a*c*d, b*c*d │ │ │ │ +000313a0: 2c20 6320 642c 2061 2a64 202c 2062 2a64 , c d, a*d , b*d │ │ │ │ +000313b0: 202c 2063 2a64 202c 2020 7c0a 7c20 2020 , c*d , |.| │ │ │ │ +000313c0: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d -------------- │ │ │ │ 000313d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000313e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000313f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031400: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ -00031410: 3a20 7461 696c 4d6f 6e6f 6d69 616c 7328 : tailMonomials( │ │ │ │ -00031420: 4d2c 2062 5e32 2c20 416c 6c53 7461 6e64 M, b^2, AllStand │ │ │ │ -00031430: 6172 643d 3e74 7275 6529 2020 2020 2020 ard=>true) │ │ │ │ +00031400: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ +00031410: 2020 2033 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00031420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031450: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00031460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031460: 2020 6420 7d7d 2020 2020 2020 2020 2020 d }} │ │ │ │ 00031470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000314a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000314b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000314c0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000314d0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +000314c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000314d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000314e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000314f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ -00031500: 3d20 7b61 2a62 2c20 612a 632c 2062 2a63 = {a*b, a*c, b*c │ │ │ │ -00031510: 2c20 6320 2c20 612a 642c 2062 2a64 2c20 , c , a*d, b*d, │ │ │ │ -00031520: 632a 642c 2064 207d 2020 2020 2020 2020 c*d, d } │ │ │ │ +000314f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ +00031500: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +00031510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031540: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00031550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031590: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ -000315a0: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ -000315b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031540: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00031550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031590: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ +000315a0: 3a20 7461 696c 4d6f 6e6f 6d69 616c 7328 : tailMonomials( │ │ │ │ +000315b0: 4d2c 2062 5e32 2920 2020 2020 2020 2020 M, b^2) │ │ │ │ 000315c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000315d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000315e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -000315f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031630: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 ----------+..See │ │ │ │ -00031640: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -00031650: 2020 2a20 2a6e 6f74 6520 7374 616e 6461 * *note standa │ │ │ │ -00031660: 7264 4d6f 6e6f 6d69 616c 733a 2073 7461 rdMonomials: sta │ │ │ │ -00031670: 6e64 6172 644d 6f6e 6f6d 6961 6c73 2c20 ndardMonomials, │ │ │ │ -00031680: 2d2d 2063 6f6d 7075 7465 7320 7374 616e -- computes stan │ │ │ │ -00031690: 6461 7264 206d 6f6e 6f6d 6961 6c73 0a20 dard monomials. │ │ │ │ -000316a0: 202a 202a 6e6f 7465 2073 6d61 6c6c 6572 * *note smaller │ │ │ │ -000316b0: 4d6f 6e6f 6d69 616c 733a 2073 6d61 6c6c Monomials: small │ │ │ │ -000316c0: 6572 4d6f 6e6f 6d69 616c 732c 202d 2d20 erMonomials, -- │ │ │ │ -000316d0: 7265 7475 726e 7320 7468 6520 7374 616e returns the stan │ │ │ │ -000316e0: 6461 7264 206d 6f6e 6f6d 6961 6c73 0a20 dard monomials. │ │ │ │ -000316f0: 2020 2073 6d61 6c6c 6572 2062 7574 206f smaller but o │ │ │ │ -00031700: 6620 7468 6520 7361 6d65 2064 6567 7265 f the same degre │ │ │ │ -00031710: 6520 6173 2067 6976 656e 206d 6f6e 6f6d e as given monom │ │ │ │ -00031720: 6961 6c28 7329 0a0a 5761 7973 2074 6f20 ial(s)..Ways to │ │ │ │ -00031730: 7573 6520 7461 696c 4d6f 6e6f 6d69 616c use tailMonomial │ │ │ │ -00031740: 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d s:.============= │ │ │ │ -00031750: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -00031760: 202a 2022 7461 696c 4d6f 6e6f 6d69 616c * "tailMonomial │ │ │ │ -00031770: 7328 4964 6561 6c29 220a 2020 2a20 2274 s(Ideal)". * "t │ │ │ │ -00031780: 6169 6c4d 6f6e 6f6d 6961 6c73 2849 6465 ailMonomials(Ide │ │ │ │ -00031790: 616c 2c52 696e 6745 6c65 6d65 6e74 2922 al,RingElement)" │ │ │ │ -000317a0: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -000317b0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -000317c0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -000317d0: 6563 7420 2a6e 6f74 6520 7461 696c 4d6f ect *note tailMo │ │ │ │ -000317e0: 6e6f 6d69 616c 733a 2074 6169 6c4d 6f6e nomials: tailMon │ │ │ │ -000317f0: 6f6d 6961 6c73 2c20 6973 2061 202a 6e6f omials, is a *no │ │ │ │ -00031800: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ -00031810: 6f6e 2077 6974 680a 6f70 7469 6f6e 733a on with.options: │ │ │ │ -00031820: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ -00031830: 6574 686f 6446 756e 6374 696f 6e57 6974 ethodFunctionWit │ │ │ │ -00031840: 684f 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d hOptions,...---- │ │ │ │ -00031850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -000318a0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -000318b0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -000318c0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -000318d0: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -000318e0: 322d 312e 3236 2e30 352b 6473 2f4d 322f 2-1.26.05+ds/M2/ │ │ │ │ -000318f0: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -00031900: 6573 2f0a 4772 6f65 626e 6572 5374 7261 es/.GroebnerStra │ │ │ │ -00031910: 7461 2e6d 323a 3536 363a 302e 0a1f 0a54 ta.m2:566:0....T │ │ │ │ -00031920: 6167 2054 6162 6c65 3a0a 4e6f 6465 3a20 ag Table:.Node: │ │ │ │ -00031930: 546f 707f 3235 370a 4e6f 6465 3a20 416c Top.257.Node: Al │ │ │ │ -00031940: 6c53 7461 6e64 6172 647f 3530 3738 390a lStandard.50789. │ │ │ │ -00031950: 4e6f 6465 3a20 6669 6e64 5765 6967 6874 Node: findWeight │ │ │ │ -00031960: 436f 6e73 7472 6169 6e74 737f 3532 3734 Constraints.5274 │ │ │ │ -00031970: 300a 4e6f 6465 3a20 6669 6e64 5765 6967 0.Node: findWeig │ │ │ │ -00031980: 6874 5665 6374 6f72 7f36 3135 3530 0a4e htVector.61550.N │ │ │ │ -00031990: 6f64 653a 2067 726f 6562 6e65 7246 616d ode: groebnerFam │ │ │ │ -000319a0: 696c 797f 3636 3236 300a 4e6f 6465 3a20 ily.66260.Node: │ │ │ │ -000319b0: 6772 6f65 626e 6572 5374 7261 7475 6d7f groebnerStratum. │ │ │ │ -000319c0: 3131 3033 3137 0a4e 6f64 653a 206c 696e 110317.Node: lin │ │ │ │ -000319d0: 6561 7250 6172 747f 3131 3935 3833 0a4e earPart.119583.N │ │ │ │ -000319e0: 6f64 653a 204d 696e 696d 616c 697a 657f ode: Minimalize. │ │ │ │ -000319f0: 3132 3136 3834 0a4e 6f64 653a 206e 6f6e 121684.Node: non │ │ │ │ -00031a00: 6d69 6e69 6d61 6c4d 6170 737f 3132 3330 minimalMaps.1230 │ │ │ │ -00031a10: 3932 0a4e 6f64 653a 2072 616e 646f 6d50 92.Node: randomP │ │ │ │ -00031a20: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ -00031a30: 7269 6574 795f 6c70 4964 6561 6c5f 7270 riety_lpIdeal_rp │ │ │ │ -00031a40: 7f31 3439 3832 360a 4e6f 6465 3a20 7261 .149826.Node: ra │ │ │ │ -00031a50: 6e64 6f6d 506f 696e 7473 4f6e 5261 7469 ndomPointsOnRati │ │ │ │ -00031a60: 6f6e 616c 5661 7269 6574 795f 6c70 4964 onalVariety_lpId │ │ │ │ -00031a70: 6561 6c5f 636d 5a5a 5f72 707f 3137 3337 eal_cmZZ_rp.1737 │ │ │ │ -00031a80: 3433 0a4e 6f64 653a 2073 6d61 6c6c 6572 43.Node: smaller │ │ │ │ -00031a90: 4d6f 6e6f 6d69 616c 737f 3138 3934 3936 Monomials.189496 │ │ │ │ -00031aa0: 0a4e 6f64 653a 2073 7461 6e64 6172 644d .Node: standardM │ │ │ │ -00031ab0: 6f6e 6f6d 6961 6c73 7f31 3933 3233 390a onomials.193239. │ │ │ │ -00031ac0: 4e6f 6465 3a20 7461 696c 4d6f 6e6f 6d69 Node: tailMonomi │ │ │ │ -00031ad0: 616c 737f 3139 3734 3637 0a1f 0a45 6e64 als.197467...End │ │ │ │ -00031ae0: 2054 6167 2054 6162 6c65 0a Tag Table. │ │ │ │ +000315e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000315f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031630: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00031640: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +00031650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031660: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00031670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031680: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ +00031690: 3d20 7b61 2a63 2c20 622a 632c 2063 202c = {a*c, b*c, c , │ │ │ │ +000316a0: 2061 2a64 2c20 622a 642c 2063 2a64 2c20 a*d, b*d, c*d, │ │ │ │ +000316b0: 6420 7d20 2020 2020 2020 2020 2020 2020 d } │ │ │ │ +000316c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000316d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000316e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000316f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031720: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ +00031730: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +00031740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031770: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00031780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000317a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000317b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000317c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ +000317d0: 3a20 7461 696c 4d6f 6e6f 6d69 616c 7328 : tailMonomials( │ │ │ │ +000317e0: 4d2c 2062 5e32 2c20 416c 6c53 7461 6e64 M, b^2, AllStand │ │ │ │ +000317f0: 6172 643d 3e74 7275 6529 2020 2020 2020 ard=>true) │ │ │ │ +00031800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031810: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00031820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031860: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00031870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031880: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00031890: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +000318a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000318b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ +000318c0: 3d20 7b61 2a62 2c20 612a 632c 2062 2a63 = {a*b, a*c, b*c │ │ │ │ +000318d0: 2c20 6320 2c20 612a 642c 2062 2a64 2c20 , c , a*d, b*d, │ │ │ │ +000318e0: 632a 642c 2064 207d 2020 2020 2020 2020 c*d, d } │ │ │ │ +000318f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031900: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00031910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031950: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ +00031960: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +00031970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000319a0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +000319b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000319c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000319d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000319e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000319f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 ----------+..See │ │ │ │ +00031a00: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ +00031a10: 2020 2a20 2a6e 6f74 6520 7374 616e 6461 * *note standa │ │ │ │ +00031a20: 7264 4d6f 6e6f 6d69 616c 733a 2073 7461 rdMonomials: sta │ │ │ │ +00031a30: 6e64 6172 644d 6f6e 6f6d 6961 6c73 2c20 ndardMonomials, │ │ │ │ +00031a40: 2d2d 2063 6f6d 7075 7465 7320 7374 616e -- computes stan │ │ │ │ +00031a50: 6461 7264 206d 6f6e 6f6d 6961 6c73 0a20 dard monomials. │ │ │ │ +00031a60: 202a 202a 6e6f 7465 2073 6d61 6c6c 6572 * *note smaller │ │ │ │ +00031a70: 4d6f 6e6f 6d69 616c 733a 2073 6d61 6c6c Monomials: small │ │ │ │ +00031a80: 6572 4d6f 6e6f 6d69 616c 732c 202d 2d20 erMonomials, -- │ │ │ │ +00031a90: 7265 7475 726e 7320 7468 6520 7374 616e returns the stan │ │ │ │ +00031aa0: 6461 7264 206d 6f6e 6f6d 6961 6c73 0a20 dard monomials. │ │ │ │ +00031ab0: 2020 2073 6d61 6c6c 6572 2062 7574 206f smaller but o │ │ │ │ +00031ac0: 6620 7468 6520 7361 6d65 2064 6567 7265 f the same degre │ │ │ │ +00031ad0: 6520 6173 2067 6976 656e 206d 6f6e 6f6d e as given monom │ │ │ │ +00031ae0: 6961 6c28 7329 0a0a 5761 7973 2074 6f20 ial(s)..Ways to │ │ │ │ +00031af0: 7573 6520 7461 696c 4d6f 6e6f 6d69 616c use tailMonomial │ │ │ │ +00031b00: 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d s:.============= │ │ │ │ +00031b10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +00031b20: 202a 2022 7461 696c 4d6f 6e6f 6d69 616c * "tailMonomial │ │ │ │ +00031b30: 7328 4964 6561 6c29 220a 2020 2a20 2274 s(Ideal)". * "t │ │ │ │ +00031b40: 6169 6c4d 6f6e 6f6d 6961 6c73 2849 6465 ailMonomials(Ide │ │ │ │ +00031b50: 616c 2c52 696e 6745 6c65 6d65 6e74 2922 al,RingElement)" │ │ │ │ +00031b60: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ +00031b70: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ +00031b80: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ +00031b90: 6563 7420 2a6e 6f74 6520 7461 696c 4d6f ect *note tailMo │ │ │ │ +00031ba0: 6e6f 6d69 616c 733a 2074 6169 6c4d 6f6e nomials: tailMon │ │ │ │ +00031bb0: 6f6d 6961 6c73 2c20 6973 2061 202a 6e6f omials, is a *no │ │ │ │ +00031bc0: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ +00031bd0: 6f6e 2077 6974 680a 6f70 7469 6f6e 733a on with.options: │ │ │ │ +00031be0: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +00031bf0: 6574 686f 6446 756e 6374 696f 6e57 6974 ethodFunctionWit │ │ │ │ +00031c00: 684f 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d hOptions,...---- │ │ │ │ +00031c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +00031c60: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +00031c70: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +00031c80: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +00031c90: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00031ca0: 322d 312e 3236 2e30 352b 6473 2f4d 322f 2-1.26.05+ds/M2/ │ │ │ │ +00031cb0: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +00031cc0: 6573 2f0a 4772 6f65 626e 6572 5374 7261 es/.GroebnerStra │ │ │ │ +00031cd0: 7461 2e6d 323a 3536 363a 302e 0a1f 0a54 ta.m2:566:0....T │ │ │ │ +00031ce0: 6167 2054 6162 6c65 3a0a 4e6f 6465 3a20 ag Table:.Node: │ │ │ │ +00031cf0: 546f 707f 3235 370a 4e6f 6465 3a20 416c Top.257.Node: Al │ │ │ │ +00031d00: 6c53 7461 6e64 6172 647f 3530 3738 390a lStandard.50789. │ │ │ │ +00031d10: 4e6f 6465 3a20 6669 6e64 5765 6967 6874 Node: findWeight │ │ │ │ +00031d20: 436f 6e73 7472 6169 6e74 737f 3532 3734 Constraints.5274 │ │ │ │ +00031d30: 300a 4e6f 6465 3a20 6669 6e64 5765 6967 0.Node: findWeig │ │ │ │ +00031d40: 6874 5665 6374 6f72 7f36 3135 3530 0a4e htVector.61550.N │ │ │ │ +00031d50: 6f64 653a 2067 726f 6562 6e65 7246 616d ode: groebnerFam │ │ │ │ +00031d60: 696c 797f 3636 3236 300a 4e6f 6465 3a20 ily.66260.Node: │ │ │ │ +00031d70: 6772 6f65 626e 6572 5374 7261 7475 6d7f groebnerStratum. │ │ │ │ +00031d80: 3131 3033 3137 0a4e 6f64 653a 206c 696e 110317.Node: lin │ │ │ │ +00031d90: 6561 7250 6172 747f 3131 3935 3833 0a4e earPart.119583.N │ │ │ │ +00031da0: 6f64 653a 204d 696e 696d 616c 697a 657f ode: Minimalize. │ │ │ │ +00031db0: 3132 3136 3834 0a4e 6f64 653a 206e 6f6e 121684.Node: non │ │ │ │ +00031dc0: 6d69 6e69 6d61 6c4d 6170 737f 3132 3330 minimalMaps.1230 │ │ │ │ +00031dd0: 3932 0a4e 6f64 653a 2072 616e 646f 6d50 92.Node: randomP │ │ │ │ +00031de0: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ +00031df0: 7269 6574 795f 6c70 4964 6561 6c5f 7270 riety_lpIdeal_rp │ │ │ │ +00031e00: 7f31 3530 3738 360a 4e6f 6465 3a20 7261 .150786.Node: ra │ │ │ │ +00031e10: 6e64 6f6d 506f 696e 7473 4f6e 5261 7469 ndomPointsOnRati │ │ │ │ +00031e20: 6f6e 616c 5661 7269 6574 795f 6c70 4964 onalVariety_lpId │ │ │ │ +00031e30: 6561 6c5f 636d 5a5a 5f72 707f 3137 3437 eal_cmZZ_rp.1747 │ │ │ │ +00031e40: 3033 0a4e 6f64 653a 2073 6d61 6c6c 6572 03.Node: smaller │ │ │ │ +00031e50: 4d6f 6e6f 6d69 616c 737f 3139 3034 3536 Monomials.190456 │ │ │ │ +00031e60: 0a4e 6f64 653a 2073 7461 6e64 6172 644d .Node: standardM │ │ │ │ +00031e70: 6f6e 6f6d 6961 6c73 7f31 3934 3139 390a onomials.194199. │ │ │ │ +00031e80: 4e6f 6465 3a20 7461 696c 4d6f 6e6f 6d69 Node: tailMonomi │ │ │ │ +00031e90: 616c 737f 3139 3834 3237 0a1f 0a45 6e64 als.198427...End │ │ │ │ +00031ea0: 2054 6167 2054 6162 6c65 0a Tag Table. │ │ ├── ./usr/share/info/GroebnerWalk.info.gz │ │ │ ├── GroebnerWalk.info │ │ │ │ @@ -207,16 +207,16 @@ │ │ │ │ 00000ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00000d10: 7c69 3520 3a20 656c 6170 7365 6454 696d |i5 : elapsedTim │ │ │ │ 00000d20: 6520 6762 2049 3220 2020 2020 2020 2020 e gb I2 │ │ │ │ 00000d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00000d50: 2020 2020 2020 7c0a 7c20 2d2d 2033 2e32 |.| -- 3.2 │ │ │ │ -00000d60: 3137 3135 7320 656c 6170 7365 6420 2020 1715s elapsed │ │ │ │ +00000d50: 2020 2020 2020 7c0a 7c20 2d2d 2032 2e32 |.| -- 2.2 │ │ │ │ +00000d60: 3332 3136 7320 656c 6170 7365 6420 2020 3216s elapsed │ │ │ │ 00000d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000d90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00000da0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00000db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -249,15 +249,15 @@ │ │ │ │ 00000f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000fa0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2065 -------+.|i6 : e │ │ │ │ 00000fb0: 6c61 7073 6564 5469 6d65 2067 726f 6562 lapsedTime groeb │ │ │ │ 00000fc0: 6e65 7257 616c 6b28 6762 2049 312c 2052 nerWalk(gb I1, R │ │ │ │ 00000fd0: 3229 2020 2020 2020 2020 2020 2020 2020 2) │ │ │ │ 00000fe0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00000ff0: 7c20 2d2d 2032 2e34 3833 3934 7320 656c | -- 2.48394s el │ │ │ │ +00000ff0: 7c20 2d2d 2031 2e36 3935 3534 7320 656c | -- 1.69554s el │ │ │ │ 00001000: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00001010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001030: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00001040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/HolonomicSystems.info.gz │ │ │ ├── HolonomicSystems.info │ │ │ │ @@ -4008,36 +4008,36 @@ │ │ │ │ 0000fa70: 7272 656e 7420 636f 6566 6669 6369 656e rrent coefficien │ │ │ │ 0000fa80: 7420 7269 6e67 206f 7220 2020 2020 2020 t ring or │ │ │ │ 0000fa90: 207c 0a7c 436f 6e76 6572 7469 6e67 2074 |.|Converting t │ │ │ │ 0000faa0: 6f20 4e61 6976 6520 616c 676f 7269 7468 o Naive algorith │ │ │ │ 0000fab0: 6d2e 2020 2020 2020 2020 2020 2020 2020 m. │ │ │ │ 0000fac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fae0: 207c 0a7c 202d 2d20 2e30 3030 3030 3337 |.| -- .0000037 │ │ │ │ -0000faf0: 3037 7320 656c 6170 7365 6420 2020 2020 07s elapsed │ │ │ │ +0000fae0: 207c 0a7c 202d 2d20 2e30 3030 3030 3735 |.| -- .0000075 │ │ │ │ +0000faf0: 3038 7320 656c 6170 7365 6420 2020 2020 08s elapsed │ │ │ │ 0000fb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fb30: 207c 0a7c 202d 2d20 2e30 3030 3030 3333 |.| -- .0000033 │ │ │ │ -0000fb40: 3436 7320 656c 6170 7365 6420 2020 2020 46s elapsed │ │ │ │ +0000fb30: 207c 0a7c 202d 2d20 2e30 3030 3030 3733 |.| -- .0000073 │ │ │ │ +0000fb40: 3635 7320 656c 6170 7365 6420 2020 2020 65s elapsed │ │ │ │ 0000fb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fb80: 207c 0a7c 202d 2d20 2e30 3030 3030 3733 |.| -- .0000073 │ │ │ │ -0000fb90: 3734 7320 656c 6170 7365 6420 2020 2020 74s elapsed │ │ │ │ +0000fb80: 207c 0a7c 202d 2d20 2e30 3030 3031 3034 |.| -- .0000104 │ │ │ │ +0000fb90: 3938 7320 656c 6170 7365 6420 2020 2020 98s elapsed │ │ │ │ 0000fba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fbd0: 207c 0a7c 202d 2d20 2e30 3030 3030 3238 |.| -- .0000028 │ │ │ │ -0000fbe0: 3635 7320 656c 6170 7365 6420 2020 2020 65s elapsed │ │ │ │ +0000fbd0: 207c 0a7c 202d 2d20 2e30 3030 3030 3535 |.| -- .0000055 │ │ │ │ +0000fbe0: 3835 7320 656c 6170 7365 6420 2020 2020 85s elapsed │ │ │ │ 0000fbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fc20: 207c 0a7c 202d 2d20 2e30 3030 3030 3138 |.| -- .0000018 │ │ │ │ -0000fc30: 3833 7320 656c 6170 7365 6420 2020 2020 83s elapsed │ │ │ │ +0000fc20: 207c 0a7c 202d 2d20 2e30 3030 3030 3636 |.| -- .0000066 │ │ │ │ +0000fc30: 3234 7320 656c 6170 7365 6420 2020 2020 24s elapsed │ │ │ │ 0000fc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc70: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000fc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5568,15 +5568,15 @@ │ │ │ │ 00015bf0: 2063 6f65 6666 6963 6965 6e74 2072 696e coefficient rin │ │ │ │ 00015c00: 6720 6f72 2069 6e68 6f6d 6f67 7c0a 7c43 g or inhomog|.|C │ │ │ │ 00015c10: 6f6e 7665 7274 696e 6720 746f 204e 6169 onverting to Nai │ │ │ │ 00015c20: 7665 2061 6c67 6f72 6974 686d 2e20 2020 ve algorithm. │ │ │ │ 00015c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015c50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00015c60: 2d2d 202e 3030 3030 3034 3238 3873 2065 -- .000004288s e │ │ │ │ +00015c60: 2d2d 202e 3030 3030 3037 3133 3473 2065 -- .000007134s e │ │ │ │ 00015c70: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 00015c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015ca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00015cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5688,15 +5688,15 @@ │ │ │ │ 00016370: 2063 6f65 6666 6963 6965 6e74 2072 696e coefficient rin │ │ │ │ 00016380: 6720 6f72 2069 6e68 6f6d 6f67 7c0a 7c43 g or inhomog|.|C │ │ │ │ 00016390: 6f6e 7665 7274 696e 6720 746f 204e 6169 onverting to Nai │ │ │ │ 000163a0: 7665 2061 6c67 6f72 6974 686d 2e20 2020 ve algorithm. │ │ │ │ 000163b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000163c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000163d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000163e0: 2d2d 202e 3030 3030 3033 3635 3773 2065 -- .000003657s e │ │ │ │ +000163e0: 2d2d 202e 3030 3030 3037 3231 3573 2065 -- .000007215s e │ │ │ │ 000163f0: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 00016400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016420: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00016430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/HyperplaneArrangements.info.gz │ │ │ ├── HyperplaneArrangements.info │ │ │ │ @@ -7210,16 +7210,16 @@ │ │ │ │ 0001c290: 3134 203a 2063 4127 2720 3d20 7472 696d 14 : cA'' = trim │ │ │ │ 0001c2a0: 2063 6f6e 6528 412c 2078 2920 2020 2020 cone(A, x) │ │ │ │ 0001c2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c2c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0001c2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c300: 2020 7c0a 7c6f 3134 203d 207b 7820 2d20 |.|o14 = {x - │ │ │ │ -0001c310: 792c 2079 2c20 787d 2020 2020 2020 2020 y, y, x} │ │ │ │ +0001c300: 2020 7c0a 7c6f 3134 203d 207b 792c 2078 |.|o14 = {y, x │ │ │ │ +0001c310: 2c20 7820 2d20 797d 2020 2020 2020 2020 , x - y} │ │ │ │ 0001c320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c330: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0001c340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c370: 2020 2020 2020 2020 7c0a 7c6f 3134 203a |.|o14 : │ │ │ │ 0001c380: 2048 7970 6572 706c 616e 6520 4172 7261 Hyperplane Arra │ │ │ │ @@ -9841,16 +9841,16 @@ │ │ │ │ 00026700: 312c 312c 312c 317d 2c31 2920 2020 2020 1,1,1,1},1) │ │ │ │ 00026710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026720: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00026730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026770: 2020 2020 7c0a 7c6f 3320 3d20 287b 7a2c |.|o3 = ({z, │ │ │ │ -00026780: 2078 2c20 7820 2d20 7a7d 2c20 7b31 2c20 x, x - z}, {1, │ │ │ │ +00026770: 2020 2020 7c0a 7c6f 3320 3d20 287b 7820 |.|o3 = ({x │ │ │ │ +00026780: 2d20 7a2c 207a 2c20 787d 2c20 7b31 2c20 - z, z, x}, {1, │ │ │ │ 00026790: 312c 2031 7d29 2020 2020 2020 2020 2020 1, 1}) │ │ │ │ 000267a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000267b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000267c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000267d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000267e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000267f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9901,15 +9901,15 @@ │ │ │ │ 00026ac0: 2064 6966 6665 7265 6e74 206d 756c 7469 different multi │ │ │ │ 00026ad0: 706c 6963 6974 6965 737c 0a7c 2020 2020 plicities|.| │ │ │ │ 00026ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b20: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ -00026b30: 7b7a 2c20 782c 2078 202d 207a 7d20 2020 {z, x, x - z} │ │ │ │ +00026b30: 7b78 202d 207a 2c20 7a2c 2078 7d20 2020 {x - z, z, x} │ │ │ │ 00026b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00026b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9957,17 +9957,17 @@ │ │ │ │ 00026e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026e50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 --------+.|i8 : │ │ │ │ 00026e60: 2841 2727 2c6d 2727 2920 3d20 6575 6c65 (A'',m'') = eule │ │ │ │ 00026e70: 7252 6573 7472 6963 7469 6f6e 2841 2c6d rRestriction(A,m │ │ │ │ 00026e80: 2c33 2920 207c 0a7c 2020 2020 2020 2020 ,3) |.| │ │ │ │ 00026e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026eb0: 2020 7c0a 7c6f 3820 3d20 287b 7a2c 2079 |.|o8 = ({z, y │ │ │ │ -00026ec0: 2c20 7920 2d20 7a7d 2c20 7b32 2c20 332c , y - z}, {2, 3, │ │ │ │ -00026ed0: 2031 7d29 2020 2020 2020 2020 2020 207c 1}) | │ │ │ │ +00026eb0: 2020 7c0a 7c6f 3820 3d20 287b 7920 2d20 |.|o8 = ({y - │ │ │ │ +00026ec0: 7a2c 207a 2c20 797d 2c20 7b31 2c20 322c z, z, y}, {1, 2, │ │ │ │ +00026ed0: 2033 7d29 2020 2020 2020 2020 2020 207c 3}) | │ │ │ │ 00026ee0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00026ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026f00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 00026f10: 3820 3a20 5365 7175 656e 6365 2020 2020 8 : Sequence │ │ │ │ 00026f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026f30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 00026f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -10057,22 +10057,22 @@ │ │ │ │ 00027480: 312c 312c 312c 312c 312c 317d 2c30 2920 1,1,1,1,1,1},0) │ │ │ │ 00027490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000274a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000274b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000274c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000274d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000274e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -000274f0: 3320 3d20 287b 7820 202b 2078 2020 2b20 3 = ({x + x + │ │ │ │ -00027500: 7820 2c20 7820 202b 2078 202c 2078 202c x , x + x , x , │ │ │ │ -00027510: 2078 2020 2b20 7820 2c20 7820 2c20 7820 x + x , x , x │ │ │ │ +000274f0: 3320 3d20 287b 7820 2c20 7820 202b 2078 3 = ({x , x + x │ │ │ │ +00027500: 202c 2078 202c 2078 202c 2078 2020 2b20 , x , x , x + │ │ │ │ +00027510: 7820 202b 2078 202c 2078 2020 2b20 7820 x + x , x + x │ │ │ │ 00027520: 7d2c 207b 312c 2031 2c20 312c 2031 2c20 }, {1, 1, 1, 1, │ │ │ │ 00027530: 312c 2031 7d29 7c0a 7c20 2020 2020 2020 1, 1})|.| │ │ │ │ -00027540: 2020 3220 2020 2033 2020 2020 3420 2020 2 3 4 │ │ │ │ -00027550: 3220 2020 2034 2020 2032 2020 2033 2020 2 4 2 3 │ │ │ │ -00027560: 2020 3420 2020 3320 2020 3420 2020 2020 4 3 4 │ │ │ │ +00027540: 2020 3220 2020 3320 2020 2034 2020 2033 2 3 4 3 │ │ │ │ +00027550: 2020 2034 2020 2032 2020 2020 3320 2020 4 2 3 │ │ │ │ +00027560: 2034 2020 2032 2020 2020 3420 2020 2020 4 2 4 │ │ │ │ 00027570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027580: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00027590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000275a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000275b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000275c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 000275d0: 3133 203a 2053 6571 7565 6e63 6520 2020 13 : Sequence │ │ │ │ @@ -18274,16 +18274,16 @@ │ │ │ │ 00047610: 2720 3d20 7472 696d 2041 2020 2020 2020 ' = trim A │ │ │ │ 00047620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047640: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00047650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047680: 207c 0a7c 6f33 203d 207b 792c 2078 2c20 |.|o3 = {y, x, │ │ │ │ -00047690: 7820 2b20 797d 2020 2020 2020 2020 2020 x + y} │ │ │ │ +00047680: 207c 0a7c 6f33 203d 207b 7820 2b20 792c |.|o3 = {x + y, │ │ │ │ +00047690: 2079 2c20 787d 2020 2020 2020 2020 2020 y, x} │ │ │ │ 000476a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000476b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000476c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000476d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000476e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000476f0: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ 00047700: 203a 2048 7970 6572 706c 616e 6520 4172 : Hyperplane Ar │ │ │ │ @@ -19190,21 +19190,21 @@ │ │ │ │ 0004af50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004af60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004af70: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0004af80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004afc0: 207c 0a7c 6f36 203d 207b 7820 202b 2078 |.|o6 = {x + x │ │ │ │ -0004afd0: 202c 2078 202c 2078 207d 2020 2020 2020 , x , x } │ │ │ │ +0004afc0: 207c 0a7c 6f36 203d 207b 7820 2c20 7820 |.|o6 = {x , x │ │ │ │ +0004afd0: 2c20 7820 202b 2078 207d 2020 2020 2020 , x + x } │ │ │ │ 0004afe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004b010: 207c 0a7c 2020 2020 2020 2031 2020 2020 |.| 1 │ │ │ │ -0004b020: 3220 2020 3220 2020 3120 2020 2020 2020 2 2 1 │ │ │ │ +0004b010: 207c 0a7c 2020 2020 2020 2032 2020 2031 |.| 2 1 │ │ │ │ +0004b020: 2020 2031 2020 2020 3220 2020 2020 2020 1 2 │ │ │ │ 0004b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b060: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0004b070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/IntegralClosure.info.gz │ │ │ ├── IntegralClosure.info │ │ │ │ @@ -4491,16 +4491,16 @@ │ │ │ │ 000118a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000118b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ 000118c0: 7469 6d65 2052 2720 3d20 696e 7465 6772 time R' = integr │ │ │ │ 000118d0: 616c 436c 6f73 7572 6520 5220 2020 2020 alClosure R │ │ │ │ 000118e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000118f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011900: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00011910: 7365 6420 302e 3336 3630 3431 7320 2863 sed 0.366041s (c │ │ │ │ -00011920: 7075 293b 2030 2e32 3838 3435 3473 2028 pu); 0.288454s ( │ │ │ │ +00011910: 7365 6420 302e 3436 3932 3832 7320 2863 sed 0.469282s (c │ │ │ │ +00011920: 7075 293b 2030 2e33 3837 3231 3673 2028 pu); 0.387216s ( │ │ │ │ 00011930: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00011940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011950: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00011960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4981,16 +4981,16 @@ │ │ │ │ 00013740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013750: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a --------+.|i10 : │ │ │ │ 00013760: 2074 696d 6520 5227 203d 2069 6e74 6567 time R' = integ │ │ │ │ 00013770: 7261 6c43 6c6f 7375 7265 2852 2c20 5374 ralClosure(R, St │ │ │ │ 00013780: 7261 7465 6779 203d 3e20 5261 6469 6361 rategy => Radica │ │ │ │ 00013790: 6c29 2020 2020 2020 2020 2020 2020 2020 l) │ │ │ │ 000137a0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000137b0: 7365 6420 302e 3437 3831 3738 7320 2863 sed 0.478178s (c │ │ │ │ -000137c0: 7075 293b 2030 2e33 3131 3530 3573 2028 pu); 0.311505s ( │ │ │ │ +000137b0: 7365 6420 302e 3535 3438 3735 7320 2863 sed 0.554875s (c │ │ │ │ +000137c0: 7075 293b 2030 2e33 3731 3231 3673 2028 pu); 0.371216s ( │ │ │ │ 000137d0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 000137e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000137f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00013800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5471,17 +5471,17 @@ │ │ │ │ 000155e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000155f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a --------+.|i16 : │ │ │ │ 00015600: 2074 696d 6520 5227 203d 2069 6e74 6567 time R' = integ │ │ │ │ 00015610: 7261 6c43 6c6f 7375 7265 2852 2c20 5374 ralClosure(R, St │ │ │ │ 00015620: 7261 7465 6779 203d 3e20 416c 6c43 6f64 rategy => AllCod │ │ │ │ 00015630: 696d 656e 7369 6f6e 7329 2020 2020 2020 imensions) │ │ │ │ 00015640: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00015650: 7365 6420 302e 3337 3035 3835 7320 2863 sed 0.370585s (c │ │ │ │ -00015660: 7075 293b 2030 2e32 3837 3735 7320 2874 pu); 0.28775s (t │ │ │ │ -00015670: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00015650: 7365 6420 302e 3433 3638 3432 7320 2863 sed 0.436842s (c │ │ │ │ +00015660: 7075 293b 2030 2e33 3538 3331 3573 2028 pu); 0.358315s ( │ │ │ │ +00015670: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00015680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015690: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000156a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000156b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000156c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000156d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000156e0: 2020 2020 2020 2020 7c0a 7c6f 3136 203d |.|o16 = │ │ │ │ @@ -5916,17 +5916,17 @@ │ │ │ │ 000171b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000171c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a --------+.|i21 : │ │ │ │ 000171d0: 2074 696d 6520 5227 203d 2069 6e74 6567 time R' = integ │ │ │ │ 000171e0: 7261 6c43 6c6f 7375 7265 2852 2c20 5374 ralClosure(R, St │ │ │ │ 000171f0: 7261 7465 6779 203d 3e20 5369 6d70 6c69 rategy => Simpli │ │ │ │ 00017200: 6679 4672 6163 7469 6f6e 7329 2020 2020 fyFractions) │ │ │ │ 00017210: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00017220: 7365 6420 302e 3533 3532 3335 7320 2863 sed 0.535235s (c │ │ │ │ -00017230: 7075 293b 2030 2e33 3832 3335 3273 2028 pu); 0.382352s ( │ │ │ │ -00017240: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00017220: 7365 6420 302e 3537 3233 7320 2863 7075 sed 0.5723s (cpu │ │ │ │ +00017230: 293b 2030 2e34 3038 3437 3773 2028 7468 ); 0.408477s (th │ │ │ │ +00017240: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00017250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017260: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00017270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000172a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000172b0: 2020 2020 2020 2020 7c0a 7c6f 3231 203d |.|o21 = │ │ │ │ @@ -6361,17 +6361,17 @@ │ │ │ │ 00018d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018d90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3236 203a --------+.|i26 : │ │ │ │ 00018da0: 2074 696d 6520 5227 203d 2069 6e74 6567 time R' = integ │ │ │ │ 00018db0: 7261 6c43 6c6f 7375 7265 2028 522c 2053 ralClosure (R, S │ │ │ │ 00018dc0: 7472 6174 6567 7920 3d3e 2052 6164 6963 trategy => Radic │ │ │ │ 00018dd0: 616c 436f 6469 6d31 2920 2020 2020 2020 alCodim1) │ │ │ │ 00018de0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00018df0: 7365 6420 302e 3830 3435 3937 7320 2863 sed 0.804597s (c │ │ │ │ -00018e00: 7075 293b 2030 2e35 3631 3732 7320 2874 pu); 0.56172s (t │ │ │ │ -00018e10: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00018df0: 7365 6420 302e 3934 3333 3034 7320 2863 sed 0.943304s (c │ │ │ │ +00018e00: 7075 293b 2030 2e36 3731 3034 3173 2028 pu); 0.671041s ( │ │ │ │ +00018e10: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00018e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00018e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e80: 2020 2020 2020 2020 7c0a 7c6f 3236 203d |.|o26 = │ │ │ │ @@ -6806,17 +6806,17 @@ │ │ │ │ 0001a950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a960: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3331 203a --------+.|i31 : │ │ │ │ 0001a970: 2074 696d 6520 5227 203d 2069 6e74 6567 time R' = integ │ │ │ │ 0001a980: 7261 6c43 6c6f 7375 7265 2028 522c 2053 ralClosure (R, S │ │ │ │ 0001a990: 7472 6174 6567 7920 3d3e 2056 6173 636f trategy => Vasco │ │ │ │ 0001a9a0: 6e63 656c 6f73 2920 2020 2020 2020 2020 ncelos) │ │ │ │ 0001a9b0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -0001a9c0: 7365 6420 302e 3338 3334 3631 7320 2863 sed 0.383461s (c │ │ │ │ -0001a9d0: 7075 293b 2030 2e33 3031 3132 3173 2028 pu); 0.301121s ( │ │ │ │ -0001a9e0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +0001a9c0: 7365 6420 302e 3432 3333 3338 7320 2863 sed 0.423338s (c │ │ │ │ +0001a9d0: 7075 293b 2030 2e33 3439 3431 7320 2874 pu); 0.34941s (t │ │ │ │ +0001a9e0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 0001a9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa00: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa50: 2020 2020 2020 2020 7c0a 7c6f 3331 203d |.|o31 = │ │ │ │ @@ -7223,17 +7223,17 @@ │ │ │ │ 0001c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c390: 2d2d 2d2d 2b0a 7c69 3336 203a 2074 696d ----+.|i36 : tim │ │ │ │ 0001c3a0: 6520 5227 203d 2069 6e74 6567 7261 6c43 e R' = integralC │ │ │ │ 0001c3b0: 6c6f 7375 7265 2052 2020 2020 2020 2020 losure R │ │ │ │ 0001c3c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001c3d0: 0a7c 202d 2d20 7573 6564 2030 2e30 3436 .| -- used 0.046 │ │ │ │ -0001c3e0: 3332 3037 7320 2863 7075 293b 2030 2e30 3207s (cpu); 0.0 │ │ │ │ -0001c3f0: 3436 3332 3139 7320 2874 6872 6561 6429 463219s (thread) │ │ │ │ +0001c3d0: 0a7c 202d 2d20 7573 6564 2030 2e30 3534 .| -- used 0.054 │ │ │ │ +0001c3e0: 3031 3831 7320 2863 7075 293b 2030 2e30 0181s (cpu); 0.0 │ │ │ │ +0001c3f0: 3534 3031 3632 7320 2874 6872 6561 6429 540162s (thread) │ │ │ │ 0001c400: 3b20 3073 2028 6763 2920 7c0a 7c20 2020 ; 0s (gc) |.| │ │ │ │ 0001c410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c440: 2020 2020 207c 0a7c 6f33 3620 3d20 5227 |.|o36 = R' │ │ │ │ 0001c450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7413,17 +7413,17 @@ │ │ │ │ 0001cf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cf50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cf60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3431 203a --------+.|i41 : │ │ │ │ 0001cf70: 2074 696d 6520 5227 203d 2069 6e74 6567 time R' = integ │ │ │ │ 0001cf80: 7261 6c43 6c6f 7375 7265 2852 2c20 5374 ralClosure(R, St │ │ │ │ 0001cf90: 7261 7465 6779 203d 3e20 5261 6469 6361 rategy => Radica │ │ │ │ 0001cfa0: 6c29 207c 0a7c 202d 2d20 7573 6564 2030 l) |.| -- used 0 │ │ │ │ -0001cfb0: 2e31 3633 3434 3973 2028 6370 7529 3b20 .163449s (cpu); │ │ │ │ -0001cfc0: 302e 3037 3634 3530 3673 2028 7468 7265 0.0764506s (thre │ │ │ │ -0001cfd0: 6164 293b 2030 7320 2867 6329 2020 7c0a ad); 0s (gc) |. │ │ │ │ +0001cfb0: 2e31 3632 3630 3473 2028 6370 7529 3b20 .162604s (cpu); │ │ │ │ +0001cfc0: 302e 3038 3837 3539 7320 2874 6872 6561 0.088759s (threa │ │ │ │ +0001cfd0: 6429 3b20 3073 2028 6763 2920 2020 7c0a d); 0s (gc) |. │ │ │ │ 0001cfe0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0001cff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d010: 2020 2020 2020 2020 207c 0a7c 6f34 3120 |.|o41 │ │ │ │ 0001d020: 3d20 5227 2020 2020 2020 2020 2020 2020 = R' │ │ │ │ 0001d030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7555,16 +7555,16 @@ │ │ │ │ 0001d820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d840: 2b0a 7c69 3436 203a 2074 696d 6520 5227 +.|i46 : time R' │ │ │ │ 0001d850: 203d 2069 6e74 6567 7261 6c43 6c6f 7375 = integralClosu │ │ │ │ 0001d860: 7265 2852 2c20 5374 7261 7465 6779 203d re(R, Strategy = │ │ │ │ 0001d870: 3e20 416c 6c43 6f64 696d 656e 7369 6f6e > AllCodimension │ │ │ │ 0001d880: 7329 7c0a 7c20 2d2d 2075 7365 6420 302e s)|.| -- used 0. │ │ │ │ -0001d890: 3036 3231 3038 3673 2028 6370 7529 3b20 0621086s (cpu); │ │ │ │ -0001d8a0: 302e 3036 3231 3038 3373 2028 7468 7265 0.0621083s (thre │ │ │ │ +0001d890: 3037 3733 3636 3373 2028 6370 7529 3b20 0773663s (cpu); │ │ │ │ +0001d8a0: 302e 3037 3733 3631 3273 2028 7468 7265 0.0773612s (thre │ │ │ │ 0001d8b0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 0001d8c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0001d8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d900: 2020 2020 2020 7c0a 7c6f 3436 203d 2052 |.|o46 = R │ │ │ │ 0001d910: 2720 2020 2020 2020 2020 2020 2020 2020 ' │ │ │ │ @@ -7698,17 +7698,17 @@ │ │ │ │ 0001e110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e130: 2d2d 2b0a 7c69 3531 203a 2074 696d 6520 --+.|i51 : time │ │ │ │ 0001e140: 5227 203d 2069 6e74 6567 7261 6c43 6c6f R' = integralClo │ │ │ │ 0001e150: 7375 7265 2028 522c 2053 7472 6174 6567 sure (R, Strateg │ │ │ │ 0001e160: 7920 3d3e 2052 6164 6963 616c 436f 6469 y => RadicalCodi │ │ │ │ 0001e170: 6d31 297c 0a7c 202d 2d20 7573 6564 2030 m1)|.| -- used 0 │ │ │ │ -0001e180: 2e31 3631 3236 7320 2863 7075 293b 2030 .16126s (cpu); 0 │ │ │ │ -0001e190: 2e30 3939 3934 3233 7320 2874 6872 6561 .0999423s (threa │ │ │ │ -0001e1a0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +0001e180: 2e31 3732 3531 3373 2028 6370 7529 3b20 .172513s (cpu); │ │ │ │ +0001e190: 302e 3039 3136 3131 3673 2028 7468 7265 0.0916116s (thre │ │ │ │ +0001e1a0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 0001e1b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0001e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e1f0: 2020 2020 207c 0a7c 6f35 3120 3d20 5227 |.|o51 = R' │ │ │ │ 0001e200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7841,17 +7841,17 @@ │ │ │ │ 0001ea00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0001ea30: 7c69 3536 203a 2074 696d 6520 5227 203d |i56 : time R' = │ │ │ │ 0001ea40: 2069 6e74 6567 7261 6c43 6c6f 7375 7265 integralClosure │ │ │ │ 0001ea50: 2028 522c 2053 7472 6174 6567 7920 3d3e (R, Strategy => │ │ │ │ 0001ea60: 2056 6173 636f 6e63 656c 6f73 297c 0a7c Vasconcelos)|.| │ │ │ │ -0001ea70: 202d 2d20 7573 6564 2030 2e30 3431 3231 -- used 0.04121 │ │ │ │ -0001ea80: 3031 7320 2863 7075 293b 2030 2e30 3431 01s (cpu); 0.041 │ │ │ │ -0001ea90: 3231 3133 7320 2874 6872 6561 6429 3b20 2113s (thread); │ │ │ │ +0001ea70: 202d 2d20 7573 6564 2030 2e30 3533 3038 -- used 0.05308 │ │ │ │ +0001ea80: 3439 7320 2863 7075 293b 2030 2e30 3532 49s (cpu); 0.052 │ │ │ │ +0001ea90: 3837 3536 7320 2874 6872 6561 6429 3b20 8756s (thread); │ │ │ │ 0001eaa0: 3073 2028 6763 2920 2020 2020 7c0a 7c20 0s (gc) |.| │ │ │ │ 0001eab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ead0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eae0: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ 0001eaf0: 3620 3d20 5227 2020 2020 2020 2020 2020 6 = R' │ │ │ │ 0001eb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8362,16 +8362,16 @@ │ │ │ │ 00020a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020aa0: 2d2d 2d2b 0a7c 6936 3720 3a20 7469 6d65 ---+.|i67 : time │ │ │ │ 00020ab0: 2052 2720 3d20 696e 7465 6772 616c 436c R' = integralCl │ │ │ │ 00020ac0: 6f73 7572 6528 522c 2053 7472 6174 6567 osure(R, Strateg │ │ │ │ 00020ad0: 7920 3d3e 2052 6164 6963 616c 2920 2020 y => Radical) │ │ │ │ 00020ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020af0: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -00020b00: 2e30 3632 3635 3634 7320 2863 7075 293b .0626564s (cpu); │ │ │ │ -00020b10: 2030 2e30 3632 3635 3633 7320 2874 6872 0.0626563s (thr │ │ │ │ +00020b00: 2e30 3732 3737 3831 7320 2863 7075 293b .0727781s (cpu); │ │ │ │ +00020b10: 2030 2e30 3732 3737 3836 7320 2874 6872 0.0727786s (thr │ │ │ │ 00020b20: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00020b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020b40: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00020b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8857,16 +8857,16 @@ │ │ │ │ 00022980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022990: 2d2d 2d2b 0a7c 6937 3820 3a20 7469 6d65 ---+.|i78 : time │ │ │ │ 000229a0: 2052 2720 3d20 696e 7465 6772 616c 436c R' = integralCl │ │ │ │ 000229b0: 6f73 7572 6528 522c 2053 7472 6174 6567 osure(R, Strateg │ │ │ │ 000229c0: 7920 3d3e 2052 6164 6963 616c 2920 2020 y => Radical) │ │ │ │ 000229d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000229e0: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -000229f0: 2e34 3939 3336 3773 2028 6370 7529 3b20 .499367s (cpu); │ │ │ │ -00022a00: 302e 3332 3036 3933 7320 2874 6872 6561 0.320693s (threa │ │ │ │ +000229f0: 2e35 3433 3637 3873 2028 6370 7529 3b20 .543678s (cpu); │ │ │ │ +00022a00: 302e 3337 3237 3537 7320 2874 6872 6561 0.372757s (threa │ │ │ │ 00022a10: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00022a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a30: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00022a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8989,17 +8989,17 @@ │ │ │ │ 000231c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000231d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000231e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000231f0: 6938 3220 3a20 7469 6d65 2052 2720 3d20 i82 : time R' = │ │ │ │ 00023200: 696e 7465 6772 616c 436c 6f73 7572 6528 integralClosure( │ │ │ │ 00023210: 522c 2053 7472 6174 6567 7920 3d3e 2041 R, Strategy => A │ │ │ │ 00023220: 6c6c 436f 6469 6d65 6e73 696f 6e73 297c llCodimensions)| │ │ │ │ -00023230: 0a7c 202d 2d20 7573 6564 2030 2e34 3736 .| -- used 0.476 │ │ │ │ -00023240: 3330 3573 2028 6370 7529 3b20 302e 3330 305s (cpu); 0.30 │ │ │ │ -00023250: 3835 3835 7320 2874 6872 6561 6429 3b20 8585s (thread); │ │ │ │ +00023230: 0a7c 202d 2d20 7573 6564 2030 2e35 3139 .| -- used 0.519 │ │ │ │ +00023240: 3636 3673 2028 6370 7529 3b20 302e 3333 666s (cpu); 0.33 │ │ │ │ +00023250: 3735 3637 7320 2874 6872 6561 6429 3b20 7567s (thread); │ │ │ │ 00023260: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00023270: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00023280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000232a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000232b0: 2020 207c 0a7c 6f38 3220 3d20 5227 2020 |.|o82 = R' │ │ │ │ 000232c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9122,41 +9122,41 @@ │ │ │ │ 00023a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00023a20: 6938 3620 3a20 7469 6d65 2052 2720 3d20 i86 : time R' = │ │ │ │ 00023a30: 696e 7465 6772 616c 436c 6f73 7572 6520 integralClosure │ │ │ │ 00023a40: 2852 2c20 5374 7261 7465 6779 203d 3e20 (R, Strategy => │ │ │ │ 00023a50: 5261 6469 6361 6c43 6f64 696d 312c 2056 RadicalCodim1, V │ │ │ │ 00023a60: 6572 626f 7369 7479 203d 3e20 207c 0a7c erbosity => |.| │ │ │ │ 00023a70: 205b 6a61 636f 6269 616e 2074 696d 6520 [jacobian time │ │ │ │ -00023a80: 2e30 3030 3535 3332 3637 2073 6563 2023 .000553267 sec # │ │ │ │ +00023a80: 2e30 3030 3634 3337 3837 2073 6563 2023 .000643787 sec # │ │ │ │ 00023a90: 6d69 6e6f 7273 2034 5d20 2020 2020 2020 minors 4] │ │ │ │ 00023aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ab0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00023ac0: 696e 7465 6772 616c 2063 6c6f 7375 7265 integral closure │ │ │ │ 00023ad0: 206e 7661 7273 2034 206e 756d 6765 6e73 nvars 4 numgens │ │ │ │ 00023ae0: 2031 2069 7320 5332 2063 6f64 696d 2031 1 is S2 codim 1 │ │ │ │ 00023af0: 2063 6f64 696d 4a20 3220 2020 2020 2020 codimJ 2 │ │ │ │ 00023b00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00023b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00023b60: 205b 7374 6570 2030 3a20 2020 7469 6d65 [step 0: time │ │ │ │ -00023b70: 202e 3130 3039 2073 6563 2020 2366 7261 .1009 sec #fra │ │ │ │ -00023b80: 6374 696f 6e73 2036 5d20 2020 2020 2020 ctions 6] │ │ │ │ +00023b70: 202e 3133 3331 3337 2073 6563 2020 2366 .133137 sec #f │ │ │ │ +00023b80: 7261 6374 696f 6e73 2036 5d20 2020 2020 ractions 6] │ │ │ │ 00023b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ba0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00023bb0: 205b 7374 6570 2031 3a20 2020 7469 6d65 [step 1: time │ │ │ │ -00023bc0: 202e 3236 3033 3237 2073 6563 2020 2366 .260327 sec #f │ │ │ │ +00023bc0: 202e 3238 3138 3633 2073 6563 2020 2366 .281863 sec #f │ │ │ │ 00023bd0: 7261 6374 696f 6e73 2036 5d20 2020 2020 ractions 6] │ │ │ │ 00023be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023c00: 202d 2d20 7573 6564 2030 2e33 3635 3035 -- used 0.36505 │ │ │ │ -00023c10: 3873 2028 6370 7529 3b20 302e 3236 3539 8s (cpu); 0.2659 │ │ │ │ -00023c20: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ +00023c00: 202d 2d20 7573 6564 2030 2e34 3139 3539 -- used 0.41959 │ │ │ │ +00023c10: 3573 2028 6370 7529 3b20 302e 3331 3530 5s (cpu); 0.3150 │ │ │ │ +00023c20: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ 00023c30: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00023c40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00023c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -9301,40 +9301,40 @@ │ │ │ │ 00024540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024550: 2b0a 7c69 3930 203a 2074 696d 6520 5227 +.|i90 : time R' │ │ │ │ 00024560: 203d 2069 6e74 6567 7261 6c43 6c6f 7375 = integralClosu │ │ │ │ 00024570: 7265 2028 522c 2053 7472 6174 6567 7920 re (R, Strategy │ │ │ │ 00024580: 3d3e 2056 6173 636f 6e63 656c 6f73 2c20 => Vasconcelos, │ │ │ │ 00024590: 5665 7262 6f73 6974 7920 3d3e 2031 297c Verbosity => 1)| │ │ │ │ 000245a0: 0a7c 205b 6a61 636f 6269 616e 2074 696d .| [jacobian tim │ │ │ │ -000245b0: 6520 2e30 3030 3534 3033 3333 2073 6563 e .000540333 sec │ │ │ │ +000245b0: 6520 2e30 3030 3733 3937 3333 2073 6563 e .000739733 sec │ │ │ │ 000245c0: 2023 6d69 6e6f 7273 2034 5d20 2020 2020 #minors 4] │ │ │ │ 000245d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000245e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000245f0: 7c69 6e74 6567 7261 6c20 636c 6f73 7572 |integral closur │ │ │ │ 00024600: 6520 6e76 6172 7320 3420 6e75 6d67 656e e nvars 4 numgen │ │ │ │ 00024610: 7320 3120 6973 2053 3220 636f 6469 6d20 s 1 is S2 codim │ │ │ │ 00024620: 3120 636f 6469 6d4a 2032 2020 2020 2020 1 codimJ 2 │ │ │ │ 00024630: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00024640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024680: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00024690: 5b73 7465 7020 303a 2020 2074 696d 6520 [step 0: time │ │ │ │ -000246a0: 2e32 3339 3636 3520 7365 6320 2023 6672 .239665 sec #fr │ │ │ │ +000246a0: 2e32 3536 3239 3420 7365 6320 2023 6672 .256294 sec #fr │ │ │ │ 000246b0: 6163 7469 6f6e 7320 365d 2020 2020 2020 actions 6] │ │ │ │ 000246c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000246d0: 2020 2020 2020 2020 2020 207c 0a7c 205b |.| [ │ │ │ │ 000246e0: 7374 6570 2031 3a20 2020 7469 6d65 202e step 1: time . │ │ │ │ -000246f0: 3235 3331 3938 2073 6563 2020 2366 7261 253198 sec #fra │ │ │ │ +000246f0: 3330 3638 3733 2073 6563 2020 2366 7261 306873 sec #fra │ │ │ │ 00024700: 6374 696f 6e73 2036 5d20 2020 2020 2020 ctions 6] │ │ │ │ 00024710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024720: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00024730: 2075 7365 6420 302e 3439 3635 3934 7320 used 0.496594s │ │ │ │ -00024740: 2863 7075 293b 2030 2e33 3136 3736 3373 (cpu); 0.316763s │ │ │ │ +00024730: 2075 7365 6420 302e 3536 3736 3939 7320 used 0.567699s │ │ │ │ +00024740: 2863 7075 293b 2030 2e33 3633 3336 3973 (cpu); 0.363369s │ │ │ │ 00024750: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 00024760: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 00024770: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00024780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000247a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000247b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9478,42 +9478,42 @@ │ │ │ │ 00025050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025060: 2d2d 2d2b 0a7c 6939 3420 3a20 7469 6d65 ---+.|i94 : time │ │ │ │ 00025070: 2052 2720 3d20 696e 7465 6772 616c 436c R' = integralCl │ │ │ │ 00025080: 6f73 7572 6520 2852 2c20 5374 7261 7465 osure (R, Strate │ │ │ │ 00025090: 6779 203d 3e20 7b56 6173 636f 6e63 656c gy => {Vasconcel │ │ │ │ 000250a0: 6f73 2c20 2020 2020 2020 2020 2020 2020 os, │ │ │ │ 000250b0: 2020 207c 0a7c 205b 6a61 636f 6269 616e |.| [jacobian │ │ │ │ -000250c0: 2074 696d 6520 2e30 3030 3837 3234 3936 time .000872496 │ │ │ │ +000250c0: 2074 696d 6520 2e30 3030 3835 3031 3433 time .000850143 │ │ │ │ 000250d0: 2073 6563 2023 6d69 6e6f 7273 2031 5d20 sec #minors 1] │ │ │ │ 000250e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000250f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025100: 2020 207c 0a7c 696e 7465 6772 616c 2063 |.|integral c │ │ │ │ 00025110: 6c6f 7375 7265 206e 7661 7273 2034 206e losure nvars 4 n │ │ │ │ 00025120: 756d 6765 6e73 2031 2069 7320 5332 2063 umgens 1 is S2 c │ │ │ │ 00025130: 6f64 696d 2031 2063 6f64 696d 4a20 3220 odim 1 codimJ 2 │ │ │ │ 00025140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025150: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00025160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000251a0: 2020 207c 0a7c 205b 7374 6570 2030 3a20 |.| [step 0: │ │ │ │ -000251b0: 2020 7469 6d65 202e 3236 3834 3334 2073 time .268434 s │ │ │ │ +000251b0: 2020 7469 6d65 202e 3330 3135 3538 2073 time .301558 s │ │ │ │ 000251c0: 6563 2020 2366 7261 6374 696f 6e73 2036 ec #fractions 6 │ │ │ │ 000251d0: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ 000251e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000251f0: 2020 207c 0a7c 205b 7374 6570 2031 3a20 |.| [step 1: │ │ │ │ -00025200: 2020 7469 6d65 202e 3730 3636 3335 2073 time .706635 s │ │ │ │ +00025200: 2020 7469 6d65 202e 3832 3635 3833 2073 time .826583 s │ │ │ │ 00025210: 6563 2020 2366 7261 6374 696f 6e73 2036 ec #fractions 6 │ │ │ │ 00025220: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ 00025230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025240: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -00025250: 2e39 3739 3631 3973 2028 6370 7529 3b20 .979619s (cpu); │ │ │ │ -00025260: 302e 3538 3735 3731 7320 2874 6872 6561 0.587571s (threa │ │ │ │ -00025270: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +00025240: 2020 207c 0a7c 202d 2d20 7573 6564 2031 |.| -- used 1 │ │ │ │ +00025250: 2e31 3332 3837 7320 2863 7075 293b 2030 .13287s (cpu); 0 │ │ │ │ +00025260: 2e37 3534 3136 3973 2028 7468 7265 6164 .754169s (thread │ │ │ │ +00025270: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 00025280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025290: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000252a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000252b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000252c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000252d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000252e0: 2020 207c 0a7c 6f39 3420 3d20 5227 2020 |.|o94 = R' │ │ │ │ @@ -10313,15 +10313,15 @@ │ │ │ │ 00028480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028490: 2b0a 7c69 3220 3a20 7469 6d65 2052 2720 +.|i2 : time R' │ │ │ │ 000284a0: 3d20 696e 7465 6772 616c 436c 6f73 7572 = integralClosur │ │ │ │ 000284b0: 6528 522c 2056 6572 626f 7369 7479 203d e(R, Verbosity = │ │ │ │ 000284c0: 3e20 3229 2020 2020 2020 2020 2020 2020 > 2) │ │ │ │ 000284d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000284e0: 7c0a 7c20 5b6a 6163 6f62 6961 6e20 7469 |.| [jacobian ti │ │ │ │ -000284f0: 6d65 202e 3030 3035 3430 3932 3420 7365 me .000540924 se │ │ │ │ +000284f0: 6d65 202e 3030 3037 3438 3332 3220 7365 me .000748322 se │ │ │ │ 00028500: 6320 236d 696e 6f72 7320 335d 2020 2020 c #minors 3] │ │ │ │ 00028510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028530: 7c0a 7c69 6e74 6567 7261 6c20 636c 6f73 |.|integral clos │ │ │ │ 00028540: 7572 6520 6e76 6172 7320 3320 6e75 6d67 ure nvars 3 numg │ │ │ │ 00028550: 656e 7320 3120 6973 2053 3220 636f 6469 ens 1 is S2 codi │ │ │ │ 00028560: 6d20 3120 636f 6469 6d4a 2032 2020 2020 m 1 codimJ 2 │ │ │ │ @@ -10334,181 +10334,181 @@ │ │ │ │ 000285d0: 7c0a 7c20 5b73 7465 7020 303a 2020 2020 |.| [step 0: │ │ │ │ 000285e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000285f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028620: 7c0a 7c20 2020 2020 2072 6164 6963 616c |.| radical │ │ │ │ 00028630: 2028 7573 6520 6d69 6e70 7269 6d65 7329 (use minprimes) │ │ │ │ -00028640: 202e 3030 3233 3734 3036 2073 6563 6f6e .00237406 secon │ │ │ │ +00028640: 202e 3030 3239 3133 3536 2073 6563 6f6e .00291356 secon │ │ │ │ 00028650: 6473 2020 2020 2020 2020 2020 2020 2020 ds │ │ │ │ 00028660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028670: 7c0a 7c20 2020 2020 2069 646c 697a 6572 |.| idlizer │ │ │ │ -00028680: 313a 2020 2e30 3037 3234 3035 3220 7365 1: .00724052 se │ │ │ │ +00028680: 313a 2020 2e30 3039 3235 3236 3320 7365 1: .00925263 se │ │ │ │ 00028690: 636f 6e64 7320 2020 2020 2020 2020 2020 conds │ │ │ │ 000286a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000286b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000286c0: 7c0a 7c20 2020 2020 2069 646c 697a 6572 |.| idlizer │ │ │ │ -000286d0: 323a 2020 2e30 3037 3936 3835 3620 7365 2: .00796856 se │ │ │ │ +000286d0: 323a 2020 2e30 3039 3839 3733 3920 7365 2: .00989739 se │ │ │ │ 000286e0: 636f 6e64 7320 2020 2020 2020 2020 2020 conds │ │ │ │ 000286f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028710: 7c0a 7c20 2020 2020 206d 696e 7072 6573 |.| minpres │ │ │ │ -00028720: 3a20 2020 2e30 3037 3534 3539 3820 7365 : .00754598 se │ │ │ │ +00028720: 3a20 2020 2e30 3039 3034 3238 3420 7365 : .00904284 se │ │ │ │ 00028730: 636f 6e64 7320 2020 2020 2020 2020 2020 conds │ │ │ │ 00028740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028760: 7c0a 7c20 2074 696d 6520 2e30 3335 3636 |.| time .03566 │ │ │ │ -00028770: 3137 2073 6563 2020 2366 7261 6374 696f 17 sec #fractio │ │ │ │ +00028760: 7c0a 7c20 2074 696d 6520 2e30 3434 3433 |.| time .04443 │ │ │ │ +00028770: 3231 2073 6563 2020 2366 7261 6374 696f 21 sec #fractio │ │ │ │ 00028780: 6e73 2034 5d20 2020 2020 2020 2020 2020 ns 4] │ │ │ │ 00028790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000287a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000287b0: 7c0a 7c20 5b73 7465 7020 313a 2020 2020 |.| [step 1: │ │ │ │ 000287c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000287d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000287e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000287f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028800: 7c0a 7c20 2020 2020 2072 6164 6963 616c |.| radical │ │ │ │ 00028810: 2028 7573 6520 6d69 6e70 7269 6d65 7329 (use minprimes) │ │ │ │ -00028820: 202e 3030 3231 3630 3534 2073 6563 6f6e .00216054 secon │ │ │ │ +00028820: 202e 3030 3237 3338 3736 2073 6563 6f6e .00273876 secon │ │ │ │ 00028830: 6473 2020 2020 2020 2020 2020 2020 2020 ds │ │ │ │ 00028840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028850: 7c0a 7c20 2020 2020 2069 646c 697a 6572 |.| idlizer │ │ │ │ -00028860: 313a 2020 2e30 3130 3733 3934 2073 6563 1: .0107394 sec │ │ │ │ +00028860: 313a 2020 2e30 3133 3439 3437 2073 6563 1: .0134947 sec │ │ │ │ 00028870: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 00028880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000288a0: 7c0a 7c20 2020 2020 2069 646c 697a 6572 |.| idlizer │ │ │ │ -000288b0: 323a 2020 2e30 3039 3738 3030 3420 7365 2: .00978004 se │ │ │ │ -000288c0: 636f 6e64 7320 2020 2020 2020 2020 2020 conds │ │ │ │ +000288b0: 323a 2020 2e30 3132 3331 3931 2073 6563 2: .0123191 sec │ │ │ │ +000288c0: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 000288d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000288e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000288f0: 7c0a 7c20 2020 2020 206d 696e 7072 6573 |.| minpres │ │ │ │ -00028900: 3a20 2020 2e30 3131 3037 3935 2073 6563 : .0110795 sec │ │ │ │ +00028900: 3a20 2020 2e30 3133 3536 3235 2073 6563 : .0135625 sec │ │ │ │ 00028910: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 00028920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028940: 7c0a 7c20 2074 696d 6520 2e30 3433 3931 |.| time .04391 │ │ │ │ -00028950: 3235 2073 6563 2020 2366 7261 6374 696f 25 sec #fractio │ │ │ │ +00028940: 7c0a 7c20 2074 696d 6520 2e30 3534 3638 |.| time .05468 │ │ │ │ +00028950: 3231 2073 6563 2020 2366 7261 6374 696f 21 sec #fractio │ │ │ │ 00028960: 6e73 2034 5d20 2020 2020 2020 2020 2020 ns 4] │ │ │ │ 00028970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028990: 7c0a 7c20 5b73 7465 7020 323a 2020 2020 |.| [step 2: │ │ │ │ 000289a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000289b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000289c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000289d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000289e0: 7c0a 7c20 2020 2020 2072 6164 6963 616c |.| radical │ │ │ │ 000289f0: 2028 7573 6520 6d69 6e70 7269 6d65 7329 (use minprimes) │ │ │ │ -00028a00: 202e 3030 3232 3135 3537 2073 6563 6f6e .00221557 secon │ │ │ │ +00028a00: 202e 3030 3238 3830 3536 2073 6563 6f6e .00288056 secon │ │ │ │ 00028a10: 6473 2020 2020 2020 2020 2020 2020 2020 ds │ │ │ │ 00028a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a30: 7c0a 7c20 2020 2020 2069 646c 697a 6572 |.| idlizer │ │ │ │ -00028a40: 313a 2020 2e31 3335 3936 2073 6563 6f6e 1: .13596 secon │ │ │ │ -00028a50: 6473 2020 2020 2020 2020 2020 2020 2020 ds │ │ │ │ +00028a40: 313a 2020 2e31 3137 3634 3920 7365 636f 1: .117649 seco │ │ │ │ +00028a50: 6e64 7320 2020 2020 2020 2020 2020 2020 nds │ │ │ │ 00028a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a80: 7c0a 7c20 2020 2020 2069 646c 697a 6572 |.| idlizer │ │ │ │ -00028a90: 323a 2020 2e30 3130 3337 3333 2073 6563 2: .0103733 sec │ │ │ │ +00028a90: 323a 2020 2e30 3132 3034 3334 2073 6563 2: .0120434 sec │ │ │ │ 00028aa0: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 00028ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ad0: 7c0a 7c20 2020 2020 206d 696e 7072 6573 |.| minpres │ │ │ │ -00028ae0: 3a20 2020 2e30 3039 3138 3335 2073 6563 : .0091835 sec │ │ │ │ +00028ae0: 3a20 2020 2e30 3131 3633 3738 2073 6563 : .0116378 sec │ │ │ │ 00028af0: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 00028b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b20: 7c0a 7c20 2074 696d 6520 2e31 3638 3339 |.| time .16839 │ │ │ │ -00028b30: 3720 7365 6320 2023 6672 6163 7469 6f6e 7 sec #fraction │ │ │ │ +00028b20: 7c0a 7c20 2074 696d 6520 2e31 3537 3437 |.| time .15747 │ │ │ │ +00028b30: 3620 7365 6320 2023 6672 6163 7469 6f6e 6 sec #fraction │ │ │ │ 00028b40: 7320 355d 2020 2020 2020 2020 2020 2020 s 5] │ │ │ │ 00028b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b70: 7c0a 7c20 5b73 7465 7020 333a 2020 2020 |.| [step 3: │ │ │ │ 00028b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028bc0: 7c0a 7c20 2020 2020 2072 6164 6963 616c |.| radical │ │ │ │ 00028bd0: 2028 7573 6520 6d69 6e70 7269 6d65 7329 (use minprimes) │ │ │ │ -00028be0: 202e 3030 3235 3139 3938 2073 6563 6f6e .00251998 secon │ │ │ │ +00028be0: 202e 3030 3239 3134 3836 2073 6563 6f6e .00291486 secon │ │ │ │ 00028bf0: 6473 2020 2020 2020 2020 2020 2020 2020 ds │ │ │ │ 00028c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c10: 7c0a 7c20 2020 2020 2069 646c 697a 6572 |.| idlizer │ │ │ │ -00028c20: 313a 2020 2e30 3133 3333 3632 2073 6563 1: .0133362 sec │ │ │ │ +00028c20: 313a 2020 2e30 3135 3037 3139 2073 6563 1: .0150719 sec │ │ │ │ 00028c30: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 00028c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c60: 7c0a 7c20 2020 2020 2069 646c 697a 6572 |.| idlizer │ │ │ │ -00028c70: 323a 2020 2e30 3134 3335 3234 2073 6563 2: .0143524 sec │ │ │ │ +00028c70: 323a 2020 2e30 3137 3035 3739 2073 6563 2: .0170579 sec │ │ │ │ 00028c80: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 00028c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028cb0: 7c0a 7c20 2020 2020 206d 696e 7072 6573 |.| minpres │ │ │ │ -00028cc0: 3a20 2020 2e30 3136 3437 3820 7365 636f : .016478 seco │ │ │ │ -00028cd0: 6e64 7320 2020 2020 2020 2020 2020 2020 nds │ │ │ │ +00028cc0: 3a20 2020 2e30 3139 3436 3832 2073 6563 : .0194682 sec │ │ │ │ +00028cd0: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 00028ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d00: 7c0a 7c20 2074 696d 6520 2e30 3539 3430 |.| time .05940 │ │ │ │ -00028d10: 3439 2073 6563 2020 2366 7261 6374 696f 49 sec #fractio │ │ │ │ +00028d00: 7c0a 7c20 2074 696d 6520 2e30 3639 3032 |.| time .06902 │ │ │ │ +00028d10: 3137 2073 6563 2020 2366 7261 6374 696f 17 sec #fractio │ │ │ │ 00028d20: 6e73 2035 5d20 2020 2020 2020 2020 2020 ns 5] │ │ │ │ 00028d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d50: 7c0a 7c20 5b73 7465 7020 343a 2020 2020 |.| [step 4: │ │ │ │ 00028d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028da0: 7c0a 7c20 2020 2020 2072 6164 6963 616c |.| radical │ │ │ │ 00028db0: 2028 7573 6520 6d69 6e70 7269 6d65 7329 (use minprimes) │ │ │ │ -00028dc0: 202e 3030 3233 3839 3632 2073 6563 6f6e .00238962 secon │ │ │ │ -00028dd0: 6473 2020 2020 2020 2020 2020 2020 2020 ds │ │ │ │ +00028dc0: 202e 3030 3331 3037 3720 7365 636f 6e64 .0031077 second │ │ │ │ +00028dd0: 7320 2020 2020 2020 2020 2020 2020 2020 s │ │ │ │ 00028de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028df0: 7c0a 7c20 2020 2020 2069 646c 697a 6572 |.| idlizer │ │ │ │ -00028e00: 313a 2020 2e30 3038 3834 3536 3520 7365 1: .00884565 se │ │ │ │ -00028e10: 636f 6e64 7320 2020 2020 2020 2020 2020 conds │ │ │ │ +00028e00: 313a 2020 2e30 3131 3132 3136 2073 6563 1: .0111216 sec │ │ │ │ +00028e10: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 00028e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e40: 7c0a 7c20 2020 2020 2069 646c 697a 6572 |.| idlizer │ │ │ │ -00028e50: 323a 2020 2e30 3136 3437 3131 2073 6563 2: .0164711 sec │ │ │ │ +00028e50: 323a 2020 2e30 3139 3530 3539 2073 6563 2: .0195059 sec │ │ │ │ 00028e60: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 00028e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e90: 7c0a 7c20 2020 2020 206d 696e 7072 6573 |.| minpres │ │ │ │ -00028ea0: 3a20 2020 2e30 3131 3738 3433 2073 6563 : .0117843 sec │ │ │ │ +00028ea0: 3a20 2020 2e30 3134 3638 3337 2073 6563 : .0146837 sec │ │ │ │ 00028eb0: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 00028ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ee0: 7c0a 7c20 2074 696d 6520 2e30 3531 3931 |.| time .05191 │ │ │ │ -00028ef0: 3536 2073 6563 2020 2366 7261 6374 696f 56 sec #fractio │ │ │ │ +00028ee0: 7c0a 7c20 2074 696d 6520 2e30 3634 3630 |.| time .06460 │ │ │ │ +00028ef0: 3933 2073 6563 2020 2366 7261 6374 696f 93 sec #fractio │ │ │ │ 00028f00: 6e73 2035 5d20 2020 2020 2020 2020 2020 ns 5] │ │ │ │ 00028f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028f30: 7c0a 7c20 5b73 7465 7020 353a 2020 2020 |.| [step 5: │ │ │ │ 00028f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028f80: 7c0a 7c20 2020 2020 2072 6164 6963 616c |.| radical │ │ │ │ 00028f90: 2028 7573 6520 6d69 6e70 7269 6d65 7329 (use minprimes) │ │ │ │ -00028fa0: 202e 3030 3235 3335 3235 2073 6563 6f6e .00253525 secon │ │ │ │ +00028fa0: 202e 3030 3330 3333 3935 2073 6563 6f6e .00303395 secon │ │ │ │ 00028fb0: 6473 2020 2020 2020 2020 2020 2020 2020 ds │ │ │ │ 00028fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028fd0: 7c0a 7c20 2020 2020 2069 646c 697a 6572 |.| idlizer │ │ │ │ -00028fe0: 313a 2020 2e30 3039 3134 3639 2073 6563 1: .0091469 sec │ │ │ │ +00028fe0: 313a 2020 2e30 3131 3834 3139 2073 6563 1: .0118419 sec │ │ │ │ 00028ff0: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 00029000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029020: 7c0a 7c20 2074 696d 6520 2e30 3138 3632 |.| time .01862 │ │ │ │ -00029030: 3537 2073 6563 2020 2366 7261 6374 696f 57 sec #fractio │ │ │ │ +00029020: 7c0a 7c20 2074 696d 6520 2e30 3234 3932 |.| time .02492 │ │ │ │ +00029030: 3934 2073 6563 2020 2366 7261 6374 696f 94 sec #fractio │ │ │ │ 00029040: 6e73 2035 5d20 2020 2020 2020 2020 2020 ns 5] │ │ │ │ 00029050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029070: 7c0a 7c20 2d2d 2075 7365 6420 302e 3338 |.| -- used 0.38 │ │ │ │ -00029080: 3138 3235 7320 2863 7075 293b 2030 2e33 1825s (cpu); 0.3 │ │ │ │ -00029090: 3032 3639 7320 2874 6872 6561 6429 3b20 0269s (thread); │ │ │ │ -000290a0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +00029070: 7c0a 7c20 2d2d 2075 7365 6420 302e 3432 |.| -- used 0.42 │ │ │ │ +00029080: 3130 3932 7320 2863 7075 293b 2030 2e33 1092s (cpu); 0.3 │ │ │ │ +00029090: 3531 3731 3973 2028 7468 7265 6164 293b 51719s (thread); │ │ │ │ +000290a0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 000290b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000290c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000290d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000290e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000290f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029110: 7c0a 7c6f 3220 3d20 5227 2020 2020 2020 |.|o2 = R' │ │ │ │ @@ -11006,18 +11006,18 @@ │ │ │ │ 0002afd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002afe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 0002aff0: 6934 203a 2074 696d 6520 696e 7465 6772 i4 : time integr │ │ │ │ 0002b000: 616c 436c 6f73 7572 6520 4a20 2020 2020 alClosure J │ │ │ │ 0002b010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b030: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002b040: 202d 2d20 7573 6564 2030 2e39 3338 3530 -- used 0.93850 │ │ │ │ -0002b050: 3973 2028 6370 7529 3b20 302e 3638 3539 9s (cpu); 0.6859 │ │ │ │ -0002b060: 3131 7320 2874 6872 6561 6429 3b20 3073 11s (thread); 0s │ │ │ │ -0002b070: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0002b040: 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7472 alClosure(J, Str │ │ │ │ 0002b380: 6174 6567 793d 3e7b 5261 6469 6361 6c43 ategy=>{RadicalC │ │ │ │ 0002b390: 6f64 696d 317d 2920 2020 2020 2020 2020 odim1}) │ │ │ │ 0002b3a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002b3b0: 202d 2d20 7573 6564 2030 2e36 3338 3739 -- used 0.63879 │ │ │ │ -0002b3c0: 3273 2028 6370 7529 3b20 302e 3436 3536 2s (cpu); 0.4656 │ │ │ │ -0002b3d0: 3338 7320 2874 6872 6561 6429 3b20 3073 38s (thread); 0s │ │ │ │ -0002b3e0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0002b3b0: 202d 2d20 7573 6564 2031 2e30 3836 3434 -- used 1.08644 │ │ │ │ +0002b3c0: 7320 2863 7075 293b 2030 2e35 3730 3430 s (cpu); 0.57040 │ │ │ │ +0002b3d0: 3873 2028 7468 7265 6164 293b 2030 7320 8s (thread); 0s │ │ │ │ +0002b3e0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0002b3f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0002b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b440: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0002b450: 2020 2020 2020 2020 2020 2020 2032 2032 2 2 │ │ ├── ./usr/share/info/InvariantRing.info.gz │ │ │ ├── InvariantRing.info │ │ │ │ @@ -3513,15 +3513,15 @@ │ │ │ │ 0000db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000db90: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2065 -------+.|i5 : e │ │ │ │ 0000dba0: 6c61 7073 6564 5469 6d65 2065 7175 6976 lapsedTime equiv │ │ │ │ 0000dbb0: 6172 6961 6e74 4869 6c62 6572 7453 6572 ariantHilbertSer │ │ │ │ 0000dbc0: 6965 7328 542c 204f 7264 6572 203d 3e20 ies(T, Order => │ │ │ │ 0000dbd0: 3529 2020 2020 2020 2020 2020 2020 2020 5) │ │ │ │ 0000dbe0: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -0000dbf0: 3036 3932 3935 3373 2065 6c61 7073 6564 0692953s elapsed │ │ │ │ +0000dbf0: 3033 3438 3938 3573 2065 6c61 7073 6564 0348985s elapsed │ │ │ 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2020 2020 │ │ │ │ 0001d100: 2020 207c 0a7c 2020 2020 2020 2032 2020 |.| 2 │ │ │ │ -0001d110: 2020 3220 2020 3220 2020 3320 2020 2020 2 2 3 │ │ │ │ +0001d110: 2032 2020 2020 3220 2020 3320 2020 2020 2 2 3 │ │ │ │ 0001d120: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0001d130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d150: 2020 207c 0a7c 6f34 203d 207b 7820 202b |.|o4 = {x + │ │ │ │ -0001d160: 2079 202c 207a 202c 2078 2079 202d 2078 y , z , x y - x │ │ │ │ +0001d150: 2020 207c 0a7c 6f34 203d 207b 7a20 2c20 |.|o4 = {z , │ │ │ │ +0001d160: 7820 202b 2079 202c 2078 2079 202d 2078 x + y , x y - x │ │ │ │ 0001d170: 2a79 207d 2020 2020 2020 2020 2020 2020 *y } │ │ │ │ 0001d180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d1a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7465,16 +7465,16 @@ │ │ │ │ 0001d280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d290: 2d2d 2d2b 0a7c 6935 203a 2074 696d 6520 ---+.|i5 : time │ │ │ │ 0001d2a0: 5032 3d70 7269 6d61 7279 496e 7661 7269 P2=primaryInvari │ │ │ │ 0001d2b0: 616e 7473 2843 3478 4332 2c44 6164 653d ants(C4xC2,Dade= │ │ │ │ 0001d2c0: 3e74 7275 6529 2020 2020 2020 2020 2020 >true) │ │ │ │ 0001d2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d2e0: 2020 207c 0a7c 202d 2d20 7573 6564 2031 |.| -- used 1 │ │ │ │ -0001d2f0: 2e32 3936 3231 7320 2863 7075 293b 2030 .29621s (cpu); 0 │ │ │ │ -0001d300: 2e35 3839 3831 3273 2028 7468 7265 6164 .589812s (thread │ │ │ │ +0001d2f0: 2e32 3233 3935 7320 2863 7075 293b 2030 .22395s (cpu); 0 │ │ │ │ +0001d300: 2e34 3837 3837 3573 2028 7468 7265 6164 .487875s (thread │ │ │ │ 0001d310: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 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2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e950: 2020 207c 0a7c 202d 2d20 7573 6564 2032 |.| -- used 2 │ │ │ │ -0001e960: 2e38 3836 3132 7320 2863 7075 293b 2031 .88612s (cpu); 1 │ │ │ │ -0001e970: 2e33 3639 3138 7320 2874 6872 6561 6429 .36918s (thread) │ │ │ │ -0001e980: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +0001e960: 2e38 3538 3232 7320 2863 7075 293b 2031 .85822s (cpu); 1 │ │ │ │ +0001e970: 2e32 3436 3173 2028 7468 7265 6164 293b .2461s (thread); │ │ │ │ +0001e980: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 0001e990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e9a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001e9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e9f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ @@ -9092,15 +9092,15 @@ │ │ │ │ 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2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9182,16 +9182,16 @@ │ │ │ │ 00023dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023df0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00023e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e40: 7c0a 7c20 2d2d 202e 3534 3234 3134 7320 |.| -- .542414s │ │ │ │ -00023e50: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ +00023e40: 7c0a 7c20 2d2d 202e 3334 3734 3873 2065 |.| -- .34748s e │ │ │ │ +00023e50: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 00023e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e90: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00023ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9712,16 +9712,16 @@ │ │ │ │ 00025ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025f10: 2d2d 2d2d 2d2b 0a7c 6934 203a 2065 6c61 -----+.|i4 : ela │ │ │ │ 00025f20: 7073 6564 5469 6d65 2069 6e76 6172 6961 psedTime invaria │ │ │ │ 00025f30: 6e74 7320 5334 2020 2020 2020 2020 2020 nts S4 │ │ │ │ 00025f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025f60: 2020 2020 207c 0a7c 202d 2d20 2e37 3234 |.| -- .724 │ │ │ │ -00025f70: 3035 3373 2065 6c61 7073 6564 2020 2020 053s elapsed │ │ │ │ +00025f60: 2020 2020 207c 0a7c 202d 2d20 2e34 3534 |.| -- .454 │ │ │ │ +00025f70: 3832 3473 2065 6c61 7073 6564 2020 2020 824s elapsed │ │ │ │ 00025f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025fb0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00025fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9777,16 +9777,16 @@ │ │ │ │ 00026300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026320: 2d2d 2d2d 2d2b 0a7c 6935 203a 2065 6c61 -----+.|i5 : ela │ │ │ │ 00026330: 7073 6564 5469 6d65 2069 6e76 6172 6961 psedTime invaria │ │ │ │ 00026340: 6e74 7328 5334 2c20 5374 7261 7465 6779 nts(S4, Strategy │ │ │ │ 00026350: 203d 3e20 224c 696e 6561 7241 6c67 6562 => "LinearAlgeb │ │ │ │ 00026360: 7261 2229 2020 2020 2020 2020 2020 2020 ra") │ │ │ │ -00026370: 2020 2020 207c 0a7c 202d 2d20 2e30 3737 |.| -- .077 │ │ 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2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d9d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0003d9e0: 2020 2020 2020 2020 2020 2020 3320 2020 3 │ │ │ │ -0003d9f0: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ -0003da00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0003da10: 6f31 203a 204d 6174 7269 7820 5a5a 2020 o1 : Matrix ZZ │ │ │ │ -0003da20: 3c2d 2d20 5a5a 2020 2020 2020 2020 2020 <-- ZZ │ │ │ │ -0003da30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003da40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0003da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003da60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003da70: 2d2b 0a7c 6932 203a 2042 3d6d 6174 7269 -+.|i2 : B=matri │ │ │ │ -0003da80: 787b 7b30 2c31 2c30 7d2c 7b31 2c30 2c30 x{{0,1,0},{1,0,0 │ │ │ │ -0003da90: 7d2c 7b30 2c30 2c31 7d7d 3b20 2020 2020 },{0,0,1}}; │ │ │ │ -0003daa0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003dab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dad0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0003dae0: 2020 2020 2020 3320 2020 2020 2020 3320 3 3 │ │ │ │ -0003daf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003db00: 2020 2020 2020 207c 0a7c 6f32 203a 204d |.|o2 : M │ │ │ │ -0003db10: 6174 7269 7820 5a5a 2020 3c2d 2d20 5a5a atrix ZZ <-- ZZ │ │ │ │ -0003db20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003db30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -0003db40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003db50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003db60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ -0003db70: 203a 2053 333d 6669 6e69 7465 4163 7469 : S3=finiteActi │ │ │ │ -0003db80: 6f6e 287b 412c 427d 2c51 515b 782c 792c on({A,B},QQ[x,y, │ │ │ │ -0003db90: 7a5d 2920 2020 2020 2020 2020 207c 0a7c z]) |.| │ │ │ │ -0003dba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dbc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003dbd0: 0a7c 6f33 203d 2051 515b 782e 2e7a 5d20 .|o3 = QQ[x..z] │ │ │ │ -0003dbe0: 3c2d 207b 7c20 3020 3120 3020 7c2c 207c <- {| 0 1 0 |, | │ │ │ │ -0003dbf0: 2030 2031 2030 207c 7d20 2020 2020 2020 0 1 0 |} │ │ │ │ -0003dc00: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003dc10: 2020 2020 2020 7c20 3020 3020 3120 7c20 | 0 0 1 | │ │ │ │ -0003dc20: 207c 2031 2030 2030 207c 2020 2020 2020 | 1 0 0 | │ │ │ │ -0003dc30: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003dc40: 2020 2020 2020 2020 7c20 3120 3020 3020 | 1 0 0 │ │ │ │ -0003dc50: 7c20 207c 2030 2030 2031 207c 2020 2020 | | 0 0 1 | │ │ │ │ -0003dc60: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0003dc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dc90: 2020 2020 2020 207c 0a7c 6f33 203a 2046 |.|o3 : F │ │ │ │ -0003dca0: 696e 6974 6547 726f 7570 4163 7469 6f6e initeGroupAction │ │ │ │ -0003dcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dcc0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -0003dcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003dce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003dcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ -0003dd00: 203a 2070 7269 6d61 7279 496e 7661 7269 : primaryInvari │ │ │ │ -0003dd10: 616e 7473 2053 3320 2020 2020 2020 2020 ants S3 │ │ │ │ -0003dd20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0003dd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003d9c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003d9d0: 2020 2020 3320 2020 2020 2020 3320 2020 3 3 │ │ │ │ +0003d9e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0003d9f0: 6f31 203a 204d 6174 7269 7820 5a5a 2020 o1 : Matrix ZZ │ │ │ │ +0003da00: 3c2d 2d20 5a5a 2020 2020 2020 2020 2020 <-- ZZ │ │ │ │ +0003da10: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0003da20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003da40: 2d2b 0a7c 6932 203a 2042 3d6d 6174 7269 -+.|i2 : B=matri │ │ │ │ +0003da50: 787b 7b30 2c31 2c30 7d2c 7b31 2c30 2c30 x{{0,1,0},{1,0,0 │ │ │ │ +0003da60: 7d2c 7b30 2c30 2c31 7d7d 3b7c 0a7c 2020 },{0,0,1}};|.| │ │ │ │ +0003da70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003da80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003da90: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003daa0: 2020 2020 2020 3320 2020 2020 2020 3320 3 3 │ │ │ │ +0003dab0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003dac0: 0a7c 6f32 203a 204d 6174 7269 7820 5a5a .|o2 : Matrix ZZ │ │ │ │ +0003dad0: 2020 3c2d 2d20 5a5a 2020 2020 2020 2020 <-- ZZ │ │ │ │ +0003dae0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0003daf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003db00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003db10: 2d2d 2d2b 0a7c 6933 203a 2053 333d 6669 ---+.|i3 : S3=fi │ │ │ │ +0003db20: 6e69 7465 4163 7469 6f6e 287b 412c 427d niteAction({A,B} │ │ │ │ +0003db30: 2c51 515b 782c 792c 7a5d 2920 207c 0a7c ,QQ[x,y,z]) |.| │ │ │ │ +0003db40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003db50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003db60: 2020 2020 2020 207c 0a7c 6f33 203d 2051 |.|o3 = Q │ │ │ │ +0003db70: 515b 782e 2e7a 5d20 3c2d 207b 7c20 3020 Q[x..z] <- {| 0 │ │ │ │ +0003db80: 3120 3020 7c2c 207c 2030 2031 2030 207c 1 0 |, | 0 1 0 | │ │ │ │ +0003db90: 7d7c 0a7c 2020 2020 2020 2020 2020 2020 }|.| │ │ │ │ +0003dba0: 2020 2020 2020 7c20 3020 3020 3120 7c20 | 0 0 1 | │ │ │ │ +0003dbb0: 207c 2031 2030 2030 207c 207c 0a7c 2020 | 1 0 0 | |.| │ │ │ │ +0003dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dbd0: 7c20 3120 3020 3020 7c20 207c 2030 2030 | 1 0 0 | | 0 0 │ │ │ │ +0003dbe0: 2031 207c 207c 0a7c 2020 2020 2020 2020 1 | |.| │ │ │ │ +0003dbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dc00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003dc10: 0a7c 6f33 203a 2046 696e 6974 6547 726f .|o3 : FiniteGro │ │ │ │ +0003dc20: 7570 4163 7469 6f6e 2020 2020 2020 2020 upAction │ │ │ │ +0003dc30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0003dc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003dc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003dc60: 2d2d 2d2b 0a7c 6934 203a 2070 7269 6d61 ---+.|i4 : prima │ │ │ │ +0003dc70: 7279 496e 7661 7269 616e 7473 2053 3320 ryInvariants S3 │ │ │ │ +0003dc80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0003dc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dcb0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0003dcc0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +0003dcd0: 2032 2020 2020 3220 2020 2020 2020 2020 2 2 │ │ │ │ +0003dce0: 207c 0a7c 6f34 203d 207b 7820 2b20 7920 |.|o4 = {x + y │ │ │ │ +0003dcf0: 2b20 7a2c 2078 2020 2b20 7920 202b 207a + z, x + y + z │ │ │ │ +0003dd00: 202c 2078 2a79 2a7a 7d20 207c 0a7c 2020 , x*y*z} |.| │ │ │ │ +0003dd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dd30: 2020 2020 207c 0a7c 6f34 203a 204c 6973 |.|o4 : Lis │ │ │ │ +0003dd40: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ 0003dd50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003dd60: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003dd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dd80: 2020 2020 2033 2020 2020 3320 2020 2033 3 3 3 │ │ │ │ -0003dd90: 207c 0a7c 6f34 203d 207b 7820 2b20 7920 |.|o4 = {x + y │ │ │ │ -0003dda0: 2b20 7a2c 2078 2a79 202b 2078 2a7a 202b + z, x*y + x*z + │ │ │ │ -0003ddb0: 2079 2a7a 2c20 7820 202b 2079 2020 2b20 y*z, x + y + │ │ │ │ -0003ddc0: 7a20 7d7c 0a7c 2020 2020 2020 2020 2020 z }|.| │ │ │ │ -0003ddd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ddf0: 2020 2020 207c 0a7c 6f34 203a 204c 6973 |.|o4 : Lis │ │ │ │ -0003de00: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -0003de10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003de20: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0003dd60: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0003dd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003dd80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 2020 2020 ---------+.. │ │ │ │ +0003dd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dda0: 2020 2020 2020 2020 2020 2020 2020 2053 S │ │ │ │ +0003ddb0: 330a 4265 6c6f 772c 2074 6865 2069 6e76 3.Below, the inv │ │ │ │ +0003ddc0: 6172 6961 6e74 2072 696e 6720 5151 5b78 ariant ring QQ[x │ │ │ │ +0003ddd0: 2c79 2c7a 5d20 2020 6973 2063 616c 6375 ,y,z] is calcu │ │ │ │ +0003dde0: 6c61 7465 6420 7769 7468 204b 2062 6569 lated with K bei │ │ │ │ +0003ddf0: 6e67 2074 6865 2066 6965 6c64 2077 6974 ng the field wit │ │ │ │ +0003de00: 680a 3130 3120 656c 656d 656e 7473 2e0a h.101 elements.. │ │ │ │ +0003de10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0003de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003de40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003de50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 2020 2020 ---------+.. │ │ │ │ -0003de60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003de70: 2020 2020 2020 2020 2020 2020 2020 2053 S │ │ │ │ -0003de80: 330a 4265 6c6f 772c 2074 6865 2069 6e76 3.Below, the inv │ │ │ │ -0003de90: 6172 6961 6e74 2072 696e 6720 5151 5b78 ariant ring QQ[x │ │ │ │ -0003dea0: 2c79 2c7a 5d20 2020 6973 2063 616c 6375 ,y,z] is calcu │ │ │ │ -0003deb0: 6c61 7465 6420 7769 7468 204b 2062 6569 lated with K bei │ │ │ │ -0003dec0: 6e67 2074 6865 2066 6965 6c64 2077 6974 ng the field wit │ │ │ │ -0003ded0: 680a 3130 3120 656c 656d 656e 7473 2e0a h.101 elements.. │ │ │ │ -0003dee0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0003def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003df00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003df10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003df20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0003df30: 0a7c 6935 203a 204b 3d47 4628 3130 3129 .|i5 : K=GF(101) │ │ │ │ -0003df40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003df50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003de50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0003de60: 0a7c 6935 203a 204b 3d47 4628 3130 3129 .|i5 : K=GF(101) │ │ │ │ +0003de70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003de80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003de90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dea0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003deb0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0003dec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ded0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003def0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003df00: 0a7c 6f35 203d 204b 2020 2020 2020 2020 .|o5 = K │ │ │ │ +0003df10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003df30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003df40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003df50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0003df60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003df70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003df80: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003df90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dfc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003dfd0: 0a7c 6f35 203d 204b 2020 2020 2020 2020 .|o5 = K │ │ │ │ -0003dfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e010: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e020: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003e030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e060: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e070: 0a7c 6f35 203a 2047 616c 6f69 7346 6965 .|o5 : GaloisFie │ │ │ │ -0003e080: 6c64 2020 2020 2020 2020 2020 2020 2020 ld │ │ │ │ -0003e090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003df70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003df80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003df90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003dfa0: 0a7c 6f35 203a 2047 616c 6f69 7346 6965 .|o5 : GaloisFie │ │ │ │ +0003dfb0: 6c64 2020 2020 2020 2020 2020 2020 2020 ld │ │ │ │ +0003dfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dfe0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003dff0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0003e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0003e040: 0a7c 6936 203a 2053 333d 6669 6e69 7465 .|i6 : S3=finite │ │ │ │ +0003e050: 4163 7469 6f6e 287b 412c 427d 2c4b 5b78 Action({A,B},K[x │ │ │ │ +0003e060: 2c79 2c7a 5d29 2020 2020 2020 2020 2020 ,y,z]) │ │ │ │ +0003e070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e080: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003e090: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0003e0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e0b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e0c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0003e0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0003e110: 0a7c 6936 203a 2053 333d 6669 6e69 7465 .|i6 : S3=finite │ │ │ │ -0003e120: 4163 7469 6f6e 287b 412c 427d 2c4b 5b78 Action({A,B},K[x │ │ │ │ -0003e130: 2c79 2c7a 5d29 2020 2020 2020 2020 2020 ,y,z]) │ │ │ │ -0003e140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e150: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e160: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003e170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e1a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e1b0: 0a7c 6f36 203d 204b 5b78 2e2e 7a5d 203c .|o6 = K[x..z] < │ │ │ │ -0003e1c0: 2d20 7b7c 2030 2031 2030 207c 2c20 7c20 - {| 0 1 0 |, | │ │ │ │ -0003e1d0: 3020 3120 3020 7c7d 2020 2020 2020 2020 0 1 0 |} │ │ │ │ +0003e0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e0d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003e0e0: 0a7c 6f36 203d 204b 5b78 2e2e 7a5d 203c .|o6 = K[x..z] < │ │ │ │ +0003e0f0: 2d20 7b7c 2030 2031 2030 207c 2c20 7c20 - {| 0 1 0 |, | │ │ │ │ +0003e100: 3020 3120 3020 7c7d 2020 2020 2020 2020 0 1 0 |} │ │ │ │ +0003e110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e120: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003e130: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0003e140: 2020 207c 2030 2030 2031 207c 2020 7c20 | 0 0 1 | | │ │ │ │ +0003e150: 3120 3020 3020 7c20 2020 2020 2020 2020 1 0 0 | │ │ │ │ +0003e160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e170: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003e180: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0003e190: 2020 207c 2031 2030 2030 207c 2020 7c20 | 1 0 0 | | │ │ │ │ +0003e1a0: 3020 3020 3120 7c20 2020 2020 2020 2020 0 0 1 | │ │ │ │ +0003e1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e1c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003e1d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0003e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e1f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e200: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003e210: 2020 207c 2030 2030 2031 207c 2020 7c20 | 0 0 1 | | │ │ │ │ -0003e220: 3120 3020 3020 7c20 2020 2020 2020 2020 1 0 0 | │ │ │ │ -0003e230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e240: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e250: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003e260: 2020 207c 2031 2030 2030 207c 2020 7c20 | 1 0 0 | | │ │ │ │ -0003e270: 3020 3020 3120 7c20 2020 2020 2020 2020 0 0 1 | │ │ │ │ -0003e280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e290: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e2a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003e2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e2c0: 2020 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---------------- │ │ │ │ +0003e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0003e2c0: 0a7c 6937 203a 2070 7269 6d61 7279 496e .|i7 : primaryIn │ │ │ │ +0003e2d0: 7661 7269 616e 7473 2853 332c 4461 6465 variants(S3,Dade │ │ │ │ +0003e2e0: 3d3e 7472 7565 2920 2020 2020 2020 2020 =>true) │ │ │ │ +0003e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e300: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003e310: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0003e320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e330: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e340: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0003e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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--------- │ │ │ │ 0003e9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0003e9d0: 0a7c 2020 2020 2020 2020 3420 3220 2020 .| 4 2 │ │ │ │ -0003e9e0: 2020 2033 2033 2020 2020 2020 3220 2020 3 3 2 │ │ │ │ -0003e9f0: 3320 2020 2020 2020 2032 2033 2020 2020 3 2 3 │ │ │ │ -0003ea00: 2020 3320 3320 2020 2020 2032 2034 2020 3 3 2 4 │ │ │ │ -0003ea10: 2020 2020 2020 2020 3420 2020 2020 207c 4 | │ │ │ │ -0003ea20: 0a7c 2020 2020 2031 3379 207a 2020 2d20 .| 13y z - │ │ │ │ -0003ea30: 3231 7820 7a20 202b 2031 3478 2079 2a7a 21x z + 14x y*z │ │ │ │ -0003ea40: 2020 2b20 3134 782a 7920 7a20 202d 2032 + 14x*y z - 2 │ │ │ │ -0003ea50: 3179 207a 2020 2b20 3133 7820 7a20 202b 1y z + 13x z + │ │ │ │ -0003ea60: 2034 3878 2a79 2a7a 2020 2b20 2020 207c 48x*y*z + | │ │ │ │ -0003ea70: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0003ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ 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U │ │ │ │ -0003f2d0: 7369 6e67 2061 2067 726f 756e 6420 6669 sing a ground fi │ │ │ │ -0003f2e0: 656c 6420 736d 616c 6c65 7220 7468 616e eld smaller than │ │ │ │ -0003f2f0: 2074 6869 7320 7275 6e73 2074 6865 2072 this runs the r │ │ │ │ -0003f300: 6973 6b20 6f66 2074 6865 0a61 6c67 6f72 isk of the.algor │ │ │ │ -0003f310: 6974 686d 2067 6574 7469 6e67 2073 7475 ithm getting stu │ │ │ │ -0003f320: 636b 2069 6e20 616e 2069 6e66 696e 6974 ck in an infinit │ │ │ │ -0003f330: 6520 6c6f 6f70 3b20 2a6e 6f74 6520 7072 e loop; *note pr │ │ │ │ -0003f340: 696d 6172 7949 6e76 6172 6961 6e74 733a imaryInvariants: │ │ │ │ -0003f350: 0a70 7269 6d61 7279 496e 7661 7269 616e .primaryInvarian │ │ │ │ -0003f360: 7473 2c20 6469 7370 6c61 7973 2061 2077 ts, displays a w │ │ │ │ -0003f370: 6172 6e69 6e67 206d 6573 7361 6765 2061 arning message a │ │ │ │ -0003f380: 736b 696e 6720 7468 6520 7573 6572 2077 sking the user w │ │ │ │ -0003f390: 6865 7468 6572 2074 6865 7920 7769 7368 hether they wish │ │ │ │ -0003f3a0: 0a74 6f20 636f 6e74 696e 7565 2077 6974 .to continue wit │ │ │ │ -0003f3b0: 6820 7468 6520 636f 6d70 7574 6174 696f h the computatio │ │ │ │ -0003f3c0: 6e20 696e 2074 6869 7320 6361 7365 2e20 n in this case. │ │ │ │ -0003f3d0: 5365 6520 2a6e 6f74 6520 6873 6f70 2061 See *note hsop a │ │ │ │ -0003f3e0: 6c67 6f72 6974 686d 733a 2068 736f 700a lgorithms: hsop. │ │ │ │ -0003f3f0: 616c 676f 7269 7468 6d73 2c20 666f 7220 algorithms, for │ │ │ │ -0003f400: 6120 6469 7363 7573 7369 6f6e 206f 6e20 a discussion on │ │ │ │ -0003f410: 7468 6520 4461 6465 2061 6c67 6f72 6974 the Dade algorit │ │ │ │ -0003f420: 686d 2e0a 0a57 6179 7320 746f 2075 7365 hm...Ways to use │ │ │ │ -0003f430: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ -0003f440: 7473 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ts:.============ │ │ │ │ -0003f450: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0003f460: 3d3d 0a0a 2020 2a20 2270 7269 6d61 7279 ==.. * "primary │ │ │ │ -0003f470: 496e 7661 7269 616e 7473 2846 696e 6974 Invariants(Finit │ │ │ │ -0003f480: 6547 726f 7570 4163 7469 6f6e 2922 0a0a eGroupAction)".. │ │ │ │ -0003f490: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -0003f4a0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -0003f4b0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -0003f4c0: 7420 2a6e 6f74 6520 7072 696d 6172 7949 t *note primaryI │ │ │ │ -0003f4d0: 6e76 6172 6961 6e74 733a 2070 7269 6d61 nvariants: prima │ │ │ │ -0003f4e0: 7279 496e 7661 7269 616e 7473 2c20 6973 ryInvariants, is │ │ │ │ -0003f4f0: 2061 202a 6e6f 7465 206d 6574 686f 640a a *note method. │ │ │ │ -0003f500: 6675 6e63 7469 6f6e 2077 6974 6820 6f70 function with op │ │ │ │ -0003f510: 7469 6f6e 733a 2028 4d61 6361 756c 6179 tions: (Macaulay │ │ │ │ -0003f520: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ -0003f530: 696f 6e57 6974 684f 7074 696f 6e73 2c2e ionWithOptions,. │ │ │ │ -0003f540: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ -0003f550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f590: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -0003f5a0: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -0003f5b0: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -0003f5c0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -0003f5d0: 6361 756c 6179 322d 312e 3236 2e30 352b caulay2-1.26.05+ │ │ │ │ -0003f5e0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -0003f5f0: 7061 636b 6167 6573 2f0a 496e 7661 7269 packages/.Invari │ │ │ │ -0003f600: 616e 7452 696e 672f 4861 7765 7344 6f63 antRing/HawesDoc │ │ │ │ -0003f610: 2e6d 323a 3137 353a 302e 0a1f 0a46 696c .m2:175:0....Fil │ │ │ │ -0003f620: 653a 2049 6e76 6172 6961 6e74 5269 6e67 e: InvariantRing │ │ │ │ -0003f630: 2e69 6e66 6f2c 204e 6f64 653a 2070 7269 .info, Node: pri │ │ │ │ -0003f640: 6d61 7279 496e 7661 7269 616e 7473 5f6c maryInvariants_l │ │ │ │ -0003f650: 705f 7064 5f70 645f 7064 5f63 6d44 6164 p_pd_pd_pd_cmDad │ │ │ │ -0003f660: 653d 3e5f 7064 5f70 645f 7064 5f72 702c e=>_pd_pd_pd_rp, │ │ │ │ -0003f670: 204e 6578 743a 2070 7269 6d61 7279 496e Next: primaryIn │ │ │ │ -0003f680: 7661 7269 616e 7473 5f6c 705f 7064 5f70 variants_lp_pd_p │ │ │ │ -0003f690: 645f 7064 5f63 6d44 6567 7265 6556 6563 d_pd_cmDegreeVec │ │ │ │ -0003f6a0: 746f 723d 3e5f 7064 5f70 645f 7064 5f72 tor=>_pd_pd_pd_r │ │ │ │ -0003f6b0: 702c 2050 7265 763a 2070 7269 6d61 7279 p, Prev: primary │ │ │ │ -0003f6c0: 496e 7661 7269 616e 7473 2c20 5570 3a20 Invariants, Up: │ │ │ │ -0003f6d0: 546f 700a 0a70 7269 6d61 7279 496e 7661 Top..primaryInva │ │ │ │ -0003f6e0: 7269 616e 7473 282e 2e2e 2c44 6164 653d riants(...,Dade= │ │ │ │ -0003f6f0: 3e2e 2e2e 2920 2d2d 2061 6e20 6f70 7469 >...) -- an opti │ │ │ │ -0003f700: 6f6e 616c 2061 7267 756d 656e 7420 666f onal argument fo │ │ │ │ -0003f710: 7220 7072 696d 6172 7949 6e76 6172 6961 r primaryInvaria │ │ │ │ -0003f720: 6e74 7320 6465 7465 726d 696e 696e 6720 nts determining │ │ │ │ -0003f730: 7768 6574 6865 7220 746f 2075 7365 2074 whether to use t │ │ │ │ -0003f740: 6865 2044 6164 6520 616c 676f 7269 7468 he Dade algorith │ │ │ │ -0003f750: 6d0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a m.************** │ │ │ │ -0003f760: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0003f770: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0003f780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0003f790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0003f7a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0003f7b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 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U │ │ │ │ +0003f200: 7369 6e67 2061 2067 726f 756e 6420 6669 sing a ground fi │ │ │ │ +0003f210: 656c 6420 736d 616c 6c65 7220 7468 616e eld smaller than │ │ │ │ +0003f220: 2074 6869 7320 7275 6e73 2074 6865 2072 this runs the r │ │ │ │ +0003f230: 6973 6b20 6f66 2074 6865 0a61 6c67 6f72 isk of the.algor │ │ │ │ +0003f240: 6974 686d 2067 6574 7469 6e67 2073 7475 ithm getting stu │ │ │ │ +0003f250: 636b 2069 6e20 616e 2069 6e66 696e 6974 ck in an infinit │ │ │ │ +0003f260: 6520 6c6f 6f70 3b20 2a6e 6f74 6520 7072 e loop; *note pr │ │ │ │ +0003f270: 696d 6172 7949 6e76 6172 6961 6e74 733a imaryInvariants: │ │ │ │ +0003f280: 0a70 7269 6d61 7279 496e 7661 7269 616e .primaryInvarian │ │ │ │ +0003f290: 7473 2c20 6469 7370 6c61 7973 2061 2077 ts, displays a w │ │ │ │ +0003f2a0: 6172 6e69 6e67 206d 6573 7361 6765 2061 arning message a │ │ │ │ +0003f2b0: 736b 696e 6720 7468 6520 7573 6572 2077 sking the user w │ │ │ │ +0003f2c0: 6865 7468 6572 2074 6865 7920 7769 7368 hether they wish │ │ │ │ +0003f2d0: 0a74 6f20 636f 6e74 696e 7565 2077 6974 .to continue wit │ │ │ │ +0003f2e0: 6820 7468 6520 636f 6d70 7574 6174 696f h the computatio │ │ │ │ +0003f2f0: 6e20 696e 2074 6869 7320 6361 7365 2e20 n in this case. │ │ │ │ +0003f300: 5365 6520 2a6e 6f74 6520 6873 6f70 2061 See *note hsop a │ │ │ │ +0003f310: 6c67 6f72 6974 686d 733a 2068 736f 700a lgorithms: hsop. │ │ │ │ +0003f320: 616c 676f 7269 7468 6d73 2c20 666f 7220 algorithms, for │ │ │ │ +0003f330: 6120 6469 7363 7573 7369 6f6e 206f 6e20 a discussion on │ │ │ │ +0003f340: 7468 6520 4461 6465 2061 6c67 6f72 6974 the Dade algorit │ │ │ │ +0003f350: 686d 2e0a 0a57 6179 7320 746f 2075 7365 hm...Ways to use │ │ │ │ +0003f360: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ +0003f370: 7473 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ts:.============ │ │ │ │ +0003f380: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0003f390: 3d3d 0a0a 2020 2a20 2270 7269 6d61 7279 ==.. * "primary │ │ │ │ +0003f3a0: 496e 7661 7269 616e 7473 2846 696e 6974 Invariants(Finit │ │ │ │ +0003f3b0: 6547 726f 7570 4163 7469 6f6e 2922 0a0a eGroupAction)".. │ │ │ │ +0003f3c0: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ +0003f3d0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ +0003f3e0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ +0003f3f0: 7420 2a6e 6f74 6520 7072 696d 6172 7949 t *note primaryI │ │ │ │ +0003f400: 6e76 6172 6961 6e74 733a 2070 7269 6d61 nvariants: prima │ │ │ │ +0003f410: 7279 496e 7661 7269 616e 7473 2c20 6973 ryInvariants, is │ │ │ │ +0003f420: 2061 202a 6e6f 7465 206d 6574 686f 640a a *note method. │ │ │ │ +0003f430: 6675 6e63 7469 6f6e 2077 6974 6820 6f70 function with op │ │ │ │ +0003f440: 7469 6f6e 733a 2028 4d61 6361 756c 6179 tions: (Macaulay │ │ │ │ +0003f450: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ +0003f460: 696f 6e57 6974 684f 7074 696f 6e73 2c2e ionWithOptions,. │ │ │ │ +0003f470: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +0003f480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003f490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003f4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003f4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003f4c0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +0003f4d0: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +0003f4e0: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +0003f4f0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +0003f500: 6361 756c 6179 322d 312e 3236 2e30 352b caulay2-1.26.05+ │ │ │ │ +0003f510: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +0003f520: 7061 636b 6167 6573 2f0a 496e 7661 7269 packages/.Invari │ │ │ │ +0003f530: 616e 7452 696e 672f 4861 7765 7344 6f63 antRing/HawesDoc │ │ │ │ +0003f540: 2e6d 323a 3137 353a 302e 0a1f 0a46 696c .m2:175:0....Fil │ │ │ │ +0003f550: 653a 2049 6e76 6172 6961 6e74 5269 6e67 e: InvariantRing │ │ │ │ +0003f560: 2e69 6e66 6f2c 204e 6f64 653a 2070 7269 .info, Node: pri │ │ │ │ +0003f570: 6d61 7279 496e 7661 7269 616e 7473 5f6c maryInvariants_l │ │ │ │ +0003f580: 705f 7064 5f70 645f 7064 5f63 6d44 6164 p_pd_pd_pd_cmDad │ │ │ │ +0003f590: 653d 3e5f 7064 5f70 645f 7064 5f72 702c e=>_pd_pd_pd_rp, │ │ │ │ +0003f5a0: 204e 6578 743a 2070 7269 6d61 7279 496e Next: primaryIn │ │ │ │ +0003f5b0: 7661 7269 616e 7473 5f6c 705f 7064 5f70 variants_lp_pd_p │ │ │ │ +0003f5c0: 645f 7064 5f63 6d44 6567 7265 6556 6563 d_pd_cmDegreeVec │ │ │ │ +0003f5d0: 746f 723d 3e5f 7064 5f70 645f 7064 5f72 tor=>_pd_pd_pd_r │ │ │ │ +0003f5e0: 702c 2050 7265 763a 2070 7269 6d61 7279 p, Prev: primary │ │ │ │ +0003f5f0: 496e 7661 7269 616e 7473 2c20 5570 3a20 Invariants, Up: │ │ │ │ +0003f600: 546f 700a 0a70 7269 6d61 7279 496e 7661 Top..primaryInva │ │ │ │ +0003f610: 7269 616e 7473 282e 2e2e 2c44 6164 653d riants(...,Dade= │ │ │ │ +0003f620: 3e2e 2e2e 2920 2d2d 2061 6e20 6f70 7469 >...) -- an opti │ │ │ │ +0003f630: 6f6e 616c 2061 7267 756d 656e 7420 666f onal argument fo │ │ │ │ +0003f640: 7220 7072 696d 6172 7949 6e76 6172 6961 r primaryInvaria │ │ │ │ +0003f650: 6e74 7320 6465 7465 726d 696e 696e 6720 nts determining │ │ │ │ +0003f660: 7768 6574 6865 7220 746f 2075 7365 2074 whether to use t │ │ │ │ +0003f670: 6865 2044 6164 6520 616c 676f 7269 7468 he Dade algorith │ │ │ │ +0003f680: 6d0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a m.************** │ │ │ │ +0003f690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0003f6a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0003f6b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0003f6c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0003f6d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0003f6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0003f6f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +0003f700: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +0003f710: 2020 2020 7072 696d 6172 7949 6e76 6172 primaryInvar │ │ │ │ +0003f720: 6961 6e74 7320 470a 2020 2a20 496e 7075 iants G. * Inpu │ │ │ │ +0003f730: 7473 3a0a 2020 2020 2020 2a20 472c 2061 ts:. * G, a │ │ │ │ +0003f740: 6e20 696e 7374 616e 6365 206f 6620 7468 n instance of th │ │ │ │ +0003f750: 6520 7479 7065 202a 6e6f 7465 2046 696e e type *note Fin │ │ │ │ +0003f760: 6974 6547 726f 7570 4163 7469 6f6e 3a20 iteGroupAction: │ │ │ │ +0003f770: 4669 6e69 7465 4772 6f75 7041 6374 696f FiniteGroupActio │ │ │ │ +0003f780: 6e2c 0a20 202a 204f 7574 7075 7473 3a0a n,. * Outputs:. │ │ │ │ +0003f790: 2020 2020 2020 2a20 6120 2a6e 6f74 6520 * a *note │ │ │ │ +0003f7a0: 6c69 7374 3a20 284d 6163 6175 6c61 7932 list: (Macaulay2 │ │ │ │ +0003f7b0: 446f 6329 4c69 7374 2c2c 2020 636f 6e73 Doc)List,, cons │ │ │ │ +0003f7c0: 6973 7469 6e67 206f 6620 6120 686f 6d6f isting of a homo │ │ │ │ +0003f7d0: 6765 6e65 6f75 7320 7379 7374 656d 0a20 geneous system. │ │ │ │ +0003f7e0: 2020 2020 2020 206f 6620 7061 7261 6d65 of parame │ │ │ │ +0003f7f0: 7465 7273 2028 6873 6f70 2920 666f 7220 ters (hsop) for │ │ │ │ +0003f800: 7468 6520 696e 7661 7269 616e 7420 7269 the invariant ri │ │ │ │ +0003f810: 6e67 206f 6620 7468 6520 6772 6f75 7020 ng of the group │ │ │ │ +0003f820: 6163 7469 6f6e 0a0a 4465 7363 7269 7074 action..Descript │ │ │ │ +0003f830: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +0003f840: 0a2a 6e6f 7465 2044 6164 653a 2070 7269 .*note Dade: pri │ │ │ │ +0003f850: 6d61 7279 496e 7661 7269 616e 7473 5f6c maryInvariants_l │ │ │ │ +0003f860: 705f 7064 5f70 645f 7064 5f63 6d44 6164 p_pd_pd_pd_cmDad │ │ │ │ +0003f870: 653d 3e5f 7064 5f70 645f 7064 5f72 702c e=>_pd_pd_pd_rp, │ │ │ │ +0003f880: 2074 616b 6573 202a 6e6f 7465 0a42 6f6f takes *note.Boo │ │ │ │ +0003f890: 6c65 616e 3a20 284d 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6d61 7279 en *note primary │ │ │ │ +0003f970: 496e 7661 7269 616e 7473 3a20 7072 696d Invariants: prim │ │ │ │ +0003f980: 6172 7949 6e76 6172 6961 6e74 732c 2077 aryInvariants, w │ │ │ │ +0003f990: 696c 6c20 7573 650a 7468 6520 4461 6465 ill use.the Dade │ │ │ │ +0003f9a0: 2061 6c67 6f72 6974 686d 2074 6f20 6361 algorithm to ca │ │ │ │ +0003f9b0: 6c63 756c 6174 6520 6120 686f 6d6f 6765 lculate a homoge │ │ │ │ +0003f9c0: 6e65 6f75 7320 7379 7374 656d 206f 6620 neous system of │ │ │ │ +0003f9d0: 7061 7261 6d65 7465 7273 2028 6873 6f70 parameters (hsop │ │ │ │ +0003f9e0: 2920 666f 720a 7468 6520 696e 7661 7269 ) for.the invari │ │ │ │ +0003f9f0: 616e 7420 7269 6e67 206f 6620 6120 6669 ant ring of a fi │ │ │ │ +0003fa00: 6e69 7465 2067 726f 7570 2e0a 0a54 6865 nite group...The │ │ │ │ +0003fa10: 2065 7861 6d70 6c65 2062 656c 6f77 2063 example below c │ │ │ │ +0003fa20: 6f6d 7075 7465 7320 7468 6520 696e 7661 omputes the inva │ │ │ │ +0003fa30: 7269 616e 7420 7269 6e67 206f 6620 5333 riant 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2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00040a30: 207c 0a7c 2020 2020 2039 3131 3334 7820 |.| 91134x │ │ │ │ +00040a40: 7920 7a20 2b20 3334 3530 3678 2a79 207a y z + 34506x*y z │ │ │ │ +00040a50: 202b 2033 3838 3879 207a 202b 2031 3533 + 3888y z + 153 │ │ │ │ +00040a60: 3237 7820 7a20 202b 2039 3131 3334 7820 27x z + 91134x │ │ │ │ +00040a70: 792a 7a20 202b 2020 2020 2020 2020 2020 y*z + │ │ │ │ +00040a80: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ 00040a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00040ab0: 2d7c 0a7c 2020 2020 2020 2020 2020 2032 -|.| 2 │ │ │ │ -00040ac0: 2033 2020 2020 2020 2020 2020 2020 3420 3 4 │ │ │ │ -00040ad0: 2020 2020 2020 2020 3520 2020 2020 2020 5 │ │ │ │ -00040ae0: 2020 2034 2032 2020 2020 2020 2020 2033 4 2 3 │ │ │ │ -00040af0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00040b00: 207c 0a7c 2020 2020 2039 3131 3334 7820 |.| 91134x │ │ │ │ -00040b10: 7920 7a20 2b20 3334 3530 3678 2a79 207a y z + 34506x*y z │ │ │ │ -00040b20: 202b 2033 3838 3879 207a 202b 2031 3533 + 3888y z + 153 │ │ │ │ -00040b30: 3237 7820 7a20 202b 2039 3131 3334 7820 27x z + 91134x │ │ │ │ -00040b40: 792a 7a20 202b 2020 2020 2020 2020 2020 y*z + │ │ │ │ -00040b50: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00040b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00040b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00040ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00040ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00040ad0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ +00040ae0: 3220 3220 3220 2020 2020 2020 2020 2020 2 2 2 │ │ │ │ +00040af0: 3320 3220 2020 2020 2020 2020 3420 3220 3 2 4 2 │ │ │ │ +00040b00: 2020 2020 2020 2020 3320 3320 2020 2020 3 3 │ │ │ │ +00040b10: 2020 2020 3220 2020 3320 2020 2020 2020 2 3 │ │ │ │ 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2020 2b20 3931 3133 3478 2a79 y z + 91134x*y │ │ │ │ -00040c10: 207a 2020 2b20 3135 3332 3779 207a 2020 z + 15327y z │ │ │ │ -00040c20: 2b20 3234 3432 3678 207a 2020 2b20 3931 + 24426x z + 91 │ │ │ │ -00040c30: 3133 3478 2079 2a7a 2020 2b20 2020 2020 134x y*z + │ │ │ │ -00040c40: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00040c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00040c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00040ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00040bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00040bc0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ +00040bd0: 2032 2033 2020 2020 2020 2020 2033 2033 2 3 3 3 │ │ │ │ +00040be0: 2020 2020 2020 2020 2032 2034 2020 2020 2 4 │ │ │ │ +00040bf0: 2020 2020 2020 2020 2034 2020 2020 2020 4 │ │ │ │ +00040c00: 2020 2032 2034 2020 2020 2020 2020 2020 2 4 │ │ │ │ +00040c10: 357c 0a7c 2020 2020 2039 3131 3334 782a 5|.| 91134x* │ │ │ │ +00040c20: 7920 7a20 202b 2032 3434 3236 7920 7a20 y z + 24426y z │ │ │ │ +00040c30: 202b 2031 3533 3237 7820 7a20 202b 2033 + 15327x z + 3 │ │ │ │ +00040c40: 3435 3036 782a 792a 7a20 202b 2031 3533 4506x*y*z + 153 │ │ │ │ +00040c50: 3237 7920 7a20 202b 2033 3838 3878 2a7a 27y z + 3888x*z │ │ │ │ +00040c60: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ 00040c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00040c90: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ -00040ca0: 2032 2033 2020 2020 2020 2020 2033 2033 2 3 3 3 │ │ │ │ -00040cb0: 2020 2020 2020 2020 2032 2034 2020 2020 2 4 │ │ │ │ -00040cc0: 2020 2020 2020 2020 2034 2020 2020 2020 4 │ │ │ │ -00040cd0: 2020 2032 2034 2020 2020 2020 2020 2020 2 4 │ │ │ │ -00040ce0: 357c 0a7c 2020 2020 2039 3131 3334 782a 5|.| 91134x* │ │ │ │ -00040cf0: 7920 7a20 202b 2032 3434 3236 7920 7a20 y z 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│ │ +000414e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000414f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041520: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00041530: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +00041540: 2020 2032 2020 2020 3220 2020 3220 2020 2 2 2 │ │ │ │ +00041550: 2020 2020 3220 2020 2032 2020 2020 2032 2 2 2 │ │ │ │ +00041560: 2020 2020 2020 2032 2020 2020 2020 3220 2 2 │ │ │ │ +00041570: 7c0a 7c6f 3520 3d20 7b78 202b 2079 202b |.|o5 = {x + y + │ │ │ │ +00041580: 207a 2c20 7820 202b 2079 2020 2b20 7a20 z, x + y + z │ │ │ │ +00041590: 2c20 7820 7920 2b20 782a 7920 202b 2078 , x y + x*y + x │ │ │ │ +000415a0: 207a 202b 2079 207a 202b 2078 2a7a 2020 z + y z + x*z │ │ │ │ +000415b0: 2b20 792a 7a20 7d7c 0a7c 2020 2020 2020 + y*z }|.| │ │ │ │ 000415c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000415d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000415e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000415f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00041600: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -00041610: 2020 2032 2020 2020 3220 2020 3220 2020 2 2 2 │ │ │ │ -00041620: 2020 2020 3220 2020 2032 2020 2020 2032 2 2 2 │ │ │ │ -00041630: 2020 2020 2020 2032 2020 2020 2020 3220 2 2 │ │ │ │ -00041640: 7c0a 7c6f 3520 3d20 7b78 202b 2079 202b |.|o5 = {x + y + │ │ │ │ -00041650: 207a 2c20 7820 202b 2079 2020 2b20 7a20 z, x + y + z │ │ │ │ -00041660: 2c20 7820 7920 2b20 782a 7920 202b 2078 , x y + x*y + x │ │ │ │ -00041670: 207a 202b 2079 207a 202b 2078 2a7a 2020 z + y z + x*z │ │ │ │ -00041680: 2b20 792a 7a20 7d7c 0a7c 2020 2020 2020 + y*z }|.| │ │ │ │ +000415f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00041600: 7c6f 3520 3a20 4c69 7374 2020 2020 2020 |o5 : List │ │ │ │ +00041610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041640: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00041650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a20 ------------+.. │ │ │ │ 00041690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000416a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000416b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000416c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000416d0: 7c6f 3520 3a20 4c69 7374 2020 2020 2020 |o5 : List │ │ │ │ -000416e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000416f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041710: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000416b0: 2020 5333 0a42 656c 6f77 2c20 7468 6520 S3.Below, the │ │ │ │ +000416c0: 696e 7661 7269 616e 7420 7269 6e67 2051 invariant ring Q │ │ │ │ +000416d0: 515b 782c 792c 7a5d 2020 2069 7320 6361 Q[x,y,z] is ca │ │ │ │ +000416e0: 6c63 756c 6174 6564 2077 6974 6820 4b20 lculated with K │ │ │ │ +000416f0: 6265 696e 6720 7468 6520 6669 656c 6420 being the field │ │ │ │ +00041700: 7769 7468 0a31 3031 2065 6c65 6d65 6e74 with.101 element │ │ │ │ +00041710: 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d s...+----------- │ │ │ │ 00041720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a20 ------------+.. │ │ │ │ -00041760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041780: 2020 5333 0a42 656c 6f77 2c20 7468 6520 S3.Below, the │ │ │ │ -00041790: 696e 7661 7269 616e 7420 7269 6e67 2051 invariant ring Q │ │ │ │ -000417a0: 515b 782c 792c 7a5d 2020 2069 7320 6361 Q[x,y,z] is ca │ │ │ │ -000417b0: 6c63 756c 6174 6564 2077 6974 6820 4b20 lculated with K │ │ │ │ -000417c0: 6265 696e 6720 7468 6520 6669 656c 6420 being the field │ │ │ │ -000417d0: 7769 7468 0a31 3031 2065 6c65 6d65 6e74 with.101 element │ │ │ │ -000417e0: 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d s...+----------- │ │ │ │ -000417f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041830: 2d2d 2b0a 7c69 3620 3a20 4b3d 4746 2831 --+.|i6 : K=GF(1 │ │ │ │ -00041840: 3031 2920 2020 2020 2020 2020 2020 2020 01) │ │ │ │ -00041850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 2020 2020 2020 2020 │ │ │ │ -00041920: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00041930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041970: 2020 7c0a 7c6f 3620 3a20 4761 6c6f 6973 |.|o6 : Galois │ │ │ │ -00041980: 4669 656c 6420 2020 2020 2020 2020 2020 Field │ │ │ │ -00041990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000418f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00041900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041940: 2d2d 2b0a 7c69 3720 3a20 5333 3d66 696e --+.|i7 : S3=fin │ │ │ │ +00041950: 6974 6541 6374 696f 6e28 7b41 2c42 7d2c iteAction({A,B}, │ │ │ │ +00041960: 4b5b 782c 792c 7a5d 2920 2020 2020 2020 K[x,y,z]) │ │ │ │ +00041970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041990: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000419a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000419b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000419c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -000419d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000419e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000419f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041a10: 2d2d 2b0a 7c69 3720 3a20 5333 3d66 696e --+.|i7 : S3=fin │ │ │ │ -00041a20: 6974 6541 6374 696f 6e28 7b41 2c42 7d2c iteAction({A,B}, │ │ │ │ -00041a30: 4b5b 782c 792c 7a5d 2920 2020 2020 2020 K[x,y,z]) │ │ │ │ -00041a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041a60: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000419c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000419d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000419e0: 2020 7c0a 7c6f 3720 3d20 4b5b 782e 2e7a |.|o7 = K[x..z │ │ │ │ +000419f0: 5d20 3c2d 207b 7c20 3020 3120 3020 7c2c ] <- {| 0 1 0 |, │ │ │ │ +00041a00: 207c 2030 2031 2030 207c 7d20 2020 2020 | 0 1 0 |} │ │ │ │ +00041a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041a30: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00041a40: 2020 2020 2020 7c20 3020 3020 3120 7c20 | 0 0 1 | │ │ │ │ +00041a50: 207c 2031 2030 2030 207c 2020 2020 2020 | 1 0 0 | │ │ │ │ +00041a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041ab0: 2020 7c0a 7c6f 3720 3d20 4b5b 782e 2e7a |.|o7 = K[x..z │ │ │ │ -00041ac0: 5d20 3c2d 207b 7c20 3020 3120 3020 7c2c ] <- {| 0 1 0 |, │ │ │ │ -00041ad0: 207c 2030 2031 2030 207c 7d20 2020 2020 | 0 1 0 |} │ │ │ │ +00041a80: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00041a90: 2020 2020 2020 7c20 3120 3020 3020 7c20 | 1 0 0 | │ │ │ │ +00041aa0: 207c 2030 2030 2031 207c 2020 2020 2020 | 0 0 1 | │ │ │ │ +00041ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041ad0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00041ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041b00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00041b10: 2020 2020 2020 7c20 3020 3020 3120 7c20 | 0 0 1 | │ │ │ │ -00041b20: 207c 2031 2030 2030 207c 2020 2020 2020 | 1 0 0 | │ │ │ │ -00041b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041b20: 2020 7c0a 7c6f 3720 3a20 4669 6e69 7465 |.|o7 : Finite │ │ │ │ +00041b30: 4772 6f75 7041 6374 696f 6e20 2020 2020 GroupAction │ │ │ │ 00041b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041b50: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00041b60: 2020 2020 2020 7c20 3120 3020 3020 7c20 | 1 0 0 | │ │ │ │ -00041b70: 207c 2030 2030 2031 207c 2020 2020 2020 | 0 0 1 | │ │ │ │ -00041b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041ba0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00041bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041bf0: 2020 7c0a 7c6f 3720 3a20 4669 6e69 7465 |.|o7 : Finite │ │ │ │ -00041c00: 4772 6f75 7041 6374 696f 6e20 2020 2020 GroupAction │ │ │ │ -00041c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041b70: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00041b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041bc0: 2d2d 2b0a 7c69 3820 3a20 7072 696d 6172 --+.|i8 : primar │ │ │ │ +00041bd0: 7949 6e76 6172 6961 6e74 7328 5333 2c44 yInvariants(S3,D │ │ │ │ +00041be0: 6164 653d 3e74 7275 6529 2020 2020 2020 ade=>true) │ │ │ │ +00041bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041c10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00041c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041c40: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00041c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041c90: 2d2d 2b0a 7c69 3820 3a20 7072 696d 6172 --+.|i8 : primar │ │ │ │ -00041ca0: 7949 6e76 6172 6961 6e74 7328 5333 2c44 yInvariants(S3,D │ │ │ │ -00041cb0: 6164 653d 3e74 7275 6529 2020 2020 2020 ade=>true) │ │ │ │ -00041cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041ce0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00041cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041d30: 2020 7c0a 7c20 2020 2020 2020 2020 3620 |.| 6 │ │ │ │ -00041d40: 2020 2020 2035 2020 2020 2020 2034 2032 5 4 2 │ │ │ │ -00041d50: 2020 2020 2020 3320 3320 2020 2020 2032 3 3 2 │ │ │ │ -00041d60: 2034 2020 2020 2020 2020 3520 2020 2020 4 5 │ │ │ │ -00041d70: 2036 2020 2020 2020 3520 2020 2020 2020 6 5 │ │ │ │ -00041d80: 2020 7c0a 7c6f 3820 3d20 7b32 3278 2020 |.|o8 = {22x │ │ │ │ -00041d90: 2d20 3230 7820 7920 2d20 3139 7820 7920 - 20x y - 19x y │ │ │ │ -00041da0: 202d 2035 3078 2079 2020 2d20 3139 7820 - 50x y - 19x │ │ │ │ -00041db0: 7920 202d 2032 3078 2a79 2020 2b20 3232 y - 20x*y + 22 │ │ │ │ -00041dc0: 7920 202d 2032 3078 207a 202b 2020 2020 y - 20x z + │ │ │ │ -00041dd0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00041de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041c60: 2020 7c0a 7c20 2020 2020 2020 2020 3620 |.| 6 │ │ │ │ +00041c70: 2020 2020 2035 2020 2020 2020 2034 2032 5 4 2 │ │ │ │ +00041c80: 2020 2020 2020 3320 3320 2020 2020 2032 3 3 2 │ │ │ │ +00041c90: 2034 2020 2020 2020 2020 3520 2020 2020 4 5 │ │ │ │ +00041ca0: 2036 2020 2020 2020 3520 2020 2020 2020 6 5 │ │ │ │ +00041cb0: 2020 7c0a 7c6f 3820 3d20 7b32 3278 2020 |.|o8 = {22x │ │ │ │ +00041cc0: 2d20 3230 7820 7920 2d20 3139 7820 7920 - 20x y - 19x y │ │ │ │ +00041cd0: 202d 2035 3078 2079 2020 2d20 3139 7820 - 50x y - 19x │ │ │ │ +00041ce0: 7920 202d 2032 3078 2a79 2020 2b20 3232 y - 20x*y + 22 │ │ │ │ +00041cf0: 7920 202d 2032 3078 207a 202b 2020 2020 y - 20x z + │ │ │ │ +00041d00: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +00041d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041d50: 2d2d 7c0a 7c20 2020 2020 2020 2034 2020 --|.| 4 │ │ │ │ +00041d60: 2020 2020 2020 2033 2032 2020 2020 2020 3 2 │ │ │ │ +00041d70: 2032 2033 2020 2020 2020 2020 2034 2020 2 3 4 │ │ │ │ +00041d80: 2020 2020 2035 2020 2020 2020 2034 2032 5 4 2 │ │ │ │ +00041d90: 2020 2020 2020 3320 2020 3220 2020 2020 3 2 │ │ │ │ +00041da0: 2020 7c0a 7c20 2020 2020 3131 7820 792a |.| 11x y* │ │ │ │ +00041db0: 7a20 2d20 3239 7820 7920 7a20 2d20 3239 z - 29x y z - 29 │ │ │ │ +00041dc0: 7820 7920 7a20 2b20 3131 782a 7920 7a20 x y z + 11x*y z │ │ │ │ +00041dd0: 2d20 3230 7920 7a20 2d20 3139 7820 7a20 - 20y z - 19x z │ │ │ │ +00041de0: 202d 2032 3978 2079 2a7a 2020 2b20 2020 - 29x y*z + │ │ │ │ +00041df0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ 00041e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041e20: 2d2d 7c0a 7c20 2020 2020 2020 2034 2020 --|.| 4 │ │ │ │ -00041e30: 2020 2020 2020 2033 2032 2020 2020 2020 3 2 │ │ │ │ -00041e40: 2032 2033 2020 2020 2020 2020 2034 2020 2 3 4 │ │ │ │ -00041e50: 2020 2020 2035 2020 2020 2020 2034 2032 5 4 2 │ │ │ │ -00041e60: 2020 2020 2020 3320 2020 3220 2020 2020 3 2 │ │ │ │ -00041e70: 2020 7c0a 7c20 2020 2020 3131 7820 792a |.| 11x y* │ │ │ │ -00041e80: 7a20 2d20 3239 7820 7920 7a20 2d20 3239 z - 29x y z - 29 │ │ │ │ -00041e90: 7820 7920 7a20 2b20 3131 782a 7920 7a20 x y z + 11x*y z │ │ │ │ -00041ea0: 2d20 3230 7920 7a20 2d20 3139 7820 7a20 - 20y z - 19x z │ │ │ │ -00041eb0: 202d 2032 3978 2079 2a7a 2020 2b20 2020 - 29x y*z + │ │ │ │ -00041ec0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00041ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041e40: 2d2d 7c0a 7c20 2020 2020 2020 2032 2032 --|.| 2 2 │ │ │ │ +00041e50: 2032 2020 2020 2020 2020 3320 3220 2020 2 3 2 │ │ │ │ +00041e60: 2020 2034 2032 2020 2020 2020 3320 3320 4 2 3 3 │ │ │ │ +00041e70: 2020 2020 2032 2020 2033 2020 2020 2020 2 3 │ │ │ │ +00041e80: 2020 3220 3320 2020 2020 2033 2033 2020 2 3 3 3 │ │ │ │ +00041e90: 2020 7c0a 7c20 2020 2020 3132 7820 7920 |.| 12x y │ │ │ │ +00041ea0: 7a20 202d 2032 3978 2a79 207a 2020 2d20 z - 29x*y z - │ │ │ │ +00041eb0: 3139 7920 7a20 202d 2035 3078 207a 2020 19y z - 50x z │ │ │ │ +00041ec0: 2d20 3239 7820 792a 7a20 202d 2032 3978 - 29x y*z - 29x │ │ │ │ +00041ed0: 2a79 207a 2020 2d20 3530 7920 7a20 202d *y z - 50y z - │ │ │ │ +00041ee0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ 00041ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041f10: 2d2d 7c0a 7c20 2020 2020 2020 2032 2032 --|.| 2 2 │ │ │ │ -00041f20: 2032 2020 2020 2020 2020 3320 3220 2020 2 3 2 │ │ │ │ -00041f30: 2020 2034 2032 2020 2020 2020 3320 3320 4 2 3 3 │ │ │ │ -00041f40: 2020 2020 2032 2020 2033 2020 2020 2020 2 3 │ │ │ │ -00041f50: 2020 3220 3320 2020 2020 2033 2033 2020 2 3 3 3 │ │ │ │ -00041f60: 2020 7c0a 7c20 2020 2020 3132 7820 7920 |.| 12x y │ │ │ │ -00041f70: 7a20 202d 2032 3978 2a79 207a 2020 2d20 z - 29x*y z - │ │ │ │ -00041f80: 3139 7920 7a20 202d 2035 3078 207a 2020 19y z - 50x z │ │ │ │ -00041f90: 2d20 3239 7820 792a 7a20 202d 2032 3978 - 29x y*z - 29x │ │ │ │ -00041fa0: 2a79 207a 2020 2d20 3530 7920 7a20 202d *y z - 50y z - │ │ │ │ -00041fb0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00041fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041f30: 2d2d 7c0a 7c20 2020 2020 2020 2032 2034 --|.| 2 4 │ │ │ │ +00041f40: 2020 2020 2020 2020 2020 3420 2020 2020 4 │ │ │ │ +00041f50: 2032 2034 2020 2020 2020 2020 3520 2020 2 4 5 │ │ │ │ +00041f60: 2020 2020 2035 2020 2020 2020 3620 2020 5 6 │ │ │ │ +00041f70: 2036 2020 2020 2020 3520 2020 2020 2034 6 5 4 │ │ │ │ +00041f80: 2032 7c0a 7c20 2020 2020 3139 7820 7a20 2|.| 19x z │ │ │ │ +00041f90: 202b 2031 3178 2a79 2a7a 2020 2d20 3139 + 11x*y*z - 19 │ │ │ │ +00041fa0: 7920 7a20 202d 2032 3078 2a7a 2020 2d20 y z - 20x*z - │ │ │ │ +00041fb0: 3230 792a 7a20 202b 2032 327a 202c 2036 20y*z + 22z , 6 │ │ │ │ +00041fc0: 7820 202b 2033 3778 2079 202b 2032 7820 x + 37x y + 2x │ │ │ │ +00041fd0: 7920 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d y |.| ------ │ │ │ │ 00041fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042000: 2d2d 7c0a 7c20 2020 2020 2020 2032 2034 --|.| 2 4 │ │ │ │ -00042010: 2020 2020 2020 2020 2020 3420 2020 2020 4 │ │ │ │ -00042020: 2032 2034 2020 2020 2020 2020 3520 2020 2 4 5 │ │ │ │ -00042030: 2020 2020 2035 2020 2020 2020 3620 2020 5 6 │ │ │ │ -00042040: 2036 2020 2020 2020 3520 2020 2020 2034 6 5 4 │ │ │ │ -00042050: 2032 7c0a 7c20 2020 2020 3139 7820 7a20 2|.| 19x z │ │ │ │ -00042060: 202b 2031 3178 2a79 2a7a 2020 2d20 3139 + 11x*y*z - 19 │ │ │ │ -00042070: 7920 7a20 202d 2032 3078 2a7a 2020 2d20 y z - 20x*z - │ │ │ │ -00042080: 3230 792a 7a20 202b 2032 327a 202c 2036 20y*z + 22z , 6 │ │ │ │ -00042090: 7820 202b 2033 3778 2079 202b 2032 7820 x + 37x y + 2x │ │ │ │ -000420a0: 7920 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d y |.| ------ │ │ │ │ -000420b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000420c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042020: 2d2d 7c0a 7c20 2020 2020 2020 2020 2033 --|.| 3 │ │ │ │ +00042030: 2033 2020 2020 2032 2034 2020 2020 2020 3 2 4 │ │ │ │ +00042040: 2020 3520 2020 2020 3620 2020 2020 2035 5 6 5 │ │ │ │ +00042050: 2020 2020 2020 2034 2020 2020 2020 2020 4 │ │ │ │ +00042060: 2033 2032 2020 2020 2020 2032 2033 2020 3 2 2 3 │ │ │ │ +00042070: 2020 7c0a 7c20 2020 2020 2b20 3234 7820 |.| + 24x │ │ │ │ +00042080: 7920 202b 2032 7820 7920 202b 2033 3778 y + 2x y + 37x │ │ │ │ +00042090: 2a79 2020 2b20 3679 2020 2b20 3337 7820 *y + 6y + 37x │ │ │ │ +000420a0: 7a20 2b20 3133 7820 792a 7a20 2b20 3239 z + 13x y*z + 29 │ │ │ │ +000420b0: 7820 7920 7a20 2b20 3239 7820 7920 7a20 x y z + 29x y z │ │ │ │ +000420c0: 2b20 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d + |.| ------ │ │ │ │ 000420d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000420e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000420f0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2033 --|.| 3 │ │ │ │ -00042100: 2033 2020 2020 2032 2034 2020 2020 2020 3 2 4 │ │ │ │ -00042110: 2020 3520 2020 2020 3620 2020 2020 2035 5 6 5 │ │ │ │ -00042120: 2020 2020 2020 2034 2020 2020 2020 2020 4 │ │ │ │ -00042130: 2033 2032 2020 2020 2020 2032 2033 2020 3 2 2 3 │ │ │ │ -00042140: 2020 7c0a 7c20 2020 2020 2b20 3234 7820 |.| + 24x │ │ │ │ -00042150: 7920 202b 2032 7820 7920 202b 2033 3778 y + 2x y + 37x │ │ │ │ -00042160: 2a79 2020 2b20 3679 2020 2b20 3337 7820 *y + 6y + 37x │ │ │ │ -00042170: 7a20 2b20 3133 7820 792a 7a20 2b20 3239 z + 13x y*z + 29 │ │ │ │ -00042180: 7820 7920 7a20 2b20 3239 7820 7920 7a20 x y z + 29x y z │ │ │ │ -00042190: 2b20 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d + |.| ------ │ │ │ │ -000421a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000421b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000420f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042110: 2d2d 7c0a 7c20 2020 2020 2020 2020 2034 --|.| 4 │ │ │ │ +00042120: 2020 2020 2020 2035 2020 2020 2020 3420 5 4 │ │ │ │ +00042130: 3220 2020 2020 2033 2020 2032 2020 2020 2 3 2 │ │ │ │ +00042140: 2020 3220 3220 3220 2020 2020 2020 2033 2 2 2 3 │ │ │ │ +00042150: 2032 2020 2020 2034 2032 2020 2020 2020 2 4 2 │ │ │ │ +00042160: 2020 7c0a 7c20 2020 2020 3133 782a 7920 |.| 13x*y │ │ │ │ +00042170: 7a20 2b20 3337 7920 7a20 2b20 3278 207a z + 37y z + 2x z │ │ │ │ +00042180: 2020 2b20 3239 7820 792a 7a20 202b 2031 + 29x y*z + 1 │ │ │ │ +00042190: 3578 2079 207a 2020 2b20 3239 782a 7920 5x y z + 29x*y │ │ │ │ +000421a0: 7a20 202b 2032 7920 7a20 202b 2020 2020 z + 2y z + │ │ │ │ +000421b0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ 000421c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000421d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000421e0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2034 --|.| 4 │ │ │ │ -000421f0: 2020 2020 2020 2035 2020 2020 2020 3420 5 4 │ │ │ │ -00042200: 3220 2020 2020 2033 2020 2032 2020 2020 2 3 2 │ │ │ │ -00042210: 2020 3220 3220 3220 2020 2020 2020 2033 2 2 2 3 │ │ │ │ -00042220: 2032 2020 2020 2034 2032 2020 2020 2020 2 4 2 │ │ │ │ -00042230: 2020 7c0a 7c20 2020 2020 3133 782a 7920 |.| 13x*y │ │ │ │ -00042240: 7a20 2b20 3337 7920 7a20 2b20 3278 207a z + 37y z + 2x z │ │ │ │ -00042250: 2020 2b20 3239 7820 792a 7a20 202b 2031 + 29x y*z + 1 │ │ │ │ -00042260: 3578 2079 207a 2020 2b20 3239 782a 7920 5x y z + 29x*y │ │ │ │ -00042270: 7a20 202b 2032 7920 7a20 202b 2020 2020 z + 2y z + │ │ │ │ -00042280: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00042290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000422a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000421e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000421f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042200: 2d2d 7c0a 7c20 2020 2020 2020 2033 2033 --|.| 3 3 │ │ │ │ +00042210: 2020 2020 2020 3220 2020 3320 2020 2020 2 3 │ │ │ │ +00042220: 2020 2032 2033 2020 2020 2020 3320 3320 2 3 3 3 │ │ │ │ +00042230: 2020 2020 3220 3420 2020 2020 2020 2020 2 4 │ │ │ │ +00042240: 2034 2020 2020 2032 2034 2020 2020 2020 4 2 4 │ │ │ │ +00042250: 2020 7c0a 7c20 2020 2020 3234 7820 7a20 |.| 24x z │ │ │ │ +00042260: 202b 2032 3978 2079 2a7a 2020 2b20 3239 + 29x y*z + 29 │ │ │ │ +00042270: 782a 7920 7a20 202b 2032 3479 207a 2020 x*y z + 24y z │ │ │ │ +00042280: 2b20 3278 207a 2020 2b20 3133 782a 792a + 2x z + 13x*y* │ │ │ │ +00042290: 7a20 202b 2032 7920 7a20 202b 2020 2020 z + 2y z + │ │ │ │ +000422a0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ 000422b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000422c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000422d0: 2d2d 7c0a 7c20 2020 2020 2020 2033 2033 --|.| 3 3 │ │ │ │ -000422e0: 2020 2020 2020 3220 2020 3320 2020 2020 2 3 │ │ │ │ -000422f0: 2020 2032 2033 2020 2020 2020 3320 3320 2 3 3 3 │ │ │ │ -00042300: 2020 2020 3220 3420 2020 2020 2020 2020 2 4 │ │ │ │ -00042310: 2034 2020 2020 2032 2034 2020 2020 2020 4 2 4 │ │ │ │ -00042320: 2020 7c0a 7c20 2020 2020 3234 7820 7a20 |.| 24x z │ │ │ │ -00042330: 202b 2032 3978 2079 2a7a 2020 2b20 3239 + 29x y*z + 29 │ │ │ │ -00042340: 782a 7920 7a20 202b 2032 3479 207a 2020 x*y z + 24y z │ │ │ │ -00042350: 2b20 3278 207a 2020 2b20 3133 782a 792a + 2x z + 13x*y* │ │ │ │ -00042360: 7a20 202b 2032 7920 7a20 202b 2020 2020 z + 2y z + │ │ │ │ -00042370: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00042380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000422d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000422e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000422f0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2035 --|.| 5 │ │ │ │ +00042300: 2020 2020 2020 2020 3520 2020 2020 3620 5 6 │ │ │ │ +00042310: 2020 2020 3620 2020 2020 2035 2020 2020 6 5 │ │ │ │ +00042320: 2020 2034 2032 2020 2020 2020 3320 3320 4 2 3 3 │ │ │ │ +00042330: 2020 2020 2032 2034 2020 2020 2020 2020 2 4 │ │ │ │ +00042340: 3520 7c0a 7c20 2020 2020 3337 782a 7a20 5 |.| 37x*z │ │ │ │ +00042350: 202b 2033 3779 2a7a 2020 2b20 367a 202c + 37y*z + 6z , │ │ │ │ +00042360: 2032 3178 2020 2d20 3432 7820 7920 2b20 21x - 42x y + │ │ │ │ +00042370: 3433 7820 7920 202d 2034 3478 2079 2020 43x y - 44x y │ │ │ │ +00042380: 2b20 3433 7820 7920 202d 2034 3278 2a79 + 43x y - 42x*y │ │ │ │ +00042390: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ 000423a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000423b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000423c0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2035 --|.| 5 │ │ │ │ -000423d0: 2020 2020 2020 2020 3520 2020 2020 3620 5 6 │ │ │ │ -000423e0: 2020 2020 3620 2020 2020 2035 2020 2020 6 5 │ │ │ │ -000423f0: 2020 2034 2032 2020 2020 2020 3320 3320 4 2 3 3 │ │ │ │ -00042400: 2020 2020 2032 2034 2020 2020 2020 2020 2 4 │ │ │ │ -00042410: 3520 7c0a 7c20 2020 2020 3337 782a 7a20 5 |.| 37x*z │ │ │ │ -00042420: 202b 2033 3779 2a7a 2020 2b20 367a 202c + 37y*z + 6z , │ │ │ │ -00042430: 2032 3178 2020 2d20 3432 7820 7920 2b20 21x - 42x y + │ │ │ │ -00042440: 3433 7820 7920 202d 2034 3478 2079 2020 43x y - 44x y │ │ │ │ -00042450: 2b20 3433 7820 7920 202d 2034 3278 2a79 + 43x y - 42x*y │ │ │ │ -00042460: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00042470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000423c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000423d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000423e0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2036 --|.| 6 │ │ │ │ +000423f0: 2020 2020 2020 3520 2020 2020 2020 3420 5 4 │ │ │ │ +00042400: 2020 2020 2020 2033 2032 2020 2020 2020 3 2 │ │ │ │ +00042410: 3220 3320 2020 2020 2020 2020 3420 2020 2 3 4 │ │ │ │ +00042420: 2020 2020 3520 2020 2020 2020 3420 3220 5 4 2 │ │ │ │ +00042430: 2020 7c0a 7c20 2020 2020 2b20 3231 7920 |.| + 21y │ │ │ │ +00042440: 202d 2034 3278 207a 202b 2033 3378 2079 - 42x z + 33x y │ │ │ │ +00042450: 2a7a 202b 2039 7820 7920 7a20 2b20 3978 *z + 9x y z + 9x │ │ │ │ +00042460: 2079 207a 202b 2033 3378 2a79 207a 202d y z + 33x*y z - │ │ │ │ +00042470: 2034 3279 207a 202b 2034 3378 207a 2020 42y z + 43x z │ │ │ │ +00042480: 2b20 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d + |.| ------ │ │ │ │ 00042490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000424a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000424b0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2036 --|.| 6 │ │ │ │ -000424c0: 2020 2020 2020 3520 2020 2020 2020 3420 5 4 │ │ │ │ -000424d0: 2020 2020 2020 2033 2032 2020 2020 2020 3 2 │ │ │ │ -000424e0: 3220 3320 2020 2020 2020 2020 3420 2020 2 3 4 │ │ │ │ -000424f0: 2020 2020 3520 2020 2020 2020 3420 3220 5 4 2 │ │ │ │ -00042500: 2020 7c0a 7c20 2020 2020 2b20 3231 7920 |.| + 21y │ │ │ │ -00042510: 202d 2034 3278 207a 202b 2033 3378 2079 - 42x z + 33x y │ │ │ │ -00042520: 2a7a 202b 2039 7820 7920 7a20 2b20 3978 *z + 9x y z + 9x │ │ │ │ -00042530: 2079 207a 202b 2033 3378 2a79 207a 202d y z + 33x*y z - │ │ │ │ -00042540: 2034 3279 207a 202b 2034 3378 207a 2020 42y z + 43x z │ │ │ │ -00042550: 2b20 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d + |.| ------ │ │ │ │ -00042560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000424b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000424c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000424d0: 2d2d 7c0a 7c20 2020 2020 2020 3320 2020 --|.| 3 │ │ │ │ +000424e0: 3220 2020 2020 2032 2032 2032 2020 2020 2 2 2 2 │ │ │ │ +000424f0: 2020 2033 2032 2020 2020 2020 3420 3220 3 2 4 2 │ │ │ │ +00042500: 2020 2020 2033 2033 2020 2020 2032 2020 3 3 2 │ │ │ │ +00042510: 2033 2020 2020 2020 2032 2033 2020 2020 3 2 3 │ │ │ │ +00042520: 2020 7c0a 7c20 2020 2020 3978 2079 2a7a |.| 9x y*z │ │ │ │ +00042530: 2020 2b20 3230 7820 7920 7a20 202b 2039 + 20x y z + 9 │ │ │ │ +00042540: 782a 7920 7a20 202b 2034 3379 207a 2020 x*y z + 43y z │ │ │ │ +00042550: 2d20 3434 7820 7a20 202b 2039 7820 792a - 44x z + 9x y* │ │ │ │ +00042560: 7a20 202b 2039 782a 7920 7a20 202d 2020 z + 9x*y z - │ │ │ │ +00042570: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 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6375 6c61 7465 2061 2068 736f .calculate a hso │ │ │ │ -000429b0: 7020 666f 7220 7468 6520 696e 7661 7269 p for the invari │ │ │ │ -000429c0: 616e 7420 7269 6e67 206f 7665 7220 6120 ant ring over a │ │ │ │ -000429d0: 6669 6e69 7465 2066 6965 6c64 2062 7920 finite field by │ │ │ │ -000429e0: 7573 696e 6720 7468 6520 4461 6465 0a20 using the Dade. │ │ │ │ -000429f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042a30: 2020 2020 200a 616c 676f 7269 7468 6d2e .algorithm. │ │ │ │ -00042a40: 2055 7365 7273 2073 686f 756c 6420 656e Users should en │ │ │ │ -00042a50: 7465 7220 7468 6520 6669 6e69 7465 2066 ter the finite f │ │ │ │ -00042a60: 6965 6c64 2061 7320 6120 2a6e 6f74 6520 ield as a *note │ │ │ │ -00042a70: 4761 6c6f 6973 4669 656c 643a 0a20 2020 GaloisField:. │ │ │ │ -00042a80: 2020 2020 2020 2020 2020 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2074 6861 ed to ensure tha │ │ │ │ -00042b80: 7420 7468 6520 6772 6f75 6e64 2066 6965 t the ground fie │ │ │ │ -00042b90: 6c64 2068 6173 0a20 2020 2020 2020 2020 ld has. │ │ │ │ -00042ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042bb0: 2020 206e 2d31 0a63 6172 6469 6e61 6c69 n-1.cardinali │ │ │ │ -00042bc0: 7479 2067 7265 6174 6572 2074 6861 6e20 ty greater than │ │ │ │ -00042bd0: 7c47 7c20 2020 2c20 7768 6572 6520 6e20 |G| , where n │ │ │ │ -00042be0: 6973 2074 6865 206e 756d 6265 7220 6f66 is the number of │ │ │ │ -00042bf0: 2076 6172 6961 626c 6573 2069 6e20 7468 variables in th │ │ │ │ -00042c00: 650a 706f 6c79 6e6f 6d69 616c 2072 696e e.polynomial rin │ │ │ │ -00042c10: 6720 522e 2055 7369 6e67 2061 2067 726f g R. 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See *note │ │ │ │ -00042d20: 6873 6f70 2061 6c67 6f72 6974 686d 733a hsop algorithms: │ │ │ │ -00042d30: 2068 736f 700a 616c 676f 7269 7468 6d73 hsop.algorithms │ │ │ │ -00042d40: 2c20 666f 7220 6120 6469 7363 7573 7369 , for a discussi │ │ │ │ -00042d50: 6f6e 206f 6e20 7468 6520 4461 6465 2061 on on the Dade a │ │ │ │ -00042d60: 6c67 6f72 6974 686d 2e0a 0a53 6565 2061 lgorithm...See a │ │ │ │ -00042d70: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -00042d80: 2a20 2a6e 6f74 6520 6873 6f70 2061 6c67 * *note hsop alg │ │ │ │ -00042d90: 6f72 6974 686d 733a 2068 736f 7020 616c orithms: hsop al │ │ │ │ -00042da0: 676f 7269 7468 6d73 2c20 2d2d 2061 6e20 gorithms, -- an │ │ │ │ -00042db0: 6f76 6572 7669 6577 206f 6620 7468 6520 overview of the │ │ │ │ -00042dc0: 616c 676f 7269 7468 6d73 0a20 2020 2075 algorithms. u │ │ │ │ -00042dd0: 7365 6420 696e 2070 7269 6d61 7279 496e sed in primaryIn │ │ │ │ -00042de0: 7661 7269 616e 7473 0a20 202a 202a 6e6f variants. * *no │ │ │ │ -00042df0: 7465 2070 7269 6d61 7279 496e 7661 7269 te primaryInvari │ │ │ │ -00042e00: 616e 7473 3a20 7072 696d 6172 7949 6e76 ants: primaryInv │ │ │ │ -00042e10: 6172 6961 6e74 732c 202d 2d20 636f 6d70 ariants, -- comp │ │ │ │ -00042e20: 7574 6573 2061 206c 6973 7420 6f66 2070 utes a list of p │ │ │ │ -00042e30: 7269 6d61 7279 0a20 2020 2069 6e76 6172 rimary. invar │ │ │ │ -00042e40: 6961 6e74 7320 666f 7220 7468 6520 696e iants for the in │ │ │ │ -00042e50: 7661 7269 616e 7420 7269 6e67 206f 6620 variant ring of │ │ │ │ -00042e60: 6120 6669 6e69 7465 2067 726f 7570 0a0a a finite group.. │ │ │ │ -00042e70: 4675 6e63 7469 6f6e 7320 7769 7468 206f Functions with o │ │ │ │ -00042e80: 7074 696f 6e61 6c20 6172 6775 6d65 6e74 ptional argument │ │ │ │ -00042e90: 206e 616d 6564 2044 6164 653a 0a3d 3d3d named Dade:.=== │ │ │ │ -00042ea0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00042eb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00042ec0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a =========.. * * │ │ │ │ -00042ed0: 6e6f 7465 2070 7269 6d61 7279 496e 7661 note primaryInva │ │ │ │ -00042ee0: 7269 616e 7473 282e 2e2e 2c44 6164 653d riants(...,Dade= │ │ │ │ -00042ef0: 3e2e 2e2e 293a 0a20 2020 2070 7269 6d61 >...):. prima │ │ │ │ -00042f00: 7279 496e 7661 7269 616e 7473 5f6c 705f ryInvariants_lp_ │ │ │ │ -00042f10: 7064 5f70 645f 7064 5f63 6d44 6164 653d pd_pd_pd_cmDade= │ │ │ │ -00042f20: 3e5f 7064 5f70 645f 7064 5f72 702c 202d >_pd_pd_pd_rp, - │ │ │ │ -00042f30: 2d20 616e 206f 7074 696f 6e61 6c20 6172 - an optional ar │ │ │ │ -00042f40: 6775 6d65 6e74 0a20 2020 2066 6f72 2070 gument. for p │ │ │ │ -00042f50: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ -00042f60: 2064 6574 6572 6d69 6e69 6e67 2077 6865 determining whe │ │ │ │ -00042f70: 7468 6572 2074 6f20 7573 6520 7468 6520 ther to use the │ │ │ │ -00042f80: 4461 6465 2061 6c67 6f72 6974 686d 0a0a Dade algorithm.. │ │ │ │ -00042f90: 4675 7274 6865 7220 696e 666f 726d 6174 Further informat │ │ │ │ -00042fa0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ion.============ │ │ │ │ -00042fb0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2044 6566 =======.. * Def │ │ │ │ -00042fc0: 6175 6c74 2076 616c 7565 3a20 2a6e 6f74 ault value: *not │ │ │ │ -00042fd0: 6520 6661 6c73 653a 2028 4d61 6361 756c e false: (Macaul │ │ │ │ -00042fe0: 6179 3244 6f63 2966 616c 7365 2c0a 2020 ay2Doc)false,. │ │ │ │ -00042ff0: 2a20 4675 6e63 7469 6f6e 3a20 2a6e 6f74 * Function: *not │ │ │ │ -00043000: 6520 7072 696d 6172 7949 6e76 6172 6961 e primaryInvaria │ │ │ │ -00043010: 6e74 733a 2070 7269 6d61 7279 496e 7661 nts: primaryInva │ │ │ │ -00043020: 7269 616e 7473 2c20 2d2d 2063 6f6d 7075 riants, -- compu │ │ │ │ -00043030: 7465 7320 6120 6c69 7374 206f 660a 2020 tes a list of. │ │ │ │ -00043040: 2020 7072 696d 6172 7920 696e 7661 7269 primary invari │ │ │ │ -00043050: 616e 7473 2066 6f72 2074 6865 2069 6e76 ants for the inv │ │ │ │ -00043060: 6172 6961 6e74 2072 696e 6720 6f66 2061 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-00043140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043180: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -00043190: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -000431a0: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -000431b0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -000431c0: 6361 756c 6179 322d 312e 3236 2e30 352b caulay2-1.26.05+ │ │ │ │ -000431d0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -000431e0: 7061 636b 6167 6573 2f0a 496e 7661 7269 packages/.Invari │ │ │ │ -000431f0: 616e 7452 696e 672f 4861 7765 7344 6f63 antRing/HawesDoc │ │ │ │ -00043200: 2e6d 323a 3234 343a 302e 0a1f 0a46 696c .m2:244:0....Fil │ │ │ │ -00043210: 653a 2049 6e76 6172 6961 6e74 5269 6e67 e: InvariantRing │ │ │ │ -00043220: 2e69 6e66 6f2c 204e 6f64 653a 2070 7269 .info, Node: pri │ │ │ │ -00043230: 6d61 7279 496e 7661 7269 616e 7473 5f6c maryInvariants_l │ │ │ │ -00043240: 705f 7064 5f70 645f 7064 5f63 6d44 6567 p_pd_pd_pd_cmDeg │ │ │ │ -00043250: 7265 6556 6563 746f 723d 3e5f 7064 5f70 reeVector=>_pd_p │ │ │ │ -00043260: 645f 7064 5f72 702c 204e 6578 743a 2072 d_pd_rp, Next: r │ │ │ │ -00043270: 616e 6b5f 6c70 4469 6167 6f6e 616c 4163 ank_lpDiagonalAc │ │ │ │ -00043280: 7469 6f6e 5f72 702c 2050 7265 763a 2070 tion_rp, Prev: p │ │ │ │ -00043290: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ -000432a0: 5f6c 705f 7064 5f70 645f 7064 5f63 6d44 _lp_pd_pd_pd_cmD │ │ │ │ -000432b0: 6164 653d 3e5f 7064 5f70 645f 7064 5f72 ade=>_pd_pd_pd_r │ │ │ │ -000432c0: 702c 2055 703a 2054 6f70 0a0a 7072 696d p, Up: Top..prim │ │ │ │ -000432d0: 6172 7949 6e76 6172 6961 6e74 7328 2e2e aryInvariants(.. │ │ │ │ -000432e0: 2e2c 4465 6772 6565 5665 6374 6f72 3d3e .,DegreeVector=> │ │ │ │ -000432f0: 2e2e 2e29 202d 2d20 616e 206f 7074 696f ...) -- an optio │ │ │ │ -00043300: 6e61 6c20 6172 6775 6d65 6e74 2066 6f72 nal argument for │ │ │ │ +000428d0: 0a63 616c 6375 6c61 7465 2061 2068 736f .calculate a hso │ │ │ │ +000428e0: 7020 666f 7220 7468 6520 696e 7661 7269 p for the invari │ │ │ │ +000428f0: 616e 7420 7269 6e67 206f 7665 7220 6120 ant ring over a │ │ │ │ +00042900: 6669 6e69 7465 2066 6965 6c64 2062 7920 finite field by │ │ │ │ +00042910: 7573 696e 6720 7468 6520 4461 6465 0a20 using the Dade. │ │ │ │ +00042920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042960: 2020 2020 200a 616c 676f 7269 7468 6d2e .algorithm. │ │ │ │ +00042970: 2055 7365 7273 2073 686f 756c 6420 656e Users should en │ │ │ │ +00042980: 7465 7220 7468 6520 6669 6e69 7465 2066 ter the finite f │ │ │ │ +00042990: 6965 6c64 2061 7320 6120 2a6e 6f74 6520 ield as a *note │ │ │ │ +000429a0: 4761 6c6f 6973 4669 656c 643a 0a20 2020 GaloisField:. │ │ │ │ +000429b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000429c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000429d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000429e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000429f0: 200a 284d 6163 6175 6c61 7932 446f 6329 .(Macaulay2Doc) │ │ │ │ +00042a00: 4761 6c6f 6973 4669 656c 642c 206f 7220 GaloisField, or │ │ │ │ +00042a10: 6120 7175 6f74 6965 6e74 2066 6965 6c64 a quotient field │ │ │ │ +00042a20: 206f 6620 7468 6520 666f 726d 202a 6e6f of the form *no │ │ │ │ +00042a30: 7465 205a 5a3a 0a20 2020 2020 2020 2020 te ZZ:. │ │ │ │ +00042a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042a70: 2020 2020 2020 2020 2020 2020 2020 0a28 .( │ │ │ │ +00042a80: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ +00042a90: 2f70 2061 6e64 2061 7265 2061 6476 6973 /p and are advis │ │ │ │ +00042aa0: 6564 2074 6f20 656e 7375 7265 2074 6861 ed to ensure tha │ │ │ │ +00042ab0: 7420 7468 6520 6772 6f75 6e64 2066 6965 t the ground fie │ │ │ │ +00042ac0: 6c64 2068 6173 0a20 2020 2020 2020 2020 ld has. │ │ │ │ +00042ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042ae0: 2020 206e 2d31 0a63 6172 6469 6e61 6c69 n-1.cardinali │ │ │ │ +00042af0: 7479 2067 7265 6174 6572 2074 6861 6e20 ty greater than │ │ │ │ +00042b00: 7c47 7c20 2020 2c20 7768 6572 6520 6e20 |G| , where n │ │ │ │ +00042b10: 6973 2074 6865 206e 756d 6265 7220 6f66 is the number of │ │ │ │ +00042b20: 2076 6172 6961 626c 6573 2069 6e20 7468 variables in th │ │ │ │ +00042b30: 650a 706f 6c79 6e6f 6d69 616c 2072 696e e.polynomial rin │ │ │ │ +00042b40: 6720 522e 2055 7369 6e67 2061 2067 726f g R. Using a gro │ │ │ │ +00042b50: 756e 6420 6669 656c 6420 736d 616c 6c65 und field smalle │ │ │ │ +00042b60: 7220 7468 616e 2074 6869 7320 7275 6e73 r than this runs │ │ │ │ +00042b70: 2074 6865 2072 6973 6b20 6f66 2074 6865 the risk of the │ │ │ │ +00042b80: 0a61 6c67 6f72 6974 686d 2067 6574 7469 .algorithm getti │ │ │ │ +00042b90: 6e67 2073 7475 636b 2069 6e20 616e 2069 ng stuck in an i │ │ │ │ +00042ba0: 6e66 696e 6974 6520 6c6f 6f70 3b20 2a6e nfinite loop; *n │ │ │ │ +00042bb0: 6f74 6520 7072 696d 6172 7949 6e76 6172 ote primaryInvar │ │ │ │ +00042bc0: 6961 6e74 733a 0a70 7269 6d61 7279 496e iants:.primaryIn │ │ │ │ +00042bd0: 7661 7269 616e 7473 2c20 6469 7370 6c61 variants, displa │ │ │ │ +00042be0: 7973 2061 2077 6172 6e69 6e67 206d 6573 ys a warning mes │ │ │ │ +00042bf0: 7361 6765 2061 736b 696e 6720 7468 6520 sage asking the │ │ │ │ +00042c00: 7573 6572 2077 6865 7468 6572 2074 6865 user whether the │ │ │ │ +00042c10: 7920 7769 7368 0a74 6f20 636f 6e74 696e y wish.to contin │ │ │ │ +00042c20: 7565 2077 6974 6820 7468 6520 636f 6d70 ue with the comp │ │ │ │ +00042c30: 7574 6174 696f 6e20 696e 2074 6869 7320 utation in this │ │ │ │ +00042c40: 6361 7365 2e20 5365 6520 2a6e 6f74 6520 case. See *note │ │ │ │ +00042c50: 6873 6f70 2061 6c67 6f72 6974 686d 733a hsop algorithms: │ │ │ │ +00042c60: 2068 736f 700a 616c 676f 7269 7468 6d73 hsop.algorithms │ │ │ │ +00042c70: 2c20 666f 7220 6120 6469 7363 7573 7369 , for a discussi │ │ │ │ +00042c80: 6f6e 206f 6e20 7468 6520 4461 6465 2061 on on the Dade a │ │ │ │ +00042c90: 6c67 6f72 6974 686d 2e0a 0a53 6565 2061 lgorithm...See a │ │ │ │ +00042ca0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ +00042cb0: 2a20 2a6e 6f74 6520 6873 6f70 2061 6c67 * *note hsop alg │ │ │ │ +00042cc0: 6f72 6974 686d 733a 2068 736f 7020 616c orithms: hsop al │ │ │ │ +00042cd0: 676f 7269 7468 6d73 2c20 2d2d 2061 6e20 gorithms, -- an │ │ │ │ +00042ce0: 6f76 6572 7669 6577 206f 6620 7468 6520 overview of the │ │ │ │ +00042cf0: 616c 676f 7269 7468 6d73 0a20 2020 2075 algorithms. u │ │ │ │ +00042d00: 7365 6420 696e 2070 7269 6d61 7279 496e sed in primaryIn │ │ │ │ +00042d10: 7661 7269 616e 7473 0a20 202a 202a 6e6f variants. * *no │ │ │ │ +00042d20: 7465 2070 7269 6d61 7279 496e 7661 7269 te primaryInvari │ │ │ │ +00042d30: 616e 7473 3a20 7072 696d 6172 7949 6e76 ants: primaryInv │ │ │ │ +00042d40: 6172 6961 6e74 732c 202d 2d20 636f 6d70 ariants, -- comp │ │ │ │ +00042d50: 7574 6573 2061 206c 6973 7420 6f66 2070 utes a list of p │ │ │ │ +00042d60: 7269 6d61 7279 0a20 2020 2069 6e76 6172 rimary. invar │ │ │ │ +00042d70: 6961 6e74 7320 666f 7220 7468 6520 696e iants for the in │ │ │ │ +00042d80: 7661 7269 616e 7420 7269 6e67 206f 6620 variant ring of │ │ │ │ +00042d90: 6120 6669 6e69 7465 2067 726f 7570 0a0a a finite group.. │ │ │ │ +00042da0: 4675 6e63 7469 6f6e 7320 7769 7468 206f Functions with o │ │ │ │ +00042db0: 7074 696f 6e61 6c20 6172 6775 6d65 6e74 ptional argument │ │ │ │ +00042dc0: 206e 616d 6564 2044 6164 653a 0a3d 3d3d named Dade:.=== │ │ │ │ +00042dd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00042de0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00042df0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a =========.. * * │ │ │ │ +00042e00: 6e6f 7465 2070 7269 6d61 7279 496e 7661 note primaryInva │ │ │ │ +00042e10: 7269 616e 7473 282e 2e2e 2c44 6164 653d riants(...,Dade= │ │ │ │ +00042e20: 3e2e 2e2e 293a 0a20 2020 2070 7269 6d61 >...):. prima │ │ │ │ +00042e30: 7279 496e 7661 7269 616e 7473 5f6c 705f ryInvariants_lp_ │ │ │ │ +00042e40: 7064 5f70 645f 7064 5f63 6d44 6164 653d pd_pd_pd_cmDade= │ │ │ │ +00042e50: 3e5f 7064 5f70 645f 7064 5f72 702c 202d >_pd_pd_pd_rp, - │ │ │ │ +00042e60: 2d20 616e 206f 7074 696f 6e61 6c20 6172 - an optional ar │ │ │ │ +00042e70: 6775 6d65 6e74 0a20 2020 2066 6f72 2070 gument. for p │ │ │ │ +00042e80: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ +00042e90: 2064 6574 6572 6d69 6e69 6e67 2077 6865 determining whe │ │ │ │ +00042ea0: 7468 6572 2074 6f20 7573 6520 7468 6520 ther to use the │ │ │ │ +00042eb0: 4461 6465 2061 6c67 6f72 6974 686d 0a0a Dade algorithm.. │ │ │ │ +00042ec0: 4675 7274 6865 7220 696e 666f 726d 6174 Further informat │ │ │ │ +00042ed0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ion.============ │ │ │ │ +00042ee0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2044 6566 =======.. * Def │ │ │ │ +00042ef0: 6175 6c74 2076 616c 7565 3a20 2a6e 6f74 ault value: *not │ │ │ │ +00042f00: 6520 6661 6c73 653a 2028 4d61 6361 756c e false: (Macaul │ │ │ │ +00042f10: 6179 3244 6f63 2966 616c 7365 2c0a 2020 ay2Doc)false,. │ │ │ │ +00042f20: 2a20 4675 6e63 7469 6f6e 3a20 2a6e 6f74 * Function: *not │ │ │ │ +00042f30: 6520 7072 696d 6172 7949 6e76 6172 6961 e primaryInvaria │ │ │ │ +00042f40: 6e74 733a 2070 7269 6d61 7279 496e 7661 nts: primaryInva │ │ │ │ +00042f50: 7269 616e 7473 2c20 2d2d 2063 6f6d 7075 riants, -- compu │ │ │ │ +00042f60: 7465 7320 6120 6c69 7374 206f 660a 2020 tes a list of. │ │ │ │ +00042f70: 2020 7072 696d 6172 7920 696e 7661 7269 primary invari │ │ │ │ +00042f80: 616e 7473 2066 6f72 2074 6865 2069 6e76 ants for the inv │ │ │ │ +00042f90: 6172 6961 6e74 2072 696e 6720 6f66 2061 ariant ring of a │ │ │ │ +00042fa0: 2066 696e 6974 6520 6772 6f75 700a 2020 finite group. │ │ │ │ +00042fb0: 2a20 4f70 7469 6f6e 206b 6579 3a20 2a6e * Option key: *n │ │ │ │ +00042fc0: 6f74 6520 4461 6465 3a20 7072 696d 6172 ote Dade: primar │ │ │ │ +00042fd0: 7949 6e76 6172 6961 6e74 735f 6c70 5f70 yInvariants_lp_p │ │ │ │ +00042fe0: 645f 7064 5f70 645f 636d 4461 6465 3d3e d_pd_pd_cmDade=> │ │ │ │ +00042ff0: 5f70 645f 7064 5f70 645f 7270 2c0a 2020 _pd_pd_pd_rp,. │ │ │ │ +00043000: 2020 2d2d 2061 6e20 6f70 7469 6f6e 616c -- an optional │ │ │ │ +00043010: 2061 7267 756d 656e 7420 666f 7220 7072 argument for pr │ │ │ │ +00043020: 696d 6172 7949 6e76 6172 6961 6e74 7320 imaryInvariants │ │ │ │ +00043030: 6465 7465 726d 696e 696e 6720 7768 6574 determining whet │ │ │ │ +00043040: 6865 7220 746f 2075 7365 0a20 2020 2074 her to use. t │ │ │ │ +00043050: 6865 2044 6164 6520 616c 676f 7269 7468 he Dade algorith │ │ │ │ +00043060: 6d0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d m.-------------- │ │ │ │ +00043070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000430a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000430b0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +000430c0: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +000430d0: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +000430e0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +000430f0: 6361 756c 6179 322d 312e 3236 2e30 352b caulay2-1.26.05+ │ │ │ │ +00043100: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +00043110: 7061 636b 6167 6573 2f0a 496e 7661 7269 packages/.Invari │ │ │ │ +00043120: 616e 7452 696e 672f 4861 7765 7344 6f63 antRing/HawesDoc │ │ │ │ +00043130: 2e6d 323a 3234 343a 302e 0a1f 0a46 696c .m2:244:0....Fil │ │ │ │ +00043140: 653a 2049 6e76 6172 6961 6e74 5269 6e67 e: InvariantRing │ │ │ │ +00043150: 2e69 6e66 6f2c 204e 6f64 653a 2070 7269 .info, Node: pri │ │ │ │ +00043160: 6d61 7279 496e 7661 7269 616e 7473 5f6c maryInvariants_l │ │ │ │ +00043170: 705f 7064 5f70 645f 7064 5f63 6d44 6567 p_pd_pd_pd_cmDeg │ │ │ │ +00043180: 7265 6556 6563 746f 723d 3e5f 7064 5f70 reeVector=>_pd_p │ │ │ │ +00043190: 645f 7064 5f72 702c 204e 6578 743a 2072 d_pd_rp, Next: r │ │ │ │ +000431a0: 616e 6b5f 6c70 4469 6167 6f6e 616c 4163 ank_lpDiagonalAc │ │ │ │ +000431b0: 7469 6f6e 5f72 702c 2050 7265 763a 2070 tion_rp, Prev: p │ │ │ │ +000431c0: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ +000431d0: 5f6c 705f 7064 5f70 645f 7064 5f63 6d44 _lp_pd_pd_pd_cmD │ │ │ │ +000431e0: 6164 653d 3e5f 7064 5f70 645f 7064 5f72 ade=>_pd_pd_pd_r │ │ │ │ +000431f0: 702c 2055 703a 2054 6f70 0a0a 7072 696d p, Up: Top..prim │ │ │ │ +00043200: 6172 7949 6e76 6172 6961 6e74 7328 2e2e aryInvariants(.. │ │ │ │ +00043210: 2e2c 4465 6772 6565 5665 6374 6f72 3d3e .,DegreeVector=> │ │ │ │ +00043220: 2e2e 2e29 202d 2d20 616e 206f 7074 696f ...) -- an optio │ │ │ │ +00043230: 6e61 6c20 6172 6775 6d65 6e74 2066 6f72 nal argument for │ │ │ │ +00043240: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ +00043250: 7473 2074 6861 7420 6669 6e64 7320 696e ts that finds in │ │ │ │ +00043260: 7661 7269 616e 7473 206f 6620 6365 7274 variants of cert │ │ │ │ +00043270: 6169 6e20 6465 6772 6565 730a 2a2a 2a2a ain degrees.**** │ │ │ │ +00043280: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00043290: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000432a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000432b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000432c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000432d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000432e0: 2a2a 2a2a 2a2a 2a2a 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If it │ │ │ │ +00043720: 2069 7320 6b6e 6f77 6e20 7468 6174 2061 is known that a │ │ │ │ +00043730: 2068 736f 7020 6578 6973 7473 2066 6f72 hsop exists for │ │ │ │ +00043740: 2061 2063 6572 7461 696e 0a63 6f6c 6c65 a certain.colle │ │ │ │ +00043750: 6374 696f 6e20 6f66 2064 6567 7265 6573 ction of degrees │ │ │ │ +00043760: 2c20 7468 6973 2063 616e 2062 6520 6173 , this can be as │ │ │ │ +00043770: 7369 676e 6564 2c20 6173 2061 202a 6e6f signed, as a *no │ │ │ │ +00043780: 7465 204c 6973 743a 0a28 4d61 6361 756c te List:.(Macaul │ │ │ │ +00043790: 6179 3244 6f63 294c 6973 742c 2c20 746f ay2Doc)List,, to │ │ │ │ +000437a0: 2074 6865 206f 7074 696f 6e61 6c20 6172 the optional ar │ │ │ │ +000437b0: 6775 6d65 6e74 202a 6e6f 7465 2044 6567 gument *note Deg │ │ │ │ +000437c0: 7265 6556 6563 746f 723a 0a70 7269 6d61 reeVector:.prima │ │ │ │ +000437d0: 7279 496e 7661 7269 616e 7473 5f6c 705f ryInvariants_lp_ │ │ │ │ +000437e0: 7064 5f70 645f 7064 5f63 6d44 6567 7265 pd_pd_pd_cmDegre │ │ │ │ +000437f0: 6556 6563 746f 723d 3e5f 7064 5f70 645f eVector=>_pd_pd_ │ │ │ │ +00043800: 7064 5f72 702c 2e20 2a6e 6f74 650a 7072 pd_rp,. *note.pr │ │ │ │ +00043810: 696d 6172 7949 6e76 6172 6961 6e74 733a imaryInvariants: │ │ │ │ +00043820: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ +00043830: 7473 2c20 7769 6c6c 2074 6865 6e20 6f75 ts, will then ou │ │ │ │ +00043840: 7470 7574 2061 2068 736f 7020 636f 7272 tput a hsop corr │ │ │ │ +00043850: 6573 706f 6e64 696e 6720 746f 0a74 6869 esponding to.thi │ │ │ │ +00043860: 7320 6c69 7374 206f 6620 6465 6772 6565 s list of degree │ │ │ │ +00043870: 732e 2049 6620 686f 7765 7665 7220 6e6f s. If however no │ │ │ │ +00043880: 2073 7563 6820 6873 6f70 2065 7869 7374 such hsop exist │ │ │ │ +00043890: 732c 202a 6e6f 7465 2070 7269 6d61 7279 s, *note primary │ │ │ │ +000438a0: 496e 7661 7269 616e 7473 3a0a 7072 696d Invariants:.prim │ │ │ │ +000438b0: 6172 7949 6e76 6172 6961 6e74 732c 206f aryInvariants, o │ │ │ │ +000438c0: 7574 7075 7473 2061 6e20 6572 726f 7220 utputs an error │ │ │ │ +000438d0: 6d65 7373 6167 652e 0a0a 4e6f 7465 2074 message...Note t │ │ │ │ +000438e0: 6861 7420 7468 6520 2a6e 6f74 6520 4c69 hat the *note Li │ │ │ │ +000438f0: 7374 3a20 284d 6163 6175 6c61 7932 446f st: (Macaulay2Do │ │ │ │ +00043900: 6329 4c69 7374 2c20 6173 7369 676e 6564 c)List, assigned │ │ │ │ +00043910: 2074 6f20 2a6e 6f74 6520 4465 6772 6565 to *note Degree │ │ │ │ +00043920: 5665 6374 6f72 3a0a 7072 696d 6172 7949 Vector:.primaryI │ │ │ │ +00043930: 6e76 6172 6961 6e74 735f 6c70 5f70 645f nvariants_lp_pd_ │ │ │ │ +00043940: 7064 5f70 645f 636d 4465 6772 6565 5665 pd_pd_cmDegreeVe │ │ │ │ +00043950: 6374 6f72 3d3e 5f70 645f 7064 5f70 645f ctor=>_pd_pd_pd_ │ │ │ │ +00043960: 7270 2c20 6973 2069 676e 6f72 6564 2069 rp, is ignored i │ │ │ │ +00043970: 6620 2a6e 6f74 650a 4461 6465 3a20 7072 f *note.Dade: pr │ │ │ │ +00043980: 696d 6172 7949 6e76 6172 6961 6e74 735f imaryInvariants_ │ │ │ │ +00043990: 6c70 5f70 645f 7064 5f70 645f 636d 4461 lp_pd_pd_pd_cmDa │ │ │ │ +000439a0: 6465 3d3e 5f70 645f 7064 5f70 645f 7270 de=>_pd_pd_pd_rp │ │ │ │ +000439b0: 2c20 6973 2073 6574 2074 6f20 2a6e 6f74 , is set to *not │ │ │ │ +000439c0: 6520 7472 7565 3a0a 284d 6163 6175 6c61 e true:.(Macaula │ │ │ │ +000439d0: 7932 446f 6329 7472 7565 2c2e 0a0a 2b2d y2Doc)true,...+- │ │ │ │ +000439e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000439f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00043a10: 0a7c 6931 203a 2041 3d6d 6174 7269 787b .|i1 : A=matrix{ │ │ │ │ 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│ -00043ba0: 2020 2020 2020 2020 2020 2020 2033 2020 3 │ │ │ │ -00043bb0: 2020 2020 2033 2020 2020 2020 2020 2020 3 │ │ │ │ +00043b70: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00043b80: 2020 2020 2033 2020 2020 2020 2033 2020 3 3 │ │ │ │ +00043b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043ba0: 2020 2020 2020 207c 0a7c 6f32 203a 204d |.|o2 : M │ │ │ │ +00043bb0: 6174 7269 7820 5a5a 2020 3c2d 2d20 5a5a atrix ZZ <-- ZZ │ │ │ │ 00043bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043be0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00043bf0: 3120 3a20 4d61 7472 6978 205a 5a20 203c 1 : Matrix ZZ < │ │ │ │ -00043c00: 2d2d 205a 5a20 2020 2020 2020 2020 2020 -- ZZ │ │ │ │ -00043c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043c30: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00043c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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│ +00043e50: 202b 2079 2020 2b20 7a20 2c20 7820 7920 + y + z , x y │ │ │ │ +00043e60: 202b 2078 207a 2020 2b20 7920 7a20 7d20 + x z + y z } │ │ │ │ +00043e70: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00043e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00043ec0: 3320 3d20 5151 5b78 2e2e 7a5d 203c 2d20 3 = QQ[x..z] <- │ │ │ │ -00043ed0: 7b7c 2030 2031 2030 207c 2c20 7c20 3020 {| 0 1 0 |, | 0 │ │ │ │ -00043ee0: 3120 3020 7c7d 2020 2020 2020 2020 2020 1 0 |} │ │ │ │ -00043ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043f00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00043f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043f20: 207c 2030 2030 2031 207c 2020 7c20 3120 | 0 0 1 | | 1 │ │ │ │ -00043f30: 3020 3020 7c20 2020 2020 2020 2020 2020 0 0 | │ │ │ │ +00043ea0: 2020 2020 7c0a 7c6f 3420 3a20 4c69 7374 |.|o4 : List │ │ │ │ +00043eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043ed0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00043ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 6176 ----------+..Cav │ │ │ │ +00043f10: 6561 740a 3d3d 3d3d 3d3d 0a0a 2020 2020 eat.======.. │ │ │ │ +00043f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043f50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00043f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043f70: 207c 2031 2030 2030 207c 2020 7c20 3020 | 1 0 0 | | 0 │ │ │ │ -00043f80: 3020 3120 7c20 2020 2020 2020 2020 2020 0 1 | │ │ │ │ -00043f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00043fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043f60: 2020 2020 2020 200a 4375 7272 656e 746c .Currentl │ │ │ │ +00043f70: 7920 7573 6572 7320 6361 6e20 6f6e 6c79 y users can only │ │ │ │ +00043f80: 2075 7365 202a 6e6f 7465 2070 7269 6d61 use *note prima │ │ │ │ +00043f90: 7279 496e 7661 7269 616e 7473 3a20 7072 ryInvariants: pr │ │ │ │ +00043fa0: 696d 6172 7949 6e76 6172 6961 6e74 732c imaryInvariants, │ │ │ │ +00043fb0: 2074 6f0a 2020 2020 2020 2020 2020 2020 to. │ │ │ │ 00043fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043ff0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00044000: 3320 3a20 4669 6e69 7465 4772 6f75 7041 3 : FiniteGroupA │ │ │ │ -00044010: 6374 696f 6e20 2020 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│ │ +00044000: 200a 6361 6c63 756c 6174 6520 6120 6873 .calculate a hs │ │ │ │ +00044010: 6f70 2066 6f72 2074 6865 2069 6e76 6172 op for the invar │ │ │ │ +00044020: 6961 6e74 2072 696e 6720 6f76 6572 2061 iant ring over a │ │ │ │ +00044030: 2066 696e 6974 6520 6669 656c 6420 6279 finite field by │ │ │ │ +00044040: 2075 7369 6e67 2074 6865 2044 6164 650a using the Dade. │ │ │ │ +00044050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044090: 2020 2020 2020 0a61 6c67 6f72 6974 686d .algorithm │ │ │ │ +000440a0: 2e20 5573 6572 7320 7368 6f75 6c64 2065 . Users should e │ │ │ │ +000440b0: 6e74 6572 2074 6865 2066 696e 6974 6520 nter the finite │ │ │ │ +000440c0: 6669 656c 6420 6173 2061 202a 6e6f 7465 field as a *note │ │ │ │ +000440d0: 2047 616c 6f69 7346 6965 6c64 3a0a 2020 GaloisField:. │ │ │ │ +000440e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000440f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044130: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00044140: 2020 2020 2020 3320 2020 2033 2020 2020 3 3 │ │ │ │ -00044150: 3320 2020 3220 2020 2020 2020 3220 2020 3 2 2 │ │ │ │ -00044160: 2032 2020 2020 2032 2020 2020 2020 2032 2 2 2 │ │ │ │ -00044170: 2020 2020 2020 3220 2020 3320 2020 2020 2 3 │ │ │ │ -00044180: 2020 3320 2020 2033 2020 2020 7c0a 7c6f 3 3 |.|o │ │ │ │ -00044190: 3420 3d20 7b78 2020 2b20 7920 202b 207a 4 = {x + y + z │ │ │ │ -000441a0: 202c 2078 2079 202b 2078 2a79 2020 2b20 , x y + x*y + │ │ │ │ -000441b0: 7820 7a20 2b20 7920 7a20 2b20 782a 7a20 x z + y z + x*z │ │ │ │ -000441c0: 202b 2079 2a7a 202c 2078 2079 202b 2078 + y*z , x y + x │ │ │ │ -000441d0: 2a79 2020 2b20 7820 7a20 2b20 7c0a 7c20 *y + x z + |.| │ │ │ │ -000441e0: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ -000441f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ -00044230: 2020 2020 2033 2020 2020 2020 2033 2020 3 3 │ │ │ │ -00044240: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ -00044250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044270: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00044280: 2020 2020 7920 7a20 2b20 782a 7a20 202b y z + x*z + │ │ │ │ -00044290: 2079 2a7a 207d 2020 2020 2020 2020 2020 y*z } │ │ │ │ -000442a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000442b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000442c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000442d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000442e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000442f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044310: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00044320: 3420 3a20 4c69 7374 2020 2020 2020 2020 4 : List │ │ │ │ -00044330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044360: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00044370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000443a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000443b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 ------------+..C │ │ │ │ -000443c0: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 2020 aveat.======.. │ │ │ │ -000443d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000443e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000443f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044410: 2020 2020 2020 2020 200a 4375 7272 656e .Curren │ │ │ │ -00044420: 746c 7920 7573 6572 7320 6361 6e20 6f6e tly users can on │ │ │ │ -00044430: 6c79 2075 7365 202a 6e6f 7465 2070 7269 ly use *note pri │ │ │ │ -00044440: 6d61 7279 496e 7661 7269 616e 7473 3a20 maryInvariants: │ │ │ │ -00044450: 7072 696d 6172 7949 6e76 6172 6961 6e74 primaryInvariant │ │ │ │ -00044460: 732c 2074 6f0a 2020 2020 2020 2020 2020 s, to. │ │ │ │ -00044470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000444a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000444b0: 2020 200a 6361 6c63 756c 6174 6520 6120 .calculate a │ │ │ │ -000444c0: 6873 6f70 2066 6f72 2074 6865 2069 6e76 hsop for the inv │ │ │ │ -000444d0: 6172 6961 6e74 2072 696e 6720 6f76 6572 ariant ring over │ │ │ │ -000444e0: 2061 2066 696e 6974 6520 6669 656c 6420 a finite field │ │ │ │ -000444f0: 6279 2075 7369 6e67 2074 6865 2044 6164 by using the Dad │ │ │ │ -00044500: 650a 2020 2020 2020 2020 2020 2020 2020 e. │ │ │ │ -00044510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044540: 2020 2020 2020 2020 0a61 6c67 6f72 6974 .algorit │ │ │ │ -00044550: 686d 2e20 5573 6572 7320 7368 6f75 6c64 hm. Users should │ │ │ │ -00044560: 2065 6e74 6572 2074 6865 2066 696e 6974 enter the finit │ │ │ │ -00044570: 6520 6669 656c 6420 6173 2061 202a 6e6f e field as a *no │ │ │ │ -00044580: 7465 2047 616c 6f69 7346 6965 6c64 3a0a te GaloisField:. │ │ │ │ -00044590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000445a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000445b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000445c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000445d0: 2020 2020 0a28 4d61 6361 756c 6179 3244 .(Macaulay2D │ │ │ │ -000445e0: 6f63 2947 616c 6f69 7346 6965 6c64 2c20 oc)GaloisField, │ │ │ │ -000445f0: 6f72 2061 2071 756f 7469 656e 7420 6669 or a quotient fi │ │ │ │ -00044600: 656c 6420 6f66 2074 6865 2066 6f72 6d20 eld of the form │ │ │ │ -00044610: 2a6e 6f74 6520 5a5a 3a0a 2020 2020 2020 *note ZZ:. │ │ │ │ -00044620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044660: 200a 284d 6163 6175 6c61 7932 446f 6329 .(Macaulay2Doc) │ │ │ │ -00044670: 5a5a 2c2f 7020 616e 6420 6172 6520 6164 ZZ,/p and are ad │ │ │ │ -00044680: 7669 7365 6420 746f 2065 6e73 7572 6520 vised to ensure │ │ │ │ -00044690: 7468 6174 2074 6865 2067 726f 756e 6420 that the ground │ │ │ │ -000446a0: 6669 656c 6420 6861 730a 2020 2020 2020 field has. │ │ │ │ -000446b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000446c0: 2020 2020 2020 6e2d 310a 6361 7264 696e n-1.cardin │ │ │ │ -000446d0: 616c 6974 7920 6772 6561 7465 7220 7468 ality greater th │ │ │ │ -000446e0: 616e 207c 477c 2020 202c 2077 6865 7265 an |G| , where │ │ │ │ -000446f0: 206e 2069 7320 7468 6520 6e75 6d62 6572 n is the number │ │ │ │ -00044700: 206f 6620 7661 7269 6162 6c65 7320 696e of variables in │ │ │ │ -00044710: 2074 6865 0a70 6f6c 796e 6f6d 6961 6c20 the.polynomial │ │ │ │ -00044720: 7269 6e67 2052 2e20 5573 696e 6720 6120 ring R. Using a │ │ │ │ -00044730: 6772 6f75 6e64 2066 6965 6c64 2073 6d61 ground field sma │ │ │ │ -00044740: 6c6c 6572 2074 6861 6e20 7468 6973 2072 ller than this r │ │ │ │ -00044750: 756e 7320 7468 6520 7269 736b 206f 6620 uns the risk of │ │ │ │ -00044760: 7468 650a 616c 676f 7269 7468 6d20 6765 the.algorithm ge │ │ │ │ -00044770: 7474 696e 6720 7374 7563 6b20 696e 2061 tting stuck in a │ │ │ │ -00044780: 6e20 696e 6669 6e69 7465 206c 6f6f 703b n infinite loop; │ │ │ │ -00044790: 202a 6e6f 7465 2070 7269 6d61 7279 496e *note primaryIn │ │ │ │ -000447a0: 7661 7269 616e 7473 3a0a 7072 696d 6172 variants:.primar │ │ │ │ -000447b0: 7949 6e76 6172 6961 6e74 732c 2064 6973 yInvariants, dis │ │ │ │ -000447c0: 706c 6179 7320 6120 7761 726e 696e 6720 plays a warning │ │ │ │ -000447d0: 6d65 7373 6167 6520 6173 6b69 6e67 2074 message asking t │ │ │ │ -000447e0: 6865 2075 7365 7220 7768 6574 6865 7220 he user whether │ │ │ │ -000447f0: 7468 6579 2077 6973 680a 746f 2063 6f6e they wish.to con │ │ │ │ -00044800: 7469 6e75 6520 7769 7468 2074 6865 2063 tinue with the c │ │ │ │ -00044810: 6f6d 7075 7461 7469 6f6e 2069 6e20 7468 omputation in th │ │ │ │ -00044820: 6973 2063 6173 652e 2053 6565 202a 6e6f is case. See *no │ │ │ │ -00044830: 7465 2068 736f 7020 616c 676f 7269 7468 te hsop algorith │ │ │ │ -00044840: 6d73 3a20 6873 6f70 0a61 6c67 6f72 6974 ms: hsop.algorit │ │ │ │ -00044850: 686d 732c 2066 6f72 2061 2064 6973 6375 hms, for a discu │ │ │ │ -00044860: 7373 696f 6e20 6f6e 2074 6865 2044 6164 ssion on the Dad │ │ │ │ -00044870: 6520 616c 676f 7269 7468 6d2e 0a0a 5365 e algorithm...Se │ │ │ │ -00044880: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ -00044890: 0a20 202a 202a 6e6f 7465 2070 7269 6d61 . * *note prima │ │ │ │ -000448a0: 7279 496e 7661 7269 616e 7473 3a20 7072 ryInvariants: pr │ │ │ │ -000448b0: 696d 6172 7949 6e76 6172 6961 6e74 732c imaryInvariants, │ │ │ │ -000448c0: 202d 2d20 636f 6d70 7574 6573 2061 206c -- computes a l │ │ │ │ -000448d0: 6973 7420 6f66 2070 7269 6d61 7279 0a20 ist of primary. │ │ │ │ -000448e0: 2020 2069 6e76 6172 6961 6e74 7320 666f invariants fo │ │ │ │ -000448f0: 7220 7468 6520 696e 7661 7269 616e 7420 r the invariant │ │ │ │ -00044900: 7269 6e67 206f 6620 6120 6669 6e69 7465 ring of a finite │ │ │ │ -00044910: 2067 726f 7570 0a0a 4675 6e63 7469 6f6e group..Function │ │ │ │ -00044920: 7320 7769 7468 206f 7074 696f 6e61 6c20 s with optional │ │ │ │ -00044930: 6172 6775 6d65 6e74 206e 616d 6564 2044 argument named D │ │ │ │ -00044940: 6567 7265 6556 6563 746f 723a 0a3d 3d3d egreeVector:.=== │ │ │ │ -00044950: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00044960: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00044970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00044980: 3d0a 0a20 202a 2022 6869 726f 6e61 6b61 =.. * "hironaka │ │ │ │ -00044990: 4465 636f 6d70 6f73 6974 696f 6e28 2e2e Decomposition(.. │ │ │ │ -000449a0: 2e2c 4465 6772 6565 5665 6374 6f72 3d3e .,DegreeVector=> │ │ │ │ -000449b0: 2e2e 2e29 220a 2020 2a20 2a6e 6f74 6520 ...)". * *note │ │ │ │ -000449c0: 7072 696d 6172 7949 6e76 6172 6961 6e74 primaryInvariant │ │ │ │ -000449d0: 7328 2e2e 2e2c 4465 6772 6565 5665 6374 s(...,DegreeVect │ │ │ │ -000449e0: 6f72 3d3e 2e2e 2e29 3a0a 2020 2020 7072 or=>...):. pr │ │ │ │ -000449f0: 696d 6172 7949 6e76 6172 6961 6e74 735f imaryInvariants_ │ │ │ │ -00044a00: 6c70 5f70 645f 7064 5f70 645f 636d 4465 lp_pd_pd_pd_cmDe │ │ │ │ -00044a10: 6772 6565 5665 6374 6f72 3d3e 5f70 645f greeVector=>_pd_ │ │ │ │ -00044a20: 7064 5f70 645f 7270 2c20 2d2d 2061 6e20 pd_pd_rp, -- an │ │ │ │ -00044a30: 6f70 7469 6f6e 616c 0a20 2020 2061 7267 optional. arg │ │ │ │ -00044a40: 756d 656e 7420 666f 7220 7072 696d 6172 ument for primar │ │ │ │ -00044a50: 7949 6e76 6172 6961 6e74 7320 7468 6174 yInvariants that │ │ │ │ -00044a60: 2066 696e 6473 2069 6e76 6172 6961 6e74 finds invariant │ │ │ │ -00044a70: 7320 6f66 2063 6572 7461 696e 2064 6567 s of certain deg │ │ │ │ -00044a80: 7265 6573 0a0a 4675 7274 6865 7220 696e rees..Further in │ │ │ │ -00044a90: 666f 726d 6174 696f 6e0a 3d3d 3d3d 3d3d formation.====== │ │ │ │ -00044aa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -00044ab0: 202a 2044 6566 6175 6c74 2076 616c 7565 * Default value │ │ │ │ -00044ac0: 3a20 300a 2020 2a20 4675 6e63 7469 6f6e : 0. * Function │ │ │ │ -00044ad0: 3a20 2a6e 6f74 6520 7072 696d 6172 7949 : *note primaryI │ │ │ │ -00044ae0: 6e76 6172 6961 6e74 733a 2070 7269 6d61 nvariants: prima │ │ │ │ -00044af0: 7279 496e 7661 7269 616e 7473 2c20 2d2d ryInvariants, -- │ │ │ │ -00044b00: 2063 6f6d 7075 7465 7320 6120 6c69 7374 computes a list │ │ │ │ -00044b10: 206f 660a 2020 2020 7072 696d 6172 7920 of. primary │ │ │ │ -00044b20: 696e 7661 7269 616e 7473 2066 6f72 2074 invariants for t │ │ │ │ -00044b30: 6865 2069 6e76 6172 6961 6e74 2072 696e he invariant rin │ │ │ │ -00044b40: 6720 6f66 2061 2066 696e 6974 6520 6772 g of a finite gr │ │ │ │ -00044b50: 6f75 700a 2020 2a20 4f70 7469 6f6e 206b oup. * Option k │ │ │ │ -00044b60: 6579 3a20 2a6e 6f74 6520 4465 6772 6565 ey: *note Degree │ │ │ │ -00044b70: 5665 6374 6f72 3a0a 2020 2020 7072 696d Vector:. prim │ │ │ │ -00044b80: 6172 7949 6e76 6172 6961 6e74 735f 6c70 aryInvariants_lp │ │ │ │ -00044b90: 5f70 645f 7064 5f70 645f 636d 4465 6772 _pd_pd_pd_cmDegr │ │ │ │ -00044ba0: 6565 5665 6374 6f72 3d3e 5f70 645f 7064 eeVector=>_pd_pd │ │ │ │ -00044bb0: 5f70 645f 7270 2c20 2d2d 2061 6e20 6f70 _pd_rp, -- an op │ │ │ │ -00044bc0: 7469 6f6e 616c 0a20 2020 2061 7267 756d tional. argum │ │ │ │ -00044bd0: 656e 7420 666f 7220 7072 696d 6172 7949 ent for primaryI │ │ │ │ -00044be0: 6e76 6172 6961 6e74 7320 7468 6174 2066 nvariants that f │ │ │ │ -00044bf0: 696e 6473 2069 6e76 6172 6961 6e74 7320 inds invariants │ │ │ │ -00044c00: 6f66 2063 6572 7461 696e 2064 6567 7265 of certain degre │ │ │ │ -00044c10: 6573 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d es.------------- │ │ │ │ -00044c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044c60: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -00044c70: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -00044c80: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -00044c90: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -00044ca0: 6163 6175 6c61 7932 2d31 2e32 362e 3035 acaulay2-1.26.05 │ │ │ │ -00044cb0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -00044cc0: 2f70 6163 6b61 6765 732f 0a49 6e76 6172 /packages/.Invar │ │ │ │ -00044cd0: 6961 6e74 5269 6e67 2f48 6177 6573 446f iantRing/HawesDo │ │ │ │ -00044ce0: 632e 6d32 3a32 3938 3a30 2e0a 1f0a 4669 c.m2:298:0....Fi │ │ │ │ -00044cf0: 6c65 3a20 496e 7661 7269 616e 7452 696e le: InvariantRin │ │ │ │ -00044d00: 672e 696e 666f 2c20 4e6f 6465 3a20 7261 g.info, Node: ra │ │ │ │ -00044d10: 6e6b 5f6c 7044 6961 676f 6e61 6c41 6374 nk_lpDiagonalAct │ │ │ │ -00044d20: 696f 6e5f 7270 2c20 4e65 7874 3a20 7265 ion_rp, Next: re │ │ │ │ -00044d30: 6c61 7469 6f6e 735f 6c70 4669 6e69 7465 lations_lpFinite │ │ │ │ -00044d40: 4772 6f75 7041 6374 696f 6e5f 7270 2c20 GroupAction_rp, │ │ │ │ -00044d50: 5072 6576 3a20 7072 696d 6172 7949 6e76 Prev: primaryInv │ │ │ │ -00044d60: 6172 6961 6e74 735f 6c70 5f70 645f 7064 ariants_lp_pd_pd │ │ │ │ -00044d70: 5f70 645f 636d 4465 6772 6565 5665 6374 _pd_cmDegreeVect │ │ │ │ -00044d80: 6f72 3d3e 5f70 645f 7064 5f70 645f 7270 or=>_pd_pd_pd_rp │ │ │ │ -00044d90: 2c20 5570 3a20 546f 700a 0a72 616e 6b28 , Up: Top..rank( │ │ │ │ -00044da0: 4469 6167 6f6e 616c 4163 7469 6f6e 2920 DiagonalAction) │ │ │ │ -00044db0: 2d2d 206f 6620 6120 6469 6167 6f6e 616c -- of a diagonal │ │ │ │ -00044dc0: 2061 6374 696f 6e0a 2a2a 2a2a 2a2a 2a2a action.******** │ │ │ │ -00044dd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00044de0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00044df0: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469 ****.. * Functi │ │ │ │ -00044e00: 6f6e 3a20 2a6e 6f74 6520 7261 6e6b 3a20 on: *note rank: │ │ │ │ -00044e10: 284d 6163 6175 6c61 7932 446f 6329 7261 (Macaulay2Doc)ra │ │ │ │ -00044e20: 6e6b 2c0a 2020 2a20 5573 6167 653a 200a nk,. * Usage: . │ │ │ │ -00044e30: 2020 2020 2020 2020 7261 6e6b 2044 0a20 rank D. │ │ │ │ -00044e40: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ -00044e50: 202a 2044 2c20 616e 2069 6e73 7461 6e63 * D, an instanc │ │ │ │ -00044e60: 6520 6f66 2074 6865 2074 7970 6520 2a6e e of the type *n │ │ │ │ -00044e70: 6f74 6520 4469 6167 6f6e 616c 4163 7469 ote DiagonalActi │ │ │ │ -00044e80: 6f6e 3a20 4469 6167 6f6e 616c 4163 7469 on: DiagonalActi │ │ │ │ -00044e90: 6f6e 2c0a 2020 2a20 4f75 7470 7574 733a on,. * Outputs: │ │ │ │ -00044ea0: 0a20 2020 2020 202a 2061 6e20 2a6e 6f74 . * an *not │ │ │ │ -00044eb0: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ -00044ec0: 756c 6179 3244 6f63 295a 5a2c 2c20 7468 ulay2Doc)ZZ,, th │ │ │ │ -00044ed0: 6520 7261 6e6b 206f 6620 7468 6520 746f e rank of the to │ │ │ │ -00044ee0: 7275 7320 6661 6374 6f72 206f 6620 610a rus factor of a. │ │ │ │ -00044ef0: 2020 2020 2020 2020 6469 6167 6f6e 616c diagonal │ │ │ │ -00044f00: 2061 6374 696f 6e0a 0a44 6573 6372 6970 action..Descrip │ │ │ │ -00044f10: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -00044f20: 0a0a 5468 6973 2066 756e 6374 696f 6e20 ..This function │ │ │ │ -00044f30: 6973 2070 726f 7669 6465 6420 6279 2074 is provided by t │ │ │ │ -00044f40: 6865 2070 6163 6b61 6765 202a 6e6f 7465 he package *note │ │ │ │ -00044f50: 2049 6e76 6172 6961 6e74 5269 6e67 3a20 InvariantRing: │ │ │ │ -00044f60: 546f 702c 2e20 0a0a 5573 6520 7468 6973 Top,. ..Use this │ │ │ │ -00044f70: 2066 756e 6374 696f 6e20 746f 2072 6563 function to rec │ │ │ │ -00044f80: 6f76 6572 2074 6865 2072 616e 6b20 6f66 over the rank of │ │ │ │ -00044f90: 2074 6865 2074 6f72 7573 2066 6163 746f the torus facto │ │ │ │ -00044fa0: 7220 6f66 2061 2064 6961 676f 6e61 6c20 r of a diagonal │ │ │ │ -00044fb0: 6163 7469 6f6e 2e0a 0a54 6865 2066 6f6c action...The fol │ │ │ │ -00044fc0: 6c6f 7769 6e67 2065 7861 6d70 6c65 2064 lowing example d │ │ │ │ -00044fd0: 6566 696e 6573 2061 6e20 6163 7469 6f6e efines an action │ │ │ │ -00044fe0: 206f 6620 6120 7477 6f2d 6469 6d65 6e73 of a two-dimens │ │ │ │ -00044ff0: 696f 6e61 6c20 746f 7275 7320 6f6e 2061 ional torus on a │ │ │ │ -00045000: 0a70 6f6c 796e 6f6d 6961 6c20 7269 6e67 .polynomial ring │ │ │ │ -00045010: 2069 6e20 666f 7572 2076 6172 6961 626c in four variabl │ │ │ │ -00045020: 6573 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d es...+---------- │ │ │ │ -00045030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00045050: 6931 203a 2052 203d 2051 515b 785f 312e i1 : R = QQ[x_1. │ │ │ │ -00045060: 2e78 5f34 5d20 2020 2020 2020 2020 2020 .x_4] │ │ │ │ -00045070: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00045080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000450a0: 207c 0a7c 6f31 203d 2052 2020 2020 2020 |.|o1 = R │ │ │ │ -000450b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000450c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000450d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000450e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000450f0: 2020 2020 207c 0a7c 6f31 203a 2050 6f6c |.|o1 : Pol │ │ │ │ -00045100: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ -00045110: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00045120: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00045130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045140: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -00045150: 2057 203d 206d 6174 7269 787b 7b30 2c31 W = matrix{{0,1 │ │ │ │ -00045160: 2c2d 312c 317d 2c7b 312c 302c 2d31 2c2d ,-1,1},{1,0,-1,- │ │ │ │ -00045170: 317d 7d7c 0a7c 2020 2020 2020 2020 2020 1}}|.| │ │ │ │ -00045180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045190: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000451a0: 6f32 203d 207c 2030 2031 202d 3120 3120 o2 = | 0 1 -1 1 │ │ │ │ -000451b0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000451c0: 2020 2020 2020 207c 0a7c 2020 2020 207c |.| | │ │ │ │ -000451d0: 2031 2030 202d 3120 2d31 207c 2020 2020 1 0 -1 -1 | │ │ │ │ -000451e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000451f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00045200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045210: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00045220: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -00045230: 2020 2020 3420 2020 2020 2020 2020 2020 4 │ │ │ │ -00045240: 2020 2020 207c 0a7c 6f32 203a 204d 6174 |.|o2 : Mat │ │ │ │ -00045250: 7269 7820 5a5a 2020 3c2d 2d20 5a5a 2020 rix ZZ 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-------------+.| │ │ │ │ -00045440: 6934 203a 2072 616e 6b20 5420 2020 2020 i4 : rank T │ │ │ │ -00045450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045460: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00045470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045490: 207c 0a7c 6f34 203d 2032 2020 2020 2020 |.|o4 = 2 │ │ │ │ -000454a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000454b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -000454c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000454d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000454e0: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f -----+..See also │ │ │ │ -000454f0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ -00045500: 6e6f 7465 2044 6961 676f 6e61 6c41 6374 note DiagonalAct │ │ │ │ -00045510: 696f 6e3a 2044 6961 676f 6e61 6c41 6374 ion: DiagonalAct │ │ │ │ -00045520: 696f 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7932 +ds/M2/Macaulay2 │ │ │ │ -000456d0: 2f70 6163 6b61 6765 732f 0a49 6e76 6172 /packages/.Invar │ │ │ │ -000456e0: 6961 6e74 5269 6e67 2f41 6265 6c69 616e iantRing/Abelian │ │ │ │ -000456f0: 4772 6f75 7073 446f 632e 6d32 3a33 3834 GroupsDoc.m2:384 │ │ │ │ -00045700: 3a30 2e0a 1f0a 4669 6c65 3a20 496e 7661 :0....File: Inva │ │ │ │ -00045710: 7269 616e 7452 696e 672e 696e 666f 2c20 riantRing.info, │ │ │ │ -00045720: 4e6f 6465 3a20 7265 6c61 7469 6f6e 735f Node: relations_ │ │ │ │ -00045730: 6c70 4669 6e69 7465 4772 6f75 7041 6374 lpFiniteGroupAct │ │ │ │ -00045740: 696f 6e5f 7270 2c20 4e65 7874 3a20 7265 ion_rp, Next: re │ │ │ │ -00045750: 796e 6f6c 6473 4f70 6572 6174 6f72 2c20 ynoldsOperator, │ │ │ │ -00045760: 5072 6576 3a20 7261 6e6b 5f6c 7044 6961 Prev: rank_lpDia │ │ │ │ -00045770: 676f 6e61 6c41 6374 696f 6e5f 7270 2c20 gonalAction_rp, │ │ │ │ -00045780: 5570 3a20 546f 700a 0a72 656c 6174 696f Up: Top..relatio │ │ │ │ -00045790: 6e73 2846 696e 6974 6547 726f 7570 4163 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And each │ │ │ │ -00045a30: 0a72 656c 6174 696f 6e20 6973 2072 6570 .relation is rep │ │ │ │ -00045a40: 7265 7365 6e74 6564 2062 7920 6120 6c69 resented by a li │ │ │ │ -00045a50: 7374 206f 6620 7477 6f20 776f 7264 7320 st of two words │ │ │ │ -00045a60: 7468 6174 2065 7175 6174 6573 2074 6865 that equates the │ │ │ │ -00045a70: 2067 726f 7570 2065 6c65 6d65 6e74 0a72 group element.r │ │ │ │ -00045a80: 6570 7265 7365 6e74 6564 2062 7920 7468 epresented by th │ │ │ │ -00045a90: 6f73 6520 7477 6f20 776f 7264 732e 0a0a ose two words... │ │ │ │ -00045aa0: 5468 6520 666f 6c6c 6f77 696e 6720 6578 The following ex │ │ │ │ -00045ab0: 616d 706c 6520 6465 6669 6e65 7320 7468 ample defines th │ │ │ │ -00045ac0: 6520 7065 726d 7574 6174 696f 6e20 6163 e permutation ac │ │ │ │ -00045ad0: 7469 6f6e 206f 6620 6120 7379 6d6d 6574 tion of a symmet │ │ │ │ -00045ae0: 7269 6320 6772 6f75 7020 6f6e 0a74 6872 ric group on.thr │ │ │ │ -00045af0: 6565 2065 6c65 6d65 6e74 7320 7573 696e ee elements usin │ │ │ │ -00045b00: 6720 7468 7265 6520 7472 616e 7370 6f73 g three transpos │ │ │ │ -00045b10: 6974 696f 6e73 2e0a 0a2b 2d2d 2d2d 2d2d itions...+------ │ │ │ │ -00045b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045b60: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 -------+.|i1 : R │ │ │ │ -00045b70: 203d 2051 515b 785f 312e 2e78 5f33 5d20 = QQ[x_1..x_3] │ │ │ │ -00045b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045bb0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00045bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044120: 2020 0a28 4d61 6361 756c 6179 3244 6f63 .(Macaulay2Doc │ │ │ │ +00044130: 2947 616c 6f69 7346 6965 6c64 2c20 6f72 )GaloisField, or │ │ │ │ +00044140: 2061 2071 756f 7469 656e 7420 6669 656c a quotient fiel │ │ │ │ +00044150: 6420 6f66 2074 6865 2066 6f72 6d20 2a6e d of the form *n │ │ │ │ +00044160: 6f74 6520 5a5a 3a0a 2020 2020 2020 2020 ote ZZ:. │ │ │ │ +00044170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000441a0: 2020 2020 2020 2020 2020 2020 2020 200a . │ │ │ │ +000441b0: 284d 6163 6175 6c61 7932 446f 6329 5a5a (Macaulay2Doc)ZZ │ │ │ │ +000441c0: 2c2f 7020 616e 6420 6172 6520 6164 7669 ,/p and are advi │ │ │ │ +000441d0: 7365 6420 746f 2065 6e73 7572 6520 7468 sed to ensure th │ │ │ │ +000441e0: 6174 2074 6865 2067 726f 756e 6420 6669 at the ground fi │ │ │ │ +000441f0: 656c 6420 6861 730a 2020 2020 2020 2020 eld has. │ │ │ │ +00044200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044210: 2020 2020 6e2d 310a 6361 7264 696e 616c n-1.cardinal │ │ │ │ +00044220: 6974 7920 6772 6561 7465 7220 7468 616e ity greater than │ │ │ │ +00044230: 207c 477c 2020 202c 2077 6865 7265 206e |G| , where n │ │ │ │ +00044240: 2069 7320 7468 6520 6e75 6d62 6572 206f is the number o │ │ │ │ +00044250: 6620 7661 7269 6162 6c65 7320 696e 2074 f variables in t │ │ │ │ +00044260: 6865 0a70 6f6c 796e 6f6d 6961 6c20 7269 he.polynomial ri │ │ │ │ +00044270: 6e67 2052 2e20 5573 696e 6720 6120 6772 ng R. Using a gr │ │ │ │ +00044280: 6f75 6e64 2066 6965 6c64 2073 6d61 6c6c ound field small │ │ │ │ +00044290: 6572 2074 6861 6e20 7468 6973 2072 756e er than this run │ │ │ │ +000442a0: 7320 7468 6520 7269 736b 206f 6620 7468 s the risk of th │ │ │ │ +000442b0: 650a 616c 676f 7269 7468 6d20 6765 7474 e.algorithm gett │ │ │ │ +000442c0: 696e 6720 7374 7563 6b20 696e 2061 6e20 ing stuck in an │ │ │ │ +000442d0: 696e 6669 6e69 7465 206c 6f6f 703b 202a infinite loop; * │ │ │ │ +000442e0: 6e6f 7465 2070 7269 6d61 7279 496e 7661 note primaryInva │ │ │ │ +000442f0: 7269 616e 7473 3a0a 7072 696d 6172 7949 riants:.primaryI │ │ │ │ +00044300: 6e76 6172 6961 6e74 732c 2064 6973 706c nvariants, displ │ │ │ │ +00044310: 6179 7320 6120 7761 726e 696e 6720 6d65 ays a warning me │ │ │ │ +00044320: 7373 6167 6520 6173 6b69 6e67 2074 6865 ssage asking the │ │ │ │ +00044330: 2075 7365 7220 7768 6574 6865 7220 7468 user whether th │ │ │ │ +00044340: 6579 2077 6973 680a 746f 2063 6f6e 7469 ey wish.to conti │ │ │ │ +00044350: 6e75 6520 7769 7468 2074 6865 2063 6f6d nue with the com │ │ │ │ +00044360: 7075 7461 7469 6f6e 2069 6e20 7468 6973 putation in this │ │ │ │ +00044370: 2063 6173 652e 2053 6565 202a 6e6f 7465 case. See *note │ │ │ │ +00044380: 2068 736f 7020 616c 676f 7269 7468 6d73 hsop algorithms │ │ │ │ +00044390: 3a20 6873 6f70 0a61 6c67 6f72 6974 686d : hsop.algorithm │ │ │ │ +000443a0: 732c 2066 6f72 2061 2064 6973 6375 7373 s, for a discuss │ │ │ │ +000443b0: 696f 6e20 6f6e 2074 6865 2044 6164 6520 ion on the Dade │ │ │ │ +000443c0: 616c 676f 7269 7468 6d2e 0a0a 5365 6520 algorithm...See │ │ │ │ +000443d0: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 also.========.. │ │ │ │ +000443e0: 202a 202a 6e6f 7465 2070 7269 6d61 7279 * *note primary │ │ │ │ +000443f0: 496e 7661 7269 616e 7473 3a20 7072 696d Invariants: prim │ │ │ │ +00044400: 6172 7949 6e76 6172 6961 6e74 732c 202d aryInvariants, - │ │ │ │ +00044410: 2d20 636f 6d70 7574 6573 2061 206c 6973 - computes a lis │ │ │ │ +00044420: 7420 6f66 2070 7269 6d61 7279 0a20 2020 t of primary. │ │ │ │ +00044430: 2069 6e76 6172 6961 6e74 7320 666f 7220 invariants for │ │ │ │ +00044440: 7468 6520 696e 7661 7269 616e 7420 7269 the invariant ri │ │ │ │ +00044450: 6e67 206f 6620 6120 6669 6e69 7465 2067 ng of a finite g │ │ │ │ +00044460: 726f 7570 0a0a 4675 6e63 7469 6f6e 7320 roup..Functions │ │ │ │ +00044470: 7769 7468 206f 7074 696f 6e61 6c20 6172 with optional ar │ │ │ │ +00044480: 6775 6d65 6e74 206e 616d 6564 2044 6567 gument named Deg │ │ │ │ +00044490: 7265 6556 6563 746f 723a 0a3d 3d3d 3d3d reeVector:.===== │ │ │ │ +000444a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000444b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000444c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +000444d0: 0a20 202a 2022 6869 726f 6e61 6b61 4465 . * "hironakaDe │ │ │ │ +000444e0: 636f 6d70 6f73 6974 696f 6e28 2e2e 2e2c composition(..., │ │ │ │ +000444f0: 4465 6772 6565 5665 6374 6f72 3d3e 2e2e DegreeVector=>.. │ │ │ │ +00044500: 2e29 220a 2020 2a20 2a6e 6f74 6520 7072 .)". * *note pr │ │ │ │ +00044510: 696d 6172 7949 6e76 6172 6961 6e74 7328 imaryInvariants( │ │ │ │ +00044520: 2e2e 2e2c 4465 6772 6565 5665 6374 6f72 ...,DegreeVector │ │ │ │ +00044530: 3d3e 2e2e 2e29 3a0a 2020 2020 7072 696d =>...):. prim │ │ │ │ +00044540: 6172 7949 6e76 6172 6961 6e74 735f 6c70 aryInvariants_lp │ │ │ │ +00044550: 5f70 645f 7064 5f70 645f 636d 4465 6772 _pd_pd_pd_cmDegr │ │ │ │ +00044560: 6565 5665 6374 6f72 3d3e 5f70 645f 7064 eeVector=>_pd_pd │ │ │ │ +00044570: 5f70 645f 7270 2c20 2d2d 2061 6e20 6f70 _pd_rp, -- an op │ │ │ │ +00044580: 7469 6f6e 616c 0a20 2020 2061 7267 756d tional. argum │ │ │ │ +00044590: 656e 7420 666f 7220 7072 696d 6172 7949 ent for primaryI │ │ │ │ +000445a0: 6e76 6172 6961 6e74 7320 7468 6174 2066 nvariants that f │ │ │ │ +000445b0: 696e 6473 2069 6e76 6172 6961 6e74 7320 inds invariants │ │ │ │ +000445c0: 6f66 2063 6572 7461 696e 2064 6567 7265 of certain degre │ │ │ │ +000445d0: 6573 0a0a 4675 7274 6865 7220 696e 666f es..Further info │ │ │ │ +000445e0: 726d 6174 696f 6e0a 3d3d 3d3d 3d3d 3d3d rmation.======== │ │ │ │ +000445f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +00044600: 2044 6566 6175 6c74 2076 616c 7565 3a20 Default value: │ │ │ │ +00044610: 300a 2020 2a20 4675 6e63 7469 6f6e 3a20 0. * Function: │ │ │ │ +00044620: 2a6e 6f74 6520 7072 696d 6172 7949 6e76 *note primaryInv │ │ │ │ +00044630: 6172 6961 6e74 733a 2070 7269 6d61 7279 ariants: primary │ │ │ │ +00044640: 496e 7661 7269 616e 7473 2c20 2d2d 2063 Invariants, -- c │ │ │ │ +00044650: 6f6d 7075 7465 7320 6120 6c69 7374 206f omputes a list o │ │ │ │ +00044660: 660a 2020 2020 7072 696d 6172 7920 696e f. primary in │ │ │ │ +00044670: 7661 7269 616e 7473 2066 6f72 2074 6865 variants for the │ │ │ │ +00044680: 2069 6e76 6172 6961 6e74 2072 696e 6720 invariant ring │ │ │ │ +00044690: 6f66 2061 2066 696e 6974 6520 6772 6f75 of a finite grou │ │ │ │ +000446a0: 700a 2020 2a20 4f70 7469 6f6e 206b 6579 p. * Option key │ │ │ │ +000446b0: 3a20 2a6e 6f74 6520 4465 6772 6565 5665 : *note DegreeVe │ │ │ │ +000446c0: 6374 6f72 3a0a 2020 2020 7072 696d 6172 ctor:. primar │ │ │ │ +000446d0: 7949 6e76 6172 6961 6e74 735f 6c70 5f70 yInvariants_lp_p │ │ │ │ +000446e0: 645f 7064 5f70 645f 636d 4465 6772 6565 d_pd_pd_cmDegree │ │ │ │ +000446f0: 5665 6374 6f72 3d3e 5f70 645f 7064 5f70 Vector=>_pd_pd_p │ │ │ │ +00044700: 645f 7270 2c20 2d2d 2061 6e20 6f70 7469 d_rp, -- an opti │ │ │ │ +00044710: 6f6e 616c 0a20 2020 2061 7267 756d 656e onal. argumen │ │ │ │ +00044720: 7420 666f 7220 7072 696d 6172 7949 6e76 t for primaryInv │ │ │ │ +00044730: 6172 6961 6e74 7320 7468 6174 2066 696e ariants that fin │ │ │ │ +00044740: 6473 2069 6e76 6172 6961 6e74 7320 6f66 ds invariants of │ │ │ │ +00044750: 2063 6572 7461 696e 2064 6567 7265 6573 certain degrees │ │ │ │ +00044760: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +00044770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00044780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00044790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000447a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000447b0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +000447c0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +000447d0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +000447e0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +000447f0: 6175 6c61 7932 2d31 2e32 362e 3035 2b64 aulay2-1.26.05+d │ │ │ │ +00044800: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00044810: 6163 6b61 6765 732f 0a49 6e76 6172 6961 ackages/.Invaria │ │ │ │ +00044820: 6e74 5269 6e67 2f48 6177 6573 446f 632e ntRing/HawesDoc. │ │ │ │ +00044830: 6d32 3a32 3938 3a30 2e0a 1f0a 4669 6c65 m2:298:0....File │ │ │ │ +00044840: 3a20 496e 7661 7269 616e 7452 696e 672e : InvariantRing. │ │ │ │ +00044850: 696e 666f 2c20 4e6f 6465 3a20 7261 6e6b info, Node: rank │ │ │ │ +00044860: 5f6c 7044 6961 676f 6e61 6c41 6374 696f _lpDiagonalActio │ │ │ │ +00044870: 6e5f 7270 2c20 4e65 7874 3a20 7265 6c61 n_rp, Next: rela │ │ │ │ +00044880: 7469 6f6e 735f 6c70 4669 6e69 7465 4772 tions_lpFiniteGr │ │ │ │ +00044890: 6f75 7041 6374 696f 6e5f 7270 2c20 5072 oupAction_rp, Pr │ │ │ │ +000448a0: 6576 3a20 7072 696d 6172 7949 6e76 6172 ev: primaryInvar │ │ │ │ +000448b0: 6961 6e74 735f 6c70 5f70 645f 7064 5f70 iants_lp_pd_pd_p │ │ │ │ +000448c0: 645f 636d 4465 6772 6565 5665 6374 6f72 d_cmDegreeVector │ │ │ │ +000448d0: 3d3e 5f70 645f 7064 5f70 645f 7270 2c20 =>_pd_pd_pd_rp, │ │ │ │ +000448e0: 5570 3a20 546f 700a 0a72 616e 6b28 4469 Up: Top..rank(Di │ │ │ │ +000448f0: 6167 6f6e 616c 4163 7469 6f6e 2920 2d2d agonalAction) -- │ │ │ │ +00044900: 206f 6620 6120 6469 6167 6f6e 616c 2061 of a diagonal a │ │ │ │ +00044910: 6374 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a ction.********** │ │ │ │ +00044920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00044930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00044940: 2a2a 0a0a 2020 2a20 4675 6e63 7469 6f6e **.. * Function │ │ │ │ +00044950: 3a20 2a6e 6f74 6520 7261 6e6b 3a20 284d : *note rank: (M │ │ │ │ +00044960: 6163 6175 6c61 7932 446f 6329 7261 6e6b acaulay2Doc)rank │ │ │ │ +00044970: 2c0a 2020 2a20 5573 6167 653a 200a 2020 ,. * Usage: . │ │ │ │ +00044980: 2020 2020 2020 7261 6e6b 2044 0a20 202a rank D. * │ │ │ │ +00044990: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +000449a0: 2044 2c20 616e 2069 6e73 7461 6e63 6520 D, an instance │ │ │ │ +000449b0: 6f66 2074 6865 2074 7970 6520 2a6e 6f74 of the type *not │ │ │ │ +000449c0: 6520 4469 6167 6f6e 616c 4163 7469 6f6e e DiagonalAction │ │ │ │ +000449d0: 3a20 4469 6167 6f6e 616c 4163 7469 6f6e : DiagonalAction │ │ │ │ +000449e0: 2c0a 2020 2a20 4f75 7470 7574 733a 0a20 ,. * Outputs:. │ │ │ │ +000449f0: 2020 2020 202a 2061 6e20 2a6e 6f74 6520 * an *note │ │ │ │ +00044a00: 696e 7465 6765 723a 2028 4d61 6361 756c integer: (Macaul │ │ │ │ +00044a10: 6179 3244 6f63 295a 5a2c 2c20 7468 6520 ay2Doc)ZZ,, the │ │ │ │ +00044a20: 7261 6e6b 206f 6620 7468 6520 746f 7275 rank of the toru │ │ │ │ +00044a30: 7320 6661 6374 6f72 206f 6620 610a 2020 s factor of a. │ │ │ │ +00044a40: 2020 2020 2020 6469 6167 6f6e 616c 2061 diagonal a │ │ │ │ +00044a50: 6374 696f 6e0a 0a44 6573 6372 6970 7469 ction..Descripti │ │ │ │ +00044a60: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +00044a70: 5468 6973 2066 756e 6374 696f 6e20 6973 This function is │ │ │ │ +00044a80: 2070 726f 7669 6465 6420 6279 2074 6865 provided by the │ │ │ │ +00044a90: 2070 6163 6b61 6765 202a 6e6f 7465 2049 package *note I │ │ │ │ +00044aa0: 6e76 6172 6961 6e74 5269 6e67 3a20 546f nvariantRing: To │ │ │ │ +00044ab0: 702c 2e20 0a0a 5573 6520 7468 6973 2066 p,. ..Use this f │ │ │ │ +00044ac0: 756e 6374 696f 6e20 746f 2072 6563 6f76 unction to recov │ │ │ │ +00044ad0: 6572 2074 6865 2072 616e 6b20 6f66 2074 er the rank of t │ │ │ │ +00044ae0: 6865 2074 6f72 7573 2066 6163 746f 7220 he torus factor │ │ │ │ +00044af0: 6f66 2061 2064 6961 676f 6e61 6c20 6163 of a diagonal ac │ │ │ │ +00044b00: 7469 6f6e 2e0a 0a54 6865 2066 6f6c 6c6f tion...The follo │ │ │ │ +00044b10: 7769 6e67 2065 7861 6d70 6c65 2064 6566 wing example def │ │ │ │ +00044b20: 696e 6573 2061 6e20 6163 7469 6f6e 206f ines an action o │ │ │ │ +00044b30: 6620 6120 7477 6f2d 6469 6d65 6e73 696f f a two-dimensio │ │ │ │ +00044b40: 6e61 6c20 746f 7275 7320 6f6e 2061 0a70 nal torus on a.p │ │ │ │ +00044b50: 6f6c 796e 6f6d 6961 6c20 7269 6e67 2069 olynomial ring i │ │ │ │ +00044b60: 6e20 666f 7572 2076 6172 6961 626c 6573 n four variables │ │ │ │ +00044b70: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ +00044b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00044b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +00044ba0: 203a 2052 203d 2051 515b 785f 312e 2e78 : R = QQ[x_1..x │ │ │ │ +00044bb0: 5f34 5d20 2020 2020 2020 2020 2020 2020 _4] │ │ │ │ +00044bc0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00044bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044be0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00044bf0: 0a7c 6f31 203d 2052 2020 2020 2020 2020 .|o1 = R │ │ │ │ +00044c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044c10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00044c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044c40: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e |.|o1 : Polyn │ │ │ │ +00044c50: 6f6d 6961 6c52 696e 6720 2020 2020 2020 omialRing │ │ │ │ +00044c60: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00044c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00044c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00044c90: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2057 -------+.|i2 : W │ │ │ │ +00044ca0: 203d 206d 6174 7269 787b 7b30 2c31 2c2d = matrix{{0,1,- │ │ │ │ +00044cb0: 312c 317d 2c7b 312c 302c 2d31 2c2d 317d 1,1},{1,0,-1,-1} │ │ │ │ +00044cc0: 7d7c 0a7c 2020 2020 2020 2020 2020 2020 }|.| │ │ │ │ +00044cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044ce0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +00044cf0: 203d 207c 2030 2031 202d 3120 3120 207c = | 0 1 -1 1 | │ │ │ │ +00044d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044d10: 2020 2020 207c 0a7c 2020 2020 207c 2031 |.| | 1 │ │ │ │ +00044d20: 2030 202d 3120 2d31 207c 2020 2020 2020 0 -1 -1 | │ │ │ │ +00044d30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00044d40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00044d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044d60: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00044d70: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00044d80: 2020 3420 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +00044d90: 2020 207c 0a7c 6f32 203a 204d 6174 7269 |.|o2 : Matri │ │ │ │ +00044da0: 7820 5a5a 2020 3c2d 2d20 5a5a 2020 2020 x ZZ <-- ZZ │ │ │ │ +00044db0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00044dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00044dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00044de0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2054 -------+.|i3 : T │ │ │ │ +00044df0: 203d 2064 6961 676f 6e61 6c41 6374 696f = diagonalActio │ │ │ │ +00044e00: 6e28 572c 2052 2920 2020 2020 2020 2020 n(W, R) │ │ │ │ +00044e10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00044e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044e30: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00044e40: 2020 2020 2020 2020 2020 202a 2032 2020 * 2 │ │ │ │ +00044e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044e60: 2020 2020 207c 0a7c 6f33 203d 2052 203c |.|o3 = R < │ │ │ │ +00044e70: 2d20 2851 5120 2920 2076 6961 2020 2020 - (QQ ) via │ │ │ │ +00044e80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00044e90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00044ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044eb0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00044ec0: 207c 2030 2031 202d 3120 3120 207c 2020 | 0 1 -1 1 | │ │ │ │ +00044ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044ee0: 2020 207c 0a7c 2020 2020 207c 2031 2030 |.| | 1 0 │ │ │ │ +00044ef0: 202d 3120 2d31 207c 2020 2020 2020 2020 -1 -1 | │ │ │ │ +00044f00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00044f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044f30: 2020 2020 2020 207c 0a7c 6f33 203a 2044 |.|o3 : D │ │ │ │ +00044f40: 6961 676f 6e61 6c41 6374 696f 6e20 2020 iagonalAction │ │ │ │ +00044f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044f60: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00044f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00044f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ +00044f90: 203a 2072 616e 6b20 5420 2020 2020 2020 : rank T │ │ │ │ +00044fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044fb0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00044fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044fd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00044fe0: 0a7c 6f34 203d 2032 2020 2020 2020 2020 .|o4 = 2 │ │ │ │ +00044ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045000: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00045010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045030: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ +00045040: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +00045050: 7465 2044 6961 676f 6e61 6c41 6374 696f te DiagonalActio │ │ │ │ +00045060: 6e3a 2044 6961 676f 6e61 6c41 6374 696f n: DiagonalActio │ │ │ │ +00045070: 6e2c 202d 2d20 7468 6520 636c 6173 7320 n, -- the class │ │ │ │ +00045080: 6f66 2061 6c6c 2064 6961 676f 6e61 6c20 of all diagonal │ │ │ │ +00045090: 6163 7469 6f6e 730a 2020 2a20 2a6e 6f74 actions. * *not │ │ │ │ +000450a0: 6520 6469 6167 6f6e 616c 4163 7469 6f6e e diagonalAction │ │ │ │ +000450b0: 3a20 6469 6167 6f6e 616c 4163 7469 6f6e : diagonalAction │ │ │ │ +000450c0: 2c20 2d2d 2064 6961 676f 6e61 6c20 6772 , -- diagonal gr │ │ │ │ +000450d0: 6f75 7020 6163 7469 6f6e 2076 6961 2077 oup action via w │ │ │ │ +000450e0: 6569 6768 7473 0a0a 5761 7973 2074 6f20 eights..Ways to │ │ │ │ +000450f0: 7573 6520 7468 6973 206d 6574 686f 643a use this method: │ │ │ │ +00045100: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00045110: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a =========.. * * │ │ │ │ +00045120: 6e6f 7465 2072 616e 6b28 4469 6167 6f6e note rank(Diagon │ │ │ │ +00045130: 616c 4163 7469 6f6e 293a 2072 616e 6b5f alAction): rank_ │ │ │ │ +00045140: 6c70 4469 6167 6f6e 616c 4163 7469 6f6e lpDiagonalAction │ │ │ │ +00045150: 5f72 702c 202d 2d20 6f66 2061 2064 6961 _rp, -- of a dia │ │ │ │ +00045160: 676f 6e61 6c0a 2020 2020 6163 7469 6f6e gonal. action │ │ │ │ +00045170: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +00045180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000451a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000451b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000451c0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +000451d0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +000451e0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +000451f0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00045200: 6175 6c61 7932 2d31 2e32 362e 3035 2b64 aulay2-1.26.05+d │ │ │ │ +00045210: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00045220: 6163 6b61 6765 732f 0a49 6e76 6172 6961 ackages/.Invaria │ │ │ │ +00045230: 6e74 5269 6e67 2f41 6265 6c69 616e 4772 ntRing/AbelianGr │ │ │ │ +00045240: 6f75 7073 446f 632e 6d32 3a33 3834 3a30 oupsDoc.m2:384:0 │ │ │ │ +00045250: 2e0a 1f0a 4669 6c65 3a20 496e 7661 7269 ....File: Invari │ │ │ │ +00045260: 616e 7452 696e 672e 696e 666f 2c20 4e6f antRing.info, No │ │ │ │ +00045270: 6465 3a20 7265 6c61 7469 6f6e 735f 6c70 de: relations_lp │ │ │ │ +00045280: 4669 6e69 7465 4772 6f75 7041 6374 696f FiniteGroupActio │ │ │ │ +00045290: 6e5f 7270 2c20 4e65 7874 3a20 7265 796e n_rp, Next: reyn │ │ │ │ +000452a0: 6f6c 6473 4f70 6572 6174 6f72 2c20 5072 oldsOperator, Pr │ │ │ │ +000452b0: 6576 3a20 7261 6e6b 5f6c 7044 6961 676f ev: rank_lpDiago │ │ │ │ +000452c0: 6e61 6c41 6374 696f 6e5f 7270 2c20 5570 nalAction_rp, Up │ │ │ │ +000452d0: 3a20 546f 700a 0a72 656c 6174 696f 6e73 : Top..relations │ │ │ │ +000452e0: 2846 696e 6974 6547 726f 7570 4163 7469 (FiniteGroupActi │ │ │ │ +000452f0: 6f6e 2920 2d2d 2072 656c 6174 696f 6e73 on) -- relations │ │ │ │ +00045300: 206f 6620 6120 6669 6e69 7465 2067 726f of a finite gro │ │ │ │ +00045310: 7570 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a up.************* │ │ │ │ +00045320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00045330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00045340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +00045350: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e * Function: *n │ │ │ │ +00045360: 6f74 6520 7265 6c61 7469 6f6e 733a 2028 ote relations: ( │ │ │ │ +00045370: 4d61 6361 756c 6179 3244 6f63 2972 656c Macaulay2Doc)rel │ │ │ │ +00045380: 6174 696f 6e73 2c0a 2020 2a20 5573 6167 ations,. * Usag │ │ │ │ +00045390: 653a 200a 2020 2020 2020 2020 7265 6c61 e: . rela │ │ │ │ +000453a0: 7469 6f6e 7320 470a 2020 2a20 496e 7075 tions G. * Inpu │ │ │ │ +000453b0: 7473 3a0a 2020 2020 2020 2a20 472c 2061 ts:. * G, a │ │ │ │ +000453c0: 6e20 696e 7374 616e 6365 206f 6620 7468 n instance of th │ │ │ │ +000453d0: 6520 7479 7065 202a 6e6f 7465 2046 696e e type *note Fin │ │ │ │ +000453e0: 6974 6547 726f 7570 4163 7469 6f6e 3a20 iteGroupAction: │ │ │ │ +000453f0: 4669 6e69 7465 4772 6f75 7041 6374 696f FiniteGroupActio │ │ │ │ +00045400: 6e2c 2c0a 2020 2020 2020 2020 7468 6520 n,,. the │ │ │ │ +00045410: 6163 7469 6f6e 206f 6620 6120 6669 6e69 action of a fini │ │ │ │ +00045420: 7465 2067 726f 7570 0a20 202a 204f 7574 te group. * Out │ │ │ │ +00045430: 7075 7473 3a0a 2020 2020 2020 2a20 6120 puts:. * a │ │ │ │ +00045440: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ +00045450: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ +00045460: 2061 206c 6973 7420 6f66 2072 656c 6174 a list of relat │ │ │ │ +00045470: 696f 6e73 206f 6620 7468 6520 6772 6f75 ions of the grou │ │ │ │ +00045480: 700a 0a44 6573 6372 6970 7469 6f6e 0a3d p..Description.= │ │ │ │ +00045490: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 ==========..This │ │ │ │ +000454a0: 2066 756e 6374 696f 6e20 6973 2070 726f function is pro │ │ │ │ +000454b0: 7669 6465 6420 6279 2074 6865 2070 6163 vided by the pac │ │ │ │ +000454c0: 6b61 6765 202a 6e6f 7465 2049 6e76 6172 kage *note Invar │ │ │ │ +000454d0: 6961 6e74 5269 6e67 3a20 546f 702c 2e20 iantRing: Top,. │ │ │ │ +000454e0: 0a0a 5573 6520 7468 6973 2066 756e 6374 ..Use this funct │ │ │ │ +000454f0: 696f 6e20 746f 2067 6574 2074 6865 2072 ion to get the r │ │ │ │ +00045500: 656c 6174 696f 6e73 2061 6d6f 6e67 2065 elations among e │ │ │ │ +00045510: 6c65 6d65 6e74 7320 6f66 2061 2067 726f lements of a gro │ │ │ │ +00045520: 7570 2e20 4561 6368 2065 6c65 6d65 6e74 up. Each element │ │ │ │ +00045530: 0a69 7320 7265 7072 6573 656e 7465 6420 .is represented │ │ │ │ +00045540: 6279 2061 2077 6f72 6420 6f66 206d 696e by a word of min │ │ │ │ +00045550: 696d 616c 206c 656e 6774 6820 696e 2074 imal length in t │ │ │ │ +00045560: 6865 2043 6f78 7465 7220 6765 6e65 7261 he Coxter genera │ │ │ │ +00045570: 746f 7273 2e20 416e 6420 6561 6368 0a72 tors. And each.r │ │ │ │ +00045580: 656c 6174 696f 6e20 6973 2072 6570 7265 elation is repre │ │ │ │ +00045590: 7365 6e74 6564 2062 7920 6120 6c69 7374 sented by a list │ │ │ │ +000455a0: 206f 6620 7477 6f20 776f 7264 7320 7468 of two words th │ │ │ │ +000455b0: 6174 2065 7175 6174 6573 2074 6865 2067 at equates the g │ │ │ │ +000455c0: 726f 7570 2065 6c65 6d65 6e74 0a72 6570 roup element.rep │ │ │ │ +000455d0: 7265 7365 6e74 6564 2062 7920 7468 6f73 resented by thos │ │ │ │ +000455e0: 6520 7477 6f20 776f 7264 732e 0a0a 5468 e two words...Th │ │ │ │ +000455f0: 6520 666f 6c6c 6f77 696e 6720 6578 616d e following exam │ │ │ │ +00045600: 706c 6520 6465 6669 6e65 7320 7468 6520 ple defines the │ │ │ │ +00045610: 7065 726d 7574 6174 696f 6e20 6163 7469 permutation acti │ │ │ │ +00045620: 6f6e 206f 6620 6120 7379 6d6d 6574 7269 on of a symmetri │ │ │ │ +00045630: 6320 6772 6f75 7020 6f6e 0a74 6872 6565 c group on.three │ │ │ │ +00045640: 2065 6c65 6d65 6e74 7320 7573 696e 6720 elements using │ │ │ │ +00045650: 7468 7265 6520 7472 616e 7370 6f73 6974 three transposit │ │ │ │ +00045660: 696f 6e73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d ions...+-------- │ │ │ │ +00045670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000456a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000456b0: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d -----+.|i1 : R = │ │ │ │ +000456c0: 2051 515b 785f 312e 2e78 5f33 5d20 2020 QQ[x_1..x_3] │ │ │ │ +000456d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000456e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000456f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045700: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00045710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045750: 2020 2020 207c 0a7c 6f31 203d 2052 2020 |.|o1 = R │ │ │ │ +00045760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000457a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000457b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000457c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000457d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000457e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000457f0: 2020 2020 207c 0a7c 6f31 203a 2050 6f6c |.|o1 : Pol │ │ │ │ +00045800: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ +00045810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045840: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00045850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045890: 2d2d 2d2d 2d2b 0a7c 6932 203a 204c 203d -----+.|i2 : L = │ │ │ │ +000458a0: 207b 6d61 7472 6978 207b 7b30 2c31 2c30 {matrix {{0,1,0 │ │ │ │ +000458b0: 7d2c 7b31 2c30 2c30 7d2c 7b30 2c30 2c31 },{1,0,0},{0,0,1 │ │ │ │ +000458c0: 7d7d 2c20 6d61 7472 6978 207b 7b30 2c30 }}, matrix {{0,0 │ │ │ │ +000458d0: 2c31 7d2c 7b30 2c31 2c30 7d2c 7b31 2c30 ,1},{0,1,0},{1,0 │ │ │ │ +000458e0: 2c30 7d7d 2c7c 0a7c 2020 2020 2020 2020 ,0}},|.| │ │ │ │ +000458f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045930: 2020 2020 207c 0a7c 6f32 203d 207b 7c20 |.|o2 = {| │ │ │ │ +00045940: 3020 3120 3020 7c2c 207c 2030 2030 2031 0 1 0 |, | 0 0 1 │ │ │ │ +00045950: 207c 2c20 7c20 3120 3020 3020 7c7d 2020 |, | 1 0 0 |} │ │ │ │ +00045960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045980: 2020 2020 207c 0a7c 2020 2020 2020 7c20 |.| | │ │ │ │ +00045990: 3120 3020 3020 7c20 207c 2030 2031 2030 1 0 0 | | 0 1 0 │ │ │ │ +000459a0: 207c 2020 7c20 3020 3020 3120 7c20 2020 | | 0 0 1 | │ │ │ │ +000459b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000459c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000459d0: 2020 2020 207c 0a7c 2020 2020 2020 7c20 |.| | │ │ │ │ +000459e0: 3020 3020 3120 7c20 207c 2031 2030 2030 0 0 1 | | 1 0 0 │ │ │ │ +000459f0: 207c 2020 7c20 3020 3120 3020 7c20 2020 | | 0 1 0 | │ │ │ │ +00045a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045a20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00045a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045a70: 2020 2020 207c 0a7c 6f32 203a 204c 6973 |.|o2 : Lis │ │ │ │ +00045a80: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00045a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045ac0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d |.|-------- │ │ │ │ +00045ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045b10: 2d2d 2d2d 2d7c 0a7c 6d61 7472 6978 207b -----|.|matrix { │ │ │ │ +00045b20: 7b31 2c30 2c30 7d2c 7b30 2c30 2c31 7d2c {1,0,0},{0,0,1}, │ │ │ │ +00045b30: 7b30 2c31 2c30 7d7d 207d 2020 2020 2020 {0,1,0}} } │ │ │ │ +00045b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045b60: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00045b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045bb0: 2d2d 2d2d 2d2b 0a7c 6933 203a 2047 203d -----+.|i3 : G = │ │ │ │ +00045bc0: 2066 696e 6974 6541 6374 696f 6e28 4c2c finiteAction(L, │ │ │ │ +00045bd0: 2052 2920 2020 2020 2020 2020 2020 2020 R) │ │ │ │ 00045be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045c00: 2020 2020 2020 207c 0a7c 6f31 203d 2052 |.|o1 = R │ │ │ │ +00045c00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00045c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045c50: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00045c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045c50: 2020 2020 207c 0a7c 6f33 203d 2052 203c |.|o3 = R < │ │ │ │ +00045c60: 2d20 7b7c 2030 2031 2030 207c 2c20 7c20 - {| 0 1 0 |, | │ │ │ │ +00045c70: 3020 3020 3120 7c2c 207c 2031 2030 2030 0 0 1 |, | 1 0 0 │ │ │ │ +00045c80: 207c 7d20 2020 2020 2020 2020 2020 2020 |} │ │ │ │ 00045c90: 2020 2020 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│ │ +00045de0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00045df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045e30: 2d2d 2d2d 2d2b 0a7c 6934 203a 2072 656c -----+.|i4 : rel │ │ │ │ +00045e40: 6174 696f 6e73 2047 2020 2020 2020 2020 ations G │ │ │ │ +00045e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045e80: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00045e90: 7c20 3020 3020 3120 7c20 207c 2031 2030 | 0 0 1 | | 1 0 │ │ │ │ -00045ea0: 2030 207c 2020 7c20 3020 3120 3020 7c20 0 | | 0 1 0 | │ │ │ │ +00045e80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00045e90: 2020 2020 2020 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2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045fc0: 2d2d 2d2d 2d2d 2d7c 0a7c 6d61 7472 6978 -------|.|matrix │ │ │ │ -00045fd0: 207b 7b31 2c30 2c30 7d2c 7b30 2c30 2c31 {{1,0,0},{0,0,1 │ │ │ │ -00045fe0: 7d2c 7b30 2c31 2c30 7d7d 207d 2020 2020 },{0,1,0}} } │ │ │ │ -00045ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046010: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00046020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046060: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2047 -------+.|i3 : G │ │ │ │ -00046070: 203d 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0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00046070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000460a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000460b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000460c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000460b0: 2020 2020 207c 0a7c 6f34 203a 204c 6973 |.|o4 : Lis │ │ │ │ +000460c0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ 000460d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000460e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000460f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046100: 2020 2020 2020 207c 0a7c 6f33 203d 2052 |.|o3 = R │ │ │ │ -00046110: 203c 2d20 7b7c 2030 2031 2030 207c 2c20 <- {| 0 1 0 |, │ │ │ │ -00046120: 7c20 3020 3020 3120 7c2c 207c 2031 2030 | 0 0 1 |, | 1 0 │ │ │ │ -00046130: 2030 207c 7d20 2020 2020 2020 2020 2020 0 |} │ │ │ │ -00046140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046150: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00046160: 2020 2020 207c 2031 2030 2030 207c 2020 | 1 0 0 | │ │ │ │ -00046170: 7c20 3020 3120 3020 7c20 207c 2030 2030 | 0 1 0 | | 0 0 │ │ │ │ -00046180: 2031 207c 2020 2020 2020 2020 2020 2020 1 | │ │ │ │ -00046190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000461a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000461b0: 2020 2020 207c 2030 2030 2031 207c 2020 | 0 0 1 | │ │ │ │ -000461c0: 7c20 3120 3020 3020 7c20 207c 2030 2031 | 1 0 0 | | 0 1 │ │ │ │ -000461d0: 2030 207c 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ -000461e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000461f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00046200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046240: 2020 2020 2020 207c 0a7c 6f33 203a 2046 |.|o3 : F │ │ │ │ -00046250: 696e 6974 6547 726f 7570 4163 7469 6f6e initeGroupAction │ │ │ │ -00046260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046290: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -000462a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000462b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000462c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000462d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000462e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2072 -------+.|i4 : r │ │ │ │ -000462f0: 656c 6174 696f 6e73 2047 2020 2020 2020 elations G │ │ │ │ -00046300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046330: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00046340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046380: 2020 2020 2020 207c 0a7c 6f34 203d 207b |.|o4 = { │ │ │ │ -00046390: 7b7b 7d2c 207b 312c 2031 7d7d 2c20 7b7b {{}, {1, 1}}, {{ │ │ │ │ -000463a0: 7d2c 207b 322c 2032 7d7d 2c20 7b7b 317d }, {2, 2}}, {{1} │ │ │ │ -000463b0: 2c20 7b30 2c20 312c 2032 7d7d 2c20 7b7b , {0, 1, 2}}, {{ │ │ │ │ -000463c0: 317d 2c20 7b30 2c20 322c 2030 7d7d 2c20 1}, {0, 2, 0}}, │ │ │ │ -000463d0: 7b7b 7d2c 2020 207c 0a7c 2020 2020 202d {{}, |.| - │ │ │ │ -000463e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000463f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046410: 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7b7b 327d 2c20 , 1, 0}}, {{2}, │ │ │ │ -000464f0: 7b30 2c20 322c 2031 7d7d 7d20 2020 2020 {0, 2, 1}}} │ │ │ │ -00046500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046510: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00046520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046560: 2020 2020 2020 207c 0a7c 6f34 203a 204c |.|o4 : L │ │ │ │ -00046570: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ -00046580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000465a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000465b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -000465c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000465d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000465e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000465f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046600: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ -00046610: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -00046620: 202a 6e6f 7465 2067 726f 7570 3a20 6772 *note group: gr │ │ │ │ -00046630: 6f75 702c 202d 2d20 6c69 7374 2061 6c6c oup, -- list all │ │ │ │ -00046640: 2065 6c65 6d65 6e74 7320 6f66 2074 6865 elements of the │ │ │ │ -00046650: 2067 726f 7570 206f 6620 6120 6669 6e69 group of a fini │ │ │ │ -00046660: 7465 2067 726f 7570 0a20 2020 2061 6374 te group. act │ │ │ │ -00046670: 696f 6e0a 2020 2a20 2a6e 6f74 6520 7363 ion. * *note sc │ │ │ │ -00046680: 6872 6569 6572 4772 6170 683a 2073 6368 hreierGraph: sch │ │ │ │ -00046690: 7265 6965 7247 7261 7068 2c20 2d2d 2053 reierGraph, -- S │ │ │ │ -000466a0: 6368 7265 6965 7220 6772 6170 6820 6f66 chreier graph of │ │ │ │ -000466b0: 2061 2066 696e 6974 6520 6772 6f75 700a a finite group. │ │ │ │ -000466c0: 2020 2a20 2a6e 6f74 6520 776f 7264 733a * *note words: │ │ │ │ -000466d0: 2077 6f72 6473 2c20 2d2d 2061 7373 6f63 words, -- assoc │ │ │ │ -000466e0: 6961 7465 2061 2077 6f72 6420 696e 2074 iate a word in t │ │ │ │ -000466f0: 6865 2067 656e 6572 6174 6f72 7320 6f66 he generators of │ │ │ │ -00046700: 2061 2067 726f 7570 2074 6f0a 2020 2020 a group to. │ │ │ │ -00046710: 6561 6368 2065 6c65 6d65 6e74 0a0a 5761 each element..Wa │ │ │ │ -00046720: 7973 2074 6f20 7573 6520 7468 6973 206d ys to use this m │ │ │ │ -00046730: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d ethod:.========= │ │ │ │ -00046740: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00046750: 0a20 202a 202a 6e6f 7465 2072 656c 6174 . * *note relat │ │ │ │ -00046760: 696f 6e73 2846 696e 6974 6547 726f 7570 ions(FiniteGroup │ │ │ │ -00046770: 4163 7469 6f6e 293a 2072 656c 6174 696f Action): relatio │ │ │ │ -00046780: 6e73 5f6c 7046 696e 6974 6547 726f 7570 ns_lpFiniteGroup │ │ │ │ -00046790: 4163 7469 6f6e 5f72 702c 202d 2d0a 2020 Action_rp, --. │ │ │ │ -000467a0: 2020 7265 6c61 7469 6f6e 7320 6f66 2061 relations of a │ │ │ │ -000467b0: 2066 696e 6974 6520 6772 6f75 700a 2d2d finite group.-- │ │ │ │ -000467c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000467d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000467e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000467f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -00046810: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -00046820: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -00046830: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -00046840: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -00046850: 6179 322d 312e 3236 2e30 352b 6473 2f4d ay2-1.26.05+ds/M │ │ │ │ -00046860: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -00046870: 6167 6573 2f0a 496e 7661 7269 616e 7452 ages/.InvariantR │ │ │ │ -00046880: 696e 672f 4669 6e69 7465 4772 6f75 7073 ing/FiniteGroups │ │ │ │ -00046890: 446f 632e 6d32 3a33 3838 3a30 2e0a 1f0a Doc.m2:388:0.... │ │ │ │ -000468a0: 4669 6c65 3a20 496e 7661 7269 616e 7452 File: InvariantR │ │ │ │ -000468b0: 696e 672e 696e 666f 2c20 4e6f 6465 3a20 ing.info, Node: │ │ │ │ -000468c0: 7265 796e 6f6c 6473 4f70 6572 6174 6f72 reynoldsOperator │ │ │ │ -000468d0: 2c20 4e65 7874 3a20 7269 6e67 5f6c 7047 , Next: ring_lpG │ │ │ │ -000468e0: 726f 7570 4163 7469 6f6e 5f72 702c 2050 roupAction_rp, P │ │ │ │ -000468f0: 7265 763a 2072 656c 6174 696f 6e73 5f6c rev: relations_l │ │ │ │ -00046900: 7046 696e 6974 6547 726f 7570 4163 7469 pFiniteGroupActi │ │ │ │ -00046910: 6f6e 5f72 702c 2055 703a 2054 6f70 0a0a on_rp, Up: Top.. │ │ │ │ -00046920: 7265 796e 6f6c 6473 4f70 6572 6174 6f72 reynoldsOperator │ │ │ │ -00046930: 202d 2d20 7468 6520 696d 6167 6520 6f66 -- the image of │ │ │ │ -00046940: 2061 2070 6f6c 796e 6f6d 6961 6c20 756e a polynomial un │ │ │ │ -00046950: 6465 7220 7468 6520 5265 796e 6f6c 6473 der the Reynolds │ │ │ │ -00046960: 206f 7065 7261 746f 720a 2a2a 2a2a 2a2a operator.****** │ │ │ │ -00046970: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00046980: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00046990: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000469a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000469b0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -000469c0: 0a20 2020 2020 2020 2072 6579 6e6f 6c64 . reynold │ │ │ │ -000469d0: 734f 7065 7261 746f 7228 662c 2047 292c sOperator(f, G), │ │ │ │ -000469e0: 2072 6579 6e6f 6c64 734f 7065 7261 746f reynoldsOperato │ │ │ │ -000469f0: 7228 662c 2044 290a 2020 2a20 496e 7075 r(f, D). * Inpu │ │ │ │ -00046a00: 7473 3a0a 2020 2020 2020 2a20 662c 2061 ts:. * f, a │ │ │ │ -00046a10: 202a 6e6f 7465 2072 696e 6720 656c 656d *note ring elem │ │ │ │ -00046a20: 656e 743a 2028 4d61 6361 756c 6179 3244 ent: (Macaulay2D │ │ │ │ -00046a30: 6f63 2952 696e 6745 6c65 6d65 6e74 2c2c oc)RingElement,, │ │ │ │ -00046a40: 2061 2070 6f6c 796e 6f6d 6961 6c20 696e a polynomial in │ │ │ │ -00046a50: 0a20 2020 2020 2020 2074 6865 2070 6f6c . the pol │ │ │ │ -00046a60: 796e 6f6d 6961 6c20 7269 6e67 206f 6620 ynomial ring of │ │ │ │ -00046a70: 7468 6520 6769 7665 6e20 6772 6f75 7020 the given group │ │ │ │ -00046a80: 6163 7469 6f6e 0a20 2020 2020 202a 2047 action. * G │ │ │ │ -00046a90: 2c20 616e 2069 6e73 7461 6e63 6520 6f66 , an instance of │ │ │ │ -00046aa0: 2074 6865 2074 7970 6520 2a6e 6f74 6520 the type *note │ │ │ │ -00046ab0: 4669 6e69 7465 4772 6f75 7041 6374 696f FiniteGroupActio │ │ │ │ -00046ac0: 6e3a 2046 696e 6974 6547 726f 7570 4163 n: FiniteGroupAc │ │ │ │ -00046ad0: 7469 6f6e 2c0a 2020 2020 2020 2a20 442c tion,. * D, │ │ │ │ -00046ae0: 2061 6e20 696e 7374 616e 6365 206f 6620 an instance of │ │ │ │ -00046af0: 7468 6520 7479 7065 202a 6e6f 7465 2044 the type *note D │ │ │ │ -00046b00: 6961 676f 6e61 6c41 6374 696f 6e3a 2044 iagonalAction: D │ │ │ │ -00046b10: 6961 676f 6e61 6c41 6374 696f 6e2c 0a20 iagonalAction,. │ │ │ │ -00046b20: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ -00046b30: 2020 2a20 6120 2a6e 6f74 6520 7269 6e67 * a *note ring │ │ │ │ -00046b40: 2065 6c65 6d65 6e74 3a20 284d 6163 6175 element: (Macau │ │ │ │ -00046b50: 6c61 7932 446f 6329 5269 6e67 456c 656d lay2Doc)RingElem │ │ │ │ -00046b60: 656e 742c 2c20 7468 6520 696e 7661 7269 ent,, the invari │ │ │ │ -00046b70: 616e 740a 2020 2020 2020 2020 706f 6c79 ant. poly │ │ │ │ -00046b80: 6e6f 6d69 616c 2077 6869 6368 2069 7320 nomial which is │ │ │ │ -00046b90: 7468 6520 696d 6167 6520 6f66 2074 6865 the image of the │ │ │ │ -00046ba0: 2067 6976 656e 2070 6f6c 796e 6f6d 6961 given polynomia │ │ │ │ -00046bb0: 6c20 756e 6465 7220 7468 650a 2020 2020 l under the. │ │ │ │ -00046bc0: 2020 2020 5265 796e 6f6c 6473 206f 7065 Reynolds ope │ │ │ │ -00046bd0: 7261 746f 7220 6f66 2074 6865 2067 6976 rator of the giv │ │ │ │ -00046be0: 656e 2066 696e 6974 6520 6772 6f75 7020 en finite group │ │ │ │ -00046bf0: 6163 7469 6f6e 206f 7220 7468 6520 6769 action or the gi │ │ │ │ -00046c00: 7665 6e20 746f 7275 730a 2020 2020 2020 ven torus. │ │ │ │ -00046c10: 2020 6163 7469 6f6e 0a0a 4465 7363 7269 action..Descri │ │ │ │ -00046c20: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -00046c30: 3d0a 0a54 6869 7320 6675 6e63 7469 6f6e =..This function │ │ │ │ -00046c40: 2069 7320 7072 6f76 6964 6564 2062 7920 is provided by │ │ │ │ -00046c50: 7468 6520 7061 636b 6167 6520 2a6e 6f74 the package *not │ │ │ │ -00046c60: 6520 496e 7661 7269 616e 7452 696e 673a e InvariantRing: │ │ │ │ -00046c70: 2054 6f70 2c2e 200a 0a54 6865 2066 6f6c Top,. ..The fol │ │ │ │ -00046c80: 6c6f 7769 6e67 2065 7861 6d70 6c65 2063 lowing example c │ │ │ │ -00046c90: 6f6d 7075 7465 7320 7468 6520 696d 6167 omputes the imag │ │ │ │ -00046ca0: 6520 6f66 2061 2070 6f6c 796e 6f6d 6961 e of a polynomia │ │ │ │ -00046cb0: 6c20 756e 6465 7220 7468 6520 5265 796e l under the Reyn │ │ │ │ -00046cc0: 6f6c 6473 0a6f 7065 7261 746f 7220 666f olds.operator fo │ │ │ │ -00046cd0: 7220 6120 6379 636c 6963 2070 6572 6d75 r a cyclic permu │ │ │ │ -00046ce0: 7461 7469 6f6e 206f 6620 7468 6520 7661 tation of the va │ │ │ │ -00046cf0: 7269 6162 6c65 732e 0a0a 2b2d 2d2d 2d2d riables...+----- │ │ │ │ -00046d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00046d40: 3120 3a20 5220 3d20 5a5a 2f33 5b78 5f30 1 : R = ZZ/3[x_0 │ │ │ │ -00046d50: 2e2e 785f 365d 2020 2020 2020 2020 2020 ..x_6] │ │ │ │ -00046d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046d80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00046d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046dc0: 2020 2020 7c0a 7c6f 3120 3d20 5220 2020 |.|o1 = R │ │ │ │ -00046dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046100: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00046110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046150: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f -----+..See also │ │ │ │ +00046160: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +00046170: 6e6f 7465 2067 726f 7570 3a20 6772 6f75 note group: grou │ │ │ │ +00046180: 702c 202d 2d20 6c69 7374 2061 6c6c 2065 p, -- list all e │ │ │ │ +00046190: 6c65 6d65 6e74 7320 6f66 2074 6865 2067 lements of the g │ │ │ │ +000461a0: 726f 7570 206f 6620 6120 6669 6e69 7465 roup of a finite │ │ │ │ +000461b0: 2067 726f 7570 0a20 2020 2061 6374 696f group. actio │ │ │ │ +000461c0: 6e0a 2020 2a20 2a6e 6f74 6520 7363 6872 n. * *note schr │ │ │ │ +000461d0: 6569 6572 4772 6170 683a 2073 6368 7265 eierGraph: schre │ │ │ │ +000461e0: 6965 7247 7261 7068 2c20 2d2d 2053 6368 ierGraph, -- Sch │ │ │ │ +000461f0: 7265 6965 7220 6772 6170 6820 6f66 2061 reier graph of a │ │ │ │ +00046200: 2066 696e 6974 6520 6772 6f75 700a 2020 finite group. │ │ │ │ +00046210: 2a20 2a6e 6f74 6520 776f 7264 733a 2077 * *note words: w │ │ │ │ +00046220: 6f72 6473 2c20 2d2d 2061 7373 6f63 6961 ords, -- associa │ │ │ │ +00046230: 7465 2061 2077 6f72 6420 696e 2074 6865 te a word in the │ │ │ │ +00046240: 2067 656e 6572 6174 6f72 7320 6f66 2061 generators of a │ │ │ │ +00046250: 2067 726f 7570 2074 6f0a 2020 2020 6561 group to. ea │ │ │ │ +00046260: 6368 2065 6c65 6d65 6e74 0a0a 5761 7973 ch element..Ways │ │ │ │ +00046270: 2074 6f20 7573 6520 7468 6973 206d 6574 to use this met │ │ │ │ +00046280: 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d hod:.=========== │ │ │ │ +00046290: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +000462a0: 202a 202a 6e6f 7465 2072 656c 6174 696f * *note relatio │ │ │ │ +000462b0: 6e73 2846 696e 6974 6547 726f 7570 4163 ns(FiniteGroupAc │ │ │ │ +000462c0: 7469 6f6e 293a 2072 656c 6174 696f 6e73 tion): relations │ │ │ │ +000462d0: 5f6c 7046 696e 6974 6547 726f 7570 4163 _lpFiniteGroupAc │ │ │ │ +000462e0: 7469 6f6e 5f72 702c 202d 2d0a 2020 2020 tion_rp, --. │ │ │ │ +000462f0: 7265 6c61 7469 6f6e 7320 6f66 2061 2066 relations of a f │ │ │ │ +00046300: 696e 6974 6520 6772 6f75 700a 2d2d 2d2d inite group.---- │ │ │ │ +00046310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +00046360: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +00046370: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +00046380: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +00046390: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +000463a0: 322d 312e 3236 2e30 352b 6473 2f4d 322f 2-1.26.05+ds/M2/ │ │ │ │ +000463b0: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +000463c0: 6573 2f0a 496e 7661 7269 616e 7452 696e es/.InvariantRin │ │ │ │ +000463d0: 672f 4669 6e69 7465 4772 6f75 7073 446f g/FiniteGroupsDo │ │ │ │ +000463e0: 632e 6d32 3a33 3838 3a30 2e0a 1f0a 4669 c.m2:388:0....Fi │ │ │ │ +000463f0: 6c65 3a20 496e 7661 7269 616e 7452 696e le: InvariantRin │ │ │ │ +00046400: 672e 696e 666f 2c20 4e6f 6465 3a20 7265 g.info, Node: re │ │ │ │ +00046410: 796e 6f6c 6473 4f70 6572 6174 6f72 2c20 ynoldsOperator, │ │ │ │ +00046420: 4e65 7874 3a20 7269 6e67 5f6c 7047 726f Next: ring_lpGro │ │ │ │ +00046430: 7570 4163 7469 6f6e 5f72 702c 2050 7265 upAction_rp, Pre │ │ │ │ +00046440: 763a 2072 656c 6174 696f 6e73 5f6c 7046 v: relations_lpF │ │ │ │ +00046450: 696e 6974 6547 726f 7570 4163 7469 6f6e initeGroupAction │ │ │ │ +00046460: 5f72 702c 2055 703a 2054 6f70 0a0a 7265 _rp, Up: Top..re │ │ │ │ +00046470: 796e 6f6c 6473 4f70 6572 6174 6f72 202d ynoldsOperator - │ │ │ │ +00046480: 2d20 7468 6520 696d 6167 6520 6f66 2061 - the image of a │ │ │ │ +00046490: 2070 6f6c 796e 6f6d 6961 6c20 756e 6465 polynomial unde │ │ │ │ +000464a0: 7220 7468 6520 5265 796e 6f6c 6473 206f r the Reynolds o │ │ │ │ +000464b0: 7065 7261 746f 720a 2a2a 2a2a 2a2a 2a2a perator.******** │ │ │ │ +000464c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000464d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000464e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000464f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00046500: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +00046510: 2020 2020 2020 2072 6579 6e6f 6c64 734f reynoldsO │ │ │ │ +00046520: 7065 7261 746f 7228 662c 2047 292c 2072 perator(f, G), r │ │ │ │ +00046530: 6579 6e6f 6c64 734f 7065 7261 746f 7228 eynoldsOperator( │ │ │ │ +00046540: 662c 2044 290a 2020 2a20 496e 7075 7473 f, D). * Inputs │ │ │ │ +00046550: 3a0a 2020 2020 2020 2a20 662c 2061 202a :. * f, a * │ │ │ │ +00046560: 6e6f 7465 2072 696e 6720 656c 656d 656e note ring elemen │ │ │ │ +00046570: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ +00046580: 2952 696e 6745 6c65 6d65 6e74 2c2c 2061 )RingElement,, a │ │ │ │ +00046590: 2070 6f6c 796e 6f6d 6961 6c20 696e 0a20 polynomial in. │ │ │ │ +000465a0: 2020 2020 2020 2074 6865 2070 6f6c 796e the polyn │ │ │ │ +000465b0: 6f6d 6961 6c20 7269 6e67 206f 6620 7468 omial ring of th │ │ │ │ +000465c0: 6520 6769 7665 6e20 6772 6f75 7020 6163 e given group ac │ │ │ │ +000465d0: 7469 6f6e 0a20 2020 2020 202a 2047 2c20 tion. * G, │ │ │ │ +000465e0: 616e 2069 6e73 7461 6e63 6520 6f66 2074 an instance of t │ │ │ │ +000465f0: 6865 2074 7970 6520 2a6e 6f74 6520 4669 he type *note Fi │ │ │ │ +00046600: 6e69 7465 4772 6f75 7041 6374 696f 6e3a niteGroupAction: │ │ │ │ +00046610: 2046 696e 6974 6547 726f 7570 4163 7469 FiniteGroupActi │ │ │ │ +00046620: 6f6e 2c0a 2020 2020 2020 2a20 442c 2061 on,. * D, a │ │ │ │ +00046630: 6e20 696e 7374 616e 6365 206f 6620 7468 n instance of th │ │ │ │ +00046640: 6520 7479 7065 202a 6e6f 7465 2044 6961 e type *note Dia │ │ │ │ +00046650: 676f 6e61 6c41 6374 696f 6e3a 2044 6961 gonalAction: Dia │ │ │ │ +00046660: 676f 6e61 6c41 6374 696f 6e2c 0a20 202a gonalAction,. * │ │ │ │ +00046670: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ +00046680: 2a20 6120 2a6e 6f74 6520 7269 6e67 2065 * a *note ring e │ │ │ │ +00046690: 6c65 6d65 6e74 3a20 284d 6163 6175 6c61 lement: (Macaula │ │ │ │ +000466a0: 7932 446f 6329 5269 6e67 456c 656d 656e y2Doc)RingElemen │ │ │ │ +000466b0: 742c 2c20 7468 6520 696e 7661 7269 616e t,, the invarian │ │ │ │ +000466c0: 740a 2020 2020 2020 2020 706f 6c79 6e6f t. polyno │ │ │ │ +000466d0: 6d69 616c 2077 6869 6368 2069 7320 7468 mial which is th │ │ │ │ +000466e0: 6520 696d 6167 6520 6f66 2074 6865 2067 e image of the g │ │ │ │ +000466f0: 6976 656e 2070 6f6c 796e 6f6d 6961 6c20 iven polynomial │ │ │ │ +00046700: 756e 6465 7220 7468 650a 2020 2020 2020 under the. │ │ │ │ +00046710: 2020 5265 796e 6f6c 6473 206f 7065 7261 Reynolds opera │ │ │ │ +00046720: 746f 7220 6f66 2074 6865 2067 6976 656e tor of the given │ │ │ │ +00046730: 2066 696e 6974 6520 6772 6f75 7020 6163 finite group ac │ │ │ │ +00046740: 7469 6f6e 206f 7220 7468 6520 6769 7665 tion or the give │ │ │ │ +00046750: 6e20 746f 7275 730a 2020 2020 2020 2020 n torus. │ │ │ │ +00046760: 6163 7469 6f6e 0a0a 4465 7363 7269 7074 action..Descript │ │ │ │ +00046770: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +00046780: 0a54 6869 7320 6675 6e63 7469 6f6e 2069 .This function i │ │ │ │ +00046790: 7320 7072 6f76 6964 6564 2062 7920 7468 s provided by th │ │ │ │ +000467a0: 6520 7061 636b 6167 6520 2a6e 6f74 6520 e package *note │ │ │ │ +000467b0: 496e 7661 7269 616e 7452 696e 673a 2054 InvariantRing: T │ │ │ │ +000467c0: 6f70 2c2e 200a 0a54 6865 2066 6f6c 6c6f op,. ..The follo │ │ │ │ +000467d0: 7769 6e67 2065 7861 6d70 6c65 2063 6f6d wing example com │ │ │ │ +000467e0: 7075 7465 7320 7468 6520 696d 6167 6520 putes the image │ │ │ │ +000467f0: 6f66 2061 2070 6f6c 796e 6f6d 6961 6c20 of a polynomial │ │ │ │ +00046800: 756e 6465 7220 7468 6520 5265 796e 6f6c under the 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2020 P, R) │ │ │ │ +00046dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046dd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00046de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046e00: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00046e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046e40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00046e50: 3120 3a20 506f 6c79 6e6f 6d69 616c 5269 1 : PolynomialRi │ │ │ │ -00046e60: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ -00046e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046e90: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00046ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046eb0: 2d2d 2d2d 2d2d 2d2d 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2020 2020 7c20 3120 3020 3020 |.| | 1 0 0 │ │ │ │ -00046fb0: 3020 3020 3020 3020 7c20 2020 2020 2020 0 0 0 0 | │ │ │ │ -00046fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046fe0: 2020 2020 7c0a 7c20 2020 2020 7c20 3020 |.| | 0 │ │ │ │ -00046ff0: 3120 3020 3020 3020 3020 3020 7c20 2020 1 0 0 0 0 0 | │ │ │ │ +00046e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046e10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00046e20: 7c6f 3320 3d20 5220 3c2d 207b 7c20 3020 |o3 = R <- {| 0 │ │ │ │ +00046e30: 3020 3020 3020 3020 3020 3120 7c7d 2020 0 0 0 0 0 1 |} │ │ │ │ +00046e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046e60: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00046e70: 7c20 3120 3020 3020 3020 3020 3020 3020 | 1 0 0 0 0 0 0 │ │ │ │ +00046e80: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00046e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046ea0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00046eb0: 2020 2020 7c20 3020 3120 3020 3020 3020 | 0 1 0 0 0 │ │ │ │ +00046ec0: 3020 3020 7c20 2020 2020 2020 2020 2020 0 0 | │ │ │ │ +00046ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046ee0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00046ef0: 2020 2020 2020 2020 7c20 3020 3020 3120 | 0 0 1 │ │ │ │ +00046f00: 3020 3020 3020 3020 7c20 2020 2020 2020 0 0 0 0 | │ │ │ │ +00046f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046f20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00046f30: 7c20 2020 2020 2020 2020 2020 7c20 3020 | | 0 │ │ │ │ +00046f40: 3020 3020 3120 3020 3020 3020 7c20 2020 0 0 1 0 0 0 | │ │ │ │ +00046f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046f70: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00046f80: 7c20 3020 3020 3020 3020 3120 3020 3020 | 0 0 0 0 1 0 0 │ │ │ │ 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2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000470a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000470b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000470c0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 7265 ------+.|i4 : re │ │ │ │ +000470d0: 796e 6f6c 6473 4f70 6572 6174 6f72 2878 ynoldsOperator(x │ │ │ │ +000470e0: 5f30 2a78 5f31 2a78 5f32 5e32 2c20 4337 _0*x_1*x_2^2, C7 │ │ │ │ +000470f0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00047100: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00047110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047130: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00047140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047170: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00047180: 2020 2020 2020 2020 2020 2020 2037 2020 7 │ │ │ │ -00047190: 2020 2020 2037 2020 2020 2020 2020 2020 7 │ │ │ │ -000471a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000471b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000471c0: 7c0a 7c6f 3220 3a20 4d61 7472 6978 205a |.|o2 : Matrix Z │ │ │ │ -000471d0: 5a20 203c 2d2d 205a 5a20 2020 2020 2020 Z <-- ZZ │ │ │ │ -000471e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000471f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047200: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00047210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047240: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ -00047250: 4337 203d 2066 696e 6974 6541 6374 696f C7 = finiteActio │ │ │ │ -00047260: 6e28 502c 2052 2920 2020 2020 2020 2020 n(P, R) │ │ │ │ +00047130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047140: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00047150: 7c20 2020 2020 2020 2020 2032 2020 2020 | 2 │ │ │ │ +00047160: 2020 2020 3220 2020 2020 2020 2032 2020 2 2 │ │ │ │ +00047170: 2020 2020 2020 3220 2020 2020 2032 2020 2 2 │ │ │ │ +00047180: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00047190: 2032 7c0a 7c6f 3420 3d20 7820 7820 7820 2|.|o4 = x x x │ │ │ │ +000471a0: 202b 2078 2078 2078 2020 2b20 7820 7820 + x x x + x x │ │ │ │ +000471b0: 7820 202b 2078 2078 2078 2020 2b20 7820 x + x x x + x │ │ │ │ +000471c0: 7820 7820 202b 2078 2078 2078 2020 2b20 x x + x x x + │ │ │ │ +000471d0: 7820 7820 7820 7c0a 7c20 2020 2020 2030 x x x |.| 0 │ │ │ │ +000471e0: 2031 2032 2020 2020 3120 3220 3320 2020 1 2 1 2 3 │ │ │ │ +000471f0: 2032 2033 2034 2020 2020 3320 3420 3520 2 3 4 3 4 5 │ │ │ │ +00047200: 2020 2030 2031 2036 2020 2020 3020 3520 0 1 6 0 5 │ │ │ │ +00047210: 3620 2020 2034 2035 2036 7c0a 7c20 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2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000476b0: 7c69 3720 3a20 5420 3d20 6469 6167 6f6e |i7 : T = diagon │ │ │ │ +000476c0: 616c 4163 7469 6f6e 2857 2c20 5229 2020 alAction(W, R) │ │ │ │ 000476d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000476e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000476e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000476f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047710: 7c0a 7c6f 3420 3a20 5220 2020 2020 2020 |.|o4 : R │ │ │ │ -00047720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047710: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00047720: 2020 2020 2020 2020 2020 2a20 3220 2020 * 2 │ │ │ │ 00047730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047750: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00047760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047790: 2d2d 2d2d 2d2d 2d2d 2b0a 0a48 6572 6520 --------+..Here │ │ │ │ -000477a0: 6973 2061 6e20 6578 616d 706c 6520 636f is an example co │ │ │ │ -000477b0: 6d70 7574 696e 6720 7468 6520 696d 6167 mputing the imag │ │ │ │ -000477c0: 6520 6f66 2061 2070 6f6c 796e 6f6d 6961 e of a polynomia │ │ │ │ -000477d0: 6c20 756e 6465 7220 7468 6520 5265 796e l under the Reyn │ │ │ │ -000477e0: 6f6c 6473 0a6f 7065 7261 746f 7220 666f olds.operator fo │ │ │ │ -000477f0: 7220 6120 7477 6f2d 6469 6d65 6e73 696f r a two-dimensio │ │ │ │ -00047800: 6e61 6c20 746f 7275 7320 6163 7469 6e67 nal torus acting │ │ │ │ -00047810: 206f 6e20 706f 6c79 6e6f 6d69 616c 2072 on polynomial r │ │ │ │ -00047820: 696e 6720 696e 2066 6f75 720a 7661 7269 ing in four.vari │ │ │ │ -00047830: 6162 6c65 733a 0a0a 2b2d 2d2d 2d2d 2d2d ables:..+------- │ │ │ │ -00047840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00047870: 3520 3a20 5220 3d20 5151 5b78 5f31 2e2e 5 : R = QQ[x_1.. │ │ │ │ -00047880: 785f 345d 2020 2020 2020 2020 2020 2020 x_4] │ │ │ │ -00047890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000478a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -000478b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000478c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000478d0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ -000478e0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ -000478f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047900: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00047910: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00047750: 7c0a 7c6f 3720 3d20 5220 3c2d 2028 5151 |.|o7 = R <- (QQ │ │ │ │ +00047760: 2029 2020 7669 6120 2020 2020 2020 2020 ) via │ │ │ │ +00047770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047780: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00047790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000477a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000477b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000477c0: 2020 2020 7c20 3020 3120 2d31 2031 2020 | 0 1 -1 1 │ │ │ │ +000477d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000477e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000477f0: 2020 7c0a 7c20 2020 2020 7c20 3120 3020 |.| | 1 0 │ │ │ │ +00047800: 2d31 202d 3120 7c20 2020 2020 2020 2020 -1 -1 | │ │ │ │ +00047810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047820: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00047830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047850: 2020 2020 2020 2020 2020 2020 2020 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7c0a 7c6f 3520 3a20 506f 6c79 |.|o5 : Poly │ │ │ │ -00047950: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ -00047960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047970: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00047980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000479a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000479b0: 2b0a 7c69 3620 3a20 5720 3d20 6d61 7472 +.|i6 : W = matr │ │ │ │ -000479c0: 6978 7b7b 302c 312c 2d31 2c31 7d2c 207b ix{{0,1,-1,1}, { │ │ │ │ -000479d0: 312c 302c 2d31 2c2d 317d 7d20 2020 2020 1,0,-1,-1}} │ │ │ │ -000479e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00047930: 2020 2020 2020 7c0a 7c6f 3820 3d20 7820 |.|o8 = x │ │ │ │ +00047940: 7820 7820 2020 2020 2020 2020 2020 2020 x x │ │ │ │ +00047950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047960: 2020 2020 2020 2020 2020 2020 7c0a 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2020 -1 -1 | │ │ │ │ -00047a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047a80: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00047a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047ab0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00047ac0: 7c20 2020 2020 2020 2020 2020 2020 2032 | 2 │ │ │ │ -00047ad0: 2020 2020 2020 2034 2020 2020 2020 2020 4 │ │ │ │ -00047ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047af0: 2020 2020 7c0a 7c6f 3620 3a20 4d61 7472 |.|o6 : Matr │ │ │ │ -00047b00: 6978 205a 5a20 203c 2d2d 205a 5a20 2020 ix ZZ <-- ZZ │ │ │ │ -00047b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047b20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00047b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047b60: 2b0a 7c69 3720 3a20 5420 3d20 6469 6167 +.|i7 : T = diag │ │ │ │ -00047b70: 6f6e 616c 4163 7469 6f6e 2857 2c20 5229 onalAction(W, R) │ │ │ │ -00047b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047b90: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00047ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047bc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00047bd0: 2020 2020 2020 2020 2020 2020 2a20 3220 * 2 │ │ │ │ -00047be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047c00: 2020 7c0a 7c6f 3720 3d20 5220 3c2d 2028 |.|o7 = R <- ( │ │ │ │ -00047c10: 5151 2029 2020 7669 6120 2020 2020 2020 QQ ) via │ │ │ │ -00047c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047c30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00047c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047c60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00047c70: 7c20 2020 2020 7c20 3020 3120 2d31 2031 | | 0 1 -1 1 │ │ │ │ -00047c80: 2020 7c20 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00047c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047ca0: 2020 2020 7c0a 7c20 2020 2020 7c20 3120 |.| | 1 │ │ │ │ -00047cb0: 3020 2d31 202d 3120 7c20 2020 2020 2020 0 -1 -1 | │ │ │ │ -00047cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047cd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00047ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047d10: 7c0a 7c6f 3720 3a20 4469 6167 6f6e 616c |.|o7 : Diagonal │ │ │ │ -00047d20: 4163 7469 6f6e 2020 2020 2020 2020 2020 Action │ │ │ │ -00047d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047d40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -00047d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00047d80: 3820 3a20 7265 796e 6f6c 6473 4f70 6572 8 : reynoldsOper │ │ │ │ -00047d90: 6174 6f72 2878 5f31 2a78 5f32 2a78 5f33 ator(x_1*x_2*x_3 │ │ │ │ -00047da0: 202b 2078 5f31 2a78 5f32 2a78 5f34 2c20 + x_1*x_2*x_4, │ │ │ │ -00047db0: 5429 7c0a 7c20 2020 2020 2020 2020 2020 T)|.| │ │ │ │ -00047dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047de0: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20 |.|o8 = │ │ │ │ -00047df0: 7820 7820 7820 2020 2020 2020 2020 2020 x x x │ │ │ │ -00047e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047e10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00047e20: 7c20 2020 2020 2031 2032 2033 2020 2020 | 1 2 3 │ │ │ │ -00047e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047e50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00047e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047e80: 2020 2020 2020 2020 2020 7c0a 7c6f 3820 |.|o8 │ │ │ │ -00047e90: 3a20 5220 2020 2020 2020 2020 2020 2020 : R │ │ │ │ -00047ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047ec0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00047a00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00047a10: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00047a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047a40: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075 ----+..Ways to u │ │ │ │ +00047a50: 7365 2072 6579 6e6f 6c64 734f 7065 7261 se reynoldsOpera │ │ │ │ +00047a60: 746f 723a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tor:.=========== │ │ │ │ +00047a70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00047a80: 3d3d 0a0a 2020 2a20 2272 6579 6e6f 6c64 ==.. * "reynold │ │ │ │ +00047a90: 734f 7065 7261 746f 7228 5269 6e67 456c sOperator(RingEl │ │ │ │ +00047aa0: 656d 656e 742c 4469 6167 6f6e 616c 4163 ement,DiagonalAc │ │ │ │ +00047ab0: 7469 6f6e 2922 0a20 202a 2022 7265 796e tion)". * "reyn │ │ │ │ +00047ac0: 6f6c 6473 4f70 6572 6174 6f72 2852 696e oldsOperator(Rin │ │ │ │ +00047ad0: 6745 6c65 6d65 6e74 2c46 696e 6974 6547 gElement,FiniteG │ │ │ │ +00047ae0: 726f 7570 4163 7469 6f6e 2922 0a0a 466f roupAction)"..Fo │ │ │ │ +00047af0: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00047b00: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00047b10: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00047b20: 2a6e 6f74 6520 7265 796e 6f6c 6473 4f70 *note reynoldsOp │ │ │ │ +00047b30: 6572 6174 6f72 3a20 7265 796e 6f6c 6473 erator: reynolds │ │ │ │ +00047b40: 4f70 6572 6174 6f72 2c20 6973 2061 202a Operator, is a * │ │ │ │ +00047b50: 6e6f 7465 206d 6574 686f 640a 6675 6e63 note method.func │ │ │ │ +00047b60: 7469 6f6e 3a20 284d 6163 6175 6c61 7932 tion: (Macaulay2 │ │ │ │ +00047b70: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ +00047b80: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ +00047b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047bd0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +00047be0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +00047bf0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +00047c00: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +00047c10: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +00047c20: 2e30 352b 6473 2f4d 322f 4d61 6361 756c .05+ds/M2/Macaul │ │ │ │ +00047c30: 6179 322f 7061 636b 6167 6573 2f0a 496e ay2/packages/.In │ │ │ │ +00047c40: 7661 7269 616e 7452 696e 672f 496e 7661 variantRing/Inva │ │ │ │ +00047c50: 7269 616e 7473 446f 632e 6d32 3a39 3035 riantsDoc.m2:905 │ │ │ │ +00047c60: 3a30 2e0a 1f0a 4669 6c65 3a20 496e 7661 :0....File: Inva │ │ │ │ +00047c70: 7269 616e 7452 696e 672e 696e 666f 2c20 riantRing.info, │ │ │ │ +00047c80: 4e6f 6465 3a20 7269 6e67 5f6c 7047 726f Node: ring_lpGro │ │ │ │ +00047c90: 7570 4163 7469 6f6e 5f72 702c 204e 6578 upAction_rp, Nex │ │ │ │ +00047ca0: 743a 2052 696e 674f 6649 6e76 6172 6961 t: RingOfInvaria │ │ │ │ +00047cb0: 6e74 732c 2050 7265 763a 2072 6579 6e6f nts, Prev: reyno │ │ │ │ +00047cc0: 6c64 734f 7065 7261 746f 722c 2055 703a ldsOperator, Up: │ │ │ │ +00047cd0: 2054 6f70 0a0a 7269 6e67 2847 726f 7570 Top..ring(Group │ │ │ │ +00047ce0: 4163 7469 6f6e 2920 2d2d 2074 6865 2070 Action) -- the p │ │ │ │ +00047cf0: 6f6c 796e 6f6d 6961 6c20 7269 6e67 2062 olynomial ring b │ │ │ │ +00047d00: 6569 6e67 2061 6374 6564 2075 706f 6e0a eing acted upon. │ │ │ │ +00047d10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00047d20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00047d30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00047d40: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 *********.. * F │ │ │ │ +00047d50: 756e 6374 696f 6e3a 202a 6e6f 7465 2072 unction: *note r │ │ │ │ +00047d60: 696e 673a 2028 4d61 6361 756c 6179 3244 ing: (Macaulay2D │ │ │ │ +00047d70: 6f63 2972 696e 672c 0a20 202a 2055 7361 oc)ring,. * Usa │ │ │ │ +00047d80: 6765 3a20 0a20 2020 2020 2020 2072 696e ge: . rin │ │ │ │ +00047d90: 6720 470a 2020 2a20 496e 7075 7473 3a0a g G. * Inputs:. │ │ │ │ +00047da0: 2020 2020 2020 2a20 472c 2061 6e20 696e * G, an in │ │ │ │ +00047db0: 7374 616e 6365 206f 6620 7468 6520 7479 stance of the ty │ │ │ │ +00047dc0: 7065 202a 6e6f 7465 2047 726f 7570 4163 pe *note GroupAc │ │ │ │ +00047dd0: 7469 6f6e 3a20 4772 6f75 7041 6374 696f tion: GroupActio │ │ │ │ +00047de0: 6e2c 2c20 6120 6772 6f75 700a 2020 2020 n,, a group. │ │ │ │ +00047df0: 2020 2020 6163 7469 6f6e 206f 6e20 6120 action on a │ │ │ │ +00047e00: 706f 6c79 6e6f 6d69 616c 2072 696e 670a polynomial ring. │ │ │ │ +00047e10: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ +00047e20: 2020 202a 2061 202a 6e6f 7465 2072 696e * a *note rin │ │ │ │ +00047e30: 673a 2028 4d61 6361 756c 6179 3244 6f63 g: (Macaulay2Doc │ │ │ │ +00047e40: 2952 696e 672c 2c20 7468 6520 706f 6c79 )Ring,, the poly │ │ │ │ +00047e50: 6e6f 6d69 616c 2072 696e 6720 6265 696e nomial ring bein │ │ │ │ +00047e60: 6720 6163 7465 6420 7570 6f6e 0a0a 4465 g acted upon..De │ │ │ │ +00047e70: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +00047e80: 3d3d 3d3d 3d0a 0a54 6869 7320 6675 6e63 =====..This func │ │ │ │ +00047e90: 7469 6f6e 2069 7320 7072 6f76 6964 6564 tion is provided │ │ │ │ +00047ea0: 2062 7920 7468 6520 7061 636b 6167 6520 by the package │ │ │ │ +00047eb0: 2a6e 6f74 6520 496e 7661 7269 616e 7452 *note InvariantR │ │ │ │ +00047ec0: 696e 673a 2054 6f70 2c2e 0a0a 2b2d 2d2d ing: Top,...+--- │ │ │ │ 00047ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00047ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047ef0: 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320 746f ------+..Ways to │ │ │ │ -00047f00: 2075 7365 2072 6579 6e6f 6c64 734f 7065 use reynoldsOpe │ │ │ │ -00047f10: 7261 746f 723a 0a3d 3d3d 3d3d 3d3d 3d3d rator:.========= │ │ │ │ -00047f20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00047f30: 3d3d 3d3d 0a0a 2020 2a20 2272 6579 6e6f ====.. * "reyno │ │ │ │ -00047f40: 6c64 734f 7065 7261 746f 7228 5269 6e67 ldsOperator(Ring │ │ │ │ -00047f50: 456c 656d 656e 742c 4469 6167 6f6e 616c Element,Diagonal │ │ │ │ -00047f60: 4163 7469 6f6e 2922 0a20 202a 2022 7265 Action)". * "re │ │ │ │ -00047f70: 796e 6f6c 6473 4f70 6572 6174 6f72 2852 ynoldsOperator(R │ │ │ │ -00047f80: 696e 6745 6c65 6d65 6e74 2c46 696e 6974 ingElement,Finit │ │ │ │ -00047f90: 6547 726f 7570 4163 7469 6f6e 2922 0a0a eGroupAction)".. │ │ │ │ -00047fa0: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -00047fb0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -00047fc0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -00047fd0: 7420 2a6e 6f74 6520 7265 796e 6f6c 6473 t *note reynolds │ │ │ │ -00047fe0: 4f70 6572 6174 6f72 3a20 7265 796e 6f6c Operator: reynol │ │ │ │ -00047ff0: 6473 4f70 6572 6174 6f72 2c20 6973 2061 dsOperator, is a │ │ │ │ -00048000: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ -00048010: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61 nction: (Macaula │ │ │ │ -00048020: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ -00048030: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ -00048040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047f00: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +00047f10: 5220 3d20 5151 5b78 5f31 2e2e 785f 345d R = QQ[x_1..x_4] │ │ │ │ +00047f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047f40: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00047f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047f80: 2020 2020 7c0a 7c6f 3120 3d20 5220 2020 |.|o1 = R │ │ │ │ +00047f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047fc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00047fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048000: 7c0a 7c6f 3120 3a20 506f 6c79 6e6f 6d69 |.|o1 : Polynomi │ │ │ │ +00048010: 616c 5269 6e67 2020 2020 2020 2020 2020 alRing │ │ │ │ +00048020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048030: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00048040: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00048050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00048060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048080: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -00048090: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -000480a0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -000480b0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -000480c0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -000480d0: 3236 2e30 352b 6473 2f4d 322f 4d61 6361 26.05+ds/M2/Maca │ │ │ │ -000480e0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ -000480f0: 496e 7661 7269 616e 7452 696e 672f 496e InvariantRing/In │ │ │ │ -00048100: 7661 7269 616e 7473 446f 632e 6d32 3a39 variantsDoc.m2:9 │ │ │ │ -00048110: 3035 3a30 2e0a 1f0a 4669 6c65 3a20 496e 05:0....File: In │ │ │ │ -00048120: 7661 7269 616e 7452 696e 672e 696e 666f variantRing.info │ │ │ │ -00048130: 2c20 4e6f 6465 3a20 7269 6e67 5f6c 7047 , Node: ring_lpG │ │ │ │ -00048140: 726f 7570 4163 7469 6f6e 5f72 702c 204e roupAction_rp, N │ │ │ │ -00048150: 6578 743a 2052 696e 674f 6649 6e76 6172 ext: RingOfInvar │ │ │ │ -00048160: 6961 6e74 732c 2050 7265 763a 2072 6579 iants, Prev: rey │ │ │ │ -00048170: 6e6f 6c64 734f 7065 7261 746f 722c 2055 noldsOperator, U │ │ │ │ -00048180: 703a 2054 6f70 0a0a 7269 6e67 2847 726f p: Top..ring(Gro │ │ │ │ -00048190: 7570 4163 7469 6f6e 2920 2d2d 2074 6865 upAction) -- the │ │ │ │ -000481a0: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ -000481b0: 2062 6569 6e67 2061 6374 6564 2075 706f being acted upo │ │ │ │ -000481c0: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a n.************** │ │ │ │ -000481d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000481e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000481f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -00048200: 2046 756e 6374 696f 6e3a 202a 6e6f 7465 Function: *note │ │ │ │ -00048210: 2072 696e 673a 2028 4d61 6361 756c 6179 ring: (Macaulay │ │ │ │ -00048220: 3244 6f63 2972 696e 672c 0a20 202a 2055 2Doc)ring,. * U │ │ │ │ -00048230: 7361 6765 3a20 0a20 2020 2020 2020 2072 sage: . r │ │ │ │ -00048240: 696e 6720 470a 2020 2a20 496e 7075 7473 ing G. * Inputs │ │ │ │ -00048250: 3a0a 2020 2020 2020 2a20 472c 2061 6e20 :. * G, an │ │ │ │ -00048260: 696e 7374 616e 6365 206f 6620 7468 6520 instance of the │ │ │ │ -00048270: 7479 7065 202a 6e6f 7465 2047 726f 7570 type *note Group │ │ │ │ -00048280: 4163 7469 6f6e 3a20 4772 6f75 7041 6374 Action: GroupAct │ │ │ │ -00048290: 696f 6e2c 2c20 6120 6772 6f75 700a 2020 ion,, a group. │ │ │ │ -000482a0: 2020 2020 2020 6163 7469 6f6e 206f 6e20 action on │ │ │ │ -000482b0: 6120 706f 6c79 6e6f 6d69 616c 2072 696e a polynomial rin │ │ │ │ -000482c0: 670a 2020 2a20 4f75 7470 7574 733a 0a20 g. * Outputs:. │ │ │ │ -000482d0: 2020 2020 202a 2061 202a 6e6f 7465 2072 * a *note r │ │ │ │ -000482e0: 696e 673a 2028 4d61 6361 756c 6179 3244 ing: (Macaulay2D │ │ │ │ -000482f0: 6f63 2952 696e 672c 2c20 7468 6520 706f oc)Ring,, the po │ │ │ │ -00048300: 6c79 6e6f 6d69 616c 2072 696e 6720 6265 lynomial ring be │ │ │ │ -00048310: 696e 6720 6163 7465 6420 7570 6f6e 0a0a ing acted upon.. │ │ │ │ -00048320: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -00048330: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 6675 =======..This fu │ │ │ │ -00048340: 6e63 7469 6f6e 2069 7320 7072 6f76 6964 nction is provid │ │ │ │ -00048350: 6564 2062 7920 7468 6520 7061 636b 6167 ed by the packag │ │ │ │ -00048360: 6520 2a6e 6f74 6520 496e 7661 7269 616e e *note Invarian │ │ │ │ -00048370: 7452 696e 673a 2054 6f70 2c2e 0a0a 2b2d tRing: Top,...+- │ │ │ │ -00048380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000483a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000483b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ -000483c0: 3a20 5220 3d20 5151 5b78 5f31 2e2e 785f : R = QQ[x_1..x_ │ │ │ │ -000483d0: 345d 2020 2020 2020 2020 2020 2020 2020 4] │ │ │ │ -000483e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000483f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00048400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048430: 2020 2020 2020 7c0a 7c6f 3120 3d20 5220 |.|o1 = R │ │ │ │ -00048440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048470: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00048480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000484a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000484b0: 2020 7c0a 7c6f 3120 3a20 506f 6c79 6e6f |.|o1 : Polyno │ │ │ │ -000484c0: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ -000484d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000484e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000484f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00048500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00048530: 7c69 3220 3a20 5420 3d20 6469 6167 6f6e |i2 : T = diagon │ │ │ │ -00048540: 616c 4163 7469 6f6e 286d 6174 7269 7820 alAction(matrix │ │ │ │ -00048550: 7b7b 302c 312c 2d31 2c31 7d2c 7b31 2c30 {{0,1,-1,1},{1,0 │ │ │ │ -00048560: 2c2d 312c 2d31 7d7d 2c20 5229 7c0a 7c20 ,-1,-1}}, R)|.| │ │ │ │ -00048570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000485a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -000485b0: 2020 2020 2020 2020 2020 2a20 3220 2020 * 2 │ │ │ │ -000485c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000485d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000485e0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ -000485f0: 5220 3c2d 2028 5151 2029 2020 7669 6120 R <- (QQ ) via │ │ │ │ -00048600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048620: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00048630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048660: 2020 2020 7c0a 7c20 2020 2020 7c20 3020 |.| | 0 │ │ │ │ -00048670: 3120 2d31 2031 2020 7c20 2020 2020 2020 1 -1 1 | │ │ │ │ -00048680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000486a0: 2020 7c0a 7c20 2020 2020 7c20 3120 3020 |.| | 1 0 │ │ │ │ -000486b0: 2d31 202d 3120 7c20 2020 2020 2020 2020 -1 -1 | │ │ │ │ -000486c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000486d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000486e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000486f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048710: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00048720: 7c6f 3220 3a20 4469 6167 6f6e 616c 4163 |o2 : DiagonalAc │ │ │ │ -00048730: 7469 6f6e 2020 2020 2020 2020 2020 2020 tion │ │ │ │ -00048740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048750: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00048760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048790: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ -000487a0: 3a20 7269 6e67 2054 203d 3d3d 2052 2020 : ring T === R │ │ │ │ -000487b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000487c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000487d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000487e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000487f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048810: 2020 2020 2020 7c0a 7c6f 3320 3d20 7472 |.|o3 = tr │ │ │ │ -00048820: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ -00048830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048850: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00048860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048890: 2d2d 2b0a 0a57 6179 7320 746f 2075 7365 --+..Ways to use │ │ │ │ -000488a0: 2074 6869 7320 6d65 7468 6f64 3a0a 3d3d this method:.== │ │ │ │ -000488b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000488c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ -000488d0: 6520 7269 6e67 2847 726f 7570 4163 7469 e ring(GroupActi │ │ │ │ -000488e0: 6f6e 293a 2072 696e 675f 6c70 4772 6f75 on): ring_lpGrou │ │ │ │ -000488f0: 7041 6374 696f 6e5f 7270 2c20 2d2d 2074 pAction_rp, -- t │ │ │ │ -00048900: 6865 2070 6f6c 796e 6f6d 6961 6c20 7269 he polynomial ri │ │ │ │ -00048910: 6e67 0a20 2020 2062 6569 6e67 2061 6374 ng. being act │ │ │ │ -00048920: 6564 2075 706f 6e0a 2d2d 2d2d 2d2d 2d2d ed upon.-------- │ │ │ │ -00048930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048970: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -00048980: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -00048990: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -000489a0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -000489b0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -000489c0: 3236 2e30 352b 6473 2f4d 322f 4d61 6361 26.05+ds/M2/Maca │ │ │ │ -000489d0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ -000489e0: 496e 7661 7269 616e 7452 696e 672f 496e InvariantRing/In │ │ │ │ -000489f0: 7661 7269 616e 7452 696e 6744 6f63 2e6d variantRingDoc.m │ │ │ │ -00048a00: 323a 3139 353a 302e 0a1f 0a46 696c 653a 2:195:0....File: │ │ │ │ -00048a10: 2049 6e76 6172 6961 6e74 5269 6e67 2e69 InvariantRing.i │ │ │ │ -00048a20: 6e66 6f2c 204e 6f64 653a 2052 696e 674f nfo, Node: RingO │ │ │ │ -00048a30: 6649 6e76 6172 6961 6e74 732c 204e 6578 fInvariants, Nex │ │ │ │ -00048a40: 743a 2073 6368 7265 6965 7247 7261 7068 t: schreierGraph │ │ │ │ -00048a50: 2c20 5072 6576 3a20 7269 6e67 5f6c 7047 , Prev: ring_lpG │ │ │ │ -00048a60: 726f 7570 4163 7469 6f6e 5f72 702c 2055 roupAction_rp, U │ │ │ │ -00048a70: 703a 2054 6f70 0a0a 5269 6e67 4f66 496e p: Top..RingOfIn │ │ │ │ -00048a80: 7661 7269 616e 7473 202d 2d20 7468 6520 variants -- the │ │ │ │ -00048a90: 636c 6173 7320 6f66 2074 6865 2072 696e class of the rin │ │ │ │ -00048aa0: 6773 206f 6620 696e 7661 7269 616e 7473 gs of invariants │ │ │ │ -00048ab0: 2075 6e64 6572 2074 6865 2061 6374 696f under the actio │ │ │ │ -00048ac0: 6e20 6f66 2061 2066 696e 6974 6520 6772 n of a finite gr │ │ │ │ -00048ad0: 6f75 702c 2061 6e20 4162 656c 6961 6e20 oup, an Abelian │ │ │ │ -00048ae0: 6772 6f75 7020 6f72 2061 206c 696e 6561 group or a linea │ │ │ │ -00048af0: 726c 7920 7265 6475 6374 6976 6520 6772 rly reductive gr │ │ │ │ -00048b00: 6f75 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a oup.************ │ │ │ │ -00048b10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00048b20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00048b30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00048b40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00048b50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00048b60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00048b70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00048b80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ -00048b90: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ -00048ba0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 2063 ========..This c │ │ │ │ -00048bb0: 6c61 7373 2069 7320 7072 6f76 6964 6564 lass is provided │ │ │ │ -00048bc0: 2062 7920 7468 6520 7061 636b 6167 6520 by the package │ │ │ │ -00048bd0: 2a6e 6f74 6520 496e 7661 7269 616e 7452 *note InvariantR │ │ │ │ -00048be0: 696e 673a 2054 6f70 2c2e 0a0a 5269 6e67 ing: Top,...Ring │ │ │ │ -00048bf0: 4f66 496e 7661 7269 616e 7473 2069 7320 OfInvariants is │ │ │ │ -00048c00: 7468 6520 636c 6173 7320 6f66 2072 696e the class of rin │ │ │ │ -00048c10: 6773 206f 6620 696e 7661 7269 616e 7473 gs of invariants │ │ │ │ -00048c20: 2077 6865 6e20 6120 6669 6e69 7465 2067 when a finite g │ │ │ │ -00048c30: 726f 7570 2c20 616e 0a41 6265 6c69 616e roup, an.Abelian │ │ │ │ -00048c40: 2067 726f 7570 206f 7220 6120 6c69 6e65 group or a line │ │ │ │ -00048c50: 6172 6c79 2072 6564 7563 7469 7665 2067 arly reductive g │ │ │ │ -00048c60: 726f 7570 2061 6374 7320 6f6e 2061 2070 roup acts on a p │ │ │ │ -00048c70: 6f6c 796e 6f6d 6961 6c20 7269 6e67 2e0a olynomial ring.. │ │ │ │ -00048c80: 0a46 756e 6374 696f 6e73 2061 6e64 206d .Functions and m │ │ │ │ -00048c90: 6574 686f 6473 2072 6574 7572 6e69 6e67 ethods returning │ │ │ │ -00048ca0: 2061 6e20 6f62 6a65 6374 206f 6620 636c an object of cl │ │ │ │ -00048cb0: 6173 7320 5269 6e67 4f66 496e 7661 7269 ass RingOfInvari │ │ │ │ -00048cc0: 616e 7473 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d ants:.========== │ │ │ │ -00048cd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00048ce0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00048cf0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00048d00: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -00048d10: 2269 6e76 6172 6961 6e74 5269 6e67 2847 "invariantRing(G │ │ │ │ -00048d20: 726f 7570 4163 7469 6f6e 2922 202d 2d20 roupAction)" -- │ │ │ │ -00048d30: 7365 6520 2a6e 6f74 6520 696e 7661 7269 see *note invari │ │ │ │ -00048d40: 616e 7452 696e 673a 2069 6e76 6172 6961 antRing: invaria │ │ │ │ -00048d50: 6e74 5269 6e67 2c20 2d2d 0a20 2020 2074 ntRing, --. t │ │ │ │ -00048d60: 6865 2072 696e 6720 6f66 2069 6e76 6172 he ring of invar │ │ │ │ -00048d70: 6961 6e74 7320 6f66 2061 2067 726f 7570 iants of a group │ │ │ │ -00048d80: 2061 6374 696f 6e0a 2020 2a20 2250 6f6c action. * "Pol │ │ │ │ -00048d90: 796e 6f6d 6961 6c52 696e 6720 5e20 4772 ynomialRing ^ Gr │ │ │ │ -00048da0: 6f75 7041 6374 696f 6e22 202d 2d20 7365 oupAction" -- se │ │ │ │ -00048db0: 6520 2a6e 6f74 6520 696e 7661 7269 616e e *note invarian │ │ │ │ -00048dc0: 7452 696e 673a 2069 6e76 6172 6961 6e74 tRing: invariant │ │ │ │ -00048dd0: 5269 6e67 2c0a 2020 2020 2d2d 2074 6865 Ring,. -- the │ │ │ │ -00048de0: 2072 696e 6720 6f66 2069 6e76 6172 6961 ring of invaria │ │ │ │ -00048df0: 6e74 7320 6f66 2061 2067 726f 7570 2061 nts of a group a │ │ │ │ -00048e00: 6374 696f 6e0a 2020 2a20 2251 756f 7469 ction. * "Quoti │ │ │ │ -00048e10: 656e 7452 696e 6720 5e20 4c69 6e65 6172 entRing ^ Linear │ │ │ │ -00048e20: 6c79 5265 6475 6374 6976 6541 6374 696f lyReductiveActio │ │ │ │ -00048e30: 6e22 202d 2d20 7365 6520 2a6e 6f74 6520 n" -- see *note │ │ │ │ -00048e40: 696e 7661 7269 616e 7452 696e 673a 0a20 invariantRing:. │ │ │ │ -00048e50: 2020 2069 6e76 6172 6961 6e74 5269 6e67 invariantRing │ │ │ │ -00048e60: 2c20 2d2d 2074 6865 2072 696e 6720 6f66 , -- the ring of │ │ │ │ -00048e70: 2069 6e76 6172 6961 6e74 7320 6f66 2061 invariants of a │ │ │ │ -00048e80: 2067 726f 7570 2061 6374 696f 6e0a 0a4d group action..M │ │ │ │ -00048e90: 6574 686f 6473 2074 6861 7420 7573 6520 ethods that use │ │ │ │ -00048ea0: 616e 206f 626a 6563 7420 6f66 2063 6c61 an object of cla │ │ │ │ -00048eb0: 7373 2052 696e 674f 6649 6e76 6172 6961 ss RingOfInvaria │ │ │ │ -00048ec0: 6e74 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d nts:.=========== │ │ │ │ -00048ed0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00048ee0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00048ef0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -00048f00: 2261 6374 696f 6e28 5269 6e67 4f66 496e "action(RingOfIn │ │ │ │ -00048f10: 7661 7269 616e 7473 2922 202d 2d20 7365 variants)" -- se │ │ │ │ -00048f20: 6520 2a6e 6f74 6520 6163 7469 6f6e 3a20 e *note action: │ │ │ │ -00048f30: 6163 7469 6f6e 2c20 2d2d 2074 6865 2067 action, -- the g │ │ │ │ -00048f40: 726f 7570 2061 6374 696f 6e0a 2020 2020 roup action. │ │ │ │ -00048f50: 7468 6174 2070 726f 6475 6365 6420 6120 that produced a │ │ │ │ -00048f60: 7269 6e67 206f 6620 696e 7661 7269 616e ring of invarian │ │ │ │ -00048f70: 7473 0a20 202a 202a 6e6f 7465 2061 6d62 ts. * *note amb │ │ │ │ -00048f80: 6965 6e74 2852 696e 674f 6649 6e76 6172 ient(RingOfInvar │ │ │ │ -00048f90: 6961 6e74 7329 3a20 616d 6269 656e 745f iants): ambient_ │ │ │ │ -00048fa0: 6c70 5269 6e67 4f66 496e 7661 7269 616e lpRingOfInvarian │ │ │ │ -00048fb0: 7473 5f72 702c 202d 2d20 7468 650a 2020 ts_rp, -- the. │ │ │ │ -00048fc0: 2020 616d 6269 656e 7420 706f 6c79 6e6f ambient polyno │ │ │ │ -00048fd0: 6d69 616c 2072 696e 6720 7768 6572 6520 mial ring where │ │ │ │ -00048fe0: 7468 6520 6772 6f75 7020 6163 7465 6420 the group acted │ │ │ │ -00048ff0: 7570 6f6e 0a20 202a 2022 6465 6669 6e69 upon. * "defini │ │ │ │ -00049000: 6e67 4964 6561 6c28 5269 6e67 4f66 496e ngIdeal(RingOfIn │ │ │ │ -00049010: 7661 7269 616e 7473 2922 202d 2d20 7365 variants)" -- se │ │ │ │ -00049020: 6520 2a6e 6f74 6520 6465 6669 6e69 6e67 e *note defining │ │ │ │ -00049030: 4964 6561 6c3a 0a20 2020 2064 6566 696e Ideal:. defin │ │ │ │ -00049040: 696e 6749 6465 616c 2c20 2d2d 2070 7265 ingIdeal, -- pre │ │ │ │ -00049050: 7365 6e74 6174 696f 6e20 6f66 2061 2072 sentation of a r │ │ │ │ -00049060: 696e 6720 6f66 2069 6e76 6172 6961 6e74 ing of invariant │ │ │ │ -00049070: 7320 6173 2070 6f6c 796e 6f6d 6961 6c20 s as polynomial │ │ │ │ -00049080: 7269 6e67 0a20 2020 206d 6f64 756c 6f20 ring. modulo │ │ │ │ -00049090: 7468 6520 6465 6669 6e69 6e67 2069 6465 the defining ide │ │ │ │ -000490a0: 616c 0a20 202a 202a 6e6f 7465 2067 656e al. * *note gen │ │ │ │ -000490b0: 6572 6174 6f72 7328 5269 6e67 4f66 496e erators(RingOfIn │ │ │ │ -000490c0: 7661 7269 616e 7473 293a 2067 656e 6572 variants): gener │ │ │ │ -000490d0: 6174 6f72 735f 6c70 5269 6e67 4f66 496e ators_lpRingOfIn │ │ │ │ -000490e0: 7661 7269 616e 7473 5f72 702c 202d 2d0a variants_rp, --. │ │ │ │ -000490f0: 2020 2020 7468 6520 6765 6e65 7261 746f the generato │ │ │ │ -00049100: 7273 2066 6f72 2061 2072 696e 6720 6f66 rs for a ring of │ │ │ │ -00049110: 2069 6e76 6172 6961 6e74 730a 2020 2a20 invariants. * │ │ │ │ -00049120: 2a6e 6f74 6520 6869 6c62 6572 7453 6572 *note hilbertSer │ │ │ │ -00049130: 6965 7328 5269 6e67 4f66 496e 7661 7269 ies(RingOfInvari │ │ │ │ -00049140: 616e 7473 293a 2068 696c 6265 7274 5365 ants): hilbertSe │ │ │ │ -00049150: 7269 6573 5f6c 7052 696e 674f 6649 6e76 ries_lpRingOfInv │ │ │ │ -00049160: 6172 6961 6e74 735f 7270 2c0a 2020 2020 ariants_rp,. │ │ │ │ -00049170: 2d2d 2048 696c 6265 7274 2073 6572 6965 -- Hilbert serie │ │ │ │ -00049180: 7320 6f66 2074 6865 2069 6e76 6172 6961 s of the invaria │ │ │ │ -00049190: 6e74 2072 696e 670a 2020 2a20 2a6e 6f74 nt ring. * *not │ │ │ │ -000491a0: 6520 6e65 7428 5269 6e67 4f66 496e 7661 e net(RingOfInva │ │ │ │ -000491b0: 7269 616e 7473 293a 206e 6574 5f6c 7052 riants): net_lpR │ │ │ │ -000491c0: 696e 674f 6649 6e76 6172 6961 6e74 735f ingOfInvariants_ │ │ │ │ -000491d0: 7270 2c20 2d2d 2066 6f72 6d61 7420 666f rp, -- format fo │ │ │ │ -000491e0: 720a 2020 2020 7072 696e 7469 6e67 2c20 r. printing, │ │ │ │ -000491f0: 6173 2061 206e 6574 0a0a 466f 7220 7468 as a net..For th │ │ │ │ -00049200: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -00049210: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00049220: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -00049230: 6520 5269 6e67 4f66 496e 7661 7269 616e e RingOfInvarian │ │ │ │ -00049240: 7473 3a20 5269 6e67 4f66 496e 7661 7269 ts: RingOfInvari │ │ │ │ -00049250: 616e 7473 2c20 6973 2061 202a 6e6f 7465 ants, is a *note │ │ │ │ -00049260: 2074 7970 653a 0a28 4d61 6361 756c 6179 type:.(Macaulay │ │ │ │ -00049270: 3244 6f63 2954 7970 652c 2c20 7769 7468 2Doc)Type,, with │ │ │ │ -00049280: 2061 6e63 6573 746f 7220 636c 6173 7365 ancestor classe │ │ │ │ -00049290: 7320 2a6e 6f74 6520 4861 7368 5461 626c s *note HashTabl │ │ │ │ -000492a0: 653a 0a28 4d61 6361 756c 6179 3244 6f63 e:.(Macaulay2Doc │ │ │ │ -000492b0: 2948 6173 6854 6162 6c65 2c20 3c20 2a6e )HashTable, < *n │ │ │ │ -000492c0: 6f74 6520 5468 696e 673a 2028 4d61 6361 ote Thing: (Maca │ │ │ │ -000492d0: 756c 6179 3244 6f63 2954 6869 6e67 2c2e ulay2Doc)Thing,. │ │ │ │ -000492e0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ -000492f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049330: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -00049340: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -00049350: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -00049360: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -00049370: 6361 756c 6179 322d 312e 3236 2e30 352b caulay2-1.26.05+ │ │ │ │ -00049380: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -00049390: 7061 636b 6167 6573 2f0a 496e 7661 7269 packages/.Invari │ │ │ │ -000493a0: 616e 7452 696e 672f 496e 7661 7269 616e antRing/Invarian │ │ │ │ -000493b0: 7473 446f 632e 6d32 3a39 3532 3a30 2e0a tsDoc.m2:952:0.. │ │ │ │ -000493c0: 1f0a 4669 6c65 3a20 496e 7661 7269 616e ..File: Invarian │ │ │ │ -000493d0: 7452 696e 672e 696e 666f 2c20 4e6f 6465 tRing.info, Node │ │ │ │ -000493e0: 3a20 7363 6872 6569 6572 4772 6170 682c : schreierGraph, │ │ │ │ -000493f0: 204e 6578 743a 2073 6563 6f6e 6461 7279 Next: secondary │ │ │ │ -00049400: 496e 7661 7269 616e 7473 2c20 5072 6576 Invariants, Prev │ │ │ │ -00049410: 3a20 5269 6e67 4f66 496e 7661 7269 616e : RingOfInvarian │ │ │ │ -00049420: 7473 2c20 5570 3a20 546f 700a 0a73 6368 ts, Up: Top..sch │ │ │ │ -00049430: 7265 6965 7247 7261 7068 202d 2d20 5363 reierGraph -- Sc │ │ │ │ -00049440: 6872 6569 6572 2067 7261 7068 206f 6620 hreier graph of │ │ │ │ -00049450: 6120 6669 6e69 7465 2067 726f 7570 0a2a a finite group.* │ │ │ │ -00049460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00049470: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00049480: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00049490: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ -000494a0: 2020 2020 2020 7363 6872 6569 6572 4772 schreierGr │ │ │ │ -000494b0: 6170 6820 470a 2020 2a20 496e 7075 7473 aph G. * Inputs │ │ │ │ -000494c0: 3a0a 2020 2020 2020 2a20 472c 2061 6e20 :. * G, an │ │ │ │ -000494d0: 696e 7374 616e 6365 206f 6620 7468 6520 instance of the │ │ │ │ -000494e0: 7479 7065 202a 6e6f 7465 2046 696e 6974 type *note Finit │ │ │ │ -000494f0: 6547 726f 7570 4163 7469 6f6e 3a20 4669 eGroupAction: Fi │ │ │ │ -00049500: 6e69 7465 4772 6f75 7041 6374 696f 6e2c niteGroupAction, │ │ │ │ -00049510: 2c0a 2020 2020 2020 2020 6120 6669 6e69 ,. a fini │ │ │ │ -00049520: 7465 2067 726f 7570 2061 6374 696f 6e0a te group action. │ │ │ │ -00049530: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -00049540: 2020 202a 2061 202a 6e6f 7465 2068 6173 * a *note has │ │ │ │ -00049550: 6820 7461 626c 653a 2028 4d61 6361 756c h table: (Macaul │ │ │ │ -00049560: 6179 3244 6f63 2948 6173 6854 6162 6c65 ay2Doc)HashTable │ │ │ │ -00049570: 2c2c 2072 6570 7265 7365 6e74 696e 6720 ,, representing │ │ │ │ -00049580: 7468 6520 5363 6872 6569 6572 0a20 2020 the Schreier. │ │ │ │ -00049590: 2020 2020 2067 7261 7068 206f 6620 7468 graph of th │ │ │ │ -000495a0: 6520 6772 6f75 700a 0a44 6573 6372 6970 e group..Descrip │ │ │ │ -000495b0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -000495c0: 0a0a 5468 6973 2066 756e 6374 696f 6e20 ..This function │ │ │ │ -000495d0: 6973 2070 726f 7669 6465 6420 6279 2074 is provided by t │ │ │ │ -000495e0: 6865 2070 6163 6b61 6765 202a 6e6f 7465 he package *note │ │ │ │ -000495f0: 2049 6e76 6172 6961 6e74 5269 6e67 3a20 InvariantRing: │ │ │ │ -00049600: 546f 702c 2e0a 0a46 6f72 2061 2066 696e Top,...For a fin │ │ │ │ -00049610: 6974 6520 6772 6f75 7020 6163 7469 6f6e ite group action │ │ │ │ -00049620: 2c20 7765 2066 6f72 6d20 6120 2a6e 6f74 , we form a *not │ │ │ │ -00049630: 6520 4861 7368 5461 626c 653a 2028 4d61 e HashTable: (Ma │ │ │ │ -00049640: 6361 756c 6179 3244 6f63 2948 6173 6854 caulay2Doc)HashT │ │ │ │ -00049650: 6162 6c65 2c0a 7768 6f73 6520 6b65 7973 able,.whose keys │ │ │ │ -00049660: 2061 7265 2074 6865 2067 656e 6572 6174 are the generat │ │ │ │ -00049670: 6f72 7320 7072 6f76 6964 6564 2062 7920 ors provided by │ │ │ │ -00049680: 7468 6520 7573 6572 2e20 5468 6520 7661 the user. The va │ │ │ │ -00049690: 6c75 6520 636f 7272 6573 706f 6e64 696e lue correspondin │ │ │ │ -000496a0: 6720 746f 0a61 2067 656e 6572 6174 6f72 g to.a generator │ │ │ │ -000496b0: 2067 2069 7320 6120 2a6e 6f74 6520 4861 g is a *note Ha │ │ │ │ -000496c0: 7368 5461 626c 653a 2028 4d61 6361 756c shTable: (Macaul │ │ │ │ -000496d0: 6179 3244 6f63 2948 6173 6854 6162 6c65 ay2Doc)HashTable │ │ │ │ -000496e0: 2c20 636f 6e74 6169 6e69 6e67 2061 6c6c , containing all │ │ │ │ -000496f0: 0a70 6169 7273 2061 203d 3e20 6220 7375 .pairs a => b su │ │ │ │ -00049700: 6368 2074 6861 7420 612a 6720 3d3d 2062 ch that a*g == b │ │ │ │ -00049710: 2e20 5468 6973 2072 6570 7265 7365 6e74 . This represent │ │ │ │ -00049720: 7320 7468 6520 5363 6872 6569 6572 2067 s the Schreier g │ │ │ │ -00049730: 7261 7068 206f 6620 7468 650a 6772 6f75 raph of the.grou │ │ │ │ -00049740: 7020 7265 6c61 7469 7665 2074 6f20 7468 p relative to th │ │ │ │ -00049750: 6520 6765 6e65 7261 7469 6e67 2073 6574 e generating set │ │ │ │ -00049760: 2070 726f 7669 6465 6420 6279 2074 6865 provided by the │ │ │ │ -00049770: 2075 7365 722e 0a0a 5468 6520 666f 6c6c user...The foll │ │ │ │ -00049780: 6f77 696e 6720 6578 616d 706c 6520 6465 owing example de │ │ │ │ -00049790: 6669 6e65 7320 7468 6520 7065 726d 7574 fines the permut │ │ │ │ -000497a0: 6174 696f 6e20 6163 7469 6f6e 206f 6620 ation action of │ │ │ │ -000497b0: 6120 7379 6d6d 6574 7269 6320 6772 6f75 a symmetric grou │ │ │ │ -000497c0: 7020 6f6e 0a74 6872 6565 2065 6c65 6d65 p on.three eleme │ │ │ │ -000497d0: 6e74 7320 7573 696e 6720 6f6e 6c79 2074 nts using only t │ │ │ │ -000497e0: 776f 2067 656e 6572 6174 6f72 732c 2061 wo generators, a │ │ │ │ -000497f0: 2074 7261 6e73 706f 7369 7469 6f6e 2061 transposition a │ │ │ │ -00049800: 6e64 2061 2033 2d63 7963 6c65 2e0a 0a2b nd a 3-cycle...+ │ │ │ │ -00049810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00049860: 6931 203a 2052 203d 2051 515b 785f 312e i1 : R = QQ[x_1. │ │ │ │ -00049870: 2e78 5f33 5d20 2020 2020 2020 2020 2020 .x_3] │ │ │ │ -00049880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00049890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000498a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000498b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000498c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00048080: 3220 3a20 5420 3d20 6469 6167 6f6e 616c 2 : T = diagonal │ │ │ │ +00048090: 4163 7469 6f6e 286d 6174 7269 7820 7b7b Action(matrix {{ │ │ │ │ +000480a0: 302c 312c 2d31 2c31 7d2c 7b31 2c30 2c2d 0,1,-1,1},{1,0,- │ │ │ │ +000480b0: 312c 2d31 7d7d 2c20 5229 7c0a 7c20 2020 1,-1}}, R)|.| │ │ │ │ +000480c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000480d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000480e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000480f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00048100: 2020 2020 2020 2020 2a20 3220 2020 2020 * 2 │ │ │ │ +00048110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048130: 2020 2020 2020 7c0a 7c6f 3220 3d20 5220 |.|o2 = R │ │ │ │ +00048140: 3c2d 2028 5151 2029 2020 7669 6120 2020 <- (QQ ) via │ │ │ │ +00048150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048170: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00048180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000481a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000481b0: 2020 7c0a 7c20 2020 2020 7c20 3020 3120 |.| | 0 1 │ │ │ │ +000481c0: 2d31 2031 2020 7c20 2020 2020 2020 2020 -1 1 | │ │ │ │ +000481d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000481e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000481f0: 7c0a 7c20 2020 2020 7c20 3120 3020 2d31 |.| | 1 0 -1 │ │ │ │ +00048200: 202d 3120 7c20 2020 2020 2020 2020 2020 -1 | │ │ │ │ +00048210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048220: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00048230: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00048240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048260: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00048270: 3220 3a20 4469 6167 6f6e 616c 4163 7469 2 : DiagonalActi │ │ │ │ +00048280: 6f6e 2020 2020 2020 2020 2020 2020 2020 on │ │ │ │ +00048290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000482a0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +000482b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000482c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000482d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000482e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ +000482f0: 7269 6e67 2054 203d 3d3d 2052 2020 2020 ring T === R │ │ │ │ +00048300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048320: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00048330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048360: 2020 2020 7c0a 7c6f 3320 3d20 7472 7565 |.|o3 = true │ │ │ │ +00048370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000483a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000483b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000483c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000483d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000483e0: 2b0a 0a57 6179 7320 746f 2075 7365 2074 +..Ways to use t │ │ │ │ +000483f0: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d his method:.==== │ │ │ │ +00048400: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00048410: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ +00048420: 7269 6e67 2847 726f 7570 4163 7469 6f6e ring(GroupAction │ │ │ │ +00048430: 293a 2072 696e 675f 6c70 4772 6f75 7041 ): ring_lpGroupA │ │ │ │ +00048440: 6374 696f 6e5f 7270 2c20 2d2d 2074 6865 ction_rp, -- the │ │ │ │ +00048450: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ +00048460: 0a20 2020 2062 6569 6e67 2061 6374 6564 . being acted │ │ │ │ +00048470: 2075 706f 6e0a 2d2d 2d2d 2d2d 2d2d 2d2d upon.---------- │ │ │ │ +00048480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000484a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000484b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000484c0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +000484d0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +000484e0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +000484f0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +00048500: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +00048510: 2e30 352b 6473 2f4d 322f 4d61 6361 756c .05+ds/M2/Macaul │ │ │ │ +00048520: 6179 322f 7061 636b 6167 6573 2f0a 496e ay2/packages/.In │ │ │ │ +00048530: 7661 7269 616e 7452 696e 672f 496e 7661 variantRing/Inva │ │ │ │ +00048540: 7269 616e 7452 696e 6744 6f63 2e6d 323a riantRingDoc.m2: │ │ │ │ +00048550: 3139 353a 302e 0a1f 0a46 696c 653a 2049 195:0....File: I │ │ │ │ +00048560: 6e76 6172 6961 6e74 5269 6e67 2e69 6e66 nvariantRing.inf │ │ │ │ +00048570: 6f2c 204e 6f64 653a 2052 696e 674f 6649 o, Node: RingOfI │ │ │ │ +00048580: 6e76 6172 6961 6e74 732c 204e 6578 743a nvariants, Next: │ │ │ │ +00048590: 2073 6368 7265 6965 7247 7261 7068 2c20 schreierGraph, │ │ │ │ +000485a0: 5072 6576 3a20 7269 6e67 5f6c 7047 726f Prev: ring_lpGro │ │ │ │ +000485b0: 7570 4163 7469 6f6e 5f72 702c 2055 703a upAction_rp, Up: │ │ │ │ +000485c0: 2054 6f70 0a0a 5269 6e67 4f66 496e 7661 Top..RingOfInva │ │ │ │ +000485d0: 7269 616e 7473 202d 2d20 7468 6520 636c riants -- the cl │ │ │ │ +000485e0: 6173 7320 6f66 2074 6865 2072 696e 6773 ass of the rings │ │ │ │ +000485f0: 206f 6620 696e 7661 7269 616e 7473 2075 of invariants u │ │ │ │ +00048600: 6e64 6572 2074 6865 2061 6374 696f 6e20 nder the action │ │ │ │ +00048610: 6f66 2061 2066 696e 6974 6520 6772 6f75 of a finite grou │ │ │ │ +00048620: 702c 2061 6e20 4162 656c 6961 6e20 6772 p, an Abelian gr │ │ │ │ +00048630: 6f75 7020 6f72 2061 206c 696e 6561 726c oup or a linearl │ │ │ │ +00048640: 7920 7265 6475 6374 6976 6520 6772 6f75 y reductive grou │ │ │ │ +00048650: 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a p.************** │ │ │ │ +00048660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00048670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00048680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00048690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000486a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000486b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000486c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000486d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 *************..D │ │ │ │ +000486e0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +000486f0: 3d3d 3d3d 3d3d 0a0a 5468 6973 2063 6c61 ======..This cla │ │ │ │ +00048700: 7373 2069 7320 7072 6f76 6964 6564 2062 ss is provided b │ │ │ │ +00048710: 7920 7468 6520 7061 636b 6167 6520 2a6e y the package *n │ │ │ │ +00048720: 6f74 6520 496e 7661 7269 616e 7452 696e ote InvariantRin │ │ │ │ +00048730: 673a 2054 6f70 2c2e 0a0a 5269 6e67 4f66 g: Top,...RingOf │ │ │ │ +00048740: 496e 7661 7269 616e 7473 2069 7320 7468 Invariants is th │ │ │ │ +00048750: 6520 636c 6173 7320 6f66 2072 696e 6773 e class of rings │ │ │ │ +00048760: 206f 6620 696e 7661 7269 616e 7473 2077 of invariants w │ │ │ │ +00048770: 6865 6e20 6120 6669 6e69 7465 2067 726f hen a finite gro │ │ │ │ +00048780: 7570 2c20 616e 0a41 6265 6c69 616e 2067 up, an.Abelian g │ │ │ │ +00048790: 726f 7570 206f 7220 6120 6c69 6e65 6172 roup or a linear │ │ │ │ +000487a0: 6c79 2072 6564 7563 7469 7665 2067 726f ly reductive gro │ │ │ │ +000487b0: 7570 2061 6374 7320 6f6e 2061 2070 6f6c up acts on a pol │ │ │ │ +000487c0: 796e 6f6d 6961 6c20 7269 6e67 2e0a 0a46 ynomial ring...F │ │ │ │ +000487d0: 756e 6374 696f 6e73 2061 6e64 206d 6574 unctions and met │ │ │ │ +000487e0: 686f 6473 2072 6574 7572 6e69 6e67 2061 hods returning a │ │ │ │ +000487f0: 6e20 6f62 6a65 6374 206f 6620 636c 6173 n object of clas │ │ │ │ +00048800: 7320 5269 6e67 4f66 496e 7661 7269 616e s RingOfInvarian │ │ │ │ +00048810: 7473 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ts:.============ │ │ │ │ +00048820: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00048830: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00048840: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00048850: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269 ========.. * "i │ │ │ │ +00048860: 6e76 6172 6961 6e74 5269 6e67 2847 726f nvariantRing(Gro │ │ │ │ +00048870: 7570 4163 7469 6f6e 2922 202d 2d20 7365 upAction)" -- se │ │ │ │ +00048880: 6520 2a6e 6f74 6520 696e 7661 7269 616e e *note invarian │ │ │ │ +00048890: 7452 696e 673a 2069 6e76 6172 6961 6e74 tRing: invariant │ │ │ │ +000488a0: 5269 6e67 2c20 2d2d 0a20 2020 2074 6865 Ring, --. the │ │ │ │ +000488b0: 2072 696e 6720 6f66 2069 6e76 6172 6961 ring of invaria │ │ │ │ +000488c0: 6e74 7320 6f66 2061 2067 726f 7570 2061 nts of a group a │ │ │ │ +000488d0: 6374 696f 6e0a 2020 2a20 2250 6f6c 796e ction. * "Polyn │ │ │ │ +000488e0: 6f6d 6961 6c52 696e 6720 5e20 4772 6f75 omialRing ^ Grou │ │ │ │ +000488f0: 7041 6374 696f 6e22 202d 2d20 7365 6520 pAction" -- see │ │ │ │ +00048900: 2a6e 6f74 6520 696e 7661 7269 616e 7452 *note invariantR │ │ │ │ +00048910: 696e 673a 2069 6e76 6172 6961 6e74 5269 ing: invariantRi │ │ │ │ +00048920: 6e67 2c0a 2020 2020 2d2d 2074 6865 2072 ng,. -- the r │ │ │ │ +00048930: 696e 6720 6f66 2069 6e76 6172 6961 6e74 ing of invariant │ │ │ │ +00048940: 7320 6f66 2061 2067 726f 7570 2061 6374 s of a group act │ │ │ │ +00048950: 696f 6e0a 2020 2a20 2251 756f 7469 656e ion. * "Quotien │ │ │ │ +00048960: 7452 696e 6720 5e20 4c69 6e65 6172 6c79 tRing ^ Linearly │ │ │ │ +00048970: 5265 6475 6374 6976 6541 6374 696f 6e22 ReductiveAction" │ │ │ │ +00048980: 202d 2d20 7365 6520 2a6e 6f74 6520 696e -- see *note in │ │ │ │ +00048990: 7661 7269 616e 7452 696e 673a 0a20 2020 variantRing:. │ │ │ │ +000489a0: 2069 6e76 6172 6961 6e74 5269 6e67 2c20 invariantRing, │ │ │ │ +000489b0: 2d2d 2074 6865 2072 696e 6720 6f66 2069 -- the ring of i │ │ │ │ +000489c0: 6e76 6172 6961 6e74 7320 6f66 2061 2067 nvariants of a g │ │ │ │ +000489d0: 726f 7570 2061 6374 696f 6e0a 0a4d 6574 roup action..Met │ │ │ │ +000489e0: 686f 6473 2074 6861 7420 7573 6520 616e hods that use an │ │ │ │ +000489f0: 206f 626a 6563 7420 6f66 2063 6c61 7373 object of class │ │ │ │ +00048a00: 2052 696e 674f 6649 6e76 6172 6961 6e74 RingOfInvariant │ │ │ │ +00048a10: 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d s:.============= │ │ │ │ +00048a20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00048a30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00048a40: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2261 ========.. * "a │ │ │ │ +00048a50: 6374 696f 6e28 5269 6e67 4f66 496e 7661 ction(RingOfInva │ │ │ │ +00048a60: 7269 616e 7473 2922 202d 2d20 7365 6520 riants)" -- see │ │ │ │ +00048a70: 2a6e 6f74 6520 6163 7469 6f6e 3a20 6163 *note action: ac │ │ │ │ +00048a80: 7469 6f6e 2c20 2d2d 2074 6865 2067 726f tion, -- the gro │ │ │ │ +00048a90: 7570 2061 6374 696f 6e0a 2020 2020 7468 up action. th │ │ │ │ +00048aa0: 6174 2070 726f 6475 6365 6420 6120 7269 at produced a ri │ │ │ │ +00048ab0: 6e67 206f 6620 696e 7661 7269 616e 7473 ng of invariants │ │ │ │ +00048ac0: 0a20 202a 202a 6e6f 7465 2061 6d62 6965 . * *note ambie │ │ │ │ +00048ad0: 6e74 2852 696e 674f 6649 6e76 6172 6961 nt(RingOfInvaria │ │ │ │ +00048ae0: 6e74 7329 3a20 616d 6269 656e 745f 6c70 nts): ambient_lp │ │ │ │ +00048af0: 5269 6e67 4f66 496e 7661 7269 616e 7473 RingOfInvariants │ │ │ │ +00048b00: 5f72 702c 202d 2d20 7468 650a 2020 2020 _rp, -- the. │ │ │ │ +00048b10: 616d 6269 656e 7420 706f 6c79 6e6f 6d69 ambient polynomi │ │ │ │ +00048b20: 616c 2072 696e 6720 7768 6572 6520 7468 al ring where th │ │ │ │ +00048b30: 6520 6772 6f75 7020 6163 7465 6420 7570 e group acted up │ │ │ │ +00048b40: 6f6e 0a20 202a 2022 6465 6669 6e69 6e67 on. * "defining │ │ │ │ +00048b50: 4964 6561 6c28 5269 6e67 4f66 496e 7661 Ideal(RingOfInva │ │ │ │ +00048b60: 7269 616e 7473 2922 202d 2d20 7365 6520 riants)" -- see │ │ │ │ +00048b70: 2a6e 6f74 6520 6465 6669 6e69 6e67 4964 *note definingId │ │ │ │ +00048b80: 6561 6c3a 0a20 2020 2064 6566 696e 696e eal:. definin │ │ │ │ +00048b90: 6749 6465 616c 2c20 2d2d 2070 7265 7365 gIdeal, -- prese │ │ │ │ +00048ba0: 6e74 6174 696f 6e20 6f66 2061 2072 696e ntation of a rin │ │ │ │ +00048bb0: 6720 6f66 2069 6e76 6172 6961 6e74 7320 g of invariants │ │ │ │ +00048bc0: 6173 2070 6f6c 796e 6f6d 6961 6c20 7269 as polynomial ri │ │ │ │ +00048bd0: 6e67 0a20 2020 206d 6f64 756c 6f20 7468 ng. modulo th │ │ │ │ +00048be0: 6520 6465 6669 6e69 6e67 2069 6465 616c e defining ideal │ │ │ │ +00048bf0: 0a20 202a 202a 6e6f 7465 2067 656e 6572 . * *note gener │ │ │ │ +00048c00: 6174 6f72 7328 5269 6e67 4f66 496e 7661 ators(RingOfInva │ │ │ │ +00048c10: 7269 616e 7473 293a 2067 656e 6572 6174 riants): generat │ │ │ │ +00048c20: 6f72 735f 6c70 5269 6e67 4f66 496e 7661 ors_lpRingOfInva │ │ │ │ +00048c30: 7269 616e 7473 5f72 702c 202d 2d0a 2020 riants_rp, --. │ │ │ │ +00048c40: 2020 7468 6520 6765 6e65 7261 746f 7273 the generators │ │ │ │ +00048c50: 2066 6f72 2061 2072 696e 6720 6f66 2069 for a ring of i │ │ │ │ +00048c60: 6e76 6172 6961 6e74 730a 2020 2a20 2a6e nvariants. * *n │ │ │ │ +00048c70: 6f74 6520 6869 6c62 6572 7453 6572 6965 ote hilbertSerie │ │ │ │ +00048c80: 7328 5269 6e67 4f66 496e 7661 7269 616e s(RingOfInvarian │ │ │ │ +00048c90: 7473 293a 2068 696c 6265 7274 5365 7269 ts): hilbertSeri │ │ │ │ +00048ca0: 6573 5f6c 7052 696e 674f 6649 6e76 6172 es_lpRingOfInvar │ │ │ │ +00048cb0: 6961 6e74 735f 7270 2c0a 2020 2020 2d2d iants_rp,. -- │ │ │ │ +00048cc0: 2048 696c 6265 7274 2073 6572 6965 7320 Hilbert series │ │ │ │ +00048cd0: 6f66 2074 6865 2069 6e76 6172 6961 6e74 of the invariant │ │ │ │ +00048ce0: 2072 696e 670a 2020 2a20 2a6e 6f74 6520 ring. * *note │ │ │ │ +00048cf0: 6e65 7428 5269 6e67 4f66 496e 7661 7269 net(RingOfInvari │ │ │ │ +00048d00: 616e 7473 293a 206e 6574 5f6c 7052 696e ants): net_lpRin │ │ │ │ +00048d10: 674f 6649 6e76 6172 6961 6e74 735f 7270 gOfInvariants_rp │ │ │ │ +00048d20: 2c20 2d2d 2066 6f72 6d61 7420 666f 720a , -- format for. │ │ │ │ +00048d30: 2020 2020 7072 696e 7469 6e67 2c20 6173 printing, as │ │ │ │ +00048d40: 2061 206e 6574 0a0a 466f 7220 7468 6520 a net..For the │ │ │ │ +00048d50: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ +00048d60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +00048d70: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +00048d80: 5269 6e67 4f66 496e 7661 7269 616e 7473 RingOfInvariants │ │ │ │ +00048d90: 3a20 5269 6e67 4f66 496e 7661 7269 616e : RingOfInvarian │ │ │ │ +00048da0: 7473 2c20 6973 2061 202a 6e6f 7465 2074 ts, is a *note t │ │ │ │ +00048db0: 7970 653a 0a28 4d61 6361 756c 6179 3244 ype:.(Macaulay2D │ │ │ │ +00048dc0: 6f63 2954 7970 652c 2c20 7769 7468 2061 oc)Type,, with a │ │ │ │ +00048dd0: 6e63 6573 746f 7220 636c 6173 7365 7320 ncestor classes │ │ │ │ +00048de0: 2a6e 6f74 6520 4861 7368 5461 626c 653a *note HashTable: │ │ │ │ +00048df0: 0a28 4d61 6361 756c 6179 3244 6f63 2948 .(Macaulay2Doc)H │ │ │ │ +00048e00: 6173 6854 6162 6c65 2c20 3c20 2a6e 6f74 ashTable, < *not │ │ │ │ +00048e10: 6520 5468 696e 673a 2028 4d61 6361 756c e Thing: (Macaul │ │ │ │ +00048e20: 6179 3244 6f63 2954 6869 6e67 2c2e 0a0a ay2Doc)Thing,... │ │ │ │ +00048e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +00048e80: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +00048e90: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +00048ea0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +00048eb0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +00048ec0: 756c 6179 322d 312e 3236 2e30 352b 6473 ulay2-1.26.05+ds │ │ │ │ +00048ed0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +00048ee0: 636b 6167 6573 2f0a 496e 7661 7269 616e ckages/.Invarian │ │ │ │ +00048ef0: 7452 696e 672f 496e 7661 7269 616e 7473 tRing/Invariants │ │ │ │ +00048f00: 446f 632e 6d32 3a39 3532 3a30 2e0a 1f0a Doc.m2:952:0.... │ │ │ │ +00048f10: 4669 6c65 3a20 496e 7661 7269 616e 7452 File: InvariantR │ │ │ │ +00048f20: 696e 672e 696e 666f 2c20 4e6f 6465 3a20 ing.info, Node: │ │ │ │ +00048f30: 7363 6872 6569 6572 4772 6170 682c 204e schreierGraph, N │ │ │ │ +00048f40: 6578 743a 2073 6563 6f6e 6461 7279 496e ext: secondaryIn │ │ │ │ +00048f50: 7661 7269 616e 7473 2c20 5072 6576 3a20 variants, Prev: │ │ │ │ +00048f60: 5269 6e67 4f66 496e 7661 7269 616e 7473 RingOfInvariants │ │ │ │ +00048f70: 2c20 5570 3a20 546f 700a 0a73 6368 7265 , Up: Top..schre │ │ │ │ +00048f80: 6965 7247 7261 7068 202d 2d20 5363 6872 ierGraph -- Schr │ │ │ │ +00048f90: 6569 6572 2067 7261 7068 206f 6620 6120 eier graph of a │ │ │ │ +00048fa0: 6669 6e69 7465 2067 726f 7570 0a2a 2a2a finite group.*** │ │ │ │ +00048fb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00048fc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00048fd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +00048fe0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +00048ff0: 2020 2020 7363 6872 6569 6572 4772 6170 schreierGrap │ │ │ │ +00049000: 6820 470a 2020 2a20 496e 7075 7473 3a0a h G. * Inputs:. │ │ │ │ +00049010: 2020 2020 2020 2a20 472c 2061 6e20 696e * G, an in │ │ │ │ +00049020: 7374 616e 6365 206f 6620 7468 6520 7479 stance of the ty │ │ │ │ +00049030: 7065 202a 6e6f 7465 2046 696e 6974 6547 pe *note FiniteG │ │ │ │ +00049040: 726f 7570 4163 7469 6f6e 3a20 4669 6e69 roupAction: Fini │ │ │ │ +00049050: 7465 4772 6f75 7041 6374 696f 6e2c 2c0a teGroupAction,,. │ │ │ │ +00049060: 2020 2020 2020 2020 6120 6669 6e69 7465 a finite │ │ │ │ +00049070: 2067 726f 7570 2061 6374 696f 6e0a 2020 group action. │ │ │ │ +00049080: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +00049090: 202a 2061 202a 6e6f 7465 2068 6173 6820 * a *note hash │ │ │ │ +000490a0: 7461 626c 653a 2028 4d61 6361 756c 6179 table: (Macaulay │ │ │ │ +000490b0: 3244 6f63 2948 6173 6854 6162 6c65 2c2c 2Doc)HashTable,, │ │ │ │ +000490c0: 2072 6570 7265 7365 6e74 696e 6720 7468 representing th │ │ │ │ +000490d0: 6520 5363 6872 6569 6572 0a20 2020 2020 e Schreier. │ │ │ │ +000490e0: 2020 2067 7261 7068 206f 6620 7468 6520 graph of the │ │ │ │ +000490f0: 6772 6f75 700a 0a44 6573 6372 6970 7469 group..Descripti │ │ │ │ +00049100: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +00049110: 5468 6973 2066 756e 6374 696f 6e20 6973 This function is │ │ │ │ +00049120: 2070 726f 7669 6465 6420 6279 2074 6865 provided by the │ │ │ │ +00049130: 2070 6163 6b61 6765 202a 6e6f 7465 2049 package *note I │ │ │ │ +00049140: 6e76 6172 6961 6e74 5269 6e67 3a20 546f nvariantRing: To │ │ │ │ +00049150: 702c 2e0a 0a46 6f72 2061 2066 696e 6974 p,...For a finit │ │ │ │ +00049160: 6520 6772 6f75 7020 6163 7469 6f6e 2c20 e group action, │ │ │ │ +00049170: 7765 2066 6f72 6d20 6120 2a6e 6f74 6520 we form a *note │ │ │ │ +00049180: 4861 7368 5461 626c 653a 2028 4d61 6361 HashTable: (Maca │ │ │ │ +00049190: 756c 6179 3244 6f63 2948 6173 6854 6162 ulay2Doc)HashTab │ │ │ │ +000491a0: 6c65 2c0a 7768 6f73 6520 6b65 7973 2061 le,.whose keys a │ │ │ │ +000491b0: 7265 2074 6865 2067 656e 6572 6174 6f72 re the generator │ │ │ │ +000491c0: 7320 7072 6f76 6964 6564 2062 7920 7468 s provided by th │ │ │ │ +000491d0: 6520 7573 6572 2e20 5468 6520 7661 6c75 e user. The valu │ │ │ │ +000491e0: 6520 636f 7272 6573 706f 6e64 696e 6720 e corresponding │ │ │ │ +000491f0: 746f 0a61 2067 656e 6572 6174 6f72 2067 to.a generator g │ │ │ │ +00049200: 2069 7320 6120 2a6e 6f74 6520 4861 7368 is a *note Hash │ │ │ │ +00049210: 5461 626c 653a 2028 4d61 6361 756c 6179 Table: (Macaulay │ │ │ │ +00049220: 3244 6f63 2948 6173 6854 6162 6c65 2c20 2Doc)HashTable, │ │ │ │ +00049230: 636f 6e74 6169 6e69 6e67 2061 6c6c 0a70 containing all.p │ │ │ │ +00049240: 6169 7273 2061 203d 3e20 6220 7375 6368 airs a => b such │ │ │ │ +00049250: 2074 6861 7420 612a 6720 3d3d 2062 2e20 that a*g == b. │ │ │ │ +00049260: 5468 6973 2072 6570 7265 7365 6e74 7320 This represents │ │ │ │ +00049270: 7468 6520 5363 6872 6569 6572 2067 7261 the Schreier gra │ │ │ │ +00049280: 7068 206f 6620 7468 650a 6772 6f75 7020 ph of the.group │ │ │ │ +00049290: 7265 6c61 7469 7665 2074 6f20 7468 6520 relative to the │ │ │ │ +000492a0: 6765 6e65 7261 7469 6e67 2073 6574 2070 generating set p │ │ │ │ +000492b0: 726f 7669 6465 6420 6279 2074 6865 2075 rovided by the u │ │ │ │ +000492c0: 7365 722e 0a0a 5468 6520 666f 6c6c 6f77 ser...The follow │ │ │ │ +000492d0: 696e 6720 6578 616d 706c 6520 6465 6669 ing example defi │ │ │ │ +000492e0: 6e65 7320 7468 6520 7065 726d 7574 6174 nes the permutat │ │ │ │ +000492f0: 696f 6e20 6163 7469 6f6e 206f 6620 6120 ion action of a │ │ │ │ +00049300: 7379 6d6d 6574 7269 6320 6772 6f75 7020 symmetric group │ │ │ │ +00049310: 6f6e 0a74 6872 6565 2065 6c65 6d65 6e74 on.three element │ │ │ │ +00049320: 7320 7573 696e 6720 6f6e 6c79 2074 776f s using only two │ │ │ │ +00049330: 2067 656e 6572 6174 6f72 732c 2061 2074 generators, a t │ │ │ │ +00049340: 7261 6e73 706f 7369 7469 6f6e 2061 6e64 ransposition and │ │ │ │ +00049350: 2061 2033 2d63 7963 6c65 2e0a 0a2b 2d2d a 3-cycle...+-- │ │ │ │ +00049360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049380: 2d2d 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2020 │ │ │ │ +00049480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049490: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000494a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000494b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000494c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000494d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000494e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +000494f0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ +00049500: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ +00049510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049530: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00049540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049570: 2d2d 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2020 │ │ │ │ 00049920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00049930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00049940: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00049950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00049960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049940: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +00049950: 203d 2052 203c 2d20 7b7c 2030 2031 2030 = R <- {| 0 1 0 │ │ │ │ +00049960: 207c 2c20 7c20 3020 3020 3120 7c7d 2020 |, | 0 0 1 |} │ │ │ │ 00049970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00049980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00049990: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000499a0: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52 o1 : PolynomialR │ │ │ │ -000499b0: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ +00049990: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000499a0: 2020 2020 2020 2020 207c 2031 2030 2030 | 1 0 0 │ │ │ │ +000499b0: 207c 2020 7c20 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2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0004ab20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ab30: 2020 2020 2020 2020 2020 2020 2020 7c20 | │ │ │ │ -0004ab40: 3020 3120 3020 7c20 2020 207c 2030 2031 0 1 0 | | 0 1 │ │ │ │ -0004ab50: 2030 207c 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ -0004ab60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0004ab70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ab80: 2020 2020 2020 2020 2020 2020 2020 7c20 | │ │ │ │ -0004ab90: 3020 3020 3120 7c20 2020 207c 2031 2030 0 0 1 | | 1 0 │ │ │ │ -0004aba0: 2030 207c 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ -0004abb0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0004abc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a750: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ +0004a760: 203a 2048 6173 6854 6162 6c65 2020 2020 : HashTable │ │ │ │ +0004a770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a7a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0004a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5761 -----------+..Wa │ │ │ │ +0004a800: 7973 2074 6f20 7573 6520 7363 6872 6569 ys to use schrei │ │ │ │ +0004a810: 6572 4772 6170 683a 0a3d 3d3d 3d3d 3d3d erGraph:.======= │ │ │ │ +0004a820: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0004a830: 3d3d 3d0a 0a20 202a 2022 7363 6872 6569 ===.. * "schrei │ │ │ │ +0004a840: 6572 4772 6170 6828 4669 6e69 7465 4772 erGraph(FiniteGr │ │ │ │ +0004a850: 6f75 7041 6374 696f 6e29 220a 0a46 6f72 oupAction)"..For │ │ │ │ +0004a860: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +0004a870: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0004a880: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +0004a890: 6e6f 7465 2073 6368 7265 6965 7247 7261 note schreierGra │ │ │ │ +0004a8a0: 7068 3a20 7363 6872 6569 6572 4772 6170 ph: schreierGrap │ │ │ │ +0004a8b0: 682c 2069 7320 6120 2a6e 6f74 6520 6d65 h, is a *note me │ │ │ │ +0004a8c0: 7468 6f64 2066 756e 6374 696f 6e20 7769 thod function wi │ │ │ │ +0004a8d0: 7468 0a6f 7074 696f 6e73 3a20 284d 6163 th.options: (Mac │ │ │ │ +0004a8e0: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ +0004a8f0: 4675 6e63 7469 6f6e 5769 7468 4f70 7469 FunctionWithOpti │ │ │ │ +0004a900: 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ons,...--------- │ │ │ │ +0004a910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a950: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +0004a960: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +0004a970: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +0004a980: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +0004a990: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +0004a9a0: 362e 3035 2b64 732f 4d32 2f4d 6163 6175 6.05+ds/M2/Macau │ │ │ │ +0004a9b0: 6c61 7932 2f70 6163 6b61 6765 732f 0a49 lay2/packages/.I │ │ │ │ +0004a9c0: 6e76 6172 6961 6e74 5269 6e67 2f46 696e nvariantRing/Fin │ │ │ │ +0004a9d0: 6974 6547 726f 7570 7344 6f63 2e6d 323a iteGroupsDoc.m2: │ │ │ │ +0004a9e0: 3236 323a 302e 0a1f 0a46 696c 653a 2049 262:0....File: I │ │ │ │ +0004a9f0: 6e76 6172 6961 6e74 5269 6e67 2e69 6e66 nvariantRing.inf │ │ │ │ +0004aa00: 6f2c 204e 6f64 653a 2073 6563 6f6e 6461 o, Node: seconda │ │ │ │ +0004aa10: 7279 496e 7661 7269 616e 7473 2c20 4e65 ryInvariants, Ne │ │ │ │ +0004aa20: 7874 3a20 7365 636f 6e64 6172 7949 6e76 xt: secondaryInv │ │ │ │ +0004aa30: 6172 6961 6e74 735f 6c70 5f70 645f 7064 ariants_lp_pd_pd │ │ │ │ +0004aa40: 5f70 645f 636d 5072 696e 7444 6567 7265 _pd_cmPrintDegre │ │ │ │ +0004aa50: 6550 6f6c 796e 6f6d 6961 6c3d 3e5f 7064 ePolynomial=>_pd │ │ │ │ +0004aa60: 5f70 645f 7064 5f72 702c 2050 7265 763a _pd_pd_rp, Prev: │ │ │ │ +0004aa70: 2073 6368 7265 6965 7247 7261 7068 2c20 schreierGraph, │ │ │ │ +0004aa80: 5570 3a20 546f 700a 0a73 6563 6f6e 6461 Up: Top..seconda │ │ │ │ +0004aa90: 7279 496e 7661 7269 616e 7473 202d 2d20 ryInvariants -- │ │ │ │ +0004aaa0: 636f 6d70 7574 6573 2073 6563 6f6e 6461 computes seconda │ │ │ │ +0004aab0: 7279 2069 6e76 6172 6961 6e74 7320 666f ry invariants fo │ │ │ │ +0004aac0: 7220 7468 6520 696e 7661 7269 616e 7420 r the invariant │ │ │ │ +0004aad0: 7269 6e67 206f 6620 6120 6669 6e69 7465 ring of a finite │ │ │ │ +0004aae0: 2067 726f 7570 0a2a 2a2a 2a2a 2a2a 2a2a group.********* │ │ │ │ +0004aaf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004ab00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004ab10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004ab20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004ab30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004ab40: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ +0004ab50: 200a 2020 2020 2020 2020 7365 636f 6e64 . second │ │ │ │ +0004ab60: 6172 7949 6e76 6172 6961 6e74 7328 502c aryInvariants(P, │ │ │ │ +0004ab70: 4729 0a20 202a 2049 6e70 7574 733a 0a20 G). * Inputs:. │ │ │ │ +0004ab80: 2020 2020 202a 2050 2c20 6120 2a6e 6f74 * P, a *not │ │ │ │ +0004ab90: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ +0004aba0: 7932 446f 6329 4c69 7374 2c2c 2061 206c y2Doc)List,, a l │ │ │ │ +0004abb0: 6973 7420 6f66 2070 7269 6d61 7279 2069 ist of primary i │ │ │ │ +0004abc0: 6e76 6172 6961 6e74 7320 696e 206e 0a20 nvariants in n. │ │ │ │ 0004abd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004abe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004abf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ac00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0004ac10: 6f34 203a 2048 6173 6854 6162 6c65 2020 o4 : HashTable │ │ │ │ -0004ac20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ac30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ac40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ac50: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0004ac60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004ac70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004ac80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004ac90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004aca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -0004acb0: 5761 7973 2074 6f20 7573 6520 7363 6872 Ways to use schr │ │ │ │ -0004acc0: 6569 6572 4772 6170 683a 0a3d 3d3d 3d3d eierGraph:.===== │ │ │ │ -0004acd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0004ace0: 3d3d 3d3d 3d0a 0a20 202a 2022 7363 6872 =====.. * "schr │ │ │ │ -0004acf0: 6569 6572 4772 6170 6828 4669 6e69 7465 eierGraph(Finite │ │ │ │ -0004ad00: 4772 6f75 7041 6374 696f 6e29 220a 0a46 GroupAction)"..F │ │ │ │ -0004ad10: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -0004ad20: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -0004ad30: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -0004ad40: 202a 6e6f 7465 2073 6368 7265 6965 7247 *note schreierG │ │ │ │ -0004ad50: 7261 7068 3a20 7363 6872 6569 6572 4772 raph: schreierGr │ │ │ │ -0004ad60: 6170 682c 2069 7320 6120 2a6e 6f74 6520 aph, is a *note │ │ │ │ -0004ad70: 6d65 7468 6f64 2066 756e 6374 696f 6e20 method function │ │ │ │ -0004ad80: 7769 7468 0a6f 7074 696f 6e73 3a20 284d with.options: (M │ │ │ │ -0004ad90: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -0004ada0: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70 odFunctionWithOp │ │ │ │ -0004adb0: 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d tions,...------- │ │ │ │ -0004adc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004add0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004ade0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004adf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004ae00: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -0004ae10: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -0004ae20: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -0004ae30: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -0004ae40: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -0004ae50: 2e32 362e 3035 2b64 732f 4d32 2f4d 6163 .26.05+ds/M2/Mac │ │ │ │ -0004ae60: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -0004ae70: 0a49 6e76 6172 6961 6e74 5269 6e67 2f46 .InvariantRing/F │ │ │ │ -0004ae80: 696e 6974 6547 726f 7570 7344 6f63 2e6d initeGroupsDoc.m │ │ │ │ -0004ae90: 323a 3236 323a 302e 0a1f 0a46 696c 653a 2:262:0....File: │ │ │ │ -0004aea0: 2049 6e76 6172 6961 6e74 5269 6e67 2e69 InvariantRing.i │ │ │ │ -0004aeb0: 6e66 6f2c 204e 6f64 653a 2073 6563 6f6e nfo, Node: secon │ │ │ │ -0004aec0: 6461 7279 496e 7661 7269 616e 7473 2c20 daryInvariants, │ │ │ │ -0004aed0: 4e65 7874 3a20 7365 636f 6e64 6172 7949 Next: secondaryI │ │ │ │ -0004aee0: 6e76 6172 6961 6e74 735f 6c70 5f70 645f nvariants_lp_pd_ │ │ │ │ -0004aef0: 7064 5f70 645f 636d 5072 696e 7444 6567 pd_pd_cmPrintDeg │ │ │ │ -0004af00: 7265 6550 6f6c 796e 6f6d 6961 6c3d 3e5f reePolynomial=>_ │ │ │ │ -0004af10: 7064 5f70 645f 7064 5f72 702c 2050 7265 pd_pd_pd_rp, Pre │ │ │ │ -0004af20: 763a 2073 6368 7265 6965 7247 7261 7068 v: schreierGraph │ │ │ │ -0004af30: 2c20 5570 3a20 546f 700a 0a73 6563 6f6e , Up: Top..secon │ │ │ │ -0004af40: 6461 7279 496e 7661 7269 616e 7473 202d daryInvariants - │ │ │ │ -0004af50: 2d20 636f 6d70 7574 6573 2073 6563 6f6e - computes secon │ │ │ │ -0004af60: 6461 7279 2069 6e76 6172 6961 6e74 7320 dary invariants │ │ │ │ -0004af70: 666f 7220 7468 6520 696e 7661 7269 616e for the invarian │ │ │ │ -0004af80: 7420 7269 6e67 206f 6620 6120 6669 6e69 t ring of a fini │ │ │ │ -0004af90: 7465 2067 726f 7570 0a2a 2a2a 2a2a 2a2a te group.******* │ │ │ │ -0004afa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004afb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004afc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004afd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004afe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004aff0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -0004b000: 653a 200a 2020 2020 2020 2020 7365 636f e: . seco │ │ │ │ -0004b010: 6e64 6172 7949 6e76 6172 6961 6e74 7328 ndaryInvariants( │ │ │ │ -0004b020: 502c 4729 0a20 202a 2049 6e70 7574 733a P,G). * Inputs: │ │ │ │ -0004b030: 0a20 2020 2020 202a 2050 2c20 6120 2a6e . * P, a *n │ │ │ │ -0004b040: 6f74 6520 6c69 7374 3a20 284d 6163 6175 ote list: (Macau │ │ │ │ -0004b050: 6c61 7932 446f 6329 4c69 7374 2c2c 2061 lay2Doc)List,, a │ │ │ │ -0004b060: 206c 6973 7420 6f66 2070 7269 6d61 7279 list of primary │ │ │ │ -0004b070: 2069 6e76 6172 6961 6e74 7320 696e 206e invariants in n │ │ │ │ -0004b080: 0a20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -0004b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004b0a0: 2020 2020 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K[f ,. │ │ │ │ +0004b020: 2e2e 2c66 205d 2d6d 6f64 756c 6520 7769 ..,f ]-module wi │ │ │ │ +0004b030: 7468 2062 6173 6973 2053 0a20 2020 2020 th basis S. │ │ │ │ +0004b040: 2020 2020 2020 3120 2020 2020 206e 0a0a 1 n.. │ │ │ │ +0004b050: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +0004b060: 3d3d 3d3d 3d3d 3d0a 0a54 6865 2065 7861 =======..The exa │ │ │ │ +0004b070: 6d70 6c65 2062 656c 6f77 2063 6f6d 7075 mple below compu │ │ │ │ +0004b080: 7465 7320 7468 6520 7365 636f 6e64 6172 tes the secondar │ │ │ │ +0004b090: 7920 696e 7661 7269 616e 7473 2066 6f72 y invariants for │ │ │ │ +0004b0a0: 2074 6865 2064 6968 6564 7261 6c20 6772 the dihedral gr │ │ │ │ +0004b0b0: 6f75 7020 7769 7468 0a36 2065 6c65 6d65 oup with.6 eleme │ │ │ │ +0004b0c0: 6e74 732c 2067 6976 656e 2061 2073 6574 nts, given a set │ │ │ │ +0004b0d0: 206f 6620 7072 696d 6172 7920 696e 7661 of primary inva │ │ │ │ +0004b0e0: 7269 616e 7473 502e 0a0a 2b2d 2d2d 2d2d riantsP...+----- │ │ │ │ +0004b0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004b100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0004b110: 6931 203a 204b 3d74 6f46 6965 6c64 2851 i1 : K=toField(Q │ │ │ │ +0004b120: 515b 615d 2f28 615e 322b 612b 3129 293b Q[a]/(a^2+a+1)); 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K[f │ │ │ │ -0004b4d0: 2c2e 2e2e 2c66 205d 2d6d 6f64 756c 6520 ,...,f ]-module │ │ │ │ -0004b4e0: 7769 7468 2062 6173 6973 2053 0a20 2020 with basis S. │ │ │ │ -0004b4f0: 2020 2020 2020 2020 3120 2020 2020 206e 1 n │ │ │ │ -0004b500: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -0004b510: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 2065 =========..The e │ │ │ │ -0004b520: 7861 6d70 6c65 2062 656c 6f77 2063 6f6d xample below com │ │ │ │ -0004b530: 7075 7465 7320 7468 6520 7365 636f 6e64 putes the second │ │ │ │ -0004b540: 6172 7920 696e 7661 7269 616e 7473 2066 ary invariants f │ │ │ │ -0004b550: 6f72 2074 6865 2064 6968 6564 7261 6c20 or the dihedral │ │ │ │ -0004b560: 6772 6f75 7020 7769 7468 0a36 2065 6c65 group with.6 ele │ │ │ │ -0004b570: 6d65 6e74 732c 2067 6976 656e 2061 2073 ments, given a s │ │ │ │ -0004b580: 6574 206f 6620 7072 696d 6172 7920 696e et of primary in │ │ │ │ -0004b590: 7661 7269 616e 7473 502e 0a0a 2b2d 2d2d variantsP...+--- │ │ │ │ -0004b5a0: 2d2d 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2032 2020 2 2 │ │ │ │ -0004b770: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ -0004b780: 203a 204d 6174 7269 7820 4b20 203c 2d2d : Matrix K <-- │ │ │ │ -0004b790: 204b 2020 2020 2020 2020 2020 2020 2020 K │ │ │ │ -0004b7a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0004b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004b7c0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2044 363d -----+.|i5 : D6= │ │ │ │ -0004b7d0: 6669 6e69 7465 4163 7469 6f6e 287b 412c finiteAction({A, │ │ │ │ -0004b7e0: 427d 2c52 2920 2020 2020 7c0a 7c20 2020 B},R) |.| │ │ │ │ -0004b7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004b800: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0004b810: 0a7c 6f35 203d 2052 203c 2d20 7b7c 2061 .|o5 = R <- {| a │ │ │ │ -0004b820: 2030 2020 2020 7c2c 207c 2030 2031 207c 0 |, | 0 1 | │ │ │ │ -0004b830: 7d20 2020 7c0a 7c20 2020 2020 2020 2020 } |.| │ │ │ │ -0004b840: 2020 7c20 3020 2d61 2d31 207c 2020 7c20 | 0 -a-1 | | │ │ │ │ -0004b850: 3120 3020 7c20 2020 207c 0a7c 2020 2020 1 0 | |.| │ │ │ │ -0004b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004b870: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0004b880: 7c6f 3520 3a20 4669 6e69 7465 4772 6f75 |o5 : FiniteGrou │ │ │ │ -0004b890: 7041 6374 696f 6e20 2020 2020 2020 2020 pAction │ │ │ │ -0004b8a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0004b8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004b8c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ -0004b8d0: 503d 7b78 5e33 2b79 5e33 2c2d 2878 5e33 P={x^3+y^3,-(x^3 │ │ │ │ -0004b8e0: 2d79 5e33 295e 327d 3b20 2020 207c 0a2b -y^3)^2}; |.+ │ │ │ │ -0004b8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004b900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004b910: 2d2d 2b0a 7c69 3720 3a20 7365 636f 6e64 --+.|i7 : second │ │ │ │ -0004b920: 6172 7949 6e76 6172 6961 6e74 7328 502c aryInvariants(P, │ │ │ │ -0004b930: 4436 2920 2020 207c 0a7c 2020 2020 2020 D6) |.| │ │ │ │ -0004b940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004b950: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0004b960: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -0004b970: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0004b980: 207c 0a7c 6f37 203d 207b 312c 2078 2a79 |.|o7 = {1, x*y │ │ │ │ -0004b990: 2c20 7820 7920 7d20 2020 2020 2020 2020 , x y } │ │ │ │ -0004b9a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0004b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004b9c0: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -0004b9d0: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ -0004b9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004b9f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0004ba00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004ba10: 2d2d 2d2d 2d2b 0a0a 5468 6973 2066 756e -----+..This fun │ │ │ │ -0004ba20: 6374 696f 6e20 6973 2070 726f 7669 6465 ction is provide │ │ │ │ -0004ba30: 6420 6279 2074 6865 2070 6163 6b61 6765 d by the package │ │ │ │ -0004ba40: 202a 6e6f 7465 2049 6e76 6172 6961 6e74 *note Invariant │ │ │ │ -0004ba50: 5269 6e67 3a20 546f 702c 2e0a 0a43 6176 Ring: Top,...Cav │ │ │ │ -0004ba60: 6561 740a 3d3d 3d3d 3d3d 0a0a 4375 7272 eat.======..Curr │ │ │ │ -0004ba70: 656e 746c 792c 2061 2075 7365 7220 6e65 ently, a user ne │ │ │ │ -0004ba80: 6564 7320 746f 2065 6e73 7572 6520 7468 eds to ensure th │ │ │ │ -0004ba90: 6174 2074 6865 2061 6c6c 2070 7269 6d61 at the all prima │ │ │ │ -0004baa0: 7279 2069 6e76 6172 6961 6e74 7320 6172 ry invariants ar │ │ │ │ -0004bab0: 6520 6465 6669 6e65 640a 7769 7468 2063 e defined.with c │ │ │ │ -0004bac0: 6f65 6666 6963 6965 6e74 7320 696e 2061 oefficients in a │ │ │ │ -0004bad0: 2072 696e 6720 7468 6174 204d 6163 6175 ring that Macau │ │ │ │ -0004bae0: 6c61 7932 2072 6563 6f67 6e69 7365 7320 lay2 recognises │ │ │ │ -0004baf0: 6173 2061 2063 6861 7261 6374 6572 6973 as a characteris │ │ │ │ -0004bb00: 7469 6320 7a65 726f 0a66 6965 6c64 2028 tic zero.field ( │ │ │ │ -0004bb10: 7365 6520 2a6e 6f74 6520 746f 4669 656c see *note toFiel │ │ │ │ -0004bb20: 643a 2028 4d61 6361 756c 6179 3244 6f63 d: (Macaulay2Doc │ │ │ │ -0004bb30: 2974 6f46 6965 6c64 5f6c 7052 696e 675f )toField_lpRing_ │ │ │ │ -0004bb40: 7270 2c20 666f 7220 6120 7761 7920 746f rp, for a way to │ │ │ │ -0004bb50: 2064 6f0a 7468 6973 292e 0a0a 4e6f 7465 do.this)...Note │ │ │ │ -0004bb60: 2061 6c73 6f20 7468 6174 2074 6865 2066 also that the f │ │ │ │ -0004bb70: 756e 6374 696f 6e20 2a6e 6f74 6520 7365 unction *note se │ │ │ │ -0004bb80: 636f 6e64 6172 7949 6e76 6172 6961 6e74 condaryInvariant │ │ │ │ -0004bb90: 733a 2073 6563 6f6e 6461 7279 496e 7661 s: secondaryInva │ │ │ │ -0004bba0: 7269 616e 7473 2c0a 6f6e 6c79 2077 6f72 riants,.only wor │ │ │ │ -0004bbb0: 6b73 2077 6865 6e20 5220 6973 2064 6566 ks when R is def │ │ │ │ -0004bbc0: 696e 6564 206f 7665 7220 6120 6669 656c ined over a fiel │ │ │ │ -0004bbd0: 6420 6f66 2063 6861 7261 6374 6572 6973 d of characteris │ │ │ │ -0004bbe0: 7469 6320 7a65 726f 2e0a 0a57 6179 7320 tic zero...Ways │ │ │ │ -0004bbf0: 746f 2075 7365 2073 6563 6f6e 6461 7279 to use secondary │ │ │ │ -0004bc00: 496e 7661 7269 616e 7473 3a0a 3d3d 3d3d Invariants:.==== │ │ │ │ -0004bc10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0004bc20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ -0004bc30: 2a20 2273 6563 6f6e 6461 7279 496e 7661 * "secondaryInva │ │ │ │ -0004bc40: 7269 616e 7473 284c 6973 742c 4669 6e69 riants(List,Fini │ │ │ │ -0004bc50: 7465 4772 6f75 7041 6374 696f 6e29 220a teGroupAction)". │ │ │ │ -0004bc60: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ -0004bc70: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ -0004bc80: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ -0004bc90: 6374 202a 6e6f 7465 2073 6563 6f6e 6461 ct *note seconda │ │ │ │ -0004bca0: 7279 496e 7661 7269 616e 7473 3a20 7365 ryInvariants: se │ │ │ │ -0004bcb0: 636f 6e64 6172 7949 6e76 6172 6961 6e74 condaryInvariant │ │ │ │ -0004bcc0: 732c 2069 7320 6120 2a6e 6f74 6520 6d65 s, is a *note me │ │ │ │ -0004bcd0: 7468 6f64 0a66 756e 6374 696f 6e20 7769 thod.function wi │ │ │ │ -0004bce0: 7468 206f 7074 696f 6e73 3a20 284d 6163 th options: (Mac │ │ │ │ -0004bcf0: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -0004bd00: 4675 6e63 7469 6f6e 5769 7468 4f70 7469 FunctionWithOpti │ │ │ │ -0004bd10: 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ons,...--------- │ │ │ │ -0004bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004bd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004bd60: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -0004bd70: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -0004bd80: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -0004bd90: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -0004bda0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -0004bdb0: 362e 3035 2b64 732f 4d32 2f4d 6163 6175 6.05+ds/M2/Macau │ │ │ │ -0004bdc0: 6c61 7932 2f70 6163 6b61 6765 732f 0a49 lay2/packages/.I │ │ │ │ -0004bdd0: 6e76 6172 6961 6e74 5269 6e67 2f48 6177 nvariantRing/Haw │ │ │ │ -0004bde0: 6573 446f 632e 6d32 3a34 3837 3a30 2e0a esDoc.m2:487:0.. │ │ │ │ -0004bdf0: 1f0a 4669 6c65 3a20 496e 7661 7269 616e ..File: Invarian │ │ │ │ -0004be00: 7452 696e 672e 696e 666f 2c20 4e6f 6465 tRing.info, Node │ │ │ │ -0004be10: 3a20 7365 636f 6e64 6172 7949 6e76 6172 : secondaryInvar │ │ │ │ -0004be20: 6961 6e74 735f 6c70 5f70 645f 7064 5f70 iants_lp_pd_pd_p │ │ │ │ -0004be30: 645f 636d 5072 696e 7444 6567 7265 6550 d_cmPrintDegreeP │ │ │ │ -0004be40: 6f6c 796e 6f6d 6961 6c3d 3e5f 7064 5f70 olynomial=>_pd_p │ │ │ │ -0004be50: 645f 7064 5f72 702c 204e 6578 743a 2077 d_pd_rp, Next: w │ │ │ │ -0004be60: 6569 6768 7473 2c20 5072 6576 3a20 7365 eights, Prev: se │ │ │ │ -0004be70: 636f 6e64 6172 7949 6e76 6172 6961 6e74 condaryInvariant │ │ │ │ -0004be80: 732c 2055 703a 2054 6f70 0a0a 7365 636f s, Up: Top..seco │ │ │ │ -0004be90: 6e64 6172 7949 6e76 6172 6961 6e74 7328 ndaryInvariants( │ │ │ │ -0004bea0: 2e2e 2e2c 5072 696e 7444 6567 7265 6550 ...,PrintDegreeP │ │ │ │ -0004beb0: 6f6c 796e 6f6d 6961 6c3d 3e2e 2e2e 2920 olynomial=>...) │ │ │ │ -0004bec0: 2d2d 2061 6e20 6f70 7469 6f6e 616c 2061 -- an optional a │ │ │ │ -0004bed0: 7267 756d 656e 7420 666f 7220 7365 636f rgument for seco │ │ │ │ -0004bee0: 6e64 6172 7949 6e76 6172 6961 6e74 7320 ndaryInvariants │ │ │ │ -0004bef0: 7468 6174 2064 6574 6572 6d69 6e65 7320 that determines │ │ │ │ -0004bf00: 7468 6520 7072 696e 7469 6e67 206f 6620 the printing of │ │ │ │ -0004bf10: 616e 2069 6e66 6f72 6d61 7469 7665 2070 an informative p │ │ │ │ -0004bf20: 6f6c 796e 6f6d 6961 6c0a 2a2a 2a2a 2a2a olynomial.****** │ │ │ │ -0004bf30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004bf40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004bf50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004bf60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004bf70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004bf80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004bf90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004bfa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004bfb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004bfc0: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ -0004bfd0: 6765 3a20 0a20 2020 2020 2020 2073 6563 ge: . sec │ │ │ │ -0004bfe0: 6f6e 6461 7279 496e 7661 7269 616e 7473 ondaryInvariants │ │ │ │ -0004bff0: 2850 2c47 290a 2020 2a20 496e 7075 7473 (P,G). * Inputs │ │ │ │ -0004c000: 3a0a 2020 2020 2020 2a20 502c 2061 202a :. * P, a * │ │ │ │ -0004c010: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ -0004c020: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ -0004c030: 6120 6c69 7374 206f 6620 7072 696d 6172 a list of primar │ │ │ │ -0004c040: 7920 696e 7661 7269 616e 7473 2069 6e20 y invariants in │ │ │ │ -0004c050: 6e0a 2020 2020 2020 2020 2020 2020 2020 n. │ │ │ │ -0004c060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004c080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004c0a0: 200a 2020 2020 2020 2020 7661 7269 6162 . variab │ │ │ │ -0004c0b0: 6c65 7320 6620 2c2e 2e2e 2c66 2020 666f les f ,...,f fo │ │ │ │ -0004c0c0: 7220 7468 6520 696e 7661 7269 616e 7420 r the invariant │ │ │ │ -0004c0d0: 7269 6e67 206f 6620 4720 6465 6669 6e65 ring of G define │ │ │ │ -0004c0e0: 6420 6f76 6572 2061 2066 6965 6c64 204b d over a field K │ │ │ │ -0004c0f0: 0a20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -0004c100: 2020 2020 3120 2020 2020 206e 0a20 2020 1 n. │ │ │ │ -0004c110: 2020 2020 206f 6620 6368 6172 6163 7465 of characte │ │ │ │ -0004c120: 7269 7374 6963 207a 6572 6f0a 2020 2020 ristic zero. │ │ │ │ -0004c130: 2020 2a20 472c 2061 6e20 696e 7374 616e * G, an instan │ │ │ │ -0004c140: 6365 206f 6620 7468 6520 7479 7065 202a ce of the type * │ │ │ │ -0004c150: 6e6f 7465 2046 696e 6974 6547 726f 7570 note FiniteGroup │ │ │ │ -0004c160: 4163 7469 6f6e 3a20 4669 6e69 7465 4772 Action: FiniteGr │ │ │ │ -0004c170: 6f75 7041 6374 696f 6e2c 0a20 202a 204f oupAction,. * O │ │ │ │ -0004c180: 7574 7075 7473 3a0a 2020 2020 2020 2020 utputs:. │ │ │ │ +0004b350: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0004b360: 6f35 203d 2052 203c 2d20 7b7c 2061 2030 o5 = R <- {| a 0 │ │ │ │ +0004b370: 2020 2020 7c2c 207c 2030 2031 207c 7d20 |, | 0 1 |} │ │ │ │ +0004b380: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004b390: 7c20 3020 2d61 2d31 207c 2020 7c20 3120 | 0 -a-1 | | 1 │ │ │ │ +0004b3a0: 3020 7c20 2020 207c 0a7c 2020 2020 2020 0 | |.| │ │ │ │ +0004b3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004b3c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0004b3d0: 3520 3a20 4669 6e69 7465 4772 6f75 7041 5 : FiniteGroupA │ │ │ │ +0004b3e0: 6374 696f 6e20 2020 2020 2020 2020 2020 ction │ │ │ │ +0004b3f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0004b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004b410: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 503d ------+.|i6 : P= │ │ │ │ +0004b420: 7b78 5e33 2b79 5e33 2c2d 2878 5e33 2d79 {x^3+y^3,-(x^3-y │ │ │ │ +0004b430: 5e33 295e 327d 3b20 2020 207c 0a2b 2d2d ^3)^2}; |.+-- │ │ │ │ +0004b440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004b450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004b460: 2b0a 7c69 3720 3a20 7365 636f 6e64 6172 +.|i7 : secondar │ │ │ │ +0004b470: 7949 6e76 6172 6961 6e74 7328 502c 4436 yInvariants(P,D6 │ │ │ │ +0004b480: 2920 2020 207c 0a7c 2020 2020 2020 2020 ) |.| │ │ │ │ +0004b490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004b4a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0004b4b0: 2020 2020 2020 2020 2020 2020 3220 3220 2 2 │ │ │ │ +0004b4c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0004b4d0: 0a7c 6f37 203d 207b 312c 2078 2a79 2c20 .|o7 = {1, x*y, │ │ │ │ +0004b4e0: 7820 7920 7d20 2020 2020 2020 2020 2020 x y } │ │ │ │ +0004b4f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0004b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004b510: 2020 2020 2020 2020 207c 0a7c 6f37 203a |.|o7 : │ │ │ │ +0004b520: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ +0004b530: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0004b540: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0004b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004b560: 2d2d 2d2b 0a0a 5468 6973 2066 756e 6374 ---+..This funct │ │ │ │ +0004b570: 696f 6e20 6973 2070 726f 7669 6465 6420 ion is provided │ │ │ │ +0004b580: 6279 2074 6865 2070 6163 6b61 6765 202a by the package * │ │ │ │ +0004b590: 6e6f 7465 2049 6e76 6172 6961 6e74 5269 note InvariantRi │ │ │ │ +0004b5a0: 6e67 3a20 546f 702c 2e0a 0a43 6176 6561 ng: Top,...Cavea │ │ │ │ +0004b5b0: 740a 3d3d 3d3d 3d3d 0a0a 4375 7272 656e t.======..Curren │ │ │ │ +0004b5c0: 746c 792c 2061 2075 7365 7220 6e65 6564 tly, a user need │ │ │ │ +0004b5d0: 7320 746f 2065 6e73 7572 6520 7468 6174 s to ensure that │ │ │ │ +0004b5e0: 2074 6865 2061 6c6c 2070 7269 6d61 7279 the all primary │ │ │ │ +0004b5f0: 2069 6e76 6172 6961 6e74 7320 6172 6520 invariants are │ │ │ │ +0004b600: 6465 6669 6e65 640a 7769 7468 2063 6f65 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G │ │ │ │ +0004c8b0: 2020 2020 6420 2020 2020 2020 2020 2020 d │ │ │ │ +0004c8c0: 640a 4828 5320 2c54 292a 2831 2d54 2031 d.H(S ,T)*(1-T 1 │ │ │ │ +0004c8d0: 292a 2e2e 2e2a 2831 2d54 206e 292e 0a0a )*...*(1-T n)... │ │ │ │ +0004c8e0: 5468 6520 6578 616d 706c 6520 6265 6c6f The example belo │ │ │ │ +0004c8f0: 7720 636f 6d70 7574 6573 2074 6865 2073 w computes the s │ │ │ │ +0004c900: 6563 6f6e 6461 7279 2069 6e76 6172 6961 econdary invaria │ │ │ │ +0004c910: 6e74 7320 666f 7220 7468 6520 6469 6865 nts for the dihe │ │ │ │ +0004c920: 6472 616c 2067 726f 7570 2077 6974 680a dral group with. │ │ │ │ +0004c930: 3620 656c 656d 656e 7473 2c20 6769 7665 6 elements, give │ │ │ │ +0004c940: 6e20 6120 7365 7420 6f66 2070 7269 6d61 n a set of prima │ │ │ │ +0004c950: 7279 2069 6e76 6172 6961 6e74 7320 502e ry invariants P. │ │ │ │ +0004c960: 2054 6865 206f 7074 696f 6e61 6c20 6172 The optional ar │ │ │ │ +0004c970: 6775 6d65 6e74 202a 6e6f 7465 0a50 7269 gument *note.Pri │ │ │ │ +0004c980: 6e74 4465 6772 6565 506f 6c79 6e6f 6d69 ntDegreePolynomi │ │ │ │ +0004c990: 616c 3a20 7365 636f 6e64 6172 7949 6e76 al: secondaryInv │ │ │ │ +0004c9a0: 6172 6961 6e74 735f 6c70 5f70 645f 7064 ariants_lp_pd_pd │ │ │ │ +0004c9b0: 5f70 645f 636d 5072 696e 7444 6567 7265 _pd_cmPrintDegre │ │ │ │ +0004c9c0: 6550 6f6c 796e 6f6d 6961 6c0a 3d3e 5f70 ePolynomial.=>_p │ │ │ │ +0004c9d0: 645f 7064 5f70 645f 7270 2c20 6973 2073 d_pd_pd_rp, is s │ │ │ │ +0004c9e0: 6574 2074 6f20 2a6e 6f74 6520 7472 7565 et to *note true │ │ │ │ +0004c9f0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0004ca00: 7472 7565 2c20 696e 206f 7264 6572 2074 true, in order t │ │ │ │ +0004ca10: 6f20 7365 6520 7768 6963 680a 6465 6772 o see which.degr │ │ │ │ +0004ca20: 6565 7320 7468 6520 7365 636f 6e64 6172 ees the secondar │ │ │ │ +0004ca30: 7920 696e 7661 7269 616e 7473 2073 686f y invariants sho │ │ │ │ +0004ca40: 756c 6420 6861 7665 2e0a 0a2b 2d2d 2d2d uld have...+---- │ │ │ │ +0004ca50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004ca80: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 4b3d ------+.|i1 : K= │ │ │ │ +0004ca90: 746f 4669 656c 6428 5151 5b61 5d2f 2861 toField(QQ[a]/(a │ │ │ │ +0004caa0: 5e32 2b61 2b31 2929 3b20 2020 2020 2020 ^2+a+1)); │ │ │ │ +0004cab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cac0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0004cad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004cae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004caf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004cb00: 2b0a 7c69 3220 3a20 523d 4b5b 782c 795d +.|i2 : R=K[x,y] │ │ │ │ +0004cb10: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +0004cb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cb30: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0004cb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004cb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004cb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004cb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ +0004cb80: 3a20 413d 6d61 7472 6978 7b7b 612c 307d : A=matrix{{a,0} │ │ │ │ +0004cb90: 2c7b 302c 615e 327d 7d3b 2020 2020 2020 ,{0,a^2}}; │ │ │ │ 0004cba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004cbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004cbc0: 2020 2020 2020 2047 2020 2020 2020 2020 G │ │ │ │ -0004cbd0: 2020 2020 2020 2020 2020 2020 200a 696e .in │ │ │ │ -0004cbe0: 7661 7269 616e 7473 206f 6620 6465 6772 variants of degr │ │ │ │ -0004cbf0: 6565 7320 6420 2c2e 2e2e 2c64 2020 666f ees d ,...,d fo │ │ │ │ -0004cc00: 7220 616e 2069 6e76 6172 6961 6e74 2072 r an invariant r │ │ │ │ -0004cc10: 696e 6720 5320 206f 6620 6120 6669 6e69 ing S of a fini │ │ │ │ -0004cc20: 7465 2067 726f 7570 2047 2c0a 2020 2020 te group G,. │ │ │ │ -0004cc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004cc40: 2020 2031 2020 2020 2020 6e0a 2020 2020 1 n. │ │ │ │ -0004cc50: 2020 2047 2020 2020 2020 2020 2020 2020 G │ │ │ │ -0004cc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004cc70: 2020 2020 2020 2020 2020 2020 2020 2047 G │ │ │ │ -0004cc80: 2020 2020 2020 2020 2020 2020 0a61 6e64 .and │ │ │ │ -0004cc90: 2048 2853 202c 5429 6465 6e6f 7465 7320 H(S ,T)denotes │ │ │ │ -0004cca0: 7468 6520 4d6f 6c69 656e 2028 4869 6c62 the Molien (Hilb │ │ │ │ -0004ccb0: 6572 7429 2073 6572 6965 7320 6f66 2053 ert) series of S │ │ │ │ -0004ccc0: 202c 2074 6865 6e20 2a6e 6f74 650a 2020 , then *note. │ │ │ │ +0004cbb0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0004cbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cbf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0004cc00: 2020 2020 3220 2020 2020 2032 2020 2020 2 2 │ │ │ │ +0004cc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cc30: 207c 0a7c 6f33 203a 204d 6174 7269 7820 |.|o3 : Matrix │ │ │ │ +0004cc40: 4b20 203c 2d2d 204b 2020 2020 2020 2020 K <-- K │ │ │ │ +0004cc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cc60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0004cc70: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0004cc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004cc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004cca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ +0004ccb0: 203a 2042 3d73 7562 286d 6174 7269 787b : B=sub(matrix{ │ │ │ │ +0004ccc0: 7b30 2c31 7d2c 7b31 2c30 7d7d 2c4b 293b {0,1},{1,0}},K); │ │ │ │ 0004ccd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004cce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cce0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0004ccf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004cd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004cd10: 2020 200a 7365 636f 6e64 6172 7949 6e76 .secondaryInv │ │ │ │ -0004cd20: 6172 6961 6e74 733a 2073 6563 6f6e 6461 ariants: seconda │ │ │ │ -0004cd30: 7279 496e 7661 7269 616e 7473 2c20 7769 ryInvariants, wi │ │ │ │ -0004cd40: 6c6c 2063 6f6d 7075 7465 2074 6865 2070 ll compute the p │ │ │ │ -0004cd50: 6f6c 796e 6f6d 6961 6c0a 2020 2047 2020 olynomial. G │ │ │ │ -0004cd60: 2020 2020 2020 6420 2020 2020 2020 2020 d │ │ │ │ -0004cd70: 2020 640a 4828 5320 2c54 292a 2831 2d54 d.H(S ,T)*(1-T │ │ │ │ -0004cd80: 2031 292a 2e2e 2e2a 2831 2d54 206e 292e 1)*...*(1-T n). │ │ │ │ -0004cd90: 0a0a 5468 6520 6578 616d 706c 6520 6265 ..The example be │ │ │ │ -0004cda0: 6c6f 7720 636f 6d70 7574 6573 2074 6865 low computes the │ │ │ │ -0004cdb0: 2073 6563 6f6e 6461 7279 2069 6e76 6172 secondary invar │ │ │ │ -0004cdc0: 6961 6e74 7320 666f 7220 7468 6520 6469 iants for the di │ │ │ │ -0004cdd0: 6865 6472 616c 2067 726f 7570 2077 6974 hedral group wit │ │ │ │ -0004cde0: 680a 3620 656c 656d 656e 7473 2c20 6769 h.6 elements, gi │ │ │ │ -0004cdf0: 7665 6e20 6120 7365 7420 6f66 2070 7269 ven a set of pri │ │ │ │ -0004ce00: 6d61 7279 2069 6e76 6172 6961 6e74 7320 mary invariants │ │ │ │ -0004ce10: 502e 2054 6865 206f 7074 696f 6e61 6c20 P. The optional │ │ │ │ -0004ce20: 6172 6775 6d65 6e74 202a 6e6f 7465 0a50 argument *note.P │ │ │ │ -0004ce30: 7269 6e74 4465 6772 6565 506f 6c79 6e6f rintDegreePolyno │ │ │ │ -0004ce40: 6d69 616c 3a20 7365 636f 6e64 6172 7949 mial: secondaryI │ │ │ │ -0004ce50: 6e76 6172 6961 6e74 735f 6c70 5f70 645f nvariants_lp_pd_ │ │ │ │ -0004ce60: 7064 5f70 645f 636d 5072 696e 7444 6567 pd_pd_cmPrintDeg │ │ │ │ -0004ce70: 7265 6550 6f6c 796e 6f6d 6961 6c0a 3d3e reePolynomial.=> │ │ │ │ -0004ce80: 5f70 645f 7064 5f70 645f 7270 2c20 6973 _pd_pd_pd_rp, is │ │ │ │ -0004ce90: 2073 6574 2074 6f20 2a6e 6f74 6520 7472 set to *note tr │ │ │ │ -0004cea0: 7565 3a20 284d 6163 6175 6c61 7932 446f ue: (Macaulay2Do │ │ │ │ -0004ceb0: 6329 7472 7565 2c20 696e 206f 7264 6572 c)true, in order │ │ │ │ -0004cec0: 2074 6f20 7365 6520 7768 6963 680a 6465 to see which.de │ │ │ │ -0004ced0: 6772 6565 7320 7468 6520 7365 636f 6e64 grees the second │ │ │ │ -0004cee0: 6172 7920 696e 7661 7269 616e 7473 2073 ary invariants s │ │ │ │ -0004cef0: 686f 756c 6420 6861 7665 2e0a 0a2b 2d2d hould have...+-- │ │ │ │ -0004cf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004cf10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004cf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004cf30: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -0004cf40: 4b3d 746f 4669 656c 6428 5151 5b61 5d2f K=toField(QQ[a]/ │ │ │ │ -0004cf50: 2861 5e32 2b61 2b31 2929 3b20 2020 2020 (a^2+a+1)); │ │ │ │ -0004cf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004cf70: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0004cf80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004cf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004cfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004cfb0: 2d2d 2b0a 7c69 3220 3a20 523d 4b5b 782c --+.|i2 : R=K[x, │ │ │ │ -0004cfc0: 795d 3b20 2020 2020 2020 2020 2020 2020 y]; │ │ │ │ -0004cfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004cfe0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0004cff0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0004d000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0004d030: 3320 3a20 413d 6d61 7472 6978 7b7b 612c 3 : A=matrix{{a, │ │ │ │ -0004d040: 307d 2c7b 302c 615e 327d 7d3b 2020 2020 0},{0,a^2}}; │ │ │ │ +0004cd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cd20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0004cd30: 2020 2020 2032 2020 2020 2020 3220 2020 2 2 │ │ │ │ +0004cd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cd60: 2020 7c0a 7c6f 3420 3a20 4d61 7472 6978 |.|o4 : Matrix │ │ │ │ +0004cd70: 204b 2020 3c2d 2d20 4b20 2020 2020 2020 K <-- K │ │ │ │ +0004cd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cd90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0004cda0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0004cdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004cdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0004cde0: 3520 3a20 4436 3d66 696e 6974 6541 6374 5 : D6=finiteAct │ │ │ │ +0004cdf0: 696f 6e28 7b41 2c42 7d2c 5229 2020 2020 ion({A,B},R) │ │ │ │ +0004ce00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ce10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0004ce20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ce30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ce40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ce50: 2020 2020 2020 7c0a 7c6f 3520 3d20 5220 |.|o5 = R │ │ │ │ +0004ce60: 3c2d 207b 7c20 6120 3020 2020 207c 2c20 <- {| a 0 |, │ │ │ │ +0004ce70: 7c20 3020 3120 7c7d 2020 2020 2020 2020 | 0 1 |} │ │ │ │ +0004ce80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ce90: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004cea0: 207c 2030 202d 612d 3120 7c20 207c 2031 | 0 -a-1 | | 1 │ │ │ │ +0004ceb0: 2030 207c 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ +0004cec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ced0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004cee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cf00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0004cf10: 6f35 203a 2046 696e 6974 6547 726f 7570 o5 : FiniteGroup │ │ │ │ +0004cf20: 4163 7469 6f6e 2020 2020 2020 2020 2020 Action │ │ │ │ +0004cf30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cf40: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0004cf50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004cf60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004cf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004cf80: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2050 -------+.|i6 : P │ │ │ │ +0004cf90: 3d7b 785e 332b 795e 332c 2d28 785e 332d ={x^3+y^3,-(x^3- │ │ │ │ +0004cfa0: 795e 3329 5e32 7d3b 2020 2020 2020 2020 y^3)^2}; │ │ │ │ +0004cfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004cfc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0004cfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004cff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d000: 2d2b 0a7c 6937 203a 2073 6563 6f6e 6461 -+.|i7 : seconda │ │ │ │ +0004d010: 7279 496e 7661 7269 616e 7473 2850 2c44 ryInvariants(P,D │ │ │ │ +0004d020: 362c 5072 696e 7444 6567 7265 6550 6f6c 6,PrintDegreePol │ │ │ │ +0004d030: 796e 6f6d 6961 6c3d 3e74 7275 6529 7c0a ynomial=>true)|. │ │ │ │ +0004d040: 7c20 3420 2020 2032 2020 2020 2020 2020 | 4 2 │ │ │ │ 0004d050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d060: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0004d070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d070: 2020 2020 2020 2020 2020 207c 0a7c 7420 |.|t │ │ │ │ +0004d080: 202b 2074 2020 2b20 3120 2020 2020 2020 + t + 1 │ │ │ │ 0004d090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d0a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0004d0b0: 2020 2020 2020 3220 2020 2020 2032 2020 2 2 │ │ │ │ +0004d0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d0b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0004d0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004d0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d0e0: 2020 207c 0a7c 6f33 203a 204d 6174 7269 |.|o3 : Matri │ │ │ │ -0004d0f0: 7820 4b20 203c 2d2d 204b 2020 2020 2020 x K <-- K │ │ │ │ -0004d100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d0f0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0004d100: 2020 2020 2020 2032 2032 2020 2020 2020 2 2 │ │ │ │ 0004d110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d120: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0004d130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0004d160: 6934 203a 2042 3d73 7562 286d 6174 7269 i4 : B=sub(matri │ │ │ │ -0004d170: 787b 7b30 2c31 7d2c 7b31 2c30 7d7d 2c4b x{{0,1},{1,0}},K │ │ │ │ -0004d180: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ -0004d190: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0004d1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d130: 2020 7c0a 7c6f 3720 3d20 7b31 2c20 782a |.|o7 = {1, x* │ │ │ │ +0004d140: 792c 2078 2079 207d 2020 2020 2020 2020 y, x y } │ │ │ │ +0004d150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d160: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0004d170: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0004d180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d1a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0004d1b0: 3720 3a20 4c69 7374 2020 2020 2020 2020 7 : List │ │ │ │ 0004d1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d1d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0004d1e0: 2020 2020 2020 2032 2020 2020 2020 3220 2 2 │ │ │ │ -0004d1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d210: 2020 2020 7c0a 7c6f 3420 3a20 4d61 7472 |.|o4 : Matr │ │ │ │ -0004d220: 6978 204b 2020 3c2d 2d20 4b20 2020 2020 ix K <-- K │ │ │ │ -0004d230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d250: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0004d260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0004d290: 7c69 3520 3a20 4436 3d66 696e 6974 6541 |i5 : D6=finiteA │ │ │ │ -0004d2a0: 6374 696f 6e28 7b41 2c42 7d2c 5229 2020 ction({A,B},R) │ │ │ │ -0004d2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d2c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0004d2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d300: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ -0004d310: 5220 3c2d 207b 7c20 6120 3020 2020 207c R <- {| a 0 | │ │ │ │ -0004d320: 2c20 7c20 3020 3120 7c7d 2020 2020 2020 , | 0 1 |} │ │ │ │ -0004d330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d340: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0004d350: 2020 207c 2030 202d 612d 3120 7c20 207c | 0 -a-1 | | │ │ │ │ -0004d360: 2031 2030 207c 2020 2020 2020 2020 2020 1 0 | │ │ │ │ -0004d370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d380: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0004d390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d3b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0004d3c0: 0a7c 6f35 203a 2046 696e 6974 6547 726f .|o5 : FiniteGro │ │ │ │ -0004d3d0: 7570 4163 7469 6f6e 2020 2020 2020 2020 upAction │ │ │ │ -0004d3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d3f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0004d400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d430: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ -0004d440: 2050 3d7b 785e 332b 795e 332c 2d28 785e P={x^3+y^3,-(x^ │ │ │ │ -0004d450: 332d 795e 3329 5e32 7d3b 2020 2020 2020 3-y^3)^2}; │ │ │ │ -0004d460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d470: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -0004d480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d4b0: 2d2d 2d2b 0a7c 6937 203a 2073 6563 6f6e ---+.|i7 : secon │ │ │ │ -0004d4c0: 6461 7279 496e 7661 7269 616e 7473 2850 daryInvariants(P │ │ │ │ -0004d4d0: 2c44 362c 5072 696e 7444 6567 7265 6550 ,D6,PrintDegreeP │ │ │ │ -0004d4e0: 6f6c 796e 6f6d 6961 6c3d 3e74 7275 6529 olynomial=>true) │ │ │ │ -0004d4f0: 7c0a 7c20 3420 2020 2032 2020 2020 2020 |.| 4 2 │ │ │ │ -0004d500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d520: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0004d530: 7420 202b 2074 2020 2b20 3120 2020 2020 t + t + 1 │ │ │ │ -0004d540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d560: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0004d570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d5a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0004d5b0: 2020 2020 2020 2020 2032 2032 2020 2020 2 2 │ │ │ │ -0004d5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d5e0: 2020 2020 7c0a 7c6f 3720 3d20 7b31 2c20 |.|o7 = {1, │ │ │ │ -0004d5f0: 782a 792c 2078 2079 207d 2020 2020 2020 x*y, x y } │ │ │ │ -0004d600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d620: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0004d630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d650: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0004d660: 7c6f 3720 3a20 4c69 7374 2020 2020 2020 |o7 : List │ │ │ │ -0004d670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d690: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0004d6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d1e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0004d1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d220: 2d2d 2d2d 2d2d 2b0a 0a43 6176 6561 740a ------+..Caveat. │ │ │ │ +0004d230: 3d3d 3d3d 3d3d 0a0a 4375 7272 656e 746c ======..Currentl │ │ │ │ +0004d240: 792c 2061 2075 7365 7220 6e65 6564 7320 y, a user needs │ │ │ │ +0004d250: 746f 2065 6e73 7572 6520 7468 6174 2074 to ensure that t │ │ │ │ +0004d260: 6865 2061 6c6c 2070 7269 6d61 7279 2069 he all primary i │ │ │ │ +0004d270: 6e76 6172 6961 6e74 7320 6172 6520 6465 nvariants are de │ │ │ │ +0004d280: 6669 6e65 640a 7769 7468 2063 6f65 6666 fined.with coeff │ │ │ │ +0004d290: 6963 6965 6e74 7320 696e 2061 2072 696e icients in a rin │ │ │ │ +0004d2a0: 6720 7468 6174 204d 6163 6175 6c61 7932 g that Macaulay2 │ │ │ │ +0004d2b0: 2072 6563 6f67 6e69 7365 7320 6173 2061 recognises as a │ │ │ │ +0004d2c0: 2063 6861 7261 6374 6572 6973 7469 6320 characteristic │ │ │ │ +0004d2d0: 7a65 726f 0a66 6965 6c64 2028 7365 6520 zero.field (see │ │ │ │ +0004d2e0: 2a6e 6f74 6520 746f 4669 656c 643a 2028 *note toField: ( │ │ │ │ +0004d2f0: 4d61 6361 756c 6179 3244 6f63 2974 6f46 Macaulay2Doc)toF │ │ │ │ +0004d300: 6965 6c64 5f6c 7052 696e 675f 7270 2c20 ield_lpRing_rp, │ │ │ │ +0004d310: 666f 7220 6120 7761 7920 746f 2064 6f0a for a way to do. │ │ │ │ +0004d320: 7468 6973 292e 0a0a 4675 6e63 7469 6f6e this)...Function │ │ │ │ +0004d330: 7320 7769 7468 206f 7074 696f 6e61 6c20 s with optional │ │ │ │ +0004d340: 6172 6775 6d65 6e74 206e 616d 6564 2050 argument named P │ │ │ │ +0004d350: 7269 6e74 4465 6772 6565 506f 6c79 6e6f rintDegreePolyno │ │ │ │ +0004d360: 6d69 616c 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d mial:.========== │ │ │ │ +0004d370: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0004d380: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0004d390: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0004d3a0: 3d3d 3d0a 0a20 202a 2022 6869 726f 6e61 ===.. * "hirona │ │ │ │ +0004d3b0: 6b61 4465 636f 6d70 6f73 6974 696f 6e28 kaDecomposition( │ │ │ │ +0004d3c0: 2e2e 2e2c 5072 696e 7444 6567 7265 6550 ...,PrintDegreeP │ │ │ │ +0004d3d0: 6f6c 796e 6f6d 6961 6c3d 3e2e 2e2e 2922 olynomial=>...)" │ │ │ │ +0004d3e0: 0a20 202a 202a 6e6f 7465 2073 6563 6f6e . * *note secon │ │ │ │ +0004d3f0: 6461 7279 496e 7661 7269 616e 7473 282e daryInvariants(. │ │ │ │ +0004d400: 2e2e 2c50 7269 6e74 4465 6772 6565 506f ..,PrintDegreePo │ │ │ │ +0004d410: 6c79 6e6f 6d69 616c 3d3e 2e2e 2e29 3a0a lynomial=>...):. │ │ │ │ +0004d420: 2020 2020 7365 636f 6e64 6172 7949 6e76 secondaryInv │ │ │ │ +0004d430: 6172 6961 6e74 735f 6c70 5f70 645f 7064 ariants_lp_pd_pd │ │ │ │ +0004d440: 5f70 645f 636d 5072 696e 7444 6567 7265 _pd_cmPrintDegre │ │ │ │ +0004d450: 6550 6f6c 796e 6f6d 6961 6c3d 3e5f 7064 ePolynomial=>_pd │ │ │ │ +0004d460: 5f70 645f 7064 5f72 702c 202d 2d0a 2020 _pd_pd_rp, --. │ │ │ │ +0004d470: 2020 616e 206f 7074 696f 6e61 6c20 6172 an optional ar │ │ │ │ +0004d480: 6775 6d65 6e74 2066 6f72 2073 6563 6f6e gument for secon │ │ │ │ +0004d490: 6461 7279 496e 7661 7269 616e 7473 2074 daryInvariants t │ │ │ │ +0004d4a0: 6861 7420 6465 7465 726d 696e 6573 2074 hat determines t │ │ │ │ +0004d4b0: 6865 2070 7269 6e74 696e 670a 2020 2020 he printing. │ │ │ │ +0004d4c0: 6f66 2061 6e20 696e 666f 726d 6174 6976 of an informativ │ │ │ │ +0004d4d0: 6520 706f 6c79 6e6f 6d69 616c 0a0a 4675 e polynomial..Fu │ │ │ │ +0004d4e0: 7274 6865 7220 696e 666f 726d 6174 696f rther informatio │ │ │ │ +0004d4f0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d n.============== │ │ │ │ +0004d500: 3d3d 3d3d 3d0a 0a20 202a 2044 6566 6175 =====.. * Defau │ │ │ │ +0004d510: 6c74 2076 616c 7565 3a20 2a6e 6f74 6520 lt value: *note │ │ │ │ +0004d520: 6661 6c73 653a 2028 4d61 6361 756c 6179 false: (Macaulay │ │ │ │ +0004d530: 3244 6f63 2966 616c 7365 2c0a 2020 2a20 2Doc)false,. * │ │ │ │ +0004d540: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520 Function: *note │ │ │ │ +0004d550: 7365 636f 6e64 6172 7949 6e76 6172 6961 secondaryInvaria │ │ │ │ +0004d560: 6e74 733a 2073 6563 6f6e 6461 7279 496e nts: secondaryIn │ │ │ │ +0004d570: 7661 7269 616e 7473 2c20 2d2d 2063 6f6d variants, -- com │ │ │ │ +0004d580: 7075 7465 730a 2020 2020 7365 636f 6e64 putes. second │ │ │ │ +0004d590: 6172 7920 696e 7661 7269 616e 7473 2066 ary invariants f │ │ │ │ +0004d5a0: 6f72 2074 6865 2069 6e76 6172 6961 6e74 or the invariant │ │ │ │ +0004d5b0: 2072 696e 6720 6f66 2061 2066 696e 6974 ring of a finit │ │ │ │ +0004d5c0: 6520 6772 6f75 700a 2020 2a20 4f70 7469 e group. * Opti │ │ │ │ +0004d5d0: 6f6e 206b 6579 3a20 2a6e 6f74 6520 5072 on key: *note Pr │ │ │ │ +0004d5e0: 696e 7444 6567 7265 6550 6f6c 796e 6f6d intDegreePolynom │ │ │ │ +0004d5f0: 6961 6c3a 0a20 2020 2073 6563 6f6e 6461 ial:. seconda │ │ │ │ +0004d600: 7279 496e 7661 7269 616e 7473 5f6c 705f ryInvariants_lp_ │ │ │ │ +0004d610: 7064 5f70 645f 7064 5f63 6d50 7269 6e74 pd_pd_pd_cmPrint │ │ │ │ +0004d620: 4465 6772 6565 506f 6c79 6e6f 6d69 616c DegreePolynomial │ │ │ │ +0004d630: 3d3e 5f70 645f 7064 5f70 645f 7270 2c20 =>_pd_pd_pd_rp, │ │ │ │ +0004d640: 2d2d 0a20 2020 2061 6e20 6f70 7469 6f6e --. an option │ │ │ │ +0004d650: 616c 2061 7267 756d 656e 7420 666f 7220 al argument for │ │ │ │ +0004d660: 7365 636f 6e64 6172 7949 6e76 6172 6961 secondaryInvaria │ │ │ │ +0004d670: 6e74 7320 7468 6174 2064 6574 6572 6d69 nts that determi │ │ │ │ +0004d680: 6e65 7320 7468 6520 7072 696e 7469 6e67 nes the printing │ │ │ │ +0004d690: 0a20 2020 206f 6620 616e 2069 6e66 6f72 . of an infor │ │ │ │ +0004d6a0: 6d61 7469 7665 2070 6f6c 796e 6f6d 6961 mative polynomia │ │ │ │ +0004d6b0: 6c0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d l.-------------- │ │ │ │ 0004d6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d6d0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 6176 6561 --------+..Cavea │ │ │ │ -0004d6e0: 740a 3d3d 3d3d 3d3d 0a0a 4375 7272 656e t.======..Curren │ │ │ │ -0004d6f0: 746c 792c 2061 2075 7365 7220 6e65 6564 tly, a user need │ │ │ │ -0004d700: 7320 746f 2065 6e73 7572 6520 7468 6174 s to ensure that │ │ │ │ -0004d710: 2074 6865 2061 6c6c 2070 7269 6d61 7279 the all primary │ │ │ │ -0004d720: 2069 6e76 6172 6961 6e74 7320 6172 6520 invariants are │ │ │ │ -0004d730: 6465 6669 6e65 640a 7769 7468 2063 6f65 defined.with coe │ │ │ │ -0004d740: 6666 6963 6965 6e74 7320 696e 2061 2072 fficients in a r │ │ │ │ -0004d750: 696e 6720 7468 6174 204d 6163 6175 6c61 ing that Macaula │ │ │ │ -0004d760: 7932 2072 6563 6f67 6e69 7365 7320 6173 y2 recognises as │ │ │ │ -0004d770: 2061 2063 6861 7261 6374 6572 6973 7469 a characteristi │ │ │ │ -0004d780: 6320 7a65 726f 0a66 6965 6c64 2028 7365 c zero.field (se │ │ │ │ -0004d790: 6520 2a6e 6f74 6520 746f 4669 656c 643a e *note toField: │ │ │ │ -0004d7a0: 2028 4d61 6361 756c 6179 3244 6f63 2974 (Macaulay2Doc)t │ │ │ │ -0004d7b0: 6f46 6965 6c64 5f6c 7052 696e 675f 7270 oField_lpRing_rp │ │ │ │ -0004d7c0: 2c20 666f 7220 6120 7761 7920 746f 2064 , for a way to d │ │ │ │ -0004d7d0: 6f0a 7468 6973 292e 0a0a 4675 6e63 7469 o.this)...Functi │ │ │ │ -0004d7e0: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ -0004d7f0: 6c20 6172 6775 6d65 6e74 206e 616d 6564 l argument named │ │ │ │ -0004d800: 2050 7269 6e74 4465 6772 6565 506f 6c79 PrintDegreePoly │ │ │ │ -0004d810: 6e6f 6d69 616c 3a0a 3d3d 3d3d 3d3d 3d3d nomial:.======== │ │ │ │ -0004d820: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0004d830: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0004d840: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0004d850: 3d3d 3d3d 3d0a 0a20 202a 2022 6869 726f =====.. * "hiro │ │ │ │ -0004d860: 6e61 6b61 4465 636f 6d70 6f73 6974 696f nakaDecompositio │ │ │ │ -0004d870: 6e28 2e2e 2e2c 5072 696e 7444 6567 7265 n(...,PrintDegre │ │ │ │ -0004d880: 6550 6f6c 796e 6f6d 6961 6c3d 3e2e 2e2e ePolynomial=>... │ │ │ │ -0004d890: 2922 0a20 202a 202a 6e6f 7465 2073 6563 )". * *note sec │ │ │ │ -0004d8a0: 6f6e 6461 7279 496e 7661 7269 616e 7473 ondaryInvariants │ │ │ │ -0004d8b0: 282e 2e2e 2c50 7269 6e74 4465 6772 6565 (...,PrintDegree │ │ │ │ -0004d8c0: 506f 6c79 6e6f 6d69 616c 3d3e 2e2e 2e29 Polynomial=>...) │ │ │ │ -0004d8d0: 3a0a 2020 2020 7365 636f 6e64 6172 7949 :. secondaryI │ │ │ │ -0004d8e0: 6e76 6172 6961 6e74 735f 6c70 5f70 645f nvariants_lp_pd_ │ │ │ │ -0004d8f0: 7064 5f70 645f 636d 5072 696e 7444 6567 pd_pd_cmPrintDeg │ │ │ │ -0004d900: 7265 6550 6f6c 796e 6f6d 6961 6c3d 3e5f reePolynomial=>_ │ │ │ │ -0004d910: 7064 5f70 645f 7064 5f72 702c 202d 2d0a pd_pd_pd_rp, --. │ │ │ │ -0004d920: 2020 2020 616e 206f 7074 696f 6e61 6c20 an optional │ │ │ │ -0004d930: 6172 6775 6d65 6e74 2066 6f72 2073 6563 argument for sec │ │ │ │ -0004d940: 6f6e 6461 7279 496e 7661 7269 616e 7473 ondaryInvariants │ │ │ │ -0004d950: 2074 6861 7420 6465 7465 726d 696e 6573 that determines │ │ │ │ -0004d960: 2074 6865 2070 7269 6e74 696e 670a 2020 the printing. │ │ │ │ -0004d970: 2020 6f66 2061 6e20 696e 666f 726d 6174 of an informat │ │ │ │ -0004d980: 6976 6520 706f 6c79 6e6f 6d69 616c 0a0a ive polynomial.. │ │ │ │ -0004d990: 4675 7274 6865 7220 696e 666f 726d 6174 Further informat │ │ │ │ -0004d9a0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ion.============ │ │ │ │ -0004d9b0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2044 6566 =======.. * Def │ │ │ │ -0004d9c0: 6175 6c74 2076 616c 7565 3a20 2a6e 6f74 ault value: *not │ │ │ │ -0004d9d0: 6520 6661 6c73 653a 2028 4d61 6361 756c e false: (Macaul │ │ │ │ -0004d9e0: 6179 3244 6f63 2966 616c 7365 2c0a 2020 ay2Doc)false,. │ │ │ │ -0004d9f0: 2a20 4675 6e63 7469 6f6e 3a20 2a6e 6f74 * Function: *not │ │ │ │ -0004da00: 6520 7365 636f 6e64 6172 7949 6e76 6172 e secondaryInvar │ │ │ │ -0004da10: 6961 6e74 733a 2073 6563 6f6e 6461 7279 iants: secondary │ │ │ │ -0004da20: 496e 7661 7269 616e 7473 2c20 2d2d 2063 Invariants, -- c │ │ │ │ -0004da30: 6f6d 7075 7465 730a 2020 2020 7365 636f omputes. seco │ │ │ │ -0004da40: 6e64 6172 7920 696e 7661 7269 616e 7473 ndary invariants │ │ │ │ -0004da50: 2066 6f72 2074 6865 2069 6e76 6172 6961 for the invaria │ │ │ │ -0004da60: 6e74 2072 696e 6720 6f66 2061 2066 696e nt ring of a fin │ │ │ │ -0004da70: 6974 6520 6772 6f75 700a 2020 2a20 4f70 ite group. * Op │ │ │ │ -0004da80: 7469 6f6e 206b 6579 3a20 2a6e 6f74 6520 tion key: *note │ │ │ │ -0004da90: 5072 696e 7444 6567 7265 6550 6f6c 796e PrintDegreePolyn │ │ │ │ -0004daa0: 6f6d 6961 6c3a 0a20 2020 2073 6563 6f6e omial:. secon │ │ │ │ -0004dab0: 6461 7279 496e 7661 7269 616e 7473 5f6c daryInvariants_l │ │ │ │ -0004dac0: 705f 7064 5f70 645f 7064 5f63 6d50 7269 p_pd_pd_pd_cmPri │ │ │ │ -0004dad0: 6e74 4465 6772 6565 506f 6c79 6e6f 6d69 ntDegreePolynomi │ │ │ │ -0004dae0: 616c 3d3e 5f70 645f 7064 5f70 645f 7270 al=>_pd_pd_pd_rp │ │ │ │ -0004daf0: 2c20 2d2d 0a20 2020 2061 6e20 6f70 7469 , --. an opti │ │ │ │ -0004db00: 6f6e 616c 2061 7267 756d 656e 7420 666f onal argument fo │ │ │ │ -0004db10: 7220 7365 636f 6e64 6172 7949 6e76 6172 r secondaryInvar │ │ │ │ -0004db20: 6961 6e74 7320 7468 6174 2064 6574 6572 iants that deter │ │ │ │ -0004db30: 6d69 6e65 7320 7468 6520 7072 696e 7469 mines the printi │ │ │ │ -0004db40: 6e67 0a20 2020 206f 6620 616e 2069 6e66 ng. of an inf │ │ │ │ -0004db50: 6f72 6d61 7469 7665 2070 6f6c 796e 6f6d ormative polynom │ │ │ │ -0004db60: 6961 6c0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ial.------------ │ │ │ │ -0004db70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004db90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004dba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004dbb0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -0004dbc0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -0004dbd0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -0004dbe0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -0004dbf0: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -0004dc00: 352b 6473 2f4d 322f 4d61 6361 756c 6179 5+ds/M2/Macaulay │ │ │ │ -0004dc10: 322f 7061 636b 6167 6573 2f0a 496e 7661 2/packages/.Inva │ │ │ │ -0004dc20: 7269 616e 7452 696e 672f 4861 7765 7344 riantRing/HawesD │ │ │ │ -0004dc30: 6f63 2e6d 323a 3535 353a 302e 0a1f 0a46 oc.m2:555:0....F │ │ │ │ -0004dc40: 696c 653a 2049 6e76 6172 6961 6e74 5269 ile: InvariantRi │ │ │ │ -0004dc50: 6e67 2e69 6e66 6f2c 204e 6f64 653a 2077 ng.info, Node: w │ │ │ │ -0004dc60: 6569 6768 7473 2c20 4e65 7874 3a20 776f eights, Next: wo │ │ │ │ -0004dc70: 7264 732c 2050 7265 763a 2073 6563 6f6e rds, Prev: secon │ │ │ │ -0004dc80: 6461 7279 496e 7661 7269 616e 7473 5f6c daryInvariants_l │ │ │ │ -0004dc90: 705f 7064 5f70 645f 7064 5f63 6d50 7269 p_pd_pd_pd_cmPri │ │ │ │ -0004dca0: 6e74 4465 6772 6565 506f 6c79 6e6f 6d69 ntDegreePolynomi │ │ │ │ -0004dcb0: 616c 3d3e 5f70 645f 7064 5f70 645f 7270 al=>_pd_pd_pd_rp │ │ │ │ -0004dcc0: 2c20 5570 3a20 546f 700a 0a77 6569 6768 , Up: Top..weigh │ │ │ │ -0004dcd0: 7473 202d 2d20 6f66 2061 2064 6961 676f ts -- of a diago │ │ │ │ -0004dce0: 6e61 6c20 6163 7469 6f6e 0a2a 2a2a 2a2a nal action.***** │ │ │ │ -0004dcf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004dd00: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -0004dd10: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -0004dd20: 7765 6967 6874 7320 440a 2020 2a20 496e weights D. * In │ │ │ │ -0004dd30: 7075 7473 3a0a 2020 2020 2020 2a20 442c puts:. * D, │ │ │ │ -0004dd40: 2061 6e20 696e 7374 616e 6365 206f 6620 an instance of │ │ │ │ -0004dd50: 7468 6520 7479 7065 202a 6e6f 7465 2044 the type *note D │ │ │ │ -0004dd60: 6961 676f 6e61 6c41 6374 696f 6e3a 2044 iagonalAction: D │ │ │ │ -0004dd70: 6961 676f 6e61 6c41 6374 696f 6e2c 0a20 iagonalAction,. │ │ │ │ -0004dd80: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ -0004dd90: 2020 2a20 6120 2a6e 6f74 6520 6d61 7472 * a *note matr │ │ │ │ -0004dda0: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ -0004ddb0: 6329 4d61 7472 6978 2c2c 2074 6865 2077 c)Matrix,, the w │ │ │ │ -0004ddc0: 6569 6768 7420 6d61 7472 6978 206f 6620 eight matrix of │ │ │ │ -0004ddd0: 7468 6520 6772 6f75 700a 2020 2020 2020 the group. │ │ │ │ -0004dde0: 2020 6163 7469 6f6e 0a0a 4465 7363 7269 action..Descri │ │ │ │ -0004ddf0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -0004de00: 3d0a 0a54 6869 7320 6675 6e63 7469 6f6e =..This function │ │ │ │ -0004de10: 2069 7320 7072 6f76 6964 6564 2062 7920 is provided by │ │ │ │ -0004de20: 7468 6520 7061 636b 6167 6520 2a6e 6f74 the package *not │ │ │ │ -0004de30: 6520 496e 7661 7269 616e 7452 696e 673a e InvariantRing: │ │ │ │ -0004de40: 2054 6f70 2c2e 200a 0a55 7365 2074 6869 Top,. ..Use thi │ │ │ │ -0004de50: 7320 6675 6e63 7469 6f6e 2074 6f20 7265 s function to re │ │ │ │ -0004de60: 636f 7665 7220 7468 6520 7765 6967 6874 cover the weight │ │ │ │ -0004de70: 206d 6174 7269 7820 6f66 2061 2064 6961 matrix of a dia │ │ │ │ -0004de80: 676f 6e61 6c20 6163 7469 6f6e 206f 6e20 gonal action on │ │ │ │ -0004de90: 610a 706f 6c79 6e6f 6d69 616c 2072 696e a.polynomial rin │ │ │ │ -0004dea0: 672e 2046 6f72 2061 2064 6961 676f 6e61 g. For a diagona │ │ │ │ -0004deb0: 6c20 6163 7469 6f6e 206f 6e20 6120 706f l action on a po │ │ │ │ -0004dec0: 6c79 6e6f 6d69 616c 2072 696e 6720 2024 lynomial ring $ │ │ │ │ -0004ded0: 6b5b 785f 312c 205c 646f 7473 2c0a 785f k[x_1, \dots,.x_ │ │ │ │ -0004dee0: 6e5d 2420 2c20 7468 6520 2024 6a24 202d n]$ , the $j$ - │ │ │ │ -0004def0: 7468 2063 6f6c 756d 6e20 6f66 2074 6865 th column of the │ │ │ │ -0004df00: 2077 6569 6768 7420 6d61 7472 6978 2069 weight matrix i │ │ │ │ -0004df10: 7320 7468 6520 7765 6967 6874 206f 6620 s the weight of │ │ │ │ -0004df20: 7468 6520 7661 7269 6162 6c65 0a24 785f the variable.$x_ │ │ │ │ -0004df30: 6a24 202e 0a0a 5468 6520 666f 6c6c 6f77 j$ ...The follow │ │ │ │ -0004df40: 696e 6720 6578 616d 706c 6520 6465 6669 ing example defi │ │ │ │ -0004df50: 6e65 7320 616e 2061 6374 696f 6e20 6f66 nes an action of │ │ │ │ -0004df60: 2061 2074 776f 2d64 696d 656e 7369 6f6e a two-dimension │ │ │ │ -0004df70: 616c 2074 6f72 7573 206f 6e20 610a 706f al torus on a.po │ │ │ │ -0004df80: 6c79 6e6f 6d69 616c 2072 696e 6720 696e lynomial ring in │ │ │ │ -0004df90: 2066 6f75 7220 7661 7269 6162 6c65 732e four variables. │ │ │ │ -0004dfa0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ -0004dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004dfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ -0004dfd0: 3a20 5220 3d20 5151 5b78 5f31 2e2e 785f : R = QQ[x_1..x_ │ │ │ │ -0004dfe0: 345d 2020 2020 2020 2020 2020 2020 2020 4] │ │ │ │ -0004dff0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0004e000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e010: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0004e020: 7c6f 3120 3d20 5220 2020 2020 2020 2020 |o1 = R │ │ │ │ -0004e030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e040: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0004e050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e070: 2020 7c0a 7c6f 3120 3a20 506f 6c79 6e6f |.|o1 : Polyno │ │ │ │ -0004e080: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ -0004e090: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0004d6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d700: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +0004d710: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +0004d720: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +0004d730: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +0004d740: 6361 756c 6179 322d 312e 3236 2e30 352b caulay2-1.26.05+ │ │ │ │ +0004d750: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +0004d760: 7061 636b 6167 6573 2f0a 496e 7661 7269 packages/.Invari │ │ │ │ +0004d770: 616e 7452 696e 672f 4861 7765 7344 6f63 antRing/HawesDoc │ │ │ │ +0004d780: 2e6d 323a 3535 353a 302e 0a1f 0a46 696c .m2:555:0....Fil │ │ │ │ +0004d790: 653a 2049 6e76 6172 6961 6e74 5269 6e67 e: InvariantRing │ │ │ │ +0004d7a0: 2e69 6e66 6f2c 204e 6f64 653a 2077 6569 .info, Node: wei │ │ │ │ +0004d7b0: 6768 7473 2c20 4e65 7874 3a20 776f 7264 ghts, Next: word │ │ │ │ +0004d7c0: 732c 2050 7265 763a 2073 6563 6f6e 6461 s, Prev: seconda │ │ │ │ +0004d7d0: 7279 496e 7661 7269 616e 7473 5f6c 705f ryInvariants_lp_ │ │ │ │ +0004d7e0: 7064 5f70 645f 7064 5f63 6d50 7269 6e74 pd_pd_pd_cmPrint │ │ │ │ +0004d7f0: 4465 6772 6565 506f 6c79 6e6f 6d69 616c DegreePolynomial │ │ │ │ +0004d800: 3d3e 5f70 645f 7064 5f70 645f 7270 2c20 =>_pd_pd_pd_rp, │ │ │ │ +0004d810: 5570 3a20 546f 700a 0a77 6569 6768 7473 Up: Top..weights │ │ │ │ +0004d820: 202d 2d20 6f66 2061 2064 6961 676f 6e61 -- of a diagona │ │ │ │ +0004d830: 6c20 6163 7469 6f6e 0a2a 2a2a 2a2a 2a2a l action.******* │ │ │ │ +0004d840: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004d850: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ +0004d860: 6167 653a 200a 2020 2020 2020 2020 7765 age: . we │ │ │ │ +0004d870: 6967 6874 7320 440a 2020 2a20 496e 7075 ights D. * Inpu │ │ │ │ +0004d880: 7473 3a0a 2020 2020 2020 2a20 442c 2061 ts:. * D, a │ │ │ │ +0004d890: 6e20 696e 7374 616e 6365 206f 6620 7468 n instance of th │ │ │ │ +0004d8a0: 6520 7479 7065 202a 6e6f 7465 2044 6961 e type *note Dia │ │ │ │ +0004d8b0: 676f 6e61 6c41 6374 696f 6e3a 2044 6961 gonalAction: Dia │ │ │ │ +0004d8c0: 676f 6e61 6c41 6374 696f 6e2c 0a20 202a gonalAction,. * │ │ │ │ +0004d8d0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ +0004d8e0: 2a20 6120 2a6e 6f74 6520 6d61 7472 6978 * a *note matrix │ │ │ │ +0004d8f0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0004d900: 4d61 7472 6978 2c2c 2074 6865 2077 6569 Matrix,, the wei │ │ │ │ +0004d910: 6768 7420 6d61 7472 6978 206f 6620 7468 ght matrix of th │ │ │ │ +0004d920: 6520 6772 6f75 700a 2020 2020 2020 2020 e group. │ │ │ │ +0004d930: 6163 7469 6f6e 0a0a 4465 7363 7269 7074 action..Descript │ │ │ │ +0004d940: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +0004d950: 0a54 6869 7320 6675 6e63 7469 6f6e 2069 .This function i │ │ │ │ +0004d960: 7320 7072 6f76 6964 6564 2062 7920 7468 s provided by th │ │ │ │ +0004d970: 6520 7061 636b 6167 6520 2a6e 6f74 6520 e package *note │ │ │ │ +0004d980: 496e 7661 7269 616e 7452 696e 673a 2054 InvariantRing: T │ │ │ │ +0004d990: 6f70 2c2e 200a 0a55 7365 2074 6869 7320 op,. ..Use this │ │ │ │ +0004d9a0: 6675 6e63 7469 6f6e 2074 6f20 7265 636f function to reco │ │ │ │ +0004d9b0: 7665 7220 7468 6520 7765 6967 6874 206d ver the weight m │ │ │ │ +0004d9c0: 6174 7269 7820 6f66 2061 2064 6961 676f atrix of a diago │ │ │ │ +0004d9d0: 6e61 6c20 6163 7469 6f6e 206f 6e20 610a nal action on a. │ │ │ │ +0004d9e0: 706f 6c79 6e6f 6d69 616c 2072 696e 672e polynomial ring. │ │ │ │ +0004d9f0: 2046 6f72 2061 2064 6961 676f 6e61 6c20 For a diagonal │ │ │ │ +0004da00: 6163 7469 6f6e 206f 6e20 6120 706f 6c79 action on a poly │ │ │ │ +0004da10: 6e6f 6d69 616c 2072 696e 6720 2024 6b5b nomial ring $k[ │ │ │ │ +0004da20: 785f 312c 205c 646f 7473 2c0a 785f 6e5d x_1, \dots,.x_n] │ │ │ │ +0004da30: 2420 2c20 7468 6520 2024 6a24 202d 7468 $ , the $j$ -th │ │ │ │ +0004da40: 2063 6f6c 756d 6e20 6f66 2074 6865 2077 column of the w │ │ │ │ +0004da50: 6569 6768 7420 6d61 7472 6978 2069 7320 eight matrix is │ │ │ │ +0004da60: 7468 6520 7765 6967 6874 206f 6620 7468 the weight of th │ │ │ │ +0004da70: 6520 7661 7269 6162 6c65 0a24 785f 6a24 e variable.$x_j$ │ │ │ │ +0004da80: 202e 0a0a 5468 6520 666f 6c6c 6f77 696e ...The followin │ │ │ │ +0004da90: 6720 6578 616d 706c 6520 6465 6669 6e65 g example define │ │ │ │ +0004daa0: 7320 616e 2061 6374 696f 6e20 6f66 2061 s an action of a │ │ │ │ +0004dab0: 2074 776f 2d64 696d 656e 7369 6f6e 616c two-dimensional │ │ │ │ +0004dac0: 2074 6f72 7573 206f 6e20 610a 706f 6c79 torus on a.poly │ │ │ │ +0004dad0: 6e6f 6d69 616c 2072 696e 6720 696e 2066 nomial ring in f │ │ │ │ +0004dae0: 6f75 7220 7661 7269 6162 6c65 732e 0a0a our variables... │ │ │ │ +0004daf0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0004db00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004db10: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +0004db20: 5220 3d20 5151 5b78 5f31 2e2e 785f 345d R = QQ[x_1..x_4] │ │ │ │ +0004db30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004db40: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004db50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004db60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0004db70: 3120 3d20 5220 2020 2020 2020 2020 2020 1 = R │ │ │ │ +0004db80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004db90: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0004dba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004dbc0: 7c0a 7c6f 3120 3a20 506f 6c79 6e6f 6d69 |.|o1 : Polynomi │ │ │ │ +0004dbd0: 616c 5269 6e67 2020 2020 2020 2020 2020 alRing │ │ │ │ +0004dbe0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0004dbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004dc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004dc10: 2d2d 2d2d 2b0a 7c69 3220 3a20 5720 3d20 ----+.|i2 : W = │ │ │ │ +0004dc20: 6d61 7472 6978 7b7b 302c 312c 2d31 2c31 matrix{{0,1,-1,1 │ │ │ │ +0004dc30: 7d2c 7b31 2c30 2c2d 312c 2d31 7d7d 7c0a },{1,0,-1,-1}}|. │ │ │ │ +0004dc40: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0004dc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004dc60: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ +0004dc70: 7c20 3020 3120 2d31 2031 2020 7c20 2020 | 0 1 -1 1 | │ │ │ │ +0004dc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004dc90: 2020 7c0a 7c20 2020 2020 7c20 3120 3020 |.| | 1 0 │ │ │ │ +0004dca0: 2d31 202d 3120 7c20 2020 2020 2020 2020 -1 -1 | │ │ │ │ +0004dcb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0004dcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004dcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004dce0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0004dcf0: 2020 2020 2020 2032 2020 2020 2020 2034 2 4 │ │ │ │ +0004dd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004dd10: 7c0a 7c6f 3220 3a20 4d61 7472 6978 205a |.|o2 : Matrix Z │ │ │ │ +0004dd20: 5a20 203c 2d2d 205a 5a20 2020 2020 2020 Z <-- ZZ │ │ │ │ +0004dd30: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0004dd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004dd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004dd60: 2d2d 2d2d 2b0a 7c69 3320 3a20 5420 3d20 ----+.|i3 : T = │ │ │ │ +0004dd70: 6469 6167 6f6e 616c 4163 7469 6f6e 2857 diagonalAction(W │ │ │ │ +0004dd80: 2c20 5229 2020 2020 2020 2020 2020 7c0a , R) |. │ │ │ │ +0004dd90: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0004dda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ddb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0004ddc0: 2020 2020 2020 2020 2a20 3220 2020 2020 * 2 │ │ │ │ +0004ddd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004dde0: 2020 7c0a 7c6f 3320 3d20 5220 3c2d 2028 |.|o3 = R <- ( │ │ │ │ +0004ddf0: 5151 2029 2020 7669 6120 2020 2020 2020 QQ ) via │ │ │ │ +0004de00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0004de10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004de20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004de30: 2020 2020 2020 7c0a 7c20 2020 2020 7c20 |.| | │ │ │ │ +0004de40: 3020 3120 2d31 2031 2020 7c20 2020 2020 0 1 -1 1 | │ │ │ │ +0004de50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004de60: 7c0a 7c20 2020 2020 7c20 3120 3020 2d31 |.| | 1 0 -1 │ │ │ │ +0004de70: 202d 3120 7c20 2020 2020 2020 2020 2020 -1 | │ │ │ │ +0004de80: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0004de90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004dea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004deb0: 2020 2020 7c0a 7c6f 3320 3a20 4469 6167 |.|o3 : Diag │ │ │ │ +0004dec0: 6f6e 616c 4163 7469 6f6e 2020 2020 2020 onalAction │ │ │ │ +0004ded0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0004dee0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0004def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004df00: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ +0004df10: 7765 6967 6874 7320 5420 2020 2020 2020 weights T │ │ │ │ +0004df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004df30: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004df40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004df50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0004df60: 3420 3d20 287c 2030 2031 202d 3120 3120 4 = (| 0 1 -1 1 │ │ │ │ +0004df70: 207c 2c20 3029 2020 2020 2020 2020 2020 |, 0) │ │ │ │ +0004df80: 2020 2020 2020 7c0a 7c20 2020 2020 207c |.| | │ │ │ │ +0004df90: 2031 2030 202d 3120 2d31 207c 2020 2020 1 0 -1 -1 | │ │ │ │ +0004dfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004dfb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004dfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004dfd0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ +0004dfe0: 3a20 5365 7175 656e 6365 2020 2020 2020 : Sequence │ │ │ │ +0004dff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004e000: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0004e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0004e030: 0a48 6572 6520 6973 2061 6e20 6578 616d .Here is an exam │ │ │ │ +0004e040: 706c 6520 6f66 2061 2070 726f 6475 6374 ple of a product │ │ │ │ +0004e050: 206f 6620 7477 6f20 6379 636c 6963 2067 of two cyclic g │ │ │ │ +0004e060: 726f 7570 7320 6f66 206f 7264 6572 2033 roups of order 3 │ │ │ │ +0004e070: 2061 6374 696e 6720 6f6e 2061 0a70 6f6c acting on a.pol │ │ │ │ +0004e080: 796e 6f6d 6961 6c20 7269 6e67 2069 6e20 ynomial ring in │ │ │ │ +0004e090: 3320 7661 7269 6162 6c65 732e 0a0a 2b2d 3 variables...+- │ │ │ │ 0004e0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e0c0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5720 ------+.|i2 : W │ │ │ │ -0004e0d0: 3d20 6d61 7472 6978 7b7b 302c 312c 2d31 = matrix{{0,1,-1 │ │ │ │ -0004e0e0: 2c31 7d2c 7b31 2c30 2c2d 312c 2d31 7d7d ,1},{1,0,-1,-1}} │ │ │ │ -0004e0f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0004e100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e110: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ -0004e120: 3d20 7c20 3020 3120 2d31 2031 2020 7c20 = | 0 1 -1 1 | │ │ │ │ +0004e0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0004e0c0: 0a7c 6935 203a 2052 203d 2051 515b 785f .|i5 : R = QQ[x_ │ │ │ │ +0004e0d0: 312e 2e78 5f33 5d20 2020 2020 2020 2020 1..x_3] │ │ │ │ +0004e0e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004e0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004e100: 2020 2020 207c 0a7c 6f35 203d 2052 2020 |.|o5 = R │ │ │ │ +0004e110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004e120: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0004e130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e140: 2020 2020 7c0a 7c20 2020 2020 7c20 3120 |.| | 1 │ │ │ │ -0004e150: 3020 2d31 202d 3120 7c20 2020 2020 2020 0 -1 -1 | │ │ │ │ -0004e160: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0004e170: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0004e180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e190: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0004e1a0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -0004e1b0: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -0004e1c0: 2020 7c0a 7c6f 3220 3a20 4d61 7472 6978 |.|o2 : Matrix │ │ │ │ -0004e1d0: 205a 5a20 203c 2d2d 205a 5a20 2020 2020 ZZ <-- ZZ │ │ │ │ -0004e1e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0004e1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e210: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 5420 ------+.|i3 : T │ │ │ │ -0004e220: 3d20 6469 6167 6f6e 616c 4163 7469 6f6e = diagonalAction │ │ │ │ -0004e230: 2857 2c20 5229 2020 2020 2020 2020 2020 (W, R) │ │ │ │ -0004e240: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0004e250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e260: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0004e270: 2020 2020 2020 2020 2020 2a20 3220 2020 * 2 │ │ │ │ -0004e280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e290: 2020 2020 7c0a 7c6f 3320 3d20 5220 3c2d |.|o3 = R <- │ │ │ │ -0004e2a0: 2028 5151 2029 2020 7669 6120 2020 2020 (QQ ) via │ │ │ │ -0004e2b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0004e2c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0004e2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e2e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0004e2f0: 7c20 3020 3120 2d31 2031 2020 7c20 2020 | 0 1 -1 1 | │ │ │ │ +0004e140: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ +0004e150: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ +0004e160: 6720 2020 2020 2020 2020 2020 2020 7c0a g |. │ │ │ │ +0004e170: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0004e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e190: 2d2b 0a7c 6936 203a 2064 203d 207b 332c -+.|i6 : d = {3, │ │ │ │ +0004e1a0: 337d 2020 2020 2020 2020 2020 2020 2020 3} │ │ │ │ +0004e1b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0004e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004e1d0: 2020 2020 2020 207c 0a7c 6f36 203d 207b |.|o6 = { │ │ │ │ +0004e1e0: 332c 2033 7d20 2020 2020 2020 2020 2020 3, 3} │ │ │ │ +0004e1f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0004e200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004e210: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0004e220: 6f36 203a 204c 6973 7420 2020 2020 2020 o6 : List │ │ │ │ +0004e230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004e240: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0004e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e260: 2d2d 2d2b 0a7c 6937 203a 2057 203d 206d ---+.|i7 : W = m │ │ │ │ +0004e270: 6174 7269 787b 7b31 2c30 2c31 7d2c 7b30 atrix{{1,0,1},{0 │ │ │ │ +0004e280: 2c31 2c31 7d7d 7c0a 7c20 2020 2020 2020 ,1,1}}|.| │ │ │ │ +0004e290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004e2a0: 2020 2020 2020 2020 207c 0a7c 6f37 203d |.|o7 = │ │ │ │ +0004e2b0: 207c 2031 2030 2031 207c 2020 2020 2020 | 1 0 1 | │ │ │ │ +0004e2c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0004e2d0: 2020 2020 7c20 3020 3120 3120 7c20 2020 | 0 1 1 | │ │ │ │ +0004e2e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0004e2f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0004e300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e310: 2020 7c0a 7c20 2020 2020 7c20 3120 3020 |.| | 1 0 │ │ │ │ -0004e320: 2d31 202d 3120 7c20 2020 2020 2020 2020 -1 -1 | │ │ │ │ -0004e330: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0004e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e360: 2020 2020 2020 7c0a 7c6f 3320 3a20 4469 |.|o3 : Di │ │ │ │ -0004e370: 6167 6f6e 616c 4163 7469 6f6e 2020 2020 agonalAction │ │ │ │ -0004e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e390: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0004e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ -0004e3c0: 3a20 7765 6967 6874 7320 5420 2020 2020 : weights T │ │ │ │ -0004e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004e310: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004e320: 2020 2032 2020 2020 2020 2033 2020 2020 2 3 │ │ │ │ +0004e330: 2020 2020 207c 0a7c 6f37 203a 204d 6174 |.|o7 : Mat │ │ │ │ +0004e340: 7269 7820 5a5a 2020 3c2d 2d20 5a5a 2020 rix ZZ <-- ZZ │ │ │ │ +0004e350: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0004e360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ +0004e380: 203a 2041 203d 2064 6961 676f 6e61 6c41 : A = diagonalA │ │ │ │ +0004e390: 6374 696f 6e28 572c 2064 2c20 5229 7c0a ction(W, d, R)|. │ │ │ │ +0004e3a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0004e3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004e3c0: 207c 0a7c 6f38 203d 2052 203c 2d20 5a5a |.|o8 = R <- ZZ │ │ │ │ +0004e3d0: 2f33 2078 205a 5a2f 3320 7669 6120 2020 /3 x ZZ/3 via │ │ │ │ 0004e3e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0004e3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e400: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0004e410: 7c6f 3420 3d20 287c 2030 2031 202d 3120 |o4 = (| 0 1 -1 │ │ │ │ -0004e420: 3120 207c 2c20 3029 2020 2020 2020 2020 1 |, 0) │ │ │ │ -0004e430: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0004e440: 207c 2031 2030 202d 3120 2d31 207c 2020 | 1 0 -1 -1 | │ │ │ │ +0004e400: 2020 2020 2020 207c 0a7c 2020 2020 207c |.| | │ │ │ │ +0004e410: 2031 2030 2031 207c 2020 2020 2020 2020 1 0 1 | │ │ │ │ +0004e420: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0004e430: 2020 7c20 3020 3120 3120 7c20 2020 2020 | 0 1 1 | │ │ │ │ +0004e440: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0004e450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e460: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0004e470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e480: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0004e490: 3420 3a20 5365 7175 656e 6365 2020 2020 4 : Sequence │ │ │ │ -0004e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e4b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -0004e4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e4e0: 2b0a 0a48 6572 6520 6973 2061 6e20 6578 +..Here is an ex │ │ │ │ -0004e4f0: 616d 706c 6520 6f66 2061 2070 726f 6475 ample of a produ │ │ │ │ -0004e500: 6374 206f 6620 7477 6f20 6379 636c 6963 ct of two cyclic │ │ │ │ -0004e510: 2067 726f 7570 7320 6f66 206f 7264 6572 groups of order │ │ │ │ -0004e520: 2033 2061 6374 696e 6720 6f6e 2061 0a70 3 acting on a.p │ │ │ │ -0004e530: 6f6c 796e 6f6d 6961 6c20 7269 6e67 2069 olynomial ring i │ │ │ │ -0004e540: 6e20 3320 7661 7269 6162 6c65 732e 0a0a n 3 variables... │ │ │ │ -0004e550: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0004e560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e570: 2d2b 0a7c 6935 203a 2052 203d 2051 515b -+.|i5 : R = QQ[ │ │ │ │ -0004e580: 785f 312e 2e78 5f33 5d20 2020 2020 2020 x_1..x_3] │ │ │ │ -0004e590: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0004e5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e5b0: 2020 2020 2020 207c 0a7c 6f35 203d 2052 |.|o5 = R │ │ │ │ -0004e5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e5d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0004e5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e5f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0004e600: 6f35 203a 2050 6f6c 796e 6f6d 6961 6c52 o5 : PolynomialR │ │ │ │ -0004e610: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -0004e620: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0004e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e640: 2d2d 2d2b 0a7c 6936 203a 2064 203d 207b ---+.|i6 : d = { │ │ │ │ -0004e650: 332c 337d 2020 2020 2020 2020 2020 2020 3,3} │ │ │ │ -0004e660: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0004e670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e680: 2020 2020 2020 2020 207c 0a7c 6f36 203d |.|o6 = │ │ │ │ -0004e690: 207b 332c 2033 7d20 2020 2020 2020 2020 {3, 3} │ │ │ │ -0004e6a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0004e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e6c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0004e6d0: 0a7c 6f36 203a 204c 6973 7420 2020 2020 .|o6 : List │ │ │ │ -0004e6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e6f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -0004e700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e710: 2d2d 2d2d 2d2b 0a7c 6937 203a 2057 203d -----+.|i7 : W = │ │ │ │ -0004e720: 206d 6174 7269 787b 7b31 2c30 2c31 7d2c matrix{{1,0,1}, │ │ │ │ -0004e730: 7b30 2c31 2c31 7d7d 7c0a 7c20 2020 2020 {0,1,1}}|.| │ │ │ │ -0004e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e750: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -0004e760: 203d 207c 2031 2030 2031 207c 2020 2020 = | 1 0 1 | │ │ │ │ -0004e770: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0004e780: 7c20 2020 2020 7c20 3020 3120 3120 7c20 | | 0 1 1 | │ │ │ │ -0004e790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e7a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0004e7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e7c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0004e7d0: 2020 2020 2032 2020 2020 2020 2033 2020 2 3 │ │ │ │ -0004e7e0: 2020 2020 2020 207c 0a7c 6f37 203a 204d |.|o7 : M │ │ │ │ -0004e7f0: 6174 7269 7820 5a5a 2020 3c2d 2d20 5a5a atrix ZZ <-- ZZ │ │ │ │ -0004e800: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -0004e810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0004e830: 6938 203a 2041 203d 2064 6961 676f 6e61 i8 : A = diagona │ │ │ │ -0004e840: 6c41 6374 696f 6e28 572c 2064 2c20 5229 lAction(W, d, R) │ │ │ │ -0004e850: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0004e860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e870: 2020 207c 0a7c 6f38 203d 2052 203c 2d20 |.|o8 = R <- │ │ │ │ -0004e880: 5a5a 2f33 2078 205a 5a2f 3320 7669 6120 ZZ/3 x ZZ/3 via │ │ │ │ -0004e890: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0004e8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e8b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0004e8c0: 207c 2031 2030 2031 207c 2020 2020 2020 | 1 0 1 | │ │ │ │ -0004e8d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0004e8e0: 2020 2020 7c20 3020 3120 3120 7c20 2020 | 0 1 1 | │ │ │ │ -0004e8f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0004e900: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0004e910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e920: 2020 7c0a 7c6f 3820 3a20 4469 6167 6f6e |.|o8 : Diagon │ │ │ │ -0004e930: 616c 4163 7469 6f6e 2020 2020 2020 2020 alAction │ │ │ │ -0004e940: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0004e950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e960: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 --------+.|i9 : │ │ │ │ -0004e970: 7765 6967 6874 7320 4120 2020 2020 2020 weights A │ │ │ │ -0004e980: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0004e990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004e9a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0004e9b0: 7c6f 3920 3d20 2830 2c20 7c20 3120 3020 |o9 = (0, | 1 0 │ │ │ │ -0004e9c0: 3120 7c29 2020 2020 2020 2020 2020 2020 1 |) │ │ │ │ -0004e9d0: 207c 0a7c 2020 2020 2020 2020 207c 2030 |.| | 0 │ │ │ │ -0004e9e0: 2031 2031 207c 2020 2020 2020 2020 2020 1 1 | │ │ │ │ -0004e9f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0004ea00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ea10: 2020 2020 2020 207c 0a7c 6f39 203a 2053 |.|o9 : S │ │ │ │ -0004ea20: 6571 7565 6e63 6520 2020 2020 2020 2020 equence │ │ │ │ -0004ea30: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -0004ea40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004ea50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -0004ea60: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ -0004ea70: 3d0a 0a20 202a 202a 6e6f 7465 2044 6961 =.. * *note Dia │ │ │ │ -0004ea80: 676f 6e61 6c41 6374 696f 6e3a 2044 6961 gonalAction: Dia │ │ │ │ -0004ea90: 676f 6e61 6c41 6374 696f 6e2c 202d 2d20 gonalAction, -- │ │ │ │ -0004eaa0: 7468 6520 636c 6173 7320 6f66 2061 6c6c the class of all │ │ │ │ -0004eab0: 2064 6961 676f 6e61 6c20 6163 7469 6f6e diagonal action │ │ │ │ -0004eac0: 730a 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-----------+..Se │ │ │ │ +0004e5b0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ +0004e5c0: 0a20 202a 202a 6e6f 7465 2044 6961 676f . * *note Diago │ │ │ │ +0004e5d0: 6e61 6c41 6374 696f 6e3a 2044 6961 676f nalAction: Diago │ │ │ │ +0004e5e0: 6e61 6c41 6374 696f 6e2c 202d 2d20 7468 nalAction, -- th │ │ │ │ +0004e5f0: 6520 636c 6173 7320 6f66 2061 6c6c 2064 e class of all d │ │ │ │ +0004e600: 6961 676f 6e61 6c20 6163 7469 6f6e 730a iagonal actions. │ │ │ │ +0004e610: 2020 2a20 2a6e 6f74 6520 6469 6167 6f6e * *note diagon │ │ │ │ +0004e620: 616c 4163 7469 6f6e 3a20 6469 6167 6f6e alAction: diagon │ │ │ │ +0004e630: 616c 4163 7469 6f6e 2c20 2d2d 2064 6961 alAction, -- dia │ │ │ │ +0004e640: 676f 6e61 6c20 6772 6f75 7020 6163 7469 gonal group acti │ │ │ │ +0004e650: 6f6e 2076 6961 2077 6569 6768 7473 0a0a on via weights.. │ │ │ │ +0004e660: 5761 7973 2074 6f20 7573 6520 7765 6967 Ways to use weig │ │ │ │ +0004e670: 6874 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 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with op │ │ │ │ -0004fde0: 7469 6f6e 733a 0a28 4d61 6361 756c 6179 tions:.(Macaulay │ │ │ │ -0004fdf0: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ -0004fe00: 696f 6e57 6974 684f 7074 696f 6e73 2c2e ionWithOptions,. │ │ │ │ -0004fe10: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ -0004fe20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004fe30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004fe40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004fe50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004fe60: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -0004fe70: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -0004fe80: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -0004fe90: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -0004fea0: 6361 756c 6179 322d 312e 3236 2e30 352b caulay2-1.26.05+ │ │ │ │ -0004feb0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -0004fec0: 7061 636b 6167 6573 2f0a 496e 7661 7269 packages/.Invari │ │ │ │ -0004fed0: 616e 7452 696e 672f 4669 6e69 7465 4772 antRing/FiniteGr │ │ │ │ -0004fee0: 6f75 7073 446f 632e 6d32 3a32 3938 3a30 oupsDoc.m2:298:0 │ │ │ │ -0004fef0: 2e0a 1f0a 5461 6720 5461 626c 653a 0a4e ....Tag Table:.N │ │ │ │ -0004ff00: 6f64 653a 2054 6f70 7f32 3339 0a4e 6f64 ode: Top.239.Nod │ │ │ │ -0004ff10: 653a 2061 6374 696f 6e7f 3231 3034 340a e: action.21044. │ │ │ │ -0004ff20: 4e6f 6465 3a20 6163 7469 6f6e 4d61 7472 Node: actionMatr │ │ │ │ -0004ff30: 6978 7f32 3434 3334 0a4e 6f64 653a 2061 ix.24434.Node: a │ │ │ │ -0004ff40: 6d62 6965 6e74 5f6c 7052 696e 674f 6649 mbient_lpRingOfI │ │ │ │ -0004ff50: 6e76 6172 6961 6e74 735f 7270 7f32 3833 nvariants_rp.283 │ │ │ │ -0004ff60: 3132 0a4e 6f64 653a 2063 7963 6c69 6346 12.Node: cyclicF │ │ │ │ -0004ff70: 6163 746f 7273 7f33 3134 3234 0a4e 6f64 actors.31424.Nod │ │ │ │ -0004ff80: 653a 2064 6566 696e 696e 6749 6465 616c e: definingIdeal │ │ │ │ -0004ff90: 7f33 3431 3039 0a4e 6f64 653a 2064 6567 .34109.Node: deg │ │ │ │ -0004ffa0: 7265 6573 5269 6e67 5f6c 7044 6961 676f reesRing_lpDiago │ │ │ │ -0004ffb0: 6e61 6c41 6374 696f 6e5f 7270 7f33 3733 nalAction_rp.373 │ │ │ │ -0004ffc0: 3434 0a4e 6f64 653a 2044 6961 676f 6e61 44.Node: Diagona │ │ │ │ -0004ffd0: 6c41 6374 696f 6e7f 3430 3438 350a 4e6f lAction.40485.No │ │ │ │ -0004ffe0: 6465 3a20 6469 6167 6f6e 616c 4163 7469 de: diagonalActi │ │ │ │ -0004fff0: 6f6e 7f34 3335 3738 0a4e 6f64 653a 2064 on.43578.Node: d │ │ │ │ -00050000: 696d 5f6c 7047 726f 7570 4163 7469 6f6e im_lpGroupAction │ │ │ │ -00050010: 5f72 707f 3530 3936 360a 4e6f 6465 3a20 _rp.50966.Node: │ │ │ │ -00050020: 6571 7569 7661 7269 616e 7448 696c 6265 equivariantHilbe │ │ │ │ -00050030: 7274 7f35 3333 3135 0a4e 6f64 653a 2065 rt.53315.Node: e │ │ │ │ -00050040: 7175 6976 6172 6961 6e74 4869 6c62 6572 quivariantHilber │ │ │ │ -00050050: 7453 6572 6965 737f 3538 3935 360a 4e6f tSeries.58956.No │ │ │ │ -00050060: 6465 3a20 6669 6e69 7465 4163 7469 6f6e de: finiteAction │ │ │ │ -00050070: 7f36 3737 3934 0a4e 6f64 653a 2046 696e .67794.Node: Fin │ │ │ │ -00050080: 6974 6547 726f 7570 4163 7469 6f6e 7f37 iteGroupAction.7 │ │ │ │ -00050090: 3136 3534 0a4e 6f64 653a 2067 656e 6572 1654.Node: gener │ │ │ │ -000500a0: 6174 6f72 735f 6c70 4669 6e69 7465 4772 ators_lpFiniteGr │ │ │ │ -000500b0: 6f75 7041 6374 696f 6e5f 7270 7f37 3538 oupAction_rp.758 │ │ │ │ -000500c0: 3537 0a4e 6f64 653a 2067 656e 6572 6174 57.Node: generat │ │ │ │ -000500d0: 6f72 735f 6c70 5269 6e67 4f66 496e 7661 ors_lpRingOfInva │ │ │ │ -000500e0: 7269 616e 7473 5f72 707f 3739 3838 340a riants_rp.79884. │ │ │ │ -000500f0: 4e6f 6465 3a20 6772 6f75 707f 3832 3935 Node: group.8295 │ │ │ │ -00050100: 310a 4e6f 6465 3a20 4772 6f75 7041 6374 1.Node: GroupAct │ │ │ │ -00050110: 696f 6e7f 3836 3833 320a 4e6f 6465 3a20 ion.86832.Node: │ │ │ │ -00050120: 6772 6f75 7049 6465 616c 7f39 3030 3431 groupIdeal.90041 │ │ │ │ -00050130: 0a4e 6f64 653a 2068 696c 6265 7274 4964 .Node: hilbertId │ │ │ │ -00050140: 6561 6c7f 3933 3834 320a 4e6f 6465 3a20 eal.93842.Node: │ │ │ │ -00050150: 6869 6c62 6572 7453 6572 6965 735f 6c70 hilbertSeries_lp │ │ │ │ -00050160: 5269 6e67 4f66 496e 7661 7269 616e 7473 RingOfInvariants │ │ │ │ -00050170: 5f72 707f 3130 3231 3035 0a4e 6f64 653a _rp.102105.Node: │ │ │ │ -00050180: 2068 6972 6f6e 616b 6144 6563 6f6d 706f hironakaDecompo │ │ │ │ -00050190: 7369 7469 6f6e 7f31 3035 3232 390a 4e6f sition.105229.No │ │ │ │ -000501a0: 6465 3a20 6873 6f70 2061 6c67 6f72 6974 de: hsop algorit │ │ │ │ -000501b0: 686d 737f 3131 3036 3633 0a4e 6f64 653a hms.110663.Node: │ │ │ │ -000501c0: 2069 6e76 6172 6961 6e74 5269 6e67 7f31 invariantRing.1 │ │ │ │ -000501d0: 3333 3530 350a 4e6f 6465 3a20 696e 7661 33505.Node: inva │ │ │ │ -000501e0: 7269 616e 7473 7f31 3337 3930 350a 4e6f riants.137905.No │ │ │ │ -000501f0: 6465 3a20 696e 7661 7269 616e 7473 5f6c de: invariants_l │ │ │ │ -00050200: 705f 7064 5f70 645f 7064 5f63 6d44 6567 p_pd_pd_pd_cmDeg │ │ │ │ -00050210: 7265 6542 6f75 6e64 3d3e 5f70 645f 7064 reeBound=>_pd_pd │ │ │ │ -00050220: 5f70 645f 7270 7f31 3432 3838 380a 4e6f _pd_rp.142888.No │ │ │ │ -00050230: 6465 3a20 696e 7661 7269 616e 7473 5f6c de: invariants_l │ │ │ │ -00050240: 705f 7064 5f70 645f 7064 5f63 6d44 6567 p_pd_pd_pd_cmDeg │ │ │ │ -00050250: 7265 654c 696d 6974 3d3e 5f70 645f 7064 reeLimit=>_pd_pd │ │ │ │ -00050260: 5f70 645f 7270 7f31 3439 3334 370a 4e6f _pd_rp.149347.No │ │ │ │ -00050270: 6465 3a20 696e 7661 7269 616e 7473 5f6c de: invariants_l │ │ │ │ -00050280: 705f 7064 5f70 645f 7064 5f63 6d53 7472 p_pd_pd_pd_cmStr │ │ │ │ -00050290: 6174 6567 793d 3e5f 7064 5f70 645f 7064 ategy=>_pd_pd_pd │ │ │ │ -000502a0: 5f72 707f 3135 3239 3339 0a4e 6f64 653a _rp.152939.Node: │ │ │ │ -000502b0: 2069 6e76 6172 6961 6e74 735f 6c70 5f70 invariants_lp_p │ │ │ │ -000502c0: 645f 7064 5f70 645f 636d 5375 6272 696e d_pd_pd_cmSubrin │ │ │ │ -000502d0: 674c 696d 6974 3d3e 5f70 645f 7064 5f70 gLimit=>_pd_pd_p │ │ │ │ -000502e0: 645f 7270 7f31 3633 3231 340a 4e6f 6465 d_rp.163214.Node │ │ │ │ -000502f0: 3a20 696e 7661 7269 616e 7473 5f6c 705f : invariants_lp_ │ │ │ │ -00050300: 7064 5f70 645f 7064 5f63 6d55 7365 436f pd_pd_pd_cmUseCo │ │ │ │ -00050310: 6566 6669 6369 656e 7452 696e 673d 3e5f efficientRing=>_ │ │ │ │ -00050320: 7064 5f70 645f 7064 5f72 707f 3136 3533 pd_pd_pd_rp.1653 │ │ │ │ -00050330: 3737 0a4e 6f64 653a 2069 6e76 6172 6961 77.Node: invaria │ │ │ │ -00050340: 6e74 735f 6c70 5f70 645f 7064 5f70 645f nts_lp_pd_pd_pd_ │ │ │ │ -00050350: 636d 5573 6550 6f6c 7968 6564 7261 3d3e cmUsePolyhedra=> │ │ │ │ -00050360: 5f70 645f 7064 5f70 645f 7270 7f31 3639 _pd_pd_pd_rp.169 │ │ │ │ -00050370: 3638 390a 4e6f 6465 3a20 696e 7661 7269 689.Node: invari │ │ │ │ -00050380: 616e 7473 5f6c 7044 6961 676f 6e61 6c41 ants_lpDiagonalA │ │ │ │ -00050390: 6374 696f 6e5f 7270 7f31 3731 3931 390a ction_rp.171919. │ │ │ │ -000503a0: 4e6f 6465 3a20 696e 7661 7269 616e 7473 Node: invariants │ │ │ │ -000503b0: 5f6c 7046 696e 6974 6547 726f 7570 4163 _lpFiniteGroupAc │ │ │ │ -000503c0: 7469 6f6e 5f72 707f 3138 3132 3834 0a4e tion_rp.181284.N │ │ │ │ -000503d0: 6f64 653a 2069 6e76 6172 6961 6e74 735f ode: invariants_ │ │ │ │ -000503e0: 6c70 4669 6e69 7465 4772 6f75 7041 6374 lpFiniteGroupAct │ │ │ │ -000503f0: 696f 6e5f 636d 5a5a 5f72 707f 3138 3931 ion_cmZZ_rp.1891 │ │ │ │ -00050400: 3136 0a4e 6f64 653a 2069 6e76 6172 6961 16.Node: invaria │ │ │ │ -00050410: 6e74 735f 6c70 4c69 6e65 6172 6c79 5265 nts_lpLinearlyRe │ │ │ │ -00050420: 6475 6374 6976 6541 6374 696f 6e5f 7270 ductiveAction_rp │ │ │ │ -00050430: 7f31 3934 3434 360a 4e6f 6465 3a20 696e .194446.Node: in │ │ │ │ -00050440: 7661 7269 616e 7473 5f6c 704c 696e 6561 variants_lpLinea │ │ │ │ -00050450: 726c 7952 6564 7563 7469 7665 4163 7469 rlyReductiveActi │ │ │ │ -00050460: 6f6e 5f63 6d5a 5a5f 7270 7f32 3031 3530 on_cmZZ_rp.20150 │ │ │ │ -00050470: 330a 4e6f 6465 3a20 6973 4162 656c 6961 3.Node: isAbelia │ │ │ │ -00050480: 6e7f 3230 3934 3836 0a4e 6f64 653a 2069 n.209486.Node: i │ │ │ │ -00050490: 7349 6e76 6172 6961 6e74 7f32 3135 3131 sInvariant.21511 │ │ │ │ -000504a0: 390a 4e6f 6465 3a20 4c69 6e65 6172 6c79 9.Node: Linearly │ │ │ │ -000504b0: 5265 6475 6374 6976 6541 6374 696f 6e7f ReductiveAction. │ │ │ │ -000504c0: 3232 3635 3236 0a4e 6f64 653a 206c 696e 226526.Node: lin │ │ │ │ -000504d0: 6561 726c 7952 6564 7563 7469 7665 4163 earlyReductiveAc │ │ │ │ -000504e0: 7469 6f6e 7f32 3239 3838 370a 4e6f 6465 tion.229887.Node │ │ │ │ -000504f0: 3a20 6d6f 6c69 656e 5365 7269 6573 7f32 : molienSeries.2 │ │ │ │ -00050500: 3336 3530 340a 4e6f 6465 3a20 6e65 745f 36504.Node: net_ │ │ │ │ -00050510: 6c70 5269 6e67 4f66 496e 7661 7269 616e lpRingOfInvarian │ │ │ │ -00050520: 7473 5f72 707f 3233 3839 3831 0a4e 6f64 ts_rp.238981.Nod │ │ │ │ -00050530: 653a 206e 756d 6765 6e73 5f6c 7044 6961 e: numgens_lpDia │ │ │ │ -00050540: 676f 6e61 6c41 6374 696f 6e5f 7270 7f32 gonalAction_rp.2 │ │ │ │ -00050550: 3339 3837 370a 4e6f 6465 3a20 6e75 6d67 39877.Node: numg │ │ │ │ -00050560: 656e 735f 6c70 4669 6e69 7465 4772 6f75 ens_lpFiniteGrou │ │ │ │ -00050570: 7041 6374 696f 6e5f 7270 7f32 3432 3438 pAction_rp.24248 │ │ │ │ -00050580: 330a 4e6f 6465 3a20 7065 726d 7574 6174 3.Node: permutat │ │ │ │ -00050590: 696f 6e4d 6174 7269 787f 3234 3632 3736 ionMatrix.246276 │ │ │ │ -000505a0: 0a4e 6f64 653a 2070 7269 6d61 7279 496e .Node: primaryIn │ │ │ │ -000505b0: 7661 7269 616e 7473 7f32 3530 3334 320a variants.250342. │ │ │ │ -000505c0: 4e6f 6465 3a20 7072 696d 6172 7949 6e76 Node: primaryInv │ │ │ │ -000505d0: 6172 6961 6e74 735f 6c70 5f70 645f 7064 ariants_lp_pd_pd │ │ │ │ -000505e0: 5f70 645f 636d 4461 6465 3d3e 5f70 645f _pd_cmDade=>_pd_ │ │ │ │ -000505f0: 7064 5f70 645f 7270 7f32 3539 3631 310a pd_pd_rp.259611. │ │ │ │ -00050600: 4e6f 6465 3a20 7072 696d 6172 7949 6e76 Node: primaryInv │ │ │ │ -00050610: 6172 6961 6e74 735f 6c70 5f70 645f 7064 ariants_lp_pd_pd │ │ │ │ -00050620: 5f70 645f 636d 4465 6772 6565 5665 6374 _pd_cmDegreeVect │ │ │ │ -00050630: 6f72 3d3e 5f70 645f 7064 5f70 645f 7270 or=>_pd_pd_pd_rp │ │ │ │ -00050640: 7f32 3734 3935 350a 4e6f 6465 3a20 7261 .274955.Node: ra │ │ │ │ -00050650: 6e6b 5f6c 7044 6961 676f 6e61 6c41 6374 nk_lpDiagonalAct │ │ │ │ -00050660: 696f 6e5f 7270 7f32 3831 3833 360a 4e6f ion_rp.281836.No │ │ │ │ -00050670: 6465 3a20 7265 6c61 7469 6f6e 735f 6c70 de: relations_lp │ │ │ │ -00050680: 4669 6e69 7465 4772 6f75 7041 6374 696f FiniteGroupActio │ │ │ │ -00050690: 6e5f 7270 7f32 3834 3432 300a 4e6f 6465 n_rp.284420.Node │ │ │ │ -000506a0: 3a20 7265 796e 6f6c 6473 4f70 6572 6174 : reynoldsOperat │ │ │ │ -000506b0: 6f72 7f32 3838 3932 360a 4e6f 6465 3a20 or.288926.Node: │ │ │ │ -000506c0: 7269 6e67 5f6c 7047 726f 7570 4163 7469 ring_lpGroupActi │ │ │ │ -000506d0: 6f6e 5f72 707f 3239 3531 3930 0a4e 6f64 on_rp.295190.Nod │ │ │ │ -000506e0: 653a 2052 696e 674f 6649 6e76 6172 6961 e: RingOfInvaria │ │ │ │ -000506f0: 6e74 737f 3239 3734 3831 0a4e 6f64 653a nts.297481.Node: │ │ │ │ -00050700: 2073 6368 7265 6965 7247 7261 7068 7f32 schreierGraph.2 │ │ │ │ -00050710: 3939 3936 380a 4e6f 6465 3a20 7365 636f 99968.Node: seco │ │ │ │ -00050720: 6e64 6172 7949 6e76 6172 6961 6e74 737f ndaryInvariants. │ │ │ │ -00050730: 3330 3638 3431 0a4e 6f64 653a 2073 6563 306841.Node: sec │ │ │ │ -00050740: 6f6e 6461 7279 496e 7661 7269 616e 7473 ondaryInvariants │ │ │ │ -00050750: 5f6c 705f 7064 5f70 645f 7064 5f63 6d50 _lp_pd_pd_pd_cmP │ │ │ │ -00050760: 7269 6e74 4465 6772 6565 506f 6c79 6e6f rintDegreePolyno │ │ │ │ -00050770: 6d69 616c 3d3e 5f70 645f 7064 5f70 645f mial=>_pd_pd_pd_ │ │ │ │ -00050780: 7270 7f33 3130 3736 380a 4e6f 6465 3a20 rp.310768.Node: │ │ │ │ -00050790: 7765 6967 6874 737f 3331 3835 3235 0a4e weights.318525.N │ │ │ │ -000507a0: 6f64 653a 2077 6f72 6473 7f33 3232 3735 ode: words.32275 │ │ │ │ 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6f75 702e 0a0a 5365 6520 616c 736f group...See also │ │ │ │ +0004f7d0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +0004f7e0: 6e6f 7465 2067 726f 7570 3a20 6772 6f75 note group: grou │ │ │ │ +0004f7f0: 702c 202d 2d20 6c69 7374 2061 6c6c 2065 p, -- list all e │ │ │ │ +0004f800: 6c65 6d65 6e74 7320 6f66 2074 6865 2067 lements of the g │ │ │ │ +0004f810: 726f 7570 206f 6620 6120 6669 6e69 7465 roup of a finite │ │ │ │ +0004f820: 2067 726f 7570 0a20 2020 2061 6374 696f group. actio │ │ │ │ +0004f830: 6e0a 2020 2a20 2a6e 6f74 6520 7363 6872 n. * *note schr │ │ │ │ +0004f840: 6569 6572 4772 6170 683a 2073 6368 7265 eierGraph: schre │ │ │ │ +0004f850: 6965 7247 7261 7068 2c20 2d2d 2053 6368 ierGraph, -- Sch │ │ │ │ +0004f860: 7265 6965 7220 6772 6170 6820 6f66 2061 reier graph of a │ │ │ │ +0004f870: 2066 696e 6974 6520 6772 6f75 700a 0a57 finite group..W │ │ │ │ +0004f880: 6179 7320 746f 2075 7365 2077 6f72 6473 ays to use words │ │ │ │ +0004f890: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +0004f8a0: 3d3d 3d3d 0a0a 2020 2a20 2277 6f72 6473 ====.. * "words │ │ │ │ +0004f8b0: 2846 696e 6974 6547 726f 7570 4163 7469 (FiniteGroupActi │ │ │ │ +0004f8c0: 6f6e 2922 0a0a 466f 7220 7468 6520 7072 on)"..For the pr │ │ │ │ +0004f8d0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +0004f8e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +0004f8f0: 206f 626a 6563 7420 2a6e 6f74 6520 776f object *note wo │ │ │ │ +0004f900: 7264 733a 2077 6f72 6473 2c20 6973 2061 rds: words, is a │ │ │ │ +0004f910: 202a 6e6f 7465 206d 6574 686f 6420 6675 *note method fu │ │ │ │ +0004f920: 6e63 7469 6f6e 2077 6974 6820 6f70 7469 nction with opti │ │ │ │ +0004f930: 6f6e 733a 0a28 4d61 6361 756c 6179 3244 ons:.(Macaulay2D │ │ │ │ +0004f940: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ +0004f950: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a nWithOptions,... │ │ │ │ +0004f960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004f970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004f980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +0004f9b0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +0004f9c0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +0004f9d0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +0004f9e0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +0004f9f0: 756c 6179 322d 312e 3236 2e30 352b 6473 ulay2-1.26.05+ds │ │ │ │ +0004fa00: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +0004fa10: 636b 6167 6573 2f0a 496e 7661 7269 616e ckages/.Invarian │ │ │ │ +0004fa20: 7452 696e 672f 4669 6e69 7465 4772 6f75 tRing/FiniteGrou │ │ │ │ +0004fa30: 7073 446f 632e 6d32 3a32 3938 3a30 2e0a psDoc.m2:298:0.. │ │ │ │ +0004fa40: 1f0a 5461 6720 5461 626c 653a 0a4e 6f64 ..Tag Table:.Nod │ │ │ │ +0004fa50: 653a 2054 6f70 7f32 3339 0a4e 6f64 653a e: Top.239.Node: │ │ │ │ +0004fa60: 2061 6374 696f 6e7f 3231 3034 340a 4e6f action.21044.No │ │ │ │ +0004fa70: 6465 3a20 6163 7469 6f6e 4d61 7472 6978 de: actionMatrix │ │ │ │ +0004fa80: 7f32 3434 3334 0a4e 6f64 653a 2061 6d62 .24434.Node: amb │ │ │ │ +0004fa90: 6965 6e74 5f6c 7052 696e 674f 6649 6e76 ient_lpRingOfInv │ │ │ │ +0004faa0: 6172 6961 6e74 735f 7270 7f32 3833 3132 ariants_rp.28312 │ │ │ │ +0004fab0: 0a4e 6f64 653a 2063 7963 6c69 6346 6163 .Node: cyclicFac │ │ │ │ +0004fac0: 746f 7273 7f33 3134 3234 0a4e 6f64 653a tors.31424.Node: │ │ │ │ +0004fad0: 2064 6566 696e 696e 6749 6465 616c 7f33 definingIdeal.3 │ │ │ │ +0004fae0: 3431 3039 0a4e 6f64 653a 2064 6567 7265 4109.Node: degre │ │ │ │ +0004faf0: 6573 5269 6e67 5f6c 7044 6961 676f 6e61 esRing_lpDiagona │ │ │ │ +0004fb00: 6c41 6374 696f 6e5f 7270 7f33 3733 3434 lAction_rp.37344 │ │ │ │ +0004fb10: 0a4e 6f64 653a 2044 6961 676f 6e61 6c41 .Node: DiagonalA │ │ │ │ +0004fb20: 6374 696f 6e7f 3430 3438 350a 4e6f 6465 ction.40485.Node │ │ │ │ +0004fb30: 3a20 6469 6167 6f6e 616c 4163 7469 6f6e : diagonalAction │ │ │ │ +0004fb40: 7f34 3335 3738 0a4e 6f64 653a 2064 696d .43578.Node: dim │ │ │ │ +0004fb50: 5f6c 7047 726f 7570 4163 7469 6f6e 5f72 _lpGroupAction_r │ │ │ │ +0004fb60: 707f 3530 3936 360a 4e6f 6465 3a20 6571 p.50966.Node: eq │ │ │ │ +0004fb70: 7569 7661 7269 616e 7448 696c 6265 7274 uivariantHilbert │ │ │ │ +0004fb80: 7f35 3333 3135 0a4e 6f64 653a 2065 7175 .53315.Node: equ │ │ │ │ +0004fb90: 6976 6172 6961 6e74 4869 6c62 6572 7453 ivariantHilbertS │ │ │ │ +0004fba0: 6572 6965 737f 3538 3935 360a 4e6f 6465 eries.58956.Node │ │ │ │ +0004fbb0: 3a20 6669 6e69 7465 4163 7469 6f6e 7f36 : finiteAction.6 │ │ │ │ +0004fbc0: 3737 3934 0a4e 6f64 653a 2046 696e 6974 7794.Node: Finit │ │ │ │ +0004fbd0: 6547 726f 7570 4163 7469 6f6e 7f37 3136 eGroupAction.716 │ │ │ │ +0004fbe0: 3534 0a4e 6f64 653a 2067 656e 6572 6174 54.Node: generat │ │ │ │ +0004fbf0: 6f72 735f 6c70 4669 6e69 7465 4772 6f75 ors_lpFiniteGrou │ │ │ │ +0004fc00: 7041 6374 696f 6e5f 7270 7f37 3538 3537 pAction_rp.75857 │ │ │ │ +0004fc10: 0a4e 6f64 653a 2067 656e 6572 6174 6f72 .Node: generator │ │ │ │ +0004fc20: 735f 6c70 5269 6e67 4f66 496e 7661 7269 s_lpRingOfInvari │ │ │ │ +0004fc30: 616e 7473 5f72 707f 3739 3838 340a 4e6f ants_rp.79884.No │ │ │ │ +0004fc40: 6465 3a20 6772 6f75 707f 3832 3935 310a de: group.82951. │ │ │ │ +0004fc50: 4e6f 6465 3a20 4772 6f75 7041 6374 696f Node: GroupActio │ │ │ │ +0004fc60: 6e7f 3836 3833 320a 4e6f 6465 3a20 6772 n.86832.Node: gr │ │ │ │ +0004fc70: 6f75 7049 6465 616c 7f39 3030 3431 0a4e oupIdeal.90041.N │ │ │ │ +0004fc80: 6f64 653a 2068 696c 6265 7274 4964 6561 ode: hilbertIdea │ │ │ │ +0004fc90: 6c7f 3933 3834 320a 4e6f 6465 3a20 6869 l.93842.Node: hi │ │ │ │ +0004fca0: 6c62 6572 7453 6572 6965 735f 6c70 5269 lbertSeries_lpRi │ │ │ │ +0004fcb0: 6e67 4f66 496e 7661 7269 616e 7473 5f72 ngOfInvariants_r │ │ │ │ +0004fcc0: 707f 3130 3231 3035 0a4e 6f64 653a 2068 p.102105.Node: h │ │ │ │ +0004fcd0: 6972 6f6e 616b 6144 6563 6f6d 706f 7369 ironakaDecomposi │ │ │ │ +0004fce0: 7469 6f6e 7f31 3035 3232 390a 4e6f 6465 tion.105229.Node │ │ │ │ +0004fcf0: 3a20 6873 6f70 2061 6c67 6f72 6974 686d : hsop algorithm │ │ │ │ +0004fd00: 737f 3131 3036 3633 0a4e 6f64 653a 2069 s.110663.Node: i │ │ │ │ +0004fd10: 6e76 6172 6961 6e74 5269 6e67 7f31 3333 nvariantRing.133 │ │ │ │ +0004fd20: 3530 350a 4e6f 6465 3a20 696e 7661 7269 505.Node: invari │ │ │ │ +0004fd30: 616e 7473 7f31 3337 3930 350a 4e6f 6465 ants.137905.Node │ │ │ │ +0004fd40: 3a20 696e 7661 7269 616e 7473 5f6c 705f : invariants_lp_ │ │ │ │ +0004fd50: 7064 5f70 645f 7064 5f63 6d44 6567 7265 pd_pd_pd_cmDegre │ │ │ │ +0004fd60: 6542 6f75 6e64 3d3e 5f70 645f 7064 5f70 eBound=>_pd_pd_p │ │ │ │ +0004fd70: 645f 7270 7f31 3432 3838 380a 4e6f 6465 d_rp.142888.Node │ │ │ │ +0004fd80: 3a20 696e 7661 7269 616e 7473 5f6c 705f : invariants_lp_ │ │ │ │ +0004fd90: 7064 5f70 645f 7064 5f63 6d44 6567 7265 pd_pd_pd_cmDegre │ │ │ │ +0004fda0: 654c 696d 6974 3d3e 5f70 645f 7064 5f70 eLimit=>_pd_pd_p │ │ │ │ +0004fdb0: 645f 7270 7f31 3439 3334 370a 4e6f 6465 d_rp.149347.Node │ │ │ │ +0004fdc0: 3a20 696e 7661 7269 616e 7473 5f6c 705f : invariants_lp_ │ │ │ │ +0004fdd0: 7064 5f70 645f 7064 5f63 6d53 7472 6174 pd_pd_pd_cmStrat │ │ │ │ +0004fde0: 6567 793d 3e5f 7064 5f70 645f 7064 5f72 egy=>_pd_pd_pd_r │ │ │ │ +0004fdf0: 707f 3135 3239 3339 0a4e 6f64 653a 2069 p.152939.Node: i │ │ │ │ +0004fe00: 6e76 6172 6961 6e74 735f 6c70 5f70 645f nvariants_lp_pd_ │ │ │ │ +0004fe10: 7064 5f70 645f 636d 5375 6272 696e 674c pd_pd_cmSubringL │ │ │ │ +0004fe20: 696d 6974 3d3e 5f70 645f 7064 5f70 645f imit=>_pd_pd_pd_ │ │ │ │ +0004fe30: 7270 7f31 3633 3231 340a 4e6f 6465 3a20 rp.163214.Node: │ │ │ │ +0004fe40: 696e 7661 7269 616e 7473 5f6c 705f 7064 invariants_lp_pd │ │ │ │ +0004fe50: 5f70 645f 7064 5f63 6d55 7365 436f 6566 _pd_pd_cmUseCoef │ │ │ │ +0004fe60: 6669 6369 656e 7452 696e 673d 3e5f 7064 ficientRing=>_pd │ │ │ │ +0004fe70: 5f70 645f 7064 5f72 707f 3136 3533 3737 _pd_pd_rp.165377 │ │ │ │ +0004fe80: 0a4e 6f64 653a 2069 6e76 6172 6961 6e74 .Node: invariant │ │ │ │ +0004fe90: 735f 6c70 5f70 645f 7064 5f70 645f 636d s_lp_pd_pd_pd_cm │ │ │ │ +0004fea0: 5573 6550 6f6c 7968 6564 7261 3d3e 5f70 UsePolyhedra=>_p │ │ │ │ +0004feb0: 645f 7064 5f70 645f 7270 7f31 3639 3638 d_pd_pd_rp.16968 │ │ │ │ +0004fec0: 390a 4e6f 6465 3a20 696e 7661 7269 616e 9.Node: invarian │ │ │ │ +0004fed0: 7473 5f6c 7044 6961 676f 6e61 6c41 6374 ts_lpDiagonalAct │ │ │ │ +0004fee0: 696f 6e5f 7270 7f31 3731 3931 390a 4e6f ion_rp.171919.No │ │ │ │ +0004fef0: 6465 3a20 696e 7661 7269 616e 7473 5f6c de: invariants_l │ │ │ │ +0004ff00: 7046 696e 6974 6547 726f 7570 4163 7469 pFiniteGroupActi │ │ │ │ +0004ff10: 6f6e 5f72 707f 3138 3132 3834 0a4e 6f64 on_rp.181284.Nod │ │ │ │ +0004ff20: 653a 2069 6e76 6172 6961 6e74 735f 6c70 e: invariants_lp │ │ │ │ +0004ff30: 4669 6e69 7465 4772 6f75 7041 6374 696f FiniteGroupActio │ │ │ │ +0004ff40: 6e5f 636d 5a5a 5f72 707f 3138 3931 3136 n_cmZZ_rp.189116 │ │ │ │ +0004ff50: 0a4e 6f64 653a 2069 6e76 6172 6961 6e74 .Node: invariant │ │ │ │ +0004ff60: 735f 6c70 4c69 6e65 6172 6c79 5265 6475 s_lpLinearlyRedu │ │ │ │ +0004ff70: 6374 6976 6541 6374 696f 6e5f 7270 7f31 ctiveAction_rp.1 │ │ │ │ +0004ff80: 3934 3434 360a 4e6f 6465 3a20 696e 7661 94446.Node: inva │ │ │ │ +0004ff90: 7269 616e 7473 5f6c 704c 696e 6561 726c riants_lpLinearl │ │ │ │ +0004ffa0: 7952 6564 7563 7469 7665 4163 7469 6f6e yReductiveAction │ │ │ │ +0004ffb0: 5f63 6d5a 5a5f 7270 7f32 3031 3530 330a _cmZZ_rp.201503. │ │ │ │ +0004ffc0: 4e6f 6465 3a20 6973 4162 656c 6961 6e7f Node: isAbelian. │ │ │ │ +0004ffd0: 3230 3934 3836 0a4e 6f64 653a 2069 7349 209486.Node: isI │ │ │ │ +0004ffe0: 6e76 6172 6961 6e74 7f32 3135 3131 390a nvariant.215119. │ │ │ │ +0004fff0: 4e6f 6465 3a20 4c69 6e65 6172 6c79 5265 Node: LinearlyRe │ │ │ │ +00050000: 6475 6374 6976 6541 6374 696f 6e7f 3232 ductiveAction.22 │ │ │ │ +00050010: 3635 3236 0a4e 6f64 653a 206c 696e 6561 6526.Node: linea │ │ │ │ +00050020: 726c 7952 6564 7563 7469 7665 4163 7469 rlyReductiveActi │ │ │ │ +00050030: 6f6e 7f32 3239 3838 370a 4e6f 6465 3a20 on.229887.Node: │ │ │ │ +00050040: 6d6f 6c69 656e 5365 7269 6573 7f32 3336 molienSeries.236 │ │ │ │ +00050050: 3530 340a 4e6f 6465 3a20 6e65 745f 6c70 504.Node: net_lp │ │ │ │ +00050060: 5269 6e67 4f66 496e 7661 7269 616e 7473 RingOfInvariants │ │ │ │ +00050070: 5f72 707f 3233 3839 3831 0a4e 6f64 653a _rp.238981.Node: │ │ │ │ +00050080: 206e 756d 6765 6e73 5f6c 7044 6961 676f numgens_lpDiago │ │ │ │ +00050090: 6e61 6c41 6374 696f 6e5f 7270 7f32 3339 nalAction_rp.239 │ │ │ │ +000500a0: 3837 370a 4e6f 6465 3a20 6e75 6d67 656e 877.Node: numgen │ │ │ │ +000500b0: 735f 6c70 4669 6e69 7465 4772 6f75 7041 s_lpFiniteGroupA │ │ │ │ +000500c0: 6374 696f 6e5f 7270 7f32 3432 3438 330a ction_rp.242483. │ │ │ │ +000500d0: 4e6f 6465 3a20 7065 726d 7574 6174 696f Node: permutatio │ │ │ │ +000500e0: 6e4d 6174 7269 787f 3234 3632 3736 0a4e nMatrix.246276.N │ │ │ │ +000500f0: 6f64 653a 2070 7269 6d61 7279 496e 7661 ode: primaryInva │ │ │ │ +00050100: 7269 616e 7473 7f32 3530 3334 320a 4e6f riants.250342.No │ │ │ │ +00050110: 6465 3a20 7072 696d 6172 7949 6e76 6172 de: primaryInvar │ │ │ │ +00050120: 6961 6e74 735f 6c70 5f70 645f 7064 5f70 iants_lp_pd_pd_p │ │ │ │ +00050130: 645f 636d 4461 6465 3d3e 5f70 645f 7064 d_cmDade=>_pd_pd │ │ │ │ +00050140: 5f70 645f 7270 7f32 3539 3430 330a 4e6f _pd_rp.259403.No │ │ │ │ +00050150: 6465 3a20 7072 696d 6172 7949 6e76 6172 de: primaryInvar │ │ │ │ +00050160: 6961 6e74 735f 6c70 5f70 645f 7064 5f70 iants_lp_pd_pd_p │ │ │ │ +00050170: 645f 636d 4465 6772 6565 5665 6374 6f72 d_cmDegreeVector │ │ │ │ +00050180: 3d3e 5f70 645f 7064 5f70 645f 7270 7f32 =>_pd_pd_pd_rp.2 │ │ │ │ +00050190: 3734 3734 370a 4e6f 6465 3a20 7261 6e6b 74747.Node: rank │ │ │ │ +000501a0: 5f6c 7044 6961 676f 6e61 6c41 6374 696f _lpDiagonalActio │ │ │ │ +000501b0: 6e5f 7270 7f32 3830 3633 340a 4e6f 6465 n_rp.280634.Node │ │ │ │ +000501c0: 3a20 7265 6c61 7469 6f6e 735f 6c70 4669 : relations_lpFi │ │ │ │ +000501d0: 6e69 7465 4772 6f75 7041 6374 696f 6e5f niteGroupAction_ │ │ │ │ +000501e0: 7270 7f32 3833 3231 380a 4e6f 6465 3a20 rp.283218.Node: │ │ │ │ +000501f0: 7265 796e 6f6c 6473 4f70 6572 6174 6f72 reynoldsOperator │ │ │ │ +00050200: 7f32 3837 3732 340a 4e6f 6465 3a20 7269 .287724.Node: ri │ │ │ │ +00050210: 6e67 5f6c 7047 726f 7570 4163 7469 6f6e ng_lpGroupAction │ │ │ │ +00050220: 5f72 707f 3239 3339 3838 0a4e 6f64 653a _rp.293988.Node: │ │ │ │ +00050230: 2052 696e 674f 6649 6e76 6172 6961 6e74 RingOfInvariant │ │ │ │ +00050240: 737f 3239 3632 3739 0a4e 6f64 653a 2073 s.296279.Node: s │ │ │ │ +00050250: 6368 7265 6965 7247 7261 7068 7f32 3938 chreierGraph.298 │ │ │ │ +00050260: 3736 360a 4e6f 6465 3a20 7365 636f 6e64 766.Node: second │ │ │ │ +00050270: 6172 7949 6e76 6172 6961 6e74 737f 3330 aryInvariants.30 │ │ │ │ +00050280: 3536 3339 0a4e 6f64 653a 2073 6563 6f6e 5639.Node: secon │ │ │ │ +00050290: 6461 7279 496e 7661 7269 616e 7473 5f6c daryInvariants_l │ │ │ │ +000502a0: 705f 7064 5f70 645f 7064 5f63 6d50 7269 p_pd_pd_pd_cmPri │ │ │ │ +000502b0: 6e74 4465 6772 6565 506f 6c79 6e6f 6d69 ntDegreePolynomi │ │ │ │ +000502c0: 616c 3d3e 5f70 645f 7064 5f70 645f 7270 al=>_pd_pd_pd_rp │ │ │ │ +000502d0: 7f33 3039 3536 360a 4e6f 6465 3a20 7765 .309566.Node: we │ │ │ │ +000502e0: 6967 6874 737f 3331 3733 3233 0a4e 6f64 ights.317323.Nod │ │ │ │ +000502f0: 653a 2077 6f72 6473 7f33 3231 3535 330a e: words.321553. │ │ │ │ +00050300: 1f0a 456e 6420 5461 6720 5461 626c 650a ..End Tag Table. │ │ ├── ./usr/share/info/Isomorphism.info.gz │ │ │ ├── Isomorphism.info │ │ │ │ @@ -4544,15 +4544,15 @@ │ │ │ │ 00011bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011c00: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3320 3a20 -------+.|i23 : │ │ │ │ 00011c10: 656c 6170 7365 6454 696d 6520 6973 4973 elapsedTime isIs │ │ │ │ 00011c20: 6f6d 6f72 7068 6963 2854 312c 2054 3229 omorphic(T1, T2) │ │ │ │ 00011c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c50: 2020 2020 2020 207c 0a7c 202d 2d20 312e |.| -- 1. │ │ │ │ -00011c60: 3533 3736 7320 656c 6170 7365 6420 2020 5376s elapsed │ │ │ │ +00011c60: 3638 3132 3273 2065 6c61 7073 6564 2020 68122s elapsed │ │ │ │ 00011c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011ca0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00011cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4569,15 +4569,15 @@ │ │ │ │ 00011d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011d90: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3420 3a20 -------+.|i24 : │ │ │ │ 00011da0: 656c 6170 7365 6454 696d 6520 6973 6f6d elapsedTime isom │ │ │ │ 00011db0: 6f72 7068 6973 6d28 5431 2c20 5432 2920 orphism(T1, T2) │ │ │ │ 00011dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011de0: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -00011df0: 3030 3032 3033 3339 7320 656c 6170 7365 00020339s elapse │ │ │ │ +00011df0: 3030 3032 3237 3338 7320 656c 6170 7365 00022738s elapse │ │ │ │ 00011e00: 6420 2020 2020 2020 2020 2020 2020 2020 d │ │ │ │ 00011e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00011e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/JSON.info.gz │ │ │ ├── JSON.info │ │ │ │ @@ -456,28 +456,28 @@ │ │ │ │ 00001c70: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 206a -------+.|i9 : j │ │ │ │ 00001c80: 736f 6e46 696c 6520 3d20 7465 6d70 6f72 sonFile = tempor │ │ │ │ 00001c90: 6172 7946 696c 654e 616d 6528 2920 7c20 aryFileName() | │ │ │ │ 00001ca0: 222e 6a73 6f6e 2220 7c0a 7c20 2020 2020 ".json" |.| │ │ │ │ 00001cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001cd0: 2020 2020 2020 2020 207c 0a7c 6f39 203d |.|o9 = │ │ │ │ -00001ce0: 202f 746d 702f 4d32 2d33 3437 3131 2d30 /tmp/M2-34711-0 │ │ │ │ +00001ce0: 202f 746d 702f 4d32 2d34 3733 3538 2d30 /tmp/M2-47358-0 │ │ │ │ 00001cf0: 2f30 2e6a 736f 6e20 2020 2020 2020 2020 /0.json │ │ │ │ 00001d00: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 00001d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ 00001d40: 3020 3a20 6a73 6f6e 4669 6c65 203c 3c20 0 : jsonFile << │ │ │ │ 00001d50: 225b 312c 2032 2c20 335d 2220 3c3c 2065 "[1, 2, 3]" << e │ │ │ │ 00001d60: 6e64 6c20 3c3c 2063 6c6f 7365 7c0a 7c20 ndl << close|.| │ │ │ │ 00001d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001d90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00001da0: 6f31 3020 3d20 2f74 6d70 2f4d 322d 3334 o10 = /tmp/M2-34 │ │ │ │ -00001db0: 3731 312d 302f 302e 6a73 6f6e 2020 2020 711-0/0.json │ │ │ │ +00001da0: 6f31 3020 3d20 2f74 6d70 2f4d 322d 3437 o10 = /tmp/M2-47 │ │ │ │ +00001db0: 3335 382d 302f 302e 6a73 6f6e 2020 2020 358-0/0.json │ │ │ │ 00001dc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00001dd0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00001de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001df0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00001e00: 0a7c 6f31 3020 3a20 4669 6c65 2020 2020 .|o10 : File │ │ │ │ 00001e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/Jets.info.gz │ │ │ ├── Jets.info │ │ │ │ @@ -5260,15 +5260,15 @@ │ │ │ │ 000148b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000148c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ 000148d0: 656c 6170 7365 6454 696d 6520 6a65 7473 elapsedTime jets │ │ │ │ 000148e0: 5261 6469 6361 6c28 322c 4929 2020 2020 Radical(2,I) │ │ │ │ 000148f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014910: 2020 2020 2020 2020 7c0a 7c20 2d2d 202e |.| -- . │ │ │ │ -00014920: 3030 3231 3236 3336 7320 656c 6170 7365 00212636s elapse │ │ │ │ +00014920: 3030 3237 3939 3431 7320 656c 6170 7365 00279941s elapse │ │ │ │ 00014930: 6420 2020 2020 2020 2020 2020 2020 2020 d │ │ │ │ 00014940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014960: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00014970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5305,15 +5305,15 @@ │ │ │ │ 00014b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014b90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ 00014ba0: 656c 6170 7365 6454 696d 6520 7261 6469 elapsedTime radi │ │ │ │ 00014bb0: 6361 6c20 4a32 4920 2020 2020 2020 2020 cal J2I │ │ │ │ 00014bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014be0: 2020 2020 2020 2020 7c0a 7c20 2d2d 202e |.| -- . │ │ │ │ -00014bf0: 3335 3636 3173 2065 6c61 7073 6564 2020 35661s elapsed │ │ │ │ +00014bf0: 3236 3130 3973 2065 6c61 7073 6564 2020 26109s elapsed │ │ │ │ 00014c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00014c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8104,16 +8104,16 @@ │ │ │ │ 0001fa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fa90: 2d2d 2b0a 7c69 3820 3a20 656c 6170 7365 --+.|i8 : elapse │ │ │ │ 0001faa0: 6454 696d 6520 6a65 7473 2833 2c49 2920 dTime jets(3,I) │ │ │ │ 0001fab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fae0: 207c 0a7c 202d 2d20 2e30 3036 3937 3436 |.| -- .0069746 │ │ │ │ -0001faf0: 3873 2065 6c61 7073 6564 2020 2020 2020 8s elapsed │ │ │ │ +0001fae0: 207c 0a7c 202d 2d20 2e30 3038 3335 3432 |.| -- .0083542 │ │ │ │ +0001faf0: 3173 2065 6c61 7073 6564 2020 2020 2020 1s elapsed │ │ │ │ 0001fb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fb30: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001fb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8203,15 +8203,15 @@ │ │ │ │ 000200a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000200b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000200c0: 7c69 3131 203a 2065 6c61 7073 6564 5469 |i11 : elapsedTi │ │ │ │ 000200d0: 6d65 206a 6574 7328 332c 4929 2020 2020 me jets(3,I) │ │ │ │ 000200e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000200f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020100: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00020110: 202d 2d20 2e30 3032 3334 3030 3873 2065 -- .00234008s e │ │ │ │ +00020110: 202d 2d20 2e30 3032 3833 3034 3273 2065 -- .00283042s e │ │ │ │ 00020120: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 00020130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020150: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00020160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8243,15 +8243,15 @@ │ │ │ │ 00020320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020330: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2065 ------+.|i12 : e │ │ │ │ 00020340: 6c61 7073 6564 5469 6d65 206a 6574 7328 lapsedTime jets( │ │ │ │ 00020350: 322c 4929 2020 2020 2020 2020 2020 2020 2,I) │ │ │ │ 00020360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020380: 2020 2020 207c 0a7c 202d 2d20 2e30 3032 |.| -- .002 │ │ │ │ -00020390: 3038 3230 3573 2065 6c61 7073 6564 2020 08205s elapsed │ │ │ │ +00020390: 3630 3235 3173 2065 6c61 7073 6564 2020 60251s elapsed │ │ │ │ 000203a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000203b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000203c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000203d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000203e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000203f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8754,15 +8754,15 @@ │ │ │ │ 00022310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00022330: 0a7c 6932 3420 3a20 656c 6170 7365 6454 .|i24 : elapsedT │ │ │ │ 00022340: 696d 6520 6a65 7473 2833 2c66 2920 2020 ime jets(3,f) │ │ │ │ 00022350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022370: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00022380: 0a7c 202d 2d20 2e30 3130 3330 3431 7320 .| -- .0103041s │ │ │ │ +00022380: 0a7c 202d 2d20 2e30 3133 3233 3138 7320 .| -- .0132318s │ │ │ │ 00022390: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 000223a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000223b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000223c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 000223d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000223e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000223f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8944,15 +8944,15 @@ │ │ │ │ 00022ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00022f10: 0a7c 6932 3720 3a20 656c 6170 7365 6454 .|i27 : elapsedT │ │ │ │ 00022f20: 696d 6520 6a65 7473 2832 2c66 2920 2020 ime jets(2,f) │ │ │ │ 00022f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022f50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00022f60: 0a7c 202d 2d20 2e30 3030 3634 3339 3134 .| -- .000643914 │ │ │ │ +00022f60: 0a7c 202d 2d20 2e30 3030 3738 3431 3237 .| -- .000784127 │ │ │ │ 00022f70: 7320 656c 6170 7365 6420 2020 2020 2020 s elapsed │ │ │ │ 00022f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022fa0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00022fb0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00022fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/K3Carpets.info.gz │ │ │ ├── K3Carpets.info │ │ │ │ @@ -1860,15 +1860,15 @@ │ │ │ │ 00007430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00007440: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 204c -------+.|i4 : L │ │ │ │ 00007450: 203d 2061 6e61 6c79 7a65 5374 7261 6e64 = analyzeStrand │ │ │ │ 00007460: 2846 2c61 293b 2023 4c20 2020 2020 2020 (F,a); #L │ │ │ │ 00007470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007490: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -000074a0: 3238 3931 3373 2065 6c61 7073 6564 2020 28913s elapsed │ │ │ │ +000074a0: 3239 3839 3336 7320 656c 6170 7365 6420 298936s elapsed │ │ │ │ 000074b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000074c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000074d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000074e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000074f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1995,35 +1995,35 @@ │ │ │ │ 00007ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00007cb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 -------+.|i10 : │ │ │ │ 00007cc0: 6361 7270 6574 4265 7474 6954 6162 6c65 carpetBettiTable │ │ │ │ 00007cd0: 2861 2c62 2c33 2920 2020 2020 2020 2020 (a,b,3) │ │ │ │ 00007ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d00: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -00007d10: 3032 3332 3339 3773 2065 6c61 7073 6564 0232397s elapsed │ │ │ │ +00007d10: 3032 3534 3330 3673 2065 6c61 7073 6564 0254306s elapsed │ │ │ │ 00007d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d50: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -00007d60: 3134 3836 3231 7320 656c 6170 7365 6420 148621s elapsed │ │ │ │ +00007d60: 3037 3236 3835 7320 656c 6170 7365 6420 072685s elapsed │ │ │ │ 00007d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007da0: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -00007db0: 3338 3539 3673 2065 6c61 7073 6564 2020 38596s elapsed │ │ │ │ +00007db0: 3338 3730 3773 2065 6c61 7073 6564 2020 38707s elapsed │ │ │ │ 00007dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007df0: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -00007e00: 3133 3730 3932 7320 656c 6170 7365 6420 137092s elapsed │ │ │ │ +00007e00: 3134 3134 3773 2065 6c61 7073 6564 2020 14147s elapsed │ │ │ │ 00007e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e40: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -00007e50: 3033 3537 3339 3573 2065 6c61 7073 6564 0357395s elapsed │ │ │ │ +00007e50: 3034 3939 3236 3873 2065 6c61 7073 6564 0499268s elapsed │ │ │ │ 00007e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00007ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -3676,35 +3676,35 @@ │ │ │ │ 0000e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e5e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2065 -------+.|i2 : e │ │ │ │ 0000e5f0: 6c61 7073 6564 5469 6d65 2054 3d63 6172 lapsedTime T=car │ │ │ │ 0000e600: 7065 7442 6574 7469 5461 626c 6528 612c petBettiTable(a, │ │ │ │ 0000e610: 622c 3329 2020 2020 2020 2020 2020 2020 b,3) │ │ │ │ -0000e620: 2020 2020 7c0a 7c20 2d2d 202e 3031 3035 |.| -- .0105 │ │ │ │ -0000e630: 3430 3473 2065 6c61 7073 6564 2020 2020 404s elapsed │ │ │ │ +0000e620: 2020 2020 7c0a 7c20 2d2d 202e 3030 3332 |.| -- .0032 │ │ │ │ +0000e630: 3938 3531 7320 656c 6170 7365 6420 2020 9851s elapsed │ │ │ │ 0000e640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e660: 207c 0a7c 202d 2d20 2e30 3036 3139 3737 |.| -- .0061977 │ │ │ │ -0000e670: 3373 2065 6c61 7073 6564 2020 2020 2020 3s elapsed │ │ │ │ +0000e660: 207c 0a7c 202d 2d20 2e30 3037 3631 3333 |.| -- .0076133 │ │ │ │ +0000e670: 3773 2065 6c61 7073 6564 2020 2020 2020 7s elapsed │ │ │ │ 0000e680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e690: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0000e6a0: 7c20 2d2d 202e 3032 3331 3838 7320 656c | -- .023188s el │ │ │ │ -0000e6b0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ +0000e6a0: 7c20 2d2d 202e 3032 3837 3932 3573 2065 | -- .0287925s e │ │ │ │ +0000e6b0: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 0000e6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e6d0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -0000e6e0: 2d20 2e30 3134 3138 3235 7320 656c 6170 - .0141825s elap │ │ │ │ +0000e6e0: 2d20 2e30 3139 3835 3831 7320 656c 6170 - .0198581s elap │ │ │ │ 0000e6f0: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 0000e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e710: 2020 2020 2020 2020 7c0a 7c20 2d2d 202e |.| -- . │ │ │ │ -0000e720: 3030 3338 3032 3431 7320 656c 6170 7365 00380241s elapse │ │ │ │ +0000e720: 3030 3433 3336 3738 7320 656c 6170 7365 00433678s elapse │ │ │ │ 0000e730: 6420 2020 2020 2020 2020 2020 2020 2020 d │ │ │ │ 0000e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e750: 2020 2020 207c 0a7c 202d 2d20 2e34 3234 |.| -- .424 │ │ │ │ -0000e760: 3133 3473 2065 6c61 7073 6564 2020 2020 134s elapsed │ │ │ │ +0000e750: 2020 2020 207c 0a7c 202d 2d20 2e34 3531 |.| -- .451 │ │ │ │ +0000e760: 3032 3973 2065 6c61 7073 6564 2020 2020 029s elapsed │ │ │ │ 0000e770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e790: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000e7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e7c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000e7d0: 0a7c 2020 2020 2020 2020 2020 2020 3020 .| 0 │ │ │ │ @@ -3764,15 +3764,15 @@ │ │ │ │ 0000eb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000eb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000eb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000eb60: 2d2d 2b0a 7c69 3420 3a20 656c 6170 7365 --+.|i4 : elapse │ │ │ │ 0000eb70: 6454 696d 6520 5427 3d6d 696e 696d 616c dTime T'=minimal │ │ │ │ 0000eb80: 4265 7474 6920 4a20 2020 2020 2020 2020 Betti J │ │ │ │ 0000eb90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000eba0: 0a7c 202d 2d20 2e31 3433 3539 3673 2065 .| -- .143596s e │ │ │ │ +0000eba0: 0a7c 202d 2d20 2e32 3139 3531 3573 2065 .| -- .219515s e │ │ │ │ 0000ebb0: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 0000ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ebd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0000ebe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ebf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ec00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ec10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ @@ -3848,42 +3848,42 @@ │ │ │ │ 0000f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f0a0: 2b0a 7c69 3620 3a20 656c 6170 7365 6454 +.|i6 : elapsedT │ │ │ │ 0000f0b0: 696d 6520 683d 6361 7270 6574 4265 7474 ime h=carpetBett │ │ │ │ 0000f0c0: 6954 6162 6c65 7328 362c 3629 3b20 2020 iTables(6,6); │ │ │ │ 0000f0d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000f0e0: 202d 2d20 2e30 3034 3638 3634 3773 2065 -- .00468647s e │ │ │ │ -0000f0f0: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ +0000f0e0: 202d 2d20 2e30 3034 3836 3237 7320 656c -- .0048627s el │ │ │ │ +0000f0f0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 0000f100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f110: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -0000f120: 202e 3036 3338 3330 3873 2065 6c61 7073 .0638308s elaps │ │ │ │ +0000f120: 202e 3032 3635 3336 3673 2065 6c61 7073 .0265366s elaps │ │ │ │ 0000f130: 6564 2020 2020 2020 2020 2020 2020 2020 ed │ │ │ │ 0000f140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f150: 2020 2020 2020 207c 0a7c 202d 2d20 2e31 |.| -- .1 │ │ │ │ -0000f160: 3830 3731 7320 656c 6170 7365 6420 2020 8071s elapsed │ │ │ │ +0000f160: 3231 3935 3873 2065 6c61 7073 6564 2020 21958s elapsed │ │ │ │ 0000f170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f190: 2020 2020 7c0a 7c20 2d2d 2031 2e32 3136 |.| -- 1.216 │ │ │ │ -0000f1a0: 3136 7320 656c 6170 7365 6420 2020 2020 16s elapsed │ │ │ │ +0000f190: 2020 2020 7c0a 7c20 2d2d 2031 2e30 3738 |.| -- 1.078 │ │ │ │ +0000f1a0: 3438 7320 656c 6170 7365 6420 2020 2020 48s elapsed │ │ │ │ 0000f1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f1d0: 207c 0a7c 202d 2d20 2e34 3336 3437 3273 |.| -- .436472s │ │ │ │ +0000f1d0: 207c 0a7c 202d 2d20 2e35 3636 3738 3573 |.| -- .566785s │ │ │ │ 0000f1e0: 2065 6c61 7073 6564 2020 2020 2020 2020 elapsed │ │ │ │ 0000f1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f200: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0000f210: 7c20 2d2d 202e 3137 3532 3434 7320 656c | -- .175244s el │ │ │ │ -0000f220: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ +0000f210: 7c20 2d2d 202e 3034 3536 3530 3873 2065 | -- .0456508s e │ │ │ │ +0000f220: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 0000f230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f240: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -0000f250: 2d20 2e30 3036 3937 3833 3473 2065 6c61 - .00697834s ela │ │ │ │ +0000f250: 2d20 2e30 3038 3430 3931 3473 2065 6c61 - .00840914s ela │ │ │ │ 0000f260: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 0000f270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f280: 2020 2020 2020 2020 7c0a 7c20 2d2d 2036 |.| -- 6 │ │ │ │ -0000f290: 2e31 3935 3332 7320 656c 6170 7365 6420 .19532s elapsed │ │ │ │ +0000f290: 2e34 3033 3933 7320 656c 6170 7365 6420 .40393s elapsed │ │ │ │ 0000f2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f2c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0000f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f300: 2d2d 2b0a 7c69 3720 3a20 6361 7270 6574 --+.|i7 : carpet │ │ │ │ @@ -4108,31 +4108,31 @@ │ │ │ │ 000100b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000100c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000100d0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2068 3d63 -----+.|i2 : h=c │ │ │ │ 000100e0: 6172 7065 7442 6574 7469 5461 626c 6573 arpetBettiTables │ │ │ │ 000100f0: 2861 2c62 2920 2020 2020 2020 2020 2020 (a,b) │ │ │ │ 00010100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010110: 2020 2020 207c 0a7c 202d 2d20 2e30 3032 |.| -- .002 │ │ │ │ -00010120: 3432 3937 3973 2065 6c61 7073 6564 2020 42979s elapsed │ │ │ │ +00010120: 3830 3930 3473 2065 6c61 7073 6564 2020 80904s elapsed │ │ │ │ 00010130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010150: 2020 2020 207c 0a7c 202d 2d20 2e30 3135 |.| -- .015 │ │ │ │ -00010160: 3039 3933 7320 656c 6170 7365 6420 2020 0993s elapsed │ │ │ │ +00010150: 2020 2020 207c 0a7c 202d 2d20 2e30 3132 |.| -- .012 │ │ │ │ +00010160: 3830 3135 7320 656c 6170 7365 6420 2020 8015s elapsed │ │ │ │ 00010170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010190: 2020 2020 207c 0a7c 202d 2d20 2e30 3233 |.| -- .023 │ │ │ │ -000101a0: 3631 3434 7320 656c 6170 7365 6420 2020 6144s elapsed │ │ │ │ +00010190: 2020 2020 207c 0a7c 202d 2d20 2e30 3437 |.| -- .047 │ │ │ │ +000101a0: 3932 3934 7320 656c 6170 7365 6420 2020 9294s elapsed │ │ │ │ 000101b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000101c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000101d0: 2020 2020 207c 0a7c 202d 2d20 2e30 3134 |.| -- .014 │ │ │ │ -000101e0: 3437 3739 7320 656c 6170 7365 6420 2020 4779s elapsed │ │ │ │ +000101d0: 2020 2020 207c 0a7c 202d 2d20 2e30 3230 |.| -- .020 │ │ │ │ +000101e0: 3532 3334 7320 656c 6170 7365 6420 2020 5234s elapsed │ │ │ │ 000101f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010210: 2020 2020 207c 0a7c 202d 2d20 2e30 3033 |.| -- .003 │ │ │ │ -00010220: 3830 3132 3473 2065 6c61 7073 6564 2020 80124s elapsed │ │ │ │ +00010210: 2020 2020 207c 0a7c 202d 2d20 2e30 3034 |.| -- .004 │ │ │ │ +00010220: 3339 3837 3873 2065 6c61 7073 6564 2020 39878s elapsed │ │ │ │ 00010230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010250: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00010260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010290: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ @@ -4287,16 +4287,16 @@ │ │ │ │ 00010be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010c10: 2d2d 2d2d 2d2b 0a7c 6935 203a 2065 6c61 -----+.|i5 : ela │ │ │ │ 00010c20: 7073 6564 5469 6d65 2054 273d 6d69 6e69 psedTime T'=mini │ │ │ │ 00010c30: 6d61 6c42 6574 7469 204a 2020 2020 2020 malBetti J │ │ │ │ 00010c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010c50: 2020 2020 207c 0a7c 202d 2d20 2e31 3430 |.| -- .140 │ │ │ │ -00010c60: 3434 3873 2065 6c61 7073 6564 2020 2020 448s elapsed │ │ │ │ +00010c50: 2020 2020 207c 0a7c 202d 2d20 2e32 3132 |.| -- .212 │ │ │ │ +00010c60: 3732 3873 2065 6c61 7073 6564 2020 2020 728s elapsed │ │ │ │ 00010c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010c90: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00010ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ @@ -4375,44 +4375,44 @@ │ │ │ │ 00011160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011190: 2d2d 2d2d 2d2b 0a7c 6937 203a 2065 6c61 -----+.|i7 : ela │ │ │ │ 000111a0: 7073 6564 5469 6d65 2068 3d63 6172 7065 psedTime h=carpe │ │ │ │ 000111b0: 7442 6574 7469 5461 626c 6573 2836 2c36 tBettiTables(6,6 │ │ │ │ 000111c0: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ -000111d0: 2020 2020 207c 0a7c 202d 2d20 2e30 3034 |.| -- .004 │ │ │ │ -000111e0: 3536 3135 3773 2065 6c61 7073 6564 2020 56157s elapsed │ │ │ │ +000111d0: 2020 2020 207c 0a7c 202d 2d20 2e30 3035 |.| -- .005 │ │ │ │ +000111e0: 3131 3739 3573 2065 6c61 7073 6564 2020 11795s elapsed │ │ │ │ 000111f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011210: 2020 2020 207c 0a7c 202d 2d20 2e30 3233 |.| -- .023 │ │ │ │ -00011220: 3534 3037 7320 656c 6170 7365 6420 2020 5407s elapsed │ │ │ │ +00011210: 2020 2020 207c 0a7c 202d 2d20 2e30 3139 |.| -- .019 │ │ │ │ +00011220: 3435 3935 7320 656c 6170 7365 6420 2020 4595s elapsed │ │ │ │ 00011230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011250: 2020 2020 207c 0a7c 202d 2d20 2e31 3136 |.| -- .116 │ │ │ │ -00011260: 3732 3673 2065 6c61 7073 6564 2020 2020 726s elapsed │ │ │ │ +00011250: 2020 2020 207c 0a7c 202d 2d20 2e31 3330 |.| -- .130 │ │ │ │ +00011260: 3835 3973 2065 6c61 7073 6564 2020 2020 859s elapsed │ │ │ │ 00011270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011290: 2020 2020 207c 0a7c 202d 2d20 312e 3139 |.| -- 1.19 │ │ │ │ -000112a0: 3236 3473 2065 6c61 7073 6564 2020 2020 264s elapsed │ │ │ │ +00011290: 2020 2020 207c 0a7c 202d 2d20 312e 3031 |.| -- 1.01 │ │ │ │ +000112a0: 3932 3973 2065 6c61 7073 6564 2020 2020 929s elapsed │ │ │ │ 000112b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000112c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000112d0: 2020 2020 207c 0a7c 202d 2d20 2e35 3835 |.| -- .585 │ │ │ │ -000112e0: 3037 3973 2065 6c61 7073 6564 2020 2020 079s elapsed │ │ │ │ +000112d0: 2020 2020 207c 0a7c 202d 2d20 2e34 3530 |.| -- .450 │ │ │ │ +000112e0: 3232 3673 2065 6c61 7073 6564 2020 2020 226s elapsed │ │ │ │ 000112f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011310: 2020 2020 207c 0a7c 202d 2d20 2e30 3530 |.| -- .050 │ │ │ │ -00011320: 3031 3238 7320 656c 6170 7365 6420 2020 0128s elapsed │ │ │ │ +00011310: 2020 2020 207c 0a7c 202d 2d20 2e30 3433 |.| -- .043 │ │ │ │ +00011320: 3437 3038 7320 656c 6170 7365 6420 2020 4708s elapsed │ │ │ │ 00011330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011350: 2020 2020 207c 0a7c 202d 2d20 2e30 3036 |.| -- .006 │ │ │ │ -00011360: 3631 3933 3273 2065 6c61 7073 6564 2020 61932s elapsed │ │ │ │ +00011350: 2020 2020 207c 0a7c 202d 2d20 2e30 3038 |.| -- .008 │ │ │ │ +00011360: 3333 3534 3473 2065 6c61 7073 6564 2020 33544s elapsed │ │ │ │ 00011370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011390: 2020 2020 207c 0a7c 202d 2d20 352e 3836 |.| -- 5.86 │ │ │ │ -000113a0: 3435 3173 2065 6c61 7073 6564 2020 2020 451s elapsed │ │ │ │ +00011390: 2020 2020 207c 0a7c 202d 2d20 362e 3135 |.| -- 6.15 │ │ │ │ +000113a0: 3930 3273 2065 6c61 7073 6564 2020 2020 902s elapsed │ │ │ │ 000113b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 000113e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000113f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011410: 2d2d 2d2d 2d2b 0a7c 6938 203a 206b 6579 -----+.|i8 : key │ │ │ │ @@ -4634,163 +4634,163 @@ │ │ │ │ 00012190: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ 000121a0: 203a 2053 6571 7565 6e63 6520 2020 2020 : Sequence │ │ │ │ 000121b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000121c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000121d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000121e0: 2d2b 0a7c 6932 203a 2064 3d63 6172 7065 -+.|i2 : d=carpe │ │ │ │ 000121f0: 7444 6574 2861 2c62 2920 2020 2020 2020 tDet(a,b) │ │ │ │ -00012200: 2020 2020 7c0a 7c20 2d2d 202e 3031 3133 |.| -- .0113 │ │ │ │ -00012210: 3339 3273 2065 6c61 7073 6564 2020 2020 392s elapsed │ │ │ │ +00012200: 2020 2020 7c0a 7c20 2d2d 202e 3030 3833 |.| -- .0083 │ │ │ │ +00012210: 3138 3934 7320 656c 6170 7365 6420 2020 1894s elapsed │ │ │ │ 00012220: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -00012230: 3233 3437 3434 7320 656c 6170 7365 6420 234744s elapsed │ │ │ │ +00012230: 3134 3038 3537 7320 656c 6170 7365 6420 140857s elapsed │ │ │ │ 00012240: 2020 2020 2020 2020 2020 7c0a 7c28 6e75 |.|(nu │ │ │ │ 00012250: 6d62 6572 204f 6620 626c 6f63 6b73 2c20 mber Of blocks, │ │ │ │ 00012260: 3236 2920 2020 2020 2020 2020 207c 0a7c 26) |.| │ │ │ │ -00012270: 202d 2d20 2e30 3030 3238 3831 3738 7320 -- .000288178s │ │ │ │ +00012270: 202d 2d20 2e30 3030 3239 3733 3032 7320 -- .000297302s │ │ │ │ 00012280: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 00012290: 7c0a 7c31 2020 2020 2020 2020 2020 2020 |.|1 │ │ │ │ 000122a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000122b0: 2020 207c 0a7c 202d 2d20 2e30 3030 3137 |.| -- .00017 │ │ │ │ -000122c0: 3032 3838 7320 656c 6170 7365 6420 2020 0288s elapsed │ │ │ │ +000122b0: 2020 207c 0a7c 202d 2d20 2e30 3030 3234 |.| -- .00024 │ │ │ │ +000122c0: 3239 3573 2065 6c61 7073 6564 2020 2020 295s elapsed │ │ │ │ 000122d0: 2020 2020 2020 7c0a 7c31 2020 2020 2020 |.|1 │ │ │ │ 000122e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000122f0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00012300: 2e30 3030 3136 3533 3039 7320 656c 6170 .000165309s elap │ │ │ │ +00012300: 2e30 3030 3139 3737 3337 7320 656c 6170 .000197737s elap │ │ │ │ 00012310: 7365 6420 2020 2020 2020 2020 7c0a 7c31 sed |.|1 │ │ │ │ 00012320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012330: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00012340: 0a7c 202d 2d20 2e30 3030 3133 3233 3438 .| -- .000132348 │ │ │ │ +00012340: 0a7c 202d 2d20 2e30 3030 3139 3230 3639 .| -- .000192069 │ │ │ │ 00012350: 7320 656c 6170 7365 6420 2020 2020 2020 s elapsed │ │ │ │ 00012360: 2020 7c0a 7c31 2020 2020 2020 2020 2020 |.|1 │ │ │ │ 00012370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012380: 2020 2020 207c 0a7c 202d 2d20 2e30 3030 |.| -- .000 │ │ │ │ -00012390: 3135 3135 3933 7320 656c 6170 7365 6420 151593s elapsed │ │ │ │ +00012390: 3230 3431 3936 7320 656c 6170 7365 6420 204196s elapsed │ │ │ │ 000123a0: 2020 2020 2020 2020 7c0a 7c32 2020 2020 |.|2 │ │ │ │ 000123b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000123c0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -000123d0: 2d20 2e30 3030 3134 3532 3431 7320 656c - .000145241s el │ │ │ │ +000123d0: 2d20 2e30 3030 3232 3038 3434 7320 656c - .000220844s el │ │ │ │ 000123e0: 6170 7365 6420 2020 2020 2020 2020 7c0a apsed |. │ │ │ │ 000123f0: 7c20 3220 2020 2020 2020 2020 2020 2020 | 2 │ │ │ │ 00012400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012410: 207c 0a7c 3220 2020 2020 2020 2020 2020 |.|2 │ │ │ │ 00012420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012430: 2020 2020 7c0a 7c20 2d2d 202e 3034 3637 |.| -- .0467 │ │ │ │ -00012440: 3633 3273 2065 6c61 7073 6564 2020 2020 632s elapsed │ │ │ │ +00012430: 2020 2020 7c0a 7c20 2d2d 202e 3032 3631 |.| -- .0261 │ │ │ │ +00012440: 3737 3973 2065 6c61 7073 6564 2020 2020 779s elapsed │ │ │ │ 00012450: 2020 2020 2020 207c 0a7c 2032 2020 2020 |.| 2 │ │ │ │ 00012460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012470: 2020 2020 2020 2020 2020 7c0a 7c32 2020 |.|2 │ │ │ │ 00012480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012490: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000124a0: 202d 2d20 2e30 3030 3137 3232 3832 7320 -- .000172282s │ │ │ │ +000124a0: 202d 2d20 2e30 3030 3337 3231 3936 7320 -- .000372196s │ │ │ │ 000124b0: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 000124c0: 7c0a 7c20 3220 2020 2020 2020 2020 2020 |.| 2 │ │ │ │ 000124d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000124e0: 2020 207c 0a7c 3220 3320 2020 2020 2020 |.|2 3 │ │ │ │ 000124f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012500: 2020 2020 2020 7c0a 7c20 2d2d 202e 3030 |.| -- .00 │ │ │ │ -00012510: 3031 3333 3432 3973 2065 6c61 7073 6564 0133429s elapsed │ │ │ │ +00012510: 3032 3035 3435 3573 2065 6c61 7073 6564 0205455s elapsed │ │ │ │ 00012520: 2020 2020 2020 2020 207c 0a7c 2032 2020 |.| 2 │ │ │ │ 00012530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012540: 2020 2020 2020 2020 2020 2020 7c0a 7c32 |.|2 │ │ │ │ 00012550: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00012560: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00012570: 0a7c 202d 2d20 2e30 3030 3132 3935 3432 .| -- .000129542 │ │ │ │ +00012570: 0a7c 202d 2d20 2e30 3030 3239 3036 3331 .| -- .000290631 │ │ │ │ 00012580: 7320 656c 6170 7365 6420 2020 2020 2020 s elapsed │ │ │ │ 00012590: 2020 7c0a 7c20 3220 2020 2020 2020 2020 |.| 2 │ │ │ │ 000125a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000125b0: 2020 2020 207c 0a7c 3220 3320 2020 2020 |.|2 3 │ │ │ │ 000125c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000125d0: 2020 2020 2020 2020 7c0a 7c20 2d2d 202e |.| -- . │ │ │ │ -000125e0: 3030 3031 3238 3738 3173 2065 6c61 7073 000128781s elaps │ │ │ │ +000125e0: 3030 3031 3834 3739 3673 2065 6c61 7073 000184796s elaps │ │ │ │ 000125f0: 6564 2020 2020 2020 2020 207c 0a7c 2032 ed |.| 2 │ │ │ │ 00012600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012610: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00012620: 7c32 2020 2020 2020 2020 2020 2020 2020 |2 │ │ │ │ 00012630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012640: 207c 0a7c 202d 2d20 2e30 3030 3132 3539 |.| -- .0001259 │ │ │ │ -00012650: 3735 7320 656c 6170 7365 6420 2020 2020 75s elapsed │ │ │ │ +00012640: 207c 0a7c 202d 2d20 2e30 3030 3137 3531 |.| -- .0001751 │ │ │ │ +00012650: 3139 7320 656c 6170 7365 6420 2020 2020 19s elapsed │ │ │ │ 00012660: 2020 2020 7c0a 7c20 3220 2020 2020 2020 |.| 2 │ │ │ │ 00012670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012680: 2020 2020 2020 207c 0a7c 3220 2020 2020 |.|2 │ │ │ │ 00012690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000126a0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -000126b0: 202e 3030 3031 3336 3632 3573 2065 6c61 .000136625s ela │ │ │ │ -000126c0: 7073 6564 2020 2020 2020 2020 207c 0a7c psed |.| │ │ │ │ +000126b0: 202e 3030 3031 3639 3537 7320 656c 6170 .00016957s elap │ │ │ │ +000126c0: 7365 6420 2020 2020 2020 2020 207c 0a7c sed |.| │ │ │ │ 000126d0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 000126e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000126f0: 7c0a 7c20 2d2d 202e 3030 3031 3338 3734 |.| -- .00013874 │ │ │ │ -00012700: 3973 2065 6c61 7073 6564 2020 2020 2020 9s elapsed │ │ │ │ +000126f0: 7c0a 7c20 2d2d 202e 3030 3031 3735 3132 |.| -- .00017512 │ │ │ │ +00012700: 3373 2065 6c61 7073 6564 2020 2020 2020 3s elapsed │ │ │ │ 00012710: 2020 207c 0a7c 3220 2020 2020 2020 2020 |.|2 │ │ │ │ 00012720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012730: 2020 2020 2020 7c0a 7c20 2d2d 202e 3030 |.| -- .00 │ │ │ │ -00012740: 3031 3432 3035 3573 2065 6c61 7073 6564 0142055s elapsed │ │ │ │ +00012740: 3032 3732 3836 3773 2065 6c61 7073 6564 0272867s elapsed │ │ │ │ 00012750: 2020 2020 2020 2020 207c 0a7c 2032 2020 |.| 2 │ │ │ │ 00012760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012770: 2020 2020 2020 2020 2020 2020 7c0a 7c32 |.|2 │ │ │ │ 00012780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012790: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000127a0: 0a7c 202d 2d20 2e30 3030 3132 3837 3173 .| -- .00012871s │ │ │ │ -000127b0: 2065 6c61 7073 6564 2020 2020 2020 2020 elapsed │ │ │ │ +000127a0: 0a7c 202d 2d20 2e30 3030 3237 3532 3936 .| -- .000275296 │ │ │ │ +000127b0: 7320 656c 6170 7365 6420 2020 2020 2020 s elapsed │ │ │ │ 000127c0: 2020 7c0a 7c20 3220 2020 2020 2020 2020 |.| 2 │ │ │ │ 000127d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000127e0: 2020 2020 207c 0a7c 3220 2020 2020 2020 |.|2 │ │ │ │ 000127f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012800: 2020 2020 2020 2020 7c0a 7c20 2d2d 202e |.| -- . │ │ │ │ -00012810: 3030 3031 3433 3839 3973 2065 6c61 7073 000143899s elaps │ │ │ │ +00012810: 3030 3031 3934 3134 3173 2065 6c61 7073 000194141s elaps │ │ │ │ 00012820: 6564 2020 2020 2020 2020 207c 0a7c 2032 ed |.| 2 │ │ │ │ 00012830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012840: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00012850: 7c32 2033 2020 2020 2020 2020 2020 2020 |2 3 │ │ │ │ 00012860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012870: 207c 0a7c 202d 2d20 2e30 3030 3134 3035 |.| -- .0001405 │ │ │ │ -00012880: 3234 7320 656c 6170 7365 6420 2020 2020 24s elapsed │ │ │ │ +00012870: 207c 0a7c 202d 2d20 2e30 3030 3230 3230 |.| -- .0002020 │ │ │ │ +00012880: 3432 7320 656c 6170 7365 6420 2020 2020 42s elapsed │ │ │ │ 00012890: 2020 2020 7c0a 7c20 3220 2020 2020 2020 |.| 2 │ │ │ │ 000128a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000128b0: 2020 2020 2020 207c 0a7c 3220 3320 2020 |.|2 3 │ │ │ │ 000128c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000128d0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -000128e0: 202e 3030 3031 3430 3638 3373 2065 6c61 .000140683s ela │ │ │ │ +000128e0: 202e 3030 3032 3937 3630 3673 2065 6c61 .000297606s ela │ │ │ │ 000128f0: 7073 6564 2020 2020 2020 2020 207c 0a7c psed |.| │ │ │ │ 00012900: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00012910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012920: 7c0a 7c32 2033 2020 2020 2020 2020 2020 |.|2 3 │ │ │ │ 00012930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012940: 2020 207c 0a7c 202d 2d20 2e30 3030 3138 |.| -- .00018 │ │ │ │ -00012950: 3139 3673 2065 6c61 7073 6564 2020 2020 196s elapsed │ │ │ │ +00012940: 2020 207c 0a7c 202d 2d20 2e30 3030 3334 |.| -- .00034 │ │ │ │ +00012950: 3239 3839 7320 656c 6170 7365 6420 2020 2989s elapsed │ │ │ │ 00012960: 2020 2020 2020 7c0a 7c20 3220 2020 2020 |.| 2 │ │ │ │ 00012970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012980: 2020 2020 2020 2020 207c 0a7c 3220 2020 |.|2 │ │ │ │ 00012990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000129a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000129b0: 2d2d 202e 3030 3031 3433 3633 3973 2065 -- .000143639s e │ │ │ │ -000129c0: 6c61 7073 6564 2020 2020 2020 2020 207c lapsed | │ │ │ │ +000129b0: 2d2d 202e 3030 3032 3938 3737 7320 656c -- .00029877s el │ │ │ │ +000129c0: 6170 7365 6420 2020 2020 2020 2020 207c apsed | │ │ │ │ 000129d0: 0a7c 2032 2020 2020 2020 2020 2020 2020 .| 2 │ │ │ │ 000129e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000129f0: 2020 7c0a 7c32 2020 2020 2020 2020 2020 |.|2 │ │ │ │ 00012a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012a10: 2020 2020 207c 0a7c 202d 2d20 2e30 3034 |.| -- .004 │ │ │ │ -00012a20: 3135 3533 7320 656c 6170 7365 6420 2020 1553s elapsed │ │ │ │ +00012a10: 2020 2020 207c 0a7c 202d 2d20 2e30 3030 |.| -- .000 │ │ │ │ +00012a20: 3238 3433 3173 2065 6c61 7073 6564 2020 28431s elapsed │ │ │ │ 00012a30: 2020 2020 2020 2020 7c0a 7c32 2020 2020 |.|2 │ │ │ │ 00012a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012a50: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00012a60: 2d20 2e30 3030 3133 3632 3934 7320 656c - .000136294s el │ │ │ │ +00012a60: 2d20 2e30 3030 3236 3934 3132 7320 656c - .000269412s el │ │ │ │ 00012a70: 6170 7365 6420 2020 2020 2020 2020 7c0a apsed |. │ │ │ │ 00012a80: 7c31 2020 2020 2020 2020 2020 2020 2020 |1 │ │ │ │ 00012a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012aa0: 207c 0a7c 202d 2d20 2e30 3030 3133 3530 |.| -- .0001350 │ │ │ │ -00012ab0: 3634 7320 656c 6170 7365 6420 2020 2020 64s elapsed │ │ │ │ +00012aa0: 207c 0a7c 202d 2d20 2e30 3030 3138 3130 |.| -- .0001810 │ │ │ │ +00012ab0: 3932 7320 656c 6170 7365 6420 2020 2020 92s elapsed │ │ │ │ 00012ac0: 2020 2020 7c0a 7c31 2020 2020 2020 2020 |.|1 │ │ │ │ 00012ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012ae0: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -00012af0: 3030 3134 3030 3332 7320 656c 6170 7365 00140032s elapse │ │ │ │ +00012af0: 3030 3137 3833 3734 7320 656c 6170 7365 00178374s elapse │ │ │ │ 00012b00: 6420 2020 2020 2020 2020 7c0a 7c31 2020 d |.|1 │ │ │ │ 00012b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012b20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00012b30: 202d 2d20 2e30 3034 3436 3536 3273 2065 -- .00446562s e │ │ │ │ -00012b40: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ +00012b30: 202d 2d20 2e30 3030 3136 3537 3239 7320 -- .000165729s │ │ │ │ +00012b40: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 00012b50: 7c0a 7c31 2020 2020 2020 2020 2020 2020 |.|1 │ │ │ │ 00012b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012b70: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00012b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012b90: 2020 2020 2020 7c0a 7c6f 3220 3d20 3331 |.|o2 = 31 │ │ │ │ 00012ba0: 3331 3033 3131 3538 3738 3420 2020 2020 31031158784 │ │ │ │ 00012bb0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ @@ -4929,26 +4929,26 @@ │ │ │ │ 00013400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013410: 2020 2020 7c0a 7c6f 3120 3a20 5365 7175 |.|o1 : Sequ │ │ │ │ 00013420: 656e 6365 2020 2020 2020 2020 2020 2020 ence │ │ │ │ 00013430: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00013440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013450: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 636f ------+.|i2 : co │ │ │ │ 00013460: 6d70 7574 6542 6f75 6e64 2033 2020 2020 mputeBound 3 │ │ │ │ -00013470: 2020 2020 2020 207c 0a7c 202d 2d20 2e32 |.| -- .2 │ │ │ │ -00013480: 3030 3631 3673 2065 6c61 7073 6564 2020 00616s elapsed │ │ │ │ +00013470: 2020 2020 2020 207c 0a7c 202d 2d20 2e31 |.| -- .1 │ │ │ │ +00013480: 3635 3637 3373 2065 6c61 7073 6564 2020 65673s elapsed │ │ │ │ 00013490: 2020 2020 2020 2020 7c0a 7c20 2d2d 202e |.| -- . │ │ │ │ -000134a0: 3232 3431 3931 7320 656c 6170 7365 6420 224191s elapsed │ │ │ │ +000134a0: 3137 3035 3635 7320 656c 6170 7365 6420 170565s elapsed │ │ │ │ 000134b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000134c0: 2e32 3137 3232 3573 2065 6c61 7073 6564 .217225s elapsed │ │ │ │ +000134c0: 2e31 3735 3836 3373 2065 6c61 7073 6564 .175863s elapsed │ │ │ │ 000134d0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -000134e0: 202e 3231 3337 3273 2065 6c61 7073 6564 .21372s elapsed │ │ │ │ -000134f0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00013500: 2d20 2e32 3536 3630 3573 2065 6c61 7073 - .256605s elaps │ │ │ │ +000134e0: 202e 3231 3030 3837 7320 656c 6170 7365 .210087s elapse │ │ │ │ +000134f0: 6420 2020 2020 2020 2020 207c 0a7c 202d d |.| - │ │ │ │ +00013500: 2d20 2e32 3136 3735 3873 2065 6c61 7073 - .216758s elaps │ │ │ │ 00013510: 6564 2020 2020 2020 2020 2020 7c0a 7c20 ed |.| │ │ │ │ -00013520: 2d2d 202e 3331 3931 3536 7320 656c 6170 -- .319156s elap │ │ │ │ +00013520: 2d2d 202e 3139 3037 3231 7320 656c 6170 -- .190721s elap │ │ │ │ 00013530: 7365 6420 2020 2020 2020 2020 207c 0a7c sed |.| │ │ │ │ 00013540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013550: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00013560: 7c6f 3220 3d20 3620 2020 2020 2020 2020 |o2 = 6 │ │ │ │ 00013570: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00013580: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00013590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -6949,33 +6949,33 @@ │ │ │ │ 0001b240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b260: 2b0a 7c69 3320 3a20 683d 6465 6765 6e65 +.|i3 : h=degene │ │ │ │ 0001b270: 7261 7465 4b33 4265 7474 6954 6162 6c65 rateK3BettiTable │ │ │ │ 0001b280: 7328 612c 622c 6529 2020 2020 2020 2020 s(a,b,e) │ │ │ │ 0001b290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b2a0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -0001b2b0: 2e30 3038 3935 3835 3373 2065 6c61 7073 .00895853s elaps │ │ │ │ -0001b2c0: 6564 2020 2020 2020 2020 2020 2020 2020 ed │ │ │ │ +0001b2b0: 2e30 3033 3236 3634 7320 656c 6170 7365 .0032664s elapse │ │ │ │ +0001b2c0: 6420 2020 2020 2020 2020 2020 2020 2020 d │ │ │ │ 0001b2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b2f0: 2020 7c0a 7c20 2d2d 202e 3031 3036 3835 |.| -- .010685 │ │ │ │ -0001b300: 3273 2065 6c61 7073 6564 2020 2020 2020 2s elapsed │ │ │ │ +0001b2f0: 2020 7c0a 7c20 2d2d 202e 3030 3738 3137 |.| -- .007817 │ │ │ │ +0001b300: 3732 7320 656c 6170 7365 6420 2020 2020 72s elapsed │ │ │ │ 0001b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b330: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -0001b340: 2d20 2e30 3235 3032 3437 7320 656c 6170 - .0250247s elap │ │ │ │ +0001b340: 2d20 2e30 3239 3230 3532 7320 656c 6170 - .0292052s elap │ │ │ │ 0001b350: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 0001b360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b380: 2020 2020 7c0a 7c20 2d2d 202e 3030 3938 |.| -- .0098 │ │ │ │ -0001b390: 3839 3831 7320 656c 6170 7365 6420 2020 8981s elapsed │ │ │ │ +0001b380: 2020 2020 7c0a 7c20 2d2d 202e 3031 3338 |.| -- .0138 │ │ │ │ +0001b390: 3733 3373 2065 6c61 7073 6564 2020 2020 733s elapsed │ │ │ │ 0001b3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b3c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001b3d0: 202d 2d20 2e30 3033 3830 3737 3273 2065 -- .00380772s e │ │ │ │ +0001b3d0: 202d 2d20 2e30 3034 3631 3837 3773 2065 -- .00461877s e │ │ │ │ 0001b3e0: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 0001b3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b410: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0001b420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7131,15 +7131,15 @@ │ │ │ │ 0001bda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001bdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001bdc0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ 0001bdd0: 656c 6170 7365 6454 696d 6520 543d 206d elapsedTime T= m │ │ │ │ 0001bde0: 696e 696d 616c 4265 7474 6920 6465 6765 inimalBetti dege │ │ │ │ 0001bdf0: 6e65 7261 7465 4b33 2861 2c62 2c65 2c43 nerateK3(a,b,e,C │ │ │ │ 0001be00: 6861 7261 6374 6572 6973 7469 633d 3e35 haracteristic=>5 │ │ │ │ -0001be10: 297c 0a7c 202d 2d20 2e31 3437 3538 3673 )|.| -- .147586s │ │ │ │ +0001be10: 297c 0a7c 202d 2d20 2e32 3335 3432 3373 )|.| -- .235423s │ │ │ │ 0001be20: 2065 6c61 7073 6564 2020 2020 2020 2020 elapsed │ │ │ │ 0001be30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001be40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001be50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0001be60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001be70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001be80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7277,33 +7277,33 @@ │ │ │ │ 0001c6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 0001c6f0: 6938 203a 2068 3d64 6567 656e 6572 6174 i8 : h=degenerat │ │ │ │ 0001c700: 654b 3342 6574 7469 5461 626c 6573 2861 eK3BettiTables(a │ │ │ │ 0001c710: 2c62 2c65 2920 2020 2020 2020 2020 2020 ,b,e) │ │ │ │ 0001c720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c730: 2020 7c0a 7c20 2d2d 202e 3030 3235 3831 |.| -- .002581 │ │ │ │ -0001c740: 3038 7320 656c 6170 7365 6420 2020 2020 08s elapsed │ │ │ │ +0001c730: 2020 7c0a 7c20 2d2d 202e 3030 3330 3032 |.| -- .003002 │ │ │ │ +0001c740: 3037 7320 656c 6170 7365 6420 2020 2020 07s elapsed │ │ │ │ 0001c750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c770: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -0001c780: 3036 3836 3135 3873 2065 6c61 7073 6564 0686158s elapsed │ │ │ │ +0001c780: 3037 3435 3639 3973 2065 6c61 7073 6564 0745699s elapsed │ │ │ │ 0001c790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c7b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001c7c0: 2d2d 202e 3032 3434 3233 3673 2065 6c61 -- .0244236s ela │ │ │ │ +0001c7c0: 2d2d 202e 3032 3739 3135 3773 2065 6c61 -- .0279157s ela │ │ │ │ 0001c7d0: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 0001c7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c800: 207c 0a7c 202d 2d20 2e30 3039 3133 3835 |.| -- .0091385 │ │ │ │ -0001c810: 3273 2065 6c61 7073 6564 2020 2020 2020 2s elapsed │ │ │ │ +0001c800: 207c 0a7c 202d 2d20 2e30 3130 3433 3533 |.| -- .0104353 │ │ │ │ +0001c810: 7320 656c 6170 7365 6420 2020 2020 2020 s elapsed │ │ │ │ 0001c820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c840: 2020 2020 2020 7c0a 7c20 2d2d 202e 3030 |.| -- .00 │ │ │ │ -0001c850: 3334 3431 3235 7320 656c 6170 7365 6420 344125s elapsed │ │ │ │ +0001c850: 3432 3733 3333 7320 656c 6170 7365 6420 427333s elapsed │ │ │ │ 0001c860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c880: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0001c890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9875,16 +9875,16 @@ │ │ │ │ 00026920: 3d20 3420 2020 2020 2020 2020 2020 2020 = 4 │ │ │ │ 00026930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026940: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00026950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026960: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ 00026970: 3a20 2864 312c 6432 293d 7265 736f 6e61 : (d1,d2)=resona │ │ │ │ 00026980: 6e63 6544 6574 2861 2920 2020 2020 2020 nceDet(a) │ │ │ │ -00026990: 2020 7c0a 7c20 2d2d 202e 3032 3133 3233 |.| -- .021323 │ │ │ │ -000269a0: 3173 2065 6c61 7073 6564 2020 2020 2020 1s elapsed │ │ │ │ +00026990: 2020 7c0a 7c20 2d2d 202e 3033 3035 3233 |.| -- .030523 │ │ │ │ +000269a0: 3573 2065 6c61 7073 6564 2020 2020 2020 5s elapsed │ │ │ │ 000269b0: 2020 2020 2020 2020 2020 7c0a 7c28 6e75 |.|(nu │ │ │ │ 000269c0: 6d62 6572 206f 6620 626c 6f63 6b73 3d20 mber of blocks= │ │ │ │ 000269d0: 2c20 3138 2920 2020 2020 2020 2020 2020 , 18) │ │ │ │ 000269e0: 2020 7c0a 7c28 7369 7a65 206f 6620 7468 |.|(size of th │ │ │ │ 000269f0: 6520 6d61 7472 6963 6573 2c20 5461 6c6c e matrices, Tall │ │ │ │ 00026a00: 797b 3120 3d3e 2034 7d29 7c0a 7c20 2020 y{1 => 4})|.| │ │ │ │ 00026a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9898,15 +9898,15 @@ │ │ │ │ 00026a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026aa0: 2020 2020 2020 2020 2020 7c0a 7c74 6f74 |.|tot │ │ │ │ 00026ab0: 616c 3a20 3120 3120 2020 2020 2020 2020 al: 1 1 │ │ │ │ 00026ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026ad0: 2020 7c0a 7c20 2020 2037 3a20 3120 3120 |.| 7: 1 1 │ │ │ │ 00026ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026af0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00026b00: 202e 3030 3030 3434 3736 3473 2065 6c61 .000044764s ela │ │ │ │ +00026b00: 202e 3030 3030 3436 3934 3873 2065 6c61 .000046948s ela │ │ │ │ 00026b10: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 00026b20: 2020 7c0a 7c28 6520 2928 2d31 2920 2020 |.|(e )(-1) │ │ │ │ 00026b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b40: 2020 2020 2020 2020 2020 7c0a 7c20 2031 |.| 1 │ │ │ │ 00026b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b70: 2020 7c0a 7c20 2020 2020 2020 3020 3120 |.| 0 1 │ │ │ │ @@ -9915,16 +9915,16 @@ │ │ │ │ 00026ba0: 616c 3a20 3220 3220 2020 2020 2020 2020 al: 2 2 │ │ │ │ 00026bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026bc0: 2020 7c0a 7c20 2020 2037 3a20 3220 2e20 |.| 7: 2 . │ │ │ │ 00026bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026be0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00026bf0: 2038 3a20 2e20 3220 2020 2020 2020 2020 8: . 2 │ │ │ │ 00026c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c10: 2020 7c0a 7c20 2d2d 202e 3030 3030 3832 |.| -- .000082 │ │ │ │ -00026c20: 3438 3373 2065 6c61 7073 6564 2020 2020 483s elapsed │ │ │ │ +00026c10: 2020 7c0a 7c20 2d2d 202e 3030 3031 3038 |.| -- .000108 │ │ │ │ +00026c20: 3130 3673 2065 6c61 7073 6564 2020 2020 106s elapsed │ │ │ │ 00026c30: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00026c40: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00026c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026c60: 2020 7c0a 7c28 6520 2920 2865 2029 282d |.|(e ) (e )(- │ │ │ │ 00026c70: 3129 2020 2020 2020 2020 2020 2020 2020 1) │ │ │ │ 00026c80: 2020 2020 2020 2020 2020 7c0a 7c20 2031 |.| 1 │ │ │ │ 00026c90: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ @@ -9938,15 +9938,15 @@ │ │ │ │ 00026d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026d20: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00026d30: 2038 3a20 2e20 2e20 2020 2020 2020 2020 8: . . │ │ │ │ 00026d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026d50: 2020 7c0a 7c20 2020 2039 3a20 2e20 3220 |.| 9: . 2 │ │ │ │ 00026d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026d70: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00026d80: 202e 3030 3030 3638 3830 3873 2065 6c61 .000068808s ela │ │ │ │ +00026d80: 202e 3030 3030 3838 3230 3373 2065 6c61 .000088203s ela │ │ │ │ 00026d90: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 00026da0: 2020 7c0a 7c20 2020 2032 2020 2020 3220 |.| 2 2 │ │ │ │ 00026db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026dc0: 2020 2020 2020 2020 2020 7c0a 7c28 6520 |.|(e │ │ │ │ 00026dd0: 2920 2865 2029 2020 2020 2020 2020 2020 ) (e ) │ │ │ │ 00026de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026df0: 2020 7c0a 7c20 2031 2020 2020 3220 2020 |.| 1 2 │ │ │ │ @@ -9963,15 +9963,15 @@ │ │ │ │ 00026ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026eb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00026ec0: 2039 3a20 2e20 3120 2020 2020 2020 2020 9: . 1 │ │ │ │ 00026ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026ee0: 2020 7c0a 7c20 2020 3130 3a20 2e20 3220 |.| 10: . 2 │ │ │ │ 00026ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026f00: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00026f10: 202e 3030 3030 3736 3730 3273 2065 6c61 .000076702s ela │ │ │ │ +00026f10: 202e 3030 3031 3232 3232 3373 2065 6c61 .000122223s ela │ │ │ │ 00026f20: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 00026f30: 2020 7c0a 7c20 2020 2032 2020 2020 3420 |.| 2 4 │ │ │ │ 00026f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026f50: 2020 2020 2020 2020 2020 7c0a 7c28 6520 |.|(e │ │ │ │ 00026f60: 2920 2865 2029 2028 2d33 2920 2020 2020 ) (e ) (-3) │ │ │ │ 00026f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026f80: 2020 7c0a 7c20 2031 2020 2020 3220 2020 |.| 1 2 │ │ │ │ @@ -9990,16 +9990,16 @@ │ │ │ │ 00027050: 2039 3a20 3220 3220 2020 2020 2020 2020 9: 2 2 │ │ │ │ 00027060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027070: 2020 7c0a 7c20 2020 3130 3a20 2e20 3120 |.| 10: . 1 │ │ │ │ 00027080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027090: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000270a0: 3131 3a20 2e20 3120 2020 2020 2020 2020 11: . 1 │ │ │ │ 000270b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000270c0: 2020 7c0a 7c20 2d2d 202e 3030 3030 3931 |.| -- .000091 │ │ │ │ -000270d0: 3338 7320 656c 6170 7365 6420 2020 2020 38s elapsed │ │ │ │ +000270c0: 2020 7c0a 7c20 2d2d 202e 3030 3031 3135 |.| -- .000115 │ │ │ │ +000270d0: 3330 3373 2065 6c61 7073 6564 2020 2020 303s elapsed │ │ │ │ 000270e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000270f0: 2032 2020 2020 3420 2020 2020 2020 2020 2 4 │ │ │ │ 00027100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027110: 2020 7c0a 7c28 6520 2920 2865 2029 2028 |.|(e ) (e ) ( │ │ │ │ 00027120: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ 00027130: 2020 2020 2020 2020 2020 7c0a 7c20 2031 |.| 1 │ │ │ │ 00027140: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ @@ -10015,16 +10015,16 @@ │ │ │ │ 000271e0: 2039 3a20 3220 3120 2020 2020 2020 2020 9: 2 1 │ │ │ │ 000271f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027200: 2020 7c0a 7c20 2020 3130 3a20 3120 3220 |.| 10: 1 2 │ │ │ │ 00027210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027220: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00027230: 3131 3a20 2e20 3120 2020 2020 2020 2020 11: . 1 │ │ │ │ 00027240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027250: 2020 7c0a 7c20 2d2d 202e 3030 3030 3837 |.| -- .000087 │ │ │ │ -00027260: 3137 3373 2065 6c61 7073 6564 2020 2020 173s elapsed │ │ │ │ +00027250: 2020 7c0a 7c20 2d2d 202e 3030 3031 3032 |.| -- .000102 │ │ │ │ +00027260: 3434 3573 2065 6c61 7073 6564 2020 2020 445s elapsed │ │ │ │ 00027270: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00027280: 2032 2020 2020 3320 2020 2020 2020 2020 2 3 │ │ │ │ 00027290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000272a0: 2020 7c0a 7c28 6520 2920 2865 2029 2028 |.|(e ) (e ) ( │ │ │ │ 000272b0: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ 000272c0: 2020 2020 2020 2020 2020 7c0a 7c20 2031 |.| 1 │ │ │ │ 000272d0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ @@ -10033,15 +10033,15 @@ │ │ │ │ 00027300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027310: 2020 2020 2020 2020 2020 7c0a 7c74 6f74 |.|tot │ │ │ │ 00027320: 616c 3a20 3120 3120 2020 2020 2020 2020 al: 1 1 │ │ │ │ 00027330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027340: 2020 7c0a 7c20 2020 2039 3a20 3120 3120 |.| 9: 1 1 │ │ │ │ 00027350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027360: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00027370: 202e 3030 3030 3234 3437 3673 2065 6c61 .000024476s ela │ │ │ │ +00027370: 202e 3030 3030 3337 3432 3873 2065 6c61 .000037428s ela │ │ │ │ 00027380: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 00027390: 2020 7c0a 7c28 6520 2928 2d31 2920 2020 |.|(e )(-1) │ │ │ │ 000273a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000273b0: 2020 2020 2020 2020 2020 7c0a 7c20 2031 |.| 1 │ │ │ │ 000273c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000273d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000273e0: 2020 7c0a 7c20 2020 2020 2020 3020 3120 |.| 0 1 │ │ │ │ @@ -10050,16 +10050,16 @@ │ │ │ │ 00027410: 616c 3a20 3220 3220 2020 2020 2020 2020 al: 2 2 │ │ │ │ 00027420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027430: 2020 7c0a 7c20 2020 2039 3a20 3120 3120 |.| 9: 1 1 │ │ │ │ 00027440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027450: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00027460: 3130 3a20 3120 3120 2020 2020 2020 2020 10: 1 1 │ │ │ │ 00027470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027480: 2020 7c0a 7c20 2d2d 202e 3030 3030 3639 |.| -- .000069 │ │ │ │ -00027490: 3233 7320 656c 6170 7365 6420 2020 2020 23s elapsed │ │ │ │ +00027480: 2020 7c0a 7c20 2d2d 202e 3030 3030 3739 |.| -- .000079 │ │ │ │ +00027490: 3832 3173 2065 6c61 7073 6564 2020 2020 821s elapsed │ │ │ │ 000274a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000274b0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 000274c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000274d0: 2020 7c0a 7c28 6520 2920 2020 2020 2020 |.|(e ) │ │ │ │ 000274e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000274f0: 2020 2020 2020 2020 2020 7c0a 7c20 2031 |.| 1 │ │ │ │ 00027500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -10073,15 +10073,15 @@ │ │ │ │ 00027580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027590: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000275a0: 3130 3a20 3120 3120 2020 2020 2020 2020 10: 1 1 │ │ │ │ 000275b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 2020 2020 2020 2020 │ │ │ │ 000280d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000280e0: 3133 3a20 2e20 3220 2020 2020 2020 2020 13: . 2 │ │ │ │ 000280f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028100: 2020 7c0a 7c20 2d2d 202e 3030 3030 3731 |.| -- .000071 │ │ │ │ -00028110: 3532 3473 2065 6c61 7073 6564 2020 2020 524s elapsed │ │ │ │ +00028100: 2020 7c0a 7c20 2d2d 202e 3030 3030 3930 |.| -- .000090 │ │ │ │ +00028110: 3836 3273 2065 6c61 7073 6564 2020 2020 862s elapsed │ │ │ │ 00028120: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00028130: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00028140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028150: 2020 7c0a 7c28 6520 2920 2865 2029 282d |.|(e ) (e )(- │ │ │ │ 00028160: 3129 2020 2020 2020 2020 2020 2020 2020 1) │ │ │ │ 00028170: 2020 2020 2020 2020 2020 7c0a 7c20 2031 |.| 1 │ │ │ │ 00028180: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ @@ -10268,15 +10268,15 @@ │ │ │ │ 000281b0: 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2d2d 2b0a --------------+. │ │ │ │ 0000a190: 7c69 3520 3a20 7469 6d65 204c 4c4c 286d |i5 : time LLL(m │ │ │ │ 0000a1a0: 2c20 5374 7261 7465 6779 3d3e 436f 6865 , Strategy=>Cohe │ │ │ │ 0000a1b0: 6e54 6f70 4c65 7665 6c29 3b20 2020 2020 nTopLevel); │ │ │ │ 0000a1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a1d0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0000a1e0: 2030 2e31 3130 3633 3473 2028 6370 7529 0.110634s (cpu) │ │ │ │ -0000a1f0: 3b20 302e 3131 3036 3473 2028 7468 7265 ; 0.11064s (thre │ │ │ │ -0000a200: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ +0000a1e0: 2030 2e31 3333 3437 3973 2028 6370 7529 0.133479s (cpu) │ │ │ │ +0000a1f0: 3b20 302e 3133 3333 3031 7320 2874 6872 ; 0.133301s (thr │ │ │ │ +0000a200: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 0000a210: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0000a220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a260: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a270: 2020 2020 3530 2020 2020 2020 2034 3720 50 47 │ │ │ │ @@ -2610,17 +2610,17 @@ │ │ │ │ 0000a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a330: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ 0000a340: 7469 6d65 204c 4c4c 286d 2c20 5374 7261 time LLL(m, Stra │ │ │ │ 0000a350: 7465 6779 3d3e 7b47 6976 656e 732c 5265 tegy=>{Givens,Re │ │ │ │ 0000a360: 616c 4650 7d29 3b20 2020 2020 2020 2020 alFP}); │ │ │ │ 0000a370: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000a380: 0a7c 202d 2d20 7573 6564 2030 2e30 3131 .| -- used 0.011 │ │ │ │ -0000a390: 3039 3238 7320 2863 7075 293b 2030 2e30 0928s (cpu); 0.0 │ │ │ │ -0000a3a0: 3131 3039 3238 7320 2874 6872 6561 6429 110928s 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│ │ 0000a500: 7b47 6976 656e 732c 5265 616c 5150 7d29 {Givens,RealQP}) │ │ │ │ 0000a510: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ 0000a520: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -0000a530: 7573 6564 2030 2e30 3437 3836 3639 7320 used 0.0478669s │ │ │ │ -0000a540: 2863 7075 293b 2030 2e30 3437 3836 3773 (cpu); 0.047867s │ │ │ │ -0000a550: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -0000a560: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ +0000a530: 7573 6564 2030 2e30 3634 3736 3332 7320 used 0.0647632s │ │ │ │ +0000a540: 2863 7075 293b 2030 2e30 3634 3736 3839 (cpu); 0.0647689 │ │ │ │ +0000a550: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +0000a560: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 0000a570: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a5b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0000a5c0: 2020 2020 2020 2020 3530 2020 2020 2020 50 │ │ │ │ 0000a5d0: 2034 3720 2020 2020 2020 2020 2020 2020 47 │ │ │ │ @@ -2664,16 +2664,16 @@ │ │ │ │ 0000a670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0000a690: 3820 3a20 7469 6d65 204c 4c4c 286d 2c20 8 : time LLL(m, │ │ │ │ 0000a6a0: 5374 7261 7465 6779 3d3e 7b47 6976 656e Strategy=>{Given │ │ │ │ 0000a6b0: 732c 5265 616c 5844 7d29 3b20 2020 2020 s,RealXD}); │ │ │ │ 0000a6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a6d0: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -0000a6e0: 2e30 3537 3139 3631 7320 2863 7075 293b .0571961s (cpu); │ │ │ │ -0000a6f0: 2030 2e30 3537 3139 3634 7320 2874 6872 0.0571964s (thr │ │ │ │ +0000a6e0: 2e30 3638 3235 3832 7320 2863 7075 293b .0682582s (cpu); │ │ │ │ +0000a6f0: 2030 2e30 3638 3236 3433 7320 2874 6872 0.0682643s (thr │ │ │ │ 0000a700: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 0000a710: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0000a720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a760: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ @@ -2690,17 +2690,17 @@ │ │ │ │ 0000a810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a830: 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 7469 ------+.|i9 : ti │ │ │ │ 0000a840: 6d65 204c 4c4c 286d 2c20 5374 7261 7465 me LLL(m, Strate │ │ │ │ 0000a850: 6779 3d3e 7b47 6976 656e 732c 5265 616c gy=>{Givens,Real │ │ │ │ 0000a860: 5252 7d29 3b20 2020 2020 2020 2020 2020 RR}); │ │ │ │ 0000a870: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000a880: 202d 2d20 7573 6564 2030 2e33 3730 3135 -- used 0.37015 │ │ │ │ -0000a890: 3173 2028 6370 7529 3b20 302e 3337 3031 1s (cpu); 0.3701 │ │ │ │ -0000a8a0: 3333 7320 2874 6872 6561 6429 3b20 3073 33s (thread); 0s │ │ │ │ +0000a880: 202d 2d20 7573 6564 2030 2e34 3235 3839 -- used 0.42589 │ │ │ │ +0000a890: 3473 2028 6370 7529 3b20 302e 3432 3539 4s (cpu); 0.4259 │ │ │ │ +0000a8a0: 3031 7320 2874 6872 6561 6429 3b20 3073 01s (thread); 0s │ │ │ │ 0000a8b0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0000a8c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0000a8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a900: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000a910: 2020 2020 2020 2020 2020 2020 3530 2020 50 │ │ │ │ @@ -2717,16 +2717,16 @@ │ │ │ │ 0000a9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a9e0: 2b0a 7c69 3130 203a 2074 696d 6520 4c4c +.|i10 : time LL │ │ │ │ 0000a9f0: 4c28 6d2c 2053 7472 6174 6567 793d 3e7b L(m, Strategy=>{ │ │ │ │ 0000aa00: 424b 5a2c 4769 7665 6e73 2c52 6561 6c51 BKZ,Givens,RealQ │ │ │ │ 0000aa10: 507d 293b 2020 2020 2020 2020 2020 2020 P}); │ │ │ │ 0000aa20: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -0000aa30: 6564 2030 2e31 3633 3433 3473 2028 6370 ed 0.163434s (cp │ │ │ │ -0000aa40: 7529 3b20 302e 3136 3334 3338 7320 2874 u); 0.163438s (t │ │ │ │ +0000aa30: 6564 2030 2e31 3735 3734 3773 2028 6370 ed 0.175747s (cp │ │ │ │ +0000aa40: 7529 3b20 302e 3137 3537 3533 7320 2874 u); 0.175753s (t │ │ │ │ 0000aa50: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 0000aa60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0000aa70: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0000aa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aab0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ ├── ./usr/share/info/LatticePolytopes.info.gz │ │ │ ├── LatticePolytopes.info │ │ │ │ @@ -1197,32 +1197,32 @@ │ │ │ │ 00004ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004ad0: 2d2b 0a7c 6936 203a 2074 696d 6520 6172 -+.|i6 : time ar │ │ │ │ 00004ae0: 6549 736f 6d6f 7270 6869 6328 502c 5029 eIsomorphic(P,P) │ │ │ │ 00004af0: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ 00004b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004b20: 207c 0a7c 202d 2d20 7573 6564 2030 2e39 |.| -- used 0.9 │ │ │ │ -00004b30: 3032 3634 3673 2028 6370 7529 3b20 302e 02646s (cpu); 0. │ │ │ │ -00004b40: 3636 3438 3635 7320 2874 6872 6561 6429 664865s (thread) │ │ │ │ +00004b30: 3330 3233 3373 2028 6370 7529 3b20 302e 30233s (cpu); 0. │ │ │ │ +00004b40: 3533 3936 3632 7320 2874 6872 6561 6429 539662s (thread) │ │ │ │ 00004b50: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00004b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004b70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00004b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004bc0: 2d2b 0a7c 6937 203a 2074 696d 6520 6172 -+.|i7 : time ar │ │ │ │ 00004bd0: 6549 736f 6d6f 7270 6869 6328 502c 502c eIsomorphic(P,P, │ │ │ │ 00004be0: 736d 6f6f 7468 5465 7374 3d3e 6661 6c73 smoothTest=>fals │ │ │ │ 00004bf0: 6529 3b20 2020 2020 2020 2020 2020 2020 e); │ │ │ │ 00004c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00004c10: 207c 0a7c 202d 2d20 7573 6564 2030 2e33 |.| -- used 0.3 │ │ │ │ -00004c20: 3337 3733 3473 2028 6370 7529 3b20 302e 37734s (cpu); 0. │ │ │ │ -00004c30: 3236 3039 3538 7320 2874 6872 6561 6429 260958s (thread) │ │ │ │ -00004c40: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +00004c10: 207c 0a7c 202d 2d20 7573 6564 2030 2e35 |.| -- used 0.5 │ │ │ │ +00004c20: 3134 3435 3173 2028 6370 7529 3b20 302e 14451s (cpu); 0. │ │ │ │ +00004c30: 3330 3636 3973 2028 7468 7265 6164 293b 30669s (thread); │ │ │ │ +00004c40: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00004c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004c60: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00004c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004cb0: 2d2b 0a0a 5761 7973 2074 6f20 7573 6520 -+..Ways to use │ │ ├── ./usr/share/info/LinearTruncations.info.gz │ │ │ ├── LinearTruncations.info │ │ │ │ @@ -2195,16 +2195,16 @@ │ │ │ │ 00008920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008940: 2d2d 2d2d 2d2b 0a7c 6936 203a 2065 6c61 -----+.|i6 : ela │ │ │ │ 00008950: 7073 6564 5469 6d65 2066 696e 6452 6567 psedTime findReg │ │ │ │ 00008960: 696f 6e28 7b7b 302c 307d 2c7b 342c 347d ion({{0,0},{4,4} │ │ │ │ 00008970: 7d2c 4d2c 6629 2020 2020 2020 2020 2020 },M,f) │ │ │ │ 00008980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00008990: 2020 2020 207c 0a7c 202d 2d20 2e30 3932 |.| -- .092 │ │ │ │ -000089a0: 3538 3234 7320 656c 6170 7365 6420 2020 5824s elapsed │ │ │ │ +00008990: 2020 2020 207c 0a7c 202d 2d20 2e30 3731 |.| -- .071 │ │ │ │ +000089a0: 3533 3133 7320 656c 6170 7365 6420 2020 5313s elapsed │ │ │ │ 000089b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000089c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000089d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000089e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000089f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2230,16 +2230,16 @@ │ │ │ │ 00008b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008b70: 2d2d 2d2d 2d2b 0a7c 6937 203a 2065 6c61 -----+.|i7 : ela │ │ │ │ 00008b80: 7073 6564 5469 6d65 2066 696e 6452 6567 psedTime findReg │ │ │ │ 00008b90: 696f 6e28 7b7b 302c 307d 2c7b 342c 347d ion({{0,0},{4,4} │ │ │ │ 00008ba0: 7d2c 4d2c 662c 496e 6e65 723d 3e7b 7b31 },M,f,Inner=>{{1 │ │ │ │ 00008bb0: 2c32 7d2c 7b33 2c31 7d7d 2c4f 7574 6572 ,2},{3,1}},Outer │ │ │ │ -00008bc0: 3d3e 7b7b 317c 0a7c 202d 2d20 2e30 3933 =>{{1|.| -- .093 │ │ │ │ -00008bd0: 3230 3531 7320 656c 6170 7365 6420 2020 2051s elapsed │ │ │ │ +00008bc0: 3d3e 7b7b 317c 0a7c 202d 2d20 2e30 3331 =>{{1|.| -- .031 │ │ │ │ +00008bd0: 3034 3138 7320 656c 6170 7365 6420 2020 0418s elapsed │ │ │ │ 00008be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008c10: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00008c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4113,16 +4113,16 @@ │ │ │ │ 00010100: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ 00010110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00010140: 3620 3a20 656c 6170 7365 6454 696d 6520 6 : elapsedTime │ │ │ │ 00010150: 6c69 6e65 6172 5472 756e 6361 7469 6f6e linearTruncation │ │ │ │ 00010160: 7328 7b7b 322c 322c 327d 2c7b 342c 342c s({{2,2,2},{4,4, │ │ │ │ -00010170: 347d 7d2c 204d 297c 0a7c 202d 2d20 342e 4}}, M)|.| -- 4. │ │ │ │ -00010180: 3338 3438 3873 2065 6c61 7073 6564 2020 38488s elapsed │ │ │ │ +00010170: 347d 7d2c 204d 297c 0a7c 202d 2d20 332e 4}}, M)|.| -- 3. │ │ │ │ +00010180: 3336 3430 3673 2065 6c61 7073 6564 2020 36406s elapsed │ │ │ │ 00010190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000101a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000101b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000101c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000101d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000101e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000101f0: 6f36 203d 207b 7b34 2c20 332c 2033 7d2c o6 = {{4, 3, 3}, │ │ │ │ @@ -4139,16 +4139,16 @@ │ │ │ │ 000102a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000102b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000102c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000102d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a ---------+.|i7 : │ │ │ │ 000102e0: 2065 6c61 7073 6564 5469 6d65 206c 696e elapsedTime lin │ │ │ │ 000102f0: 6561 7254 7275 6e63 6174 696f 6e73 426f earTruncationsBo │ │ │ │ 00010300: 756e 6420 4d20 2020 2020 2020 2020 2020 und M │ │ │ │ -00010310: 2020 2020 7c0a 7c20 2d2d 202e 3032 3539 |.| -- .0259 │ │ │ │ -00010320: 3130 3873 2065 6c61 7073 6564 2020 2020 108s elapsed │ │ │ │ +00010310: 2020 2020 7c0a 7c20 2d2d 202e 3033 3130 |.| -- .0310 │ │ │ │ +00010320: 3534 3873 2065 6c61 7073 6564 2020 2020 548s elapsed │ │ │ │ 00010330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010340: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00010350: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00010360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010380: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ 00010390: 3d20 7b7b 342c 2033 2c20 337d 2c20 7b34 = {{4, 3, 3}, {4 │ │ ├── ./usr/share/info/LocalRings.info.gz │ │ │ ├── LocalRings.info │ │ │ │ @@ -2607,15 +2607,15 @@ │ │ │ │ 0000a2e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0000a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a310: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ 0000a320: 656c 6170 7365 6454 696d 6520 6869 6c62 elapsedTime hilb │ │ │ │ 0000a330: 6572 7453 616d 7565 6c46 756e 6374 696f ertSamuelFunctio │ │ │ │ 0000a340: 6e28 4d2c 2030 2c20 3629 2020 2020 2020 n(M, 0, 6) │ │ │ │ -0000a350: 7c0a 7c20 2d2d 202e 3239 3431 3335 7320 |.| -- .294135s │ │ │ │ +0000a350: 7c0a 7c20 2d2d 202e 3232 3130 3538 7320 |.| -- .221058s │ │ │ │ 0000a360: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 0000a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a380: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0000a390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a3c0: 7c0a 7c6f 3520 3d20 7b31 2c20 332c 2036 |.|o5 = {1, 3, 6 │ │ │ │ @@ -2745,15 +2745,15 @@ │ │ │ │ 0000ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ab90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 ---------+.|i11 │ │ │ │ 0000aba0: 3a20 656c 6170 7365 6454 696d 6520 6869 : elapsedTime hi │ │ │ │ 0000abb0: 6c62 6572 7453 616d 7565 6c46 756e 6374 lbertSamuelFunct │ │ │ │ 0000abc0: 696f 6e28 4e2c 2030 2c20 3529 202d 2d20 ion(N, 0, 5) -- │ │ │ │ 0000abd0: 6e2b 3120 2d2d 2030 2e30 3220 7365 636f n+1 -- 0.02 seco │ │ │ │ 0000abe0: 6e64 7320 2020 2020 207c 0a7c 202d 2d20 nds |.| -- │ │ │ │ -0000abf0: 2e30 3131 3937 3535 7320 656c 6170 7365 .0119755s elapse │ │ │ │ +0000abf0: 2e30 3134 3730 3033 7320 656c 6170 7365 .0147003s elapse │ │ │ │ 0000ac00: 6420 2020 2020 2020 2020 2020 2020 2020 d │ │ │ │ 0000ac10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0000ac40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2780,15 +2780,15 @@ │ │ │ │ 0000adb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000adc0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 ---------+.|i12 │ │ │ │ 0000add0: 3a20 656c 6170 7365 6454 696d 6520 6869 : elapsedTime hi │ │ │ │ 0000ade0: 6c62 6572 7453 616d 7565 6c46 756e 6374 lbertSamuelFunct │ │ │ │ 0000adf0: 696f 6e28 712c 204e 2c20 302c 2035 2920 ion(q, N, 0, 5) │ │ │ │ 0000ae00: 2d2d 2036 286e 2b31 2920 2d2d 2030 2e33 -- 6(n+1) -- 0.3 │ │ │ │ 0000ae10: 3220 7365 636f 6e64 737c 0a7c 202d 2d20 2 seconds|.| -- │ │ │ │ -0000ae20: 2e31 3836 3231 3773 2065 6c61 7073 6564 .186217s elapsed │ │ │ │ +0000ae20: 2e32 3333 3737 3173 2065 6c61 7073 6564 .233771s elapsed │ │ │ │ 0000ae30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae60: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0000ae70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/MRDI.info.gz │ │ │ ├── MRDI.info │ │ │ │ @@ -918,16 +918,16 @@ │ │ │ │ 00003950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003960: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00003970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000039a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000039b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -000039c0: 3720 3d20 2f74 6d70 2f4d 322d 3339 3436 7 = /tmp/M2-3946 │ │ │ │ -000039d0: 302d 302f 302e 6d72 6469 2020 2020 2020 0-0/0.mrdi │ │ │ │ +000039c0: 3720 3d20 2f74 6d70 2f4d 322d 3535 3039 7 = /tmp/M2-5509 │ │ │ │ +000039d0: 382d 302f 302e 6d72 6469 2020 2020 2020 8-0/0.mrdi │ │ │ │ 000039e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000039f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003a00: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00003a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -1107,40 +1107,40 @@ │ │ │ │ 00004520: 4d52 4449 7d20 2020 2020 2020 2020 2020 MRDI} │ │ │ │ 00004530: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00004540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004560: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ 00004570: 3220 3d20 7b30 203d 3e20 287b 4d61 6361 2 = {0 => ({Maca │ │ │ │ 00004580: 756c 6179 322c 2073 6176 654d 5244 497d ulay2, saveMRDI} │ │ │ │ -00004590: 2c20 5175 6f74 6965 6e74 5269 6e67 2920 , QuotientRing) │ │ │ │ +00004590: 2c20 5269 6e67 456c 656d 656e 7429 2020 , RingElement) │ │ │ │ 000045a0: 207d 7c0a 7c20 2020 2020 207b 3120 3d3e }|.| {1 => │ │ │ │ 000045b0: 2028 7b4d 6163 6175 6c61 7932 2c20 7361 ({Macaulay2, sa │ │ │ │ -000045c0: 7665 4d52 4449 7d2c 205a 5a29 2020 2020 veMRDI}, ZZ) │ │ │ │ +000045c0: 7665 4d52 4449 7d2c 2052 696e 6729 2020 veMRDI}, Ring) │ │ │ │ 000045d0: 2020 2020 2020 2020 7d7c 0a7c 2020 2020 }|.| │ │ │ │ 000045e0: 2020 7b32 203d 3e20 287b 4d61 6361 756c {2 => ({Macaul │ │ │ │ 000045f0: 6179 322c 2073 6176 654d 5244 497d 2c20 ay2, saveMRDI}, │ │ │ │ -00004600: 5269 6e67 456c 656d 656e 7429 2020 207d RingElement) } │ │ │ │ +00004600: 4d61 7472 6978 2920 2020 2020 2020 207d Matrix) } │ │ │ │ 00004610: 7c0a 7c20 2020 2020 207b 3320 3d3e 2028 |.| {3 => ( │ │ │ │ 00004620: 7b4d 6163 6175 6c61 7932 2c20 7361 7665 {Macaulay2, save │ │ │ │ -00004630: 4d52 4449 7d2c 2052 696e 6729 2020 2020 MRDI}, Ring) │ │ │ │ -00004640: 2020 2020 2020 7d7c 0a7c 2020 2020 2020 }|.| │ │ │ │ +00004630: 4d52 4449 7d2c 2047 616c 6f69 7346 6965 MRDI}, GaloisFie │ │ │ │ +00004640: 6c64 2920 2020 7d7c 0a7c 2020 2020 2020 ld) }|.| │ │ │ │ 00004650: 7b34 203d 3e20 287b 4d61 6361 756c 6179 {4 => ({Macaulay │ │ │ │ -00004660: 322c 2073 6176 654d 5244 497d 2c20 4d61 2, saveMRDI}, Ma │ │ │ │ -00004670: 7472 6978 2920 2020 2020 2020 207d 7c0a trix) }|. │ │ │ │ +00004660: 322c 2073 6176 654d 5244 497d 2c20 4964 2, saveMRDI}, Id │ │ │ │ +00004670: 6561 6c29 2020 2020 2020 2020 207d 7c0a eal) }|. │ │ │ │ 00004680: 7c20 2020 2020 207b 3520 3d3e 2028 7b4d | {5 => ({M │ │ │ │ 00004690: 6163 6175 6c61 7932 2c20 7361 7665 4d52 acaulay2, saveMR │ │ │ │ -000046a0: 4449 7d2c 2047 616c 6f69 7346 6965 6c64 DI}, GaloisField │ │ │ │ -000046b0: 2920 2020 7d7c 0a7c 2020 2020 2020 7b36 ) }|.| {6 │ │ │ │ +000046a0: 4449 7d2c 2050 6f6c 796e 6f6d 6961 6c52 DI}, PolynomialR │ │ │ │ +000046b0: 696e 6729 7d7c 0a7c 2020 2020 2020 7b36 ing)}|.| {6 │ │ │ │ 000046c0: 203d 3e20 287b 4d61 6361 756c 6179 322c => ({Macaulay2, │ │ │ │ -000046d0: 2073 6176 654d 5244 497d 2c20 4964 6561 saveMRDI}, Idea │ │ │ │ -000046e0: 6c29 2020 2020 2020 2020 207d 7c0a 7c20 l) }|.| │ │ │ │ +000046d0: 2073 6176 654d 5244 497d 2c20 5175 6f74 saveMRDI}, Quot │ │ │ │ +000046e0: 6965 6e74 5269 6e67 2920 207d 7c0a 7c20 ientRing) }|.| │ │ │ │ 000046f0: 2020 2020 207b 3720 3d3e 2028 7b4d 6163 {7 => ({Mac │ │ │ │ 00004700: 6175 6c61 7932 2c20 7361 7665 4d52 4449 aulay2, saveMRDI │ │ │ │ -00004710: 7d2c 2050 6f6c 796e 6f6d 6961 6c52 696e }, PolynomialRin │ │ │ │ -00004720: 6729 7d7c 0a7c 2020 2020 2020 2020 2020 g)}|.| │ │ │ │ +00004710: 7d2c 205a 5a29 2020 2020 2020 2020 2020 }, ZZ) │ │ │ │ +00004720: 2020 7d7c 0a7c 2020 2020 2020 2020 2020 }|.| │ │ │ │ 00004730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004750: 2020 2020 2020 2020 2020 7c0a 7c6f 3132 |.|o12 │ │ │ │ 00004760: 203a 204e 756d 6265 7265 6456 6572 7469 : NumberedVerti │ │ │ │ 00004770: 6361 6c4c 6973 7420 2020 2020 2020 2020 calList │ │ │ │ 00004780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004790: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ @@ -1904,16 +1904,16 @@ │ │ │ │ 000076f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007700: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00007710: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00007720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007750: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00007760: 7c6f 3130 203d 202f 746d 702f 4d32 2d33 |o10 = /tmp/M2-3 │ │ │ │ -00007770: 3934 3431 2d30 2f30 2e6d 7264 6920 2020 9441-0/0.mrdi │ │ │ │ +00007760: 7c6f 3130 203d 202f 746d 702f 4d32 2d35 |o10 = /tmp/M2-5 │ │ │ │ +00007770: 3530 3539 2d30 2f30 2e6d 7264 6920 2020 5059-0/0.mrdi │ │ │ │ 00007780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000077a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000077b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000077c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000077d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000077e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ ├── ./usr/share/info/Macaulay2Doc.info.gz │ │ │ ├── Macaulay2Doc.info │ │ │ │ @@ -4907,17 +4907,17 @@ │ │ │ │ 000132a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000132b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000132c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000132d0: 2d2d 2b0a 7c69 3539 203a 2074 696d 6520 --+.|i59 : time │ │ │ │ 000132e0: 4320 3d20 7265 736f 6c75 7469 6f6e 204d C = resolution M │ │ │ │ 000132f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013300: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013310: 7c20 2d2d 2075 7365 6420 302e 3030 3239 | -- used 0.0029 │ │ │ │ -00013320: 3338 3733 7320 2863 7075 293b 2030 2e30 3873s (cpu); 0.0 │ │ │ │ -00013330: 3032 3933 3135 3573 2028 7468 7265 6164 0293155s (thread │ │ │ │ +00013310: 7c20 2d2d 2075 7365 6420 302e 3030 3339 | -- used 0.0039 │ │ │ │ +00013320: 3632 3734 7320 2863 7075 293b 2030 2e30 6274s (cpu); 0.0 │ │ │ │ +00013330: 3033 3936 3236 3473 2028 7468 7265 6164 0396264s (thread │ │ │ │ 00013340: 293b 2030 7320 2867 6329 7c0a 7c20 2020 ); 0s (gc)|.| │ │ │ │ 00013350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013380: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00013390: 3320 2020 2020 2036 2020 2020 2020 3135 3 6 15 │ │ │ │ 000133a0: 2020 2020 2020 3138 2020 2020 2020 3620 18 6 │ │ │ │ @@ -26092,16 +26092,16 @@ │ │ │ │ 00065eb0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ 00065ec0: 2028 6320 3d20 436f 6d6d 616e 6420 2264 (c = Command "d │ │ │ │ 00065ed0: 6174 6522 3b29 2020 2020 2020 2020 7c0a ate";) |. │ │ │ │ 00065ee0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00065ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00065f00: 2d2d 2d2b 0a7c 6934 203a 2063 2020 2020 ---+.|i4 : c │ │ │ │ 00065f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065f20: 2020 2020 2020 2020 7c0a 7c4d 6f6e 204d |.|Mon M │ │ │ │ -00065f30: 6179 2031 3820 3132 3a33 323a 3539 2055 ay 18 12:32:59 U │ │ │ │ +00065f20: 2020 2020 2020 2020 7c0a 7c57 6564 204d |.|Wed M │ │ │ │ +00065f30: 6179 2032 3020 3137 3a32 323a 3032 2055 ay 20 17:22:02 U │ │ │ │ 00065f40: 5443 2032 3032 3620 2020 2020 207c 0a7c TC 2026 |.| │ │ │ │ 00065f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065f70: 2020 7c0a 7c6f 3420 3d20 3020 2020 2020 |.|o4 = 0 │ │ │ │ 00065f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065f90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00065fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -39701,16 +39701,16 @@ │ │ │ │ 0009b140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009b150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009b160: 2d2b 0a7c 6931 3120 3a20 656c 6170 7365 -+.|i11 : elapse │ │ │ │ 0009b170: 6454 696d 6520 7363 616e 2831 3030 302c dTime scan(1000, │ │ │ │ 0009b180: 2069 202d 3e20 7323 6920 3d20 695e 3229 i -> s#i = i^2) │ │ │ │ 0009b190: 202d 2d20 7175 6164 7261 7469 632c 2073 -- quadratic, s │ │ │ │ 0009b1a0: 696e 6365 2077 6520 6772 6f77 2073 2061 ince we grow s a │ │ │ │ -0009b1b0: 747c 0a7c 202d 2d20 2e30 3033 3431 3634 t|.| -- .0034164 │ │ │ │ -0009b1c0: 3273 2065 6c61 7073 6564 2020 2020 2020 2s elapsed │ │ │ │ +0009b1b0: 747c 0a7c 202d 2d20 2e30 3035 3136 3631 t|.| -- .0051661 │ │ │ │ +0009b1c0: 3973 2065 6c61 7073 6564 2020 2020 2020 9s elapsed │ │ │ │ 0009b1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009b1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009b1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009b200: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ 0009b210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009b220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009b230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -39756,16 +39756,16 @@ │ │ │ │ 0009b4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009b4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009b4d0: 2d2b 0a7c 6931 3320 3a20 656c 6170 7365 -+.|i13 : elapse │ │ │ │ 0009b4e0: 6454 696d 6520 7363 616e 2831 3030 302c dTime scan(1000, │ │ │ │ 0009b4f0: 2069 202d 3e20 7423 6920 3d20 695e 3229 i -> t#i = i^2) │ │ │ │ 0009b500: 202d 2d20 6c69 6e65 6172 2020 2020 2020 -- linear │ │ │ │ 0009b510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0009b520: 207c 0a7c 202d 2d20 2e30 3030 3435 3737 |.| -- .0004577 │ │ │ │ -0009b530: 3936 7320 656c 6170 7365 6420 2020 2020 96s elapsed │ │ │ │ +0009b520: 207c 0a7c 202d 2d20 2e30 3030 3338 3639 |.| -- .0003869 │ │ │ │ +0009b530: 3833 7320 656c 6170 7365 6420 2020 2020 83s elapsed │ │ │ │ 0009b540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009b550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009b560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009b570: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0009b580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009b590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009b5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -76284,41 +76284,41 @@ │ │ │ │ 00129fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00129fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00129fd0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00129fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00129ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0012a000: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0012a010: 7c6f 3420 3d20 7b30 203d 3e20 2863 6f64 |o4 = {0 => (cod │ │ │ │ -0012a020: 696d 2c20 4d6f 6475 6c65 2920 2020 2020 im, Module) │ │ │ │ +0012a020: 696d 2c20 4964 6561 6c29 2020 2020 2020 im, Ideal) │ │ │ │ 0012a030: 2020 207d 2020 2020 2020 2020 2020 2020 } │ │ │ │ 0012a040: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0012a050: 207b 3120 3d3e 2028 636f 6469 6d2c 2043 {1 => (codim, C │ │ │ │ -0012a060: 6f68 6572 656e 7453 6865 6166 2920 7d20 oherentSheaf) } │ │ │ │ +0012a050: 207b 3120 3d3e 2028 636f 6469 6d2c 2050 {1 => (codim, P │ │ │ │ +0012a060: 6f6c 796e 6f6d 6961 6c52 696e 6729 7d20 olynomialRing)} │ │ │ │ 0012a070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0012a080: 2020 2020 7c0a 7c20 2020 2020 7b32 203d |.| {2 = │ │ │ │ -0012a090: 3e20 2863 6f64 696d 2c20 5661 7269 6574 > (codim, Variet │ │ │ │ -0012a0a0: 7929 2020 2020 2020 207d 2020 2020 2020 y) } │ │ │ │ +0012a090: 3e20 2863 6f64 696d 2c20 4265 7474 6954 > (codim, BettiT │ │ │ │ +0012a0a0: 616c 6c79 2920 2020 207d 2020 2020 2020 ally) } │ │ │ │ 0012a0b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0012a0c0: 0a7c 2020 2020 207b 3320 3d3e 2028 636f .| {3 => (co │ │ │ │ -0012a0d0: 6469 6d2c 204d 6f6e 6f6d 6961 6c49 6465 dim, MonomialIde │ │ │ │ -0012a0e0: 616c 2920 7d20 2020 2020 2020 2020 2020 al) } │ │ │ │ +0012a0d0: 6469 6d2c 2051 756f 7469 656e 7452 696e dim, QuotientRin │ │ │ │ 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2020 2020 ) } │ │ │ │ +0012a180: 6f64 696d 2c20 5661 7269 6574 7929 2020 odim, Variety) │ │ │ │ +0012a190: 2020 2020 207d 2020 2020 2020 2020 2020 } │ │ │ │ 0012a1a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0012a1b0: 2020 207b 3720 3d3e 2028 636f 6469 6d2c {7 => (codim, │ │ │ │ -0012a1c0: 2051 756f 7469 656e 7452 696e 6729 2020 QuotientRing) │ │ │ │ +0012a1c0: 204d 6f6e 6f6d 6961 6c49 6465 616c 2920 MonomialIdeal) │ │ │ │ 0012a1d0: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ 0012a1e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0012a1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0012a200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0012a210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0012a220: 207c 0a7c 6f34 203a 204e 756d 6265 7265 |.|o4 : Numbere │ │ │ │ 0012a230: 6456 6572 7469 6361 6c4c 6973 7420 2020 dVerticalList │ │ │ │ @@ -76339,31 +76339,31 @@ │ │ │ │ 0012a320: 6f6e 5461 626c 657b 4765 6e65 7269 6320 onTable{Generic │ │ │ │ 0012a330: 3d3e 2066 616c 7365 7d29 7d20 2020 2020 => false})} │ │ │ │ 0012a340: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0012a350: 7b31 203d 3e20 284f 7074 696f 6e54 6162 {1 => (OptionTab │ │ │ │ 0012a360: 6c65 7b47 656e 6572 6963 203d 3e20 6661 le{Generic => fa │ │ │ │ 0012a370: 6c73 657d 297d 2020 2020 2020 2020 2020 lse})} │ │ │ │ 0012a380: 2020 207c 0a7c 2020 2020 207b 3220 3d3e |.| {2 => │ │ │ │ -0012a390: 2028 4f70 7469 6f6e 5461 626c 657b 4765 (OptionTable{Ge │ │ │ │ -0012a3a0: 6e65 7269 6320 3d3e 2066 616c 7365 7d29 neric => false}) │ │ │ │ +0012a390: 2028 4f70 7469 6f6e 5461 626c 657b 7d29 (OptionTable{}) │ │ │ │ +0012a3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0012a3b0: 7d20 2020 2020 2020 2020 2020 2020 7c0a } |. │ │ │ │ 0012a3c0: 7c20 2020 2020 7b33 203d 3e20 284f 7074 | {3 => (Opt │ │ │ │ 0012a3d0: 696f 6e54 6162 6c65 7b47 656e 6572 6963 ionTable{Generic │ │ │ │ 0012a3e0: 203d 3e20 6661 6c73 657d 297d 2020 2020 => false})} │ │ │ │ 0012a3f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0012a400: 207b 3420 3d3e 2028 4f70 7469 6f6e 5461 {4 => (OptionTa │ │ │ │ 0012a410: 626c 657b 4765 6e65 7269 6320 3d3e 2066 ble{Generic => f │ │ │ │ 0012a420: 616c 7365 7d29 7d20 2020 2020 2020 2020 alse})} │ │ │ │ 0012a430: 2020 2020 7c0a 7c20 2020 2020 7b35 203d |.| {5 = │ │ │ │ 0012a440: 3e20 284f 7074 696f 6e54 6162 6c65 7b47 > (OptionTable{G │ │ │ │ 0012a450: 656e 6572 6963 203d 3e20 6661 6c73 657d eneric => false} │ │ │ │ 0012a460: 297d 2020 2020 2020 2020 2020 2020 207c )} | │ │ │ │ 0012a470: 0a7c 2020 2020 207b 3620 3d3e 2028 4f70 .| {6 => (Op │ │ │ │ -0012a480: 7469 6f6e 5461 626c 657b 7d29 2020 2020 tionTable{}) │ │ │ │ -0012a490: 2020 2020 2020 2020 2020 2020 7d20 2020 } │ │ │ │ +0012a480: 7469 6f6e 5461 626c 657b 4765 6e65 7269 tionTable{Generi │ │ │ │ +0012a490: 6320 3d3e 2066 616c 7365 7d29 7d20 2020 c => false})} │ │ │ │ 0012a4a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0012a4b0: 2020 7b37 203d 3e20 284f 7074 696f 6e54 {7 => (OptionT │ │ │ │ 0012a4c0: 6162 6c65 7b47 656e 6572 6963 203d 3e20 able{Generic => │ │ │ │ 0012a4d0: 6661 6c73 657d 297d 2020 2020 2020 2020 false})} │ │ │ │ 0012a4e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0012a4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0012a500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -103405,16 +103405,16 @@ │ │ │ │ 00193ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00193ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00193ee0: 2d2d 2d2d 2d2b 0a7c 6938 203a 2074 696d -----+.|i8 : tim │ │ │ │ 00193ef0: 6520 4a20 3d20 7472 756e 6361 7465 2838 e J = truncate(8 │ │ │ │ 00193f00: 2c20 492c 204d 696e 696d 616c 4765 6e65 , I, MinimalGene │ │ │ │ 00193f10: 7261 746f 7273 203d 3e20 6661 6c73 6529 rators => false) │ │ │ │ 00193f20: 3b7c 0a7c 202d 2d20 7573 6564 2030 2e30 ;|.| -- used 0.0 │ │ │ │ -00193f30: 3038 3535 3338 3873 2028 6370 7529 3b20 0855388s (cpu); │ │ │ │ -00193f40: 302e 3030 3835 3436 3239 7320 2874 6872 0.00854629s (thr │ │ │ │ +00193f30: 3035 3532 3535 3573 2028 6370 7529 3b20 0552555s (cpu); │ │ │ │ +00193f40: 302e 3030 3535 3231 3231 7320 2874 6872 0.00552121s (thr │ │ │ │ 00193f50: 6561 6429 3b20 3073 2028 6763 297c 0a7c ead); 0s (gc)|.| │ │ │ │ 00193f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00193f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00193f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00193f90: 2020 2020 2020 2020 207c 0a7c 6f38 203a |.|o8 : │ │ │ │ 00193fa0: 2049 6465 616c 206f 6620 5220 2020 2020 Ideal of R │ │ │ │ 00193fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -103423,17 +103423,17 @@ │ │ │ │ 00193fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00193ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00194000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00194010: 2d2b 0a7c 6939 203a 2074 696d 6520 4b20 -+.|i9 : time K │ │ │ │ 00194020: 3d20 7472 756e 6361 7465 2838 2c20 492c = truncate(8, I, │ │ │ │ 00194030: 204d 696e 696d 616c 4765 6e65 7261 746f MinimalGenerato │ │ │ │ 00194040: 7273 203d 3e20 7472 7565 293b 207c 0a7c rs => true); |.| │ │ │ │ -00194050: 202d 2d20 7573 6564 2030 2e30 3637 3634 -- used 0.06764 │ │ │ │ -00194060: 3239 7320 2863 7075 293b 2030 2e30 3637 29s (cpu); 0.067 │ │ │ │ -00194070: 3635 3437 7320 2874 6872 6561 6429 3b20 6547s (thread); │ │ │ │ +00194050: 202d 2d20 7573 6564 2030 2e30 3439 3730 -- used 0.04970 │ │ │ │ +00194060: 3331 7320 2863 7075 293b 2030 2e30 3439 31s (cpu); 0.049 │ │ │ │ +00194070: 3731 3234 7320 2874 6872 6561 6429 3b20 7124s (thread); │ │ │ │ 00194080: 3073 2028 6763 2920 207c 0a7c 2020 2020 0s (gc) |.| │ │ │ │ 00194090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001940a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001940b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001940c0: 2020 2020 207c 0a7c 6f39 203a 2049 6465 |.|o9 : Ide │ │ │ │ 001940d0: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ 001940e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -104032,15 +104032,15 @@ │ │ │ │ 001965f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00196600: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ 00196610: 2d20 636f 6d70 7574 696e 6720 7064 696d - computing pdim │ │ │ │ 00196620: 2720 2020 2020 2020 2020 2020 2020 2020 ' │ │ │ │ 00196630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00196640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00196650: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00196660: 2d20 2e30 3036 3431 3230 3773 2065 6c61 - .00641207s ela │ │ │ │ +00196660: 2d20 2e30 3034 3339 3632 3973 2065 6c61 - .00439629s ela │ │ │ │ 00196670: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 00196680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00196690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001966a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 001966b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001966c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001966d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -104057,15 +104057,15 @@ │ │ │ │ 00196780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00196790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ 001967a0: 203a 2065 6c61 7073 6564 5469 6d65 2070 : elapsedTime p │ │ │ │ 001967b0: 6469 6d27 204d 2020 2020 2020 2020 2020 dim' M │ │ │ │ 001967c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001967d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001967e0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -001967f0: 2d20 2e30 3030 3030 3139 3334 7320 656c - .000001934s el │ │ │ │ +001967f0: 2d20 2e30 3030 3030 3331 3534 7320 656c - .000003154s el │ │ │ │ 00196800: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00196810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00196820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00196830: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00196840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00196850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00196860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -104560,17 +104560,17 @@ │ │ │ │ 001986f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00198700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00198710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00198720: 2d2d 2d2d 2d2b 0a7c 6932 203a 2074 696d -----+.|i2 : tim │ │ │ │ 00198730: 6520 6669 6220 3238 2020 2020 2020 2020 e fib 28 │ │ │ │ 00198740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00198750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00198760: 207c 0a7c 202d 2d20 7573 6564 2031 2e36 |.| -- used 1.6 │ │ │ │ -00198770: 3239 3035 7320 2863 7075 293b 2030 2e38 2905s (cpu); 0.8 │ │ │ │ -00198780: 3039 3334 3573 2028 7468 7265 6164 293b 09345s (thread); │ │ │ │ +00198760: 207c 0a7c 202d 2d20 7573 6564 2031 2e30 |.| -- used 1.0 │ │ │ │ +00198770: 3434 3636 7320 2863 7075 293b 2030 2e36 4466s (cpu); 0.6 │ │ │ │ +00198780: 3231 3634 3973 2028 7468 7265 6164 293b 21649s (thread); │ │ │ │ 00198790: 2030 7320 2867 6329 2020 2020 207c 0a7c 0s (gc) |.| │ │ │ │ 001987a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001987b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001987c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001987d0: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ 001987e0: 2035 3134 3232 3920 2020 2020 2020 2020 514229 │ │ │ │ 001987f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -104602,16 +104602,16 @@ │ │ │ │ 00198990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001989a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001989b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ 001989c0: 2074 696d 6520 6669 6220 3238 2020 2020 time fib 28 │ │ │ │ 001989d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001989e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001989f0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00198a00: 2036 2e38 3238 3865 2d30 3573 2028 6370 6.8288e-05s (cp │ │ │ │ -00198a10: 7529 3b20 362e 3738 3738 652d 3035 7320 u); 6.7878e-05s │ │ │ │ +00198a00: 2037 2e32 3338 3565 2d30 3573 2028 6370 7.2385e-05s (cp │ │ │ │ +00198a10: 7529 3b20 372e 3030 3833 652d 3035 7320 u); 7.0083e-05s │ │ │ │ 00198a20: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00198a30: 297c 0a7c 2020 2020 2020 2020 2020 2020 )|.| │ │ │ │ 00198a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00198a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00198a60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00198a70: 6f34 203d 2035 3134 3232 3920 2020 2020 o4 = 514229 │ │ │ │ 00198a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -104620,17 +104620,17 @@ │ │ │ │ 00198ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00198ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00198ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00198ae0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 696d -----+.|i5 : tim │ │ │ │ 00198af0: 6520 6669 6220 3238 2020 2020 2020 2020 e fib 28 │ │ │ │ 00198b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00198b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00198b20: 207c 0a7c 202d 2d20 7573 6564 2034 2e33 |.| -- used 4.3 │ │ │ │ +00198b20: 207c 0a7c 202d 2d20 7573 6564 2033 2e37 |.| -- used 3.7 │ │ │ │ 00198b30: 3438 652d 3036 7320 2863 7075 293b 2033 48e-06s (cpu); 3 │ │ │ │ -00198b40: 2e39 3938 652d 3036 7320 2874 6872 6561 .998e-06s (threa │ │ │ │ +00198b40: 2e31 3833 652d 3036 7320 2874 6872 6561 .183e-06s (threa │ │ │ │ 00198b50: 6429 3b20 3073 2028 6763 2920 207c 0a7c d); 0s (gc) |.| │ │ │ │ 00198b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00198b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00198b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00198b90: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ 00198ba0: 2035 3134 3232 3920 2020 2020 2020 2020 514229 │ │ │ │ 00198bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -107170,15 +107170,15 @@ │ │ │ │ 001a2a10: 2020 2020 2020 2020 2020 6c69 6e65 4e75 lineNu │ │ │ │ 001a2a20: 6d62 6572 203d 3e20 3220 2020 2020 2020 mber => 2 │ │ │ │ 001a2a30: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 001a2a40: 2020 2020 2020 206c 6f61 6444 6570 7468 loadDepth │ │ │ │ 001a2a50: 203d 3e20 3320 2020 2020 2020 2020 2020 => 3 │ │ │ │ 001a2a60: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 001a2a70: 2020 2020 6d61 7841 6c6c 6f77 6162 6c65 maxAllowable │ │ │ │ -001a2a80: 5468 7265 6164 7320 3d3e 2037 2020 207c Threads => 7 | │ │ │ │ +001a2a80: 5468 7265 6164 7320 3d3e 2031 3720 207c Threads => 17 | │ │ │ │ 001a2a90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 001a2aa0: 206d 6178 4578 706f 6e65 6e74 203d 3e20 maxExponent => │ │ │ │ 001a2ab0: 3130 3733 3734 3138 3233 2020 7c0a 7c20 1073741823 |.| │ │ │ │ 001a2ac0: 2020 2020 2020 2020 2020 2020 2020 6d69 mi │ │ │ │ 001a2ad0: 6e45 7870 6f6e 656e 7420 3d3e 202d 3130 nExponent => -10 │ │ │ │ 001a2ae0: 3733 3734 3138 3234 207c 0a7c 2020 2020 73741824 |.| │ │ │ │ 001a2af0: 2020 2020 2020 2020 2020 206e 756d 5442 numTB │ │ │ │ @@ -117608,16 +117608,16 @@ │ │ │ │ 001cb670: 6d62 6572 206f 6620 7365 636f 6e64 732e mber of seconds. │ │ │ │ 001cb680: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 001cb690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001cb6a0: 2b0a 7c69 3120 3a20 6265 6e63 686d 6172 +.|i1 : benchmar │ │ │ │ 001cb6b0: 6b20 2273 7172 7420 3270 3130 3030 3030 k "sqrt 2p100000 │ │ │ │ 001cb6c0: 227c 0a7c 2020 2020 2020 2020 2020 2020 "|.| │ │ │ │ 001cb6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -001cb6e0: 2020 7c0a 7c6f 3120 3d20 2e30 3030 3239 |.|o1 = .00029 │ │ │ │ -001cb6f0: 3134 3833 3132 3930 3134 3834 3831 2020 14831290148481 │ │ │ │ +001cb6e0: 2020 7c0a 7c6f 3120 3d20 2e30 3030 3335 |.|o1 = .00035 │ │ │ │ +001cb6f0: 3433 3033 3731 3839 3231 3436 3136 2020 43037189214616 │ │ │ │ 001cb700: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 001cb710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001cb720: 2020 2020 7c0a 7c6f 3120 3a20 5252 2028 |.|o1 : RR ( │ │ │ │ 001cb730: 6f66 2070 7265 6369 7369 6f6e 2035 3329 of precision 53) │ │ │ │ 001cb740: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 001cb750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001cb760: 2d2d 2d2d 2d2d 2b0a 5468 6520 736e 6970 ------+.The snip │ │ │ │ @@ -123242,40 +123242,40 @@ │ │ │ │ 001e1690: 7469 5461 6c6c 7929 2020 2020 2020 2020 tiTally) │ │ │ │ 001e16a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e16b0: 2020 2020 2020 7d7c 0a7c 2020 2020 207b }|.| { │ │ │ │ 001e16c0: 3138 203d 3e20 286d 6174 684d 4c2c 2042 18 => (mathML, B │ │ │ │ 001e16d0: 6574 7469 5461 6c6c 7929 2020 2020 2020 ettiTally) │ │ │ │ 001e16e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e16f0: 2020 2020 2020 2020 2020 2020 7d7c 0a7c }|.| │ │ │ │ -001e1700: 2020 2020 207b 3139 203d 3e20 2874 7275 {19 => (tru │ │ │ │ -001e1710: 6e63 6174 652c 2042 6574 7469 5461 6c6c ncate, BettiTall │ │ │ │ -001e1720: 792c 205a 5a2c 205a 5a29 2020 2020 2020 y, ZZ, ZZ) │ │ │ │ +001e1700: 2020 2020 207b 3139 203d 3e20 2863 6f64 {19 => (cod │ │ │ │ +001e1710: 696d 2c20 4265 7474 6954 616c 6c79 2920 im, BettiTally) │ │ │ │ +001e1720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e1730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e1740: 2020 7d7c 0a7c 2020 2020 207b 3230 203d }|.| {20 = │ │ │ │ 001e1750: 3e20 2874 7275 6e63 6174 652c 2042 6574 > (truncate, Bet │ │ │ │ -001e1760: 7469 5461 6c6c 792c 205a 5a2c 2049 6e66 tiTally, ZZ, Inf │ │ │ │ -001e1770: 696e 6974 654e 756d 6265 7229 2020 2020 initeNumber) │ │ │ │ -001e1780: 2020 2020 2020 2020 7d7c 0a7c 2020 2020 }|.| │ │ │ │ -001e1790: 207b 3231 203d 3e20 2874 7275 6e63 6174 {21 => (truncat │ │ │ │ -001e17a0: 652c 2042 6574 7469 5461 6c6c 792c 2049 e, BettiTally, I │ │ │ │ -001e17b0: 6e66 696e 6974 654e 756d 6265 722c 205a nfiniteNumber, Z │ │ │ │ -001e17c0: 5a29 2020 2020 2020 2020 2020 2020 7d7c Z) }| │ │ │ │ -001e17d0: 0a7c 2020 2020 207b 3232 203d 3e20 2863 .| {22 => (c │ │ │ │ -001e17e0: 6f64 696d 2c20 4265 7474 6954 616c 6c79 odim, BettiTally │ │ │ │ -001e17f0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +001e1760: 7469 5461 6c6c 792c 2049 6e66 696e 6974 tiTally, Infinit │ │ │ │ +001e1770: 654e 756d 6265 722c 2049 6e66 696e 6974 eNumber, Infinit │ │ │ │ +001e1780: 654e 756d 6265 7229 7d7c 0a7c 2020 2020 eNumber)}|.| │ │ │ │ +001e1790: 207b 3231 203d 3e20 2864 7561 6c2c 2042 {21 => (dual, B │ │ │ │ +001e17a0: 6574 7469 5461 6c6c 7929 2020 2020 2020 ettiTally) │ │ │ │ +001e17b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +001e17c0: 2020 2020 2020 2020 2020 2020 2020 7d7c }| │ │ │ │ +001e17d0: 0a7c 2020 2020 207b 3232 203d 3e20 2874 .| {22 => (t │ │ │ │ +001e17e0: 7275 6e63 6174 652c 2042 6574 7469 5461 runcate, BettiTa │ │ │ │ +001e17f0: 6c6c 792c 205a 5a2c 205a 5a29 2020 2020 lly, ZZ, ZZ) │ │ │ │ 001e1800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e1810: 2020 2020 7d7c 0a7c 2020 2020 207b 3233 }|.| {23 │ │ │ │ 001e1820: 203d 3e20 2874 7275 6e63 6174 652c 2042 => (truncate, B │ │ │ │ -001e1830: 6574 7469 5461 6c6c 792c 2049 6e66 696e ettiTally, Infin │ │ │ │ -001e1840: 6974 654e 756d 6265 722c 2049 6e66 696e iteNumber, Infin │ │ │ │ -001e1850: 6974 654e 756d 6265 7229 7d7c 0a7c 2020 iteNumber)}|.| │ │ │ │ -001e1860: 2020 207b 3234 203d 3e20 2864 7561 6c2c {24 => (dual, │ │ │ │ -001e1870: 2042 6574 7469 5461 6c6c 7929 2020 2020 BettiTally) │ │ │ │ -001e1880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -001e1890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +001e1830: 6574 7469 5461 6c6c 792c 205a 5a2c 2049 ettiTally, ZZ, I │ │ │ │ +001e1840: 6e66 696e 6974 654e 756d 6265 7229 2020 nfiniteNumber) │ │ │ │ +001e1850: 2020 2020 2020 2020 2020 7d7c 0a7c 2020 }|.| │ │ │ │ +001e1860: 2020 207b 3234 203d 3e20 2874 7275 6e63 {24 => (trunc │ │ │ │ +001e1870: 6174 652c 2042 6574 7469 5461 6c6c 792c ate, BettiTally, │ │ │ │ +001e1880: 2049 6e66 696e 6974 654e 756d 6265 722c InfiniteNumber, │ │ │ │ +001e1890: 205a 5a29 2020 2020 2020 2020 2020 2020 ZZ) │ │ │ │ 001e18a0: 7d7c 0a7c 2020 2020 207b 3235 203d 3e20 }|.| {25 => │ │ │ │ 001e18b0: 285e 2c20 5269 6e67 2c20 4265 7474 6954 (^, Ring, BettiT │ │ │ │ 001e18c0: 616c 6c79 2920 2020 2020 2020 2020 2020 ally) │ │ │ │ 001e18d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e18e0: 2020 2020 2020 7d7c 0a7c 2020 2020 2020 }|.| │ │ │ │ 001e18f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e1900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -123454,37 +123454,37 @@ │ │ │ │ 001e23d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e23e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 001e23f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e2400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e2410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e2420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e2430: 2020 7c0a 7c6f 3520 3d20 7b30 203d 3e20 |.|o5 = {0 => │ │ │ │ -001e2440: 282b 2c20 4d61 7472 6978 2c20 4d61 7472 (+, Matrix, Matr │ │ │ │ -001e2450: 6978 2920 2020 2020 2020 2020 2020 2020 ix) │ │ │ │ +001e2440: 2864 6966 662c 204d 6174 7269 782c 204d (diff, Matrix, M │ │ │ │ +001e2450: 6174 7269 7829 2020 2020 2020 2020 2020 atrix) │ │ │ │ 001e2460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e2470: 2020 2020 2020 7d7c 0a7c 2020 2020 207b }|.| { │ │ │ │ -001e2480: 3120 3d3e 2028 2d2c 204d 6174 7269 782c 1 => (-, Matrix, │ │ │ │ -001e2490: 204d 6174 7269 7829 2020 2020 2020 2020 Matrix) │ │ │ │ +001e2480: 3120 3d3e 2028 636f 6e74 7261 6374 2c20 1 => (contract, │ │ │ │ +001e2490: 4d61 7472 6978 2c20 4d61 7472 6978 2920 Matrix, Matrix) │ │ │ │ 001e24a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e24b0: 2020 2020 2020 2020 2020 207d 7c0a 7c20 }|.| │ │ │ │ -001e24c0: 2020 2020 7b32 203d 3e20 2863 6f6e 7472 {2 => (contr │ │ │ │ -001e24d0: 6163 742c 204d 6174 7269 782c 204d 6174 act, Matrix, Mat │ │ │ │ -001e24e0: 7269 7829 2020 2020 2020 2020 2020 2020 rix) │ │ │ │ +001e24c0: 2020 2020 7b32 203d 3e20 282b 2c20 4d61 {2 => (+, Ma │ │ │ │ +001e24d0: 7472 6978 2c20 4d61 7472 6978 2920 2020 trix, Matrix) │ │ │ │ +001e24e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e24f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e2500: 7d7c 0a7c 2020 2020 207b 3320 3d3e 2028 }|.| {3 => ( │ │ │ │ -001e2510: 6469 6666 272c 204d 6174 7269 782c 204d diff', Matrix, M │ │ │ │ -001e2520: 6174 7269 7829 2020 2020 2020 2020 2020 atrix) │ │ │ │ +001e2510: 2d2c 204d 6174 7269 782c 204d 6174 7269 -, Matrix, Matri │ │ │ │ +001e2520: 7829 2020 2020 2020 2020 2020 2020 2020 x) │ │ │ │ 001e2530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e2540: 2020 2020 207d 7c0a 7c20 2020 2020 7b34 }|.| {4 │ │ │ │ 001e2550: 203d 3e20 2863 6f6e 7472 6163 7427 2c20 => (contract', │ │ │ │ 001e2560: 4d61 7472 6978 2c20 4d61 7472 6978 2920 Matrix, Matrix) │ │ │ │ 001e2570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e2580: 2020 2020 2020 2020 2020 7d7c 0a7c 2020 }|.| │ │ │ │ -001e2590: 2020 207b 3520 3d3e 2028 6469 6666 2c20 {5 => (diff, │ │ │ │ -001e25a0: 4d61 7472 6978 2c20 4d61 7472 6978 2920 Matrix, Matrix) │ │ │ │ +001e2590: 2020 207b 3520 3d3e 2028 6469 6666 272c {5 => (diff', │ │ │ │ +001e25a0: 204d 6174 7269 782c 204d 6174 7269 7829 Matrix, Matrix) │ │ │ │ 001e25b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e25c0: 2020 2020 2020 2020 2020 2020 2020 207d } │ │ │ │ 001e25d0: 7c0a 7c20 2020 2020 7b36 203d 3e20 286d |.| {6 => (m │ │ │ │ 001e25e0: 6172 6b65 6447 422c 204d 6174 7269 782c arkedGB, Matrix, │ │ │ │ 001e25f0: 204d 6174 7269 7829 2020 2020 2020 2020 Matrix) │ │ │ │ 001e2600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e2610: 2020 2020 7d7c 0a7c 2020 2020 207b 3720 }|.| {7 │ │ │ │ @@ -123583,22 +123583,22 @@ │ │ │ │ 001e2be0: 7472 6978 2920 2020 2020 2020 2020 2020 trix) │ │ │ │ 001e2bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e2c00: 2020 7d7c 0a7c 2020 2020 207b 3239 203d }|.| {29 = │ │ │ │ 001e2c10: 3e20 2869 6e74 6572 7365 6374 2c20 4d61 > (intersect, Ma │ │ │ │ 001e2c20: 7472 6978 2c20 4d61 7472 6978 2920 2020 trix, Matrix) │ │ │ │ 001e2c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e2c40: 2020 2020 2020 207d 7c0a 7c20 2020 2020 }|.| │ │ │ │ -001e2c50: 7b33 3020 3d3e 2028 696e 7465 7273 6563 {30 => (intersec │ │ │ │ -001e2c60: 742c 204d 6174 7269 782c 204d 6174 7269 t, Matrix, Matri │ │ │ │ -001e2c70: 782c 204d 6174 7269 782c 204d 6174 7269 x, Matrix, Matri │ │ │ │ -001e2c80: 7829 2020 2020 2020 2020 2020 7d7c 0a7c x) }|.| │ │ │ │ -001e2c90: 2020 2020 207b 3331 203d 3e20 2870 756c {31 => (pul │ │ │ │ -001e2ca0: 6c62 6163 6b2c 204d 6174 7269 782c 204d lback, Matrix, M │ │ │ │ -001e2cb0: 6174 7269 7829 2020 2020 2020 2020 2020 atrix) │ │ │ │ -001e2cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +001e2c50: 7b33 3020 3d3e 2028 7075 6c6c 6261 636b {30 => (pullback │ │ │ │ +001e2c60: 2c20 4d61 7472 6978 2c20 4d61 7472 6978 , Matrix, Matrix │ │ │ │ +001e2c70: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +001e2c80: 2020 2020 2020 2020 2020 2020 7d7c 0a7c }|.| │ │ │ │ +001e2c90: 2020 2020 207b 3331 203d 3e20 2869 6e74 {31 => (int │ │ │ │ +001e2ca0: 6572 7365 6374 2c20 4d61 7472 6978 2c20 ersect, Matrix, │ │ │ │ +001e2cb0: 4d61 7472 6978 2c20 4d61 7472 6978 2c20 Matrix, Matrix, │ │ │ │ +001e2cc0: 4d61 7472 6978 2920 2020 2020 2020 2020 Matrix) │ │ │ │ 001e2cd0: 207d 7c0a 7c20 2020 2020 7b33 3220 3d3e }|.| {32 => │ │ │ │ 001e2ce0: 2028 7375 6273 7469 7475 7465 2c20 4d61 (substitute, Ma │ │ │ │ 001e2cf0: 7472 6978 2c20 4d61 7472 6978 2920 2020 trix, Matrix) │ │ │ │ 001e2d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e2d10: 2020 2020 2020 7d7c 0a7c 2020 2020 207b }|.| { │ │ │ │ 001e2d20: 3333 203d 3e20 2879 6f6e 6564 6150 726f 33 => (yonedaPro │ │ │ │ 001e2d30: 6475 6374 2c20 4d61 7472 6978 2c20 4d61 duct, Matrix, Ma │ │ │ │ @@ -124506,84 +124506,84 @@ │ │ │ │ 001e6590: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 001e65a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e65b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e65c0: 2020 2020 2020 7c0a 7c6f 3220 3d20 2372 |.|o2 = #r │ │ │ │ 001e65d0: 756e 2020 2574 696d 6520 2020 706f 7369 un %time posi │ │ │ │ 001e65e0: 7469 6f6e 2020 2020 2020 2020 2020 2020 tion │ │ │ │ 001e65f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -001e6600: 2020 2020 2031 2020 2020 2039 342e 3431 1 94.41 │ │ │ │ +001e6600: 2020 2020 2031 2020 2020 2039 332e 3238 1 93.28 │ │ │ │ 001e6610: 2020 202e 2e2f 2e2e 2f6d 322f 6d61 7472 ../../m2/matr │ │ │ │ 001e6620: 6978 312e 6d32 3a32 3733 3a34 2d32 3736 ix1.m2:273:4-276 │ │ │ │ 001e6630: 3a35 3820 7c0a 7c20 2020 2020 3120 2020 :58 |.| 1 │ │ │ │ -001e6640: 2020 3931 2e38 3220 2020 2e2e 2f2e 2e2f 91.82 ../../ │ │ │ │ +001e6640: 2020 3930 2e33 2020 2020 2e2e 2f2e 2e2f 90.3 ../../ │ │ │ │ 001e6650: 6d32 2f6d 6174 7269 7831 2e6d 323a 3237 m2/matrix1.m2:27 │ │ │ │ 001e6660: 353a 3232 2d32 3735 3a34 337c 0a7c 2020 5:22-275:43|.| │ │ │ │ -001e6670: 2020 2031 2020 2020 2034 342e 3737 2020 1 44.77 │ │ │ │ +001e6670: 2020 2031 2020 2020 2034 332e 3731 2020 1 43.71 │ │ │ │ 001e6680: 202e 2e2f 2e2e 2f6d 322f 6d61 7472 6978 ../../m2/matrix │ │ │ │ 001e6690: 312e 6d32 3a31 3833 3a32 352d 3138 333a 1.m2:183:25-183: │ │ │ │ 001e66a0: 3532 7c0a 7c20 2020 2020 3120 2020 2020 52|.| 1 │ │ │ │ -001e66b0: 3330 2e39 3920 2020 2e2e 2f2e 2e2f 6d32 30.99 ../../m2 │ │ │ │ +001e66b0: 3330 2e36 3820 2020 2e2e 2f2e 2e2f 6d32 30.68 ../../m2 │ │ │ │ 001e66c0: 2f6d 6174 7269 7831 2e6d 323a 3130 343a /matrix1.m2:104: │ │ │ │ 001e66d0: 352d 3134 363a 3732 207c 0a7c 2020 2020 5-146:72 |.| │ │ │ │ -001e66e0: 2031 2020 2020 2032 392e 3831 2020 202e 1 29.81 . │ │ │ │ +001e66e0: 2031 2020 2020 2032 392e 3533 2020 202e 1 29.53 . │ │ │ │ 001e66f0: 2e2f 2e2e 2f6d 322f 6d61 7472 6978 312e ./../m2/matrix1. │ │ │ │ 001e6700: 6d32 3a31 3330 3a31 302d 3134 353a 3136 m2:130:10-145:16 │ │ │ │ 001e6710: 7c0a 7c20 2020 2020 3120 2020 2020 3232 |.| 1 22 │ │ │ │ -001e6720: 2e33 3820 2020 2e2e 2f2e 2e2f 6d32 2f6d .38 ../../m2/m │ │ │ │ +001e6720: 2e39 3620 2020 2e2e 2f2e 2e2f 6d32 2f6d .96 ../../m2/m │ │ │ │ 001e6730: 6174 7269 7831 2e6d 323a 3137 313a 342d atrix1.m2:171:4- │ │ │ │ 001e6740: 3137 313a 3432 207c 0a7c 2020 2020 2031 171:42 |.| 1 │ │ │ │ -001e6750: 2020 2020 2032 312e 3235 2020 202e 2e2f 21.25 ../ │ │ │ │ -001e6760: 2e2e 2f6d 322f 6d61 7472 6978 312e 6d32 ../m2/matrix1.m2 │ │ │ │ -001e6770: 3a33 353a 3130 2d33 393a 3232 2020 7c0a :35:10-39:22 |. │ │ │ │ -001e6780: 7c20 2020 2020 3120 2020 2020 3231 2e30 | 1 21.0 │ │ │ │ -001e6790: 3920 2020 2e2e 2f2e 2e2f 6d32 2f73 6574 9 ../../m2/set │ │ │ │ -001e67a0: 2e6d 323a 3132 393a 352d 3132 393a 3631 .m2:129:5-129:61 │ │ │ │ -001e67b0: 2020 2020 207c 0a7c 2020 2020 2031 2020 |.| 1 │ │ │ │ -001e67c0: 2020 2033 2e33 3120 2020 202e 2e2f 2e2e 3.31 ../.. │ │ │ │ +001e6750: 2020 2020 2032 312e 3432 2020 202e 2e2f 21.42 ../ │ │ │ │ +001e6760: 2e2e 2f6d 322f 7365 742e 6d32 3a31 3239 ../m2/set.m2:129 │ │ │ │ +001e6770: 3a35 2d31 3239 3a36 3120 2020 2020 7c0a :5-129:61 |. │ │ │ │ +001e6780: 7c20 2020 2020 3120 2020 2020 3231 2e31 | 1 21.1 │ │ │ │ +001e6790: 3620 2020 2e2e 2f2e 2e2f 6d32 2f6d 6174 6 ../../m2/mat │ │ │ │ +001e67a0: 7269 7831 2e6d 323a 3335 3a31 302d 3339 rix1.m2:35:10-39 │ │ │ │ +001e67b0: 3a32 3220 207c 0a7c 2020 2020 2031 2020 :22 |.| 1 │ │ │ │ +001e67c0: 2020 2033 2e31 3420 2020 202e 2e2f 2e2e 3.14 ../.. │ │ │ │ 001e67d0: 2f6d 322f 6d61 7472 6978 312e 6d32 3a31 /m2/matrix1.m2:1 │ │ │ │ 001e67e0: 3032 3a35 2d31 3032 3a32 3920 7c0a 7c20 02:5-102:29 |.| │ │ │ │ -001e67f0: 2020 2020 3120 2020 2020 322e 3237 2020 1 2.27 │ │ │ │ +001e67f0: 2020 2020 3120 2020 2020 322e 3430 2020 1 2.40 │ │ │ │ 001e6800: 2020 2e2e 2f2e 2e2f 6d32 2f6d 6174 7269 ../../m2/matri │ │ │ │ 001e6810: 7831 2e6d 323a 3133 313a 3133 2d31 3331 x1.m2:131:13-131 │ │ │ │ 001e6820: 3a37 387c 0a7c 2020 2020 2031 2020 2020 :78|.| 1 │ │ │ │ -001e6830: 2032 2e31 3220 2020 202e 2e2f 2e2e 2f6d 2.12 ../../m │ │ │ │ +001e6830: 2031 2e39 3120 2020 202e 2e2f 2e2e 2f6d 1.91 ../../m │ │ │ │ 001e6840: 322f 6d61 7472 6978 312e 6d32 3a38 363a 2/matrix1.m2:86: │ │ │ │ 001e6850: 352d 3939 3a31 3120 2020 7c0a 7c20 2020 5-99:11 |.| │ │ │ │ -001e6860: 2020 3120 2020 2020 312e 3433 2020 2020 1 1.43 │ │ │ │ +001e6860: 2020 3120 2020 2020 312e 3439 2020 2020 1 1.49 │ │ │ │ 001e6870: 2e2e 2f2e 2e2f 6d32 2f6d 6174 7269 7831 ../../m2/matrix1 │ │ │ │ 001e6880: 2e6d 323a 3237 353a 372d 3237 353a 3136 .m2:275:7-275:16 │ │ │ │ 001e6890: 207c 0a7c 2020 2020 2031 2020 2020 2031 |.| 1 1 │ │ │ │ -001e68a0: 2e33 3520 2020 202e 2e2f 2e2e 2f6d 322f .35 ../../m2/ │ │ │ │ +001e68a0: 2e33 3220 2020 202e 2e2f 2e2e 2f6d 322f .32 ../../m2/ │ │ │ │ 001e68b0: 6d61 7472 6978 312e 6d32 3a31 3337 3a32 matrix1.m2:137:2 │ │ │ │ 001e68c0: 302d 3133 373a 3634 7c0a 7c20 2020 2020 0-137:64|.| │ │ │ │ -001e68d0: 3120 2020 2020 312e 3220 2020 2020 2e2e 1 1.2 .. │ │ │ │ +001e68d0: 3120 2020 2020 312e 3331 2020 2020 2e2e 1 1.31 .. │ │ │ │ 001e68e0: 2f2e 2e2f 6d32 2f6d 6174 7269 7831 2e6d /../m2/matrix1.m │ │ │ │ 001e68f0: 323a 3237 303a 342d 3237 313a 3733 207c 2:270:4-271:73 | │ │ │ │ -001e6900: 0a7c 2020 2020 2031 2020 2020 2031 2e31 .| 1 1.1 │ │ │ │ -001e6910: 3420 2020 202e 2e2f 2e2e 2f6d 322f 6d61 4 ../../m2/ma │ │ │ │ -001e6920: 7472 6978 312e 6d32 3a31 3031 3a35 2d31 trix1.m2:101:5-1 │ │ │ │ -001e6930: 3031 3a39 3120 7c0a 7c20 2020 2020 3120 01:91 |.| 1 │ │ │ │ -001e6940: 2020 2020 312e 3130 2020 2020 2e2e 2f2e 1.10 ../. │ │ │ │ +001e6900: 0a7c 2020 2020 2031 2020 2020 2031 2e30 .| 1 1.0 │ │ │ │ +001e6910: 3820 2020 202e 2e2f 2e2e 2f6d 322f 6d61 8 ../../m2/ma │ │ │ │ +001e6920: 7472 6978 312e 6d32 3a31 3732 3a34 2d31 trix1.m2:172:4-1 │ │ │ │ +001e6930: 3734 3a37 3420 7c0a 7c20 2020 2020 3120 74:74 |.| 1 │ │ │ │ +001e6940: 2020 2020 312e 3035 2020 2020 2e2e 2f2e 1.05 ../. │ │ │ │ 001e6950: 2e2f 6d32 2f6d 6174 7269 7831 2e6d 323a ./m2/matrix1.m2: │ │ │ │ -001e6960: 3838 3a31 302d 3838 3a34 3620 207c 0a7c 88:10-88:46 |.| │ │ │ │ -001e6970: 2020 2020 2031 2020 2020 2031 2e30 3320 1 1.03 │ │ │ │ +001e6960: 3130 313a 352d 3130 313a 3931 207c 0a7c 101:5-101:91 |.| │ │ │ │ +001e6970: 2020 2020 2031 2020 2020 202e 3938 2020 1 .98 │ │ │ │ 001e6980: 2020 202e 2e2f 2e2e 2f6d 322f 6d61 7472 ../../m2/matr │ │ │ │ -001e6990: 6978 312e 6d32 3a31 3732 3a34 2d31 3734 ix1.m2:172:4-174 │ │ │ │ -001e69a0: 3a37 3420 7c0a 7c20 2020 2020 3120 2020 :74 |.| 1 │ │ │ │ -001e69b0: 2020 2e37 3720 2020 2020 2e2e 2f2e 2e2f .77 ../../ │ │ │ │ -001e69c0: 6d32 2f6d 6f64 756c 6573 2e6d 323a 3238 m2/modules.m2:28 │ │ │ │ -001e69d0: 323a 342d 3238 323a 3532 207c 0a7c 2020 2:4-282:52 |.| │ │ │ │ -001e69e0: 2020 2032 3020 2020 202e 3632 2020 2020 20 .62 │ │ │ │ -001e69f0: 202e 2e2f 2e2e 2f6d 322f 6d61 7472 6978 ../../m2/matrix │ │ │ │ -001e6a00: 312e 6d32 3a31 3831 3a31 342d 3138 323a 1.m2:181:14-182: │ │ │ │ -001e6a10: 3637 7c0a 7c20 2020 2020 3230 2020 2020 67|.| 20 │ │ │ │ -001e6a20: 2e35 3320 2020 2020 2e2e 2f2e 2e2f 6d32 .53 ../../m2 │ │ │ │ +001e6990: 6978 312e 6d32 3a38 383a 3130 2d38 383a ix1.m2:88:10-88: │ │ │ │ +001e69a0: 3436 2020 7c0a 7c20 2020 2020 3230 2020 46 |.| 20 │ │ │ │ +001e69b0: 2020 2e39 3520 2020 2020 2e2e 2f2e 2e2f .95 ../../ │ │ │ │ +001e69c0: 6d32 2f6d 6174 7269 7831 2e6d 323a 3138 m2/matrix1.m2:18 │ │ │ │ +001e69d0: 313a 3134 2d31 3832 3a36 377c 0a7c 2020 1:14-182:67|.| │ │ │ │ +001e69e0: 2020 2031 3920 2020 202e 3731 2020 2020 19 .71 │ │ │ │ +001e69f0: 202e 2e2f 2e2e 2f6d 322f 7365 742e 6d32 ../../m2/set.m2 │ │ │ │ +001e6a00: 3a31 3239 3a33 362d 3132 393a 3431 2020 :129:36-129:41 │ │ │ │ +001e6a10: 2020 7c0a 7c20 2020 2020 3230 2020 2020 |.| 20 │ │ │ │ +001e6a20: 2e36 3820 2020 2020 2e2e 2f2e 2e2f 6d32 .68 ../../m2 │ │ │ │ 001e6a30: 2f6d 6174 7269 7831 2e6d 323a 3337 3a34 /matrix1.m2:37:4 │ │ │ │ 001e6a40: 332d 3337 3a37 3120 207c 0a7c 2020 2020 3-37:71 |.| │ │ │ │ -001e6a50: 2031 2020 2020 202e 3030 3436 7320 2065 1 .0046s e │ │ │ │ +001e6a50: 2031 2020 2020 202e 3030 3339 7320 2065 1 .0039s e │ │ │ │ 001e6a60: 6c61 7073 6564 2074 6f74 616c 2020 2020 lapsed total │ │ │ │ 001e6a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e6a80: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 001e6a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001e6aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001e6ab0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2063 -------+.|i3 : c │ │ │ │ 001e6ac0: 6f76 6572 6167 6553 756d 6d61 7279 2020 overageSummary │ │ │ │ @@ -135041,15 +135041,15 @@ │ │ │ │ 0020f800: 7265 6c6f 6164 696e 6720 4669 7273 7450 reloading FirstP │ │ │ │ 0020f810: 6163 6b61 6765 3b20 7265 6372 6561 7465 ackage; recreate │ │ │ │ 0020f820: 2069 6e73 7461 6e63 6573 206f 6620 7479 instances of ty │ │ │ │ 0020f830: 7065 7320 6672 6f6d 2074 6869 7320 2020 pes from this │ │ │ │ 0020f840: 7c0a 7c20 2d2d 2063 6170 7475 7269 6e67 |.| -- capturing │ │ │ │ 0020f850: 2063 6865 636b 2831 2c20 2246 6972 7374 check(1, "First │ │ │ │ 0020f860: 5061 636b 6167 6522 2920 2020 2020 2020 Package") │ │ │ │ -0020f870: 202d 2d20 2e31 3733 3639 3873 2065 6c61 -- .173698s ela │ │ │ │ +0020f870: 202d 2d20 2e32 3139 3038 3473 2065 6c61 -- .219084s ela │ │ │ │ 0020f880: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 0020f890: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------- │ │ │ │ 0020f8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0020f8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0020f8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0020f8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0020f8e0: 7c0a 7c70 6163 6b61 6765 2020 2020 2020 |.|package │ │ │ │ @@ -135066,21 +135066,21 @@ │ │ │ │ 0020f990: 7273 7450 6163 6b61 6765 2020 2020 2020 rstPackage │ │ │ │ 0020f9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0020f9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0020f9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0020f9d0: 7c0a 7c20 2d2d 2063 6170 7475 7269 6e67 |.| -- capturing │ │ │ │ 0020f9e0: 2063 6865 636b 2830 2c20 2246 6972 7374 check(0, "First │ │ │ │ 0020f9f0: 5061 636b 6167 6522 2920 2020 2020 2020 Package") │ │ │ │ -0020fa00: 202d 2d20 2e31 3830 3038 3873 2065 6c61 -- .180088s ela │ │ │ │ +0020fa00: 202d 2d20 2e31 3632 3834 3273 2065 6c61 -- .162842s ela │ │ │ │ 0020fa10: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 0020fa20: 7c0a 7c20 2d2d 2063 6170 7475 7269 6e67 |.| -- capturing │ │ │ │ 0020fa30: 2063 6865 636b 2831 2c20 2246 6972 7374 check(1, "First │ │ │ │ 0020fa40: 5061 636b 6167 6522 2920 2020 2020 2020 Package") │ │ │ │ -0020fa50: 202d 2d20 2e31 3739 3537 3973 2065 6c61 -- .179579s ela │ │ │ │ -0020fa60: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ +0020fa50: 202d 2d20 2e31 3633 3334 7320 656c 6170 -- .16334s elap │ │ │ │ +0020fa60: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 0020fa70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0020fa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0020fa90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0020faa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0020fab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0020fac0: 2b0a 0a41 6c74 6572 6e61 7469 7665 6c79 +..Alternatively │ │ │ │ 0020fad0: 2c20 6966 2074 6865 2070 6163 6b61 6765 , if the package │ │ │ │ @@ -135095,32 +135095,32 @@ │ │ │ │ 0020fb60: 2d2b 0a7c 6934 203a 2063 6865 636b 5f31 -+.|i4 : check_1 │ │ │ │ 0020fb70: 2022 4669 7273 7450 6163 6b61 6765 2220 "FirstPackage" │ │ │ │ 0020fb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0020fb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0020fba0: 2020 2020 207c 0a7c 202d 2d20 6361 7074 |.| -- capt │ │ │ │ 0020fbb0: 7572 696e 6720 6368 6563 6b28 312c 2022 uring check(1, " │ │ │ │ 0020fbc0: 4669 7273 7450 6163 6b61 6765 2229 2020 FirstPackage") │ │ │ │ -0020fbd0: 2020 2020 2020 2d2d 202e 3137 3838 3732 -- .178872 │ │ │ │ -0020fbe0: 7320 656c 6170 7365 647c 0a2b 2d2d 2d2d s elapsed|.+---- │ │ │ │ +0020fbd0: 2020 2020 2020 2d2d 202e 3136 3538 3273 -- .16582s │ │ │ │ +0020fbe0: 2065 6c61 7073 6564 207c 0a2b 2d2d 2d2d elapsed |.+---- │ │ │ │ 0020fbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0020fc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0020fc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0020fc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 0020fc30: 6935 203a 2063 6865 636b 2022 4669 7273 i5 : check "Firs │ │ │ │ 0020fc40: 7450 6163 6b61 6765 2220 2020 2020 2020 tPackage" │ │ │ │ 0020fc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0020fc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0020fc70: 207c 0a7c 202d 2d20 6361 7074 7572 696e |.| -- capturin │ │ │ │ 0020fc80: 6720 6368 6563 6b28 302c 2022 4669 7273 g check(0, "Firs │ │ │ │ 0020fc90: 7450 6163 6b61 6765 2229 2020 2020 2020 tPackage") │ │ │ │ -0020fca0: 2020 2d2d 202e 3137 3839 3438 7320 656c -- .178948s el │ │ │ │ +0020fca0: 2020 2d2d 202e 3136 3235 3835 7320 656c -- .162585s el │ │ │ │ 0020fcb0: 6170 7365 647c 0a7c 202d 2d20 6361 7074 apsed|.| -- capt │ │ │ │ 0020fcc0: 7572 696e 6720 6368 6563 6b28 312c 2022 uring check(1, " │ │ │ │ 0020fcd0: 4669 7273 7450 6163 6b61 6765 2229 2020 FirstPackage") │ │ │ │ -0020fce0: 2020 2020 2020 2d2d 202e 3138 3131 3636 -- .181166 │ │ │ │ +0020fce0: 2020 2020 2020 2d2d 202e 3136 3136 3632 -- .161662 │ │ │ │ 0020fcf0: 7320 656c 6170 7365 647c 0a2b 2d2d 2d2d s elapsed|.+---- │ │ │ │ 0020fd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0020fd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0020fd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0020fd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ 0020fd40: 4120 2a6e 6f74 6520 5465 7374 496e 7075 A *note TestInpu │ │ │ │ 0020fd50: 743a 2074 6573 7473 2c20 6f62 6a65 6374 t: tests, object │ │ │ │ @@ -135162,16 +135162,16 @@ │ │ │ │ 0020ff90: 7c69 3720 3a20 6368 6563 6b20 6f6f 2020 |i7 : check oo │ │ │ │ 0020ffa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0020ffb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0020ffc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0020ffd0: 2020 2020 2020 7c0a 7c20 2d2d 2063 6170 |.| -- cap │ │ │ │ 0020ffe0: 7475 7269 6e67 2063 6865 636b 2831 2c20 turing check(1, │ │ │ │ 0020fff0: 2246 6972 7374 5061 636b 6167 6522 2920 "FirstPackage") │ │ │ │ -00210000: 2020 2020 2020 202d 2d20 2e31 3739 3737 -- .17977 │ │ │ │ -00210010: 3873 2065 6c61 7073 6564 2020 2020 7c0a 8s elapsed |. │ │ │ │ +00210000: 2020 2020 2020 202d 2d20 2e31 3633 3730 -- .16370 │ │ │ │ +00210010: 3473 2065 6c61 7073 6564 2020 2020 7c0a 4s elapsed |. │ │ │ │ 00210020: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00210030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00210040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00210050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00210060: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 7465 ------+.|i8 : te │ │ │ │ 00210070: 7374 7320 2246 6972 7374 5061 636b 6167 sts "FirstPackag │ │ │ │ 00210080: 6522 2020 2020 2020 2020 2020 2020 2020 e" │ │ │ │ @@ -135207,20 +135207,20 @@ │ │ │ │ 00210260: 7c69 3920 3a20 6368 6563 6b20 6f6f 2020 |i9 : check oo │ │ │ │ 00210270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00210280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00210290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002102a0: 2020 2020 2020 7c0a 7c20 2d2d 2063 6170 |.| -- cap │ │ │ │ 002102b0: 7475 7269 6e67 2063 6865 636b 2830 2c20 turing check(0, │ │ │ │ 002102c0: 2246 6972 7374 5061 636b 6167 6522 2920 "FirstPackage") │ │ │ │ -002102d0: 2020 2020 2020 202d 2d20 2e31 3833 3735 -- .18375 │ │ │ │ -002102e0: 3773 2065 6c61 7073 6564 2020 2020 7c0a 7s elapsed |. │ │ │ │ +002102d0: 2020 2020 2020 202d 2d20 2e31 3631 3438 -- .16148 │ │ │ │ +002102e0: 3873 2065 6c61 7073 6564 2020 2020 7c0a 8s elapsed |. │ │ │ │ 002102f0: 7c20 2d2d 2063 6170 7475 7269 6e67 2063 | -- capturing c │ │ │ │ 00210300: 6865 636b 2831 2c20 2246 6972 7374 5061 heck(1, "FirstPa │ │ │ │ 00210310: 636b 6167 6522 2920 2020 2020 2020 202d ckage") - │ │ │ │ -00210320: 2d20 2e31 3831 3538 3373 2065 6c61 7073 - .181583s elaps │ │ │ │ +00210320: 2d20 2e31 3634 3830 3673 2065 6c61 7073 - .164806s elaps │ │ │ │ 00210330: 6564 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d ed |.+------- │ │ │ │ 00210340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00210350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00210360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00210370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00210380: 0a49 6620 6f6e 6c79 2061 6e20 696e 7465 .If only an inte │ │ │ │ 00210390: 6765 7220 6973 2070 6173 7365 6420 6173 ger is passed as │ │ │ │ @@ -135270,15 +135270,15 @@ │ │ │ │ 00210650: 6563 6b20 3120 2020 2020 2020 2020 2020 eck 1 │ │ │ │ 00210660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00210670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00210680: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00210690: 7c20 2d2d 2063 6170 7475 7269 6e67 2063 | -- capturing c │ │ │ │ 002106a0: 6865 636b 2831 2c20 2246 6972 7374 5061 heck(1, "FirstPa │ │ │ │ 002106b0: 636b 6167 6522 2920 2020 2020 2020 202d ckage") - │ │ │ │ -002106c0: 2d20 2e32 3838 3331 3273 2065 6c61 7073 - .288312s elaps │ │ │ │ +002106c0: 2d20 2e31 3537 3433 3773 2065 6c61 7073 - .157437s elaps │ │ │ │ 002106d0: 6564 2020 2020 207c 0a2b 2d2d 2d2d 2d2d ed |.+------ │ │ │ │ 002106e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002106f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00210700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00210710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00210720: 2b0a 0a43 6176 6561 740a 3d3d 3d3d 3d3d +..Caveat.====== │ │ │ │ 00210730: 0a0a 4375 7272 656e 746c 792c 2069 6620 ..Currently, if │ │ │ │ @@ -139210,15 +139210,15 @@ │ │ │ │ 0021fc90: 2020 2020 2020 2020 2072 6177 2064 6f63 raw doc │ │ │ │ 0021fca0: 756d 656e 7461 7469 6f6e 203d 3e20 4d75 umentation => Mu │ │ │ │ 0021fcb0: 7461 626c 6548 6173 6854 6162 6c65 7b7d tableHashTable{} │ │ │ │ 0021fcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0021fcd0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0021fce0: 2020 2020 2020 2020 2073 6f75 7263 6520 source │ │ │ │ 0021fcf0: 6469 7265 6374 6f72 7920 3d3e 202f 746d directory => /tm │ │ │ │ -0021fd00: 702f 4d32 2d31 3038 3232 2d30 2f39 342d p/M2-10822-0/94- │ │ │ │ +0021fd00: 702f 4d32 2d31 3130 3432 2d30 2f39 342d p/M2-11042-0/94- │ │ │ │ 0021fd10: 7275 6e64 6972 2f20 2020 2020 2020 2020 rundir/ │ │ │ │ 0021fd20: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0021fd30: 2020 2020 2020 2020 2073 6f75 7263 6520 source │ │ │ │ 0021fd40: 6669 6c65 203d 3e20 7374 6469 6f20 2020 file => stdio │ │ │ │ 0021fd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0021fd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0021fd70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ @@ -145297,25 +145297,25 @@ │ │ │ │ 00237900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00237910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00237920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00237930: 2d2b 0a7c 6933 203a 2065 6c61 7073 6564 -+.|i3 : elapsed │ │ │ │ 00237940: 5469 6d65 2020 2020 2020 2020 2061 7070 Time app │ │ │ │ 00237950: 6c79 2831 2e2e 3130 302c 206e 202d 3e20 ly(1..100, n -> │ │ │ │ 00237960: 736f 7274 204c 293b 7c0a 7c20 2d2d 202e sort L);|.| -- . │ │ │ │ -00237970: 3538 3030 3973 2065 6c61 7073 6564 2020 58009s elapsed │ │ │ │ +00237970: 3639 3230 3038 7320 656c 6170 7365 6420 692008s elapsed │ │ │ │ 00237980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00237990: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 002379a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 002379b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002379c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002379d0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 656c ------+.|i4 : el │ │ │ │ 002379e0: 6170 7365 6454 696d 6520 7061 7261 6c6c apsedTime parall │ │ │ │ 002379f0: 656c 4170 706c 7928 312e 2e31 3030 2c20 elApply(1..100, │ │ │ │ 00237a00: 6e20 2d3e 2073 6f72 7420 4c29 3b7c 0a7c n -> sort L);|.| │ │ │ │ -00237a10: 202d 2d20 2e32 3735 3330 3673 2065 6c61 -- .275306s ela │ │ │ │ +00237a10: 202d 2d20 2e31 3836 3932 3773 2065 6c61 -- .186927s ela │ │ │ │ 00237a20: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 00237a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00237a40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 00237a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00237a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00237a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 596f -----------+..Yo │ │ │ │ 00237a80: 7520 7769 6c6c 2068 6176 6520 746f 2074 u will have to t │ │ │ │ @@ -145396,15 +145396,15 @@ │ │ │ │ 00237f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00237f40: 2d2d 2d2d 2b0a 7c69 3620 3a20 616c 6c6f ----+.|i6 : allo │ │ │ │ 00237f50: 7761 626c 6554 6872 6561 6473 203d 206d wableThreads = m │ │ │ │ 00237f60: 6178 416c 6c6f 7761 626c 6554 6872 6561 axAllowableThrea │ │ │ │ 00237f70: 6473 7c0a 7c20 2020 2020 2020 2020 2020 ds|.| │ │ │ │ 00237f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00237f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00237fa0: 7c0a 7c6f 3620 3d20 3720 2020 2020 2020 |.|o6 = 7 │ │ │ │ +00237fa0: 7c0a 7c6f 3620 3d20 3137 2020 2020 2020 |.|o6 = 17 │ │ │ │ 00237fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00237fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00237fd0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00237fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00237ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 ------------+..T │ │ │ │ 00238000: 6f20 7275 6e20 6120 6675 6e63 7469 6f6e o run a function │ │ │ │ 00238010: 2069 6e20 616e 6f74 6865 7220 7468 7265 in another thre │ │ │ │ @@ -145530,15 +145530,15 @@ │ │ │ │ 00238790: 2074 6865 0a63 6f6d 7075 7461 7469 6f6e the.computation │ │ │ │ 002387a0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ 002387b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ 002387c0: 3120 3a20 7420 2020 2020 2020 2020 2020 1 : t │ │ │ │ 002387d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 002387e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 002387f0: 0a7c 6f31 3120 3d20 3c3c 7461 736b 2c20 .|o11 = <>|.| │ │ │ │ +00238800: 6372 6561 7465 643e 3e7c 0a7c 2020 2020 created>>|.| │ │ │ │ 00238810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00238820: 2020 207c 0a7c 6f31 3120 3a20 5461 736b |.|o11 : Task │ │ │ │ 00238830: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 00238840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00238850: 2d2d 2d2d 2d2d 2d2b 0a0a 5573 6520 2a6e -------+..Use *n │ │ │ │ 00238860: 6f74 6520 6973 5265 6164 793a 2069 7352 ote isReady: isR │ │ │ │ 00238870: 6561 6479 5f6c 7046 696c 655f 7270 2c20 eady_lpFile_rp, │ │ │ │ @@ -145647,15 +145647,15 @@ │ │ │ │ 00238ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00238ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00238f00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00238f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00238f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00238f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00238f40: 2020 2020 2020 7c0a 7c6f 3138 203d 203c |.|o18 = < │ │ │ │ -00238f50: 3c74 6173 6b2c 2072 756e 6e69 6e67 3e3e > │ │ │ │ +00238f50: 3c74 6173 6b2c 2063 7265 6174 6564 3e3e > │ │ │ │ 00238f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00238f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00238f80: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00238f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00238fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00238fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00238fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ @@ -146917,15 +146917,15 @@ │ │ │ │ 0023de40: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 6e20 ------+.|i5 : n │ │ │ │ 0023de50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023de60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023de70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0023de80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023de90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023dea0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ -0023deb0: 3536 3934 3338 2020 2020 2020 2020 2020 569438 │ │ │ │ +0023deb0: 3930 3431 3936 2020 2020 2020 2020 2020 904196 │ │ │ │ 0023dec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023ded0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 0023dee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023df00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ 0023df10: 3a20 736c 6565 7020 3120 2020 2020 2020 : sleep 1 │ │ │ │ 0023df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -146959,16 +146959,16 @@ │ │ │ │ 0023e0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023e0f0: 2d2d 2d2d 2b0a 7c69 3820 3a20 6e20 2020 ----+.|i8 : n │ │ │ │ 0023e100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e120: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0023e130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0023e150: 2020 2020 2020 7c0a 7c6f 3820 3d20 3132 |.|o8 = 12 │ │ │ │ -0023e160: 3434 3838 3120 2020 2020 2020 2020 2020 44881 │ │ │ │ +0023e150: 2020 2020 2020 7c0a 7c6f 3820 3d20 3139 |.|o8 = 19 │ │ │ │ +0023e160: 3532 3231 3520 2020 2020 2020 2020 2020 52215 │ │ │ │ 0023e170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e180: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 0023e190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023e1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023e1b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 --------+.|i9 : │ │ │ │ 0023e1c0: 6973 5265 6164 7920 7420 2020 2020 2020 isReady t │ │ │ │ 0023e1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -146987,15 +146987,15 @@ │ │ │ │ 0023e2a0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 0023e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023e2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023e2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0023e2e0: 7c69 3131 203a 2073 6c65 6570 2032 2020 |i11 : sleep 2 │ │ │ │ 0023e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e300: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0023e310: 0a7c 7374 6469 6f3a 323a 3339 3a28 3329 .|stdio:2:39:(3) │ │ │ │ +0023e310: 0a7c 7374 6469 6f3a 323a 3235 3a28 3329 .|stdio:2:25:(3) │ │ │ │ 0023e320: 3a5b 315d 3a20 6572 726f 723a 2069 6e74 :[1]: error: int │ │ │ │ 0023e330: 6572 7275 7074 6564 2020 2020 2020 2020 errupted │ │ │ │ 0023e340: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0023e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e370: 207c 0a7c 6f31 3120 3d20 3020 2020 2020 |.|o11 = 0 │ │ │ │ 0023e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -147024,15 +147024,15 @@ │ │ │ │ 0023e4f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 ---------+.|i13 │ │ │ │ 0023e500: 3a20 6e20 2020 2020 2020 2020 2020 2020 : n │ │ │ │ 0023e510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e520: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0023e530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e550: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0023e560: 3320 3d20 3132 3435 3039 3620 2020 2020 3 = 1245096 │ │ │ │ +0023e560: 3320 3d20 3139 3532 3536 3320 2020 2020 3 = 1952563 │ │ │ │ 0023e570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e580: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0023e590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023e5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 0023e5c0: 6931 3420 3a20 736c 6565 7020 3120 2020 i14 : sleep 1 │ │ │ │ 0023e5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -147048,16 +147048,16 @@ │ │ │ │ 0023e670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023e680: 2d2b 0a7c 6931 3520 3a20 6e20 2020 2020 -+.|i15 : n │ │ │ │ 0023e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e6b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0023e6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0023e6e0: 2020 207c 0a7c 6f31 3520 3d20 3132 3435 |.|o15 = 1245 │ │ │ │ -0023e6f0: 3039 3620 2020 2020 2020 2020 2020 2020 096 │ │ │ │ +0023e6e0: 2020 207c 0a7c 6f31 3520 3d20 3139 3532 |.|o15 = 1952 │ │ │ │ +0023e6f0: 3536 3320 2020 2020 2020 2020 2020 2020 563 │ │ │ │ 0023e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023e710: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0023e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023e740: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 6973 -----+.|i16 : is │ │ │ │ 0023e750: 5265 6164 7920 7420 2020 2020 2020 2020 Ready t │ │ │ │ 0023e760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -147815,15 +147815,15 @@ │ │ │ │ 00241660: 6520 7365 740a 0a44 6573 6372 6970 7469 e set..Descripti │ │ │ │ 00241670: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ 00241680: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00241690: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ 002416a0: 206d 6178 416c 6c6f 7761 626c 6554 6872 maxAllowableThr │ │ │ │ 002416b0: 6561 6473 7c0a 7c20 2020 2020 2020 2020 eads|.| │ │ │ │ 002416c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -002416d0: 0a7c 6f31 203d 2037 2020 2020 2020 2020 .|o1 = 7 │ │ │ │ +002416d0: 0a7c 6f31 203d 2031 3720 2020 2020 2020 .|o1 = 17 │ │ │ │ 002416e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 002416f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00241700: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f -----+..See also │ │ │ │ 00241710: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ 00241720: 6e6f 7465 2070 6172 616c 6c65 6c20 7072 note parallel pr │ │ │ │ 00241730: 6f67 7261 6d6d 696e 6720 7769 7468 2074 ogramming with t │ │ │ │ 00241740: 6872 6561 6473 2061 6e64 2074 6173 6b73 hreads and tasks │ │ │ │ @@ -149047,15 +149047,15 @@ │ │ │ │ 00246360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246370: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00246380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002463a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002463b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002463c0: 2020 207c 0a7c 6f33 203d 202f 746d 702f |.|o3 = /tmp/ │ │ │ │ -002463d0: 4d32 2d31 3136 3031 2d30 2f30 2020 2020 M2-11601-0/0 │ │ │ │ +002463d0: 4d32 2d31 3235 3931 2d30 2f30 2020 2020 M2-12591-0/0 │ │ │ │ 002463e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002463f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246410: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00246420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00246430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00246440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -149067,15 +149067,15 @@ │ │ │ │ 002464a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002464b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 002464c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002464d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002464e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002464f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246500: 2020 207c 0a7c 6f34 203d 202f 746d 702f |.|o4 = /tmp/ │ │ │ │ -00246510: 4d32 2d31 3136 3031 2d30 2f30 2020 2020 M2-11601-0/0 │ │ │ │ +00246510: 4d32 2d31 3235 3931 2d30 2f30 2020 2020 M2-12591-0/0 │ │ │ │ 00246520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246550: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00246560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -149227,15 +149227,15 @@ │ │ │ │ 00246ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246eb0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00246ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246f00: 2020 207c 0a7c 6f39 203d 202f 746d 702f |.|o9 = /tmp/ │ │ │ │ -00246f10: 4d32 2d31 3136 3031 2d30 2f30 2020 2020 M2-11601-0/0 │ │ │ │ +00246f10: 4d32 2d31 3235 3931 2d30 2f30 2020 2020 M2-12591-0/0 │ │ │ │ 00246f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246f50: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00246f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00246f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -149287,15 +149287,15 @@ │ │ │ │ 00247260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00247270: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00247280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00247290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002472a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002472b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002472c0: 2020 207c 0a7c 6f31 3120 3d20 2f74 6d70 |.|o11 = /tmp │ │ │ │ -002472d0: 2f4d 322d 3131 3630 312d 302f 3020 2020 /M2-11601-0/0 │ │ │ │ +002472d0: 2f4d 322d 3132 3539 312d 302f 3020 2020 /M2-12591-0/0 │ │ │ │ 002472e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002472f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00247300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00247310: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00247320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00247330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00247340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -149692,16 +149692,16 @@ │ │ │ │ 00248bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00248bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248c10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00248c20: 6f31 203d 202f 746d 702f 4d32 2d31 3137 o1 = /tmp/M2-117 │ │ │ │ -00248c30: 3736 2d30 2f30 2020 2020 2020 2020 2020 76-0/0 │ │ │ │ +00248c20: 6f31 203d 202f 746d 702f 4d32 2d31 3239 o1 = /tmp/M2-129 │ │ │ │ +00248c30: 3436 2d30 2f30 2020 2020 2020 2020 2020 46-0/0 │ │ │ │ 00248c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248c60: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 00248c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00248c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00248c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00248ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -149712,16 +149712,16 @@ │ │ │ │ 00248cf0: 2a7a 5e32 2b31 322a 795e 322a 7a5e 322b *z^2+12*y^2*z^2+ │ │ │ │ 00248d00: 785e 332b 362a 785e 322a 792b 207c 0a7c x^3+6*x^2*y+ |.| │ │ │ │ 00248d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248d50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00248d60: 6f32 203d 202f 746d 702f 4d32 2d31 3137 o2 = /tmp/M2-117 │ │ │ │ -00248d70: 3736 2d30 2f30 2020 2020 2020 2020 2020 76-0/0 │ │ │ │ +00248d60: 6f32 203d 202f 746d 702f 4d32 2d31 3239 o2 = /tmp/M2-129 │ │ │ │ +00248d70: 3436 2d30 2f30 2020 2020 2020 2020 2020 46-0/0 │ │ │ │ 00248d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248da0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00248db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00248de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -149928,15 +149928,15 @@ │ │ │ │ 00249a70: 2022 7361 6d70 6c65 203d 2032 5e31 3030 "sample = 2^100 │ │ │ │ 00249a80: 7c0a 7c20 2020 2020 7072 696e 7420 7361 |.| print sa │ │ │ │ 00249a90: 6d70 6c65 2020 2020 2020 2020 207c 0a7c mple |.| │ │ │ │ 00249aa0: 2020 2020 2022 203c 3c20 636c 6f73 6520 " << close │ │ │ │ 00249ab0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00249ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00249ad0: 2020 2020 2020 207c 0a7c 6f37 203d 202f |.|o7 = / │ │ │ │ -00249ae0: 746d 702f 4d32 2d31 3137 3736 2d30 2f30 tmp/M2-11776-0/0 │ │ │ │ +00249ae0: 746d 702f 4d32 2d31 3239 3436 2d30 2f30 tmp/M2-12946-0/0 │ │ │ │ 00249af0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00249b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00249b10: 207c 0a7c 6f37 203a 2046 696c 6520 2020 |.|o7 : File │ │ │ │ 00249b20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00249b30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00249b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a4e 6f77 -----------+.Now │ │ │ │ 00249b50: 2076 6572 6966 7920 7468 6174 2069 7420 verify that it │ │ │ │ @@ -150529,15 +150529,15 @@ │ │ │ │ 0024c000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c010: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0024c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c060: 2020 2020 207c 0a7c 6f35 203d 202f 746d |.|o5 = /tm │ │ │ │ -0024c070: 702f 4d32 2d31 3230 3235 2d30 2f30 2020 p/M2-12025-0/0 │ │ │ │ +0024c070: 702f 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4d32 2d31 3230 3235 2d30 2f30 2020 p/M2-12025-0/0 │ │ │ │ +0024c1b0: 702f 4d32 2d31 3334 3535 2d30 2f30 2020 p/M2-13455-0/0 │ │ │ │ 0024c1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c1f0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0024c200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -151503,24 +151503,24 @@ │ │ │ │ 0024fce0: 7379 7374 656d 2e0a 2b2d 2d2d 2d2d 2d2d system..+------- │ │ │ │ 0024fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0024fd00: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2066 -------+.|i1 : f │ │ │ │ 0024fd10: 6e20 3d20 7465 6d70 6f72 6172 7946 696c n = temporaryFil │ │ │ │ 0024fd20: 654e 616d 6528 2920 7c0a 7c20 2020 2020 eName() |.| │ │ │ │ 0024fd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024fd40: 2020 2020 2020 2020 207c 0a7c 6f31 203d |.|o1 = │ │ │ │ -0024fd50: 202f 746d 702f 4d32 2d31 3238 3736 2d30 /tmp/M2-12876-0 │ │ │ │ +0024fd50: 202f 746d 702f 4d32 2d31 3531 3936 2d30 /tmp/M2-15196-0 │ │ │ │ 0024fd60: 2f30 2020 2020 2020 2020 7c0a 2b2d 2d2d /0 |.+--- │ │ │ │ 0024fd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0024fd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ 0024fd90: 203a 2066 6e20 3c3c 2022 6869 2074 6865 : fn << "hi the │ │ │ │ 0024fda0: 7265 2220 3c3c 2063 6c6f 7365 7c0a 7c20 re" << close|.| │ │ │ │ 0024fdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024fdc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0024fdd0: 6f32 203d 202f 746d 702f 4d32 2d31 3238 o2 = /tmp/M2-128 │ │ │ │ -0024fde0: 3736 2d30 2f30 2020 2020 2020 2020 7c0a 76-0/0 |. │ │ │ │ +0024fdd0: 6f32 203d 202f 746d 702f 4d32 2d31 3531 o2 = /tmp/M2-151 │ │ │ │ +0024fde0: 3936 2d30 2f30 2020 2020 2020 2020 7c0a 96-0/0 |. │ │ │ │ 0024fdf0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0024fe00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0024fe10: 0a7c 6f32 203a 2046 696c 6520 2020 2020 .|o2 : File │ │ │ │ 0024fe20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024fe30: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0024fe40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0024fe50: 2d2b 0a7c 6933 203a 2069 7352 6567 756c -+.|i3 : isRegul │ │ │ │ @@ -151606,23 +151606,23 @@ │ │ │ │ 00250350: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00250360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00250370: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 666e ------+.|i2 : fn │ │ │ │ 00250380: 203d 2074 656d 706f 7261 7279 4669 6c65 = temporaryFile │ │ │ │ 00250390: 4e61 6d65 2829 207c 0a7c 2020 2020 2020 Name() |.| │ │ │ │ 002503a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002503b0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ -002503c0: 2f74 6d70 2f4d 322d 3131 3031 312d 302f /tmp/M2-11011-0/ │ │ │ │ +002503c0: 2f74 6d70 2f4d 322d 3131 3430 312d 302f /tmp/M2-11401-0/ │ │ │ │ 002503d0: 3020 2020 2020 2020 207c 0a2b 2d2d 2d2d 0 |.+---- │ │ │ │ 002503e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002503f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ 00250400: 3a20 666e 203c 3c20 2268 6920 7468 6572 : fn << "hi ther │ │ │ │ 00250410: 6522 203c 3c20 636c 6f73 657c 0a7c 2020 e" << close|.| │ │ │ │ 00250420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00250430: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00250440: 3320 3d20 2f74 6d70 2f4d 322d 3131 3031 3 = /tmp/M2-1101 │ │ │ │ +00250440: 3320 3d20 2f74 6d70 2f4d 322d 3131 3430 3 = /tmp/M2-1140 │ │ │ │ 00250450: 312d 302f 3020 2020 2020 2020 207c 0a7c 1-0/0 |.| │ │ │ │ 00250460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00250470: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00250480: 7c6f 3320 3a20 4669 6c65 2020 2020 2020 |o3 : File │ │ │ │ 00250490: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 002504a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 002504b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -151802,25 +151802,25 @@ │ │ │ │ 00250f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00250fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00250fb0: 6931 203a 2074 656d 706f 7261 7279 4669 i1 : temporaryFi │ │ │ │ 00250fc0: 6c65 4e61 6d65 2028 2920 7c20 222e 7465 leName () | ".te │ │ │ │ 00250fd0: 7822 207c 0a7c 2020 2020 2020 2020 2020 x" |.| │ │ │ │ 00250fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00250ff0: 2020 2020 2020 2020 207c 0a7c 6f31 203d |.|o1 = │ │ │ │ -00251000: 202f 746d 702f 4d32 2d31 3238 3537 2d30 /tmp/M2-12857-0 │ │ │ │ +00251000: 202f 746d 702f 4d32 2d31 3531 3537 2d30 /tmp/M2-15157-0 │ │ │ │ 00251010: 2f30 2e74 6578 2020 2020 2020 2020 207c /0.tex | │ │ │ │ 00251020: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00251030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00251040: 2d2d 2d2d 2d2b 0a7c 6932 203a 2074 656d -----+.|i2 : tem │ │ │ │ 00251050: 706f 7261 7279 4669 6c65 4e61 6d65 2028 poraryFileName ( │ │ │ │ 00251060: 2920 7c20 222e 6874 6d6c 227c 0a7c 2020 ) | ".html"|.| │ │ │ │ 00251070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00251080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00251090: 207c 0a7c 6f32 203d 202f 746d 702f 4d32 |.|o2 = /tmp/M2 │ │ │ │ -002510a0: 2d31 3238 3537 2d30 2f31 2e68 746d 6c20 -12857-0/1.html │ │ │ │ +002510a0: 2d31 3531 3537 2d30 2f31 2e68 746d 6c20 -15157-0/1.html │ │ │ │ 002510b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 002510c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002510d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ 002510e0: 5468 6973 2066 756e 6374 696f 6e20 7769 This function wi │ │ │ │ 002510f0: 6c6c 2077 6f72 6b20 756e 6465 7220 556e ll work under Un │ │ │ │ 00251100: 6978 2c20 616e 6420 616c 736f 2075 6e64 ix, and also und │ │ │ │ 00251110: 6572 2057 696e 646f 7773 2069 6620 796f er Windows if yo │ │ │ │ @@ -151984,23 +151984,23 @@ │ │ │ │ 00251af0: 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d =..+------------ │ │ │ │ 00251b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00251b10: 2d2b 0a7c 6931 203a 2066 6e20 3d20 7465 -+.|i1 : fn = te │ │ │ │ 00251b20: 6d70 6f72 6172 7946 696c 654e 616d 6528 mporaryFileName( │ │ │ │ 00251b30: 297c 0a7c 2020 2020 2020 2020 2020 2020 )|.| │ │ │ │ 00251b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00251b50: 207c 0a7c 6f31 203d 202f 746d 702f 4d32 |.|o1 = /tmp/M2 │ │ │ │ -00251b60: 2d31 3230 3434 2d30 2f30 2020 2020 2020 -12044-0/0 │ │ │ │ +00251b60: 2d31 3334 3934 2d30 2f30 2020 2020 2020 -13494-0/0 │ │ │ │ 00251b70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00251b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00251b90: 2d2b 0a7c 6932 203a 2066 203d 2066 6e20 -+.|i2 : f = fn │ │ │ │ 00251ba0: 3c3c 2022 6869 2074 6865 7265 2220 2020 << "hi there" │ │ │ │ 00251bb0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00251bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00251bd0: 207c 0a7c 6f32 203d 202f 746d 702f 4d32 |.|o2 = /tmp/M2 │ │ │ │ -00251be0: 2d31 3230 3434 2d30 2f30 2020 2020 2020 -12044-0/0 │ │ │ │ +00251be0: 2d31 3334 3934 2d30 2f30 2020 2020 2020 -13494-0/0 │ │ │ │ 00251bf0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00251c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00251c10: 207c 0a7c 6f32 203a 2046 696c 6520 2020 |.|o2 : File │ │ │ │ 00251c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00251c30: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00251c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00251c50: 2d2b 0a7c 6933 203a 2066 696c 654d 6f64 -+.|i3 : fileMod │ │ │ │ @@ -152012,15 +152012,15 @@ │ │ │ │ 00251cb0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00251cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00251cd0: 2d2b 0a7c 6934 203a 2063 6c6f 7365 2066 -+.|i4 : close f │ │ │ │ 00251ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00251cf0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00251d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00251d10: 207c 0a7c 6f34 203d 202f 746d 702f 4d32 |.|o4 = /tmp/M2 │ │ │ │ -00251d20: 2d31 3230 3434 2d30 2f30 2020 2020 2020 -12044-0/0 │ │ │ │ +00251d20: 2d31 3334 3934 2d30 2f30 2020 2020 2020 -13494-0/0 │ │ │ │ 00251d30: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00251d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00251d50: 207c 0a7c 6f34 203a 2046 696c 6520 2020 |.|o4 : File │ │ │ │ 00251d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00251d70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00251d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00251d90: 2d2b 0a7c 6935 203a 2072 656d 6f76 6546 -+.|i5 : removeF │ │ │ │ @@ -152078,23 +152078,23 @@ │ │ │ │ 002520d0: 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d ========..+----- │ │ │ │ 002520e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002520f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ 00252100: 666e 203d 2074 656d 706f 7261 7279 4669 fn = temporaryFi │ │ │ │ 00252110: 6c65 4e61 6d65 2829 7c0a 7c20 2020 2020 leName()|.| │ │ │ │ 00252120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00252130: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -00252140: 2f74 6d70 2f4d 322d 3131 3532 332d 302f /tmp/M2-11523-0/ │ │ │ │ +00252140: 2f74 6d70 2f4d 322d 3132 3433 332d 302f /tmp/M2-12433-0/ │ │ │ │ 00252150: 3020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 0 |.+----- │ │ │ │ 00252160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00252170: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ 00252180: 6620 3d20 666e 203c 3c20 2268 6920 7468 f = fn << "hi th │ │ │ │ 00252190: 6572 6522 2020 2020 7c0a 7c20 2020 2020 ere" |.| │ │ │ │ 002521a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002521b0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ -002521c0: 2f74 6d70 2f4d 322d 3131 3532 332d 302f /tmp/M2-11523-0/ │ │ │ │ +002521c0: 2f74 6d70 2f4d 322d 3132 3433 332d 302f /tmp/M2-12433-0/ │ │ │ │ 002521d0: 3020 2020 2020 2020 7c0a 7c20 2020 2020 0 |.| │ │ │ │ 002521e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002521f0: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ 00252200: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ 00252210: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 00252220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00252230: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ @@ -152118,15 +152118,15 @@ │ │ │ │ 00252350: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 00252360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00252370: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ 00252380: 636c 6f73 6520 6620 2020 2020 2020 2020 close f │ │ │ │ 00252390: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 002523a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002523b0: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ -002523c0: 2f74 6d70 2f4d 322d 3131 3532 332d 302f /tmp/M2-11523-0/ │ │ │ │ +002523c0: 2f74 6d70 2f4d 322d 3132 3433 332d 302f /tmp/M2-12433-0/ │ │ │ │ 002523d0: 3020 2020 2020 2020 7c0a 7c20 2020 2020 0 |.| │ │ │ │ 002523e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002523f0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20 |.|o6 : │ │ │ │ 00252400: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ 00252410: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 00252420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00252430: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 --------+.|i7 : │ │ │ │ @@ -152194,23 +152194,23 @@ │ │ │ │ 00252810: 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d =====..+-------- │ │ │ │ 00252820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00252830: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 666e ------+.|i1 : fn │ │ │ │ 00252840: 203d 2074 656d 706f 7261 7279 4669 6c65 = temporaryFile │ │ │ │ 00252850: 4e61 6d65 2829 207c 0a7c 2020 2020 2020 Name() |.| │ │ │ │ 00252860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00252870: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -00252880: 2f74 6d70 2f4d 322d 3131 3635 382d 302f /tmp/M2-11658-0/ │ │ │ │ +00252880: 2f74 6d70 2f4d 322d 3132 3730 382d 302f /tmp/M2-12708-0/ │ │ │ │ 00252890: 3020 2020 2020 2020 207c 0a2b 2d2d 2d2d 0 |.+---- │ │ │ │ 002528a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002528b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ 002528c0: 3a20 666e 203c 3c20 2268 6920 7468 6572 : fn << "hi ther │ │ │ │ 002528d0: 6522 203c 3c20 636c 6f73 657c 0a7c 2020 e" << close|.| │ │ │ │ 002528e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002528f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00252900: 3220 3d20 2f74 6d70 2f4d 322d 3131 3635 2 = /tmp/M2-1165 │ │ │ │ +00252900: 3220 3d20 2f74 6d70 2f4d 322d 3132 3730 2 = /tmp/M2-1270 │ │ │ │ 00252910: 382d 302f 3020 2020 2020 2020 207c 0a7c 8-0/0 |.| │ │ │ │ 00252920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00252930: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00252940: 7c6f 3220 3a20 4669 6c65 2020 2020 2020 |o2 : File │ │ │ │ 00252950: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00252960: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00252970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -152279,23 +152279,23 @@ │ │ │ │ 00252d60: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00252d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00252d80: 2b0a 7c69 3120 3a20 666e 203d 2074 656d +.|i1 : fn = tem │ │ │ │ 00252d90: 706f 7261 7279 4669 6c65 4e61 6d65 2829 poraryFileName() │ │ │ │ 00252da0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00252db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00252dc0: 2020 7c0a 7c6f 3120 3d20 2f74 6d70 2f4d |.|o1 = /tmp/M │ │ │ │ -00252dd0: 322d 3132 3636 352d 302f 3020 2020 2020 2-12665-0/0 │ │ │ │ +00252dd0: 322d 3134 3736 352d 302f 3020 2020 2020 2-14765-0/0 │ │ │ │ 00252de0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00252df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00252e00: 2d2d 2d2d 2b0a 7c69 3220 3a20 666e 203c ----+.|i2 : fn < │ │ │ │ 00252e10: 3c20 2268 6920 7468 6572 6522 203c 3c20 < "hi there" << │ │ │ │ 00252e20: 636c 6f73 657c 0a7c 2020 2020 2020 2020 close|.| │ │ │ │ 00252e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00252e40: 2020 2020 2020 7c0a 7c6f 3220 3d20 2f74 |.|o2 = /t │ │ │ │ -00252e50: 6d70 2f4d 322d 3132 3636 352d 302f 3020 mp/M2-12665-0/0 │ │ │ │ +00252e50: 6d70 2f4d 322d 3134 3736 352d 302f 3020 mp/M2-14765-0/0 │ │ │ │ 00252e60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00252e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00252e80: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ 00252e90: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ 00252ea0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 00252eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00252ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ @@ -152373,15 +152373,15 @@ │ │ │ │ 00253340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00253350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00253360: 7c69 3120 3a20 666e 203d 2074 656d 706f |i1 : fn = tempo │ │ │ │ 00253370: 7261 7279 4669 6c65 4e61 6d65 2829 207c raryFileName() | │ │ │ │ 00253380: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00253390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002533a0: 7c0a 7c6f 3120 3d20 2f74 6d70 2f4d 322d |.|o1 = /tmp/M2- │ │ │ │ -002533b0: 3131 3138 392d 302f 3020 2020 2020 2020 11189-0/0 │ │ │ │ +002533b0: 3131 3735 392d 302f 3020 2020 2020 2020 11759-0/0 │ │ │ │ 002533c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 002533d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002533e0: 2d2d 2b0a 7c69 3220 3a20 6669 6c65 4578 --+.|i2 : fileEx │ │ │ │ 002533f0: 6973 7473 2066 6e20 2020 2020 2020 2020 ists fn │ │ │ │ 00253400: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00253410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00253420: 2020 2020 7c0a 7c6f 3220 3d20 6661 6c73 |.|o2 = fals │ │ │ │ @@ -152389,15 +152389,15 @@ │ │ │ │ 00253440: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00253450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00253460: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 666e ------+.|i3 : fn │ │ │ │ 00253470: 203c 3c20 2268 6920 7468 6572 6522 203c << "hi there" < │ │ │ │ 00253480: 3c20 636c 6f73 657c 0a7c 2020 2020 2020 < close|.| │ │ │ │ 00253490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002534a0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -002534b0: 2f74 6d70 2f4d 322d 3131 3138 392d 302f /tmp/M2-11189-0/ │ │ │ │ +002534b0: 2f74 6d70 2f4d 322d 3131 3735 392d 302f /tmp/M2-11759-0/ │ │ │ │ 002534c0: 3020 2020 2020 2020 207c 0a7c 2020 2020 0 |.| │ │ │ │ 002534d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002534e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ 002534f0: 3a20 4669 6c65 2020 2020 2020 2020 2020 : File │ │ │ │ 00253500: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00253510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00253520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ @@ -152691,15 +152691,15 @@ │ │ │ │ 00254720: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00254730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00254740: 2d2d 2b0a 7c69 3120 3a20 6375 7272 656e --+.|i1 : curren │ │ │ │ 00254750: 7454 696d 6528 2920 2d20 6669 6c65 5469 tTime() - fileTi │ │ │ │ 00254760: 6d65 2022 2e22 7c0a 7c20 2020 2020 2020 me "."|.| │ │ │ │ 00254770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254780: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ -00254790: 3d20 3833 2020 2020 2020 2020 2020 2020 = 83 │ │ │ │ +00254790: 3d20 3731 2020 2020 2020 2020 2020 2020 = 71 │ │ │ │ 002547a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 002547b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 002547c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002547d0: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d --+..See also.== │ │ │ │ 002547e0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ 002547f0: 6520 6375 7272 656e 7454 696d 653a 2063 e currentTime: c │ │ │ │ 00254800: 7572 7265 6e74 5469 6d65 2c20 2d2d 2067 urrentTime, -- g │ │ │ │ @@ -152800,15 +152800,15 @@ │ │ │ │ 00254df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254e10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00254e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254e50: 2020 2020 207c 0a7c 6f31 203d 202f 746d |.|o1 = /tm │ │ │ │ -00254e60: 702f 4d32 2d31 3136 3339 2d30 2f30 2020 p/M2-11639-0/0 │ │ │ │ +00254e60: 702f 4d32 2d31 3236 3639 2d30 2f30 2020 p/M2-12669-0/0 │ │ │ │ 00254e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254e90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 00254ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00254eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00254ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00254ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ @@ -152817,15 +152817,15 @@ │ │ │ │ 00254f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254f10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00254f20: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00254f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254f60: 207c 0a7c 6f32 203d 202f 746d 702f 4d32 |.|o2 = /tmp/M2 │ │ │ │ -00254f70: 2d31 3136 3339 2d30 2f31 2020 2020 2020 -11639-0/1 │ │ │ │ +00254f70: 2d31 3236 3639 2d30 2f31 2020 2020 2020 -12669-0/1 │ │ │ │ 00254f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254fa0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 00254fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00254fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00254fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00254fe0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2073 -------+.|i3 : s │ │ │ │ @@ -152833,16 +152833,16 @@ │ │ │ │ 00255000: 203c 3c20 636c 6f73 6520 2020 2020 2020 << close │ │ │ │ 00255010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255020: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00255030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255060: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00255070: 6f33 203d 202f 746d 702f 4d32 2d31 3136 o3 = /tmp/M2-116 │ │ │ │ -00255080: 3339 2d30 2f30 2020 2020 2020 2020 2020 39-0/0 │ │ │ │ +00255070: 6f33 203d 202f 746d 702f 4d32 2d31 3236 o3 = /tmp/M2-126 │ │ │ │ +00255080: 3639 2d30 2f30 2020 2020 2020 2020 2020 69-0/0 │ │ │ │ 00255090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002550a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002550b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 002550c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002550d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002550e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002550f0: 2020 207c 0a7c 6f33 203a 2046 696c 6520 |.|o3 : File │ │ │ │ @@ -152855,16 +152855,16 @@ │ │ │ │ 00255160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255170: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ 00255180: 2063 6f70 7946 696c 6528 7372 632c 6473 copyFile(src,ds │ │ │ │ 00255190: 742c 5665 7262 6f73 653d 3e74 7275 6529 t,Verbose=>true) │ │ │ │ 002551a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002551b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 002551c0: 2d2d 2063 6f70 7969 6e67 3a20 2f74 6d70 -- copying: /tmp │ │ │ │ -002551d0: 2f4d 322d 3131 3633 392d 302f 3020 2d3e /M2-11639-0/0 -> │ │ │ │ -002551e0: 202f 746d 702f 4d32 2d31 3136 3339 2d30 /tmp/M2-11639-0 │ │ │ │ +002551d0: 2f4d 322d 3132 3636 392d 302f 3020 2d3e /M2-12669-0/0 -> │ │ │ │ +002551e0: 202f 746d 702f 4d32 2d31 3236 3639 2d30 /tmp/M2-12669-0 │ │ │ │ 002551f0: 2f31 2020 2020 2020 2020 2020 2020 207c /1 | │ │ │ │ 00255200: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00255210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255240: 2d2d 2b0a 7c69 3520 3a20 6765 7420 6473 --+.|i5 : get ds │ │ │ │ 00255250: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ @@ -152884,32 +152884,32 @@ │ │ │ │ 00255330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00255350: 7c69 3620 3a20 636f 7079 4669 6c65 2873 |i6 : copyFile(s │ │ │ │ 00255360: 7263 2c64 7374 2c56 6572 626f 7365 3d3e rc,dst,Verbose=> │ │ │ │ 00255370: 7472 7565 2c55 7064 6174 654f 6e6c 7920 true,UpdateOnly │ │ │ │ 00255380: 3d3e 2074 7275 6529 2020 2020 2020 2020 => true) │ │ │ │ 00255390: 207c 0a7c 202d 2d20 736b 6970 7069 6e67 |.| -- skipping │ │ │ │ -002553a0: 3a20 2f74 6d70 2f4d 322d 3131 3633 392d : /tmp/M2-11639- │ │ │ │ +002553a0: 3a20 2f74 6d70 2f4d 322d 3132 3636 392d : /tmp/M2-12669- │ │ │ │ 002553b0: 302f 3020 6e6f 7420 6e65 7765 7220 7468 0/0 not newer th │ │ │ │ -002553c0: 616e 202f 746d 702f 4d32 2d31 3136 3339 an /tmp/M2-11639 │ │ │ │ +002553c0: 616e 202f 746d 702f 4d32 2d31 3236 3639 an /tmp/M2-12669 │ │ │ │ 002553d0: 2d30 2f31 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d -0/1|.+--------- │ │ │ │ 002553e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002553f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255410: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2073 -------+.|i7 : s │ │ │ │ 00255420: 7263 203c 3c20 2268 6f20 7468 6572 6522 rc << "ho there" │ │ │ │ 00255430: 203c 3c20 636c 6f73 6520 2020 2020 2020 << close │ │ │ │ 00255440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255450: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00255460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255490: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -002554a0: 6f37 203d 202f 746d 702f 4d32 2d31 3136 o7 = /tmp/M2-116 │ │ │ │ -002554b0: 3339 2d30 2f30 2020 2020 2020 2020 2020 39-0/0 │ │ │ │ +002554a0: 6f37 203d 202f 746d 702f 4d32 2d31 3236 o7 = /tmp/M2-126 │ │ │ │ +002554b0: 3639 2d30 2f30 2020 2020 2020 2020 2020 69-0/0 │ │ │ │ 002554c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002554d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002554e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 002554f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255520: 2020 207c 0a7c 6f37 203a 2046 696c 6520 |.|o7 : File │ │ │ │ @@ -152922,17 +152922,17 @@ │ │ │ │ 00255590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002555a0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a ---------+.|i8 : │ │ │ │ 002555b0: 2063 6f70 7946 696c 6528 7372 632c 6473 copyFile(src,ds │ │ │ │ 002555c0: 742c 5665 7262 6f73 653d 3e74 7275 652c t,Verbose=>true, │ │ │ │ 002555d0: 5570 6461 7465 4f6e 6c79 203d 3e20 7472 UpdateOnly => tr │ │ │ │ 002555e0: 7565 2920 2020 2020 2020 2020 7c0a 7c20 ue) |.| │ │ │ │ 002555f0: 2d2d 2073 6b69 7070 696e 673a 202f 746d -- skipping: /tm │ │ │ │ -00255600: 702f 4d32 2d31 3136 3339 2d30 2f30 206e p/M2-11639-0/0 n │ │ │ │ +00255600: 702f 4d32 2d31 3236 3639 2d30 2f30 206e p/M2-12669-0/0 n │ │ │ │ 00255610: 6f74 206e 6577 6572 2074 6861 6e20 2f74 ot newer than /t │ │ │ │ -00255620: 6d70 2f4d 322d 3131 3633 392d 302f 317c mp/M2-11639-0/1| │ │ │ │ +00255620: 6d70 2f4d 322d 3132 3636 392d 302f 317c mp/M2-12669-0/1| │ │ │ │ 00255630: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00255640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255670: 2d2d 2b0a 7c69 3920 3a20 6765 7420 6473 --+.|i9 : get ds │ │ │ │ 00255680: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ 00255690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -153070,41 +153070,41 @@ │ │ │ │ 00255ed0: 0a7c 6931 203a 2073 7263 203d 2074 656d .|i1 : src = tem │ │ │ │ 00255ee0: 706f 7261 7279 4669 6c65 4e61 6d65 2829 poraryFileName() │ │ │ │ 00255ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255f00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00255f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255f30: 2020 2020 207c 0a7c 6f31 203d 202f 746d |.|o1 = /tm │ │ │ │ -00255f40: 702f 4d32 2d31 3132 3436 2d30 2f30 2020 p/M2-11246-0/0 │ │ │ │ +00255f40: 702f 4d32 2d31 3138 3736 2d30 2f30 2020 p/M2-11876-0/0 │ │ │ │ 00255f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255f60: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 00255f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ 00255fa0: 203a 2064 7374 203d 2074 656d 706f 7261 : dst = tempora │ │ │ │ 00255fb0: 7279 4669 6c65 4e61 6d65 2829 2020 2020 ryFileName() │ │ │ │ 00255fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00255fd0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00255fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00256000: 207c 0a7c 6f32 203d 202f 746d 702f 4d32 |.|o2 = /tmp/M2 │ │ │ │ -00256010: 2d31 3132 3436 2d30 2f31 2020 2020 2020 -11246-0/1 │ │ │ │ +00256010: 2d31 3138 3736 2d30 2f31 2020 2020 2020 -11876-0/1 │ │ │ │ 00256020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00256030: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 00256040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00256050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00256060: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2073 -------+.|i3 : s │ │ │ │ 00256070: 7263 203c 3c20 2268 6920 7468 6572 6522 rc << "hi there" │ │ │ │ 00256080: 203c 3c20 636c 6f73 6520 2020 2020 2020 << close │ │ │ │ 00256090: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 002560a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002560b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002560c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -002560d0: 6f33 203d 202f 746d 702f 4d32 2d31 3132 o3 = /tmp/M2-112 │ │ │ │ -002560e0: 3436 2d30 2f30 2020 2020 2020 2020 2020 46-0/0 │ │ │ │ +002560d0: 6f33 203d 202f 746d 702f 4d32 2d31 3138 o3 = /tmp/M2-118 │ │ │ │ +002560e0: 3736 2d30 2f30 2020 2020 2020 2020 2020 76-0/0 │ │ │ │ 002560f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00256100: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00256110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00256120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00256130: 2020 207c 0a7c 6f33 203a 2046 696c 6520 |.|o3 : File │ │ │ │ 00256140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00256150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -153112,16 +153112,16 @@ │ │ │ │ 00256170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00256180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00256190: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ 002561a0: 206d 6f76 6546 696c 6528 7372 632c 6473 moveFile(src,ds │ │ │ │ 002561b0: 742c 5665 7262 6f73 653d 3e74 7275 6529 t,Verbose=>true) │ │ │ │ 002561c0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 002561d0: 2d6d 6f76 696e 673a 202f 746d 702f 4d32 -moving: /tmp/M2 │ │ │ │ -002561e0: 2d31 3132 3436 2d30 2f30 202d 3e20 2f74 -11246-0/0 -> /t │ │ │ │ -002561f0: 6d70 2f4d 322d 3131 3234 362d 302f 317c mp/M2-11246-0/1| │ │ │ │ +002561e0: 2d31 3138 3736 2d30 2f30 202d 3e20 2f74 -11876-0/0 -> /t │ │ │ │ +002561f0: 6d70 2f4d 322d 3131 3837 362d 302f 317c mp/M2-11876-0/1| │ │ │ │ 00256200: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00256210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00256220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00256230: 2d2d 2b0a 7c69 3520 3a20 6765 7420 6473 --+.|i5 : get ds │ │ │ │ 00256240: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ 00256250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00256260: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ @@ -153135,20 +153135,20 @@ │ │ │ │ 002562e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002562f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00256300: 7c69 3620 3a20 6261 6b20 3d20 6d6f 7665 |i6 : bak = move │ │ │ │ 00256310: 4669 6c65 2864 7374 2c56 6572 626f 7365 File(dst,Verbose │ │ │ │ 00256320: 3d3e 7472 7565 2920 2020 2020 2020 2020 =>true) │ │ │ │ 00256330: 207c 0a7c 2d2d 6261 636b 7570 2066 696c |.|--backup fil │ │ │ │ 00256340: 6520 6372 6561 7465 643a 202f 746d 702f e created: /tmp/ │ │ │ │ -00256350: 4d32 2d31 3132 3436 2d30 2f31 2e62 616b M2-11246-0/1.bak │ │ │ │ +00256350: 4d32 2d31 3138 3736 2d30 2f31 2e62 616b M2-11876-0/1.bak │ │ │ │ 00256360: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00256370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00256380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00256390: 2020 2020 2020 207c 0a7c 6f36 203d 202f |.|o6 = / │ │ │ │ -002563a0: 746d 702f 4d32 2d31 3132 3436 2d30 2f31 tmp/M2-11246-0/1 │ │ │ │ +002563a0: 746d 702f 4d32 2d31 3138 3736 2d30 2f31 tmp/M2-11876-0/1 │ │ │ │ 002563b0: 2e62 616b 2020 2020 2020 2020 2020 2020 .bak │ │ │ │ 002563c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 002563d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002563e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002563f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00256400: 6937 203a 2072 656d 6f76 6546 696c 6520 i7 : removeFile │ │ │ │ 00256410: 6261 6b20 2020 2020 2020 2020 2020 2020 bak │ │ │ │ @@ -153363,16 +153363,16 @@ │ │ │ │ 00257120: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b n.===========..+ │ │ │ │ 00257130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00257140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00257150: 6931 203a 2066 6e20 3d20 7465 6d70 6f72 i1 : fn = tempor │ │ │ │ 00257160: 6172 7946 696c 654e 616d 6528 297c 0a7c aryFileName()|.| │ │ │ │ 00257170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00257180: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00257190: 6f31 203d 202f 746d 702f 4d32 2d31 3138 o1 = /tmp/M2-118 │ │ │ │ -002571a0: 3733 2d30 2f30 2020 2020 2020 207c 0a2b 73-0/0 |.+ │ │ │ │ +00257190: 6f31 203d 202f 746d 702f 4d32 2d31 3331 o1 = /tmp/M2-131 │ │ │ │ +002571a0: 3433 2d30 2f30 2020 2020 2020 207c 0a2b 43-0/0 |.+ │ │ │ │ 002571b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002571c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 002571d0: 6932 203a 2073 796d 6c69 6e6b 4669 6c65 i2 : symlinkFile │ │ │ │ 002571e0: 2822 7177 6572 7422 2c20 666e 297c 0a2b ("qwert", fn)|.+ │ │ │ │ 002571f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00257200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00257210: 6933 203a 2066 696c 6545 7869 7374 7320 i3 : fileExists │ │ │ │ @@ -153525,15 +153525,15 @@ │ │ │ │ 00257b40: 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d =====..+-------- │ │ │ │ 00257b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00257b60: 2d2d 2d2d 2d2b 0a7c 6931 203a 2070 203d -----+.|i1 : p = │ │ │ │ 00257b70: 2074 656d 706f 7261 7279 4669 6c65 4e61 temporaryFileNa │ │ │ │ 00257b80: 6d65 2028 297c 0a7c 2020 2020 2020 2020 me ()|.| │ │ │ │ 00257b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00257ba0: 2020 2020 207c 0a7c 6f31 203d 202f 746d |.|o1 = /tm │ │ │ │ -00257bb0: 702f 4d32 2d31 3234 3934 2d30 2f30 2020 p/M2-12494-0/0 │ │ │ │ +00257bb0: 702f 4d32 2d31 3434 3034 2d30 2f30 2020 p/M2-14404-0/0 │ │ │ │ 00257bc0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00257bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00257be0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2073 796d -----+.|i2 : sym │ │ │ │ 00257bf0: 6c69 6e6b 4669 6c65 2028 2266 6f6f 222c linkFile ("foo", │ │ │ │ 00257c00: 2070 2920 207c 0a2b 2d2d 2d2d 2d2d 2d2d p) |.+-------- │ │ │ │ 00257c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00257c20: 2d2d 2d2d 2d2b 0a7c 6933 203a 2072 6561 -----+.|i3 : rea │ │ │ │ @@ -153615,66 +153615,66 @@ │ │ │ │ 002580e0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ 002580f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00258100: 2d2d 2b0a 7c69 3120 3a20 7265 616c 7061 --+.|i1 : realpa │ │ │ │ 00258110: 7468 2022 2e22 2020 2020 2020 2020 2020 th "." │ │ │ │ 00258120: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00258130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00258140: 2020 2020 2020 7c0a 7c6f 3120 3d20 2f74 |.|o1 = /t │ │ │ │ -00258150: 6d70 2f4d 322d 3130 3832 322d 302f 3839 mp/M2-10822-0/89 │ │ │ │ +00258150: 6d70 2f4d 322d 3131 3034 322d 302f 3839 mp/M2-11042-0/89 │ │ │ │ 00258160: 2d72 756e 6469 722f 7c0a 2b2d 2d2d 2d2d -rundir/|.+----- │ │ │ │ 00258170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00258180: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ 00258190: 3a20 7020 3d20 7465 6d70 6f72 6172 7946 : p = temporaryF │ │ │ │ 002581a0: 696c 654e 616d 6528 2920 2020 7c0a 7c20 ileName() |.| │ │ │ │ 002581b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002581c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -002581d0: 7c6f 3220 3d20 2f74 6d70 2f4d 322d 3132 |o2 = /tmp/M2-12 │ │ │ │ -002581e0: 3531 332d 302f 3020 2020 2020 2020 2020 513-0/0 │ │ │ │ +002581d0: 7c6f 3220 3d20 2f74 6d70 2f4d 322d 3134 |o2 = /tmp/M2-14 │ │ │ │ +002581e0: 3434 332d 302f 3020 2020 2020 2020 2020 443-0/0 │ │ │ │ 002581f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00258200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00258210: 2d2d 2b0a 7c69 3320 3a20 7120 3d20 7465 --+.|i3 : q = te │ │ │ │ 00258220: 6d70 6f72 6172 7946 696c 654e 616d 6528 mporaryFileName( │ │ │ │ 00258230: 2920 2020 7c0a 7c20 2020 2020 2020 2020 ) |.| │ │ │ │ 00258240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00258250: 2020 2020 2020 7c0a 7c6f 3320 3d20 2f74 |.|o3 = /t │ │ │ │ -00258260: 6d70 2f4d 322d 3132 3531 332d 302f 3120 mp/M2-12513-0/1 │ │ │ │ +00258260: 6d70 2f4d 322d 3134 3434 332d 302f 3120 mp/M2-14443-0/1 │ │ │ │ 00258270: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 00258280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00258290: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ 002582a0: 3a20 7379 6d6c 696e 6b46 696c 6528 702c : symlinkFile(p, │ │ │ │ 002582b0: 7129 2020 2020 2020 2020 2020 7c0a 2b2d q) |.+- │ │ │ │ 002582c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002582d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 002582e0: 7c69 3520 3a20 7020 3c3c 2063 6c6f 7365 |i5 : p << close │ │ │ │ 002582f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00258300: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00258310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00258320: 2020 7c0a 7c6f 3520 3d20 2f74 6d70 2f4d |.|o5 = /tmp/M │ │ │ │ -00258330: 322d 3132 3531 332d 302f 3020 2020 2020 2-12513-0/0 │ │ │ │ +00258330: 322d 3134 3434 332d 302f 3020 2020 2020 2-14443-0/0 │ │ │ │ 00258340: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00258350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00258360: 2020 2020 2020 7c0a 7c6f 3520 3a20 4669 |.|o5 : Fi │ │ │ │ 00258370: 6c65 2020 2020 2020 2020 2020 2020 2020 le │ │ │ │ 00258380: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 00258390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002583a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ 002583b0: 3a20 7265 6164 6c69 6e6b 2071 2020 2020 : readlink q │ │ │ │ 002583c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 002583d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002583e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -002583f0: 7c6f 3620 3d20 2f74 6d70 2f4d 322d 3132 |o6 = /tmp/M2-12 │ │ │ │ -00258400: 3531 332d 302f 3020 2020 2020 2020 2020 513-0/0 │ │ │ │ +002583f0: 7c6f 3620 3d20 2f74 6d70 2f4d 322d 3134 |o6 = /tmp/M2-14 │ │ │ │ +00258400: 3434 332d 302f 3020 2020 2020 2020 2020 443-0/0 │ │ │ │ 00258410: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00258420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00258430: 2d2d 2b0a 7c69 3720 3a20 7265 616c 7061 --+.|i7 : realpa │ │ │ │ 00258440: 7468 2071 2020 2020 2020 2020 2020 2020 th q │ │ │ │ 00258450: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00258460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00258470: 2020 2020 2020 7c0a 7c6f 3720 3d20 2f74 |.|o7 = /t │ │ │ │ -00258480: 6d70 2f4d 322d 3132 3531 332d 302f 3020 mp/M2-12513-0/0 │ │ │ │ +00258480: 6d70 2f4d 322d 3134 3434 332d 302f 3020 mp/M2-14443-0/0 │ │ │ │ 00258490: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 002584a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002584b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 ----------+.|i8 │ │ │ │ 002584c0: 3a20 7265 6d6f 7665 4669 6c65 2070 2020 : removeFile p │ │ │ │ 002584d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 002584e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002584f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ @@ -153690,15 +153690,15 @@ │ │ │ │ 00258590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002585a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002585b0: 2b0a 7c69 3130 203a 2072 6561 6c70 6174 +.|i10 : realpat │ │ │ │ 002585c0: 6820 2222 2020 2020 2020 2020 2020 2020 h "" │ │ │ │ 002585d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 002585e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002585f0: 2020 2020 2020 7c0a 7c6f 3130 203d 202f |.|o10 = / │ │ │ │ -00258600: 746d 702f 4d32 2d31 3038 3232 2d30 2f38 tmp/M2-10822-0/8 │ │ │ │ +00258600: 746d 702f 4d32 2d31 3130 3432 2d30 2f38 tmp/M2-11042-0/8 │ │ │ │ 00258610: 392d 7275 6e64 6972 2f7c 0a2b 2d2d 2d2d 9-rundir/|.+---- │ │ │ │ 00258620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00258630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 ------------+..C │ │ │ │ 00258640: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 4576 aveat.======..Ev │ │ │ │ 00258650: 6572 7920 636f 6d70 6f6e 656e 7420 6f66 ery component of │ │ │ │ 00258660: 2074 6865 2070 6174 6820 6d75 7374 2065 the path must e │ │ │ │ 00258670: 7869 7374 2069 6e20 7468 6520 6669 6c65 xist in the file │ │ │ │ @@ -153856,42 +153856,42 @@ │ │ │ │ 00258ff0: 7220 3d20 7465 6d70 6f72 6172 7946 696c r = temporaryFil │ │ │ │ 00259000: 654e 616d 6528 2920 2020 2020 2020 2020 eName() │ │ │ │ 00259010: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00259020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259050: 2020 2020 7c0a 7c6f 3120 3d20 2f74 6d70 |.|o1 = /tmp │ │ │ │ -00259060: 2f4d 322d 3132 3233 342d 302f 3020 2020 /M2-12234-0/0 │ │ │ │ +00259060: 2f4d 322d 3133 3838 342d 302f 3020 2020 /M2-13884-0/0 │ │ │ │ 00259070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259080: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00259090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002590a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002590b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002590c0: 2d2d 2b0a 7c69 3220 3a20 6d61 6b65 4469 --+.|i2 : makeDi │ │ │ │ 002590d0: 7265 6374 6f72 7920 6469 7220 2020 2020 rectory dir │ │ │ │ 002590e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002590f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00259100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259130: 7c0a 7c6f 3220 3d20 2f74 6d70 2f4d 322d |.|o2 = /tmp/M2- │ │ │ │ -00259140: 3132 3233 342d 302f 3020 2020 2020 2020 12234-0/0 │ │ │ │ +00259140: 3133 3838 342d 302f 3020 2020 2020 2020 13884-0/0 │ │ │ │ 00259150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259160: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00259170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 002591a0: 7c69 3320 3a20 2866 6e20 3d20 6469 7220 |i3 : (fn = dir │ │ │ │ 002591b0: 7c20 222f 2220 7c20 2266 6f6f 2229 203c | "/" | "foo") < │ │ │ │ 002591c0: 3c20 2268 6920 7468 6572 6522 203c 3c20 < "hi there" << │ │ │ │ 002591d0: 636c 6f73 657c 0a7c 2020 2020 2020 2020 close|.| │ │ │ │ 002591e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002591f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259200: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00259210: 3320 3d20 2f74 6d70 2f4d 322d 3132 3233 3 = /tmp/M2-1223 │ │ │ │ +00259210: 3320 3d20 2f74 6d70 2f4d 322d 3133 3838 3 = /tmp/M2-1388 │ │ │ │ 00259220: 342d 302f 302f 666f 6f20 2020 2020 2020 4-0/0/foo │ │ │ │ 00259230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259240: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00259250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259270: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ 00259280: 3a20 4669 6c65 2020 2020 2020 2020 2020 : File │ │ │ │ @@ -153904,15 +153904,15 @@ │ │ │ │ 002592f0: 7265 6164 4469 7265 6374 6f72 7920 6469 readDirectory di │ │ │ │ 00259300: 7220 2020 2020 2020 2020 2020 2020 2020 r │ │ │ │ 00259310: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00259320: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00259330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259350: 2020 2020 2020 7c0a 7c6f 3420 3d20 7b2e |.|o4 = {. │ │ │ │ -00259360: 2c20 2e2e 2c20 666f 6f7d 2020 2020 2020 , .., foo} │ │ │ │ +00259360: 2e2c 202e 2c20 666f 6f7d 2020 2020 2020 ., ., foo} │ │ │ │ 00259370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259380: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00259390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002593a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002593b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002593c0: 2020 2020 7c0a 7c6f 3420 3a20 4c69 7374 |.|o4 : List │ │ │ │ 002593d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -154011,25 +154011,25 @@ │ │ │ │ 002599a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002599b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002599c0: 2d2b 0a7c 6931 203a 2064 6972 203d 2074 -+.|i1 : dir = t │ │ │ │ 002599d0: 656d 706f 7261 7279 4669 6c65 4e61 6d65 emporaryFileName │ │ │ │ 002599e0: 2829 2020 2020 207c 0a7c 2020 2020 2020 () |.| │ │ │ │ 002599f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259a00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00259a10: 6f31 203d 202f 746d 702f 4d32 2d31 3133 o1 = /tmp/M2-113 │ │ │ │ -00259a20: 3931 2d30 2f30 2020 2020 2020 2020 2020 91-0/0 │ │ │ │ +00259a10: 6f31 203d 202f 746d 702f 4d32 2d31 3231 o1 = /tmp/M2-121 │ │ │ │ +00259a20: 3631 2d30 2f30 2020 2020 2020 2020 2020 61-0/0 │ │ │ │ 00259a30: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00259a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259a50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ 00259a60: 206d 616b 6544 6972 6563 746f 7279 2028 makeDirectory ( │ │ │ │ 00259a70: 6469 727c 222f 612f 622f 6322 2920 207c dir|"/a/b/c") | │ │ │ │ 00259a80: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00259a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259aa0: 2020 2020 207c 0a7c 6f32 203d 202f 746d |.|o2 = /tm │ │ │ │ -00259ab0: 702f 4d32 2d31 3133 3931 2d30 2f30 2f61 p/M2-11391-0/0/a │ │ │ │ +00259ab0: 702f 4d32 2d31 3231 3631 2d30 2f30 2f61 p/M2-12161-0/0/a │ │ │ │ 00259ac0: 2f62 2f63 2020 2020 2020 207c 0a2b 2d2d /b/c |.+-- │ │ │ │ 00259ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259af0: 2d2b 0a7c 6933 203a 2072 656d 6f76 6544 -+.|i3 : removeD │ │ │ │ 00259b00: 6972 6563 746f 7279 2028 6469 727c 222f irectory (dir|"/ │ │ │ │ 00259b10: 612f 622f 6322 297c 0a2b 2d2d 2d2d 2d2d a/b/c")|.+------ │ │ │ │ 00259b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -154106,15 +154106,15 @@ │ │ │ │ 00259f90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ 00259fa0: 7020 3d20 7465 6d70 6f72 6172 7946 696c p = temporaryFil │ │ │ │ 00259fb0: 654e 616d 6528 2920 7c20 222f 2220 2020 eName() | "/" │ │ │ │ 00259fc0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00259fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259ff0: 2020 2020 2020 7c0a 7c6f 3120 3d20 2f74 |.|o1 = /t │ │ │ │ -0025a000: 6d70 2f4d 322d 3131 3431 302d 302f 302f mp/M2-11410-0/0/ │ │ │ │ +0025a000: 6d70 2f4d 322d 3132 3230 302d 302f 302f mp/M2-12200-0/0/ │ │ │ │ 0025a010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a020: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0025a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025a050: 2d2d 2d2d 2b0a 7c69 3220 3a20 6d6b 6469 ----+.|i2 : mkdi │ │ │ │ 0025a060: 7220 7020 2020 2020 2020 2020 2020 2020 r p │ │ │ │ 0025a070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -154135,15 +154135,15 @@ │ │ │ │ 0025a160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0025a170: 7c69 3420 3a20 2866 6e20 3d20 7020 7c20 |i4 : (fn = p | │ │ │ │ 0025a180: 2266 6f6f 2229 203c 3c20 2268 6920 7468 "foo") << "hi th │ │ │ │ 0025a190: 6572 6522 203c 3c20 636c 6f73 657c 0a7c ere" << close|.| │ │ │ │ 0025a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a1c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0025a1d0: 3420 3d20 2f74 6d70 2f4d 322d 3131 3431 4 = /tmp/M2-1141 │ │ │ │ +0025a1d0: 3420 3d20 2f74 6d70 2f4d 322d 3132 3230 4 = /tmp/M2-1220 │ │ │ │ 0025a1e0: 302d 302f 302f 666f 6f20 2020 2020 2020 0-0/0/foo │ │ │ │ 0025a1f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0025a200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a220: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ 0025a230: 3a20 4669 6c65 2020 2020 2020 2020 2020 : File │ │ │ │ 0025a240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -154239,40 +154239,40 @@ │ │ │ │ 0025a7e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0025a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0025a800: 0a7c 6931 203a 2064 6972 203d 2074 656d .|i1 : dir = tem │ │ │ │ 0025a810: 706f 7261 7279 4669 6c65 4e61 6d65 2829 poraryFileName() │ │ │ │ 0025a820: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0025a830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a840: 207c 0a7c 6f31 203d 202f 746d 702f 4d32 |.|o1 = /tmp/M2 │ │ │ │ -0025a850: 2d31 3130 3934 2d30 2f30 2020 2020 2020 -11094-0/0 │ │ │ │ +0025a850: 2d31 3135 3634 2d30 2f30 2020 2020 2020 -11564-0/0 │ │ │ │ 0025a860: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0025a870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025a880: 2d2d 2d2b 0a7c 6932 203a 206d 616b 6544 ---+.|i2 : makeD │ │ │ │ 0025a890: 6972 6563 746f 7279 2064 6972 2020 2020 irectory dir │ │ │ │ 0025a8a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0025a8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a8c0: 2020 2020 207c 0a7c 6f32 203d 202f 746d |.|o2 = /tm │ │ │ │ -0025a8d0: 702f 4d32 2d31 3130 3934 2d30 2f30 2020 p/M2-11094-0/0 │ │ │ │ +0025a8d0: 702f 4d32 2d31 3135 3634 2d30 2f30 2020 p/M2-11564-0/0 │ │ │ │ 0025a8e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 0025a8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025a900: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2063 -------+.|i3 : c │ │ │ │ 0025a910: 6861 6e67 6544 6972 6563 746f 7279 2064 hangeDirectory d │ │ │ │ 0025a920: 6972 2020 2020 2020 7c0a 7c20 2020 2020 ir |.| │ │ │ │ 0025a930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a940: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ -0025a950: 202f 746d 702f 4d32 2d31 3130 3934 2d30 /tmp/M2-11094-0 │ │ │ │ +0025a950: 202f 746d 702f 4d32 2d31 3135 3634 2d30 /tmp/M2-11564-0 │ │ │ │ 0025a960: 2f30 2f20 2020 2020 2020 7c0a 2b2d 2d2d /0/ |.+--- │ │ │ │ 0025a970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025a980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ 0025a990: 203a 2063 7572 7265 6e74 4469 7265 6374 : currentDirect │ │ │ │ 0025a9a0: 6f72 7928 2920 2020 2020 2020 7c0a 7c20 ory() |.| │ │ │ │ 0025a9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a9c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0025a9d0: 6f34 203d 202f 746d 702f 4d32 2d31 3130 o4 = /tmp/M2-110 │ │ │ │ -0025a9e0: 3934 2d30 2f30 2f20 2020 2020 2020 7c0a 94-0/0/ |. │ │ │ │ +0025a9d0: 6f34 203d 202f 746d 702f 4d32 2d31 3135 o4 = /tmp/M2-115 │ │ │ │ +0025a9e0: 3634 2d30 2f30 2f20 2020 2020 2020 7c0a 64-0/0/ |. │ │ │ │ 0025a9f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0025aa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0025aa10: 0a0a 4966 2064 6972 2069 7320 6f6d 6974 ..If dir is omit │ │ │ │ 0025aa20: 7465 642c 2074 6865 6e20 7468 6520 6375 ted, then the cu │ │ │ │ 0025aa30: 7272 656e 7420 776f 726b 696e 6720 6469 rrent working di │ │ │ │ 0025aa40: 7265 6374 6f72 7920 6973 2063 6861 6e67 rectory is chang │ │ │ │ 0025aa50: 6564 2074 6f20 7468 6520 7573 6572 2773 ed to the user's │ │ │ │ @@ -154399,16 +154399,16 @@ │ │ │ │ 0025b1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b1f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0025b200: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0025b210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b240: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0025b250: 7c6f 3120 3d20 2f74 6d70 2f4d 322d 3131 |o1 = /tmp/M2-11 │ │ │ │ -0025b260: 3835 342d 302f 302f 2020 2020 2020 2020 854-0/0/ │ │ │ │ +0025b250: 7c6f 3120 3d20 2f74 6d70 2f4d 322d 3133 |o1 = /tmp/M2-13 │ │ │ │ +0025b260: 3130 342d 302f 302f 2020 2020 2020 2020 104-0/0/ │ │ │ │ 0025b270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b290: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 0025b2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -154419,15 +154419,15 @@ │ │ │ │ 0025b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b330: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0025b340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b380: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ -0025b390: 3d20 2f74 6d70 2f4d 322d 3131 3835 342d = /tmp/M2-11854- │ │ │ │ +0025b390: 3d20 2f74 6d70 2f4d 322d 3133 3130 342d = /tmp/M2-13104- │ │ │ │ 0025b3a0: 302f 312f 2020 2020 2020 2020 2020 2020 0/1/ │ │ │ │ 0025b3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b3d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 0025b3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -154439,15 +154439,15 @@ │ │ │ │ 0025b460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b470: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0025b480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b4c0: 2020 2020 2020 7c0a 7c6f 3320 3d20 2f74 |.|o3 = /t │ │ │ │ -0025b4d0: 6d70 2f4d 322d 3131 3835 342d 302f 302f mp/M2-11854-0/0/ │ │ │ │ +0025b4d0: 6d70 2f4d 322d 3133 3130 342d 302f 302f mp/M2-13104-0/0/ │ │ │ │ 0025b4e0: 612f 2020 2020 2020 2020 2020 2020 2020 a/ │ │ │ │ 0025b4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b510: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0025b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -154459,15 +154459,15 @@ │ │ │ │ 0025b5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b5b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0025b5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b600: 2020 7c0a 7c6f 3420 3d20 2f74 6d70 2f4d |.|o4 = /tmp/M │ │ │ │ -0025b610: 322d 3131 3835 342d 302f 302f 622f 2020 2-11854-0/0/b/ │ │ │ │ +0025b610: 322d 3133 3130 342d 302f 302f 622f 2020 2-13104-0/0/b/ │ │ │ │ 0025b620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b650: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0025b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -154478,16 +154478,16 @@ │ │ │ │ 0025b6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b6e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0025b6f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0025b700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b730: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0025b740: 7c6f 3520 3d20 2f74 6d70 2f4d 322d 3131 |o5 = /tmp/M2-11 │ │ │ │ -0025b750: 3835 342d 302f 302f 622f 632f 2020 2020 854-0/0/b/c/ │ │ │ │ +0025b740: 7c6f 3520 3d20 2f74 6d70 2f4d 322d 3133 |o5 = /tmp/M2-13 │ │ │ │ +0025b750: 3130 342d 302f 302f 622f 632f 2020 2020 104-0/0/b/c/ │ │ │ │ 0025b760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b780: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 0025b790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -154498,15 +154498,15 @@ │ │ │ │ 0025b810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b820: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0025b830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b870: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ -0025b880: 3d20 2f74 6d70 2f4d 322d 3131 3835 342d = /tmp/M2-11854- │ │ │ │ +0025b880: 3d20 2f74 6d70 2f4d 322d 3133 3130 342d = /tmp/M2-13104- │ │ │ │ 0025b890: 302f 302f 612f 6620 2020 2020 2020 2020 0/0/a/f │ │ │ │ 0025b8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b8c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0025b8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -154528,15 +154528,15 @@ │ │ │ │ 0025b9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ba00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0025ba10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ba20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ba30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ba40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ba50: 2020 2020 7c0a 7c6f 3720 3d20 2f74 6d70 |.|o7 = /tmp │ │ │ │ -0025ba60: 2f4d 322d 3131 3835 342d 302f 302f 612f /M2-11854-0/0/a/ │ │ │ │ +0025ba60: 2f4d 322d 3133 3130 342d 302f 302f 612f /M2-13104-0/0/a/ │ │ │ │ 0025ba70: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ 0025ba80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ba90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025baa0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0025bab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -154557,16 +154557,16 @@ │ │ │ │ 0025bbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bbd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0025bbe0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0025bbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bc20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0025bc30: 7c6f 3820 3d20 2f74 6d70 2f4d 322d 3131 |o8 = /tmp/M2-11 │ │ │ │ -0025bc40: 3835 342d 302f 302f 622f 632f 6720 2020 854-0/0/b/c/g │ │ │ │ +0025bc30: 7c6f 3820 3d20 2f74 6d70 2f4d 322d 3133 |o8 = /tmp/M2-13 │ │ │ │ +0025bc40: 3130 342d 302f 302f 622f 632f 6720 2020 104-0/0/b/c/g │ │ │ │ 0025bc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bc70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0025bc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -154587,98 +154587,98 @@ │ │ │ │ 0025bda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bdb0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0025bdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025be00: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20 |.|o9 = │ │ │ │ -0025be10: 2f74 6d70 2f4d 322d 3131 3835 342d 302f /tmp/M2-11854-0/ │ │ │ │ +0025be10: 2f74 6d70 2f4d 322d 3133 3130 342d 302f /tmp/M2-13104-0/ │ │ │ │ 0025be20: 302f 2020 2020 2020 2020 2020 2020 2020 0/ │ │ │ │ 0025be30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025be40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025be50: 2020 2020 2020 207c 0a7c 2020 2020 202f |.| / │ │ │ │ -0025be60: 746d 702f 4d32 2d31 3138 3534 2d30 2f30 tmp/M2-11854-0/0 │ │ │ │ -0025be70: 2f62 2f20 2020 2020 2020 2020 2020 2020 /b/ │ │ │ │ +0025be60: 746d 702f 4d32 2d31 3331 3034 2d30 2f30 tmp/M2-13104-0/0 │ │ │ │ +0025be70: 2f61 2f20 2020 2020 2020 2020 2020 2020 /a/ │ │ │ │ 0025be80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025be90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bea0: 2020 2020 2020 7c0a 7c20 2020 2020 2f74 |.| /t │ │ │ │ -0025beb0: 6d70 2f4d 322d 3131 3835 342d 302f 302f mp/M2-11854-0/0/ │ │ │ │ -0025bec0: 622f 632f 2020 2020 2020 2020 2020 2020 b/c/ │ │ │ │ +0025beb0: 6d70 2f4d 322d 3133 3130 342d 302f 302f mp/M2-13104-0/0/ │ │ │ │ +0025bec0: 612f 6720 2020 2020 2020 2020 2020 2020 a/g │ │ │ │ 0025bed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bef0: 2020 2020 207c 0a7c 2020 2020 202f 746d |.| /tm │ │ │ │ -0025bf00: 702f 4d32 2d31 3138 3534 2d30 2f30 2f62 p/M2-11854-0/0/b │ │ │ │ -0025bf10: 2f63 2f67 2020 2020 2020 2020 2020 2020 /c/g │ │ │ │ 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│ │ 0025c600: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0025c610: 7c20 2020 2020 202f 746d 702f 4d32 2d31 | /tmp/M2-1 │ │ │ │ -0025c620: 3138 3534 2d30 2f31 2f62 2f63 2f67 2020 1854-0/1/b/c/g │ │ │ │ +0025c620: 3331 3034 2d30 2f31 2f62 2f63 2f67 2020 3104-0/1/b/c/g │ │ │ │ 0025c630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025c640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025c650: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 0025c660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025c670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025c680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025c690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -154865,31 +154865,31 @@ │ │ │ │ 0025cf00: 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ====..+--------- │ │ │ │ 0025cf10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025cf20: 2d2d 2d2d 2d2b 0a7c 6931 203a 2064 6972 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2020 │ │ │ │ 0025d0d0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0025d0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -155056,30 +155056,30 @@ │ │ │ │ 0025daf0: 654e 616d 6528 2920 7c20 222f 2220 2020 eName() | "/" │ │ │ │ 0025db00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025db10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0025db20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025db30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025db40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025db50: 2020 7c0a 7c6f 3120 3d20 2f74 6d70 2f4d |.|o1 = /tmp/M │ │ │ │ -0025db60: 322d 3131 3831 362d 302f 302f 2020 2020 2-11816-0/0/ │ │ │ │ +0025db60: 322d 3133 3032 362d 302f 302f 2020 2020 2-13026-0/0/ │ │ │ │ 0025db70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025db80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025db90: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0025dba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025dbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025dbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0025dbd0: 7c69 3220 3a20 6473 7420 3d20 7465 6d70 |i2 : dst = temp │ │ │ │ 0025dbe0: 6f72 6172 7946 696c 654e 616d 6528 2920 oraryFileName() │ │ │ │ 0025dbf0: 7c20 222f 2220 2020 2020 2020 2020 2020 | "/" │ │ │ │ 0025dc00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0025dc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dc40: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ -0025dc50: 3d20 2f74 6d70 2f4d 322d 3131 3831 362d = /tmp/M2-11816- │ │ │ │ +0025dc50: 3d20 2f74 6d70 2f4d 322d 3133 3032 362d = /tmp/M2-13026- │ │ │ │ 0025dc60: 302f 312f 2020 2020 2020 2020 2020 2020 0/1/ │ │ │ │ 0025dc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dc80: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 0025dc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025dca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025dcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025dcc0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 6d61 ------+.|i3 : ma │ │ │ │ @@ -155087,30 +155087,30 @@ │ │ │ │ 0025dce0: 7c22 612f 2229 2020 2020 2020 2020 2020 |"a/") │ │ │ │ 0025dcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dd00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0025dd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dd40: 2020 7c0a 7c6f 3320 3d20 2f74 6d70 2f4d |.|o3 = /tmp/M │ │ │ │ -0025dd50: 322d 3131 3831 362d 302f 302f 612f 2020 2-11816-0/0/a/ │ │ │ │ +0025dd50: 322d 3133 3032 362d 302f 302f 612f 2020 2-13026-0/0/a/ │ │ │ │ 0025dd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dd80: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0025dd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025dda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025ddb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0025ddc0: 7c69 3420 3a20 6d61 6b65 4469 7265 6374 |i4 : makeDirect │ │ │ │ 0025ddd0: 6f72 7920 2873 7263 7c22 622f 2229 2020 ory (src|"b/") │ │ │ │ 0025dde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ddf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0025de00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025de10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025de20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025de30: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ -0025de40: 3d20 2f74 6d70 2f4d 322d 3131 3831 362d = /tmp/M2-11816- │ │ │ │ +0025de40: 3d20 2f74 6d70 2f4d 322d 3133 3032 362d = /tmp/M2-13026- │ │ │ │ 0025de50: 302f 302f 622f 2020 2020 2020 2020 2020 0/0/b/ │ │ │ │ 0025de60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025de70: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 0025de80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025de90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025dea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025deb0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 6d61 ------+.|i5 : ma │ │ │ │ @@ -155118,30 +155118,30 @@ │ │ │ │ 0025ded0: 7c22 622f 632f 2229 2020 2020 2020 2020 |"b/c/") │ │ │ │ 0025dee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025def0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0025df00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025df10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025df30: 2020 7c0a 7c6f 3520 3d20 2f74 6d70 2f4d |.|o5 = /tmp/M │ │ │ │ -0025df40: 322d 3131 3831 362d 302f 302f 622f 632f 2-11816-0/0/b/c/ │ │ │ │ +0025df40: 322d 3133 3032 362d 302f 302f 622f 632f 2-13026-0/0/b/c/ │ │ │ │ 0025df50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025df60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025df70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0025df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0025dfb0: 7c69 3620 3a20 7372 637c 2261 2f66 2220 |i6 : src|"a/f" │ │ │ │ 0025dfc0: 3c3c 2022 6869 2074 6865 7265 2220 3c3c << "hi there" << │ │ │ │ 0025dfd0: 2063 6c6f 7365 2020 2020 2020 2020 2020 close │ │ │ │ 0025dfe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0025dff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e020: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ -0025e030: 3d20 2f74 6d70 2f4d 322d 3131 3831 362d = /tmp/M2-11816- │ │ │ │ +0025e030: 3d20 2f74 6d70 2f4d 322d 3133 3032 362d = /tmp/M2-13026- │ │ │ │ 0025e040: 302f 302f 612f 6620 2020 2020 2020 2020 0/0/a/f │ │ │ │ 0025e050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e060: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0025e070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e0a0: 2020 2020 2020 7c0a 7c6f 3620 3a20 4669 |.|o6 : Fi │ │ │ │ @@ -155156,16 +155156,16 @@ │ │ │ │ 0025e130: 2f67 2220 3c3c 2022 6869 2074 6865 7265 /g" << "hi there │ │ │ │ 0025e140: 2220 3c3c 2063 6c6f 7365 2020 2020 2020 " << close │ │ │ │ 0025e150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e160: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0025e170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e190: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0025e1a0: 7c6f 3720 3d20 2f74 6d70 2f4d 322d 3131 |o7 = /tmp/M2-11 │ │ │ │ -0025e1b0: 3831 362d 302f 302f 612f 6720 2020 2020 816-0/0/a/g │ │ │ │ +0025e1a0: 7c6f 3720 3d20 2f74 6d70 2f4d 322d 3133 |o7 = /tmp/M2-13 │ │ │ │ +0025e1b0: 3032 362d 302f 302f 612f 6720 2020 2020 026-0/0/a/g │ │ │ │ 0025e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e1d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0025e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e210: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ 0025e220: 3a20 4669 6c65 2020 2020 2020 2020 2020 : File │ │ │ │ @@ -155180,15 +155180,15 @@ │ │ │ │ 0025e2b0: 2074 6865 7265 2220 3c3c 2063 6c6f 7365 there" << close │ │ │ │ 0025e2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e2d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0025e2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e310: 2020 7c0a 7c6f 3820 3d20 2f74 6d70 2f4d |.|o8 = /tmp/M │ │ │ │ -0025e320: 322d 3131 3831 362d 302f 302f 622f 632f 2-11816-0/0/b/c/ │ │ │ │ +0025e320: 322d 3133 3032 362d 302f 302f 622f 632f 2-13026-0/0/b/c/ │ │ │ │ 0025e330: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ 0025e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e350: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0025e360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e380: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0025e390: 7c6f 3820 3a20 4669 6c65 2020 2020 2020 |o8 : File │ │ │ │ @@ -155199,25 +155199,25 @@ │ │ │ │ 0025e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 ----------+.|i9 │ │ │ │ 0025e410: 3a20 7379 6d6c 696e 6b44 6972 6563 746f : symlinkDirecto │ │ │ │ 0025e420: 7279 2873 7263 2c64 7374 2c56 6572 626f ry(src,dst,Verbo │ │ │ │ 0025e430: 7365 3d3e 7472 7565 2920 2020 2020 2020 se=>true) │ │ │ │ 0025e440: 2020 2020 2020 2020 7c0a 7c2d 2d73 796d |.|--sym │ │ │ │ -0025e450: 6c69 6e6b 696e 673a 202e 2e2f 2e2e 2f2e linking: ../../. │ │ │ │ -0025e460: 2e2f 302f 622f 632f 6720 2d3e 202f 746d ./0/b/c/g -> /tm │ │ │ │ -0025e470: 702f 4d32 2d31 3138 3136 2d30 2f31 2f62 p/M2-11816-0/1/b │ │ │ │ -0025e480: 2f63 2f67 2020 7c0a 7c2d 2d73 796d 6c69 /c/g |.|--symli │ │ │ │ +0025e450: 6c69 6e6b 696e 673a 202e 2e2f 2e2e 2f30 linking: ../../0 │ │ │ │ +0025e460: 2f61 2f67 202d 3e20 2f74 6d70 2f4d 322d /a/g -> /tmp/M2- │ │ │ │ +0025e470: 3133 3032 362d 302f 312f 612f 6720 2020 13026-0/1/a/g │ │ │ │ +0025e480: 2020 2020 2020 7c0a 7c2d 2d73 796d 6c69 |.|--symli │ │ │ │ 0025e490: 6e6b 696e 673a 202e 2e2f 2e2e 2f30 2f61 nking: ../../0/a │ │ │ │ -0025e4a0: 2f67 202d 3e20 2f74 6d70 2f4d 322d 3131 /g -> /tmp/M2-11 │ │ │ │ -0025e4b0: 3831 362d 302f 312f 612f 6720 2020 2020 816-0/1/a/g │ │ │ │ +0025e4a0: 2f66 202d 3e20 2f74 6d70 2f4d 322d 3133 /f -> /tmp/M2-13 │ │ │ │ +0025e4b0: 3032 362d 302f 312f 612f 6620 2020 2020 026-0/1/a/f │ │ │ │ 0025e4c0: 2020 2020 7c0a 7c2d 2d73 796d 6c69 6e6b |.|--symlink │ │ │ │ -0025e4d0: 696e 673a 202e 2e2f 2e2e 2f30 2f61 2f66 ing: ../../0/a/f │ │ │ │ -0025e4e0: 202d 3e20 2f74 6d70 2f4d 322d 3131 3831 -> /tmp/M2-1181 │ │ │ │ -0025e4f0: 362d 302f 312f 612f 6620 2020 2020 2020 6-0/1/a/f │ │ │ │ +0025e4d0: 696e 673a 202e 2e2f 2e2e 2f2e 2e2f 302f ing: ../../../0/ │ │ │ │ +0025e4e0: 622f 632f 6720 2d3e 202f 746d 702f 4d32 b/c/g -> /tmp/M2 │ │ │ │ +0025e4f0: 2d31 3330 3236 2d30 2f31 2f62 2f63 2f67 -13026-0/1/b/c/g │ │ │ │ 0025e500: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0025e510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025e520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025e530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025e540: 2b0a 7c69 3130 203a 2067 6574 2028 6473 +.|i10 : get (ds │ │ │ │ 0025e550: 747c 2262 2f63 2f67 2229 2020 2020 2020 t|"b/c/g") │ │ │ │ 0025e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -155234,25 +155234,25 @@ │ │ │ │ 0025e610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025e620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025e630: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a --------+.|i11 : │ │ │ │ 0025e640: 2073 796d 6c69 6e6b 4469 7265 6374 6f72 symlinkDirector │ │ │ │ 0025e650: 7928 7372 632c 6473 742c 5665 7262 6f73 y(src,dst,Verbos │ │ │ │ 0025e660: 653d 3e74 7275 652c 556e 646f 3d3e 7472 e=>true,Undo=>tr │ │ │ │ 0025e670: 7565 2920 2020 7c0a 7c2d 2d75 6e73 796d ue) |.|--unsym │ │ │ │ -0025e680: 6c69 6e6b 696e 673a 202e 2e2f 2e2e 2f2e linking: ../../. │ │ │ │ -0025e690: 2e2f 302f 622f 632f 6720 2d3e 202f 746d ./0/b/c/g -> /tm │ │ │ │ -0025e6a0: 702f 4d32 2d31 3138 3136 2d30 2f31 2f62 p/M2-11816-0/1/b │ │ │ │ -0025e6b0: 2f63 2f67 7c0a 7c2d 2d75 6e73 796d 6c69 /c/g|.|--unsymli │ │ │ │ +0025e680: 6c69 6e6b 696e 673a 202e 2e2f 2e2e 2f30 linking: ../../0 │ │ │ │ +0025e690: 2f61 2f67 202d 3e20 2f74 6d70 2f4d 322d /a/g -> /tmp/M2- │ │ │ │ +0025e6a0: 3133 3032 362d 302f 312f 612f 6720 2020 13026-0/1/a/g │ │ │ │ +0025e6b0: 2020 2020 7c0a 7c2d 2d75 6e73 796d 6c69 |.|--unsymli │ │ │ │ 0025e6c0: 6e6b 696e 673a 202e 2e2f 2e2e 2f30 2f61 nking: ../../0/a │ │ │ │ -0025e6d0: 2f67 202d 3e20 2f74 6d70 2f4d 322d 3131 /g -> /tmp/M2-11 │ │ │ │ -0025e6e0: 3831 362d 302f 312f 612f 6720 2020 2020 816-0/1/a/g │ │ │ │ +0025e6d0: 2f66 202d 3e20 2f74 6d70 2f4d 322d 3133 /f -> /tmp/M2-13 │ │ │ │ +0025e6e0: 3032 362d 302f 312f 612f 6620 2020 2020 026-0/1/a/f │ │ │ │ 0025e6f0: 2020 7c0a 7c2d 2d75 6e73 796d 6c69 6e6b |.|--unsymlink │ │ │ │ -0025e700: 696e 673a 202e 2e2f 2e2e 2f30 2f61 2f66 ing: ../../0/a/f │ │ │ │ -0025e710: 202d 3e20 2f74 6d70 2f4d 322d 3131 3831 -> /tmp/M2-1181 │ │ │ │ -0025e720: 362d 302f 312f 612f 6620 2020 2020 2020 6-0/1/a/f │ │ │ │ +0025e700: 696e 673a 202e 2e2f 2e2e 2f2e 2e2f 302f ing: ../../../0/ │ │ │ │ +0025e710: 622f 632f 6720 2d3e 202f 746d 702f 4d32 b/c/g -> /tmp/M2 │ │ │ │ +0025e720: 2d31 3330 3236 2d30 2f31 2f62 2f63 2f67 -13026-0/1/b/c/g │ │ │ │ 0025e730: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0025e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025e750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0025e770: 4e6f 7720 7765 2072 656d 6f76 6520 7468 Now we remove th │ │ │ │ 0025e780: 6520 6669 6c65 7320 616e 6420 6469 7265 e files and dire │ │ │ │ 0025e790: 6374 6f72 6965 7320 7765 2063 7265 6174 ctories we creat │ │ │ │ @@ -155375,36 +155375,36 @@ │ │ │ │ 0025eee0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ 0025eef0: 2072 756e 2022 756e 616d 6520 2d61 2220 run "uname -a" │ │ │ │ 0025ef00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ef10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ef20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ef30: 2020 2020 2020 2020 207c 0a7c 4c69 6e75 |.|Linu │ │ │ │ 0025ef40: 7820 7362 7569 6c64 2036 2e31 322e 3838 x sbuild 6.12.88 │ │ │ │ -0025ef50: 2b64 6562 3133 2d61 6d64 3634 2023 3120 +deb13-amd64 #1 │ │ │ │ -0025ef60: 534d 5020 5052 4545 4d50 545f 4459 4e41 SMP PREEMPT_DYNA │ │ │ │ -0025ef70: 4d49 4320 4465 6269 616e 2036 2e31 322e MIC Debian 6.12. │ │ │ │ -0025ef80: 3838 2d31 2020 2020 207c 0a7c 2020 2020 88-1 |.| │ │ │ │ +0025ef50: 2b64 6562 3133 2d63 6c6f 7564 2d61 6d64 +deb13-cloud-amd │ │ │ │ +0025ef60: 3634 2023 3120 534d 5020 5052 4545 4d50 64 #1 SMP PREEMP │ │ │ │ +0025ef70: 545f 4459 4e41 4d49 4320 4465 6269 616e T_DYNAMIC Debian │ │ │ │ +0025ef80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0025ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025efa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025efb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025efc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025efd0: 2020 2020 2020 2020 207c 0a7c 6f31 203d |.|o1 = │ │ │ │ 0025efe0: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ 0025eff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025f000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025f010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025f020: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d |.|---- │ │ │ │ 0025f030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0025f070: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2832 3032 ---------|.|(202 │ │ │ │ -0025f080: 362d 3035 2d31 3529 2078 3836 5f36 3420 6-05-15) x86_64 │ │ │ │ -0025f090: 474e 552f 4c69 6e75 7820 2020 2020 2020 GNU/Linux │ │ │ │ -0025f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025f070: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 362e 3132 ---------|.|6.12 │ │ │ │ +0025f080: 2e38 382d 3120 2832 3032 362d 3035 2d31 .88-1 (2026-05-1 │ │ │ │ +0025f090: 3529 2078 3836 5f36 3420 474e 552f 4c69 5) x86_64 GNU/Li │ │ │ │ +0025f0a0: 6e75 7820 2020 2020 2020 2020 2020 2020 nux │ │ │ │ 0025f0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025f0c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 0025f0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f110: 2d2d 2d2d 2d2d 2d2d 2d2b 0a54 6f20 7275 ---------+.To ru │ │ │ │ @@ -155476,294318 +155476,294304 @@ │ │ │ │ 0025f530: 2073 6565 0a77 6865 7468 6572 2069 7420 see.whether it │ │ │ │ 0025f540: 696e 636c 7564 6573 2061 206e 6577 6c69 includes a newli │ │ │ │ 0025f550: 6e65 2063 6861 7261 6374 6572 2e0a 2b2d ne character..+- │ │ │ │ 0025f560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0025f5a0: 2d2d 2d2b 0a7c 6933 203a 2070 6565 6b20 ---+.|i3 : peek │ │ │ │ -0025f5b0: 6765 7420 2221 756e 616d 6520 2d61 2220 get "!uname -a" │ │ │ │ -0025f5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025f5a0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ +0025f5b0: 2070 6565 6b20 6765 7420 2221 756e 616d peek get "!unam │ │ │ │ +0025f5c0: 6520 2d61 2220 2020 2020 2020 2020 2020 e -a" │ │ │ │ 0025f5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025f5e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0025f5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025f5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025f5f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0025f600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025f610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025f620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025f630: 207c 0a7c 6f33 203d 2022 4c69 6e75 7820 |.|o3 = "Linux │ │ │ │ -0025f640: 7362 7569 6c64 2036 2e31 322e 3838 2b64 sbuild 6.12.88+d │ │ │ │ -0025f650: 6562 3133 2d61 6d64 3634 2023 3120 534d eb13-amd64 #1 SM │ │ │ │ -0025f660: 5020 5052 4545 4d50 545f 4459 4e41 4d49 P PREEMPT_DYNAMI │ │ │ │ -0025f670: 4320 4465 6269 616e 7c0a 7c20 2020 2020 C Debian|.| │ │ │ │ -0025f680: 362e 3132 2e38 382d 3120 2832 3032 362d 6.12.88-1 (2026- │ │ │ │ -0025f690: 3035 2d31 3529 2078 3836 5f36 3420 474e 05-15) x86_64 GN │ │ │ │ -0025f6a0: 552f 4c69 6e75 785c 6e22 2020 2020 2020 U/Linux\n" │ │ │ │ -0025f6b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0025f6c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0025f6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0025f630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025f640: 2020 207c 0a7c 6f33 203d 2022 4c69 6e75 |.|o3 = "Linu │ │ │ │ +0025f650: 7820 7362 7569 6c64 2036 2e31 322e 3838 x sbuild 6.12.88 │ │ │ │ +0025f660: 2b64 6562 3133 2d63 6c6f 7564 2d61 6d64 +deb13-cloud-amd │ │ │ │ +0025f670: 3634 2023 3120 534d 5020 5052 4545 4d50 64 #1 SMP PREEMP │ │ │ │ +0025f680: 545f 4459 4e41 4d49 4320 4465 6269 616e T_DYNAMIC Debian │ │ │ │ +0025f690: 7c0a 7c20 2020 2020 362e 3132 2e38 382d |.| 6.12.88- │ │ │ │ +0025f6a0: 3120 2832 3032 362d 3035 2d31 3529 2078 1 (2026-05-15) x │ │ │ │ +0025f6b0: 3836 5f36 3420 474e 552f 4c69 6e75 785c 86_64 GNU/Linux\ │ │ │ │ +0025f6c0: 6e22 2020 2020 2020 2020 2020 2020 2020 n" │ │ │ │ +0025f6d0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 0025f6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0025f700: 2d2d 2d2d 2d2d 2b0a 4269 6469 7265 6374 ------+.Bidirect │ │ │ │ -0025f710: 696f 6e61 6c20 636f 6d6d 756e 6963 6174 ional communicat │ │ │ │ -0025f720: 696f 6e20 7769 7468 2061 2070 726f 6772 ion with a progr │ │ │ │ -0025f730: 616d 2069 7320 616c 736f 2070 6f73 7369 am is also possi │ │ │ │ -0025f740: 626c 652e 2020 5765 2075 7365 202a 6e6f ble. We use *no │ │ │ │ -0025f750: 7465 0a6f 7065 6e49 6e4f 7574 3a20 6f70 te.openInOut: op │ │ │ │ -0025f760: 656e 496e 4f75 742c 2074 6f20 6372 6561 enInOut, to crea │ │ │ │ -0025f770: 7465 2061 2066 696c 6520 7468 6174 2073 te a file that s │ │ │ │ -0025f780: 6572 7665 7320 6173 2061 2062 6964 6972 erves as a bidir │ │ │ │ -0025f790: 6563 7469 6f6e 616c 0a63 6f6e 6e65 6374 ectional.connect │ │ │ │ -0025f7a0: 696f 6e20 746f 2061 2070 726f 6772 616d ion to a program │ │ │ │ -0025f7b0: 2e20 2054 6861 7420 6669 6c65 2069 7320 . That file is │ │ │ │ -0025f7c0: 6361 6c6c 6564 2061 6e20 696e 7075 7420 called an input │ │ │ │ -0025f7d0: 6f75 7470 7574 2066 696c 652e 2020 496e output file. In │ │ │ │ -0025f7e0: 2074 6869 730a 6578 616d 706c 6520 7765 this.example we │ │ │ │ -0025f7f0: 206f 7065 6e20 6120 636f 6e6e 6563 7469 open a connecti │ │ │ │ -0025f800: 6f6e 2074 6f20 7468 6520 756e 6978 2075 on to the unix u │ │ │ │ -0025f810: 7469 6c69 7479 2067 7265 7020 616e 6420 tility grep and │ │ │ │ -0025f820: 7573 6520 6974 2074 6f20 6c6f 6361 7465 use it to locate │ │ │ │ -0025f830: 2074 6865 0a73 796d 626f 6c20 6e61 6d65 the.symbol name │ │ │ │ -0025f840: 7320 696e 204d 6163 6175 6c61 7932 2074 s in Macaulay2 t │ │ │ │ -0025f850: 6861 7420 6265 6769 6e20 7769 7468 2069 hat begin with i │ │ │ │ -0025f860: 6e2e 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d n..+------------ │ │ │ │ -0025f870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0025f880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0025f890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0025f8a0: 3420 3a20 6620 3d20 6f70 656e 496e 4f75 4 : f = openInOu │ │ │ │ -0025f8b0: 7420 2221 6772 6570 202d 4520 275e 696e t "!grep -E '^in │ │ │ │ -0025f8c0: 2722 2020 2020 2020 2020 2020 2020 2020 '" │ │ │ │ -0025f8d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0025f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025f8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0025f710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0025f720: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 4269 6469 ----------+.Bidi │ │ │ │ +0025f730: 7265 6374 696f 6e61 6c20 636f 6d6d 756e rectional commun │ │ │ │ +0025f740: 6963 6174 696f 6e20 7769 7468 2061 2070 ication with a p │ │ │ │ +0025f750: 726f 6772 616d 2069 7320 616c 736f 2070 rogram is also p │ │ │ │ +0025f760: 6f73 7369 626c 652e 2020 5765 2075 7365 ossible. We use │ │ │ │ +0025f770: 202a 6e6f 7465 0a6f 7065 6e49 6e4f 7574 *note.openInOut │ │ │ │ +0025f780: 3a20 6f70 656e 496e 4f75 742c 2074 6f20 : openInOut, to │ │ │ │ +0025f790: 6372 6561 7465 2061 2066 696c 6520 7468 create a file th │ │ │ │ +0025f7a0: 6174 2073 6572 7665 7320 6173 2061 2062 at serves as a b │ │ │ │ +0025f7b0: 6964 6972 6563 7469 6f6e 616c 0a63 6f6e idirectional.con │ │ │ │ +0025f7c0: 6e65 6374 696f 6e20 746f 2061 2070 726f nection to a pro │ │ │ │ +0025f7d0: 6772 616d 2e20 2054 6861 7420 6669 6c65 gram. That file │ │ │ │ +0025f7e0: 2069 7320 6361 6c6c 6564 2061 6e20 696e is called an in │ │ │ │ +0025f7f0: 7075 7420 6f75 7470 7574 2066 696c 652e put output file. │ │ │ │ +0025f800: 2020 496e 2074 6869 730a 6578 616d 706c In this.exampl │ │ │ │ +0025f810: 6520 7765 206f 7065 6e20 6120 636f 6e6e e we open a conn │ │ │ │ +0025f820: 6563 7469 6f6e 2074 6f20 7468 6520 756e ection to the un │ │ │ │ +0025f830: 6978 2075 7469 6c69 7479 2067 7265 7020 ix utility grep │ │ │ │ +0025f840: 616e 6420 7573 6520 6974 2074 6f20 6c6f and use it to lo │ │ │ │ +0025f850: 6361 7465 2074 6865 0a73 796d 626f 6c20 cate the.symbol │ │ │ │ +0025f860: 6e61 6d65 7320 696e 204d 6163 6175 6c61 names in Macaula │ │ │ │ +0025f870: 7932 2074 6861 7420 6265 6769 6e20 7769 y2 that begin wi │ │ │ │ +0025f880: 7468 2069 6e2e 0a2b 2d2d 2d2d 2d2d 2d2d th in..+-------- │ │ │ │ +0025f890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0025f8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0025f8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0025f8c0: 2b0a 7c69 3420 3a20 6620 3d20 6f70 656e +.|i4 : f = open │ │ │ │ +0025f8d0: 496e 4f75 7420 2221 6772 6570 202d 4520 InOut "!grep -E │ │ │ │ +0025f8e0: 275e 696e 2722 2020 2020 2020 2020 2020 '^in'" │ │ │ │ +0025f8f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0025f900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025f910: 2020 7c0a 7c6f 3420 3d20 2167 7265 7020 |.|o4 = !grep │ │ │ │ -0025f920: 2d45 2027 5e69 6e27 2020 2020 2020 2020 -E '^in' │ │ │ │ -0025f930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025f940: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0025f910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025f920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025f930: 2020 2020 2020 7c0a 7c6f 3420 3d20 2167 |.|o4 = !g │ │ │ │ +0025f940: 7265 7020 2d45 2027 5e69 6e27 2020 2020 rep -E '^in' │ │ │ │ 0025f950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025f960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025f970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025f980: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ -0025f990: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ -0025f9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025f9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025f9c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0025f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0025f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0025f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0025fa00: 7c69 3520 3a20 7363 616e 286b 6579 7320 |i5 : scan(keys │ │ │ │ -0025fa10: 436f 7265 2e44 6963 7469 6f6e 6172 792c Core.Dictionary, │ │ │ │ -0025fa20: 206b 6579 202d 3e20 6620 3c3c 206b 6579 key -> f << key │ │ │ │ -0025fa30: 203c 3c20 656e 646c 297c 0a2b 2d2d 2d2d << endl)|.+---- │ │ │ │ -0025fa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0025fa50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0025f970: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0025f980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025f990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025f9a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0025f9b0: 3420 3a20 4669 6c65 2020 2020 2020 2020 4 : File │ │ │ │ +0025f9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025f9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025f9e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0025f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0025fa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0025fa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0025fa20: 2d2d 2b0a 7c69 3520 3a20 7363 616e 286b --+.|i5 : scan(k │ │ │ │ +0025fa30: 6579 7320 436f 7265 2e44 6963 7469 6f6e eys Core.Diction │ │ │ │ +0025fa40: 6172 792c 206b 6579 202d 3e20 6620 3c3c ary, key -> f << │ │ │ │ +0025fa50: 206b 6579 203c 3c20 656e 646c 297c 0a2b key << endl)|.+ │ │ │ │ 0025fa60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0025fa70: 2d2d 2d2d 2b0a 7c69 3620 3a20 6620 3c3c ----+.|i6 : f << │ │ │ │ -0025fa80: 2063 6c6f 7365 4f75 7420 2020 2020 2020 closeOut │ │ │ │ -0025fa90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025faa0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0025fab0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0025fa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0025fa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0025fa90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ +0025faa0: 6620 3c3c 2063 6c6f 7365 4f75 7420 2020 f << closeOut │ │ │ │ +0025fab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025fac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025fad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025fae0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ -0025faf0: 3d20 2167 7265 7020 2d45 2027 5e69 6e27 = !grep -E '^in' │ │ │ │ -0025fb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025fb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025fb20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0025fad0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0025fae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025faf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025fb00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0025fb10: 7c6f 3620 3d20 2167 7265 7020 2d45 2027 |o6 = !grep -E ' │ │ │ │ +0025fb20: 5e69 6e27 2020 2020 2020 2020 2020 2020 ^in' │ │ │ │ 0025fb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025fb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0025fb40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0025fb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025fb60: 7c0a 7c6f 3620 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In thi │ │ │ │ -00260470: 7320 6578 616d 706c 6520 7765 2073 7563 s example we suc │ │ │ │ -00260480: 6365 6564 6564 2062 6563 6175 7365 0a74 ceeded because.t │ │ │ │ -00260490: 6865 2065 6e74 6972 6520 6f75 7470 7574 he entire output │ │ │ │ -002604a0: 2066 726f 6d20 6772 6570 2077 6173 2073 from grep was s │ │ │ │ -002604b0: 6d61 6c6c 6572 2074 6861 6e20 3430 3936 maller than 4096 │ │ │ │ -002604c0: 2062 7974 6573 2e20 2049 6e20 6765 6e65 bytes. In gene │ │ │ │ -002604d0: 7261 6c2c 206f 6e65 0a73 686f 756c 6420 ral, one.should │ │ │ │ -002604e0: 6265 2063 6172 6566 756c 2074 6f20 6172 be careful to ar │ │ │ │ -002604f0: 7261 6e67 6520 7468 696e 6773 2073 6f20 range things so │ │ │ │ -00260500: 7468 6174 2074 6865 2074 776f 2070 726f that the two pro │ │ │ │ -00260510: 6772 616d 7320 7461 6b65 2074 7572 6e73 grams take turns │ │ │ │ -00260520: 2075 7369 6e67 0a74 6865 2063 6f6d 6d75 using.the commu │ │ │ │ -00260530: 6e69 6361 7469 6f6e 2063 6861 6e6e 656c nication channel │ │ │ │ -00260540: 2c20 736f 2074 6861 7420 7768 656e 206f , so that when o │ │ │ │ -00260550: 6e65 2069 7320 7772 6974 696e 6720 6461 ne is writing da │ │ │ │ -00260560: 7461 2c20 7468 6520 6f74 6865 7220 6973 ta, the other is │ │ │ │ -00260570: 0a72 6561 6469 6e67 2069 742e 0a0a 0a41 .reading it....A │ │ │ │ -00260580: 2075 7365 6675 6c20 6675 6e63 7469 6f6e useful function │ │ │ │ -00260590: 2069 6e20 7468 6973 2063 6f6e 6e65 6374 in this connect │ │ │ │ -002605a0: 696f 6e20 6973 202a 6e6f 7465 2069 7352 ion is *note isR │ │ │ │ -002605b0: 6561 6479 3a20 6973 5265 6164 795f 6c70 eady: isReady_lp │ │ │ │ -002605c0: 4669 6c65 5f72 702c 2c0a 7768 6963 6820 File_rp,,.which │ │ │ │ -002605d0: 7769 6c6c 2074 656c 6c20 796f 7520 7768 will tell you wh │ │ │ │ -002605e0: 6574 6865 7220 616e 2069 6e70 7574 2066 ether an input f │ │ │ │ -002605f0: 696c 6520 6861 7320 616e 7920 696e 7075 ile has any inpu │ │ │ │ -00260600: 7420 6176 6169 6c61 626c 6520 666f 7220 t available for │ │ │ │ -00260610: 7265 6164 696e 672c 0a6f 7220 7768 6574 reading,.or whet │ │ │ │ -00260620: 6865 7220 6974 2068 6173 2061 7272 6976 her it has arriv │ │ │ │ -00260630: 6564 2061 7420 7468 6520 656e 642e 2020 ed at the end. │ │ │ │ -00260640: 5765 2069 6c6c 7573 7472 6174 6520 6974 We illustrate it │ │ │ │ -00260650: 2069 6e20 7468 6520 666f 6c6c 6f77 696e in the followin │ │ │ │ -00260660: 670a 6578 616d 706c 6520 6279 2073 696d g.example by sim │ │ │ │ -00260670: 756c 6174 696e 6720 6120 636f 6d70 7574 ulating a comput │ │ │ │ -00260680: 6174 696f 6e20 7468 6174 2074 616b 6573 ation that takes │ │ │ │ -00260690: 2035 2073 6563 6f6e 6473 2074 6f20 636f 5 seconds to co │ │ │ │ -002606a0: 6d70 6c65 7465 2c20 7072 696e 7469 6e67 mplete, printing │ │ │ │ -002606b0: 0a6f 6e65 2064 6f74 2070 6572 2073 6563 .one dot per sec │ │ │ │ -002606c0: 6f6e 6420 7768 696c 6520 7761 6974 696e ond while waitin │ │ │ │ -002606d0: 672e 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d g..+------------ │ │ │ │ -002606e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002606f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ -00260710: 203a 2066 203d 206f 7065 6e49 6e20 2221 : f = openIn "! │ │ │ │ -00260720: 736c 6565 7020 353b 2065 6368 6f20 2d6e sleep 5; echo -n │ │ │ │ -00260730: 2074 6865 2061 6e73 7765 7220 6973 2034 the answer is 4 │ │ │ │ -00260740: 2220 2020 207c 0a7c 2020 2020 2020 2020 " |.| │ │ │ │ -00260750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260770: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00260780: 0a7c 6f38 203d 2021 736c 6565 7020 353b .|o8 = !sleep 5; │ │ │ │ -00260790: 2065 6368 6f20 2d6e 2074 6865 2061 6e73 echo -n the ans │ │ │ │ -002607a0: 7765 7220 6973 2034 2020 2020 2020 2020 wer is 4 │ │ │ │ -002607b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -002607c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002607d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002602e0: 2020 2020 7c0a 7c20 2020 2020 696e 7665 |.| inve │ │ │ │ +002602f0: 7273 6520 2020 2020 2020 2020 2020 2020 rse │ │ │ │ +00260300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260310: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00260320: 0a7c 2020 2020 2069 6e73 7461 6c6c 4869 .| installHi │ │ │ │ +00260330: 6c62 6572 7446 756e 6374 696f 6e20 2020 lbertFunction │ │ │ │ +00260340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260350: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00260360: 2020 696e 7374 616e 6365 7320 2020 2020 instances │ │ │ │ +00260370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260390: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +002603a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002603b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002603c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002603d0: 2b0a 5769 7468 2074 6869 7320 666f 726d +.With this form │ │ │ │ +002603e0: 206f 6620 6269 6469 7265 6374 696f 6e61 of bidirectiona │ │ │ │ +002603f0: 6c20 636f 6d6d 756e 6963 6174 696f 6e20 l communication │ │ │ │ +00260400: 7468 6572 6520 6973 2061 6c77 6179 7320 there is always │ │ │ │ +00260410: 6120 6461 6e67 6572 206f 660a 626c 6f63 a danger of.bloc │ │ │ │ +00260420: 6b69 6e67 2c20 6265 6361 7573 6520 7468 king, because th │ │ │ │ +00260430: 6520 6275 6666 6572 7320 6173 736f 6369 e buffers associ │ │ │ │ +00260440: 6174 6564 2077 6974 6820 7468 6520 636f ated with the co │ │ │ │ +00260450: 6d6d 756e 6963 6174 696f 6e20 6368 616e mmunication chan │ │ │ │ +00260460: 6e65 6c73 0a28 7069 7065 7329 2074 7970 nels.(pipes) typ │ │ │ │ +00260470: 6963 616c 6c79 2068 6f6c 6420 6f6e 6c79 ically hold only │ │ │ │ +00260480: 2034 3039 3620 6279 7465 732e 2020 496e 4096 bytes. In │ │ │ │ +00260490: 2074 6869 7320 6578 616d 706c 6520 7765 this example we │ │ │ │ +002604a0: 2073 7563 6365 6564 6564 2062 6563 6175 succeeded becau │ │ │ │ +002604b0: 7365 0a74 6865 2065 6e74 6972 6520 6f75 se.the entire ou │ │ │ │ +002604c0: 7470 7574 2066 726f 6d20 6772 6570 2077 tput from grep w │ │ │ │ +002604d0: 6173 2073 6d61 6c6c 6572 2074 6861 6e20 as smaller than │ │ │ │ +002604e0: 3430 3936 2062 7974 6573 2e20 2049 6e20 4096 bytes. In │ │ │ │ +002604f0: 6765 6e65 7261 6c2c 206f 6e65 0a73 686f general, one.sho │ │ │ │ +00260500: 756c 6420 6265 2063 6172 6566 756c 2074 uld be careful t │ │ │ │ +00260510: 6f20 6172 7261 6e67 6520 7468 696e 6773 o arrange things │ │ │ │ +00260520: 2073 6f20 7468 6174 2074 6865 2074 776f so that the two │ │ │ │ +00260530: 2070 726f 6772 616d 7320 7461 6b65 2074 programs take t │ │ │ │ +00260540: 7572 6e73 2075 7369 6e67 0a74 6865 2063 urns using.the c │ │ │ │ +00260550: 6f6d 6d75 6e69 6361 7469 6f6e 2063 6861 ommunication cha │ │ │ │ +00260560: 6e6e 656c 2c20 736f 2074 6861 7420 7768 nnel, so that wh │ │ │ │ +00260570: 656e 206f 6e65 2069 7320 7772 6974 696e en one is writin │ │ │ │ +00260580: 6720 6461 7461 2c20 7468 6520 6f74 6865 g data, the othe │ │ │ │ +00260590: 7220 6973 0a72 6561 6469 6e67 2069 742e r is.reading it. │ │ │ │ +002605a0: 0a0a 0a41 2075 7365 6675 6c20 6675 6e63 ...A useful func │ │ │ │ +002605b0: 7469 6f6e 2069 6e20 7468 6973 2063 6f6e tion in this con │ │ │ │ +002605c0: 6e65 6374 696f 6e20 6973 202a 6e6f 7465 nection is *note │ │ │ │ +002605d0: 2069 7352 6561 6479 3a20 6973 5265 6164 isReady: isRead │ │ │ │ +002605e0: 795f 6c70 4669 6c65 5f72 702c 2c0a 7768 y_lpFile_rp,,.wh │ │ │ │ +002605f0: 6963 6820 7769 6c6c 2074 656c 6c20 796f ich will tell yo │ │ │ │ +00260600: 7520 7768 6574 6865 7220 616e 2069 6e70 u whether an inp │ │ │ │ +00260610: 7574 2066 696c 6520 6861 7320 616e 7920 ut file has any │ │ │ │ +00260620: 696e 7075 7420 6176 6169 6c61 626c 6520 input available │ │ │ │ +00260630: 666f 7220 7265 6164 696e 672c 0a6f 7220 for reading,.or │ │ │ │ +00260640: 7768 6574 6865 7220 6974 2068 6173 2061 whether it has a │ │ │ │ +00260650: 7272 6976 6564 2061 7420 7468 6520 656e rrived at the en │ │ │ │ +00260660: 642e 2020 5765 2069 6c6c 7573 7472 6174 d. We illustrat │ │ │ │ +00260670: 6520 6974 2069 6e20 7468 6520 666f 6c6c e it in the foll │ │ │ │ +00260680: 6f77 696e 670a 6578 616d 706c 6520 6279 owing.example by │ │ │ │ +00260690: 2073 696d 756c 6174 696e 6720 6120 636f simulating a co │ │ │ │ +002606a0: 6d70 7574 6174 696f 6e20 7468 6174 2074 mputation that t │ │ │ │ +002606b0: 616b 6573 2035 2073 6563 6f6e 6473 2074 akes 5 seconds t │ │ │ │ +002606c0: 6f20 636f 6d70 6c65 7465 2c20 7072 696e o complete, prin │ │ │ │ +002606d0: 7469 6e67 0a6f 6e65 2064 6f74 2070 6572 ting.one dot per │ │ │ │ +002606e0: 2073 6563 6f6e 6420 7768 696c 6520 7761 second while wa │ │ │ │ +002606f0: 6974 696e 672e 0a2b 2d2d 2d2d 2d2d 2d2d iting..+-------- │ │ │ │ +00260700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00260730: 0a7c 6938 203a 2066 203d 206f 7065 6e49 .|i8 : f = openI │ │ │ │ +00260740: 6e20 2221 736c 6565 7020 353b 2065 6368 n "!sleep 5; ech │ │ │ │ +00260750: 6f20 2d6e 2074 6865 2061 6e73 7765 7220 o -n the answer │ │ │ │ +00260760: 6973 2034 2220 2020 207c 0a7c 2020 2020 is 4" |.| │ │ │ │ +00260770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002607a0: 2020 207c 0a7c 6f38 203d 2021 736c 6565 |.|o8 = !slee │ │ │ │ +002607b0: 7020 353b 2065 6368 6f20 2d6e 2074 6865 p 5; echo -n the │ │ │ │ +002607c0: 2061 6e73 7765 7220 6973 2034 2020 2020 answer is 4 │ │ │ │ +002607d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 002607e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002607f0: 2020 207c 0a7c 6f38 203a 2046 696c 6520 |.|o8 : File │ │ │ │ +002607f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260820: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00260830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260860: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2069 -------+.|i9 : i │ │ │ │ -00260870: 7352 6561 6479 2066 2020 2020 2020 2020 sReady f │ │ │ │ -00260880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002608a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00260810: 2020 2020 2020 207c 0a7c 6f38 203a 2046 |.|o8 : F │ │ │ │ +00260820: 696c 6520 2020 2020 2020 2020 2020 2020 ile │ │ │ │ +00260830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260850: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00260860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 -----------+.|i9 │ │ │ │ +00260890: 203a 2069 7352 6561 6479 2066 2020 2020 : isReady f │ │ │ │ +002608a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002608b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002608c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002608d0: 2020 2020 2020 2020 2020 207c 0a7c 6f39 |.|o9 │ │ │ │ -002608e0: 203d 2066 616c 7365 2020 2020 2020 2020 = false │ │ │ │ -002608f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260910: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00260920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00260950: 0a7c 6931 3020 3a20 7768 696c 6520 6e6f .|i10 : while no │ │ │ │ -00260960: 7420 6973 5265 6164 7920 6620 646f 2028 t isReady f do ( │ │ │ │ -00260970: 736c 6565 7020 313b 203c 3c20 222e 2220 sleep 1; << "." │ │ │ │ -00260980: 3c3c 2066 6c75 7368 297c 0a7c 2e2e 2e2e << flush)|.|.... │ │ │ │ -00260990: 2e2e 2020 2020 2020 2020 2020 2020 2020 .. │ │ │ │ -002609a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002609b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002609c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -002609d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002609e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002609f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00260a00: 6931 3120 3a20 7265 6164 2066 2020 2020 i11 : read f │ │ │ │ -00260a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260a30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +002608c0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +002608d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002608e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002608f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00260900: 0a7c 6f39 203d 2066 616c 7365 2020 2020 .|o9 = false │ │ │ │ +00260910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260930: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00260940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260970: 2d2d 2d2b 0a7c 6931 3020 3a20 7768 696c ---+.|i10 : whil │ │ │ │ +00260980: 6520 6e6f 7420 6973 5265 6164 7920 6620 e not isReady f │ │ │ │ +00260990: 646f 2028 736c 6565 7020 313b 203c 3c20 do (sleep 1; << │ │ │ │ +002609a0: 222e 2220 3c3c 2066 6c75 7368 297c 0a7c "." << flush)|.| │ │ │ │ +002609b0: 2e2e 2e2e 2e2e 2020 2020 2020 2020 2020 ...... │ │ │ │ +002609c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002609d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002609e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +002609f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260a20: 2d2b 0a7c 6931 3120 3a20 7265 6164 2066 -+.|i11 : read f │ │ │ │ +00260a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260a50: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00260a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260a70: 207c 0a7c 6f31 3120 3d20 7468 6520 616e |.|o11 = the an │ │ │ │ -00260a80: 7377 6572 2069 7320 3420 2020 2020 2020 swer is 4 │ │ │ │ -00260a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260aa0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00260ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260ae0: 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 6973 -----+.|i12 : is │ │ │ │ -00260af0: 5265 6164 7920 6620 2020 2020 2020 2020 Ready f │ │ │ │ -00260b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260b10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00260b20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00260a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260a90: 2020 2020 207c 0a7c 6f31 3120 3d20 7468 |.|o11 = th │ │ │ │ +00260aa0: 6520 616e 7377 6572 2069 7320 3420 2020 e answer is 4 │ │ │ │ +00260ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260ac0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00260ad0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00260ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260b00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 ---------+.|i12 │ │ │ │ +00260b10: 3a20 6973 5265 6164 7920 6620 2020 2020 : isReady f │ │ │ │ +00260b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260b50: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ -00260b60: 3d20 7472 7565 2020 2020 2020 2020 2020 = true │ │ │ │ -00260b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260b90: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00260ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00260bd0: 6931 3320 3a20 6174 456e 644f 6646 696c i13 : atEndOfFil │ │ │ │ -00260be0: 6520 6620 2020 2020 2020 2020 2020 2020 e f │ │ │ │ -00260bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260c00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00260b40: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00260b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260b70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00260b80: 6f31 3220 3d20 7472 7565 2020 2020 2020 o12 = true │ │ │ │ +00260b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260bb0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00260bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260bf0: 2d2b 0a7c 6931 3320 3a20 6174 456e 644f -+.|i13 : atEndO │ │ │ │ +00260c00: 6646 696c 6520 6620 2020 2020 2020 2020 fFile f │ │ │ │ 00260c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260c20: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00260c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260c40: 207c 0a7c 6f31 3320 3d20 7472 7565 2020 |.|o13 = true │ │ │ │ +00260c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260c70: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00260c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260cb0: 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 636c -----+.|i14 : cl │ │ │ │ -00260cc0: 6f73 6520 6620 2020 2020 2020 2020 2020 ose f │ │ │ │ -00260cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260ce0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00260cf0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00260c60: 2020 2020 207c 0a7c 6f31 3320 3d20 7472 |.|o13 = tr │ │ │ │ +00260c70: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ +00260c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260c90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00260ca0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00260cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260cd0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 ---------+.|i14 │ │ │ │ +00260ce0: 3a20 636c 6f73 6520 6620 2020 2020 2020 : close f │ │ │ │ +00260cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260d20: 2020 2020 2020 2020 207c 0a7c 6f31 3420 |.|o14 │ │ │ │ -00260d30: 3d20 2173 6c65 6570 2035 3b20 6563 686f = !sleep 5; echo │ │ │ │ -00260d40: 202d 6e20 7468 6520 616e 7377 6572 2069 -n the answer i │ │ │ │ -00260d50: 7320 3420 2020 2020 2020 2020 2020 2020 s 4 │ │ │ │ -00260d60: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00260d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260d90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00260da0: 6f31 3420 3a20 4669 6c65 2020 2020 2020 o14 : File │ │ │ │ +00260d10: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00260d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260d40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00260d50: 6f31 3420 3d20 2173 6c65 6570 2035 3b20 o14 = !sleep 5; │ │ │ │ +00260d60: 6563 686f 202d 6e20 7468 6520 616e 7377 echo -n the answ │ │ │ │ +00260d70: 6572 2069 7320 3420 2020 2020 2020 2020 er is 4 │ │ │ │ +00260d80: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00260d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00260dd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00260de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260dc0: 207c 0a7c 6f31 3420 3a20 4669 6c65 2020 |.|o14 : File │ │ │ │ +00260dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260df0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00260e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00260e10: 2d2b 0a57 6520 616c 736f 2061 6c6c 6f77 -+.We also allow │ │ │ │ -00260e20: 2066 6f72 2062 6964 6972 6563 7469 6f6e for bidirection │ │ │ │ -00260e30: 616c 2063 6f6d 6d75 6e69 6361 7469 6f6e al communication │ │ │ │ -00260e40: 2074 6872 6f75 6768 2073 6f63 6b65 7473 through sockets │ │ │ │ -00260e50: 206f 7665 7220 7468 650a 696e 7465 726e over the.intern │ │ │ │ -00260e60: 6574 2e20 2053 6565 202a 6e6f 7465 206f et. See *note o │ │ │ │ -00260e70: 7065 6e49 6e4f 7574 3a20 6f70 656e 496e penInOut: openIn │ │ │ │ -00260e80: 4f75 742c 2061 6e64 202a 6e6f 7465 206f Out, and *note o │ │ │ │ -00260e90: 7065 6e4c 6973 7465 6e65 723a 0a6f 7065 penListener:.ope │ │ │ │ -00260ea0: 6e4c 6973 7465 6e65 725f 6c70 5374 7269 nListener_lpStri │ │ │ │ -00260eb0: 6e67 5f72 702c 2c20 6f72 2074 6865 206e ng_rp,, or the n │ │ │ │ -00260ec0: 6578 7420 7365 6374 696f 6e2e 0a0a 0a41 ext section....A │ │ │ │ -00260ed0: 6e6f 7468 6572 2075 7365 6675 6c20 6675 nother useful fu │ │ │ │ -00260ee0: 6e63 7469 6f6e 2069 7320 2a6e 6f74 6520 nction is *note │ │ │ │ -00260ef0: 7761 6974 3a20 7761 6974 2c2c 2077 6869 wait: wait,, whi │ │ │ │ -00260f00: 6368 2063 616e 2062 6520 7573 6564 2074 ch can be used t │ │ │ │ -00260f10: 6f20 7761 6974 2066 6f72 0a69 6e70 7574 o wait for.input │ │ │ │ -00260f20: 2074 6f20 6265 2061 7661 696c 6162 6c65 to be available │ │ │ │ -00260f30: 2066 726f 6d20 616e 7920 6f66 2061 206c from any of a l │ │ │ │ -00260f40: 6973 7420 6f66 2069 6e70 7574 2066 696c ist of input fil │ │ │ │ -00260f50: 6573 2e0a 2a20 4d65 6e75 3a0a 0a2a 2072 es..* Menu:..* r │ │ │ │ -00260f60: 756e 3a3a 2020 2020 2020 2020 2020 2020 un:: │ │ │ │ -00260f70: 2020 2020 2020 2020 2020 2020 2072 756e run │ │ │ │ -00260f80: 2061 6e20 6578 7465 726e 616c 2063 6f6d an external com │ │ │ │ -00260f90: 6d61 6e64 0a2a 2066 696e 6450 726f 6772 mand.* findProgr │ │ │ │ -00260fa0: 616d 3a3a 2020 2020 2020 2020 2020 2020 am:: │ │ │ │ -00260fb0: 2020 2020 206c 6f61 6420 6578 7465 726e load extern │ │ │ │ -00260fc0: 616c 2070 726f 6772 616d 0a2a 2072 756e al program.* run │ │ │ │ -00260fd0: 5072 6f67 7261 6d3a 3a20 2020 2020 2020 Program:: │ │ │ │ -00260fe0: 2020 2020 2020 2020 2020 2072 756e 2061 run a │ │ │ │ -00260ff0: 6e20 6578 7465 726e 616c 2070 726f 6772 n external progr │ │ │ │ -00261000: 616d 0a2a 2050 726f 6772 616d 203c 3c20 am.* Program << │ │ │ │ -00261010: 5468 696e 673a 3a20 2020 2020 2020 2020 Thing:: │ │ │ │ -00261020: 2020 2072 756e 2070 726f 6772 616d 2077 run program w │ │ │ │ -00261030: 6974 6820 696e 7075 7420 7265 6469 7265 ith input redire │ │ │ │ -00261040: 6374 696f 6e0a 2d2d 2d2d 2d2d 2d2d 2d2d ction.---------- │ │ │ │ -00261050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00261060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00260e30: 2d2d 2d2d 2d2b 0a57 6520 616c 736f 2061 -----+.We also a │ │ │ │ +00260e40: 6c6c 6f77 2066 6f72 2062 6964 6972 6563 llow for bidirec │ │ │ │ +00260e50: 7469 6f6e 616c 2063 6f6d 6d75 6e69 6361 tional communica │ │ │ │ +00260e60: 7469 6f6e 2074 6872 6f75 6768 2073 6f63 tion through soc │ │ │ │ +00260e70: 6b65 7473 206f 7665 7220 7468 650a 696e kets over the.in │ │ │ │ +00260e80: 7465 726e 6574 2e20 2053 6565 202a 6e6f ternet. See *no │ │ │ │ +00260e90: 7465 206f 7065 6e49 6e4f 7574 3a20 6f70 te openInOut: op │ │ │ │ +00260ea0: 656e 496e 4f75 742c 2061 6e64 202a 6e6f enInOut, and *no │ │ │ │ +00260eb0: 7465 206f 7065 6e4c 6973 7465 6e65 723a te openListener: │ │ │ │ +00260ec0: 0a6f 7065 6e4c 6973 7465 6e65 725f 6c70 .openListener_lp │ │ │ │ +00260ed0: 5374 7269 6e67 5f72 702c 2c20 6f72 2074 String_rp,, or t │ │ │ │ +00260ee0: 6865 206e 6578 7420 7365 6374 696f 6e2e he next section. │ │ │ │ +00260ef0: 0a0a 0a41 6e6f 7468 6572 2075 7365 6675 ...Another usefu │ │ │ │ +00260f00: 6c20 6675 6e63 7469 6f6e 2069 7320 2a6e l function is *n │ │ │ │ +00260f10: 6f74 6520 7761 6974 3a20 7761 6974 2c2c ote wait: wait,, │ │ │ │ +00260f20: 2077 6869 6368 2063 616e 2062 6520 7573 which can be us │ │ │ │ +00260f30: 6564 2074 6f20 7761 6974 2066 6f72 0a69 ed to wait for.i │ │ │ │ +00260f40: 6e70 7574 2074 6f20 6265 2061 7661 696c nput to be avail │ │ │ │ +00260f50: 6162 6c65 2066 726f 6d20 616e 7920 6f66 able from any of │ │ │ │ +00260f60: 2061 206c 6973 7420 6f66 2069 6e70 7574 a list of input │ │ │ │ +00260f70: 2066 696c 6573 2e0a 2a20 4d65 6e75 3a0a files..* Menu:. │ │ │ │ +00260f80: 0a2a 2072 756e 3a3a 2020 2020 2020 2020 .* run:: │ │ │ │ +00260f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00260fa0: 2072 756e 2061 6e20 6578 7465 726e 616c run an external │ │ │ │ +00260fb0: 2063 6f6d 6d61 6e64 0a2a 2066 696e 6450 command.* findP │ │ │ │ +00260fc0: 726f 6772 616d 3a3a 2020 2020 2020 2020 rogram:: │ │ │ │ +00260fd0: 2020 2020 2020 2020 206c 6f61 6420 6578 load ex │ │ │ │ +00260fe0: 7465 726e 616c 2070 726f 6772 616d 0a2a ternal program.* │ │ │ │ +00260ff0: 2072 756e 5072 6f67 7261 6d3a 3a20 2020 runProgram:: │ │ │ │ +00261000: 2020 2020 2020 2020 2020 2020 2020 2072 r │ │ │ │ +00261010: 756e 2061 6e20 6578 7465 726e 616c 2070 un an external p │ │ │ │ +00261020: 726f 6772 616d 0a2a 2050 726f 6772 616d rogram.* Program │ │ │ │ +00261030: 203c 3c20 5468 696e 673a 3a20 2020 2020 << Thing:: │ │ │ │ +00261040: 2020 2020 2020 2072 756e 2070 726f 6772 run progr │ │ │ │ +00261050: 616d 2077 6974 6820 696e 7075 7420 7265 am with input re │ │ │ │ +00261060: 6469 7265 6374 696f 6e0a 2d2d 2d2d 2d2d direction.------ │ │ │ │ 00261070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00261080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00261090: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -002610a0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -002610b0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -002610c0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -002610d0: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -002610e0: 2e30 352b 6473 2f4d 322f 4d61 6361 756c .05+ds/M2/Macaul │ │ │ │ -002610f0: 6179 322f 7061 636b 6167 6573 2f0a 4d61 ay2/packages/.Ma │ │ │ │ -00261100: 6361 756c 6179 3244 6f63 2f6f 765f 6669 caulay2Doc/ov_fi │ │ │ │ -00261110: 6c65 732e 6d32 3a34 3435 3a30 2e0a 1f0a les.m2:445:0.... │ │ │ │ -00261120: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ -00261130: 6f63 2e69 6e66 6f2c 204e 6f64 653a 2072 oc.info, Node: r │ │ │ │ -00261140: 756e 2c20 4e65 7874 3a20 6669 6e64 5072 un, Next: findPr │ │ │ │ -00261150: 6f67 7261 6d2c 2055 703a 2063 6f6d 6d75 ogram, Up: commu │ │ │ │ -00261160: 6e69 6361 7469 6e67 2077 6974 6820 7072 nicating with pr │ │ │ │ -00261170: 6f67 7261 6d73 0a0a 7275 6e20 2d2d 2072 ograms..run -- r │ │ │ │ -00261180: 756e 2061 6e20 6578 7465 726e 616c 2063 un an external c │ │ │ │ -00261190: 6f6d 6d61 6e64 0a2a 2a2a 2a2a 2a2a 2a2a ommand.********* │ │ │ │ -002611a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002611b0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ -002611c0: 3a20 0a20 2020 2020 2020 2072 756e 2073 : . run s │ │ │ │ -002611d0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ -002611e0: 2020 202a 2073 2c20 6120 2a6e 6f74 6520 * s, a *note │ │ │ │ -002611f0: 7374 7269 6e67 3a20 5374 7269 6e67 2c2c string: String,, │ │ │ │ -00261200: 2061 2063 6f6d 6d61 6e64 2075 6e64 6572 a command under │ │ │ │ -00261210: 7374 616e 6461 626c 6520 6279 2074 6865 standable by the │ │ │ │ -00261220: 206f 7065 7261 7469 6e67 0a20 2020 2020 operating. │ │ │ │ -00261230: 2020 2073 7973 7465 6d0a 2020 2a20 4f75 system. * Ou │ │ │ │ -00261240: 7470 7574 733a 0a20 2020 2020 202a 2061 tputs:. * a │ │ │ │ -00261250: 6e20 2a6e 6f74 6520 696e 7465 6765 723a n *note integer: │ │ │ │ -00261260: 205a 5a2c 2c20 7468 6520 7265 7475 726e ZZ,, the return │ │ │ │ -00261270: 2063 6f64 650a 0a44 6573 6372 6970 7469 code..Descripti │ │ │ │ -00261280: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00261290: 5468 6520 7072 6f63 6573 7320 6973 2072 The process is r │ │ │ │ -002612a0: 756e 2069 6e20 7468 6520 7361 6d65 2070 un in the same p │ │ │ │ -002612b0: 726f 6365 7373 2067 726f 7570 2c20 736f rocess group, so │ │ │ │ -002612c0: 2073 6967 6e61 6c73 2069 6e76 6f6b 6564 signals invoked │ │ │ │ -002612d0: 2062 7920 636f 6e74 726f 6c0a 6368 6172 by control.char │ │ │ │ -002612e0: 6163 7465 7273 2061 7420 7468 6520 7465 acters at the te │ │ │ │ -002612f0: 726d 696e 616c 2077 696c 6c20 676f 2062 rminal will go b │ │ │ │ -00261300: 6f74 6820 746f 2069 7420 616e 6420 746f oth to it and to │ │ │ │ -00261310: 204d 6163 6175 6c61 7932 2e0a 0a55 6e64 Macaulay2...Und │ │ │ │ -00261320: 6572 204c 696e 7578 2061 6e64 204d 6163 er Linux and Mac │ │ │ │ -00261330: 204f 532c 2074 6865 2072 6574 7572 6e20 OS, the return │ │ │ │ -00261340: 636f 6465 2069 7320 3235 3620 7469 6d65 code is 256 time │ │ │ │ -00261350: 7320 7468 6520 6578 6974 2073 7461 7475 s the exit statu │ │ │ │ -00261360: 7320 636f 6465 206f 660a 7468 6520 636f s code of.the co │ │ │ │ -00261370: 6d6d 616e 642c 2069 6620 7468 6520 636f mmand, if the co │ │ │ │ -00261380: 6d6d 616e 6420 6578 6974 6564 206e 6f72 mmand exited nor │ │ │ │ -00261390: 6d61 6c6c 793b 2062 7920 636f 6e76 656e mally; by conven │ │ │ │ -002613a0: 7469 6f6e 2c20 616e 2065 7869 7420 7374 tion, an exit st │ │ │ │ -002613b0: 6174 7573 2063 6f64 650a 6f66 2030 2069 atus code.of 0 i │ │ │ │ -002613c0: 6e64 6963 6174 6573 2065 7272 6f72 2066 ndicates error f │ │ │ │ -002613d0: 7265 6520 6578 6563 7574 696f 6e20 6f72 ree execution or │ │ │ │ -002613e0: 2074 6865 2042 6f6f 6c65 616e 2076 616c the Boolean val │ │ │ │ -002613f0: 7565 2074 7275 652c 2061 6e20 6578 6974 ue true, an exit │ │ │ │ -00261400: 2073 7461 7475 730a 636f 6465 206f 6620 status.code of │ │ │ │ -00261410: 3120 696e 6469 6361 7465 7320 616e 2065 1 indicates an e │ │ │ │ -00261420: 7272 6f72 206f 7220 7468 6520 426f 6f6c rror or the Bool │ │ │ │ -00261430: 6561 6e20 7661 6c75 6520 6661 6c73 653b ean value false; │ │ │ │ -00261440: 2069 6620 7468 6520 636f 6d6d 616e 640a if the command. │ │ │ │ -00261450: 7465 726d 696e 6174 6564 2069 6e20 7265 terminated in re │ │ │ │ -00261460: 7370 6f6e 7365 2074 6f20 6120 7369 676e sponse to a sign │ │ │ │ -00261470: 616c 206f 7220 6661 756c 742c 2074 6865 al or fault, the │ │ │ │ -00261480: 2073 6967 6e61 6c20 6e75 6d62 6572 2028 signal number ( │ │ │ │ -00261490: 696e 2074 6865 2072 616e 6765 0a31 2d31 in the range.1-1 │ │ │ │ -002614a0: 3236 2920 6973 2072 6574 7572 6e65 642c 26) is returned, │ │ │ │ -002614b0: 2061 6464 6564 2074 6f20 3132 3820 6966 added to 128 if │ │ │ │ -002614c0: 2061 2063 6f72 6520 6475 6d70 2077 6173 a core dump was │ │ │ │ -002614d0: 2063 7265 6174 6564 3b20 6966 2074 6865 created; if the │ │ │ │ -002614e0: 2073 6865 6c6c 0a28 636f 6d6d 616e 6420 shell.(command │ │ │ │ -002614f0: 696e 7465 7270 7265 7465 7229 2063 6f75 interpreter) cou │ │ │ │ -00261500: 6c64 206e 6f74 2062 6520 6578 6563 7574 ld not be execut │ │ │ │ -00261510: 6564 2c20 7468 656e 2031 3237 2069 7320 ed, then 127 is │ │ │ │ -00261520: 7265 7475 726e 6564 2e20 2053 6967 6e61 returned. Signa │ │ │ │ -00261530: 6c0a 6e75 6d62 6572 7320 7479 7069 6361 l.numbers typica │ │ │ │ -00261540: 6c6c 7920 696e 636c 7564 6520 3220 666f lly include 2 fo │ │ │ │ -00261550: 7220 696e 7465 7272 7570 742c 2033 2066 r interrupt, 3 f │ │ │ │ -00261560: 6f72 2071 7569 742c 2036 2066 6f72 2061 or quit, 6 for a │ │ │ │ -00261570: 626f 7274 2c20 3920 666f 7220 6b69 6c6c bort, 9 for kill │ │ │ │ -00261580: 2c0a 3131 2066 6f72 2073 6567 6d65 6e74 ,.11 for segment │ │ │ │ -00261590: 6174 696f 6e20 6661 756c 742c 2061 6e64 ation fault, and │ │ │ │ -002615a0: 2031 3520 666f 7220 7465 726d 696e 6174 15 for terminat │ │ │ │ -002615b0: 696f 6e2e 2020 466f 7220 6465 7461 696c ion. For detail │ │ │ │ -002615c0: 732c 2073 6565 2074 6865 206d 616e 0a70 s, see the man.p │ │ │ │ -002615d0: 6167 6520 6f66 2074 6865 206c 6962 6320 age of the libc │ │ │ │ -002615e0: 726f 7574 696e 6520 7379 7374 656d 2829 routine system() │ │ │ │ -002615f0: 2e0a 0a46 6f72 2074 6865 2070 726f 6772 ...For the progr │ │ │ │ -00261600: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ -00261610: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ -00261620: 6a65 6374 202a 6e6f 7465 2072 756e 3a20 ject *note run: │ │ │ │ -00261630: 7275 6e2c 2069 7320 6120 2a6e 6f74 6520 run, is a *note │ │ │ │ -00261640: 636f 6d70 696c 6564 2066 756e 6374 696f compiled functio │ │ │ │ -00261650: 6e3a 2043 6f6d 7069 6c65 6446 756e 6374 n: CompiledFunct │ │ │ │ -00261660: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ion,...--------- │ │ │ │ -00261670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00261680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00261090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002610a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002610b0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +002610c0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +002610d0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +002610e0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +002610f0: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +00261100: 312e 3236 2e30 352b 6473 2f4d 322f 4d61 1.26.05+ds/M2/Ma │ │ │ │ +00261110: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +00261120: 2f0a 4d61 6361 756c 6179 3244 6f63 2f6f /.Macaulay2Doc/o │ │ │ │ +00261130: 765f 6669 6c65 732e 6d32 3a34 3435 3a30 v_files.m2:445:0 │ │ │ │ +00261140: 2e0a 1f0a 4669 6c65 3a20 4d61 6361 756c ....File: Macaul │ │ │ │ +00261150: 6179 3244 6f63 2e69 6e66 6f2c 204e 6f64 ay2Doc.info, Nod │ │ │ │ +00261160: 653a 2072 756e 2c20 4e65 7874 3a20 6669 e: run, Next: fi │ │ │ │ +00261170: 6e64 5072 6f67 7261 6d2c 2055 703a 2063 ndProgram, Up: c │ │ │ │ +00261180: 6f6d 6d75 6e69 6361 7469 6e67 2077 6974 ommunicating wit │ │ │ │ +00261190: 6820 7072 6f67 7261 6d73 0a0a 7275 6e20 h programs..run │ │ │ │ +002611a0: 2d2d 2072 756e 2061 6e20 6578 7465 726e -- run an extern │ │ │ │ +002611b0: 616c 2063 6f6d 6d61 6e64 0a2a 2a2a 2a2a al command.***** │ │ │ │ +002611c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002611d0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +002611e0: 7361 6765 3a20 0a20 2020 2020 2020 2072 sage: . r │ │ │ │ +002611f0: 756e 2073 0a20 202a 2049 6e70 7574 733a un s. * Inputs: │ │ │ │ +00261200: 0a20 2020 2020 202a 2073 2c20 6120 2a6e . * s, a *n │ │ │ │ +00261210: 6f74 6520 7374 7269 6e67 3a20 5374 7269 ote string: Stri │ │ │ │ +00261220: 6e67 2c2c 2061 2063 6f6d 6d61 6e64 2075 ng,, a command u │ │ │ │ +00261230: 6e64 6572 7374 616e 6461 626c 6520 6279 nderstandable by │ │ │ │ +00261240: 2074 6865 206f 7065 7261 7469 6e67 0a20 the operating. │ │ │ │ +00261250: 2020 2020 2020 2073 7973 7465 6d0a 2020 system. │ │ │ │ +00261260: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +00261270: 202a 2061 6e20 2a6e 6f74 6520 696e 7465 * an *note inte │ │ │ │ +00261280: 6765 723a 205a 5a2c 2c20 7468 6520 7265 ger: ZZ,, the re │ │ │ │ +00261290: 7475 726e 2063 6f64 650a 0a44 6573 6372 turn code..Descr │ │ │ │ +002612a0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +002612b0: 3d3d 0a0a 5468 6520 7072 6f63 6573 7320 ==..The process │ │ │ │ +002612c0: 6973 2072 756e 2069 6e20 7468 6520 7361 is run in the sa │ │ │ │ +002612d0: 6d65 2070 726f 6365 7373 2067 726f 7570 me process group │ │ │ │ +002612e0: 2c20 736f 2073 6967 6e61 6c73 2069 6e76 , so signals inv │ │ │ │ +002612f0: 6f6b 6564 2062 7920 636f 6e74 726f 6c0a oked by control. │ │ │ │ +00261300: 6368 6172 6163 7465 7273 2061 7420 7468 characters at th │ │ │ │ +00261310: 6520 7465 726d 696e 616c 2077 696c 6c20 e terminal will │ │ │ │ +00261320: 676f 2062 6f74 6820 746f 2069 7420 616e go both to it an │ │ │ │ +00261330: 6420 746f 204d 6163 6175 6c61 7932 2e0a d to Macaulay2.. │ │ │ │ +00261340: 0a55 6e64 6572 204c 696e 7578 2061 6e64 .Under Linux and │ │ │ │ +00261350: 204d 6163 204f 532c 2074 6865 2072 6574 Mac OS, the ret │ │ │ │ +00261360: 7572 6e20 636f 6465 2069 7320 3235 3620 urn code is 256 │ │ │ │ +00261370: 7469 6d65 7320 7468 6520 6578 6974 2073 times the exit s │ │ │ │ +00261380: 7461 7475 7320 636f 6465 206f 660a 7468 tatus code of.th │ │ │ │ +00261390: 6520 636f 6d6d 616e 642c 2069 6620 7468 e command, if th │ │ │ │ +002613a0: 6520 636f 6d6d 616e 6420 6578 6974 6564 e command exited │ │ │ │ +002613b0: 206e 6f72 6d61 6c6c 793b 2062 7920 636f normally; by co │ │ │ │ +002613c0: 6e76 656e 7469 6f6e 2c20 616e 2065 7869 nvention, an exi │ │ │ │ +002613d0: 7420 7374 6174 7573 2063 6f64 650a 6f66 t status code.of │ │ │ │ +002613e0: 2030 2069 6e64 6963 6174 6573 2065 7272 0 indicates err │ │ │ │ +002613f0: 6f72 2066 7265 6520 6578 6563 7574 696f or free executio │ │ │ │ +00261400: 6e20 6f72 2074 6865 2042 6f6f 6c65 616e n or the Boolean │ │ │ │ +00261410: 2076 616c 7565 2074 7275 652c 2061 6e20 value true, an │ │ │ │ +00261420: 6578 6974 2073 7461 7475 730a 636f 6465 exit status.code │ │ │ │ +00261430: 206f 6620 3120 696e 6469 6361 7465 7320 of 1 indicates │ │ │ │ +00261440: 616e 2065 7272 6f72 206f 7220 7468 6520 an error or the │ │ │ │ +00261450: 426f 6f6c 6561 6e20 7661 6c75 6520 6661 Boolean value fa │ │ │ │ +00261460: 6c73 653b 2069 6620 7468 6520 636f 6d6d lse; if the comm │ │ │ │ +00261470: 616e 640a 7465 726d 696e 6174 6564 2069 and.terminated i │ │ │ │ +00261480: 6e20 7265 7370 6f6e 7365 2074 6f20 6120 n response to a │ │ │ │ +00261490: 7369 676e 616c 206f 7220 6661 756c 742c signal or fault, │ │ │ │ +002614a0: 2074 6865 2073 6967 6e61 6c20 6e75 6d62 the signal numb │ │ │ │ +002614b0: 6572 2028 696e 2074 6865 2072 616e 6765 er (in the range │ │ │ │ +002614c0: 0a31 2d31 3236 2920 6973 2072 6574 7572 .1-126) is retur │ │ │ │ +002614d0: 6e65 642c 2061 6464 6564 2074 6f20 3132 ned, added to 12 │ │ │ │ +002614e0: 3820 6966 2061 2063 6f72 6520 6475 6d70 8 if a core dump │ │ │ │ +002614f0: 2077 6173 2063 7265 6174 6564 3b20 6966 was created; if │ │ │ │ +00261500: 2074 6865 2073 6865 6c6c 0a28 636f 6d6d the shell.(comm │ │ │ │ +00261510: 616e 6420 696e 7465 7270 7265 7465 7229 and interpreter) │ │ │ │ +00261520: 2063 6f75 6c64 206e 6f74 2062 6520 6578 could not be ex │ │ │ │ +00261530: 6563 7574 6564 2c20 7468 656e 2031 3237 ecuted, then 127 │ │ │ │ +00261540: 2069 7320 7265 7475 726e 6564 2e20 2053 is returned. S │ │ │ │ +00261550: 6967 6e61 6c0a 6e75 6d62 6572 7320 7479 ignal.numbers ty │ │ │ │ +00261560: 7069 6361 6c6c 7920 696e 636c 7564 6520 pically include │ │ │ │ +00261570: 3220 666f 7220 696e 7465 7272 7570 742c 2 for interrupt, │ │ │ │ +00261580: 2033 2066 6f72 2071 7569 742c 2036 2066 3 for quit, 6 f │ │ │ │ +00261590: 6f72 2061 626f 7274 2c20 3920 666f 7220 or abort, 9 for │ │ │ │ +002615a0: 6b69 6c6c 2c0a 3131 2066 6f72 2073 6567 kill,.11 for seg │ │ │ │ +002615b0: 6d65 6e74 6174 696f 6e20 6661 756c 742c mentation fault, │ │ │ │ +002615c0: 2061 6e64 2031 3520 666f 7220 7465 726d and 15 for term │ │ │ │ +002615d0: 696e 6174 696f 6e2e 2020 466f 7220 6465 ination. For de │ │ │ │ +002615e0: 7461 696c 732c 2073 6565 2074 6865 206d tails, see the m │ │ │ │ +002615f0: 616e 0a70 6167 6520 6f66 2074 6865 206c an.page of the l │ │ │ │ +00261600: 6962 6320 726f 7574 696e 6520 7379 7374 ibc routine syst │ │ │ │ +00261610: 656d 2829 2e0a 0a46 6f72 2074 6865 2070 em()...For the p │ │ │ │ +00261620: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +00261630: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +00261640: 6520 6f62 6a65 6374 202a 6e6f 7465 2072 e object *note r │ │ │ │ +00261650: 756e 3a20 7275 6e2c 2069 7320 6120 2a6e un: run, is a *n │ │ │ │ +00261660: 6f74 6520 636f 6d70 696c 6564 2066 756e ote compiled fun │ │ │ │ +00261670: 6374 696f 6e3a 2043 6f6d 7069 6c65 6446 ction: CompiledF │ │ │ │ +00261680: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ 00261690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002616a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002616b0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -002616c0: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -002616d0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -002616e0: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -002616f0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00261700: 362e 3035 2b64 732f 4d32 2f4d 6163 6175 6.05+ds/M2/Macau │ │ │ │ -00261710: 6c61 7932 2f70 6163 6b61 6765 732f 0a4d lay2/packages/.M │ │ │ │ -00261720: 6163 6175 6c61 7932 446f 632f 6f76 5f73 acaulay2Doc/ov_s │ │ │ │ -00261730: 7973 7465 6d2e 6d32 3a38 3730 3a30 2e0a ystem.m2:870:0.. │ │ │ │ -00261740: 1f0a 4669 6c65 3a20 4d61 6361 756c 6179 ..File: Macaulay │ │ │ │ -00261750: 3244 6f63 2e69 6e66 6f2c 204e 6f64 653a 2Doc.info, Node: │ │ │ │ -00261760: 2066 696e 6450 726f 6772 616d 2c20 4e65 findProgram, Ne │ │ │ │ -00261770: 7874 3a20 7275 6e50 726f 6772 616d 2c20 xt: runProgram, │ │ │ │ -00261780: 5072 6576 3a20 7275 6e2c 2055 703a 2063 Prev: run, Up: c │ │ │ │ -00261790: 6f6d 6d75 6e69 6361 7469 6e67 2077 6974 ommunicating wit │ │ │ │ -002617a0: 6820 7072 6f67 7261 6d73 0a0a 6669 6e64 h programs..find │ │ │ │ -002617b0: 5072 6f67 7261 6d20 2d2d 206c 6f61 6420 Program -- load │ │ │ │ -002617c0: 6578 7465 726e 616c 2070 726f 6772 616d external program │ │ │ │ -002617d0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ -002617e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002617f0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ -00261800: 3a20 0a20 2020 2020 2020 2066 696e 6450 : . findP │ │ │ │ -00261810: 726f 6772 616d 206e 616d 650a 2020 2020 rogram name. │ │ │ │ -00261820: 2020 2020 6669 6e64 5072 6f67 7261 6d28 findProgram( │ │ │ │ -00261830: 6e61 6d65 2c20 636d 6429 0a20 2020 2020 name, cmd). │ │ │ │ -00261840: 2020 2066 696e 6450 726f 6772 616d 286e findProgram(n │ │ │ │ -00261850: 616d 652c 2063 6d64 7329 0a20 202a 2049 ame, cmds). * I │ │ │ │ -00261860: 6e70 7574 733a 0a20 2020 2020 202a 206e nputs:. * n │ │ │ │ -00261870: 616d 652c 2061 202a 6e6f 7465 2073 7472 ame, a *note str │ │ │ │ -00261880: 696e 673a 2053 7472 696e 672c 2c20 7468 ing: String,, th │ │ │ │ -00261890: 6520 6e61 6d65 206f 6620 7468 6520 7072 e name of the pr │ │ │ │ -002618a0: 6f67 7261 6d20 746f 206c 6f61 642e 2020 ogram to load. │ │ │ │ -002618b0: 5468 6973 0a20 2020 2020 2020 2073 686f This. sho │ │ │ │ -002618c0: 756c 6420 6d61 7463 6820 7468 6520 636f uld match the co │ │ │ │ -002618d0: 7272 6573 706f 6e64 696e 6720 6b65 7920 rresponding key │ │ │ │ -002618e0: 696e 202a 6e6f 7465 2070 726f 6772 616d in *note program │ │ │ │ -002618f0: 5061 7468 733a 0a20 2020 2020 2020 2070 Paths:. p │ │ │ │ -00261900: 726f 6772 616d 5061 7468 732c 2e0a 2020 rogramPaths,.. │ │ │ │ -00261910: 2020 2020 2a20 636d 642c 2061 202a 6e6f * cmd, a *no │ │ │ │ -00261920: 7465 2073 7472 696e 673a 2053 7472 696e te string: Strin │ │ │ │ -00261930: 672c 2c20 6120 636f 6d6d 616e 6420 746f g,, a command to │ │ │ │ -00261940: 2072 756e 2074 6861 7420 7368 6f75 6c64 run that should │ │ │ │ -00261950: 2072 6574 7572 6e20 3020 6966 0a20 2020 return 0 if. │ │ │ │ -00261960: 2020 2020 2074 6865 2070 726f 6772 616d the program │ │ │ │ -00261970: 2069 7320 7072 6573 656e 742e 0a20 2020 is present.. │ │ │ │ -00261980: 2020 202a 2063 6d64 732c 2061 202a 6e6f * cmds, a *no │ │ │ │ -00261990: 7465 206c 6973 743a 204c 6973 742c 2c20 te list: List,, │ │ │ │ -002619a0: 6120 6c69 7374 206f 6620 636f 6d6d 616e a list of comman │ │ │ │ -002619b0: 6473 2074 6f20 7275 6e20 7468 6174 2073 ds to run that s │ │ │ │ -002619c0: 686f 756c 6420 616c 6c0a 2020 2020 2020 hould all. │ │ │ │ -002619d0: 2020 7265 7475 726e 2030 2069 6620 7468 return 0 if th │ │ │ │ -002619e0: 6520 7072 6f67 7261 6d20 6973 2070 7265 e program is pre │ │ │ │ -002619f0: 7365 6e74 2e0a 2020 2a20 2a6e 6f74 6520 sent.. * *note │ │ │ │ -00261a00: 4f70 7469 6f6e 616c 2069 6e70 7574 733a Optional inputs: │ │ │ │ -00261a10: 2075 7369 6e67 2066 756e 6374 696f 6e73 using functions │ │ │ │ -00261a20: 2077 6974 6820 6f70 7469 6f6e 616c 2069 with optional i │ │ │ │ -00261a30: 6e70 7574 732c 3a0a 2020 2020 2020 2a20 nputs,:. * │ │ │ │ -00261a40: 5261 6973 6545 7272 6f72 203d 3e20 6120 RaiseError => a │ │ │ │ -00261a50: 2a6e 6f74 6520 426f 6f6c 6561 6e20 7661 *note Boolean va │ │ │ │ -00261a60: 6c75 653a 2042 6f6f 6c65 616e 2c2c 2064 lue: Boolean,, d │ │ │ │ -00261a70: 6566 6175 6c74 2076 616c 7565 2074 7275 efault value tru │ │ │ │ -00261a80: 652c 0a20 2020 2020 2020 2077 6865 7468 e,. wheth │ │ │ │ -00261a90: 6572 2074 6f20 7261 6973 6520 616e 2065 er to raise an e │ │ │ │ -00261aa0: 7272 6f72 2069 6620 7468 6520 7072 6f67 rror if the prog │ │ │ │ -00261ab0: 7261 6d20 6973 206e 6f74 2066 6f75 6e64 ram is not found │ │ │ │ -00261ac0: 2e0a 2020 2020 2020 2a20 5665 7262 6f73 .. * Verbos │ │ │ │ -00261ad0: 6520 3d3e 2061 202a 6e6f 7465 2042 6f6f e => a *note Boo │ │ │ │ -00261ae0: 6c65 616e 2076 616c 7565 3a20 426f 6f6c lean value: Bool │ │ │ │ -00261af0: 6561 6e2c 2c20 6465 6661 756c 7420 7661 ean,, default va │ │ │ │ -00261b00: 6c75 6520 6661 6c73 652c 0a20 2020 2020 lue false,. │ │ │ │ -00261b10: 2020 2077 6865 7468 6572 2074 6f20 696e whether to in │ │ │ │ -00261b20: 666f 726d 2074 6865 2075 7365 7220 6f66 form the user of │ │ │ │ -00261b30: 2065 6163 6820 7061 7468 2074 6861 7420 each path that │ │ │ │ -00261b40: 6973 2063 6865 636b 6564 2e0a 2020 2020 is checked.. │ │ │ │ -00261b50: 2020 2a20 5072 6566 6978 203d 3e20 6120 * Prefix => a │ │ │ │ -00261b60: 2a6e 6f74 6520 6c69 7374 3a20 4c69 7374 *note list: List │ │ │ │ -00261b70: 2c2c 2064 6566 6175 6c74 2076 616c 7565 ,, default value │ │ │ │ -00261b80: 207b 7d2c 2061 206c 6973 7420 6f66 2073 {}, a list of s │ │ │ │ -00261b90: 6571 7565 6e63 6573 0a20 2020 2020 2020 equences. │ │ │ │ -00261ba0: 2063 6f6e 7461 696e 696e 6720 7477 6f20 containing two │ │ │ │ -00261bb0: 7374 7269 6e67 7320 6964 656e 7469 6679 strings identify │ │ │ │ -00261bc0: 696e 6720 6120 7072 6566 6978 2074 6861 ing a prefix tha │ │ │ │ -00261bd0: 7420 6973 2061 6464 6564 2074 6f20 7468 t is added to th │ │ │ │ -00261be0: 650a 2020 2020 2020 2020 6578 6563 7574 e. execut │ │ │ │ -00261bf0: 6162 6c65 2062 696e 6172 6965 7320 6265 able binaries be │ │ │ │ -00261c00: 6c6f 6e67 696e 6720 746f 2074 6865 2070 longing to the p │ │ │ │ -00261c10: 726f 6772 616d 2062 7920 736f 6d65 2064 rogram by some d │ │ │ │ -00261c20: 6973 7472 6962 7574 696f 6e73 2e0a 2020 istributions.. │ │ │ │ -00261c30: 2020 2020 2020 5468 6573 6520 7365 7175 These sequ │ │ │ │ -00261c40: 656e 6365 7320 7368 6f75 6c64 2062 6520 ences should be │ │ │ │ -00261c50: 6f66 2074 6865 2066 6f72 6d20 2872 6567 of the form (reg │ │ │ │ -00261c60: 6578 2c20 7072 6566 6978 292c 2077 6865 ex, prefix), whe │ │ │ │ -00261c70: 7265 2072 6567 6578 2069 7320 610a 2020 re regex is a. │ │ │ │ -00261c80: 2020 2020 2020 2a6e 6f74 6520 7265 6775 *note regu │ │ │ │ -00261c90: 6c61 7220 6578 7072 6573 7369 6f6e 3a20 lar expression: │ │ │ │ -00261ca0: 7265 6775 6c61 7220 6578 7072 6573 7369 regular expressi │ │ │ │ -00261cb0: 6f6e 732c 2074 6861 7420 7368 6f75 6c64 ons, that should │ │ │ │ -00261cc0: 206d 6174 6368 2061 6c6c 0a20 2020 2020 match all. │ │ │ │ -00261cd0: 2020 2062 696e 6172 7920 6578 6563 7574 binary execut │ │ │ │ -00261ce0: 6162 6c65 7320 7468 6174 206e 6565 6420 ables that need │ │ │ │ -00261cf0: 7468 6520 7072 6566 6978 2061 6e64 2070 the prefix and p │ │ │ │ -00261d00: 7265 6669 7820 6973 2074 6865 2070 7265 refix is the pre │ │ │ │ -00261d10: 6669 780a 2020 2020 2020 2020 6974 7365 fix. itse │ │ │ │ -00261d20: 6c66 2e0a 2020 2020 2020 2a20 4164 6469 lf.. * Addi │ │ │ │ -00261d30: 7469 6f6e 616c 5061 7468 7320 3d3e 2061 tionalPaths => a │ │ │ │ -00261d40: 202a 6e6f 7465 206c 6973 743a 204c 6973 *note list: Lis │ │ │ │ -00261d50: 742c 2c20 6465 6661 756c 7420 7661 6c75 t,, default valu │ │ │ │ -00261d60: 6520 7b7d 2c20 6120 6c69 7374 206f 660a e {}, a list of. │ │ │ │ -00261d70: 2020 2020 2020 2020 7374 7269 6e67 7320 strings │ │ │ │ -00261d80: 636f 6e74 6169 6e69 6e67 2061 6e79 2070 containing any p │ │ │ │ -00261d90: 6174 6873 2074 6f20 6368 6563 6b20 666f aths to check fo │ │ │ │ -00261da0: 7220 7468 6520 7072 6f67 7261 6d20 696e r the program in │ │ │ │ -00261db0: 2061 6464 6974 696f 6e20 746f 0a20 2020 addition to. │ │ │ │ -00261dc0: 2020 2020 2074 6865 2064 6566 6175 6c74 the default │ │ │ │ -00261dd0: 206f 6e65 732e 0a20 2020 2020 202a 204d ones.. * M │ │ │ │ -00261de0: 696e 696d 756d 5665 7273 696f 6e20 3d3e inimumVersion => │ │ │ │ -00261df0: 2061 202a 6e6f 7465 2073 6571 7565 6e63 a *note sequenc │ │ │ │ -00261e00: 653a 2053 6571 7565 6e63 652c 2c20 6465 e: Sequence,, de │ │ │ │ -00261e10: 6661 756c 7420 7661 6c75 6520 6e75 6c6c fault value null │ │ │ │ -00261e20: 2c0a 2020 2020 2020 2020 636f 6e74 6169 ,. contai │ │ │ │ -00261e30: 6e69 6e67 2074 776f 2073 7472 696e 6773 ning two strings │ │ │ │ -00261e40: 2074 6865 2066 6f72 6d20 286d 696e 5665 the form (minVe │ │ │ │ -00261e50: 7273 696f 6e2c 2076 6572 7369 6f6e 436f rsion, versionCo │ │ │ │ -00261e60: 6d6d 616e 6429 2077 6865 7265 0a20 2020 mmand) where. │ │ │ │ -00261e70: 2020 2020 206d 696e 5665 7273 696f 6e20 minVersion │ │ │ │ -00261e80: 6973 2074 6865 206d 696e 696d 756d 2072 is the minimum r │ │ │ │ -00261e90: 6571 7569 7265 6420 7665 7273 696f 6e20 equired version │ │ │ │ -00261ea0: 6f66 2074 6865 2070 726f 6772 616d 2061 of the program a │ │ │ │ -00261eb0: 6e64 0a20 2020 2020 2020 2076 6572 7369 nd. versi │ │ │ │ -00261ec0: 6f6e 436f 6d6d 616e 6420 6973 2061 2073 onCommand is a s │ │ │ │ -00261ed0: 6865 6c6c 2063 6f6d 6d61 6e64 2074 6f20 hell command to │ │ │ │ -00261ee0: 6f62 7461 696e 2074 6865 2076 6572 7369 obtain the versi │ │ │ │ -00261ef0: 6f6e 206e 756d 6265 7220 6f66 2061 6e0a on number of an. │ │ │ │ -00261f00: 2020 2020 2020 2020 696e 7374 616c 6c65 installe │ │ │ │ -00261f10: 6420 7072 6f67 7261 6d2e 0a20 202a 204f d program.. * O │ │ │ │ -00261f20: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -00261f30: 616e 2069 6e73 7461 6e63 6520 6f66 2074 an instance of t │ │ │ │ -00261f40: 6865 2074 7970 6520 2a6e 6f74 6520 5072 he type *note Pr │ │ │ │ -00261f50: 6f67 7261 6d3a 2050 726f 6772 616d 2c2c ogram: Program,, │ │ │ │ -00261f60: 2074 6865 2070 726f 6772 616d 2074 6861 the program tha │ │ │ │ -00261f70: 7420 7761 730a 2020 2020 2020 2020 6c6f t was. lo │ │ │ │ -00261f80: 6164 6564 2e20 2049 6620 7468 6520 7072 aded. If the pr │ │ │ │ -00261f90: 6f67 7261 6d20 6973 206e 6f74 2066 6f75 ogram is not fou │ │ │ │ -00261fa0: 6e64 2061 6e64 2052 6169 7365 4572 726f nd and RaiseErro │ │ │ │ -00261fb0: 7220 6973 2073 6574 2074 6f20 2a6e 6f74 r is set to *not │ │ │ │ -00261fc0: 650a 2020 2020 2020 2020 6661 6c73 653a e. false: │ │ │ │ -00261fd0: 2066 616c 7365 2c2c 2074 6865 6e20 2a6e false,, then *n │ │ │ │ -00261fe0: 6f74 6520 6e75 6c6c 3a20 6e75 6c6c 2c20 ote null: null, │ │ │ │ -00261ff0: 6973 2072 6574 7572 6e65 642e 0a0a 4465 is returned...De │ │ │ │ -00262000: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ -00262010: 3d3d 3d3d 3d0a 0a54 6869 7320 6675 6e63 =====..This func │ │ │ │ -00262020: 7469 6f6e 2063 6865 636b 7320 666f 7220 tion checks for │ │ │ │ -00262030: 7468 6520 6578 6973 7465 6e63 6520 6f66 the existence of │ │ │ │ -00262040: 2061 6e20 6578 7465 726e 616c 2070 726f an external pro │ │ │ │ -00262050: 6772 616d 2062 7920 7275 6e6e 696e 6720 gram by running │ │ │ │ -00262060: 636d 640a 286f 7220 6576 6572 7920 656c cmd.(or every el │ │ │ │ -00262070: 656d 656e 7420 6f66 2063 6d64 7329 2c20 ement of cmds), │ │ │ │ -00262080: 7072 6570 656e 6465 6420 7769 7468 2076 prepended with v │ │ │ │ -00262090: 6172 696f 7573 2070 6174 6873 2069 6e20 arious paths in │ │ │ │ -002620a0: 7468 6520 666f 6c6c 6f77 696e 670a 6f72 the following.or │ │ │ │ -002620b0: 6465 723a 0a0a 2020 2a20 5468 6520 7573 der:.. * The us │ │ │ │ -002620c0: 6572 2d64 6566 696e 6564 2070 6174 6820 er-defined path │ │ │ │ -002620d0: 7370 6563 6966 6965 6420 6279 2070 726f specified by pro │ │ │ │ -002620e0: 6772 616d 5061 7468 7323 6e61 6d65 2c20 gramPaths#name, │ │ │ │ -002620f0: 6966 2069 7420 6578 6973 7473 2e0a 2020 if it exists.. │ │ │ │ -00262100: 2a20 5468 6520 7061 7468 2073 7065 6369 * The path speci │ │ │ │ -00262110: 6669 6564 2062 7920 7072 6566 6978 4469 fied by prefixDi │ │ │ │ -00262120: 7265 6374 6f72 7920 7c20 6375 7272 656e rectory | curren │ │ │ │ -00262130: 744c 6179 6f75 7423 2270 726f 6772 616d tLayout#"program │ │ │ │ -00262140: 7322 2c20 7768 6572 6520 7468 650a 2020 s", where the. │ │ │ │ -00262150: 2020 7072 6f67 7261 6d73 2073 6869 7070 programs shipp │ │ │ │ -00262160: 6564 2077 6974 6820 4d61 6361 756c 6179 ed with Macaulay │ │ │ │ -00262170: 3220 6172 6520 696e 7374 616c 6c65 642e 2 are installed. │ │ │ │ -00262180: 0a20 202a 2045 6163 6820 7061 7468 2073 . * Each path s │ │ │ │ -00262190: 7065 6369 6669 6564 2062 7920 7468 6520 pecified by the │ │ │ │ -002621a0: 4164 6469 7469 6f6e 616c 5061 7468 7320 AdditionalPaths │ │ │ │ -002621b0: 6f70 7469 6f6e 2e0a 2020 2a20 4561 6368 option.. * Each │ │ │ │ -002621c0: 2070 6174 6820 7370 6563 6966 6965 6420 path specified │ │ │ │ -002621d0: 6279 2074 6865 2075 7365 7227 7320 5041 by the user's PA │ │ │ │ -002621e0: 5448 2065 6e76 6972 6f6e 6d65 6e74 2076 TH environment v │ │ │ │ -002621f0: 6172 6961 626c 652e 0a20 202a 2054 6865 ariable.. * The │ │ │ │ -00262200: 2070 6174 6820 746f 204d 322d 6269 6e61 path to M2-bina │ │ │ │ -00262210: 7279 2e0a 0a46 6f72 2065 6163 6820 7061 ry...For each pa │ │ │ │ -00262220: 7468 2c20 616e 7920 7072 6566 6978 6573 th, any prefixes │ │ │ │ -00262230: 2073 7065 6369 6669 6564 2062 7920 7468 specified by th │ │ │ │ -00262240: 6520 5072 6566 6978 206f 7074 696f 6e20 e Prefix option │ │ │ │ -00262250: 6172 6520 6368 6563 6b65 642e 0a0a 4f6e are checked...On │ │ │ │ -00262260: 6365 2074 6869 7320 6973 2073 7563 6365 ce this is succe │ │ │ │ -00262270: 7373 6675 6c20 2869 2e65 2e2c 2063 6d64 ssful (i.e., cmd │ │ │ │ -00262280: 206f 7220 6561 6368 2065 6c65 6d65 6e74 or each element │ │ │ │ -00262290: 206f 6620 636d 6473 2072 6574 7572 6e73 of cmds returns │ │ │ │ -002622a0: 2061 2076 616c 7565 206f 660a 3029 2074 a value of.0) t │ │ │ │ -002622b0: 6865 6e20 6120 2a6e 6f74 6520 5072 6f67 hen a *note Prog │ │ │ │ -002622c0: 7261 6d3a 2050 726f 6772 616d 2c20 6f62 ram: Program, ob │ │ │ │ -002622d0: 6a65 6374 2069 7320 7265 7475 726e 6564 ject is returned │ │ │ │ -002622e0: 2e20 2049 6620 6974 2069 7320 756e 7375 . If it is unsu │ │ │ │ -002622f0: 6363 6573 7366 756c 2c0a 7468 656e 2065 ccessful,.then e │ │ │ │ -00262300: 6974 6865 7220 616e 2065 7272 6f72 2069 ither an error i │ │ │ │ -00262310: 7320 7261 6973 6564 206f 7220 2a6e 6f74 s raised or *not │ │ │ │ -00262320: 6520 6e75 6c6c 3a20 6e75 6c6c 2c20 6973 e null: null, is │ │ │ │ -00262330: 2072 6574 7572 6e65 642c 2064 6570 656e returned, depen │ │ │ │ -00262340: 6469 6e67 206f 6e0a 7468 6520 7661 6c75 ding on.the valu │ │ │ │ -00262350: 6520 6f66 2052 6169 7365 4572 726f 722e e of RaiseError. │ │ │ │ -00262360: 0a0a 4e6f 7465 2074 6861 7420 6966 2061 ..Note that if a │ │ │ │ -00262370: 2070 726f 6772 616d 2063 6f6e 7369 7374 program consist │ │ │ │ -00262380: 7320 6f66 2061 2073 696e 676c 6520 6578 s of a single ex │ │ │ │ -00262390: 6563 7574 6162 6c65 2062 696e 6172 7920 ecutable binary │ │ │ │ -002623a0: 6669 6c65 2c20 7468 656e 206e 616d 650a file, then name. │ │ │ │ -002623b0: 7368 6f75 6c64 2063 6f69 6e63 6964 6520 should coincide │ │ │ │ -002623c0: 7769 7468 2074 6865 206e 616d 6520 6f66 with the name of │ │ │ │ -002623d0: 2074 6869 7320 6669 6c65 2e0a 0a2b 2d2d this file...+-- │ │ │ │ -002623e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002623f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00262400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002616b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002616c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002616d0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +002616e0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +002616f0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +00261700: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +00261710: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +00261720: 2d31 2e32 362e 3035 2b64 732f 4d32 2f4d -1.26.05+ds/M2/M │ │ │ │ +00261730: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +00261740: 732f 0a4d 6163 6175 6c61 7932 446f 632f s/.Macaulay2Doc/ │ │ │ │ +00261750: 6f76 5f73 7973 7465 6d2e 6d32 3a38 3730 ov_system.m2:870 │ │ │ │ +00261760: 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 6361 :0....File: Maca │ │ │ │ +00261770: 756c 6179 3244 6f63 2e69 6e66 6f2c 204e ulay2Doc.info, N │ │ │ │ +00261780: 6f64 653a 2066 696e 6450 726f 6772 616d ode: findProgram │ │ │ │ +00261790: 2c20 4e65 7874 3a20 7275 6e50 726f 6772 , Next: runProgr │ │ │ │ +002617a0: 616d 2c20 5072 6576 3a20 7275 6e2c 2055 am, Prev: run, U │ │ │ │ +002617b0: 703a 2063 6f6d 6d75 6e69 6361 7469 6e67 p: communicating │ │ │ │ +002617c0: 2077 6974 6820 7072 6f67 7261 6d73 0a0a with programs.. │ │ │ │ +002617d0: 6669 6e64 5072 6f67 7261 6d20 2d2d 206c findProgram -- l │ │ │ │ +002617e0: 6f61 6420 6578 7465 726e 616c 2070 726f oad external pro │ │ │ │ +002617f0: 6772 616d 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a gram.*********** │ │ │ │ +00261800: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00261810: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +00261820: 7361 6765 3a20 0a20 2020 2020 2020 2066 sage: . f │ │ │ │ +00261830: 696e 6450 726f 6772 616d 206e 616d 650a indProgram name. │ │ │ │ +00261840: 2020 2020 2020 2020 6669 6e64 5072 6f67 findProg │ │ │ │ +00261850: 7261 6d28 6e61 6d65 2c20 636d 6429 0a20 ram(name, cmd). │ │ │ │ +00261860: 2020 2020 2020 2066 696e 6450 726f 6772 findProgr │ │ │ │ +00261870: 616d 286e 616d 652c 2063 6d64 7329 0a20 am(name, cmds). │ │ │ │ +00261880: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +00261890: 202a 206e 616d 652c 2061 202a 6e6f 7465 * name, a *note │ │ │ │ +002618a0: 2073 7472 696e 673a 2053 7472 696e 672c string: String, │ │ │ │ +002618b0: 2c20 7468 6520 6e61 6d65 206f 6620 7468 , the name of th │ │ │ │ +002618c0: 6520 7072 6f67 7261 6d20 746f 206c 6f61 e program to loa │ │ │ │ +002618d0: 642e 2020 5468 6973 0a20 2020 2020 2020 d. This. │ │ │ │ +002618e0: 2073 686f 756c 6420 6d61 7463 6820 7468 should match th │ │ │ │ +002618f0: 6520 636f 7272 6573 706f 6e64 696e 6720 e corresponding │ │ │ │ +00261900: 6b65 7920 696e 202a 6e6f 7465 2070 726f key in *note pro │ │ │ │ +00261910: 6772 616d 5061 7468 733a 0a20 2020 2020 gramPaths:. │ │ │ │ +00261920: 2020 2070 726f 6772 616d 5061 7468 732c programPaths, │ │ │ │ +00261930: 2e0a 2020 2020 2020 2a20 636d 642c 2061 .. * cmd, a │ │ │ │ +00261940: 202a 6e6f 7465 2073 7472 696e 673a 2053 *note string: S │ │ │ │ +00261950: 7472 696e 672c 2c20 6120 636f 6d6d 616e tring,, a comman │ │ │ │ +00261960: 6420 746f 2072 756e 2074 6861 7420 7368 d to run that sh │ │ │ │ +00261970: 6f75 6c64 2072 6574 7572 6e20 3020 6966 ould return 0 if │ │ │ │ +00261980: 0a20 2020 2020 2020 2074 6865 2070 726f . the pro │ │ │ │ +00261990: 6772 616d 2069 7320 7072 6573 656e 742e gram is present. │ │ │ │ +002619a0: 0a20 2020 2020 202a 2063 6d64 732c 2061 . * cmds, a │ │ │ │ +002619b0: 202a 6e6f 7465 206c 6973 743a 204c 6973 *note list: Lis │ │ │ │ +002619c0: 742c 2c20 6120 6c69 7374 206f 6620 636f t,, a list of co │ │ │ │ +002619d0: 6d6d 616e 6473 2074 6f20 7275 6e20 7468 mmands to run th │ │ │ │ +002619e0: 6174 2073 686f 756c 6420 616c 6c0a 2020 at should all. │ │ │ │ +002619f0: 2020 2020 2020 7265 7475 726e 2030 2069 return 0 i │ │ │ │ +00261a00: 6620 7468 6520 7072 6f67 7261 6d20 6973 f the program is │ │ │ │ +00261a10: 2070 7265 7365 6e74 2e0a 2020 2a20 2a6e present.. * *n │ │ │ │ +00261a20: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70 ote Optional inp │ │ │ │ +00261a30: 7574 733a 2075 7369 6e67 2066 756e 6374 uts: using funct │ │ │ │ +00261a40: 696f 6e73 2077 6974 6820 6f70 7469 6f6e ions with option │ │ │ │ +00261a50: 616c 2069 6e70 7574 732c 3a0a 2020 2020 al inputs,:. │ │ │ │ +00261a60: 2020 2a20 5261 6973 6545 7272 6f72 203d * RaiseError = │ │ │ │ +00261a70: 3e20 6120 2a6e 6f74 6520 426f 6f6c 6561 > a *note Boolea │ │ │ │ +00261a80: 6e20 7661 6c75 653a 2042 6f6f 6c65 616e n value: Boolean │ │ │ │ +00261a90: 2c2c 2064 6566 6175 6c74 2076 616c 7565 ,, default value │ │ │ │ +00261aa0: 2074 7275 652c 0a20 2020 2020 2020 2077 true,. w │ │ │ │ +00261ab0: 6865 7468 6572 2074 6f20 7261 6973 6520 hether to raise │ │ │ │ +00261ac0: 616e 2065 7272 6f72 2069 6620 7468 6520 an error if the │ │ │ │ +00261ad0: 7072 6f67 7261 6d20 6973 206e 6f74 2066 program is not f │ │ │ │ +00261ae0: 6f75 6e64 2e0a 2020 2020 2020 2a20 5665 ound.. * Ve │ │ │ │ +00261af0: 7262 6f73 6520 3d3e 2061 202a 6e6f 7465 rbose => a *note │ │ │ │ +00261b00: 2042 6f6f 6c65 616e 2076 616c 7565 3a20 Boolean value: │ │ │ │ +00261b10: 426f 6f6c 6561 6e2c 2c20 6465 6661 756c Boolean,, defaul │ │ │ │ +00261b20: 7420 7661 6c75 6520 6661 6c73 652c 0a20 t value false,. │ │ │ │ +00261b30: 2020 2020 2020 2077 6865 7468 6572 2074 whether t │ │ │ │ +00261b40: 6f20 696e 666f 726d 2074 6865 2075 7365 o inform the use │ │ │ │ +00261b50: 7220 6f66 2065 6163 6820 7061 7468 2074 r of each path t │ │ │ │ +00261b60: 6861 7420 6973 2063 6865 636b 6564 2e0a hat is checked.. │ │ │ │ +00261b70: 2020 2020 2020 2a20 5072 6566 6978 203d * Prefix = │ │ │ │ +00261b80: 3e20 6120 2a6e 6f74 6520 6c69 7374 3a20 > a *note list: │ │ │ │ +00261b90: 4c69 7374 2c2c 2064 6566 6175 6c74 2076 List,, default v │ │ │ │ +00261ba0: 616c 7565 207b 7d2c 2061 206c 6973 7420 alue {}, a list │ │ │ │ +00261bb0: 6f66 2073 6571 7565 6e63 6573 0a20 2020 of sequences. │ │ │ │ +00261bc0: 2020 2020 2063 6f6e 7461 696e 696e 6720 containing │ │ │ │ +00261bd0: 7477 6f20 7374 7269 6e67 7320 6964 656e two strings iden │ │ │ │ +00261be0: 7469 6679 696e 6720 6120 7072 6566 6978 tifying a prefix │ │ │ │ +00261bf0: 2074 6861 7420 6973 2061 6464 6564 2074 that is added t │ │ │ │ +00261c00: 6f20 7468 650a 2020 2020 2020 2020 6578 o the. ex │ │ │ │ +00261c10: 6563 7574 6162 6c65 2062 696e 6172 6965 ecutable binarie │ │ │ │ +00261c20: 7320 6265 6c6f 6e67 696e 6720 746f 2074 s belonging to t │ │ │ │ +00261c30: 6865 2070 726f 6772 616d 2062 7920 736f he program by so │ │ │ │ +00261c40: 6d65 2064 6973 7472 6962 7574 696f 6e73 me distributions │ │ │ │ +00261c50: 2e0a 2020 2020 2020 2020 5468 6573 6520 .. These │ │ │ │ +00261c60: 7365 7175 656e 6365 7320 7368 6f75 6c64 sequences should │ │ │ │ +00261c70: 2062 6520 6f66 2074 6865 2066 6f72 6d20 be of the form │ │ │ │ +00261c80: 2872 6567 6578 2c20 7072 6566 6978 292c (regex, prefix), │ │ │ │ +00261c90: 2077 6865 7265 2072 6567 6578 2069 7320 where regex is │ │ │ │ +00261ca0: 610a 2020 2020 2020 2020 2a6e 6f74 6520 a. *note │ │ │ │ +00261cb0: 7265 6775 6c61 7220 6578 7072 6573 7369 regular expressi │ │ │ │ +00261cc0: 6f6e 3a20 7265 6775 6c61 7220 6578 7072 on: regular expr │ │ │ │ +00261cd0: 6573 7369 6f6e 732c 2074 6861 7420 7368 essions, that sh │ │ │ │ +00261ce0: 6f75 6c64 206d 6174 6368 2061 6c6c 0a20 ould match all. │ │ │ │ +00261cf0: 2020 2020 2020 2062 696e 6172 7920 6578 binary ex │ │ │ │ +00261d00: 6563 7574 6162 6c65 7320 7468 6174 206e ecutables that n │ │ │ │ +00261d10: 6565 6420 7468 6520 7072 6566 6978 2061 eed the prefix a │ │ │ │ +00261d20: 6e64 2070 7265 6669 7820 6973 2074 6865 nd prefix is the │ │ │ │ +00261d30: 2070 7265 6669 780a 2020 2020 2020 2020 prefix. │ │ │ │ +00261d40: 6974 7365 6c66 2e0a 2020 2020 2020 2a20 itself.. * │ │ │ │ +00261d50: 4164 6469 7469 6f6e 616c 5061 7468 7320 AdditionalPaths │ │ │ │ +00261d60: 3d3e 2061 202a 6e6f 7465 206c 6973 743a => a *note list: │ │ │ │ +00261d70: 204c 6973 742c 2c20 6465 6661 756c 7420 List,, default │ │ │ │ +00261d80: 7661 6c75 6520 7b7d 2c20 6120 6c69 7374 value {}, a list │ │ │ │ +00261d90: 206f 660a 2020 2020 2020 2020 7374 7269 of. stri │ │ │ │ +00261da0: 6e67 7320 636f 6e74 6169 6e69 6e67 2061 ngs containing a │ │ │ │ +00261db0: 6e79 2070 6174 6873 2074 6f20 6368 6563 ny paths to chec │ │ │ │ +00261dc0: 6b20 666f 7220 7468 6520 7072 6f67 7261 k for the progra │ │ │ │ +00261dd0: 6d20 696e 2061 6464 6974 696f 6e20 746f m in addition to │ │ │ │ +00261de0: 0a20 2020 2020 2020 2074 6865 2064 6566 . the def │ │ │ │ +00261df0: 6175 6c74 206f 6e65 732e 0a20 2020 2020 ault ones.. │ │ │ │ +00261e00: 202a 204d 696e 696d 756d 5665 7273 696f * MinimumVersio │ │ │ │ +00261e10: 6e20 3d3e 2061 202a 6e6f 7465 2073 6571 n => a *note seq │ │ │ │ +00261e20: 7565 6e63 653a 2053 6571 7565 6e63 652c uence: Sequence, │ │ │ │ +00261e30: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +00261e40: 6e75 6c6c 2c0a 2020 2020 2020 2020 636f null,. co │ │ │ │ +00261e50: 6e74 6169 6e69 6e67 2074 776f 2073 7472 ntaining two str │ │ │ │ +00261e60: 696e 6773 2074 6865 2066 6f72 6d20 286d ings the form (m │ │ │ │ +00261e70: 696e 5665 7273 696f 6e2c 2076 6572 7369 inVersion, versi │ │ │ │ +00261e80: 6f6e 436f 6d6d 616e 6429 2077 6865 7265 onCommand) where │ │ │ │ +00261e90: 0a20 2020 2020 2020 206d 696e 5665 7273 . minVers │ │ │ │ +00261ea0: 696f 6e20 6973 2074 6865 206d 696e 696d ion is the minim │ │ │ │ +00261eb0: 756d 2072 6571 7569 7265 6420 7665 7273 um required vers │ │ │ │ +00261ec0: 696f 6e20 6f66 2074 6865 2070 726f 6772 ion of the progr │ │ │ │ +00261ed0: 616d 2061 6e64 0a20 2020 2020 2020 2076 am and. v │ │ │ │ +00261ee0: 6572 7369 6f6e 436f 6d6d 616e 6420 6973 ersionCommand is │ │ │ │ +00261ef0: 2061 2073 6865 6c6c 2063 6f6d 6d61 6e64 a shell command │ │ │ │ +00261f00: 2074 6f20 6f62 7461 696e 2074 6865 2076 to obtain the v │ │ │ │ +00261f10: 6572 7369 6f6e 206e 756d 6265 7220 6f66 ersion number of │ │ │ │ +00261f20: 2061 6e0a 2020 2020 2020 2020 696e 7374 an. inst │ │ │ │ +00261f30: 616c 6c65 6420 7072 6f67 7261 6d2e 0a20 alled program.. │ │ │ │ +00261f40: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +00261f50: 2020 2a20 616e 2069 6e73 7461 6e63 6520 * an instance │ │ │ │ +00261f60: 6f66 2074 6865 2074 7970 6520 2a6e 6f74 of the type *not │ │ │ │ +00261f70: 6520 5072 6f67 7261 6d3a 2050 726f 6772 e Program: Progr │ │ │ │ +00261f80: 616d 2c2c 2074 6865 2070 726f 6772 616d am,, the program │ │ │ │ +00261f90: 2074 6861 7420 7761 730a 2020 2020 2020 that was. │ │ │ │ +00261fa0: 2020 6c6f 6164 6564 2e20 2049 6620 7468 loaded. If th │ │ │ │ +00261fb0: 6520 7072 6f67 7261 6d20 6973 206e 6f74 e program is not │ │ │ │ +00261fc0: 2066 6f75 6e64 2061 6e64 2052 6169 7365 found and Raise │ │ │ │ +00261fd0: 4572 726f 7220 6973 2073 6574 2074 6f20 Error is set to │ │ │ │ +00261fe0: 2a6e 6f74 650a 2020 2020 2020 2020 6661 *note. fa │ │ │ │ +00261ff0: 6c73 653a 2066 616c 7365 2c2c 2074 6865 lse: false,, the │ │ │ │ +00262000: 6e20 2a6e 6f74 6520 6e75 6c6c 3a20 6e75 n *note null: nu │ │ │ │ +00262010: 6c6c 2c20 6973 2072 6574 7572 6e65 642e ll, is returned. │ │ │ │ +00262020: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +00262030: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 =========..This │ │ │ │ +00262040: 6675 6e63 7469 6f6e 2063 6865 636b 7320 function checks │ │ │ │ +00262050: 666f 7220 7468 6520 6578 6973 7465 6e63 for the existenc │ │ │ │ +00262060: 6520 6f66 2061 6e20 6578 7465 726e 616c e of an external │ │ │ │ +00262070: 2070 726f 6772 616d 2062 7920 7275 6e6e program by runn │ │ │ │ +00262080: 696e 6720 636d 640a 286f 7220 6576 6572 ing cmd.(or ever │ │ │ │ +00262090: 7920 656c 656d 656e 7420 6f66 2063 6d64 y element of cmd │ │ │ │ +002620a0: 7329 2c20 7072 6570 656e 6465 6420 7769 s), prepended wi │ │ │ │ +002620b0: 7468 2076 6172 696f 7573 2070 6174 6873 th various paths │ │ │ │ +002620c0: 2069 6e20 7468 6520 666f 6c6c 6f77 696e in the followin │ │ │ │ +002620d0: 670a 6f72 6465 723a 0a0a 2020 2a20 5468 g.order:.. * Th │ │ │ │ +002620e0: 6520 7573 6572 2d64 6566 696e 6564 2070 e user-defined p │ │ │ │ +002620f0: 6174 6820 7370 6563 6966 6965 6420 6279 ath specified by │ │ │ │ +00262100: 2070 726f 6772 616d 5061 7468 7323 6e61 programPaths#na │ │ │ │ +00262110: 6d65 2c20 6966 2069 7420 6578 6973 7473 me, if it exists │ │ │ │ +00262120: 2e0a 2020 2a20 5468 6520 7061 7468 2073 .. * The path s │ │ │ │ +00262130: 7065 6369 6669 6564 2062 7920 7072 6566 pecified by pref │ │ │ │ +00262140: 6978 4469 7265 6374 6f72 7920 7c20 6375 ixDirectory | cu │ │ │ │ +00262150: 7272 656e 744c 6179 6f75 7423 2270 726f rrentLayout#"pro │ │ │ │ +00262160: 6772 616d 7322 2c20 7768 6572 6520 7468 grams", where th │ │ │ │ +00262170: 650a 2020 2020 7072 6f67 7261 6d73 2073 e. programs s │ │ │ │ +00262180: 6869 7070 6564 2077 6974 6820 4d61 6361 hipped with Maca │ │ │ │ +00262190: 756c 6179 3220 6172 6520 696e 7374 616c ulay2 are instal │ │ │ │ +002621a0: 6c65 642e 0a20 202a 2045 6163 6820 7061 led.. * Each pa │ │ │ │ +002621b0: 7468 2073 7065 6369 6669 6564 2062 7920 th specified by │ │ │ │ +002621c0: 7468 6520 4164 6469 7469 6f6e 616c 5061 the AdditionalPa │ │ │ │ +002621d0: 7468 7320 6f70 7469 6f6e 2e0a 2020 2a20 ths option.. * │ │ │ │ +002621e0: 4561 6368 2070 6174 6820 7370 6563 6966 Each path specif │ │ │ │ +002621f0: 6965 6420 6279 2074 6865 2075 7365 7227 ied by the user' │ │ │ │ +00262200: 7320 5041 5448 2065 6e76 6972 6f6e 6d65 s PATH environme │ │ │ │ +00262210: 6e74 2076 6172 6961 626c 652e 0a20 202a nt variable.. * │ │ │ │ +00262220: 2054 6865 2070 6174 6820 746f 204d 322d The path to M2- │ │ │ │ +00262230: 6269 6e61 7279 2e0a 0a46 6f72 2065 6163 binary...For eac │ │ │ │ +00262240: 6820 7061 7468 2c20 616e 7920 7072 6566 h path, any pref │ │ │ │ +00262250: 6978 6573 2073 7065 6369 6669 6564 2062 ixes specified b │ │ │ │ +00262260: 7920 7468 6520 5072 6566 6978 206f 7074 y the Prefix opt │ │ │ │ +00262270: 696f 6e20 6172 6520 6368 6563 6b65 642e ion are checked. │ │ │ │ +00262280: 0a0a 4f6e 6365 2074 6869 7320 6973 2073 ..Once this is s │ │ │ │ +00262290: 7563 6365 7373 6675 6c20 2869 2e65 2e2c uccessful (i.e., │ │ │ │ +002622a0: 2063 6d64 206f 7220 6561 6368 2065 6c65 cmd or each ele │ │ │ │ +002622b0: 6d65 6e74 206f 6620 636d 6473 2072 6574 ment of cmds ret │ │ │ │ +002622c0: 7572 6e73 2061 2076 616c 7565 206f 660a urns a value of. │ │ │ │ +002622d0: 3029 2074 6865 6e20 6120 2a6e 6f74 6520 0) then a *note │ │ │ │ +002622e0: 5072 6f67 7261 6d3a 2050 726f 6772 616d Program: Program │ │ │ │ +002622f0: 2c20 6f62 6a65 6374 2069 7320 7265 7475 , object is retu │ │ │ │ +00262300: 726e 6564 2e20 2049 6620 6974 2069 7320 rned. If it is │ │ │ │ +00262310: 756e 7375 6363 6573 7366 756c 2c0a 7468 unsuccessful,.th │ │ │ │ +00262320: 656e 2065 6974 6865 7220 616e 2065 7272 en either an err │ │ │ │ +00262330: 6f72 2069 7320 7261 6973 6564 206f 7220 or is raised or │ │ │ │ +00262340: 2a6e 6f74 6520 6e75 6c6c 3a20 6e75 6c6c *note null: null │ │ │ │ +00262350: 2c20 6973 2072 6574 7572 6e65 642c 2064 , is returned, d │ │ │ │ +00262360: 6570 656e 6469 6e67 206f 6e0a 7468 6520 epending on.the │ │ │ │ +00262370: 7661 6c75 6520 6f66 2052 6169 7365 4572 value of RaiseEr │ │ │ │ +00262380: 726f 722e 0a0a 4e6f 7465 2074 6861 7420 ror...Note that │ │ │ │ +00262390: 6966 2061 2070 726f 6772 616d 2063 6f6e if a program con │ │ │ │ +002623a0: 7369 7374 7320 6f66 2061 2073 696e 676c sists of a singl │ │ │ │ +002623b0: 6520 6578 6563 7574 6162 6c65 2062 696e e executable bin │ │ │ │ +002623c0: 6172 7920 6669 6c65 2c20 7468 656e 206e ary file, then n │ │ │ │ +002623d0: 616d 650a 7368 6f75 6c64 2063 6f69 6e63 ame.should coinc │ │ │ │ +002623e0: 6964 6520 7769 7468 2074 6865 206e 616d ide with the nam │ │ │ │ +002623f0: 6520 6f66 2074 6869 7320 6669 6c65 2e0a e of this file.. │ │ │ │ +00262400: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00262410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00262420: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7072 ------+.|i1 : pr │ │ │ │ -00262430: 6f67 7261 6d50 6174 6873 2322 6766 616e ogramPaths#"gfan │ │ │ │ -00262440: 2220 3d20 222f 7061 7468 2f74 6f2f 6766 " = "/path/to/gf │ │ │ │ -00262450: 616e 2f22 2020 2020 2020 2020 2020 2020 an/" │ │ │ │ -00262460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262470: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00262420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00262430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00262440: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ +00262450: 3a20 7072 6f67 7261 6d50 6174 6873 2322 : programPaths#" │ │ │ │ +00262460: 6766 616e 2220 3d20 222f 7061 7468 2f74 gfan" = "/path/t │ │ │ │ +00262470: 6f2f 6766 616e 2f22 2020 2020 2020 2020 o/gfan/" │ │ │ │ 00262480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262490: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 002624a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002624b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -002624c0: 3120 3d20 2f70 6174 682f 746f 2f67 6661 1 = /path/to/gfa │ │ │ │ -002624d0: 6e2f 2020 2020 2020 2020 2020 2020 2020 n/ │ │ │ │ -002624e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002624f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262500: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00262510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00262520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002624b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002624c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002624d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002624e0: 7c0a 7c6f 3120 3d20 2f70 6174 682f 746f |.|o1 = /path/to │ │ │ │ +002624f0: 2f67 6661 6e2f 2020 2020 2020 2020 2020 /gfan/ │ │ │ │ +00262500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262520: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00262530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00262540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00262550: 2d2d 2b0a 7c69 3220 3a20 6766 616e 203d --+.|i2 : gfan = │ │ │ │ -00262560: 2066 696e 6450 726f 6772 616d 2822 6766 findProgram("gf │ │ │ │ -00262570: 616e 222c 2022 6766 616e 205f 7665 7273 an", "gfan _vers │ │ │ │ -00262580: 696f 6e20 2d2d 6865 6c70 222c 2056 6572 ion --help", Ver │ │ │ │ -00262590: 626f 7365 203d 3e20 7472 7565 297c 0a7c bose => true)|.| │ │ │ │ -002625a0: 202d 2d20 2f70 6174 682f 746f 2f67 6661 -- /path/to/gfa │ │ │ │ -002625b0: 6e2f 6766 616e 2064 6f65 7320 6e6f 7420 n/gfan does not │ │ │ │ -002625c0: 6578 6973 7420 2020 2020 2020 2020 2020 exist │ │ │ │ -002625d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002625e0: 2020 2020 2020 2020 7c0a 7c20 2d2d 202f |.| -- / │ │ │ │ -002625f0: 7573 722f 6c69 6265 7865 632f 4d61 6361 usr/libexec/Maca │ │ │ │ -00262600: 756c 6179 322f 6269 6e2f 6766 616e 2064 ulay2/bin/gfan d │ │ │ │ -00262610: 6f65 7320 6e6f 7420 6578 6973 7420 2020 oes not exist │ │ │ │ -00262620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262630: 2020 207c 0a7c 202d 2d20 2f75 7372 2f6c |.| -- /usr/l │ │ │ │ -00262640: 6f63 616c 2f73 6269 6e2f 6766 616e 2064 ocal/sbin/gfan d │ │ │ │ -00262650: 6f65 7320 6e6f 7420 6578 6973 7420 2020 oes not exist │ │ │ │ -00262660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262670: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00262680: 7c20 2d2d 202f 7573 722f 6c6f 6361 6c2f | -- /usr/local/ │ │ │ │ -00262690: 6269 6e2f 6766 616e 2064 6f65 7320 6e6f bin/gfan does no │ │ │ │ -002626a0: 7420 6578 6973 7420 2020 2020 2020 2020 t exist │ │ │ │ -002626b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002626c0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -002626d0: 2f75 7372 2f73 6269 6e2f 6766 616e 2064 /usr/sbin/gfan d │ │ │ │ -002626e0: 6f65 7320 6e6f 7420 6578 6973 7420 2020 oes not exist │ │ │ │ -002626f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262710: 2020 2020 7c0a 7c20 2d2d 202f 7573 722f |.| -- /usr/ │ │ │ │ -00262720: 6269 6e2f 6766 616e 2065 7869 7374 7320 bin/gfan exists │ │ │ │ -00262730: 616e 6420 6973 2065 7865 6375 7461 626c and is executabl │ │ │ │ -00262740: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ -00262750: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00262760: 0a7c 202d 2d20 7275 6e6e 696e 6720 222f .| -- running "/ │ │ │ │ -00262770: 7573 722f 6269 6e2f 6766 616e 205f 7665 usr/bin/gfan _ve │ │ │ │ -00262780: 7273 696f 6e20 2d2d 6865 6c70 223a 2020 rsion --help": │ │ │ │ -00262790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002627a0: 2020 2020 2020 2020 2020 7c0a 7c54 6869 |.|Thi │ │ │ │ -002627b0: 7320 7072 6f67 7261 6d20 7772 6974 6573 s program writes │ │ │ │ -002627c0: 206f 7574 2076 6572 7369 6f6e 2069 6e66 out version inf │ │ │ │ -002627d0: 6f72 6d61 7469 6f6e 206f 6620 7468 6520 ormation of the │ │ │ │ -002627e0: 4766 616e 2069 6e73 7461 6c6c 6174 696f Gfan installatio │ │ │ │ -002627f0: 6e2e 2020 207c 0a7c 202d 2d20 7265 7475 n. |.| -- retu │ │ │ │ -00262800: 726e 2076 616c 7565 3a20 3020 2020 2020 rn value: 0 │ │ │ │ -00262810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00262560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00262570: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 6766 ------+.|i2 : gf │ │ │ │ +00262580: 616e 203d 2066 696e 6450 726f 6772 616d an = findProgram │ │ │ │ +00262590: 2822 6766 616e 222c 2022 6766 616e 205f ("gfan", "gfan _ │ │ │ │ +002625a0: 7665 7273 696f 6e20 2d2d 6865 6c70 222c version --help", │ │ │ │ +002625b0: 2056 6572 626f 7365 203d 3e20 7472 7565 Verbose => true │ │ │ │ +002625c0: 297c 0a7c 202d 2d20 2f70 6174 682f 746f )|.| -- /path/to │ │ │ │ +002625d0: 2f67 6661 6e2f 6766 616e 2064 6f65 7320 /gfan/gfan does │ │ │ │ +002625e0: 6e6f 7420 6578 6973 7420 2020 2020 2020 not exist │ │ │ │ +002625f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262600: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00262610: 2d2d 202f 7573 722f 6c69 6265 7865 632f -- /usr/libexec/ │ │ │ │ +00262620: 4d61 6361 756c 6179 322f 6269 6e2f 6766 Macaulay2/bin/gf │ │ │ │ +00262630: 616e 2064 6f65 7320 6e6f 7420 6578 6973 an does not exis │ │ │ │ +00262640: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00262650: 2020 2020 2020 207c 0a7c 202d 2d20 2f75 |.| -- /u │ │ │ │ +00262660: 7372 2f6c 6f63 616c 2f73 6269 6e2f 6766 sr/local/sbin/gf │ │ │ │ +00262670: 616e 2064 6f65 7320 6e6f 7420 6578 6973 an does not exis │ │ │ │ +00262680: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00262690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002626a0: 2020 7c0a 7c20 2d2d 202f 7573 722f 6c6f |.| -- /usr/lo │ │ │ │ +002626b0: 6361 6c2f 6269 6e2f 6766 616e 2064 6f65 cal/bin/gfan doe │ │ │ │ +002626c0: 7320 6e6f 7420 6578 6973 7420 2020 2020 s not exist │ │ │ │ +002626d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002626e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +002626f0: 202d 2d20 2f75 7372 2f73 6269 6e2f 6766 -- /usr/sbin/gf │ │ │ │ +00262700: 616e 2064 6f65 7320 6e6f 7420 6578 6973 an does not exis │ │ │ │ +00262710: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00262720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262730: 2020 2020 2020 2020 7c0a 7c20 2d2d 202f |.| -- / │ │ │ │ +00262740: 7573 722f 6269 6e2f 6766 616e 2065 7869 usr/bin/gfan exi │ │ │ │ +00262750: 7374 7320 616e 6420 6973 2065 7865 6375 sts and is execu │ │ │ │ +00262760: 7461 626c 6520 2020 2020 2020 2020 2020 table │ │ │ │ +00262770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262780: 2020 207c 0a7c 202d 2d20 7275 6e6e 696e |.| -- runnin │ │ │ │ +00262790: 6720 222f 7573 722f 6269 6e2f 6766 616e g "/usr/bin/gfan │ │ │ │ +002627a0: 205f 7665 7273 696f 6e20 2d2d 6865 6c70 _version --help │ │ │ │ +002627b0: 223a 2020 2020 2020 2020 2020 2020 2020 ": │ │ │ │ +002627c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +002627d0: 7c54 6869 7320 7072 6f67 7261 6d20 7772 |This program wr │ │ │ │ +002627e0: 6974 6573 206f 7574 2076 6572 7369 6f6e ites out version │ │ │ │ +002627f0: 2069 6e66 6f72 6d61 7469 6f6e 206f 6620 information of │ │ │ │ +00262800: 7468 6520 4766 616e 2069 6e73 7461 6c6c the Gfan install │ │ │ │ +00262810: 6174 696f 6e2e 2020 207c 0a7c 202d 2d20 ation. |.| -- │ │ │ │ +00262820: 7265 7475 726e 2076 616c 7565 3a20 3020 return value: 0 │ │ │ │ 00262830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262840: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00262840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262860: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00262870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262880: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00262890: 203d 2067 6661 6e20 2020 2020 2020 2020 = gfan │ │ │ │ -002628a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002628b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002628a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +002628b0: 0a7c 6f32 203d 2067 6661 6e20 2020 2020 .|o2 = gfan │ │ │ │ 002628c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002628d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +002628d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002628e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002628f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002628f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00262900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262920: 207c 0a7c 6f32 203a 2050 726f 6772 616d |.|o2 : Program │ │ │ │ +00262920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262960: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00262970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00262980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00262990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00262940: 2020 2020 207c 0a7c 6f32 203a 2050 726f |.|o2 : Pro │ │ │ │ +00262950: 6772 616d 2020 2020 2020 2020 2020 2020 gram │ │ │ │ +00262960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262990: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 002629a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002629b0: 2d2d 2d2d 2d2d 2d2b 0a0a 4966 2063 6d64 -------+..If cmd │ │ │ │ -002629c0: 2069 7320 6e6f 7420 7072 6f76 6964 6564 is not provided │ │ │ │ -002629d0: 2c20 7468 656e 206e 616d 6520 6973 2072 , then name is r │ │ │ │ -002629e0: 756e 2077 6974 6820 7468 6520 636f 6d6d un with the comm │ │ │ │ -002629f0: 6f6e 202d 2d76 6572 7369 6f6e 2063 6f6d on --version com │ │ │ │ -00262a00: 6d61 6e64 206c 696e 650a 6f70 7469 6f6e mand line.option │ │ │ │ -00262a10: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ -00262a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00262a30: 0a7c 6933 203a 2066 696e 6450 726f 6772 .|i3 : findProgr │ │ │ │ -00262a40: 616d 2022 6e6f 726d 616c 697a 227c 0a7c am "normaliz"|.| │ │ │ │ -00262a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262a60: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ -00262a70: 203d 206e 6f72 6d61 6c69 7a20 2020 2020 = normaliz │ │ │ │ -00262a80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00262a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262aa0: 2020 2020 2020 207c 0a7c 6f33 203a 2050 |.|o3 : P │ │ │ │ -00262ab0: 726f 6772 616d 2020 2020 2020 2020 2020 rogram │ │ │ │ -00262ac0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00262ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00262ae0: 2d2d 2d2b 0a0a 4f6e 6520 7072 6f67 7261 ---+..One progra │ │ │ │ -00262af0: 6d20 7468 6174 2069 7320 7368 6970 7065 m that is shippe │ │ │ │ -00262b00: 6420 7769 7468 2061 2076 6172 6965 7479 d with a variety │ │ │ │ -00262b10: 206f 6620 7072 6566 6978 6573 2069 6e20 of prefixes in │ │ │ │ -00262b20: 6469 6666 6572 656e 740a 6469 7374 7269 different.distri │ │ │ │ -00262b30: 6275 7469 6f6e 7320 616e 6420 666f 7220 butions and for │ │ │ │ -00262b40: 7768 6963 6820 7468 6520 5072 6566 6978 which the Prefix │ │ │ │ -00262b50: 206f 7074 696f 6e20 6973 2075 7365 6675 option is usefu │ │ │ │ -00262b60: 6c20 6973 2054 4f50 434f 4d3a 0a0a 2b2d l is TOPCOM:..+- │ │ │ │ -00262b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00262b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00262b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002629b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002629c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002629d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4966 -----------+..If │ │ │ │ +002629e0: 2063 6d64 2069 7320 6e6f 7420 7072 6f76 cmd is not prov │ │ │ │ +002629f0: 6964 6564 2c20 7468 656e 206e 616d 6520 ided, then name │ │ │ │ +00262a00: 6973 2072 756e 2077 6974 6820 7468 6520 is run with the │ │ │ │ +00262a10: 636f 6d6d 6f6e 202d 2d76 6572 7369 6f6e common --version │ │ │ │ +00262a20: 2063 6f6d 6d61 6e64 206c 696e 650a 6f70 command line.op │ │ │ │ +00262a30: 7469 6f6e 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d tion...+-------- │ │ │ │ +00262a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00262a50: 2d2d 2d2b 0a7c 6933 203a 2066 696e 6450 ---+.|i3 : findP │ │ │ │ +00262a60: 726f 6772 616d 2022 6e6f 726d 616c 697a rogram "normaliz │ │ │ │ +00262a70: 227c 0a7c 2020 2020 2020 2020 2020 2020 "|.| │ │ │ │ +00262a80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00262a90: 0a7c 6f33 203d 206e 6f72 6d61 6c69 7a20 .|o3 = normaliz │ │ │ │ +00262aa0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00262ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262ac0: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +00262ad0: 203a 2050 726f 6772 616d 2020 2020 2020 : Program │ │ │ │ +00262ae0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00262af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00262b00: 2d2d 2d2d 2d2d 2d2b 0a0a 4f6e 6520 7072 -------+..One pr │ │ │ │ +00262b10: 6f67 7261 6d20 7468 6174 2069 7320 7368 ogram that is sh │ │ │ │ +00262b20: 6970 7065 6420 7769 7468 2061 2076 6172 ipped with a var │ │ │ │ +00262b30: 6965 7479 206f 6620 7072 6566 6978 6573 iety of prefixes │ │ │ │ +00262b40: 2069 6e20 6469 6666 6572 656e 740a 6469 in different.di │ │ │ │ +00262b50: 7374 7269 6275 7469 6f6e 7320 616e 6420 stributions and │ │ │ │ +00262b60: 666f 7220 7768 6963 6820 7468 6520 5072 for which the Pr │ │ │ │ +00262b70: 6566 6978 206f 7074 696f 6e20 6973 2075 efix option is u │ │ │ │ +00262b80: 7365 6675 6c20 6973 2054 4f50 434f 4d3a seful is TOPCOM: │ │ │ │ +00262b90: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ 00262ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00262bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00262bc0: 3420 3a20 6669 6e64 5072 6f67 7261 6d28 4 : findProgram( │ │ │ │ -00262bd0: 2274 6f70 636f 6d22 2c20 2263 7562 6520 "topcom", "cube │ │ │ │ -00262be0: 3322 2c20 5665 7262 6f73 6520 3d3e 2074 3", Verbose => t │ │ │ │ -00262bf0: 7275 652c 2050 7265 6669 7820 3d3e 207b rue, Prefix => { │ │ │ │ -00262c00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00262c10: 2020 2020 2020 2822 2e2a 222c 2022 746f (".*", "to │ │ │ │ -00262c20: 7063 6f6d 2d22 292c 2020 2020 2020 2020 pcom-"), │ │ │ │ -00262c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262c50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00262c60: 2020 2020 2020 2822 5e28 6372 6f73 737c ("^(cross| │ │ │ │ -00262c70: 6375 6265 7c63 7963 6c69 637c 6879 7065 cube|cyclic|hype │ │ │ │ -00262c80: 7273 696d 706c 6578 7c6c 6174 7469 6365 rsimplex|lattice │ │ │ │ -00262c90: 2924 222c 2022 544f 5043 4f4d 2d22 292c )$", "TOPCOM-"), │ │ │ │ -00262ca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00262cb0: 2020 2020 2020 2822 5e63 7562 6524 222c ("^cube$", │ │ │ │ -00262cc0: 2022 746f 7063 6f6d 5f22 297d 2920 2020 "topcom_")}) │ │ │ │ -00262cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262cf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00262d00: 2d2d 202f 7573 722f 6c69 6265 7865 632f -- /usr/libexec/ │ │ │ │ -00262d10: 4d61 6361 756c 6179 322f 6269 6e2f 6375 Macaulay2/bin/cu │ │ │ │ -00262d20: 6265 2064 6f65 7320 6e6f 7420 6578 6973 be does not exis │ │ │ │ -00262d30: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -00262d40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00262d50: 2d2d 202f 7573 722f 6c69 6265 7865 632f -- /usr/libexec/ │ │ │ │ -00262d60: 4d61 6361 756c 6179 322f 6269 6e2f 746f Macaulay2/bin/to │ │ │ │ -00262d70: 7063 6f6d 2d63 7562 6520 646f 6573 206e pcom-cube does n │ │ │ │ -00262d80: 6f74 2065 7869 7374 2020 2020 2020 2020 ot exist │ │ │ │ -00262d90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00262da0: 2d2d 202f 7573 722f 6c69 6265 7865 632f -- /usr/libexec/ │ │ │ │ -00262db0: 4d61 6361 756c 6179 322f 6269 6e2f 544f Macaulay2/bin/TO │ │ │ │ -00262dc0: 5043 4f4d 2d63 7562 6520 646f 6573 206e PCOM-cube does n │ │ │ │ -00262dd0: 6f74 2065 7869 7374 2020 2020 2020 2020 ot exist │ │ │ │ -00262de0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00262df0: 2d2d 202f 7573 722f 6c69 6265 7865 632f -- /usr/libexec/ │ │ │ │ -00262e00: 4d61 6361 756c 6179 322f 6269 6e2f 746f Macaulay2/bin/to │ │ │ │ -00262e10: 7063 6f6d 5f63 7562 6520 646f 6573 206e pcom_cube does n │ │ │ │ -00262e20: 6f74 2065 7869 7374 2020 2020 2020 2020 ot exist │ │ │ │ -00262e30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00262e40: 2d2d 202f 7573 722f 6c6f 6361 6c2f 7362 -- /usr/local/sb │ │ │ │ -00262e50: 696e 2f63 7562 6520 646f 6573 206e 6f74 in/cube does not │ │ │ │ -00262e60: 2065 7869 7374 2020 2020 2020 2020 2020 exist │ │ │ │ -00262e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262e80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00262e90: 2d2d 202f 7573 722f 6c6f 6361 6c2f 7362 -- /usr/local/sb │ │ │ │ -00262ea0: 696e 2f74 6f70 636f 6d2d 6375 6265 2064 in/topcom-cube d │ │ │ │ -00262eb0: 6f65 7320 6e6f 7420 6578 6973 7420 2020 oes not exist │ │ │ │ -00262ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262ed0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00262ee0: 2d2d 202f 7573 722f 6c6f 6361 6c2f 7362 -- /usr/local/sb │ │ │ │ -00262ef0: 696e 2f54 4f50 434f 4d2d 6375 6265 2064 in/TOPCOM-cube d │ │ │ │ -00262f00: 6f65 7320 6e6f 7420 6578 6973 7420 2020 oes not exist │ │ │ │ -00262f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262f20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00262f30: 2d2d 202f 7573 722f 6c6f 6361 6c2f 7362 -- /usr/local/sb │ │ │ │ -00262f40: 696e 2f74 6f70 636f 6d5f 6375 6265 2064 in/topcom_cube d │ │ │ │ -00262f50: 6f65 7320 6e6f 7420 6578 6973 7420 2020 oes not exist │ │ │ │ -00262f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262f70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00262f80: 2d2d 202f 7573 722f 6c6f 6361 6c2f 6269 -- /usr/local/bi │ │ │ │ -00262f90: 6e2f 6375 6265 2064 6f65 7320 6e6f 7420 n/cube does not │ │ │ │ -00262fa0: 6578 6973 7420 2020 2020 2020 2020 2020 exist │ │ │ │ -00262fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00262fc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00262fd0: 2d2d 202f 7573 722f 6c6f 6361 6c2f 6269 -- /usr/local/bi │ │ │ │ -00262fe0: 6e2f 746f 7063 6f6d 2d63 7562 6520 646f n/topcom-cube do │ │ │ │ -00262ff0: 6573 206e 6f74 2065 7869 7374 2020 2020 es not exist │ │ │ │ -00263000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263010: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00263020: 2d2d 202f 7573 722f 6c6f 6361 6c2f 6269 -- /usr/local/bi │ │ │ │ -00263030: 6e2f 544f 5043 4f4d 2d63 7562 6520 646f n/TOPCOM-cube do │ │ │ │ -00263040: 6573 206e 6f74 2065 7869 7374 2020 2020 es not exist │ │ │ │ -00263050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263060: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00263070: 2d2d 202f 7573 722f 6c6f 6361 6c2f 6269 -- /usr/local/bi │ │ │ │ -00263080: 6e2f 746f 7063 6f6d 5f63 7562 6520 646f n/topcom_cube do │ │ │ │ -00263090: 6573 206e 6f74 2065 7869 7374 2020 2020 es not exist │ │ │ │ -002630a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002630b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -002630c0: 2d2d 202f 7573 722f 7362 696e 2f63 7562 -- /usr/sbin/cub │ │ │ │ -002630d0: 6520 646f 6573 206e 6f74 2065 7869 7374 e does not exist │ │ │ │ -002630e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 t exist │ │ │ │ -002631e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002631f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00263200: 2d2d 202f 7573 722f 6269 6e2f 6375 6265 -- /usr/bin/cube │ │ │ │ -00263210: 2064 6f65 7320 6e6f 7420 6578 6973 7420 does not exist │ │ │ │ -00263220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263240: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00263250: 2d2d 202f 7573 722f 6269 6e2f 746f 7063 -- /usr/bin/topc │ │ │ │ -00263260: 6f6d 2d63 7562 6520 6578 6973 7473 2061 om-cube exists a │ │ │ │ -00263270: 6e64 2069 7320 6578 6563 7574 6162 6c65 nd is executable │ │ │ │ -00263280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263290: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -002632a0: 2d2d 2072 756e 6e69 6e67 2022 2f75 7372 -- running "/usr │ │ │ │ -002632b0: 2f62 696e 2f74 6f70 636f 6d2d 6375 6265 /bin/topcom-cube │ │ │ │ -002632c0: 2033 223a 2020 2020 2020 2020 2020 2020 3": │ │ │ │ -002632d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002632e0: 2020 2020 2020 2020 2020 2020 7c0a 7c5b |.|[ │ │ │ │ -002632f0: 5b30 2c30 2c30 2c31 5d2c 5b31 2c30 2c30 [0,0,0,1],[1,0,0 │ │ │ │ -00263300: 2c31 5d2c 5b30 2c31 2c30 2c31 5d2c 5b31 ,1],[0,1,0,1],[1 │ │ │ │ -00263310: 2c31 2c30 2c31 5d2c 5b30 2c30 2c31 2c31 ,1,0,1],[0,0,1,1 │ │ │ │ -00263320: 5d2c 5b31 2c30 2c31 2c31 5d2c 5b30 2c31 ],[1,0,1,1],[0,1 │ │ │ │ -00263330: 2c31 2c31 5d2c 5b31 2c31 2c31 7c0a 7c5b ,1,1],[1,1,1|.|[ │ │ │ │ -00263340: 5b37 2c36 2c35 2c34 2c33 2c32 2c31 2c30 [7,6,5,4,3,2,1,0 │ │ │ │ -00263350: 5d2c 5b36 2c37 2c34 2c35 2c32 2c33 2c30 ],[6,7,4,5,2,3,0 │ │ │ │ -00263360: 2c31 5d2c 5b34 2c36 2c35 2c37 2c30 2c32 ,1],[4,6,5,7,0,2 │ │ │ │ -00263370: 2c31 2c33 5d2c 5b30 2c34 2c32 2c36 2c31 ,1,3],[0,4,2,6,1 │ │ │ │ -00263380: 2c35 2c33 2c37 5d5d 2020 2020 7c0a 7c20 ,5,3,7]] |.| │ │ │ │ -00263390: 2d2d 2072 6574 7572 6e20 7661 6c75 653a -- return value: │ │ │ │ -002633a0: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ -002633b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002633c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002633d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00262bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00262bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00262bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00262be0: 2b0a 7c69 3420 3a20 6669 6e64 5072 6f67 +.|i4 : findProg │ │ │ │ +00262bf0: 7261 6d28 2274 6f70 636f 6d22 2c20 2263 ram("topcom", "c │ │ │ │ +00262c00: 7562 6520 3322 2c20 5665 7262 6f73 6520 ube 3", Verbose │ │ │ │ +00262c10: 3d3e 2074 7275 652c 2050 7265 6669 7820 => true, Prefix │ │ │ │ +00262c20: 3d3e 207b 2020 2020 2020 2020 2020 2020 => { │ │ │ │ +00262c30: 7c0a 7c20 2020 2020 2020 2822 2e2a 222c |.| (".*", │ │ │ │ +00262c40: 2022 746f 7063 6f6d 2d22 292c 2020 2020 "topcom-"), │ │ │ │ +00262c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262c80: 7c0a 7c20 2020 2020 2020 2822 5e28 6372 |.| ("^(cr │ │ │ │ +00262c90: 6f73 737c 6375 6265 7c63 7963 6c69 637c oss|cube|cyclic| │ │ │ │ +00262ca0: 6879 7065 7273 696d 706c 6578 7c6c 6174 hypersimplex|lat │ │ │ │ +00262cb0: 7469 6365 2924 222c 2022 544f 5043 4f4d tice)$", "TOPCOM │ │ │ │ +00262cc0: 2d22 292c 2020 2020 2020 2020 2020 2020 -"), │ │ │ │ +00262cd0: 7c0a 7c20 2020 2020 2020 2822 5e63 7562 |.| ("^cub │ │ │ │ +00262ce0: 6524 222c 2022 746f 7063 6f6d 5f22 297d e$", "topcom_")} │ │ │ │ +00262cf0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00262d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262d20: 7c0a 7c20 2d2d 202f 7573 722f 6c69 6265 |.| -- /usr/libe │ │ │ │ +00262d30: 7865 632f 4d61 6361 756c 6179 322f 6269 xec/Macaulay2/bi │ │ │ │ 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7573 722f 6c6f 6361 |.| -- /usr/loca │ │ │ │ +00262f10: 6c2f 7362 696e 2f54 4f50 434f 4d2d 6375 l/sbin/TOPCOM-cu │ │ │ │ +00262f20: 6265 2064 6f65 7320 6e6f 7420 6578 6973 be does not exis │ │ │ │ +00262f30: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00262f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262f50: 7c0a 7c20 2d2d 202f 7573 722f 6c6f 6361 |.| -- /usr/loca │ │ │ │ +00262f60: 6c2f 7362 696e 2f74 6f70 636f 6d5f 6375 l/sbin/topcom_cu │ │ │ │ +00262f70: 6265 2064 6f65 7320 6e6f 7420 6578 6973 be does not exis │ │ │ │ +00262f80: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00262f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262fa0: 7c0a 7c20 2d2d 202f 7573 722f 6c6f 6361 |.| -- /usr/loca │ │ │ │ +00262fb0: 6c2f 6269 6e2f 6375 6265 2064 6f65 7320 l/bin/cube does │ │ │ │ +00262fc0: 6e6f 7420 6578 6973 7420 2020 2020 2020 not exist │ │ │ │ +00262fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00262ff0: 7c0a 7c20 2d2d 202f 7573 722f 6c6f 6361 |.| -- /usr/loca │ │ │ │ +00263000: 6c2f 6269 6e2f 746f 7063 6f6d 2d63 7562 l/bin/topcom-cub │ │ │ │ +00263010: 6520 646f 6573 206e 6f74 2065 7869 7374 e does not exist │ │ │ │ +00263020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263040: 7c0a 7c20 2d2d 202f 7573 722f 6c6f 6361 |.| -- /usr/loca │ │ │ │ +00263050: 6c2f 6269 6e2f 544f 5043 4f4d 2d63 7562 l/bin/TOPCOM-cub │ │ │ │ +00263060: 6520 646f 6573 206e 6f74 2065 7869 7374 e does not exist │ │ │ │ +00263070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263090: 7c0a 7c20 2d2d 202f 7573 722f 6c6f 6361 |.| -- /usr/loca │ │ │ │ +002630a0: 6c2f 6269 6e2f 746f 7063 6f6d 5f63 7562 l/bin/topcom_cub │ │ │ │ +002630b0: 6520 646f 6573 206e 6f74 2065 7869 7374 e does not exist │ │ │ │ +002630c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002630d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002630e0: 7c0a 7c20 2d2d 202f 7573 722f 7362 696e |.| -- /usr/sbin │ │ │ │ +002630f0: 2f63 7562 6520 646f 6573 206e 6f74 2065 /cube does not e │ │ │ │ +00263100: 7869 7374 2020 2020 2020 2020 2020 2020 xist │ │ │ │ +00263110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263130: 7c0a 7c20 2d2d 202f 7573 722f 7362 696e |.| -- /usr/sbin │ │ │ │ +00263140: 2f74 6f70 636f 6d2d 6375 6265 2064 6f65 /topcom-cube doe │ │ │ │ +00263150: 7320 6e6f 7420 6578 6973 7420 2020 2020 s not exist │ │ │ │ +00263160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263180: 7c0a 7c20 2d2d 202f 7573 722f 7362 696e |.| -- /usr/sbin │ │ │ │ +00263190: 2f54 4f50 434f 4d2d 6375 6265 2064 6f65 /TOPCOM-cube doe │ │ │ │ +002631a0: 7320 6e6f 7420 6578 6973 7420 2020 2020 s not exist │ │ │ │ +002631b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002631c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002631d0: 7c0a 7c20 2d2d 202f 7573 722f 7362 696e |.| -- /usr/sbin │ │ │ │ +002631e0: 2f74 6f70 636f 6d5f 6375 6265 2064 6f65 /topcom_cube doe │ │ │ │ +002631f0: 7320 6e6f 7420 6578 6973 7420 2020 2020 s not exist │ │ │ │ +00263200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263220: 7c0a 7c20 2d2d 202f 7573 722f 6269 6e2f |.| -- /usr/bin/ │ │ │ │ +00263230: 6375 6265 2064 6f65 7320 6e6f 7420 6578 cube does not ex │ │ │ │ +00263240: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ +00263250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263270: 7c0a 7c20 2d2d 202f 7573 722f 6269 6e2f |.| -- /usr/bin/ │ │ │ │ +00263280: 746f 7063 6f6d 2d63 7562 6520 6578 6973 topcom-cube exis │ │ │ │ +00263290: 7473 2061 6e64 2069 7320 6578 6563 7574 ts and is execut │ │ │ │ +002632a0: 6162 6c65 2020 2020 2020 2020 2020 2020 able │ │ │ │ +002632b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002632c0: 7c0a 7c20 2d2d 2072 756e 6e69 6e67 2022 |.| -- running " │ │ │ │ +002632d0: 2f75 7372 2f62 696e 2f74 6f70 636f 6d2d /usr/bin/topcom- │ │ │ │ +002632e0: 6375 6265 2033 223a 2020 2020 2020 2020 cube 3": │ │ │ │ +002632f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263310: 7c0a 7c5b 5b30 2c30 2c30 2c31 5d2c 5b31 |.|[[0,0,0,1],[1 │ │ │ │ +00263320: 2c30 2c30 2c31 5d2c 5b30 2c31 2c30 2c31 ,0,0,1],[0,1,0,1 │ │ │ │ +00263330: 5d2c 5b31 2c31 2c30 2c31 5d2c 5b30 2c30 ],[1,1,0,1],[0,0 │ │ │ │ +00263340: 2c31 2c31 5d2c 5b31 2c30 2c31 2c31 5d2c ,1,1],[1,0,1,1], │ │ │ │ +00263350: 5b30 2c31 2c31 2c31 5d2c 5b31 2c31 2c31 [0,1,1,1],[1,1,1 │ │ │ │ +00263360: 7c0a 7c5b 5b37 2c36 2c35 2c34 2c33 2c32 |.|[[7,6,5,4,3,2 │ │ │ │ +00263370: 2c31 2c30 5d2c 5b36 2c37 2c34 2c35 2c32 ,1,0],[6,7,4,5,2 │ │ │ │ +00263380: 2c33 2c30 2c31 5d2c 5b34 2c36 2c35 2c37 ,3,0,1],[4,6,5,7 │ │ │ │ +00263390: 2c30 2c32 2c31 2c33 5d2c 5b30 2c34 2c32 ,0,2,1,3],[0,4,2 │ │ │ │ +002633a0: 2c36 2c31 2c35 2c33 2c37 5d5d 2020 2020 ,6,1,5,3,7]] │ │ │ │ +002633b0: 7c0a 7c20 2d2d 2072 6574 7572 6e20 7661 |.| -- return va │ │ │ │ +002633c0: 6c75 653a 2030 2020 2020 2020 2020 2020 lue: 0 │ │ │ │ +002633d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002633e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002633f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263400: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00263410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263420: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00263430: 3420 3d20 746f 7063 6f6d 2020 2020 2020 4 = topcom │ │ │ │ +00263420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263450: 7c0a 7c6f 3420 3d20 746f 7063 6f6d 2020 |.|o4 = topcom │ │ │ │ 00263460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263470: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00263470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002634a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002634a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 002634b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002634c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -002634d0: 3420 3a20 5072 6f67 7261 6d20 2020 2020 4 : Program │ │ │ │ +002634c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002634d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002634e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002634f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002634f0: 7c0a 7c6f 3420 3a20 5072 6f67 7261 6d20 |.|o4 : Program │ │ │ │ 00263500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263510: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -00263520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00263530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00263540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263540: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------- │ │ │ │ 00263550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00263560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2c ------------|.|, │ │ │ │ -00263570: 315d 5d20 2020 2020 2020 2020 2020 2020 1]] │ │ │ │ -00263580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263590: 7c0a 7c2c 315d 5d20 2020 2020 2020 2020 |.|,1]] │ │ │ │ 002635a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002635b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -002635c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002635d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002635e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002635b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002635c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002635d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002635e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 002635f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00263600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a4e ------------+..N │ │ │ │ -00263610: 6f74 6520 7468 6174 2077 6865 6e20 7573 ote that when us │ │ │ │ -00263620: 696e 6720 7468 6520 4d69 6e69 6d75 6d56 ing the MinimumV │ │ │ │ -00263630: 6572 7369 6f6e 206f 7074 696f 6e2c 2074 ersion option, t │ │ │ │ -00263640: 6865 2063 6f6d 6d61 6e64 2075 7365 6420 he command used │ │ │ │ -00263650: 746f 206f 6274 6169 6e20 7468 650a 6375 to obtain the.cu │ │ │ │ -00263660: 7272 656e 7420 7665 7273 696f 6e20 6e75 rrent version nu │ │ │ │ -00263670: 6d62 6572 206d 7573 7420 7265 6d6f 7665 mber must remove │ │ │ │ -00263680: 2065 7665 7279 7468 696e 6720 6578 6365 everything exce │ │ │ │ -00263690: 7074 2074 6865 2076 6572 7369 6f6e 206e pt the version n │ │ │ │ -002636a0: 756d 6265 7220 6974 7365 6c66 0a61 6e64 umber itself.and │ │ │ │ -002636b0: 2061 6e79 206c 6561 6469 6e67 206f 7220 any leading or │ │ │ │ -002636c0: 7472 6169 6c69 6e67 2077 6869 7465 7370 trailing whitesp │ │ │ │ -002636d0: 6163 652e 2020 5069 7069 6e67 2077 6974 ace. Piping wit │ │ │ │ -002636e0: 6820 7374 616e 6461 7264 2055 4e49 5820 h standard UNIX │ │ │ │ -002636f0: 7574 696c 6974 6965 730a 7375 6368 2061 utilities.such a │ │ │ │ -00263700: 7320 7365 642c 2068 6561 642c 2074 6169 s sed, head, tai │ │ │ │ -00263710: 6c2c 2063 7574 2c20 616e 6420 7472 206d l, cut, and tr m │ │ │ │ -00263720: 6179 2062 6520 7573 6566 756c 2e0a 0a2b ay be useful...+ │ │ │ │ -00263730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00263740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00263750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263630: 2b0a 0a4e 6f74 6520 7468 6174 2077 6865 +..Note that whe │ │ │ │ +00263640: 6e20 7573 696e 6720 7468 6520 4d69 6e69 n using the Mini │ │ │ │ +00263650: 6d75 6d56 6572 7369 6f6e 206f 7074 696f mumVersion optio │ │ │ │ +00263660: 6e2c 2074 6865 2063 6f6d 6d61 6e64 2075 n, the command u │ │ │ │ +00263670: 7365 6420 746f 206f 6274 6169 6e20 7468 sed to obtain th │ │ │ │ +00263680: 650a 6375 7272 656e 7420 7665 7273 696f e.current versio │ │ │ │ +00263690: 6e20 6e75 6d62 6572 206d 7573 7420 7265 n number must re │ │ │ │ +002636a0: 6d6f 7665 2065 7665 7279 7468 696e 6720 move everything │ │ │ │ +002636b0: 6578 6365 7074 2074 6865 2076 6572 7369 except the versi │ │ │ │ +002636c0: 6f6e 206e 756d 6265 7220 6974 7365 6c66 on number itself │ │ │ │ +002636d0: 0a61 6e64 2061 6e79 206c 6561 6469 6e67 .and any leading │ │ │ │ +002636e0: 206f 7220 7472 6169 6c69 6e67 2077 6869 or trailing whi │ │ │ │ +002636f0: 7465 7370 6163 652e 2020 5069 7069 6e67 tespace. Piping │ │ │ │ +00263700: 2077 6974 6820 7374 616e 6461 7264 2055 with standard U │ │ │ │ +00263710: 4e49 5820 7574 696c 6974 6965 730a 7375 NIX utilities.su │ │ │ │ +00263720: 6368 2061 7320 7365 642c 2068 6561 642c ch as sed, head, │ │ │ │ +00263730: 2074 6169 6c2c 2063 7574 2c20 616e 6420 tail, cut, and │ │ │ │ +00263740: 7472 206d 6179 2062 6520 7573 6566 756c tr may be useful │ │ │ │ +00263750: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ 00263760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00263770: 2d2d 2d2d 2d2b 0a7c 6935 203a 2066 696e -----+.|i5 : fin │ │ │ │ -00263780: 6450 726f 6772 616d 2822 6766 616e 222c dProgram("gfan", │ │ │ │ -00263790: 2022 6766 616e 205f 7665 7273 696f 6e20 "gfan _version │ │ │ │ -002637a0: 2d2d 6865 6c70 222c 2056 6572 626f 7365 --help", Verbose │ │ │ │ -002637b0: 203d 3e20 7472 7565 2c20 2020 207c 0a7c => true, |.| │ │ │ │ -002637c0: 2020 2020 2020 204d 696e 696d 756d 5665 MinimumVe │ │ │ │ -002637d0: 7273 696f 6e20 3d3e 2028 2230 2e35 222c rsion => ("0.5", │ │ │ │ -002637e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002637f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263800: 2020 2020 207c 0a7c 2020 2020 2020 2267 |.| "g │ │ │ │ -00263810: 6661 6e20 5f76 6572 7369 6f6e 207c 2068 fan _version | h │ │ │ │ -00263820: 6561 6420 2d32 207c 2074 6169 6c20 2d31 ead -2 | tail -1 │ │ │ │ -00263830: 207c 2073 6564 2027 732f 6766 616e 2f2f | sed 's/gfan// │ │ │ │ -00263840: 2722 2929 2020 2020 2020 2020 207c 0a7c '")) |.| │ │ │ │ -00263850: 202d 2d20 2f70 6174 682f 746f 2f67 6661 -- /path/to/gfa │ │ │ │ -00263860: 6e2f 6766 616e 2064 6f65 7320 6e6f 7420 n/gfan does not │ │ │ │ -00263870: 6578 6973 7420 2020 2020 2020 2020 2020 exist │ │ │ │ -00263880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263890: 2020 2020 207c 0a7c 202d 2d20 2f75 7372 |.| -- /usr │ │ │ │ -002638a0: 2f6c 6962 6578 6563 2f4d 6163 6175 6c61 /libexec/Macaula │ │ │ │ -002638b0: 7932 2f62 696e 2f67 6661 6e20 646f 6573 y2/bin/gfan does │ │ │ │ -002638c0: 206e 6f74 2065 7869 7374 2020 2020 2020 not exist │ │ │ │ -002638d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -002638e0: 202d 2d20 2f75 7372 2f6c 6f63 616c 2f73 -- /usr/local/s │ │ │ │ -002638f0: 6269 6e2f 6766 616e 2064 6f65 7320 6e6f bin/gfan does no │ │ │ │ -00263900: 7420 6578 6973 7420 2020 2020 2020 2020 t exist │ │ │ │ -00263910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263920: 2020 2020 207c 0a7c 202d 2d20 2f75 7372 |.| -- /usr │ │ │ │ -00263930: 2f6c 6f63 616c 2f62 696e 2f67 6661 6e20 /local/bin/gfan │ │ │ │ -00263940: 646f 6573 206e 6f74 2065 7869 7374 2020 does not exist │ │ │ │ -00263950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263960: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00263970: 202d 2d20 2f75 7372 2f73 6269 6e2f 6766 -- /usr/sbin/gf │ │ │ │ -00263980: 616e 2064 6f65 7320 6e6f 7420 6578 6973 an does not exis │ │ │ │ -00263990: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -002639a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002639b0: 2020 2020 207c 0a7c 202d 2d20 2f75 7372 |.| -- /usr │ │ │ │ -002639c0: 2f62 696e 2f67 6661 6e20 6578 6973 7473 /bin/gfan exists │ │ │ │ -002639d0: 2061 6e64 2069 7320 6578 6563 7574 6162 and is executab │ │ │ │ -002639e0: 6c65 2020 2020 2020 2020 2020 2020 2020 le │ │ │ │ -002639f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00263a00: 202d 2d20 7275 6e6e 696e 6720 222f 7573 -- running "/us │ │ │ │ -00263a10: 722f 6269 6e2f 6766 616e 205f 7665 7273 r/bin/gfan _vers │ │ │ │ -00263a20: 696f 6e20 2d2d 6865 6c70 223a 2020 2020 ion --help": │ │ │ │ -00263a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263a40: 2020 2020 207c 0a7c 5468 6973 2070 726f |.|This pro │ │ │ │ -00263a50: 6772 616d 2077 7269 7465 7320 6f75 7420 gram writes out │ │ │ │ -00263a60: 7665 7273 696f 6e20 696e 666f 726d 6174 version informat │ │ │ │ -00263a70: 696f 6e20 6f66 2074 6865 2047 6661 6e20 ion of the Gfan │ │ │ │ -00263a80: 696e 7374 616c 6c61 7469 6f6e 2e7c 0a7c installation.|.| │ │ │ │ -00263a90: 202d 2d20 7265 7475 726e 2076 616c 7565 -- return value │ │ │ │ -00263aa0: 3a20 3020 2020 2020 2020 2020 2020 2020 : 0 │ │ │ │ -00263ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263ad0: 2020 2020 207c 0a7c 202d 2d20 666f 756e |.| -- foun │ │ │ │ -00263ae0: 6420 7665 7273 696f 6e20 302e 3720 3e3d d version 0.7 >= │ │ │ │ -00263af0: 2030 2e35 2020 2020 2020 2020 2020 2020 0.5 │ │ │ │ -00263b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263b10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00263770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263790: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ +002637a0: 2066 696e 6450 726f 6772 616d 2822 6766 findProgram("gf │ │ │ │ +002637b0: 616e 222c 2022 6766 616e 205f 7665 7273 an", "gfan _vers │ │ │ │ +002637c0: 696f 6e20 2d2d 6865 6c70 222c 2056 6572 ion --help", Ver │ │ │ │ +002637d0: 626f 7365 203d 3e20 7472 7565 2c20 2020 bose => true, │ │ │ │ +002637e0: 207c 0a7c 2020 2020 2020 204d 696e 696d |.| Minim │ │ │ │ +002637f0: 756d 5665 7273 696f 6e20 3d3e 2028 2230 umVersion => ("0 │ │ │ │ +00263800: 2e35 222c 2020 2020 2020 2020 2020 2020 .5", │ │ │ │ +00263810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263820: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00263830: 2020 2267 6661 6e20 5f76 6572 7369 6f6e "gfan _version │ │ │ │ +00263840: 207c 2068 6561 6420 2d32 207c 2074 6169 | head -2 | tai │ │ │ │ +00263850: 6c20 2d31 207c 2073 6564 2027 732f 6766 l -1 | sed 's/gf │ │ │ │ +00263860: 616e 2f2f 2722 2929 2020 2020 2020 2020 an//'")) │ │ │ │ +00263870: 207c 0a7c 202d 2d20 2f70 6174 682f 746f |.| -- /path/to │ │ │ │ +00263880: 2f67 6661 6e2f 6766 616e 2064 6f65 7320 /gfan/gfan does │ │ │ │ +00263890: 6e6f 7420 6578 6973 7420 2020 2020 2020 not exist │ │ │ │ +002638a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002638b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ +002638c0: 2f75 7372 2f6c 6962 6578 6563 2f4d 6163 /usr/libexec/Mac │ │ │ │ +002638d0: 6175 6c61 7932 2f62 696e 2f67 6661 6e20 aulay2/bin/gfan │ │ │ │ +002638e0: 646f 6573 206e 6f74 2065 7869 7374 2020 does not exist │ │ │ │ +002638f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263900: 207c 0a7c 202d 2d20 2f75 7372 2f6c 6f63 |.| -- /usr/loc │ │ │ │ +00263910: 616c 2f73 6269 6e2f 6766 616e 2064 6f65 al/sbin/gfan doe │ │ │ │ +00263920: 7320 6e6f 7420 6578 6973 7420 2020 2020 s not exist │ │ │ │ +00263930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263940: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ +00263950: 2f75 7372 2f6c 6f63 616c 2f62 696e 2f67 /usr/local/bin/g │ │ │ │ +00263960: 6661 6e20 646f 6573 206e 6f74 2065 7869 fan does not exi │ │ │ │ +00263970: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ +00263980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263990: 207c 0a7c 202d 2d20 2f75 7372 2f73 6269 |.| -- /usr/sbi │ │ │ │ +002639a0: 6e2f 6766 616e 2064 6f65 7320 6e6f 7420 n/gfan does not │ │ │ │ +002639b0: 6578 6973 7420 2020 2020 2020 2020 2020 exist │ │ │ │ +002639c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002639d0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ +002639e0: 2f75 7372 2f62 696e 2f67 6661 6e20 6578 /usr/bin/gfan ex │ │ │ │ +002639f0: 6973 7473 2061 6e64 2069 7320 6578 6563 ists and is exec │ │ │ │ +00263a00: 7574 6162 6c65 2020 2020 2020 2020 2020 utable │ │ │ │ +00263a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263a20: 207c 0a7c 202d 2d20 7275 6e6e 696e 6720 |.| -- running │ │ │ │ +00263a30: 222f 7573 722f 6269 6e2f 6766 616e 205f "/usr/bin/gfan _ │ │ │ │ +00263a40: 7665 7273 696f 6e20 2d2d 6865 6c70 223a version --help": │ │ │ │ +00263a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263a60: 2020 2020 2020 2020 207c 0a7c 5468 6973 |.|This │ │ │ │ +00263a70: 2070 726f 6772 616d 2077 7269 7465 7320 program writes │ │ │ │ +00263a80: 6f75 7420 7665 7273 696f 6e20 696e 666f out version info │ │ │ │ +00263a90: 726d 6174 696f 6e20 6f66 2074 6865 2047 rmation of the G │ │ │ │ +00263aa0: 6661 6e20 696e 7374 616c 6c61 7469 6f6e fan installation │ │ │ │ +00263ab0: 2e7c 0a7c 202d 2d20 7265 7475 726e 2076 .|.| -- return v │ │ │ │ +00263ac0: 616c 7565 3a20 3020 2020 2020 2020 2020 alue: 0 │ │ │ │ +00263ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263af0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ +00263b00: 666f 756e 6420 7665 7273 696f 6e20 302e found version 0. │ │ │ │ +00263b10: 3720 3e3d 2030 2e35 2020 2020 2020 2020 7 >= 0.5 │ │ │ │ 00263b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263b40: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00263b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263b60: 2020 2020 207c 0a7c 6f35 203d 2067 6661 |.|o5 = gfa │ │ │ │ -00263b70: 6e20 2020 2020 2020 2020 2020 2020 2020 n │ │ │ │ -00263b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263ba0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00263b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263b80: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ +00263b90: 2067 6661 6e20 2020 2020 2020 2020 2020 gfan │ │ │ │ +00263ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263bd0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00263be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263bf0: 2020 2020 207c 0a7c 6f35 203a 2050 726f |.|o5 : Pro │ │ │ │ -00263c00: 6772 616d 2020 2020 2020 2020 2020 2020 gram │ │ │ │ -00263c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00263c30: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00263c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00263c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00263c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263c10: 2020 2020 2020 2020 207c 0a7c 6f35 203a |.|o5 : │ │ │ │ +00263c20: 2050 726f 6772 616d 2020 2020 2020 2020 Program │ │ │ │ +00263c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00263c60: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00263c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00263c80: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f -----+..See also │ │ │ │ -00263c90: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ -00263ca0: 6e6f 7465 2072 756e 5072 6f67 7261 6d3a note runProgram: │ │ │ │ -00263cb0: 2072 756e 5072 6f67 7261 6d2c 202d 2d20 runProgram, -- │ │ │ │ -00263cc0: 7275 6e20 616e 2065 7874 6572 6e61 6c20 run an external │ │ │ │ -00263cd0: 7072 6f67 7261 6d0a 2020 2a20 2a6e 6f74 program. * *not │ │ │ │ -00263ce0: 6520 7365 6172 6368 5061 7468 3a20 7365 e searchPath: se │ │ │ │ -00263cf0: 6172 6368 5061 7468 5f6c 704c 6973 745f archPath_lpList_ │ │ │ │ -00263d00: 636d 5374 7269 6e67 5f72 702c 202d 2d20 cmString_rp, -- │ │ │ │ -00263d10: 7365 6172 6368 2061 2070 6174 6820 666f search a path fo │ │ │ │ -00263d20: 7220 610a 2020 2020 6669 6c65 0a2a 204d r a. file.* M │ │ │ │ -00263d30: 656e 753a 0a0a 2a20 5072 6f67 7261 6d3a enu:..* Program: │ │ │ │ -00263d40: 3a20 2020 2020 2020 2020 2020 2020 2020 : │ │ │ │ -00263d50: 2020 2020 2020 6578 7465 726e 616c 2070 external p │ │ │ │ -00263d60: 726f 6772 616d 206f 626a 6563 740a 2a20 rogram object.* │ │ │ │ -00263d70: 7072 6f67 7261 6d50 6174 6873 3a3a 2020 programPaths:: │ │ │ │ -00263d80: 2020 2020 2020 2020 2020 2020 2020 7573 us │ │ │ │ -00263d90: 6572 2d64 6566 696e 6564 2065 7874 6572 er-defined exter │ │ │ │ -00263da0: 6e61 6c20 7072 6f67 7261 6d20 7061 7468 nal program path │ │ │ │ -00263db0: 730a 0a57 6179 7320 746f 2075 7365 2066 s..Ways to use f │ │ │ │ -00263dc0: 696e 6450 726f 6772 616d 3a0a 3d3d 3d3d indProgram:.==== │ │ │ │ -00263dd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00263de0: 3d3d 3d3d 0a0a 2020 2a20 2266 696e 6450 ====.. * "findP │ │ │ │ -00263df0: 726f 6772 616d 2853 7472 696e 6729 220a rogram(String)". │ │ │ │ -00263e00: 2020 2a20 2266 696e 6450 726f 6772 616d * "findProgram │ │ │ │ -00263e10: 2853 7472 696e 672c 4c69 7374 2922 0a20 (String,List)". │ │ │ │ -00263e20: 202a 2022 6669 6e64 5072 6f67 7261 6d28 * "findProgram( │ │ │ │ -00263e30: 5374 7269 6e67 2c53 7472 696e 6729 220a String,String)". │ │ │ │ -00263e40: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ -00263e50: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ -00263e60: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ -00263e70: 6374 202a 6e6f 7465 2066 696e 6450 726f ct *note findPro │ │ │ │ -00263e80: 6772 616d 3a20 6669 6e64 5072 6f67 7261 gram: findProgra │ │ │ │ -00263e90: 6d2c 2069 7320 6120 2a6e 6f74 6520 6d65 m, is a *note me │ │ │ │ -00263ea0: 7468 6f64 2066 756e 6374 696f 6e20 7769 thod function wi │ │ │ │ -00263eb0: 7468 0a6f 7074 696f 6e73 3a20 4d65 7468 th.options: Meth │ │ │ │ -00263ec0: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70 odFunctionWithOp │ │ │ │ -00263ed0: 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d tions,...------- │ │ │ │ -00263ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00263ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263ca0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 ---------+..See │ │ │ │ +00263cb0: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 also.========.. │ │ │ │ +00263cc0: 202a 202a 6e6f 7465 2072 756e 5072 6f67 * *note runProg │ │ │ │ +00263cd0: 7261 6d3a 2072 756e 5072 6f67 7261 6d2c ram: runProgram, │ │ │ │ +00263ce0: 202d 2d20 7275 6e20 616e 2065 7874 6572 -- run an exter │ │ │ │ +00263cf0: 6e61 6c20 7072 6f67 7261 6d0a 2020 2a20 nal program. * │ │ │ │ +00263d00: 2a6e 6f74 6520 7365 6172 6368 5061 7468 *note searchPath │ │ │ │ +00263d10: 3a20 7365 6172 6368 5061 7468 5f6c 704c : searchPath_lpL │ │ │ │ +00263d20: 6973 745f 636d 5374 7269 6e67 5f72 702c ist_cmString_rp, │ │ │ │ +00263d30: 202d 2d20 7365 6172 6368 2061 2070 6174 -- search a pat │ │ │ │ +00263d40: 6820 666f 7220 610a 2020 2020 6669 6c65 h for a. file │ │ │ │ +00263d50: 0a2a 204d 656e 753a 0a0a 2a20 5072 6f67 .* Menu:..* Prog │ │ │ │ +00263d60: 7261 6d3a 3a20 2020 2020 2020 2020 2020 ram:: │ │ │ │ +00263d70: 2020 2020 2020 2020 2020 6578 7465 726e extern │ │ │ │ +00263d80: 616c 2070 726f 6772 616d 206f 626a 6563 al program objec │ │ │ │ +00263d90: 740a 2a20 7072 6f67 7261 6d50 6174 6873 t.* programPaths │ │ │ │ +00263da0: 3a3a 2020 2020 2020 2020 2020 2020 2020 :: │ │ │ │ +00263db0: 2020 7573 6572 2d64 6566 696e 6564 2065 user-defined e │ │ │ │ +00263dc0: 7874 6572 6e61 6c20 7072 6f67 7261 6d20 xternal program │ │ │ │ +00263dd0: 7061 7468 730a 0a57 6179 7320 746f 2075 paths..Ways to u │ │ │ │ +00263de0: 7365 2066 696e 6450 726f 6772 616d 3a0a se findProgram:. │ │ │ │ +00263df0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00263e00: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2266 ========.. * "f │ │ │ │ +00263e10: 696e 6450 726f 6772 616d 2853 7472 696e indProgram(Strin │ │ │ │ +00263e20: 6729 220a 2020 2a20 2266 696e 6450 726f g)". * "findPro │ │ │ │ +00263e30: 6772 616d 2853 7472 696e 672c 4c69 7374 gram(String,List │ │ │ │ +00263e40: 2922 0a20 202a 2022 6669 6e64 5072 6f67 )". * "findProg │ │ │ │ +00263e50: 7261 6d28 5374 7269 6e67 2c53 7472 696e ram(String,Strin │ │ │ │ +00263e60: 6729 220a 0a46 6f72 2074 6865 2070 726f g)"..For the pro │ │ │ │ +00263e70: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +00263e80: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +00263e90: 6f62 6a65 6374 202a 6e6f 7465 2066 696e object *note fin │ │ │ │ +00263ea0: 6450 726f 6772 616d 3a20 6669 6e64 5072 dProgram: findPr │ │ │ │ +00263eb0: 6f67 7261 6d2c 2069 7320 6120 2a6e 6f74 ogram, is a *not │ │ │ │ +00263ec0: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +00263ed0: 6e20 7769 7468 0a6f 7074 696f 6e73 3a20 n with.options: │ │ │ │ +00263ee0: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769 MethodFunctionWi │ │ │ │ +00263ef0: 7468 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d thOptions,...--- │ │ │ │ 00263f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00263f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00263f20: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -00263f30: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -00263f40: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -00263f50: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -00263f60: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -00263f70: 2e32 362e 3035 2b64 732f 4d32 2f4d 6163 .26.05+ds/M2/Mac │ │ │ │ -00263f80: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -00263f90: 0a4d 6163 6175 6c61 7932 446f 632f 6675 .Macaulay2Doc/fu │ │ │ │ -00263fa0: 6e63 7469 6f6e 732f 6669 6e64 5072 6f67 nctions/findProg │ │ │ │ -00263fb0: 7261 6d2d 646f 632e 6d32 3a31 3734 3a30 ram-doc.m2:174:0 │ │ │ │ -00263fc0: 2e0a 1f0a 4669 6c65 3a20 4d61 6361 756c ....File: Macaul │ │ │ │ -00263fd0: 6179 3244 6f63 2e69 6e66 6f2c 204e 6f64 ay2Doc.info, Nod │ │ │ │ -00263fe0: 653a 2050 726f 6772 616d 2c20 4e65 7874 e: Program, Next │ │ │ │ -00263ff0: 3a20 7072 6f67 7261 6d50 6174 6873 2c20 : programPaths, │ │ │ │ -00264000: 5570 3a20 6669 6e64 5072 6f67 7261 6d0a Up: findProgram. │ │ │ │ -00264010: 0a50 726f 6772 616d 202d 2d20 6578 7465 .Program -- exte │ │ │ │ -00264020: 726e 616c 2070 726f 6772 616d 206f 626a rnal program obj │ │ │ │ -00264030: 6563 740a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ect.************ │ │ │ │ -00264040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00264050: 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 7074 ******..Descript │ │ │ │ -00264060: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -00264070: 0a41 2068 6173 6820 7461 626c 6520 7265 .A hash table re │ │ │ │ -00264080: 7475 726e 6564 2062 7920 2a6e 6f74 6520 turned by *note │ │ │ │ -00264090: 6669 6e64 5072 6f67 7261 6d3a 2066 696e findProgram: fin │ │ │ │ -002640a0: 6450 726f 6772 616d 2c20 7769 7468 2074 dProgram, with t │ │ │ │ -002640b0: 6865 2066 6f6c 6c6f 7769 6e67 0a73 7472 he following.str │ │ │ │ -002640c0: 696e 6773 2061 7320 6b65 7973 3a0a 0a20 ings as keys:.. │ │ │ │ -002640d0: 202a 2022 6e61 6d65 222c 2074 6865 206e * "name", the n │ │ │ │ -002640e0: 616d 6520 6f66 2074 6865 206c 6f61 6465 ame of the loade │ │ │ │ -002640f0: 6420 7072 6f67 7261 6d2e 2020 5468 6973 d program. This │ │ │ │ -00264100: 2063 6f6d 6573 2066 726f 6d20 7468 6520 comes from the │ │ │ │ -00264110: 6669 7273 7420 6172 6775 6d65 6e74 0a20 first argument. │ │ │ │ -00264120: 2020 2070 6173 7365 6420 746f 202a 6e6f passed to *no │ │ │ │ -00264130: 7465 2066 696e 6450 726f 6772 616d 3a20 te findProgram: │ │ │ │ -00264140: 6669 6e64 5072 6f67 7261 6d2c 2e20 2049 findProgram,. I │ │ │ │ -00264150: 7420 6973 2061 6c73 6f20 7768 6174 2069 t is also what i │ │ │ │ -00264160: 7320 6469 7370 6c61 7965 640a 2020 2020 s displayed. │ │ │ │ -00264170: 7768 656e 2070 7269 6e74 696e 6720 6120 when printing a │ │ │ │ -00264180: 5072 6f67 7261 6d2e 0a20 202a 2022 7061 Program.. * "pa │ │ │ │ -00264190: 7468 222c 2074 6865 2070 6174 6820 746f th", the path to │ │ │ │ -002641a0: 2074 6865 2070 726f 6772 616d 2061 7320 the program as │ │ │ │ -002641b0: 6465 7465 726d 696e 6564 2062 7920 2a6e determined by *n │ │ │ │ -002641c0: 6f74 6520 6669 6e64 5072 6f67 7261 6d3a ote findProgram: │ │ │ │ -002641d0: 0a20 2020 2066 696e 6450 726f 6772 616d . findProgram │ │ │ │ -002641e0: 2c2e 0a20 202a 2022 7072 6566 6978 222c ,.. * "prefix", │ │ │ │ -002641f0: 2061 2073 6571 7565 6e63 6520 6f66 2074 a sequence of t │ │ │ │ -00264200: 776f 2073 7472 696e 6773 2069 6465 6e74 wo strings ident │ │ │ │ -00264210: 6966 7969 6e67 2074 6865 2070 7265 6669 ifying the prefi │ │ │ │ -00264220: 7820 7072 6570 656e 6465 6420 746f 2074 x prepended to t │ │ │ │ -00264230: 6865 0a20 2020 2062 696e 6172 7920 6578 he. binary ex │ │ │ │ -00264240: 6563 7574 6162 6c65 732e 2020 5365 6520 ecutables. See │ │ │ │ -00264250: 2a6e 6f74 6520 6669 6e64 5072 6f67 7261 *note findProgra │ │ │ │ -00264260: 6d3a 2066 696e 6450 726f 6772 616d 2c2c m: findProgram,, │ │ │ │ -00264270: 2073 7065 6369 6669 6361 6c6c 7920 7468 specifically th │ │ │ │ -00264280: 650a 2020 2020 6465 7363 7269 7074 696f e. descriptio │ │ │ │ -00264290: 6e20 6f66 2074 6865 2050 7265 6669 7820 n of the Prefix │ │ │ │ -002642a0: 6f70 7469 6f6e 2c20 666f 7220 6d6f 7265 option, for more │ │ │ │ -002642b0: 2e0a 2020 2a20 2276 6572 7369 6f6e 222c .. * "version", │ │ │ │ -002642c0: 2061 2073 7472 696e 6720 636f 6e74 6169 a string contai │ │ │ │ -002642d0: 6e69 6e67 2074 6865 2076 6572 7369 6f6e ning the version │ │ │ │ -002642e0: 206e 756d 6265 7220 6f66 2074 6865 2070 number of the p │ │ │ │ -002642f0: 726f 6772 616d 2e20 204f 6e6c 790a 2020 rogram. Only. │ │ │ │ -00264300: 2020 7072 6573 656e 7420 6966 202a 6e6f present if *no │ │ │ │ -00264310: 7465 2066 696e 6450 726f 6772 616d 3a20 te findProgram: │ │ │ │ -00264320: 6669 6e64 5072 6f67 7261 6d2c 2077 6173 findProgram, was │ │ │ │ -00264330: 2063 616c 6c65 6420 7769 7468 2074 6865 called with the │ │ │ │ -00264340: 0a20 2020 204d 696e 696d 756d 5665 7273 . MinimumVers │ │ │ │ -00264350: 696f 6e20 6f70 7469 6f6e 2e0a 0a53 6565 ion option...See │ │ │ │ -00264360: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -00264370: 2020 2a20 2a6e 6f74 6520 6669 6e64 5072 * *note findPr │ │ │ │ -00264380: 6f67 7261 6d3a 2066 696e 6450 726f 6772 ogram: findProgr │ │ │ │ -00264390: 616d 2c20 2d2d 206c 6f61 6420 6578 7465 am, -- load exte │ │ │ │ -002643a0: 726e 616c 2070 726f 6772 616d 0a20 202a rnal program. * │ │ │ │ -002643b0: 202a 6e6f 7465 2072 756e 5072 6f67 7261 *note runProgra │ │ │ │ -002643c0: 6d3a 2072 756e 5072 6f67 7261 6d2c 202d m: runProgram, - │ │ │ │ -002643d0: 2d20 7275 6e20 616e 2065 7874 6572 6e61 - run an externa │ │ │ │ -002643e0: 6c20 7072 6f67 7261 6d0a 2020 2a20 2a6e l program. * *n │ │ │ │ -002643f0: 6f74 6520 7072 6f67 7261 6d50 6174 6873 ote programPaths │ │ │ │ -00264400: 3a20 7072 6f67 7261 6d50 6174 6873 2c20 : programPaths, │ │ │ │ -00264410: 2d2d 2075 7365 722d 6465 6669 6e65 6420 -- user-defined │ │ │ │ -00264420: 6578 7465 726e 616c 2070 726f 6772 616d external program │ │ │ │ -00264430: 2070 6174 6873 0a0a 4675 6e63 7469 6f6e paths..Function │ │ │ │ -00264440: 7320 616e 6420 6d65 7468 6f64 7320 7265 s and methods re │ │ │ │ -00264450: 7475 726e 696e 6720 616e 206f 626a 6563 turning an objec │ │ │ │ -00264460: 7420 6f66 2063 6c61 7373 2050 726f 6772 t of class Progr │ │ │ │ -00264470: 616d 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d am:.============ │ │ │ │ -00264480: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00264490: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -002644a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -002644b0: 0a20 202a 202a 6e6f 7465 2066 696e 6450 . * *note findP │ │ │ │ -002644c0: 726f 6772 616d 3a20 6669 6e64 5072 6f67 rogram: findProg │ │ │ │ -002644d0: 7261 6d2c 202d 2d20 6c6f 6164 2065 7874 ram, -- load ext │ │ │ │ -002644e0: 6572 6e61 6c20 7072 6f67 7261 6d0a 0a4d ernal program..M │ │ │ │ -002644f0: 6574 686f 6473 2074 6861 7420 7573 6520 ethods that use │ │ │ │ -00264500: 616e 206f 626a 6563 7420 6f66 2063 6c61 an object of cla │ │ │ │ -00264510: 7373 2050 726f 6772 616d 3a0a 3d3d 3d3d ss Program:.==== │ │ │ │ -00264520: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00264530: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00264540: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ -00264550: 6f74 6520 5072 6f67 7261 6d20 3c3c 2054 ote Program << T │ │ │ │ -00264560: 6869 6e67 3a20 5072 6f67 7261 6d20 3c3c hing: Program << │ │ │ │ -00264570: 2054 6869 6e67 2c20 2d2d 2072 756e 2070 Thing, -- run p │ │ │ │ -00264580: 726f 6772 616d 2077 6974 6820 696e 7075 rogram with inpu │ │ │ │ -00264590: 740a 2020 2020 7265 6469 7265 6374 696f t. redirectio │ │ │ │ -002645a0: 6e0a 2020 2a20 2272 756e 5072 6f67 7261 n. * "runProgra │ │ │ │ -002645b0: 6d28 5072 6f67 7261 6d2c 5374 7269 6e67 m(Program,String │ │ │ │ -002645c0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ -002645d0: 7275 6e50 726f 6772 616d 3a20 7275 6e50 runProgram: runP │ │ │ │ -002645e0: 726f 6772 616d 2c20 2d2d 2072 756e 2061 rogram, -- run a │ │ │ │ -002645f0: 6e0a 2020 2020 6578 7465 726e 616c 2070 n. external p │ │ │ │ -00264600: 726f 6772 616d 0a20 202a 2022 7275 6e50 rogram. * "runP │ │ │ │ -00264610: 726f 6772 616d 2850 726f 6772 616d 2c53 rogram(Program,S │ │ │ │ -00264620: 7472 696e 672c 5374 7269 6e67 2922 202d tring,String)" - │ │ │ │ -00264630: 2d20 7365 6520 2a6e 6f74 6520 7275 6e50 - see *note runP │ │ │ │ -00264640: 726f 6772 616d 3a20 7275 6e50 726f 6772 rogram: runProgr │ │ │ │ -00264650: 616d 2c20 2d2d 0a20 2020 2072 756e 2061 am, --. run a │ │ │ │ -00264660: 6e20 6578 7465 726e 616c 2070 726f 6772 n external progr │ │ │ │ -00264670: 616d 0a0a 466f 7220 7468 6520 7072 6f67 am..For the prog │ │ │ │ -00264680: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -00264690: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -002646a0: 626a 6563 7420 2a6e 6f74 6520 5072 6f67 bject *note Prog │ │ │ │ -002646b0: 7261 6d3a 2050 726f 6772 616d 2c20 6973 ram: Program, is │ │ │ │ -002646c0: 2061 202a 6e6f 7465 2074 7970 653a 2054 a *note type: T │ │ │ │ -002646d0: 7970 652c 2c20 7769 7468 2061 6e63 6573 ype,, with ances │ │ │ │ -002646e0: 746f 720a 636c 6173 7365 7320 2a6e 6f74 tor.classes *not │ │ │ │ -002646f0: 6520 4861 7368 5461 626c 653a 2048 6173 e HashTable: Has │ │ │ │ -00264700: 6854 6162 6c65 2c20 3c20 2a6e 6f74 6520 hTable, < *note │ │ │ │ -00264710: 5468 696e 673a 2054 6869 6e67 2c2e 0a0a Thing: Thing,... │ │ │ │ -00264720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00264730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00264740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00263f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +00263f50: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +00263f60: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +00263f70: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +00263f80: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +00263f90: 7932 2d31 2e32 362e 3035 2b64 732f 4d32 y2-1.26.05+ds/M2 │ │ │ │ +00263fa0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +00263fb0: 6765 732f 0a4d 6163 6175 6c61 7932 446f ges/.Macaulay2Do │ │ │ │ +00263fc0: 632f 6675 6e63 7469 6f6e 732f 6669 6e64 c/functions/find │ │ │ │ +00263fd0: 5072 6f67 7261 6d2d 646f 632e 6d32 3a31 Program-doc.m2:1 │ │ │ │ +00263fe0: 3734 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 74:0....File: Ma │ │ │ │ +00263ff0: 6361 756c 6179 3244 6f63 2e69 6e66 6f2c caulay2Doc.info, │ │ │ │ +00264000: 204e 6f64 653a 2050 726f 6772 616d 2c20 Node: Program, │ │ │ │ +00264010: 4e65 7874 3a20 7072 6f67 7261 6d50 6174 Next: programPat │ │ │ │ +00264020: 6873 2c20 5570 3a20 6669 6e64 5072 6f67 hs, Up: findProg │ │ │ │ +00264030: 7261 6d0a 0a50 726f 6772 616d 202d 2d20 ram..Program -- │ │ │ │ +00264040: 6578 7465 726e 616c 2070 726f 6772 616d external program │ │ │ │ +00264050: 206f 626a 6563 740a 2a2a 2a2a 2a2a 2a2a object.******** │ │ │ │ +00264060: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00264070: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 **********..Desc │ │ │ │ +00264080: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +00264090: 3d3d 3d0a 0a41 2068 6173 6820 7461 626c ===..A hash tabl │ │ │ │ +002640a0: 6520 7265 7475 726e 6564 2062 7920 2a6e e returned by *n │ │ │ │ +002640b0: 6f74 6520 6669 6e64 5072 6f67 7261 6d3a ote findProgram: │ │ │ │ +002640c0: 2066 696e 6450 726f 6772 616d 2c20 7769 findProgram, wi │ │ │ │ +002640d0: 7468 2074 6865 2066 6f6c 6c6f 7769 6e67 th the following │ │ │ │ +002640e0: 0a73 7472 696e 6773 2061 7320 6b65 7973 .strings as keys │ │ │ │ +002640f0: 3a0a 0a20 202a 2022 6e61 6d65 222c 2074 :.. * "name", t │ │ │ │ +00264100: 6865 206e 616d 6520 6f66 2074 6865 206c he name of the l │ │ │ │ +00264110: 6f61 6465 6420 7072 6f67 7261 6d2e 2020 oaded program. │ │ │ │ +00264120: 5468 6973 2063 6f6d 6573 2066 726f 6d20 This comes from │ │ │ │ +00264130: 7468 6520 6669 7273 7420 6172 6775 6d65 the first argume │ │ │ │ +00264140: 6e74 0a20 2020 2070 6173 7365 6420 746f nt. passed to │ │ │ │ +00264150: 202a 6e6f 7465 2066 696e 6450 726f 6772 *note findProgr │ │ │ │ +00264160: 616d 3a20 6669 6e64 5072 6f67 7261 6d2c am: findProgram, │ │ │ │ +00264170: 2e20 2049 7420 6973 2061 6c73 6f20 7768 . It is also wh │ │ │ │ +00264180: 6174 2069 7320 6469 7370 6c61 7965 640a at is displayed. │ │ │ │ +00264190: 2020 2020 7768 656e 2070 7269 6e74 696e when printin │ │ │ │ +002641a0: 6720 6120 5072 6f67 7261 6d2e 0a20 202a g a Program.. * │ │ │ │ +002641b0: 2022 7061 7468 222c 2074 6865 2070 6174 "path", the pat │ │ │ │ +002641c0: 6820 746f 2074 6865 2070 726f 6772 616d h to the program │ │ │ │ +002641d0: 2061 7320 6465 7465 726d 696e 6564 2062 as determined b │ │ │ │ +002641e0: 7920 2a6e 6f74 6520 6669 6e64 5072 6f67 y *note findProg │ │ │ │ +002641f0: 7261 6d3a 0a20 2020 2066 696e 6450 726f ram:. findPro │ │ │ │ +00264200: 6772 616d 2c2e 0a20 202a 2022 7072 6566 gram,.. * "pref │ │ │ │ +00264210: 6978 222c 2061 2073 6571 7565 6e63 6520 ix", a sequence │ │ │ │ +00264220: 6f66 2074 776f 2073 7472 696e 6773 2069 of two strings i │ │ │ │ +00264230: 6465 6e74 6966 7969 6e67 2074 6865 2070 dentifying the p │ │ │ │ +00264240: 7265 6669 7820 7072 6570 656e 6465 6420 refix prepended │ │ │ │ +00264250: 746f 2074 6865 0a20 2020 2062 696e 6172 to the. binar │ │ │ │ +00264260: 7920 6578 6563 7574 6162 6c65 732e 2020 y executables. │ │ │ │ +00264270: 5365 6520 2a6e 6f74 6520 6669 6e64 5072 See *note findPr │ │ │ │ +00264280: 6f67 7261 6d3a 2066 696e 6450 726f 6772 ogram: findProgr │ │ │ │ +00264290: 616d 2c2c 2073 7065 6369 6669 6361 6c6c am,, specificall │ │ │ │ +002642a0: 7920 7468 650a 2020 2020 6465 7363 7269 y the. descri │ │ │ │ +002642b0: 7074 696f 6e20 6f66 2074 6865 2050 7265 ption of the Pre │ │ │ │ +002642c0: 6669 7820 6f70 7469 6f6e 2c20 666f 7220 fix option, for │ │ │ │ +002642d0: 6d6f 7265 2e0a 2020 2a20 2276 6572 7369 more.. * "versi │ │ │ │ +002642e0: 6f6e 222c 2061 2073 7472 696e 6720 636f on", a string co │ │ │ │ +002642f0: 6e74 6169 6e69 6e67 2074 6865 2076 6572 ntaining the ver │ │ │ │ +00264300: 7369 6f6e 206e 756d 6265 7220 6f66 2074 sion number of t │ │ │ │ +00264310: 6865 2070 726f 6772 616d 2e20 204f 6e6c he program. Onl │ │ │ │ +00264320: 790a 2020 2020 7072 6573 656e 7420 6966 y. present if │ │ │ │ +00264330: 202a 6e6f 7465 2066 696e 6450 726f 6772 *note findProgr │ │ │ │ +00264340: 616d 3a20 6669 6e64 5072 6f67 7261 6d2c am: findProgram, │ │ │ │ +00264350: 2077 6173 2063 616c 6c65 6420 7769 7468 was called with │ │ │ │ +00264360: 2074 6865 0a20 2020 204d 696e 696d 756d the. Minimum │ │ │ │ +00264370: 5665 7273 696f 6e20 6f70 7469 6f6e 2e0a Version option.. │ │ │ │ +00264380: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ +00264390: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6669 ==.. * *note fi │ │ │ │ +002643a0: 6e64 5072 6f67 7261 6d3a 2066 696e 6450 ndProgram: findP │ │ │ │ +002643b0: 726f 6772 616d 2c20 2d2d 206c 6f61 6420 rogram, -- load │ │ │ │ +002643c0: 6578 7465 726e 616c 2070 726f 6772 616d external program │ │ │ │ +002643d0: 0a20 202a 202a 6e6f 7465 2072 756e 5072 . * *note runPr │ │ │ │ +002643e0: 6f67 7261 6d3a 2072 756e 5072 6f67 7261 ogram: runProgra │ │ │ │ +002643f0: 6d2c 202d 2d20 7275 6e20 616e 2065 7874 m, -- run an ext │ │ │ │ +00264400: 6572 6e61 6c20 7072 6f67 7261 6d0a 2020 ernal program. │ │ │ │ +00264410: 2a20 2a6e 6f74 6520 7072 6f67 7261 6d50 * *note programP │ │ │ │ +00264420: 6174 6873 3a20 7072 6f67 7261 6d50 6174 aths: programPat │ │ │ │ +00264430: 6873 2c20 2d2d 2075 7365 722d 6465 6669 hs, -- user-defi │ │ │ │ +00264440: 6e65 6420 6578 7465 726e 616c 2070 726f ned external pro │ │ │ │ +00264450: 6772 616d 2070 6174 6873 0a0a 4675 6e63 gram paths..Func │ │ │ │ +00264460: 7469 6f6e 7320 616e 6420 6d65 7468 6f64 tions and method │ │ │ │ +00264470: 7320 7265 7475 726e 696e 6720 616e 206f s returning an o │ │ │ │ +00264480: 626a 6563 7420 6f66 2063 6c61 7373 2050 bject of class P │ │ │ │ +00264490: 726f 6772 616d 3a0a 3d3d 3d3d 3d3d 3d3d rogram:.======== │ │ │ │ +002644a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +002644b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +002644c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +002644d0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2066 ===.. * *note f │ │ │ │ +002644e0: 696e 6450 726f 6772 616d 3a20 6669 6e64 indProgram: find │ │ │ │ +002644f0: 5072 6f67 7261 6d2c 202d 2d20 6c6f 6164 Program, -- load │ │ │ │ +00264500: 2065 7874 6572 6e61 6c20 7072 6f67 7261 external progra │ │ │ │ +00264510: 6d0a 0a4d 6574 686f 6473 2074 6861 7420 m..Methods that │ │ │ │ +00264520: 7573 6520 616e 206f 626a 6563 7420 6f66 use an object of │ │ │ │ +00264530: 2063 6c61 7373 2050 726f 6772 616d 3a0a class Program:. │ │ │ │ +00264540: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00264550: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00264560: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +00264570: 2a20 2a6e 6f74 6520 5072 6f67 7261 6d20 * *note Program │ │ │ │ +00264580: 3c3c 2054 6869 6e67 3a20 5072 6f67 7261 << Thing: Progra │ │ │ │ +00264590: 6d20 3c3c 2054 6869 6e67 2c20 2d2d 2072 m << Thing, -- r │ │ │ │ +002645a0: 756e 2070 726f 6772 616d 2077 6974 6820 un program with │ │ │ │ +002645b0: 696e 7075 740a 2020 2020 7265 6469 7265 input. redire │ │ │ │ +002645c0: 6374 696f 6e0a 2020 2a20 2272 756e 5072 ction. * "runPr │ │ │ │ +002645d0: 6f67 7261 6d28 5072 6f67 7261 6d2c 5374 ogram(Program,St │ │ │ │ +002645e0: 7269 6e67 2922 202d 2d20 7365 6520 2a6e ring)" -- see *n │ │ │ │ +002645f0: 6f74 6520 7275 6e50 726f 6772 616d 3a20 ote runProgram: │ │ │ │ +00264600: 7275 6e50 726f 6772 616d 2c20 2d2d 2072 runProgram, -- r │ │ │ │ +00264610: 756e 2061 6e0a 2020 2020 6578 7465 726e un an. extern │ │ │ │ +00264620: 616c 2070 726f 6772 616d 0a20 202a 2022 al program. * " │ │ │ │ +00264630: 7275 6e50 726f 6772 616d 2850 726f 6772 runProgram(Progr │ │ │ │ +00264640: 616d 2c53 7472 696e 672c 5374 7269 6e67 am,String,String │ │ │ │ +00264650: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ +00264660: 7275 6e50 726f 6772 616d 3a20 7275 6e50 runProgram: runP │ │ │ │ +00264670: 726f 6772 616d 2c20 2d2d 0a20 2020 2072 rogram, --. r │ │ │ │ +00264680: 756e 2061 6e20 6578 7465 726e 616c 2070 un an external p │ │ │ │ +00264690: 726f 6772 616d 0a0a 466f 7220 7468 6520 rogram..For the │ │ │ │ +002646a0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ +002646b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +002646c0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +002646d0: 5072 6f67 7261 6d3a 2050 726f 6772 616d Program: Program │ │ │ │ +002646e0: 2c20 6973 2061 202a 6e6f 7465 2074 7970 , is a *note typ │ │ │ │ +002646f0: 653a 2054 7970 652c 2c20 7769 7468 2061 e: Type,, with a │ │ │ │ +00264700: 6e63 6573 746f 720a 636c 6173 7365 7320 ncestor.classes │ │ │ │ +00264710: 2a6e 6f74 6520 4861 7368 5461 626c 653a *note HashTable: │ │ │ │ +00264720: 2048 6173 6854 6162 6c65 2c20 3c20 2a6e HashTable, < *n │ │ │ │ +00264730: 6f74 6520 5468 696e 673a 2054 6869 6e67 ote Thing: Thing │ │ │ │ +00264740: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ 00264750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00264760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -00264770: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -00264780: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -00264790: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -002647a0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -002647b0: 756c 6179 322d 312e 3236 2e30 352b 6473 ulay2-1.26.05+ds │ │ │ │ -002647c0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -002647d0: 636b 6167 6573 2f0a 4d61 6361 756c 6179 ckages/.Macaulay │ │ │ │ -002647e0: 3244 6f63 2f66 756e 6374 696f 6e73 2f66 2Doc/functions/f │ │ │ │ -002647f0: 696e 6450 726f 6772 616d 2d64 6f63 2e6d indProgram-doc.m │ │ │ │ -00264800: 323a 3330 3a30 2e0a 1f0a 4669 6c65 3a20 2:30:0....File: │ │ │ │ -00264810: 4d61 6361 756c 6179 3244 6f63 2e69 6e66 Macaulay2Doc.inf │ │ │ │ -00264820: 6f2c 204e 6f64 653a 2070 726f 6772 616d o, Node: program │ │ │ │ -00264830: 5061 7468 732c 2050 7265 763a 2050 726f Paths, Prev: Pro │ │ │ │ -00264840: 6772 616d 2c20 5570 3a20 6669 6e64 5072 gram, Up: findPr │ │ │ │ -00264850: 6f67 7261 6d0a 0a70 726f 6772 616d 5061 ogram..programPa │ │ │ │ -00264860: 7468 7320 2d2d 2075 7365 722d 6465 6669 ths -- user-defi │ │ │ │ -00264870: 6e65 6420 6578 7465 726e 616c 2070 726f ned external pro │ │ │ │ -00264880: 6772 616d 2070 6174 6873 0a2a 2a2a 2a2a gram paths.***** │ │ │ │ -00264890: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002648a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002648b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -002648c0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -002648d0: 3d3d 3d3d 3d3d 3d0a 0a41 206d 7574 6162 =======..A mutab │ │ │ │ -002648e0: 6c65 2068 6173 6820 7461 626c 6520 636f le hash table co │ │ │ │ -002648f0: 6e74 6169 6e69 6e67 2075 7365 722d 6465 ntaining user-de │ │ │ │ -00264900: 6669 6e65 6420 7061 7468 7320 746f 2065 fined paths to e │ │ │ │ -00264910: 7874 6572 6e61 6c20 7072 6f67 7261 6d73 xternal programs │ │ │ │ -00264920: 2075 7365 6420 6279 0a4d 6163 6175 6c61 used by.Macaula │ │ │ │ -00264930: 7932 2e20 2049 7473 206b 6579 7320 6172 y2. Its keys ar │ │ │ │ -00264940: 6520 7374 7269 6e67 7320 636f 6e74 6169 e strings contai │ │ │ │ -00264950: 6e69 6e67 2074 6865 206e 616d 6573 206f ning the names o │ │ │ │ -00264960: 6620 7072 6f67 7261 6d73 2061 6e64 206d f programs and m │ │ │ │ -00264970: 7573 740a 636f 696e 6369 6465 2077 6974 ust.coincide wit │ │ │ │ -00264980: 6820 7468 6520 6669 7273 7420 6172 6775 h the first argu │ │ │ │ -00264990: 6d65 6e74 206f 6620 2a6e 6f74 6520 6669 ment of *note fi │ │ │ │ -002649a0: 6e64 5072 6f67 7261 6d3a 2066 696e 6450 ndProgram: findP │ │ │ │ -002649b0: 726f 6772 616d 2c2e 0a0a 4974 2069 7320 rogram,...It is │ │ │ │ -002649c0: 6f6e 6c79 206e 6563 6573 7361 7279 2074 only necessary t │ │ │ │ -002649d0: 6f20 6465 6669 6e65 2061 2070 6174 6820 o define a path │ │ │ │ -002649e0: 696e 2074 6869 7320 7761 7920 6966 2061 in this way if a │ │ │ │ -002649f0: 2070 726f 6772 616d 2069 7320 696e 7374 program is inst │ │ │ │ -00264a00: 616c 6c65 6420 696e 0a61 206e 6f6e 2d73 alled in.a non-s │ │ │ │ -00264a10: 7461 6e64 6172 6420 6c6f 6361 7469 6f6e tandard location │ │ │ │ -00264a20: 2e20 2049 6e20 7061 7274 6963 756c 6172 . In particular │ │ │ │ -00264a30: 2c20 2a6e 6f74 6520 6669 6e64 5072 6f67 , *note findProg │ │ │ │ -00264a40: 7261 6d3a 2066 696e 6450 726f 6772 616d ram: findProgram │ │ │ │ -00264a50: 2c0a 616c 7265 6164 7920 6368 6563 6b73 ,.already checks │ │ │ │ -00264a60: 2070 7265 6669 7844 6972 6563 746f 7279 prefixDirectory │ │ │ │ -00264a70: 207c 2063 7572 7265 6e74 4c61 796f 7574 | currentLayout │ │ │ │ -00264a80: 2322 7072 6f67 7261 6d73 222c 2028 7768 #"programs", (wh │ │ │ │ -00264a90: 6572 6520 7468 6520 7072 6f67 7261 6d73 ere the programs │ │ │ │ -00264aa0: 0a73 6869 7070 6564 2077 6974 6820 4d61 .shipped with Ma │ │ │ │ -00264ab0: 6361 756c 6179 3220 6172 6520 696e 7374 caulay2 are inst │ │ │ │ -00264ac0: 616c 6c65 6429 2061 6e64 2061 6c6c 206f alled) and all o │ │ │ │ -00264ad0: 6620 7468 6520 6469 7265 6374 6f72 6965 f the directorie │ │ │ │ -00264ae0: 7320 696e 2074 6865 2075 7365 7227 730a s in the user's. │ │ │ │ -00264af0: 5041 5448 2065 6e76 6972 6f6e 6d65 6e74 PATH environment │ │ │ │ -00264b00: 2076 6172 6961 626c 652e 0a0a 4966 2079 variable...If y │ │ │ │ -00264b10: 6f75 2075 7365 2061 2070 6172 7469 6375 ou use a particu │ │ │ │ -00264b20: 6c61 7220 7072 6f67 7261 6d20 6672 6571 lar program freq │ │ │ │ -00264b30: 7565 6e74 6c79 2061 6e64 2069 7420 6973 uently and it is │ │ │ │ -00264b40: 2069 6e73 7461 6c6c 6564 2069 6e20 610a installed in a. │ │ │ │ -00264b50: 6e6f 6e2d 7374 616e 6461 7264 206c 6f63 non-standard loc │ │ │ │ -00264b60: 6174 696f 6e2c 2074 6865 6e20 6974 206d ation, then it m │ │ │ │ -00264b70: 6179 2062 6520 7573 6566 756c 2074 6f20 ay be useful to │ │ │ │ -00264b80: 6164 6420 6120 6c69 6e65 2074 6f20 796f add a line to yo │ │ │ │ -00264b90: 7572 202a 6e6f 7465 0a69 6e69 7469 616c ur *note.initial │ │ │ │ -00264ba0: 697a 6174 696f 6e20 6669 6c65 3a20 696e ization file: in │ │ │ │ -00264bb0: 6974 6961 6c69 7a61 7469 6f6e 2066 696c itialization fil │ │ │ │ -00264bc0: 652c 2064 6566 696e 696e 6720 6974 7320 e, defining its │ │ │ │ -00264bd0: 7061 7468 2069 6e20 7468 6973 2077 6179 path in this way │ │ │ │ -00264be0: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d ...See also.==== │ │ │ │ -00264bf0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -00264c00: 5072 6f67 7261 6d3a 2050 726f 6772 616d Program: Program │ │ │ │ -00264c10: 2c20 2d2d 2065 7874 6572 6e61 6c20 7072 , -- external pr │ │ │ │ -00264c20: 6f67 7261 6d20 6f62 6a65 6374 0a20 202a ogram object. * │ │ │ │ -00264c30: 202a 6e6f 7465 2070 6174 683a 2070 6174 *note path: pat │ │ │ │ -00264c40: 682c 202d 2d20 6c69 7374 206f 6620 6469 h, -- list of di │ │ │ │ -00264c50: 7265 6374 6f72 6965 7320 746f 206c 6f6f rectories to loo │ │ │ │ -00264c60: 6b20 696e 0a0a 466f 7220 7468 6520 7072 k in..For the pr │ │ │ │ -00264c70: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ -00264c80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -00264c90: 206f 626a 6563 7420 2a6e 6f74 6520 7072 object *note pr │ │ │ │ -00264ca0: 6f67 7261 6d50 6174 6873 3a20 7072 6f67 ogramPaths: prog │ │ │ │ -00264cb0: 7261 6d50 6174 6873 2c20 6973 2061 202a ramPaths, is a * │ │ │ │ -00264cc0: 6e6f 7465 206d 7574 6162 6c65 2068 6173 note mutable has │ │ │ │ -00264cd0: 6820 7461 626c 653a 0a4d 7574 6162 6c65 h table:.Mutable │ │ │ │ -00264ce0: 4861 7368 5461 626c 652c 2e0a 0a2d 2d2d HashTable,...--- │ │ │ │ -00264cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00264d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00264d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00264760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00264770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00264780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00264790: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +002647a0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +002647b0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +002647c0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +002647d0: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ +002647e0: 352b 6473 2f4d 322f 4d61 6361 756c 6179 5+ds/M2/Macaulay │ │ │ │ +002647f0: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ +00264800: 756c 6179 3244 6f63 2f66 756e 6374 696f ulay2Doc/functio │ │ │ │ +00264810: 6e73 2f66 696e 6450 726f 6772 616d 2d64 ns/findProgram-d │ │ │ │ +00264820: 6f63 2e6d 323a 3330 3a30 2e0a 1f0a 4669 oc.m2:30:0....Fi │ │ │ │ +00264830: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ +00264840: 2e69 6e66 6f2c 204e 6f64 653a 2070 726f .info, Node: pro │ │ │ │ +00264850: 6772 616d 5061 7468 732c 2050 7265 763a gramPaths, Prev: │ │ │ │ +00264860: 2050 726f 6772 616d 2c20 5570 3a20 6669 Program, Up: fi │ │ │ │ +00264870: 6e64 5072 6f67 7261 6d0a 0a70 726f 6772 ndProgram..progr │ │ │ │ +00264880: 616d 5061 7468 7320 2d2d 2075 7365 722d amPaths -- user- │ │ │ │ +00264890: 6465 6669 6e65 6420 6578 7465 726e 616c defined external │ │ │ │ +002648a0: 2070 726f 6772 616d 2070 6174 6873 0a2a program paths.* │ │ │ │ +002648b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002648c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002648d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002648e0: 2a2a 0a0a 4465 7363 7269 7074 696f 6e0a **..Description. │ │ │ │ +002648f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 206d ===========..A m │ │ │ │ +00264900: 7574 6162 6c65 2068 6173 6820 7461 626c utable hash tabl │ │ │ │ +00264910: 6520 636f 6e74 6169 6e69 6e67 2075 7365 e containing use │ │ │ │ +00264920: 722d 6465 6669 6e65 6420 7061 7468 7320 r-defined paths │ │ │ │ +00264930: 746f 2065 7874 6572 6e61 6c20 7072 6f67 to external prog │ │ │ │ +00264940: 7261 6d73 2075 7365 6420 6279 0a4d 6163 rams used by.Mac │ │ │ │ +00264950: 6175 6c61 7932 2e20 2049 7473 206b 6579 aulay2. Its key │ │ │ │ +00264960: 7320 6172 6520 7374 7269 6e67 7320 636f s are strings co │ │ │ │ +00264970: 6e74 6169 6e69 6e67 2074 6865 206e 616d ntaining the nam │ │ │ │ +00264980: 6573 206f 6620 7072 6f67 7261 6d73 2061 es of programs a │ │ │ │ +00264990: 6e64 206d 7573 740a 636f 696e 6369 6465 nd must.coincide │ │ │ │ +002649a0: 2077 6974 6820 7468 6520 6669 7273 7420 with the first │ │ │ │ +002649b0: 6172 6775 6d65 6e74 206f 6620 2a6e 6f74 argument of *not │ │ │ │ +002649c0: 6520 6669 6e64 5072 6f67 7261 6d3a 2066 e findProgram: f │ │ │ │ +002649d0: 696e 6450 726f 6772 616d 2c2e 0a0a 4974 indProgram,...It │ │ │ │ +002649e0: 2069 7320 6f6e 6c79 206e 6563 6573 7361 is only necessa │ │ │ │ +002649f0: 7279 2074 6f20 6465 6669 6e65 2061 2070 ry to define a p │ │ │ │ +00264a00: 6174 6820 696e 2074 6869 7320 7761 7920 ath in this way │ │ │ │ +00264a10: 6966 2061 2070 726f 6772 616d 2069 7320 if a program is │ │ │ │ +00264a20: 696e 7374 616c 6c65 6420 696e 0a61 206e installed in.a n │ │ │ │ +00264a30: 6f6e 2d73 7461 6e64 6172 6420 6c6f 6361 on-standard loca │ │ │ │ +00264a40: 7469 6f6e 2e20 2049 6e20 7061 7274 6963 tion. In partic │ │ │ │ +00264a50: 756c 6172 2c20 2a6e 6f74 6520 6669 6e64 ular, *note find │ │ │ │ +00264a60: 5072 6f67 7261 6d3a 2066 696e 6450 726f Program: findPro │ │ │ │ +00264a70: 6772 616d 2c0a 616c 7265 6164 7920 6368 gram,.already ch │ │ │ │ +00264a80: 6563 6b73 2070 7265 6669 7844 6972 6563 ecks prefixDirec │ │ │ │ +00264a90: 746f 7279 207c 2063 7572 7265 6e74 4c61 tory | currentLa │ │ │ │ +00264aa0: 796f 7574 2322 7072 6f67 7261 6d73 222c yout#"programs", │ │ │ │ +00264ab0: 2028 7768 6572 6520 7468 6520 7072 6f67 (where the prog │ │ │ │ +00264ac0: 7261 6d73 0a73 6869 7070 6564 2077 6974 rams.shipped wit │ │ │ │ +00264ad0: 6820 4d61 6361 756c 6179 3220 6172 6520 h Macaulay2 are │ │ │ │ +00264ae0: 696e 7374 616c 6c65 6429 2061 6e64 2061 installed) and a │ │ │ │ +00264af0: 6c6c 206f 6620 7468 6520 6469 7265 6374 ll of the direct │ │ │ │ +00264b00: 6f72 6965 7320 696e 2074 6865 2075 7365 ories in the use │ │ │ │ +00264b10: 7227 730a 5041 5448 2065 6e76 6972 6f6e r's.PATH environ │ │ │ │ +00264b20: 6d65 6e74 2076 6172 6961 626c 652e 0a0a ment variable... │ │ │ │ +00264b30: 4966 2079 6f75 2075 7365 2061 2070 6172 If you use a par │ │ │ │ +00264b40: 7469 6375 6c61 7220 7072 6f67 7261 6d20 ticular program │ │ │ │ +00264b50: 6672 6571 7565 6e74 6c79 2061 6e64 2069 frequently and i │ │ │ │ +00264b60: 7420 6973 2069 6e73 7461 6c6c 6564 2069 t is installed i │ │ │ │ +00264b70: 6e20 610a 6e6f 6e2d 7374 616e 6461 7264 n a.non-standard │ │ │ │ +00264b80: 206c 6f63 6174 696f 6e2c 2074 6865 6e20 location, then │ │ │ │ +00264b90: 6974 206d 6179 2062 6520 7573 6566 756c it may be useful │ │ │ │ +00264ba0: 2074 6f20 6164 6420 6120 6c69 6e65 2074 to add a line t │ │ │ │ +00264bb0: 6f20 796f 7572 202a 6e6f 7465 0a69 6e69 o your *note.ini │ │ │ │ +00264bc0: 7469 616c 697a 6174 696f 6e20 6669 6c65 tialization file │ │ │ │ +00264bd0: 3a20 696e 6974 6961 6c69 7a61 7469 6f6e : initialization │ │ │ │ +00264be0: 2066 696c 652c 2064 6566 696e 696e 6720 file, defining │ │ │ │ +00264bf0: 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7261 6d50 6174 6873 3a20 e programPaths: │ │ │ │ +00264cd0: 7072 6f67 7261 6d50 6174 6873 2c20 6973 programPaths, is │ │ │ │ +00264ce0: 2061 202a 6e6f 7465 206d 7574 6162 6c65 a *note mutable │ │ │ │ +00264cf0: 2068 6173 6820 7461 626c 653a 0a4d 7574 hash table:.Mut │ │ │ │ +00264d00: 6162 6c65 4861 7368 5461 626c 652c 2e0a ableHashTable,.. │ │ │ │ +00264d10: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 00264d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00264d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -00264d40: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -00264d50: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -00264d60: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -00264d70: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -00264d80: 7932 2d31 2e32 362e 3035 2b64 732f 4d32 y2-1.26.05+ds/M2 │ │ │ │ -00264d90: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -00264da0: 6765 732f 0a4d 6163 6175 6c61 7932 446f ges/.Macaulay2Do │ │ │ │ -00264db0: 632f 6675 6e63 7469 6f6e 732f 6669 6e64 c/functions/find │ │ │ │ -00264dc0: 5072 6f67 7261 6d2d 646f 632e 6d32 3a35 Program-doc.m2:5 │ │ │ │ -00264dd0: 383a 302e 0a1f 0a46 696c 653a 204d 6163 8:0....File: Mac │ │ │ │ -00264de0: 6175 6c61 7932 446f 632e 696e 666f 2c20 aulay2Doc.info, │ │ │ │ -00264df0: 4e6f 6465 3a20 7275 6e50 726f 6772 616d Node: runProgram │ │ │ │ -00264e00: 2c20 4e65 7874 3a20 5072 6f67 7261 6d20 , Next: Program │ │ │ │ -00264e10: 3c3c 2054 6869 6e67 2c20 5072 6576 3a20 << Thing, Prev: │ │ │ │ -00264e20: 6669 6e64 5072 6f67 7261 6d2c 2055 703a findProgram, Up: │ │ │ │ -00264e30: 2063 6f6d 6d75 6e69 6361 7469 6e67 2077 communicating w │ │ │ │ -00264e40: 6974 6820 7072 6f67 7261 6d73 0a0a 7275 ith programs..ru │ │ │ │ -00264e50: 6e50 726f 6772 616d 202d 2d20 7275 6e20 nProgram -- run │ │ │ │ -00264e60: 616e 2065 7874 6572 6e61 6c20 7072 6f67 an external prog │ │ │ │ -00264e70: 7261 6d0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ram.************ │ │ │ │ -00264e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00264e90: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -00264ea0: 7361 6765 3a20 0a20 2020 2020 2020 2072 sage: . r │ │ │ │ -00264eb0: 756e 5072 6f67 7261 6d28 7072 6f67 7261 unProgram(progra │ │ │ │ -00264ec0: 6d2c 2061 7267 7329 0a20 2020 2020 2020 m, args). │ │ │ │ -00264ed0: 2072 756e 5072 6f67 7261 6d28 7072 6f67 runProgram(prog │ │ │ │ -00264ee0: 7261 6d2c 2065 7865 2c20 6172 6773 290a ram, exe, args). │ │ │ │ -00264ef0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -00264f00: 2020 2a20 7072 6f67 7261 6d2c 2061 6e20 * program, an │ │ │ │ -00264f10: 696e 7374 616e 6365 206f 6620 7468 6520 instance of the │ │ │ │ -00264f20: 7479 7065 202a 6e6f 7465 2050 726f 6772 type *note Progr │ │ │ │ -00264f30: 616d 3a20 5072 6f67 7261 6d2c 2c20 7468 am: Program,, th │ │ │ │ -00264f40: 6520 7072 6f67 7261 6d0a 2020 2020 2020 e program. │ │ │ │ -00264f50: 2020 746f 2072 756e 2c20 6765 6e65 7261 to run, genera │ │ │ │ -00264f60: 7465 6420 6279 202a 6e6f 7465 2066 696e ted by *note fin │ │ │ │ -00264f70: 6450 726f 6772 616d 3a20 6669 6e64 5072 dProgram: findPr │ │ │ │ -00264f80: 6f67 7261 6d2c 2e0a 2020 2020 2020 2a20 ogram,.. * │ │ │ │ -00264f90: 6578 652c 2061 202a 6e6f 7465 2073 7472 exe, a *note str │ │ │ │ -00264fa0: 696e 673a 2053 7472 696e 672c 2c20 7468 ing: String,, th │ │ │ │ -00264fb0: 6520 7370 6563 6966 6963 2065 7865 6375 e specific execu │ │ │ │ -00264fc0: 7461 626c 6520 6669 6c65 2074 6f20 7275 table file to ru │ │ │ │ -00264fd0: 6e2e 2054 6869 730a 2020 2020 2020 2020 n. This. │ │ │ │ -00264fe0: 6973 206f 6e6c 7920 6e65 6365 7373 6172 is only necessar │ │ │ │ -00264ff0: 7920 6966 2074 6865 2070 726f 6772 616d y if the program │ │ │ │ -00265000: 2063 6f6e 7369 7374 7320 6f66 206d 756c consists of mul │ │ │ │ -00265010: 7469 706c 6520 7375 6368 2066 696c 6573 tiple such files │ │ │ │ -00265020: 2e20 2049 660a 2020 2020 2020 2020 6e6f . If. no │ │ │ │ -00265030: 7420 6769 7665 6e2c 2074 6865 6e20 7072 t given, then pr │ │ │ │ -00265040: 6f67 7261 6d23 226e 616d 6522 2069 7320 ogram#"name" is │ │ │ │ -00265050: 7573 6564 2e0a 2020 2020 2020 2a20 6172 used.. * ar │ │ │ │ -00265060: 6773 2c20 6120 2a6e 6f74 6520 7374 7269 gs, a *note stri │ │ │ │ -00265070: 6e67 3a20 5374 7269 6e67 2c2c 2074 6865 ng: String,, the │ │ │ │ -00265080: 2063 6f6d 6d61 6e64 206c 696e 6520 6172 command line ar │ │ │ │ -00265090: 6775 6d65 6e74 7320 7061 7373 6564 2074 guments passed t │ │ │ │ -002650a0: 6f20 7468 650a 2020 2020 2020 2020 7072 o the. pr │ │ │ │ -002650b0: 6f67 7261 6d2e 0a20 202a 202a 6e6f 7465 ogram.. * *note │ │ │ │ -002650c0: 204f 7074 696f 6e61 6c20 696e 7075 7473 Optional inputs │ │ │ │ -002650d0: 3a20 7573 696e 6720 6675 6e63 7469 6f6e : using function │ │ │ │ -002650e0: 7320 7769 7468 206f 7074 696f 6e61 6c20 s with optional │ │ │ │ -002650f0: 696e 7075 7473 2c3a 0a20 2020 2020 202a inputs,:. * │ │ │ │ -00265100: 204b 6565 7046 696c 6573 203d 3e20 6120 KeepFiles => a │ │ │ │ -00265110: 2a6e 6f74 6520 426f 6f6c 6561 6e20 7661 *note Boolean va │ │ │ │ -00265120: 6c75 653a 2042 6f6f 6c65 616e 2c2c 2064 lue: Boolean,, d │ │ │ │ -00265130: 6566 6175 6c74 2076 616c 7565 2066 616c efault value fal │ │ │ │ -00265140: 7365 2c0a 2020 2020 2020 2020 7768 6574 se,. whet │ │ │ │ -00265150: 6865 7220 746f 206b 6565 7020 7468 6520 her to keep the │ │ │ │ -00265160: 7465 6d70 6f72 6172 7920 6669 6c65 7320 temporary files │ │ │ │ -00265170: 636f 6e74 6169 6e69 6e67 2074 6865 2070 containing the p │ │ │ │ -00265180: 726f 6772 616d 2773 206f 7574 7075 742e rogram's output. │ │ │ │ -00265190: 0a20 2020 2020 202a 2052 6169 7365 4572 . * RaiseEr │ │ │ │ -002651a0: 726f 7220 3d3e 2061 202a 6e6f 7465 2042 ror => a *note B │ │ │ │ -002651b0: 6f6f 6c65 616e 2076 616c 7565 3a20 426f oolean value: Bo │ │ │ │ -002651c0: 6f6c 6561 6e2c 2c20 6465 6661 756c 7420 olean,, default │ │ │ │ -002651d0: 7661 6c75 6520 7472 7565 2c0a 2020 2020 value true,. │ │ │ │ -002651e0: 2020 2020 7768 6574 6865 7220 746f 2072 whether to r │ │ │ │ -002651f0: 6169 7365 2061 6e20 6572 726f 7220 6966 aise an error if │ │ │ │ -00265200: 2074 6865 2070 726f 6772 616d 2072 6574 the program ret │ │ │ │ -00265210: 7572 6e73 2061 206e 6f6e 7a65 726f 2076 urns a nonzero v │ │ │ │ -00265220: 616c 7565 2e0a 2020 2020 2020 2a20 5275 alue.. * Ru │ │ │ │ -00265230: 6e44 6972 6563 746f 7279 203d 3e20 6120 nDirectory => a │ │ │ │ -00265240: 2a6e 6f74 6520 7374 7269 6e67 3a20 5374 *note string: St │ │ │ │ -00265250: 7269 6e67 2c2c 2064 6566 6175 6c74 2076 ring,, default v │ │ │ │ -00265260: 616c 7565 206e 756c 6c2c 2074 6865 0a20 alue null, the. │ │ │ │ -00265270: 2020 2020 2020 2064 6972 6563 746f 7279 directory │ │ │ │ -00265280: 2066 726f 6d20 7768 6963 6820 746f 2072 from which to r │ │ │ │ -00265290: 756e 2074 6865 2070 726f 6772 616d 2e20 un the program. │ │ │ │ -002652a0: 2049 6620 6974 2064 6f65 7320 6e6f 7420 If it does not │ │ │ │ -002652b0: 6578 6973 742c 2074 6865 6e20 6974 0a20 exist, then it. │ │ │ │ -002652c0: 2020 2020 2020 2077 696c 6c20 6265 2063 will be c │ │ │ │ -002652d0: 7265 6174 6564 2e20 2049 6620 2a6e 6f74 reated. If *not │ │ │ │ -002652e0: 6520 6e75 6c6c 3a20 6e75 6c6c 2c2c 2074 e null: null,, t │ │ │ │ -002652f0: 6865 6e20 7468 6520 7072 6f67 7261 6d20 hen the program │ │ │ │ -00265300: 7769 6c6c 2062 6520 7275 6e0a 2020 2020 will be run. │ │ │ │ -00265310: 2020 2020 6672 6f6d 2074 6865 2063 7572 from the cur │ │ │ │ -00265320: 7265 6e74 2077 6f72 6b69 6e67 2064 6972 rent working dir │ │ │ │ -00265330: 6563 746f 7279 2028 7365 6520 2a6e 6f74 ectory (see *not │ │ │ │ -00265340: 6520 6375 7272 656e 7444 6972 6563 746f e currentDirecto │ │ │ │ -00265350: 7279 3a0a 2020 2020 2020 2020 6375 7272 ry:. curr │ │ │ │ -00265360: 656e 7444 6972 6563 746f 7279 2c29 2e0a entDirectory,).. │ │ │ │ -00265370: 2020 2020 2020 2a20 5665 7262 6f73 6520 * Verbose │ │ │ │ -00265380: 3d3e 2061 202a 6e6f 7465 2042 6f6f 6c65 => a *note Boole │ │ │ │ -00265390: 616e 2076 616c 7565 3a20 426f 6f6c 6561 an value: Boolea │ │ │ │ -002653a0: 6e2c 2c20 6465 6661 756c 7420 7661 6c75 n,, default valu │ │ │ │ -002653b0: 6520 6661 6c73 652c 0a20 2020 2020 2020 e false,. │ │ │ │ -002653c0: 2077 6865 7468 6572 2074 6f20 7072 696e whether to prin │ │ │ │ -002653d0: 7420 7468 6520 636f 6d6d 616e 6420 6c69 t the command li │ │ │ │ -002653e0: 6e65 2069 6e70 7574 2061 6e64 2074 6865 ne input and the │ │ │ │ -002653f0: 2070 726f 6772 616d 2773 206f 7574 7075 program's outpu │ │ │ │ -00265400: 742e 0a0a 4465 7363 7269 7074 696f 6e0a t...Description. │ │ │ │ -00265410: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 ===========..Thi │ │ │ │ -00265420: 7320 6d65 7468 6f64 2072 756e 7320 616e s method runs an │ │ │ │ -00265430: 2065 7874 6572 6e61 6c20 7072 6f67 7261 external progra │ │ │ │ -00265440: 6d20 7768 6963 6820 6861 7320 616c 7265 m which has alre │ │ │ │ -00265450: 6164 7920 6265 656e 206c 6f61 6465 6420 ady been loaded │ │ │ │ -00265460: 7573 696e 6720 2a6e 6f74 650a 6669 6e64 using *note.find │ │ │ │ -00265470: 5072 6f67 7261 6d3a 2066 696e 6450 726f Program: findPro │ │ │ │ -00265480: 6772 616d 2c2e 2020 5468 6520 7265 7375 gram,. The resu │ │ │ │ -00265490: 6c74 7320 6f66 2074 6869 7320 7275 6e20 lts of this run │ │ │ │ -002654a0: 6172 6520 6176 6169 6c61 626c 6520 696e are available in │ │ │ │ -002654b0: 2061 202a 6e6f 7465 0a50 726f 6772 616d a *note.Program │ │ │ │ -002654c0: 5275 6e3a 2050 726f 6772 616d 5275 6e2c Run: ProgramRun, │ │ │ │ -002654d0: 206f 626a 6563 742e 0a0a 2b2d 2d2d 2d2d object...+----- │ │ │ │ -002654e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002654f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00264d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00264d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00264d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00264d60: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +00264d70: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +00264d80: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +00264d90: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00264da0: 6175 6c61 7932 2d31 2e32 362e 3035 2b64 aulay2-1.26.05+d │ │ │ │ +00264db0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00264dc0: 6163 6b61 6765 732f 0a4d 6163 6175 6c61 ackages/.Macaula │ │ │ │ +00264dd0: 7932 446f 632f 6675 6e63 7469 6f6e 732f y2Doc/functions/ │ │ │ │ +00264de0: 6669 6e64 5072 6f67 7261 6d2d 646f 632e findProgram-doc. │ │ │ │ +00264df0: 6d32 3a35 383a 302e 0a1f 0a46 696c 653a m2:58:0....File: │ │ │ │ +00264e00: 204d 6163 6175 6c61 7932 446f 632e 696e Macaulay2Doc.in │ │ │ │ +00264e10: 666f 2c20 4e6f 6465 3a20 7275 6e50 726f fo, Node: runPro │ │ │ │ +00264e20: 6772 616d 2c20 4e65 7874 3a20 5072 6f67 gram, Next: Prog │ │ │ │ +00264e30: 7261 6d20 3c3c 2054 6869 6e67 2c20 5072 ram << Thing, Pr │ │ │ │ +00264e40: 6576 3a20 6669 6e64 5072 6f67 7261 6d2c ev: findProgram, │ │ │ │ +00264e50: 2055 703a 2063 6f6d 6d75 6e69 6361 7469 Up: communicati │ │ │ │ +00264e60: 6e67 2077 6974 6820 7072 6f67 7261 6d73 ng with programs │ │ │ │ +00264e70: 0a0a 7275 6e50 726f 6772 616d 202d 2d20 ..runProgram -- │ │ │ │ +00264e80: 7275 6e20 616e 2065 7874 6572 6e61 6c20 run an external │ │ │ │ +00264e90: 7072 6f67 7261 6d0a 2a2a 2a2a 2a2a 2a2a program.******** │ │ │ │ +00264ea0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00264eb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ +00264ec0: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ +00264ed0: 2020 2072 756e 5072 6f67 7261 6d28 7072 runProgram(pr │ │ │ │ +00264ee0: 6f67 7261 6d2c 2061 7267 7329 0a20 2020 ogram, args). │ │ │ │ +00264ef0: 2020 2020 2072 756e 5072 6f67 7261 6d28 runProgram( │ │ │ │ +00264f00: 7072 6f67 7261 6d2c 2065 7865 2c20 6172 program, exe, ar │ │ │ │ +00264f10: 6773 290a 2020 2a20 496e 7075 7473 3a0a gs). * Inputs:. │ │ │ │ +00264f20: 2020 2020 2020 2a20 7072 6f67 7261 6d2c * program, │ │ │ │ +00264f30: 2061 6e20 696e 7374 616e 6365 206f 6620 an instance of │ │ │ │ +00264f40: 7468 6520 7479 7065 202a 6e6f 7465 2050 the type *note P │ │ │ │ +00264f50: 726f 6772 616d 3a20 5072 6f67 7261 6d2c rogram: Program, │ │ │ │ +00264f60: 2c20 7468 6520 7072 6f67 7261 6d0a 2020 , the program. │ │ │ │ +00264f70: 2020 2020 2020 746f 2072 756e 2c20 6765 to run, ge │ │ │ │ +00264f80: 6e65 7261 7465 6420 6279 202a 6e6f 7465 nerated by *note │ │ │ │ +00264f90: 2066 696e 6450 726f 6772 616d 3a20 6669 findProgram: fi │ │ │ │ +00264fa0: 6e64 5072 6f67 7261 6d2c 2e0a 2020 2020 ndProgram,.. │ │ │ │ +00264fb0: 2020 2a20 6578 652c 2061 202a 6e6f 7465 * exe, a *note │ │ │ │ +00264fc0: 2073 7472 696e 673a 2053 7472 696e 672c string: String, │ │ │ │ +00264fd0: 2c20 7468 6520 7370 6563 6966 6963 2065 , the specific e │ │ │ │ +00264fe0: 7865 6375 7461 626c 6520 6669 6c65 2074 xecutable file t │ │ │ │ +00264ff0: 6f20 7275 6e2e 2054 6869 730a 2020 2020 o run. This. │ │ │ │ +00265000: 2020 2020 6973 206f 6e6c 7920 6e65 6365 is only nece │ │ │ │ +00265010: 7373 6172 7920 6966 2074 6865 2070 726f ssary if the pro │ │ │ │ +00265020: 6772 616d 2063 6f6e 7369 7374 7320 6f66 gram consists of │ │ │ │ +00265030: 206d 756c 7469 706c 6520 7375 6368 2066 multiple such f │ │ │ │ +00265040: 696c 6573 2e20 2049 660a 2020 2020 2020 iles. If. │ │ │ │ +00265050: 2020 6e6f 7420 6769 7665 6e2c 2074 6865 not given, the │ │ │ │ +00265060: 6e20 7072 6f67 7261 6d23 226e 616d 6522 n program#"name" │ │ │ │ +00265070: 2069 7320 7573 6564 2e0a 2020 2020 2020 is used.. │ │ │ │ +00265080: 2a20 6172 6773 2c20 6120 2a6e 6f74 6520 * args, a *note │ │ │ │ +00265090: 7374 7269 6e67 3a20 5374 7269 6e67 2c2c string: String,, │ │ │ │ +002650a0: 2074 6865 2063 6f6d 6d61 6e64 206c 696e the command lin │ │ │ │ +002650b0: 6520 6172 6775 6d65 6e74 7320 7061 7373 e arguments pass │ │ │ │ +002650c0: 6564 2074 6f20 7468 650a 2020 2020 2020 ed to the. │ │ │ │ +002650d0: 2020 7072 6f67 7261 6d2e 0a20 202a 202a program.. * * │ │ │ │ +002650e0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e note Optional in │ │ │ │ +002650f0: 7075 7473 3a20 7573 696e 6720 6675 6e63 puts: using func │ │ │ │ +00265100: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ +00265110: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020 nal inputs,:. │ │ │ │ +00265120: 2020 202a 204b 6565 7046 696c 6573 203d * KeepFiles = │ │ │ │ +00265130: 3e20 6120 2a6e 6f74 6520 426f 6f6c 6561 > a *note Boolea │ │ │ │ +00265140: 6e20 7661 6c75 653a 2042 6f6f 6c65 616e n value: Boolean │ │ │ │ +00265150: 2c2c 2064 6566 6175 6c74 2076 616c 7565 ,, default value │ │ │ │ +00265160: 2066 616c 7365 2c0a 2020 2020 2020 2020 false,. │ │ │ │ +00265170: 7768 6574 6865 7220 746f 206b 6565 7020 whether to keep │ │ │ │ +00265180: 7468 6520 7465 6d70 6f72 6172 7920 6669 the temporary fi │ │ │ │ +00265190: 6c65 7320 636f 6e74 6169 6e69 6e67 2074 les containing t │ │ │ │ +002651a0: 6865 2070 726f 6772 616d 2773 206f 7574 he program's out │ │ │ │ +002651b0: 7075 742e 0a20 2020 2020 202a 2052 6169 put.. * Rai │ │ │ │ +002651c0: 7365 4572 726f 7220 3d3e 2061 202a 6e6f seError => a *no │ │ │ │ +002651d0: 7465 2042 6f6f 6c65 616e 2076 616c 7565 te Boolean value │ │ │ │ +002651e0: 3a20 426f 6f6c 6561 6e2c 2c20 6465 6661 : Boolean,, defa │ │ │ │ +002651f0: 756c 7420 7661 6c75 6520 7472 7565 2c0a ult value true,. │ │ │ │ +00265200: 2020 2020 2020 2020 7768 6574 6865 7220 whether │ │ │ │ +00265210: 746f 2072 6169 7365 2061 6e20 6572 726f to raise an erro │ │ │ │ +00265220: 7220 6966 2074 6865 2070 726f 6772 616d r if the program │ │ │ │ +00265230: 2072 6574 7572 6e73 2061 206e 6f6e 7a65 returns a nonze │ │ │ │ +00265240: 726f 2076 616c 7565 2e0a 2020 2020 2020 ro value.. │ │ │ │ +00265250: 2a20 5275 6e44 6972 6563 746f 7279 203d * RunDirectory = │ │ │ │ +00265260: 3e20 6120 2a6e 6f74 6520 7374 7269 6e67 > a *note string │ │ │ │ +00265270: 3a20 5374 7269 6e67 2c2c 2064 6566 6175 : String,, defau │ │ │ │ +00265280: 6c74 2076 616c 7565 206e 756c 6c2c 2074 lt value null, t │ │ │ │ +00265290: 6865 0a20 2020 2020 2020 2064 6972 6563 he. direc │ │ │ │ +002652a0: 746f 7279 2066 726f 6d20 7768 6963 6820 tory from which │ │ │ │ +002652b0: 746f 2072 756e 2074 6865 2070 726f 6772 to run the progr │ │ │ │ +002652c0: 616d 2e20 2049 6620 6974 2064 6f65 7320 am. If it does │ │ │ │ +002652d0: 6e6f 7420 6578 6973 742c 2074 6865 6e20 not exist, then │ │ │ │ +002652e0: 6974 0a20 2020 2020 2020 2077 696c 6c20 it. will │ │ │ │ +002652f0: 6265 2063 7265 6174 6564 2e20 2049 6620 be created. If │ │ │ │ +00265300: 2a6e 6f74 6520 6e75 6c6c 3a20 6e75 6c6c *note null: null │ │ │ │ +00265310: 2c2c 2074 6865 6e20 7468 6520 7072 6f67 ,, then the prog │ │ │ │ +00265320: 7261 6d20 7769 6c6c 2062 6520 7275 6e0a ram will be run. │ │ │ │ +00265330: 2020 2020 2020 2020 6672 6f6d 2074 6865 from the │ │ │ │ +00265340: 2063 7572 7265 6e74 2077 6f72 6b69 6e67 current working │ │ │ │ +00265350: 2064 6972 6563 746f 7279 2028 7365 6520 directory (see │ │ │ │ +00265360: 2a6e 6f74 6520 6375 7272 656e 7444 6972 *note currentDir │ │ │ │ +00265370: 6563 746f 7279 3a0a 2020 2020 2020 2020 ectory:. │ │ │ │ +00265380: 6375 7272 656e 7444 6972 6563 746f 7279 currentDirectory │ │ │ │ +00265390: 2c29 2e0a 2020 2020 2020 2a20 5665 7262 ,).. * Verb │ │ │ │ +002653a0: 6f73 6520 3d3e 2061 202a 6e6f 7465 2042 ose => a *note B │ │ │ │ +002653b0: 6f6f 6c65 616e 2076 616c 7565 3a20 426f oolean value: Bo │ │ │ │ +002653c0: 6f6c 6561 6e2c 2c20 6465 6661 756c 7420 olean,, default │ │ │ │ +002653d0: 7661 6c75 6520 6661 6c73 652c 0a20 2020 value false,. │ │ │ │ +002653e0: 2020 2020 2077 6865 7468 6572 2074 6f20 whether to │ │ │ │ +002653f0: 7072 696e 7420 7468 6520 636f 6d6d 616e print the comman │ │ │ │ +00265400: 6420 6c69 6e65 2069 6e70 7574 2061 6e64 d line input and │ │ │ │ +00265410: 2074 6865 2070 726f 6772 616d 2773 206f the program's o │ │ │ │ +00265420: 7574 7075 742e 0a0a 4465 7363 7269 7074 utput...Descript │ │ │ │ +00265430: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +00265440: 0a54 6869 7320 6d65 7468 6f64 2072 756e .This method run │ │ │ │ +00265450: 7320 616e 2065 7874 6572 6e61 6c20 7072 s an external pr │ │ │ │ +00265460: 6f67 7261 6d20 7768 6963 6820 6861 7320 ogram which has │ │ │ │ +00265470: 616c 7265 6164 7920 6265 656e 206c 6f61 already been loa │ │ │ │ +00265480: 6465 6420 7573 696e 6720 2a6e 6f74 650a ded using *note. │ │ │ │ +00265490: 6669 6e64 5072 6f67 7261 6d3a 2066 696e findProgram: fin │ │ │ │ +002654a0: 6450 726f 6772 616d 2c2e 2020 5468 6520 dProgram,. The │ │ │ │ +002654b0: 7265 7375 6c74 7320 6f66 2074 6869 7320 results of this │ │ │ │ +002654c0: 7275 6e20 6172 6520 6176 6169 6c61 626c run are availabl │ │ │ │ +002654d0: 6520 696e 2061 202a 6e6f 7465 0a50 726f e in a *note.Pro │ │ │ │ +002654e0: 6772 616d 5275 6e3a 2050 726f 6772 616d gramRun: Program │ │ │ │ +002654f0: 5275 6e2c 206f 626a 6563 742e 0a0a 2b2d Run, object...+- │ │ │ │ 00265500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265510: 2d2d 2d2d 2b0a 7c69 3120 3a20 6766 616e ----+.|i1 : gfan │ │ │ │ -00265520: 203d 2066 696e 6450 726f 6772 616d 2822 = findProgram(" │ │ │ │ -00265530: 6766 616e 222c 2022 6766 616e 202d 2d68 gfan", "gfan --h │ │ │ │ -00265540: 656c 7022 2920 2020 2020 2020 2020 2020 elp") │ │ │ │ -00265550: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00265560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265580: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00265590: 3120 3d20 6766 616e 2020 2020 2020 2020 1 = gfan │ │ │ │ +00265510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265530: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +00265540: 6766 616e 203d 2066 696e 6450 726f 6772 gfan = findProgr │ │ │ │ +00265550: 616d 2822 6766 616e 222c 2022 6766 616e am("gfan", "gfan │ │ │ │ +00265560: 202d 2d68 656c 7022 2920 2020 2020 2020 --help") │ │ │ │ +00265570: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00265580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002655a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002655b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002655c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +002655b0: 7c0a 7c6f 3120 3d20 6766 616e 2020 2020 |.|o1 = gfan │ │ │ │ +002655c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002655d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002655e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002655e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 002655f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265600: 2020 2020 7c0a 7c6f 3120 3a20 5072 6f67 |.|o1 : Prog │ │ │ │ -00265610: 7261 6d20 2020 2020 2020 2020 2020 2020 ram │ │ │ │ -00265620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265640: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00265650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00265680: 3220 3a20 7275 6e50 726f 6772 616d 2867 2 : runProgram(g │ │ │ │ -00265690: 6661 6e2c 2022 5f76 6572 7369 6f6e 2229 fan, "_version") │ │ │ │ -002656a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002656b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -002656c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002656d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265620: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ +00265630: 5072 6f67 7261 6d20 2020 2020 2020 2020 Program │ │ │ │ +00265640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265660: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00265670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002656a0: 2b0a 7c69 3220 3a20 7275 6e50 726f 6772 +.|i2 : runProgr │ │ │ │ +002656b0: 616d 2867 6661 6e2c 2022 5f76 6572 7369 am(gfan, "_versi │ │ │ │ +002656c0: 6f6e 2229 2020 2020 2020 2020 2020 2020 on") │ │ │ │ +002656d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 002656e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002656f0: 2020 2020 7c0a 7c6f 3220 3d20 3020 2020 |.|o2 = 0 │ │ │ │ +002656f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00265700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265730: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00265710: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ +00265720: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +00265730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00265740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265760: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00265770: 3220 3a20 5072 6f67 7261 6d52 756e 2020 2 : ProgramRun │ │ │ │ +00265750: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00265760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00265780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002657a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -002657b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002657c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265790: 7c0a 7c6f 3220 3a20 5072 6f67 7261 6d52 |.|o2 : ProgramR │ │ │ │ +002657a0: 756e 2020 2020 2020 2020 2020 2020 2020 un │ │ │ │ +002657b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002657c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 002657d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002657e0: 2d2d 2d2d 2b0a 7c69 3320 3a20 6f6f 2322 ----+.|i3 : oo#" │ │ │ │ -002657f0: 6f75 7470 7574 2220 2020 2020 2020 2020 output" │ │ │ │ -00265800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265820: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +002657e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002657f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265800: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ +00265810: 6f6f 2322 6f75 7470 7574 2220 2020 2020 oo#"output" │ │ │ │ +00265820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00265830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265850: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00265860: 3320 3d20 4766 616e 2076 6572 7369 6f6e 3 = Gfan version │ │ │ │ -00265870: 3a20 2020 2020 2020 2020 2020 2020 2020 : │ │ │ │ -00265880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265890: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -002658a0: 6766 616e 302e 3720 2020 2020 2020 2020 gfan0.7 │ │ │ │ -002658b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002658c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002658d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00265840: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00265850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265880: 7c0a 7c6f 3320 3d20 4766 616e 2076 6572 |.|o3 = Gfan ver │ │ │ │ +00265890: 7369 6f6e 3a20 2020 2020 2020 2020 2020 sion: │ │ │ │ +002658a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002658b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +002658c0: 2020 2020 6766 616e 302e 3720 2020 2020 gfan0.7 │ │ │ │ +002658d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002658e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002658f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002658f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00265900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265910: 7c0a 7c20 2020 2020 466f 726b 6564 2066 |.| Forked f │ │ │ │ -00265920: 726f 6d20 736f 7572 6365 2074 7265 6520 rom source tree │ │ │ │ -00265930: 6f6e 3a20 2020 2020 2020 2020 2020 2020 on: │ │ │ │ -00265940: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00265950: 2020 2020 3137 3233 3437 3834 3135 204d 1723478415 M │ │ │ │ -00265960: 6f6e 2041 7567 2031 3220 3138 3a30 303a on Aug 12 18:00: │ │ │ │ -00265970: 3135 2032 3032 3420 2020 2020 2020 2020 15 2024 │ │ │ │ -00265980: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00265990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002659a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265930: 2020 2020 7c0a 7c20 2020 2020 466f 726b |.| Fork │ │ │ │ +00265940: 6564 2066 726f 6d20 736f 7572 6365 2074 ed from source t │ │ │ │ +00265950: 7265 6520 6f6e 3a20 2020 2020 2020 2020 ree on: │ │ │ │ +00265960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265970: 7c0a 7c20 2020 2020 3137 3233 3437 3834 |.| 17234784 │ │ │ │ +00265980: 3135 204d 6f6e 2041 7567 2031 3220 3138 15 Mon Aug 12 18 │ │ │ │ +00265990: 3a30 303a 3135 2032 3032 3420 2020 2020 :00:15 2024 │ │ │ │ +002659a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 002659b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002659c0: 2020 2020 7c0a 7c20 2020 2020 4c69 6e6b |.| Link │ │ │ │ -002659d0: 6564 206c 6962 7261 7269 6573 3a20 2020 ed libraries: │ │ │ │ -002659e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002659f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265a00: 7c0a 7c20 2020 2020 474d 5020 362e 332e |.| GMP 6.3. │ │ │ │ -00265a10: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ -00265a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265a30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00265a40: 2020 2020 4364 646c 6962 2020 2020 2020 Cddlib │ │ │ │ -00265a50: 2059 4553 2020 2020 2020 2020 2020 2020 YES │ │ │ │ -00265a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265a70: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00265a80: 536f 506c 6578 2020 2020 2020 2020 4e4f SoPlex NO │ │ │ │ -00265a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265ab0: 2020 2020 7c0a 7c20 2020 2020 5369 6e67 |.| Sing │ │ │ │ -00265ac0: 756c 6172 2020 2020 2020 4e4f 2020 2020 ular NO │ │ │ │ -00265ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265af0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00265b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00265b30: 3420 3a20 7275 6e50 726f 6772 616d 2867 4 : runProgram(g │ │ │ │ -00265b40: 6661 6e2c 2022 5f66 6f6f 222c 2052 6169 fan, "_foo", Rai │ │ │ │ -00265b50: 7365 4572 726f 7220 3d3e 2066 616c 7365 seError => false │ │ │ │ -00265b60: 2920 2020 2020 2020 7c0a 7c20 2020 2020 ) |.| │ │ │ │ -00265b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002659c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002659d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002659e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +002659f0: 4c69 6e6b 6564 206c 6962 7261 7269 6573 Linked libraries │ │ │ │ +00265a00: 3a20 2020 2020 2020 2020 2020 2020 2020 : │ │ │ │ +00265a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265a20: 2020 2020 7c0a 7c20 2020 2020 474d 5020 |.| GMP │ │ │ │ +00265a30: 362e 332e 3020 2020 2020 2020 2020 2020 6.3.0 │ │ │ │ +00265a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265a60: 7c0a 7c20 2020 2020 4364 646c 6962 2020 |.| Cddlib │ │ │ │ +00265a70: 2020 2020 2059 4553 2020 2020 2020 2020 YES │ │ │ │ +00265a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265a90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00265aa0: 2020 2020 536f 506c 6578 2020 2020 2020 SoPlex │ │ │ │ +00265ab0: 2020 4e4f 2020 2020 2020 2020 2020 2020 NO │ │ │ │ +00265ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265ad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00265ae0: 5369 6e67 756c 6172 2020 2020 2020 4e4f Singular NO │ │ │ │ +00265af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265b10: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00265b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265b50: 2b0a 7c69 3420 3a20 7275 6e50 726f 6772 +.|i4 : runProgr │ │ │ │ +00265b60: 616d 2867 6661 6e2c 2022 5f66 6f6f 222c am(gfan, "_foo", │ │ │ │ +00265b70: 2052 6169 7365 4572 726f 7220 3d3e 2066 RaiseError => f │ │ │ │ +00265b80: 616c 7365 2920 2020 2020 2020 7c0a 7c20 alse) |.| │ │ │ │ 00265b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265ba0: 2020 2020 7c0a 7c6f 3420 3d20 3235 3620 |.|o4 = 256 │ │ │ │ +00265ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00265bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265be0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00265bc0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ +00265bd0: 3235 3620 2020 2020 2020 2020 2020 2020 256 │ │ │ │ +00265be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00265bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265c10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00265c20: 3420 3a20 5072 6f67 7261 6d52 756e 2020 4 : ProgramRun │ │ │ │ +00265c00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00265c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00265c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265c50: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -00265c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265c40: 7c0a 7c6f 3420 3a20 5072 6f67 7261 6d52 |.|o4 : ProgramR │ │ │ │ +00265c50: 756e 2020 2020 2020 2020 2020 2020 2020 un │ │ │ │ +00265c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265c70: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00265c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265c90: 2d2d 2d2d 2b0a 7c69 3520 3a20 6f6f 2322 ----+.|i5 : oo#" │ │ │ │ -00265ca0: 6572 726f 7222 2020 2020 2020 2020 2020 error" │ │ │ │ -00265cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265cd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00265c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265cb0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ +00265cc0: 6f6f 2322 6572 726f 7222 2020 2020 2020 oo#"error" │ │ │ │ +00265cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00265ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265d00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00265d10: 3520 3d20 554e 4b4e 4f57 4e20 4f50 5449 5 = UNKNOWN OPTI │ │ │ │ -00265d20: 4f4e 3a20 5f66 6f6f 2e20 2020 2020 2020 ON: _foo. │ │ │ │ -00265d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265d40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00265d50: 5553 4520 2d2d 6865 6c70 2041 5320 4120 USE --help AS A │ │ │ │ -00265d60: 5349 4e47 4c45 204f 5054 494f 4e20 544f SINGLE OPTION TO │ │ │ │ -00265d70: 2056 4945 5720 5448 4520 4845 4c50 2054 VIEW THE HELP T │ │ │ │ -00265d80: 4558 542e 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d EXT.|.+--------- │ │ │ │ -00265d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265cf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00265d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265d30: 7c0a 7c6f 3520 3d20 554e 4b4e 4f57 4e20 |.|o5 = UNKNOWN │ │ │ │ +00265d40: 4f50 5449 4f4e 3a20 5f66 6f6f 2e20 2020 OPTION: _foo. │ │ │ │ +00265d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265d60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00265d70: 2020 2020 5553 4520 2d2d 6865 6c70 2041 USE --help A │ │ │ │ +00265d80: 5320 4120 5349 4e47 4c45 204f 5054 494f S A SINGLE OPTIO │ │ │ │ +00265d90: 4e20 544f 2056 4945 5720 5448 4520 4845 N TO VIEW THE HE │ │ │ │ +00265da0: 4c50 2054 4558 542e 7c0a 2b2d 2d2d 2d2d LP TEXT.|.+----- │ │ │ │ 00265db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265dc0: 2b0a 0a54 6865 2076 616c 7565 2063 6f72 +..The value cor │ │ │ │ -00265dd0: 7265 7370 6f6e 6469 6e67 2074 6f20 7468 responding to th │ │ │ │ -00265de0: 6520 226f 7574 7075 7422 206b 6579 206d e "output" key m │ │ │ │ -00265df0: 6179 2061 6c73 6f20 6265 206f 6274 6169 ay also be obtai │ │ │ │ -00265e00: 6e65 6420 7573 696e 6720 2a6e 6f74 650a ned using *note. │ │ │ │ -00265e10: 746f 5374 7269 6e67 3a20 746f 5374 7269 toString: toStri │ │ │ │ -00265e20: 6e67 2c2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ng,...+--------- │ │ │ │ -00265e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265e50: 2d2b 0a7c 6936 203a 2074 6f53 7472 696e -+.|i6 : toStrin │ │ │ │ -00265e60: 6720 7275 6e50 726f 6772 616d 2867 6661 g runProgram(gfa │ │ │ │ -00265e70: 6e2c 2022 5f76 6572 7369 6f6e 2229 7c0a n, "_version")|. │ │ │ │ -00265e80: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00265e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265ea0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -00265eb0: 203d 2047 6661 6e20 7665 7273 696f 6e3a = Gfan version: │ │ │ │ -00265ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265ed0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00265ee0: 6766 616e 302e 3720 2020 2020 2020 2020 gfan0.7 │ │ │ │ -00265ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265f00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00265dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265de0: 2d2d 2d2d 2b0a 0a54 6865 2076 616c 7565 ----+..The value │ │ │ │ +00265df0: 2063 6f72 7265 7370 6f6e 6469 6e67 2074 corresponding t │ │ │ │ +00265e00: 6f20 7468 6520 226f 7574 7075 7422 206b o the "output" k │ │ │ │ +00265e10: 6579 206d 6179 2061 6c73 6f20 6265 206f ey may also be o │ │ │ │ +00265e20: 6274 6169 6e65 6420 7573 696e 6720 2a6e btained using *n │ │ │ │ +00265e30: 6f74 650a 746f 5374 7269 6e67 3a20 746f ote.toString: to │ │ │ │ +00265e40: 5374 7269 6e67 2c2e 0a0a 2b2d 2d2d 2d2d String,...+----- │ │ │ │ +00265e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265e70: 2d2d 2d2d 2d2b 0a7c 6936 203a 2074 6f53 -----+.|i6 : toS │ │ │ │ +00265e80: 7472 696e 6720 7275 6e50 726f 6772 616d tring runProgram │ │ │ │ +00265e90: 2867 6661 6e2c 2022 5f76 6572 7369 6f6e (gfan, "_version │ │ │ │ +00265ea0: 2229 7c0a 7c20 2020 2020 2020 2020 2020 ")|.| │ │ │ │ +00265eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265ec0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00265ed0: 0a7c 6f36 203d 2047 6661 6e20 7665 7273 .|o6 = Gfan vers │ │ │ │ +00265ee0: 696f 6e3a 2020 2020 2020 2020 2020 2020 ion: │ │ │ │ +00265ef0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00265f00: 2020 2020 6766 616e 302e 3720 2020 2020 gfan0.7 │ │ │ │ 00265f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265f30: 2020 7c0a 7c20 2020 2020 466f 726b 6564 |.| Forked │ │ │ │ -00265f40: 2066 726f 6d20 736f 7572 6365 2074 7265 from source tre │ │ │ │ -00265f50: 6520 6f6e 3a20 2020 2020 2020 2020 207c e on: | │ │ │ │ -00265f60: 0a7c 2020 2020 2031 3732 3334 3738 3431 .| 172347841 │ │ │ │ -00265f70: 3520 4d6f 6e20 4175 6720 3132 2031 383a 5 Mon Aug 12 18: │ │ │ │ -00265f80: 3030 3a31 3520 3230 3234 2020 7c0a 7c20 00:15 2024 |.| │ │ │ │ -00265f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265fb0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00265fc0: 204c 696e 6b65 6420 6c69 6272 6172 6965 Linked librarie │ │ │ │ -00265fd0: 733a 2020 2020 2020 2020 2020 2020 2020 s: │ │ │ │ -00265fe0: 2020 2020 2020 7c0a 7c20 2020 2020 474d |.| GM │ │ │ │ -00265ff0: 5020 362e 332e 3020 2020 2020 2020 2020 P 6.3.0 │ │ │ │ -00266000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266010: 2020 207c 0a7c 2020 2020 2043 6464 6c69 |.| Cddli │ │ │ │ -00266020: 6220 2020 2020 2020 5945 5320 2020 2020 b YES │ │ │ │ -00266030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266040: 7c0a 7c20 2020 2020 536f 506c 6578 2020 |.| SoPlex │ │ │ │ -00266050: 2020 2020 2020 4e4f 2020 2020 2020 2020 NO │ │ │ │ -00266060: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00266070: 2020 2020 2053 696e 6775 6c61 7220 2020 Singular │ │ │ │ -00266080: 2020 204e 4f20 2020 2020 2020 2020 2020 NO │ │ │ │ -00266090: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -002660a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002660b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002660c0: 2d2d 2d2d 2d2d 2d2b 0a0a 4974 2069 7320 -------+..It is │ │ │ │ -002660d0: 616c 736f 2070 6f73 7369 626c 6520 746f also possible to │ │ │ │ -002660e0: 2073 6b69 7020 2a6e 6f74 6520 6669 6e64 skip *note find │ │ │ │ -002660f0: 5072 6f67 7261 6d3a 2066 696e 6450 726f Program: findPro │ │ │ │ -00266100: 6772 616d 2c20 616e 6420 6a75 7374 2070 gram, and just p │ │ │ │ -00266110: 726f 7669 6465 0a74 776f 2073 7472 696e rovide.two strin │ │ │ │ -00266120: 6773 3a20 7468 6520 6e61 6d65 206f 6620 gs: the name of │ │ │ │ -00266130: 7468 6520 7072 6f67 7261 6d20 616e 6420 the program and │ │ │ │ -00266140: 7468 6520 636f 6d6d 616e 6420 6c69 6e65 the command line │ │ │ │ -00266150: 2061 7267 756d 656e 7473 2e0a 0a2b 2d2d arguments...+-- │ │ │ │ -00266160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00266170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00266180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265f20: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00265f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265f50: 2020 2020 2020 7c0a 7c20 2020 2020 466f |.| Fo │ │ │ │ +00265f60: 726b 6564 2066 726f 6d20 736f 7572 6365 rked from source │ │ │ │ +00265f70: 2074 7265 6520 6f6e 3a20 2020 2020 2020 tree on: │ │ │ │ +00265f80: 2020 207c 0a7c 2020 2020 2031 3732 3334 |.| 17234 │ │ │ │ +00265f90: 3738 3431 3520 4d6f 6e20 4175 6720 3132 78415 Mon Aug 12 │ │ │ │ +00265fa0: 2031 383a 3030 3a31 3520 3230 3234 2020 18:00:15 2024 │ │ │ │ +00265fb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00265fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265fd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00265fe0: 2020 2020 204c 696e 6b65 6420 6c69 6272 Linked libr │ │ │ │ +00265ff0: 6172 6965 733a 2020 2020 2020 2020 2020 aries: │ │ │ │ +00266000: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00266010: 2020 474d 5020 362e 332e 3020 2020 2020 GMP 6.3.0 │ │ │ │ +00266020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266030: 2020 2020 2020 207c 0a7c 2020 2020 2043 |.| C │ │ │ │ +00266040: 6464 6c69 6220 2020 2020 2020 5945 5320 ddlib YES │ │ │ │ +00266050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266060: 2020 2020 7c0a 7c20 2020 2020 536f 506c |.| SoPl │ │ │ │ +00266070: 6578 2020 2020 2020 2020 4e4f 2020 2020 ex NO │ │ │ │ +00266080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266090: 207c 0a7c 2020 2020 2053 696e 6775 6c61 |.| Singula │ │ │ │ +002660a0: 7220 2020 2020 204e 4f20 2020 2020 2020 r NO │ │ │ │ +002660b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +002660c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +002660d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002660e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4974 -----------+..It │ │ │ │ +002660f0: 2069 7320 616c 736f 2070 6f73 7369 626c is also possibl │ │ │ │ +00266100: 6520 746f 2073 6b69 7020 2a6e 6f74 6520 e to skip *note │ │ │ │ +00266110: 6669 6e64 5072 6f67 7261 6d3a 2066 696e findProgram: fin │ │ │ │ +00266120: 6450 726f 6772 616d 2c20 616e 6420 6a75 dProgram, and ju │ │ │ │ +00266130: 7374 2070 726f 7669 6465 0a74 776f 2073 st provide.two s │ │ │ │ +00266140: 7472 696e 6773 3a20 7468 6520 6e61 6d65 trings: the name │ │ │ │ +00266150: 206f 6620 7468 6520 7072 6f67 7261 6d20 of the program │ │ │ │ +00266160: 616e 6420 7468 6520 636f 6d6d 616e 6420 and the command │ │ │ │ +00266170: 6c69 6e65 2061 7267 756d 656e 7473 2e0a line arguments.. │ │ │ │ +00266180: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00266190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002661a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ -002661b0: 203a 2072 756e 5072 6f67 7261 6d28 226e : runProgram("n │ │ │ │ -002661c0: 6f72 6d61 6c69 7a22 2c20 222d 2d76 6572 ormaliz", "--ver │ │ │ │ -002661d0: 7369 6f6e 2229 2020 2020 2020 2020 2020 sion") │ │ │ │ -002661e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002661f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +002661a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002661b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002661c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +002661d0: 0a7c 6937 203a 2072 756e 5072 6f67 7261 .|i7 : runProgra │ │ │ │ +002661e0: 6d28 226e 6f72 6d61 6c69 7a22 2c20 222d m("normaliz", "- │ │ │ │ +002661f0: 2d76 6572 7369 6f6e 2229 2020 2020 2020 -version") │ │ │ │ 00266200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266210: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00266220: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00266230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266240: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -00266250: 203d 2030 2020 2020 2020 2020 2020 2020 = 0 │ │ │ │ -00266260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266260: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00266270: 0a7c 6f37 203d 2030 2020 2020 2020 2020 .|o7 = 0 │ │ │ │ 00266280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266290: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00266290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002662a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002662b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002662c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002662b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +002662c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 002662d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002662e0: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -002662f0: 203a 2050 726f 6772 616d 5275 6e20 2020 : ProgramRun │ │ │ │ -00266300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266330: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00266340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00266350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00266360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002662e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002662f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266300: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00266310: 0a7c 6f37 203a 2050 726f 6772 616d 5275 .|o7 : ProgramRu │ │ │ │ +00266320: 6e20 2020 2020 2020 2020 2020 2020 2020 n │ │ │ │ +00266330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266350: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00266360: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00266370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00266380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ -00266390: 203a 2070 6565 6b20 6f6f 2020 2020 2020 : peek oo │ │ │ │ -002663a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002663b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00266390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002663a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +002663b0: 0a7c 6938 203a 2070 6565 6b20 6f6f 2020 .|i8 : peek oo │ │ │ │ 002663c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002663d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +002663d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002663e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002663f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002663f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00266400: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00266410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266420: 2020 2020 2020 2020 2020 207c 0a7c 6f38 |.|o8 │ │ │ │ -00266430: 203d 2050 726f 6772 616d 5275 6e7b 636f = ProgramRun{co │ │ │ │ -00266440: 6d6d 616e 6420 3d3e 202f 7573 722f 6269 mmand => /usr/bi │ │ │ │ -00266450: 6e2f 6e6f 726d 616c 697a 202d 2d76 6572 n/normaliz --ver │ │ │ │ -00266460: 7369 6f6e 2020 2020 2020 2020 2020 2020 sion │ │ │ │ -00266470: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00266480: 2020 2020 2020 2020 2020 2020 2020 6572 er │ │ │ │ -00266490: 726f 7220 3d3e 2020 2020 2020 2020 2020 ror => │ │ │ │ -002664a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002664b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002664c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -002664d0: 2020 2020 2020 2020 2020 2020 2020 6f75 ou │ │ │ │ -002664e0: 7470 7574 203d 3e20 4e6f 726d 616c 697a tput => Normaliz │ │ │ │ -002664f0: 2033 2e31 312e 3120 2020 2020 2020 2020 3.11.1 │ │ │ │ -00266500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266510: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00266420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266440: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00266450: 0a7c 6f38 203d 2050 726f 6772 616d 5275 .|o8 = ProgramRu │ │ │ │ +00266460: 6e7b 636f 6d6d 616e 6420 3d3e 202f 7573 n{command => /us │ │ │ │ +00266470: 722f 6269 6e2f 6e6f 726d 616c 697a 202d r/bin/normaliz - │ │ │ │ +00266480: 2d76 6572 7369 6f6e 2020 2020 2020 2020 -version │ │ │ │ +00266490: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +002664a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +002664b0: 2020 6572 726f 7220 3d3e 2020 2020 2020 error => │ │ │ │ +002664c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002664d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002664e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +002664f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00266500: 2020 6f75 7470 7574 203d 3e20 4e6f 726d output => Norm │ │ │ │ +00266510: 616c 697a 2033 2e31 312e 3120 2020 2020 aliz 3.11.1 │ │ │ │ 00266520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266530: 2020 2020 2020 2020 2d2d 2d2d 2d2d 2d2d -------- │ │ │ │ -00266540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00266550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00266560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -00266570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266580: 2020 2020 2020 2020 7769 7468 2070 6163 with pac │ │ │ │ -00266590: 6b61 6765 2873 2920 466c 696e 7420 652d kage(s) Flint e- │ │ │ │ -002665a0: 616e 7469 6320 6e61 7574 7920 2020 2020 antic nauty │ │ │ │ -002665b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -002665c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002665d0: 2020 2020 2020 2020 436f 7079 7269 6768 Copyrigh │ │ │ │ -002665e0: 7420 2843 2920 3230 3037 2d32 3032 3520 t (C) 2007-2025 │ │ │ │ -002665f0: 2054 6865 204e 6f72 6d61 6c69 7a20 5465 The Normaliz Te │ │ │ │ -00266600: 616d 2c20 556e 6976 6572 737c 0a7c 2020 am, Univers|.| │ │ │ │ -00266610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266620: 2020 2020 2020 2020 5468 6973 2070 726f This pro │ │ │ │ -00266630: 6772 616d 2063 6f6d 6573 2077 6974 6820 gram comes with │ │ │ │ -00266640: 4142 534f 4c55 5445 4c59 204e 4f20 5741 ABSOLUTELY NO WA │ │ │ │ -00266650: 5252 414e 5459 3b20 5468 697c 0a7c 2020 RRANTY; Thi|.| │ │ │ │ -00266660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266670: 2020 2020 2020 2020 616e 6420 796f 7520 and you │ │ │ │ -00266680: 6172 6520 7765 6c63 6f6d 6520 746f 2072 are welcome to r │ │ │ │ -00266690: 6564 6973 7472 6962 7574 6520 6974 2075 edistribute it u │ │ │ │ -002666a0: 6e64 6572 2063 6572 7461 697c 0a7c 2020 nder certai|.| │ │ │ │ -002666b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002666c0: 2020 2020 2020 2020 5365 6520 434f 5059 See COPY │ │ │ │ -002666d0: 494e 4720 666f 7220 6465 7461 696c 732e ING for details. │ │ │ │ -002666e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002666f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00266700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266530: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00266540: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00266550: 2020 2020 2020 2020 2020 2020 2d2d 2d2d ---- │ │ │ │ +00266560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00266570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00266580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00266590: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +002665a0: 2020 2020 2020 2020 2020 2020 7769 7468 with │ │ │ │ +002665b0: 2070 6163 6b61 6765 2873 2920 466c 696e package(s) Flin │ │ │ │ +002665c0: 7420 652d 616e 7469 6320 6e61 7574 7920 t e-antic nauty │ │ │ │ +002665d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +002665e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +002665f0: 2020 2020 2020 2020 2020 2020 436f 7079 Copy │ │ │ │ +00266600: 7269 6768 7420 2843 2920 3230 3037 2d32 right (C) 2007-2 │ │ │ │ +00266610: 3032 3520 2054 6865 204e 6f72 6d61 6c69 025 The Normali │ │ │ │ +00266620: 7a20 5465 616d 2c20 556e 6976 6572 737c z Team, Univers| │ │ │ │ +00266630: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00266640: 2020 2020 2020 2020 2020 2020 5468 6973 This │ │ │ │ +00266650: 2070 726f 6772 616d 2063 6f6d 6573 2077 program comes w │ │ │ │ +00266660: 6974 6820 4142 534f 4c55 5445 4c59 204e ith ABSOLUTELY N │ │ │ │ +00266670: 4f20 5741 5252 414e 5459 3b20 5468 697c O WARRANTY; Thi| │ │ │ │ +00266680: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00266690: 2020 2020 2020 2020 2020 2020 616e 6420 and │ │ │ │ +002666a0: 796f 7520 6172 6520 7765 6c63 6f6d 6520 you are welcome │ │ │ │ +002666b0: 746f 2072 6564 6973 7472 6962 7574 6520 to redistribute │ │ │ │ +002666c0: 6974 2075 6e64 6572 2063 6572 7461 697c it under certai| │ │ │ │ +002666d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +002666e0: 2020 2020 2020 2020 2020 2020 5365 6520 See │ │ │ │ +002666f0: 434f 5059 494e 4720 666f 7220 6465 7461 COPYING for deta │ │ │ │ +00266700: 696c 732e 2020 2020 2020 2020 2020 2020 ils. │ │ │ │ +00266710: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00266720: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00266730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266740: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00266750: 2020 2020 2020 2020 2020 2020 2020 7265 re │ │ │ │ -00266760: 7475 726e 2076 616c 7565 203d 3e20 3020 turn value => 0 │ │ │ │ -00266770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266790: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ -002667a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002667b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002667c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00266740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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│ │ │ -00266950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266940: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00266950: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00266960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266970: 2020 2020 2020 2020 2020 207c 0a7c 6974 |.|it │ │ │ │ -00266980: 7920 6f66 204f 736e 6162 7275 6563 6b2e y of Osnabrueck. │ │ │ │ -00266990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002669a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002669b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002669c0: 2020 2020 2020 2020 2020 207c 0a7c 7320 |.|s │ │ │ │ -002669d0: 6973 2066 7265 6520 736f 6674 7761 7265 is free software │ │ │ │ -002669e0: 2c20 2020 2020 2020 2020 2020 2020 2020 , │ │ │ │ -002669f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266a10: 2020 2020 2020 2020 2020 207c 0a7c 6e20 |.|n │ │ │ │ -00266a20: 636f 6e64 6974 696f 6e73 3b20 2020 2020 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6e73 3b20 .|n conditions; │ │ │ │ 00266a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00266a60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00266a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00266a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00266a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00266a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266a80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00266a90: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00266aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00266ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 496e -----------+..In │ │ │ │ -00266ac0: 7465 726e 616c 6c79 2c20 7468 6973 2072 ternally, this r │ │ │ │ -00266ad0: 6f75 7469 6e65 2075 7365 7320 2a6e 6f74 outine uses *not │ │ │ │ -00266ae0: 6520 7275 6e3a 2072 756e 2c2e 2041 6e6f e run: 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-00266bc0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ -00266bd0: 6f74 6520 5072 6f67 7261 6d3a 2050 726f ote Program: Pro │ │ │ │ -00266be0: 6772 616d 2c20 2d2d 2065 7874 6572 6e61 gram, -- externa │ │ │ │ -00266bf0: 6c20 7072 6f67 7261 6d20 6f62 6a65 6374 l program object │ │ │ │ -00266c00: 0a20 202a 202a 6e6f 7465 2066 696e 6450 . * *note findP │ │ │ │ -00266c10: 726f 6772 616d 3a20 6669 6e64 5072 6f67 rogram: findProg │ │ │ │ -00266c20: 7261 6d2c 202d 2d20 6c6f 6164 2065 7874 ram, -- load ext │ │ │ │ -00266c30: 6572 6e61 6c20 7072 6f67 7261 6d0a 2020 ernal program. │ │ │ │ -00266c40: 2a20 2a6e 6f74 6520 7374 6174 7573 2850 * *note status(P │ │ │ │ -00266c50: 726f 6772 616d 5275 6e29 3a20 7374 6174 rogramRun): stat │ │ │ │ -00266c60: 7573 5f6c 7050 726f 6772 616d 5275 6e5f us_lpProgramRun_ │ │ │ │ -00266c70: 7270 2c20 2d2d 2067 6574 2074 6865 2072 rp, -- get the r │ │ │ │ -00266c80: 6574 7572 6e20 7374 6174 7573 0a20 2020 eturn status. │ │ │ │ -00266c90: 206f 6620 6120 7072 6f67 7261 6d20 7275 of a program ru │ │ │ │ -00266ca0: 6e0a 2020 2a20 2a6e 6f74 6520 7275 6e3a n. * *note run: │ │ │ │ -00266cb0: 2072 756e 2c20 2d2d 2072 756e 2061 6e20 run, -- run an │ │ │ │ -00266cc0: 6578 7465 726e 616c 2063 6f6d 6d61 6e64 external command │ │ │ │ -00266cd0: 0a20 202a 202a 6e6f 7465 2067 6574 3a20 . * *note get: │ │ │ │ -00266ce0: 6765 742c 202d 2d20 6765 7420 7468 6520 get, -- get the │ │ │ │ -00266cf0: 636f 6e74 656e 7473 206f 6620 6120 6669 contents of a fi │ │ │ │ -00266d00: 6c65 0a2a 204d 656e 753a 0a0a 2a20 5072 le.* Menu:..* Pr │ │ │ │ -00266d10: 6f67 7261 6d52 756e 3a3a 2020 2020 2020 ogramRun:: │ │ │ │ -00266d20: 2020 2020 2020 2020 2020 2020 7265 7375 resu │ │ │ │ -00266d30: 6c74 206f 6620 7275 6e6e 696e 6720 616e lt of running an │ │ │ │ -00266d40: 2065 7874 6572 6e61 6c20 7072 6f67 7261 external progra │ │ │ │ -00266d50: 6d0a 0a57 6179 7320 746f 2075 7365 2072 m..Ways to use r │ │ │ │ -00266d60: 756e 5072 6f67 7261 6d3a 0a3d 3d3d 3d3d unProgram:.===== │ │ │ │ -00266d70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00266d80: 3d3d 0a0a 2020 2a20 2272 756e 5072 6f67 ==.. * "runProg │ │ │ │ -00266d90: 7261 6d28 5072 6f67 7261 6d2c 5374 7269 ram(Program,Stri │ │ │ │ -00266da0: 6e67 2922 0a20 202a 2022 7275 6e50 726f ng)". * "runPro │ │ │ │ -00266db0: 6772 616d 2850 726f 6772 616d 2c53 7472 gram(Program,Str │ │ │ │ -00266dc0: 696e 672c 5374 7269 6e67 2922 0a20 202a ing,String)". * │ │ │ │ -00266dd0: 2022 7275 6e50 726f 6772 616d 2853 7472 "runProgram(Str │ │ │ │ -00266de0: 696e 672c 5374 7269 6e67 2922 0a0a 466f ing,String)"..Fo │ │ │ │ -00266df0: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -00266e00: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -00266e10: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -00266e20: 2a6e 6f74 6520 7275 6e50 726f 6772 616d *note runProgram │ │ │ │ -00266e30: 3a20 7275 6e50 726f 6772 616d 2c20 6973 : runProgram, is │ │ │ │ -00266e40: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ -00266e50: 6675 6e63 7469 6f6e 2077 6974 680a 6f70 function with.op │ │ │ │ -00266e60: 7469 6f6e 733a 204d 6574 686f 6446 756e tions: MethodFun │ │ │ │ -00266e70: 6374 696f 6e57 6974 684f 7074 696f 6e73 ctionWithOptions │ │ │ │ -00266e80: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ -00266e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00266ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00266ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00266ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00266ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00266ae0: 0a0a 496e 7465 726e 616c 6c79 2c20 7468 ..Internally, th │ │ │ │ +00266af0: 6973 2072 6f75 7469 6e65 2075 7365 7320 is routine uses │ │ │ │ +00266b00: 2a6e 6f74 6520 7275 6e3a 2072 756e 2c2e *note run: run,. │ │ │ │ +00266b10: 2041 6e6f 7468 6572 2077 6179 2074 6f20 Another way to │ │ │ │ +00266b20: 696e 7465 7261 6374 2077 6974 680a 7072 interact with.pr │ │ │ │ +00266b30: 6f67 7261 6d73 2069 7320 746f 2070 6173 ograms is to pas │ │ │ │ +00266b40: 7320 6120 7374 7269 6e67 2062 6567 696e s a string begin │ │ │ │ +00266b50: 6e69 6e67 2077 6974 6820 2221 2220 746f ning with "!" to │ │ │ │ +00266b60: 202a 6e6f 7465 2067 6574 3a20 6765 742c *note get: get, │ │ │ │ +00266b70: 2c20 2a6e 6f74 650a 6f70 656e 496e 3a20 , *note.openIn: │ │ │ │ +00266b80: 6f70 656e 496e 5f6c 7053 7472 696e 675f openIn_lpString_ │ │ │ │ +00266b90: 7270 2c2c 202a 6e6f 7465 206f 7065 6e4f rp,, *note openO │ │ │ │ +00266ba0: 7574 3a20 6f70 656e 4f75 745f 6c70 5374 ut: openOut_lpSt │ │ │ │ +00266bb0: 7269 6e67 5f72 702c 2c20 6f72 202a 6e6f ring_rp,, or *no │ │ │ │ +00266bc0: 7465 0a6f 7065 6e49 6e4f 7574 3a20 6f70 te.openInOut: op │ │ │ │ +00266bd0: 656e 496e 4f75 742c 2e0a 0a53 6565 2061 enInOut,...See a │ │ │ │ +00266be0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ +00266bf0: 2a20 2a6e 6f74 6520 5072 6f67 7261 6d3a * *note Program: │ │ │ │ +00266c00: 2050 726f 6772 616d 2c20 2d2d 2065 7874 Program, -- ext │ │ │ │ +00266c10: 6572 6e61 6c20 7072 6f67 7261 6d20 6f62 ernal program ob │ │ │ │ +00266c20: 6a65 6374 0a20 202a 202a 6e6f 7465 2066 ject. * *note f │ │ │ │ +00266c30: 696e 6450 726f 6772 616d 3a20 6669 6e64 indProgram: find │ │ │ │ +00266c40: 5072 6f67 7261 6d2c 202d 2d20 6c6f 6164 Program, -- load │ │ │ │ +00266c50: 2065 7874 6572 6e61 6c20 7072 6f67 7261 external progra │ │ │ │ +00266c60: 6d0a 2020 2a20 2a6e 6f74 6520 7374 6174 m. * *note stat │ │ │ │ +00266c70: 7573 2850 726f 6772 616d 5275 6e29 3a20 us(ProgramRun): │ │ │ │ +00266c80: 7374 6174 7573 5f6c 7050 726f 6772 616d status_lpProgram │ │ │ │ +00266c90: 5275 6e5f 7270 2c20 2d2d 2067 6574 2074 Run_rp, -- get t │ │ │ │ +00266ca0: 6865 2072 6574 7572 6e20 7374 6174 7573 he return status │ │ │ │ +00266cb0: 0a20 2020 206f 6620 6120 7072 6f67 7261 . of a progra │ │ │ │ +00266cc0: 6d20 7275 6e0a 2020 2a20 2a6e 6f74 6520 m run. * *note │ │ │ │ +00266cd0: 7275 6e3a 2072 756e 2c20 2d2d 2072 756e run: run, -- run │ │ │ │ +00266ce0: 2061 6e20 6578 7465 726e 616c 2063 6f6d an external com │ │ │ │ +00266cf0: 6d61 6e64 0a20 202a 202a 6e6f 7465 2067 mand. * *note g │ │ │ │ +00266d00: 6574 3a20 6765 742c 202d 2d20 6765 7420 et: get, -- get │ │ │ │ +00266d10: 7468 6520 636f 6e74 656e 7473 206f 6620 the contents of │ │ │ │ +00266d20: 6120 6669 6c65 0a2a 204d 656e 753a 0a0a a file.* Menu:.. │ │ │ │ +00266d30: 2a20 5072 6f67 7261 6d52 756e 3a3a 2020 * ProgramRun:: │ │ │ │ +00266d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00266d50: 7265 7375 6c74 206f 6620 7275 6e6e 696e result of runnin │ │ │ │ +00266d60: 6720 616e 2065 7874 6572 6e61 6c20 7072 g an external pr │ │ │ │ +00266d70: 6f67 7261 6d0a 0a57 6179 7320 746f 2075 ogram..Ways to u │ │ │ │ +00266d80: 7365 2072 756e 5072 6f67 7261 6d3a 0a3d se runProgram:.= │ │ │ │ +00266d90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00266da0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2272 756e ======.. * "run │ │ │ │ +00266db0: 5072 6f67 7261 6d28 5072 6f67 7261 6d2c Program(Program, │ │ │ │ +00266dc0: 5374 7269 6e67 2922 0a20 202a 2022 7275 String)". * "ru │ │ │ │ +00266dd0: 6e50 726f 6772 616d 2850 726f 6772 616d nProgram(Program │ │ │ │ +00266de0: 2c53 7472 696e 672c 5374 7269 6e67 2922 ,String,String)" │ │ │ │ +00266df0: 0a20 202a 2022 7275 6e50 726f 6772 616d . * "runProgram │ │ │ │ +00266e00: 2853 7472 696e 672c 5374 7269 6e67 2922 (String,String)" │ │ │ │ +00266e10: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ +00266e20: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ +00266e30: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ +00266e40: 6563 7420 2a6e 6f74 6520 7275 6e50 726f ect *note runPro │ │ │ │ +00266e50: 6772 616d 3a20 7275 6e50 726f 6772 616d gram: runProgram │ │ │ │ +00266e60: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ +00266e70: 686f 6420 6675 6e63 7469 6f6e 2077 6974 hod function wit │ │ │ │ +00266e80: 680a 6f70 7469 6f6e 733a 204d 6574 686f h.options: Metho │ │ │ │ +00266e90: 6446 756e 6374 696f 6e57 6974 684f 7074 dFunctionWithOpt │ │ │ │ +00266ea0: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d ions,...-------- │ │ │ │ 00266eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00266ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00266ed0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -00266ee0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -00266ef0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -00266f00: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -00266f10: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -00266f20: 352b 6473 2f4d 322f 4d61 6361 756c 6179 5+ds/M2/Macaulay │ │ │ │ -00266f30: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ -00266f40: 756c 6179 3244 6f63 2f66 756e 6374 696f ulay2Doc/functio │ │ │ │ -00266f50: 6e73 2f72 756e 5072 6f67 7261 6d2d 646f ns/runProgram-do │ │ │ │ -00266f60: 632e 6d32 3a31 3037 3a30 2e0a 1f0a 4669 c.m2:107:0....Fi │ │ │ │ -00266f70: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ -00266f80: 2e69 6e66 6f2c 204e 6f64 653a 2050 726f .info, Node: Pro │ │ │ │ -00266f90: 6772 616d 5275 6e2c 2055 703a 2072 756e gramRun, Up: run │ │ │ │ -00266fa0: 5072 6f67 7261 6d0a 0a50 726f 6772 616d Program..Program │ │ │ │ -00266fb0: 5275 6e20 2d2d 2072 6573 756c 7420 6f66 Run -- result of │ │ │ │ -00266fc0: 2072 756e 6e69 6e67 2061 6e20 6578 7465 running an exte │ │ │ │ -00266fd0: 726e 616c 2070 726f 6772 616d 0a2a 2a2a rnal program.*** │ │ │ │ -00266fe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00266ff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00267000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00267010: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -00267020: 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 2068 6173 =========..A has │ │ │ │ -00267030: 6820 7461 626c 6520 7265 7475 726e 6564 h table returned │ │ │ │ -00267040: 2062 7920 2a6e 6f74 6520 7275 6e50 726f by *note runPro │ │ │ │ -00267050: 6772 616d 3a20 7275 6e50 726f 6772 616d gram: runProgram │ │ │ │ -00267060: 2c20 7769 7468 2074 6865 2066 6f6c 6c6f , with the follo │ │ │ │ -00267070: 7769 6e67 0a73 7472 696e 6773 2061 7320 wing.strings as │ │ │ │ -00267080: 6b65 7973 3a0a 0a20 202a 2022 636f 6d6d keys:.. * "comm │ │ │ │ -00267090: 616e 6422 2c20 7468 6520 636f 6d6d 616e and", the comman │ │ │ │ -002670a0: 6420 7468 6174 2077 6173 2075 7365 6420 d that was used │ │ │ │ -002670b0: 746f 2072 756e 2074 6865 2070 726f 6772 to run the progr │ │ │ │ -002670c0: 616d 2e0a 2020 2a20 226f 7574 7075 7422 am.. * "output" │ │ │ │ -002670d0: 2c20 7468 6520 6f75 7470 7574 206f 6620 , the output of │ │ │ │ -002670e0: 7468 6520 7072 6f67 7261 6d20 746f 2073 the program to s │ │ │ │ -002670f0: 7464 6f75 742e 0a20 202a 2022 6572 726f tdout.. * "erro │ │ │ │ -00267100: 7222 2c20 7468 6520 6f75 7470 7574 206f r", the output o │ │ │ │ -00267110: 6620 7468 6520 7072 6f67 7261 6d20 746f f the program to │ │ │ │ -00267120: 2073 7464 6572 722e 0a20 202a 2022 7265 stderr.. * "re │ │ │ │ -00267130: 7475 726e 2076 616c 7565 222c 2074 6865 turn value", the │ │ │ │ -00267140: 2072 6574 7572 6e20 7661 6c75 6520 6f66 return value of │ │ │ │ -00267150: 2074 6865 2070 726f 6772 616d 2c20 706f the program, po │ │ │ │ -00267160: 7373 6962 6c79 206d 756c 7469 706c 6965 ssibly multiplie │ │ │ │ -00267170: 6420 6279 2032 3536 0a20 2020 2028 7365 d by 256. (se │ │ │ │ -00267180: 6520 2a6e 6f74 6520 7275 6e3a 2072 756e e *note run: run │ │ │ │ -00267190: 2c29 2e20 204e 6f74 6520 7468 6174 2074 ,). Note that t │ │ │ │ -002671a0: 6869 7320 6973 2077 6861 7420 6973 2064 his is what is d │ │ │ │ -002671b0: 6973 706c 6179 6564 2077 6865 6e20 7072 isplayed when pr │ │ │ │ -002671c0: 696e 7469 6e67 2061 0a20 2020 2050 726f inting a. Pro │ │ │ │ -002671d0: 6772 616d 5275 6e20 6f62 6a65 6374 2e0a gramRun object.. │ │ │ │ -002671e0: 0a49 6e20 6164 6469 7469 6f6e 2c20 6966 .In addition, if │ │ │ │ -002671f0: 202a 6e6f 7465 2072 756e 5072 6f67 7261 *note runProgra │ │ │ │ -00267200: 6d3a 2072 756e 5072 6f67 7261 6d2c 2069 m: runProgram, i │ │ │ │ -00267210: 7320 6361 6c6c 6564 2077 6974 6820 7468 s called with th │ │ │ │ -00267220: 6520 4b65 6570 4669 6c65 730a 6f70 7469 e KeepFiles.opti │ │ │ │ -00267230: 6f6e 2073 6574 2074 6f20 2a6e 6f74 6520 on set to *note │ │ │ │ -00267240: 7472 7565 3a20 7472 7565 2c2c 2074 6865 true: true,, the │ │ │ │ -00267250: 6e20 7468 6520 666f 6c6c 6f77 696e 6720 n the following │ │ │ │ -00267260: 6b65 7973 2077 696c 6c20 6265 2070 7265 keys will be pre │ │ │ │ -00267270: 7365 6e74 3a0a 0a20 202a 2022 6f75 7470 sent:.. * "outp │ │ │ │ -00267280: 7574 2066 696c 6522 2c20 7468 6520 7061 ut file", the pa │ │ │ │ -00267290: 7468 2074 6f20 6120 6669 6c65 2063 6f6e th to a file con │ │ │ │ -002672a0: 7461 696e 696e 6720 7468 6520 6f75 7470 taining the outp │ │ │ │ -002672b0: 7574 206f 6620 7468 6520 7072 6f67 7261 ut of the progra │ │ │ │ -002672c0: 6d20 746f 0a20 2020 2073 7464 6f75 742e m to. stdout. │ │ │ │ -002672d0: 0a20 202a 2022 6572 726f 7220 6669 6c65 . * "error file │ │ │ │ -002672e0: 222c 2074 6865 2070 6174 6820 746f 2061 ", the path to a │ │ │ │ -002672f0: 2066 696c 6520 636f 6e74 6169 6e69 6e67 file containing │ │ │ │ -00267300: 2074 6865 206f 7574 7075 7420 6f66 2074 the output of t │ │ │ │ -00267310: 6865 2070 726f 6772 616d 2074 6f0a 2020 he program to. │ │ │ │ -00267320: 2020 7374 6465 7272 2e0a 0a53 6565 2061 stderr...See a │ │ │ │ -00267330: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -00267340: 2a20 2a6e 6f74 6520 7275 6e50 726f 6772 * *note runProgr │ │ │ │ -00267350: 616d 3a20 7275 6e50 726f 6772 616d 2c20 am: runProgram, │ │ │ │ -00267360: 2d2d 2072 756e 2061 6e20 6578 7465 726e -- run an extern │ │ │ │ -00267370: 616c 2070 726f 6772 616d 0a20 202a 202a al program. * * │ │ │ │ -00267380: 6e6f 7465 2050 726f 6772 616d 203c 3c20 note Program << │ │ │ │ -00267390: 5468 696e 673a 2050 726f 6772 616d 203c Thing: Program < │ │ │ │ -002673a0: 3c20 5468 696e 672c 202d 2d20 7275 6e20 < Thing, -- run │ │ │ │ -002673b0: 7072 6f67 7261 6d20 7769 7468 2069 6e70 program with inp │ │ │ │ -002673c0: 7574 0a20 2020 2072 6564 6972 6563 7469 ut. redirecti │ │ │ │ -002673d0: 6f6e 0a20 202a 202a 6e6f 7465 2050 726f on. * *note Pro │ │ │ │ -002673e0: 6772 616d 3a20 5072 6f67 7261 6d2c 202d gram: Program, - │ │ │ │ -002673f0: 2d20 6578 7465 726e 616c 2070 726f 6772 - external progr │ │ │ │ -00267400: 616d 206f 626a 6563 740a 2a20 4d65 6e75 am object.* Menu │ │ │ │ -00267410: 3a0a 0a2a 2073 7461 7475 7328 5072 6f67 :..* status(Prog │ │ │ │ -00267420: 7261 6d52 756e 293a 2073 7461 7475 735f ramRun): status_ │ │ │ │ -00267430: 6c70 5072 6f67 7261 6d52 756e 5f72 702e lpProgramRun_rp. │ │ │ │ -00267440: 2020 6765 7420 7468 6520 7265 7475 726e get the return │ │ │ │ -00267450: 2073 7461 7475 7320 6f66 2061 2070 726f status of a pro │ │ │ │ -00267460: 6772 616d 2072 756e 0a0a 4675 6e63 7469 gram run..Functi │ │ │ │ -00267470: 6f6e 7320 616e 6420 6d65 7468 6f64 7320 ons and methods │ │ │ │ -00267480: 7265 7475 726e 696e 6720 616e 206f 626a returning an obj │ │ │ │ -00267490: 6563 7420 6f66 2063 6c61 7373 2050 726f ect of class Pro │ │ │ │ -002674a0: 6772 616d 5275 6e3a 0a3d 3d3d 3d3d 3d3d gramRun:.======= │ │ │ │ -002674b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -002674c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00266ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00266ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00266ef0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +00266f00: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +00266f10: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +00266f20: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +00266f30: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00266f40: 3236 2e30 352b 6473 2f4d 322f 4d61 6361 26.05+ds/M2/Maca │ │ │ │ +00266f50: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +00266f60: 4d61 6361 756c 6179 3244 6f63 2f66 756e Macaulay2Doc/fun │ │ │ │ +00266f70: 6374 696f 6e73 2f72 756e 5072 6f67 7261 ctions/runProgra │ │ │ │ +00266f80: 6d2d 646f 632e 6d32 3a31 3037 3a30 2e0a m-doc.m2:107:0.. │ │ │ │ +00266f90: 1f0a 4669 6c65 3a20 4d61 6361 756c 6179 ..File: Macaulay │ │ │ │ +00266fa0: 3244 6f63 2e69 6e66 6f2c 204e 6f64 653a 2Doc.info, Node: │ │ │ │ +00266fb0: 2050 726f 6772 616d 5275 6e2c 2055 703a ProgramRun, Up: │ │ │ │ +00266fc0: 2072 756e 5072 6f67 7261 6d0a 0a50 726f runProgram..Pro │ │ │ │ +00266fd0: 6772 616d 5275 6e20 2d2d 2072 6573 756c gramRun -- resul │ │ │ │ +00266fe0: 7420 6f66 2072 756e 6e69 6e67 2061 6e20 t of running an │ │ │ │ +00266ff0: 6578 7465 726e 616c 2070 726f 6772 616d external program │ │ │ │ +00267000: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ +00267010: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00267020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00267030: 2a2a 2a2a 0a0a 4465 7363 7269 7074 696f ****..Descriptio │ │ │ │ +00267040: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 n.===========..A │ │ │ │ +00267050: 2068 6173 6820 7461 626c 6520 7265 7475 hash table retu │ │ │ │ +00267060: 726e 6564 2062 7920 2a6e 6f74 6520 7275 rned by *note ru │ │ │ │ +00267070: 6e50 726f 6772 616d 3a20 7275 6e50 726f nProgram: runPro │ │ │ │ +00267080: 6772 616d 2c20 7769 7468 2074 6865 2066 gram, with the f │ │ │ │ +00267090: 6f6c 6c6f 7769 6e67 0a73 7472 696e 6773 ollowing.strings │ │ │ │ +002670a0: 2061 7320 6b65 7973 3a0a 0a20 202a 2022 as keys:.. * " │ │ │ │ +002670b0: 636f 6d6d 616e 6422 2c20 7468 6520 636f command", the co │ │ │ │ +002670c0: 6d6d 616e 6420 7468 6174 2077 6173 2075 mmand that was u │ │ │ │ +002670d0: 7365 6420 746f 2072 756e 2074 6865 2070 sed to run the p │ │ │ │ +002670e0: 726f 6772 616d 2e0a 2020 2a20 226f 7574 rogram.. * "out │ │ │ │ +002670f0: 7075 7422 2c20 7468 6520 6f75 7470 7574 put", the output │ │ │ │ +00267100: 206f 6620 7468 6520 7072 6f67 7261 6d20 of the program │ │ │ │ +00267110: 746f 2073 7464 6f75 742e 0a20 202a 2022 to stdout.. * " │ │ │ │ +00267120: 6572 726f 7222 2c20 7468 6520 6f75 7470 error", the outp │ │ │ │ +00267130: 7574 206f 6620 7468 6520 7072 6f67 7261 ut of the progra │ │ │ │ +00267140: 6d20 746f 2073 7464 6572 722e 0a20 202a m to stderr.. * │ │ │ │ +00267150: 2022 7265 7475 726e 2076 616c 7565 222c "return value", │ │ │ │ +00267160: 2074 6865 2072 6574 7572 6e20 7661 6c75 the return valu │ │ │ │ +00267170: 6520 6f66 2074 6865 2070 726f 6772 616d e of the program │ │ │ │ +00267180: 2c20 706f 7373 6962 6c79 206d 756c 7469 , possibly multi │ │ │ │ +00267190: 706c 6965 6420 6279 2032 3536 0a20 2020 plied by 256. │ │ │ │ +002671a0: 2028 7365 6520 2a6e 6f74 6520 7275 6e3a (see *note run: │ │ │ │ +002671b0: 2072 756e 2c29 2e20 204e 6f74 6520 7468 run,). Note th │ │ │ │ +002671c0: 6174 2074 6869 7320 6973 2077 6861 7420 at this is what │ │ │ │ +002671d0: 6973 2064 6973 706c 6179 6564 2077 6865 is displayed whe │ │ │ │ +002671e0: 6e20 7072 696e 7469 6e67 2061 0a20 2020 n printing a. │ │ │ │ +002671f0: 2050 726f 6772 616d 5275 6e20 6f62 6a65 ProgramRun obje │ │ │ │ +00267200: 6374 2e0a 0a49 6e20 6164 6469 7469 6f6e ct...In addition │ │ │ │ +00267210: 2c20 6966 202a 6e6f 7465 2072 756e 5072 , if *note runPr │ │ │ │ +00267220: 6f67 7261 6d3a 2072 756e 5072 6f67 7261 ogram: runProgra │ │ │ │ +00267230: 6d2c 2069 7320 6361 6c6c 6564 2077 6974 m, is called wit │ │ │ │ +00267240: 6820 7468 6520 4b65 6570 4669 6c65 730a h the KeepFiles. │ │ │ │ +00267250: 6f70 7469 6f6e 2073 6574 2074 6f20 2a6e option set to *n │ │ │ │ +00267260: 6f74 6520 7472 7565 3a20 7472 7565 2c2c ote true: true,, │ │ │ │ +00267270: 2074 6865 6e20 7468 6520 666f 6c6c 6f77 then the follow │ │ │ │ +00267280: 696e 6720 6b65 7973 2077 696c 6c20 6265 ing keys will be │ │ │ │ +00267290: 2070 7265 7365 6e74 3a0a 0a20 202a 2022 present:.. * " │ │ │ │ +002672a0: 6f75 7470 7574 2066 696c 6522 2c20 7468 output file", th │ │ │ │ +002672b0: 6520 7061 7468 2074 6f20 6120 6669 6c65 e path to a file │ │ │ │ +002672c0: 2063 6f6e 7461 696e 696e 6720 7468 6520 containing the │ │ │ │ +002672d0: 6f75 7470 7574 206f 6620 7468 6520 7072 output of the pr │ │ │ │ +002672e0: 6f67 7261 6d20 746f 0a20 2020 2073 7464 ogram to. std │ │ │ │ +002672f0: 6f75 742e 0a20 202a 2022 6572 726f 7220 out.. * "error │ │ │ │ +00267300: 6669 6c65 222c 2074 6865 2070 6174 6820 file", the path │ │ │ │ +00267310: 746f 2061 2066 696c 6520 636f 6e74 6169 to a file contai │ │ │ │ +00267320: 6e69 6e67 2074 6865 206f 7574 7075 7420 ning the output │ │ │ │ +00267330: 6f66 2074 6865 2070 726f 6772 616d 2074 of the program t │ │ │ │ +00267340: 6f0a 2020 2020 7374 6465 7272 2e0a 0a53 o. stderr...S │ │ │ │ +00267350: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +00267360: 0a0a 2020 2a20 2a6e 6f74 6520 7275 6e50 .. * *note runP │ │ │ │ +00267370: 726f 6772 616d 3a20 7275 6e50 726f 6772 rogram: runProgr │ │ │ │ +00267380: 616d 2c20 2d2d 2072 756e 2061 6e20 6578 am, -- run an ex │ │ │ │ +00267390: 7465 726e 616c 2070 726f 6772 616d 0a20 ternal program. │ │ │ │ +002673a0: 202a 202a 6e6f 7465 2050 726f 6772 616d * *note Program │ │ │ │ +002673b0: 203c 3c20 5468 696e 673a 2050 726f 6772 << Thing: Progr │ │ │ │ +002673c0: 616d 203c 3c20 5468 696e 672c 202d 2d20 am << Thing, -- │ │ │ │ +002673d0: 7275 6e20 7072 6f67 7261 6d20 7769 7468 run program with │ │ │ │ +002673e0: 2069 6e70 7574 0a20 2020 2072 6564 6972 input. redir │ │ │ │ +002673f0: 6563 7469 6f6e 0a20 202a 202a 6e6f 7465 ection. * *note │ │ │ │ +00267400: 2050 726f 6772 616d 3a20 5072 6f67 7261 Program: Progra │ │ │ │ +00267410: 6d2c 202d 2d20 6578 7465 726e 616c 2070 m, -- external p │ │ │ │ +00267420: 726f 6772 616d 206f 626a 6563 740a 2a20 rogram object.* │ │ │ │ +00267430: 4d65 6e75 3a0a 0a2a 2073 7461 7475 7328 Menu:..* status( │ │ │ │ +00267440: 5072 6f67 7261 6d52 756e 293a 2073 7461 ProgramRun): sta │ │ │ │ +00267450: 7475 735f 6c70 5072 6f67 7261 6d52 756e tus_lpProgramRun │ │ │ │ +00267460: 5f72 702e 2020 6765 7420 7468 6520 7265 _rp. get the re │ │ │ │ +00267470: 7475 726e 2073 7461 7475 7320 6f66 2061 turn status of a │ │ │ │ +00267480: 2070 726f 6772 616d 2072 756e 0a0a 4675 program run..Fu │ │ │ │ +00267490: 6e63 7469 6f6e 7320 616e 6420 6d65 7468 nctions and meth │ │ │ │ +002674a0: 6f64 7320 7265 7475 726e 696e 6720 616e ods returning an │ │ │ │ +002674b0: 206f 626a 6563 7420 6f66 2063 6c61 7373 object of class │ │ │ │ +002674c0: 2050 726f 6772 616d 5275 6e3a 0a3d 3d3d ProgramRun:.=== │ │ │ │ 002674d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -002674e0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ -002674f0: 7465 2072 756e 5072 6f67 7261 6d3a 2072 te runProgram: r │ │ │ │ -00267500: 756e 5072 6f67 7261 6d2c 202d 2d20 7275 unProgram, -- ru │ │ │ │ -00267510: 6e20 616e 2065 7874 6572 6e61 6c20 7072 n an external pr │ │ │ │ -00267520: 6f67 7261 6d0a 0a4d 6574 686f 6473 2074 ogram..Methods t │ │ │ │ -00267530: 6861 7420 7573 6520 616e 206f 626a 6563 hat use an objec │ │ │ │ -00267540: 7420 6f66 2063 6c61 7373 2050 726f 6772 t of class Progr │ │ │ │ -00267550: 616d 5275 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d amRun:.========= │ │ │ │ -00267560: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00267570: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00267580: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ -00267590: 6520 7374 6174 7573 2850 726f 6772 616d e status(Program │ │ │ │ -002675a0: 5275 6e29 3a20 7374 6174 7573 5f6c 7050 Run): status_lpP │ │ │ │ -002675b0: 726f 6772 616d 5275 6e5f 7270 2c20 2d2d rogramRun_rp, -- │ │ │ │ -002675c0: 2067 6574 2074 6865 2072 6574 7572 6e20 get the return │ │ │ │ -002675d0: 7374 6174 7573 0a20 2020 206f 6620 6120 status. of a │ │ │ │ -002675e0: 7072 6f67 7261 6d20 7275 6e0a 0a46 6f72 program run..For │ │ │ │ -002675f0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -00267600: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00267610: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -00267620: 6e6f 7465 2050 726f 6772 616d 5275 6e3a note ProgramRun: │ │ │ │ -00267630: 2050 726f 6772 616d 5275 6e2c 2069 7320 ProgramRun, is │ │ │ │ -00267640: 6120 2a6e 6f74 6520 7479 7065 3a20 5479 a *note type: Ty │ │ │ │ -00267650: 7065 2c2c 2077 6974 6820 616e 6365 7374 pe,, with ancest │ │ │ │ -00267660: 6f72 0a63 6c61 7373 6573 202a 6e6f 7465 or.classes *note │ │ │ │ -00267670: 2048 6173 6854 6162 6c65 3a20 4861 7368 HashTable: Hash │ │ │ │ -00267680: 5461 626c 652c 203c 202a 6e6f 7465 2054 Table, < *note T │ │ │ │ -00267690: 6869 6e67 3a20 5468 696e 672c 2e0a 0a2d hing: Thing,...- │ │ │ │ -002676a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002676b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002676c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002674e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +002674f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00267500: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +00267510: 202a 6e6f 7465 2072 756e 5072 6f67 7261 *note runProgra │ │ │ │ +00267520: 6d3a 2072 756e 5072 6f67 7261 6d2c 202d m: runProgram, - │ │ │ │ +00267530: 2d20 7275 6e20 616e 2065 7874 6572 6e61 - run an externa │ │ │ │ +00267540: 6c20 7072 6f67 7261 6d0a 0a4d 6574 686f l program..Metho │ │ │ │ +00267550: 6473 2074 6861 7420 7573 6520 616e 206f ds that use an o │ │ │ │ +00267560: 626a 6563 7420 6f66 2063 6c61 7373 2050 bject of class P │ │ │ │ +00267570: 726f 6772 616d 5275 6e3a 0a3d 3d3d 3d3d rogramRun:.===== │ │ │ │ +00267580: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00267590: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +002675a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +002675b0: 2a6e 6f74 6520 7374 6174 7573 2850 726f *note status(Pro │ │ │ │ +002675c0: 6772 616d 5275 6e29 3a20 7374 6174 7573 gramRun): status │ │ │ │ +002675d0: 5f6c 7050 726f 6772 616d 5275 6e5f 7270 _lpProgramRun_rp │ │ │ │ +002675e0: 2c20 2d2d 2067 6574 2074 6865 2072 6574 , -- get the ret │ │ │ │ +002675f0: 7572 6e20 7374 6174 7573 0a20 2020 206f urn status. o │ │ │ │ +00267600: 6620 6120 7072 6f67 7261 6d20 7275 6e0a f a program run. │ │ │ │ +00267610: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +00267620: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +00267630: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +00267640: 6374 202a 6e6f 7465 2050 726f 6772 616d ct *note Program │ │ │ │ +00267650: 5275 6e3a 2050 726f 6772 616d 5275 6e2c Run: ProgramRun, │ │ │ │ +00267660: 2069 7320 6120 2a6e 6f74 6520 7479 7065 is a *note type │ │ │ │ +00267670: 3a20 5479 7065 2c2c 2077 6974 6820 616e : Type,, with an │ │ │ │ +00267680: 6365 7374 6f72 0a63 6c61 7373 6573 202a cestor.classes * │ │ │ │ +00267690: 6e6f 7465 2048 6173 6854 6162 6c65 3a20 note HashTable: │ │ │ │ +002676a0: 4861 7368 5461 626c 652c 203c 202a 6e6f HashTable, < *no │ │ │ │ +002676b0: 7465 2054 6869 6e67 3a20 5468 696e 672c te Thing: Thing, │ │ │ │ +002676c0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 002676d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002676e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -002676f0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -00267700: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -00267710: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -00267720: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -00267730: 6c61 7932 2d31 2e32 362e 3035 2b64 732f lay2-1.26.05+ds/ │ │ │ │ -00267740: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -00267750: 6b61 6765 732f 0a4d 6163 6175 6c61 7932 kages/.Macaulay2 │ │ │ │ -00267760: 446f 632f 6675 6e63 7469 6f6e 732f 7275 Doc/functions/ru │ │ │ │ -00267770: 6e50 726f 6772 616d 2d64 6f63 2e6d 323a nProgram-doc.m2: │ │ │ │ -00267780: 3335 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 35:0....File: Ma │ │ │ │ -00267790: 6361 756c 6179 3244 6f63 2e69 6e66 6f2c caulay2Doc.info, │ │ │ │ -002677a0: 204e 6f64 653a 2073 7461 7475 735f 6c70 Node: status_lp │ │ │ │ -002677b0: 5072 6f67 7261 6d52 756e 5f72 702c 2055 ProgramRun_rp, U │ │ │ │ -002677c0: 703a 2050 726f 6772 616d 5275 6e0a 0a73 p: ProgramRun..s │ │ │ │ -002677d0: 7461 7475 7328 5072 6f67 7261 6d52 756e tatus(ProgramRun │ │ │ │ -002677e0: 2920 2d2d 2067 6574 2074 6865 2072 6574 ) -- get the ret │ │ │ │ -002677f0: 7572 6e20 7374 6174 7573 206f 6620 6120 urn status of a │ │ │ │ -00267800: 7072 6f67 7261 6d20 7275 6e0a 2a2a 2a2a program run.**** │ │ │ │ -00267810: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00267820: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002676e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002676f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267710: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +00267720: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +00267730: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +00267740: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +00267750: 6163 6175 6c61 7932 2d31 2e32 362e 3035 acaulay2-1.26.05 │ │ │ │ +00267760: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +00267770: 2f70 6163 6b61 6765 732f 0a4d 6163 6175 /packages/.Macau │ │ │ │ +00267780: 6c61 7932 446f 632f 6675 6e63 7469 6f6e lay2Doc/function │ │ │ │ +00267790: 732f 7275 6e50 726f 6772 616d 2d64 6f63 s/runProgram-doc │ │ │ │ +002677a0: 2e6d 323a 3335 3a30 2e0a 1f0a 4669 6c65 .m2:35:0....File │ │ │ │ +002677b0: 3a20 4d61 6361 756c 6179 3244 6f63 2e69 : Macaulay2Doc.i │ │ │ │ +002677c0: 6e66 6f2c 204e 6f64 653a 2073 7461 7475 nfo, Node: statu │ │ │ │ +002677d0: 735f 6c70 5072 6f67 7261 6d52 756e 5f72 s_lpProgramRun_r │ │ │ │ +002677e0: 702c 2055 703a 2050 726f 6772 616d 5275 p, Up: ProgramRu │ │ │ │ +002677f0: 6e0a 0a73 7461 7475 7328 5072 6f67 7261 n..status(Progra │ │ │ │ +00267800: 6d52 756e 2920 2d2d 2067 6574 2074 6865 mRun) -- get the │ │ │ │ +00267810: 2072 6574 7572 6e20 7374 6174 7573 206f return status o │ │ │ │ +00267820: 6620 6120 7072 6f67 7261 6d20 7275 6e0a f a program run. │ │ │ │ 00267830: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00267840: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675 ********.. * Fu │ │ │ │ -00267850: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 7374 nction: *note st │ │ │ │ -00267860: 6174 7573 3a20 7374 6174 7573 2c0a 2020 atus: status,. │ │ │ │ -00267870: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -00267880: 2020 7374 6174 7573 2070 720a 2020 2a20 status pr. * │ │ │ │ -00267890: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -002678a0: 7072 2c20 616e 2069 6e73 7461 6e63 6520 pr, an instance │ │ │ │ -002678b0: 6f66 2074 6865 2074 7970 6520 2a6e 6f74 of the type *not │ │ │ │ -002678c0: 6520 5072 6f67 7261 6d52 756e 3a20 5072 e ProgramRun: Pr │ │ │ │ -002678d0: 6f67 7261 6d52 756e 2c2c 200a 2020 2a20 ogramRun,, . * │ │ │ │ -002678e0: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069 *note Optional i │ │ │ │ -002678f0: 6e70 7574 733a 2075 7369 6e67 2066 756e nputs: using fun │ │ │ │ -00267900: 6374 696f 6e73 2077 6974 6820 6f70 7469 ctions with opti │ │ │ │ -00267910: 6f6e 616c 2069 6e70 7574 732c 3a0a 2020 onal inputs,:. │ │ │ │ -00267920: 2020 2020 2a20 4d6f 6e6f 6d69 616c 7320 * Monomials │ │ │ │ -00267930: 286d 6973 7369 6e67 2064 6f63 756d 656e (missing documen │ │ │ │ -00267940: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20 tation) => ..., │ │ │ │ -00267950: 6465 6661 756c 7420 7661 6c75 6520 6661 default value fa │ │ │ │ -00267960: 6c73 652c 200a 2020 2020 2020 2a20 5061 lse, . * Pa │ │ │ │ -00267970: 6972 7352 656d 6169 6e69 6e67 2028 6d69 irsRemaining (mi │ │ │ │ -00267980: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ -00267990: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566 ion) => ..., def │ │ │ │ -002679a0: 6175 6c74 2076 616c 7565 2066 616c 7365 ault value false │ │ │ │ -002679b0: 2c20 0a20 2020 2020 202a 2054 6f74 616c , . * Total │ │ │ │ -002679c0: 5061 6972 7320 286d 6973 7369 6e67 2064 Pairs (missing d │ │ │ │ -002679d0: 6f63 756d 656e 7461 7469 6f6e 2920 3d3e ocumentation) => │ │ │ │ -002679e0: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ -002679f0: 6c75 6520 7472 7565 2c20 0a20 202a 204f lue true, . * O │ │ │ │ -00267a00: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -00267a10: 616e 202a 6e6f 7465 2069 6e74 6567 6572 an *note integer │ │ │ │ -00267a20: 3a20 5a5a 2c2c 200a 0a44 6573 6372 6970 : ZZ,, ..Descrip │ │ │ │ -00267a30: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -00267a40: 0a0a 4765 7420 7468 6520 7265 7475 726e ..Get the return │ │ │ │ -00267a50: 2073 7461 7475 7320 6f66 2061 2070 726f status of a pro │ │ │ │ -00267a60: 6772 616d 2072 756e 2e20 2055 7375 616c gram run. Usual │ │ │ │ -00267a70: 6c79 2c20 3020 6d65 616e 7320 7468 6174 ly, 0 means that │ │ │ │ -00267a80: 2069 7420 7761 730a 7375 6363 6573 7366 it was.successf │ │ │ │ -00267a90: 756c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ul...+---------- │ │ │ │ -00267aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267ac0: 2d2d 2d2b 0a7c 6931 203a 206e 6f72 6d61 ---+.|i1 : norma │ │ │ │ -00267ad0: 6c69 7a20 3d20 6669 6e64 5072 6f67 7261 liz = findProgra │ │ │ │ -00267ae0: 6d20 226e 6f72 6d61 6c69 7a22 2020 2020 m "normaliz" │ │ │ │ -00267af0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00267b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267b20: 2020 207c 0a7c 6f31 203d 206e 6f72 6d61 |.|o1 = norma │ │ │ │ -00267b30: 6c69 7a20 2020 2020 2020 2020 2020 2020 liz │ │ │ │ -00267b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267b50: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00267840: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00267850: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00267860: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ +00267870: 2a20 4675 6e63 7469 6f6e 3a20 2a6e 6f74 * Function: *not │ │ │ │ +00267880: 6520 7374 6174 7573 3a20 7374 6174 7573 e status: status │ │ │ │ +00267890: 2c0a 2020 2a20 5573 6167 653a 200a 2020 ,. * Usage: . │ │ │ │ +002678a0: 2020 2020 2020 7374 6174 7573 2070 720a status pr. │ │ │ │ +002678b0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +002678c0: 2020 2a20 7072 2c20 616e 2069 6e73 7461 * pr, an insta │ │ │ │ +002678d0: 6e63 6520 6f66 2074 6865 2074 7970 6520 nce of the type │ │ │ │ +002678e0: 2a6e 6f74 6520 5072 6f67 7261 6d52 756e *note ProgramRun │ │ │ │ +002678f0: 3a20 5072 6f67 7261 6d52 756e 2c2c 200a : ProgramRun,, . │ │ │ │ +00267900: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e * *note Option │ │ │ │ +00267910: 616c 2069 6e70 7574 733a 2075 7369 6e67 al inputs: using │ │ │ │ +00267920: 2066 756e 6374 696f 6e73 2077 6974 6820 functions with │ │ │ │ +00267930: 6f70 7469 6f6e 616c 2069 6e70 7574 732c optional inputs, │ │ │ │ +00267940: 3a0a 2020 2020 2020 2a20 4d6f 6e6f 6d69 :. * Monomi │ │ │ │ +00267950: 616c 7320 286d 6973 7369 6e67 2064 6f63 als (missing doc │ │ │ │ +00267960: 756d 656e 7461 7469 6f6e 2920 3d3e 202e umentation) => . │ │ │ │ +00267970: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ +00267980: 6520 6661 6c73 652c 200a 2020 2020 2020 e false, . │ │ │ │ +00267990: 2a20 5061 6972 7352 656d 6169 6e69 6e67 * PairsRemaining │ │ │ │ +002679a0: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ +002679b0: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c ntation) => ..., │ │ │ │ +002679c0: 2064 6566 6175 6c74 2076 616c 7565 2066 default value f │ │ │ │ +002679d0: 616c 7365 2c20 0a20 2020 2020 202a 2054 alse, . * T │ │ │ │ +002679e0: 6f74 616c 5061 6972 7320 286d 6973 7369 otalPairs (missi │ │ │ │ +002679f0: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ +00267a00: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c ) => ..., defaul │ │ │ │ +00267a10: 7420 7661 6c75 6520 7472 7565 2c20 0a20 t value true, . │ │ │ │ +00267a20: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +00267a30: 2020 2a20 616e 202a 6e6f 7465 2069 6e74 * an *note int │ │ │ │ +00267a40: 6567 6572 3a20 5a5a 2c2c 200a 0a44 6573 eger: ZZ,, ..Des │ │ │ │ +00267a50: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +00267a60: 3d3d 3d3d 0a0a 4765 7420 7468 6520 7265 ====..Get the re │ │ │ │ +00267a70: 7475 726e 2073 7461 7475 7320 6f66 2061 turn status of a │ │ │ │ +00267a80: 2070 726f 6772 616d 2072 756e 2e20 2055 program run. U │ │ │ │ +00267a90: 7375 616c 6c79 2c20 3020 6d65 616e 7320 sually, 0 means │ │ │ │ +00267aa0: 7468 6174 2069 7420 7761 730a 7375 6363 that it was.succ │ │ │ │ +00267ab0: 6573 7366 756c 2e0a 0a2b 2d2d 2d2d 2d2d essful...+------ │ │ │ │ +00267ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267ae0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206e -------+.|i1 : n │ │ │ │ +00267af0: 6f72 6d61 6c69 7a20 3d20 6669 6e64 5072 ormaliz = findPr │ │ │ │ +00267b00: 6f67 7261 6d20 226e 6f72 6d61 6c69 7a22 ogram "normaliz" │ │ │ │ +00267b10: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00267b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267b40: 2020 2020 2020 207c 0a7c 6f31 203d 206e |.|o1 = n │ │ │ │ +00267b50: 6f72 6d61 6c69 7a20 2020 2020 2020 2020 ormaliz │ │ │ │ 00267b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267b80: 2020 207c 0a7c 6f31 203a 2050 726f 6772 |.|o1 : Progr │ │ │ │ -00267b90: 616d 2020 2020 2020 2020 2020 2020 2020 am │ │ │ │ -00267ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267bb0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00267bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267be0: 2d2d 2d2b 0a7c 6932 203a 2073 7461 7475 ---+.|i2 : statu │ │ │ │ -00267bf0: 7320 7275 6e50 726f 6772 616d 286e 6f72 s runProgram(nor │ │ │ │ -00267c00: 6d61 6c69 7a2c 2022 2d2d 7665 7273 696f maliz, "--versio │ │ │ │ -00267c10: 6e22 297c 0a7c 2020 2020 2020 2020 2020 n")|.| │ │ │ │ -00267c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267c40: 2020 207c 0a7c 6f32 203d 2030 2020 2020 |.|o2 = 0 │ │ │ │ +00267b70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00267b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267ba0: 2020 2020 2020 207c 0a7c 6f31 203a 2050 |.|o1 : P │ │ │ │ +00267bb0: 726f 6772 616d 2020 2020 2020 2020 2020 rogram │ │ │ │ +00267bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267bd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00267be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267c00: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2073 -------+.|i2 : s │ │ │ │ +00267c10: 7461 7475 7320 7275 6e50 726f 6772 616d tatus runProgram │ │ │ │ +00267c20: 286e 6f72 6d61 6c69 7a2c 2022 2d2d 7665 (normaliz, "--ve │ │ │ │ +00267c30: 7273 696f 6e22 297c 0a7c 2020 2020 2020 rsion")|.| │ │ │ │ +00267c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00267c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267c70: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00267c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267ca0: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ -00267cb0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ -00267cc0: 7465 2066 696e 6450 726f 6772 616d 3a20 te findProgram: │ │ │ │ -00267cd0: 6669 6e64 5072 6f67 7261 6d2c 202d 2d20 findProgram, -- │ │ │ │ -00267ce0: 6c6f 6164 2065 7874 6572 6e61 6c20 7072 load external pr │ │ │ │ -00267cf0: 6f67 7261 6d0a 2020 2a20 2a6e 6f74 6520 ogram. * *note │ │ │ │ -00267d00: 7275 6e50 726f 6772 616d 3a20 7275 6e50 runProgram: runP │ │ │ │ -00267d10: 726f 6772 616d 2c20 2d2d 2072 756e 2061 rogram, -- run a │ │ │ │ -00267d20: 6e20 6578 7465 726e 616c 2070 726f 6772 n external progr │ │ │ │ -00267d30: 616d 0a0a 5761 7973 2074 6f20 7573 6520 am..Ways to use │ │ │ │ -00267d40: 7468 6973 206d 6574 686f 643a 0a3d 3d3d this method:.=== │ │ │ │ -00267d50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00267d60: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -00267d70: 2073 7461 7475 7328 5072 6f67 7261 6d52 status(ProgramR │ │ │ │ -00267d80: 756e 293a 2073 7461 7475 735f 6c70 5072 un): status_lpPr │ │ │ │ -00267d90: 6f67 7261 6d52 756e 5f72 702c 202d 2d20 ogramRun_rp, -- │ │ │ │ -00267da0: 6765 7420 7468 6520 7265 7475 726e 2073 get the return s │ │ │ │ -00267db0: 7461 7475 730a 2020 2020 6f66 2061 2070 tatus. of a p │ │ │ │ -00267dc0: 726f 6772 616d 2072 756e 0a2d 2d2d 2d2d rogram run.----- │ │ │ │ -00267dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267c60: 2020 2020 2020 207c 0a7c 6f32 203d 2030 |.|o2 = 0 │ │ │ │ +00267c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267c90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00267ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267cc0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ +00267cd0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ +00267ce0: 202a 6e6f 7465 2066 696e 6450 726f 6772 *note findProgr │ │ │ │ +00267cf0: 616d 3a20 6669 6e64 5072 6f67 7261 6d2c am: findProgram, │ │ │ │ +00267d00: 202d 2d20 6c6f 6164 2065 7874 6572 6e61 -- load externa │ │ │ │ +00267d10: 6c20 7072 6f67 7261 6d0a 2020 2a20 2a6e l program. * *n │ │ │ │ +00267d20: 6f74 6520 7275 6e50 726f 6772 616d 3a20 ote runProgram: │ │ │ │ +00267d30: 7275 6e50 726f 6772 616d 2c20 2d2d 2072 runProgram, -- r │ │ │ │ +00267d40: 756e 2061 6e20 6578 7465 726e 616c 2070 un an external p │ │ │ │ +00267d50: 726f 6772 616d 0a0a 5761 7973 2074 6f20 rogram..Ways to │ │ │ │ +00267d60: 7573 6520 7468 6973 206d 6574 686f 643a use this method: │ │ │ │ +00267d70: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00267d80: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a =========.. * * │ │ │ │ +00267d90: 6e6f 7465 2073 7461 7475 7328 5072 6f67 note status(Prog │ │ │ │ +00267da0: 7261 6d52 756e 293a 2073 7461 7475 735f ramRun): status_ │ │ │ │ +00267db0: 6c70 5072 6f67 7261 6d52 756e 5f72 702c lpProgramRun_rp, │ │ │ │ +00267dc0: 202d 2d20 6765 7420 7468 6520 7265 7475 -- get the retu │ │ │ │ +00267dd0: 726e 2073 7461 7475 730a 2020 2020 6f66 rn status. of │ │ │ │ +00267de0: 2061 2070 726f 6772 616d 2072 756e 0a2d a program run.- │ │ │ │ 00267df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00267e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267e10: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -00267e20: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -00267e30: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -00267e40: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -00267e50: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -00267e60: 2d31 2e32 362e 3035 2b64 732f 4d32 2f4d -1.26.05+ds/M2/M │ │ │ │ -00267e70: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00267e80: 732f 0a4d 6163 6175 6c61 7932 446f 632f s/.Macaulay2Doc/ │ │ │ │ -00267e90: 6675 6e63 7469 6f6e 732f 7275 6e50 726f functions/runPro │ │ │ │ -00267ea0: 6772 616d 2d64 6f63 2e6d 323a 3133 303a gram-doc.m2:130: │ │ │ │ -00267eb0: 302e 0a1f 0a46 696c 653a 204d 6163 6175 0....File: Macau │ │ │ │ -00267ec0: 6c61 7932 446f 632e 696e 666f 2c20 4e6f lay2Doc.info, No │ │ │ │ -00267ed0: 6465 3a20 5072 6f67 7261 6d20 3c3c 2054 de: Program << T │ │ │ │ -00267ee0: 6869 6e67 2c20 5072 6576 3a20 7275 6e50 hing, Prev: runP │ │ │ │ -00267ef0: 726f 6772 616d 2c20 5570 3a20 636f 6d6d rogram, Up: comm │ │ │ │ -00267f00: 756e 6963 6174 696e 6720 7769 7468 2070 unicating with p │ │ │ │ -00267f10: 726f 6772 616d 730a 0a50 726f 6772 616d rograms..Program │ │ │ │ -00267f20: 203c 3c20 5468 696e 6720 2d2d 2072 756e << Thing -- run │ │ │ │ -00267f30: 2070 726f 6772 616d 2077 6974 6820 696e program with in │ │ │ │ -00267f40: 7075 7420 7265 6469 7265 6374 696f 6e0a put redirection. │ │ │ │ -00267f50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00267f60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00267f70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00267f80: 2a2a 2a2a 2a2a 0a0a 2020 2a20 4f70 6572 ******.. * Oper │ │ │ │ -00267f90: 6174 6f72 3a20 2a6e 6f74 6520 3c3c 3a20 ator: *note <<: │ │ │ │ -00267fa0: 3c3c 2c0a 2020 2a20 5573 6167 653a 200a <<,. * Usage: . │ │ │ │ -00267fb0: 2020 2020 2020 2020 7072 6f67 203c 3c20 prog << │ │ │ │ -00267fc0: 780a 2020 2a20 496e 7075 7473 3a0a 2020 x. * Inputs:. │ │ │ │ -00267fd0: 2020 2020 2a20 7072 6f67 2c20 616e 2069 * prog, an i │ │ │ │ -00267fe0: 6e73 7461 6e63 6520 6f66 2074 6865 2074 nstance of the t │ │ │ │ -00267ff0: 7970 6520 2a6e 6f74 6520 5072 6f67 7261 ype *note Progra │ │ │ │ -00268000: 6d3a 2050 726f 6772 616d 2c2c 200a 2020 m: Program,, . │ │ │ │ -00268010: 2020 2020 2a20 782c 2061 202a 6e6f 7465 * x, a *note │ │ │ │ -00268020: 2074 6869 6e67 3a20 5468 696e 672c 2c20 thing: Thing,, │ │ │ │ -00268030: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -00268040: 2020 2020 2a20 616e 2069 6e73 7461 6e63 * an instanc │ │ │ │ -00268050: 6520 6f66 2074 6865 2074 7970 6520 2a6e e of the type *n │ │ │ │ -00268060: 6f74 6520 5072 6f67 7261 6d52 756e 3a20 ote ProgramRun: │ │ │ │ -00268070: 5072 6f67 7261 6d52 756e 2c2c 200a 0a44 ProgramRun,, ..D │ │ │ │ -00268080: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -00268090: 3d3d 3d3d 3d3d 0a0a 5772 6974 6520 7820 ======..Write x │ │ │ │ -002680a0: 746f 2061 2074 656d 706f 7261 7279 2066 to a temporary f │ │ │ │ -002680b0: 696c 6520 616e 6420 7275 6e20 7072 6f67 ile and run prog │ │ │ │ -002680c0: 2077 6974 6820 7468 6973 2066 696c 6520 with this file │ │ │ │ -002680d0: 6173 2069 6e70 7574 2075 7369 6e67 2069 as input using i │ │ │ │ -002680e0: 6e70 7574 0a72 6564 6972 6563 7469 6f6e nput.redirection │ │ │ │ -002680f0: 2028 7468 6520 3c20 6f70 6572 6174 6f72 (the < operator │ │ │ │ -00268100: 2069 6e20 6120 504f 5349 5820 7368 656c in a POSIX shel │ │ │ │ -00268110: 6c29 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d l)...+---------- │ │ │ │ -00268120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00268130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00268140: 6931 203a 204d 3220 3d20 6669 6e64 5072 i1 : M2 = findPr │ │ │ │ -00268150: 6f67 7261 6d20 224d 3222 2020 2020 2020 ogram "M2" │ │ │ │ -00268160: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00268170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268190: 207c 0a7c 6f31 203d 204d 3220 2020 2020 |.|o1 = M2 │ │ │ │ +00267e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +00267e40: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +00267e50: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +00267e60: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +00267e70: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +00267e80: 6c61 7932 2d31 2e32 362e 3035 2b64 732f lay2-1.26.05+ds/ │ │ │ │ +00267e90: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +00267ea0: 6b61 6765 732f 0a4d 6163 6175 6c61 7932 kages/.Macaulay2 │ │ │ │ +00267eb0: 446f 632f 6675 6e63 7469 6f6e 732f 7275 Doc/functions/ru │ │ │ │ +00267ec0: 6e50 726f 6772 616d 2d64 6f63 2e6d 323a nProgram-doc.m2: │ │ │ │ +00267ed0: 3133 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2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +00267fb0: 4f70 6572 6174 6f72 3a20 2a6e 6f74 6520 Operator: *note │ │ │ │ +00267fc0: 3c3c 3a20 3c3c 2c0a 2020 2a20 5573 6167 <<: <<,. * Usag │ │ │ │ +00267fd0: 653a 200a 2020 2020 2020 2020 7072 6f67 e: . prog │ │ │ │ +00267fe0: 203c 3c20 780a 2020 2a20 496e 7075 7473 << x. * Inputs │ │ │ │ +00267ff0: 3a0a 2020 2020 2020 2a20 7072 6f67 2c20 :. * prog, │ │ │ │ +00268000: 616e 2069 6e73 7461 6e63 6520 6f66 2074 an instance of t │ │ │ │ +00268010: 6865 2074 7970 6520 2a6e 6f74 6520 5072 he type *note Pr │ │ │ │ +00268020: 6f67 7261 6d3a 2050 726f 6772 616d 2c2c ogram: Program,, │ │ │ │ +00268030: 200a 2020 2020 2020 2a20 782c 2061 202a . * x, a * │ │ │ │ +00268040: 6e6f 7465 2074 6869 6e67 3a20 5468 696e note thing: Thin │ │ │ │ +00268050: 672c 2c20 0a20 202a 204f 7574 7075 7473 g,, . * Outputs │ │ │ │ +00268060: 3a0a 2020 2020 2020 2a20 616e 2069 6e73 :. * an ins │ │ │ │ +00268070: 7461 6e63 6520 6f66 2074 6865 2074 7970 tance of the typ │ │ │ │ +00268080: 6520 2a6e 6f74 6520 5072 6f67 7261 6d52 e *note ProgramR │ │ │ │ +00268090: 756e 3a20 5072 6f67 7261 6d52 756e 2c2c un: ProgramRun,, │ │ │ │ +002680a0: 200a 0a44 6573 6372 6970 7469 6f6e 0a3d ..Description.= │ │ │ │ +002680b0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5772 6974 ==========..Writ │ │ │ │ +002680c0: 6520 7820 746f 2061 2074 656d 706f 7261 e x to a tempora │ │ │ │ +002680d0: 7279 2066 696c 6520 616e 6420 7275 6e20 ry file and run │ │ │ │ +002680e0: 7072 6f67 2077 6974 6820 7468 6973 2066 prog with this f │ │ │ │ +002680f0: 696c 6520 6173 2069 6e70 7574 2075 7369 ile as input usi │ │ │ │ +00268100: 6e67 2069 6e70 7574 0a72 6564 6972 6563 ng input.redirec │ │ │ │ +00268110: 7469 6f6e 2028 7468 6520 3c20 6f70 6572 tion (the < oper │ │ │ │ +00268120: 6174 6f72 2069 6e20 6120 504f 5349 5820 ator in a POSIX │ │ │ │ +00268130: 7368 656c 6c29 2e0a 0a2b 2d2d 2d2d 2d2d shell)...+------ │ │ │ │ +00268140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268160: 2d2b 0a7c 6931 203a 204d 3220 3d20 6669 -+.|i1 : M2 = fi │ │ │ │ +00268170: 6e64 5072 6f67 7261 6d20 224d 3222 2020 ndProgram "M2" │ │ │ │ +00268180: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00268190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002681a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002681b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +002681b0: 2020 2020 207c 0a7c 6f31 203d 204d 3220 |.|o1 = M2 │ │ │ │ 002681c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002681d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002681e0: 2020 2020 207c 0a7c 6f31 203a 2050 726f |.|o1 : Pro │ │ │ │ -002681f0: 6772 616d 2020 2020 2020 2020 2020 2020 gram │ │ │ │ -00268200: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00268210: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00268220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00268230: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -00268240: 204d 3220 3c3c 2022 3220 2b20 3222 2020 M2 << "2 + 2" │ │ │ │ -00268250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268260: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00268270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268280: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00268290: 6f32 203d 2030 2020 2020 2020 2020 2020 o2 = 0 │ │ │ │ +002681d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +002681e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +002681f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268200: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ +00268210: 2050 726f 6772 616d 2020 2020 2020 2020 Program │ │ │ │ +00268220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268230: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00268240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00268260: 6932 203a 204d 3220 3c3c 2022 3220 2b20 i2 : M2 << "2 + │ │ │ │ +00268270: 3222 2020 2020 2020 2020 2020 2020 2020 2" │ │ │ │ +00268280: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00268290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002682a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002682b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +002682b0: 207c 0a7c 6f32 203d 2030 2020 2020 2020 |.|o2 = 0 │ │ │ │ 002682c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002682d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002682e0: 207c 0a7c 6f32 203a 2050 726f 6772 616d |.|o2 : Program │ │ │ │ -002682f0: 5275 6e20 2020 2020 2020 2020 2020 2020 Run │ │ │ │ -00268300: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00268310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00268320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00268330: 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 6f53 -----+.|i3 : toS │ │ │ │ -00268340: 7472 696e 6720 6f6f 2020 2020 2020 2020 tring oo │ │ │ │ -00268350: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00268360: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +002682d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +002682e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002682f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268300: 2020 2020 207c 0a7c 6f32 203a 2050 726f |.|o2 : Pro │ │ │ │ +00268310: 6772 616d 5275 6e20 2020 2020 2020 2020 gramRun │ │ │ │ +00268320: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00268330: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00268340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268350: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ +00268360: 2074 6f53 7472 696e 6720 6f6f 2020 2020 toString oo │ │ │ │ 00268370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268380: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ -00268390: 204d 6163 6175 6c61 7932 2c20 7665 7273 Macaulay2, vers │ │ │ │ -002683a0: 696f 6e20 312e 3236 2e30 3520 2020 2020 ion 1.26.05 │ │ │ │ -002683b0: 2020 207c 0a7c 2020 2020 2054 7970 6520 |.| Type │ │ │ │ -002683c0: 2268 656c 7022 2074 6f20 7365 6520 7573 "help" to see us │ │ │ │ -002683d0: 6566 756c 2063 6f6d 6d61 6e64 737c 0a7c eful commands|.| │ │ │ │ -002683e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002683f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268400: 2020 2020 2020 207c 0a7c 2020 2020 2069 |.| i │ │ │ │ -00268410: 3120 3a20 3220 2b20 3220 2020 2020 2020 1 : 2 + 2 │ │ │ │ -00268420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268430: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00268380: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00268390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002683a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +002683b0: 6f33 203d 204d 6163 6175 6c61 7932 2c20 o3 = Macaulay2, │ │ │ │ +002683c0: 7665 7273 696f 6e20 312e 3236 2e30 3520 version 1.26.05 │ │ │ │ +002683d0: 2020 2020 2020 207c 0a7c 2020 2020 2054 |.| T │ │ │ │ +002683e0: 7970 6520 2268 656c 7022 2074 6f20 7365 ype "help" to se │ │ │ │ +002683f0: 6520 7573 6566 756c 2063 6f6d 6d61 6e64 e useful command │ │ │ │ +00268400: 737c 0a7c 2020 2020 2020 2020 2020 2020 s|.| │ │ │ │ +00268410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268420: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00268430: 2020 2069 3120 3a20 3220 2b20 3220 2020 i1 : 2 + 2 │ │ │ │ 00268440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268450: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00268460: 2020 206f 3120 3d20 3420 2020 2020 2020 o1 = 4 │ │ │ │ -00268470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268480: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00268490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002684a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -002684b0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -002684c0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2066 ===.. * *note f │ │ │ │ -002684d0: 696e 6450 726f 6772 616d 3a20 6669 6e64 indProgram: find │ │ │ │ -002684e0: 5072 6f67 7261 6d2c 202d 2d20 6c6f 6164 Program, -- load │ │ │ │ -002684f0: 2065 7874 6572 6e61 6c20 7072 6f67 7261 external progra │ │ │ │ -00268500: 6d0a 2020 2a20 2a6e 6f74 6520 7275 6e50 m. * *note runP │ │ │ │ -00268510: 726f 6772 616d 3a20 7275 6e50 726f 6772 rogram: runProgr │ │ │ │ -00268520: 616d 2c20 2d2d 2072 756e 2061 6e20 6578 am, -- run an ex │ │ │ │ -00268530: 7465 726e 616c 2070 726f 6772 616d 0a20 ternal program. │ │ │ │ -00268540: 202a 202a 6e6f 7465 2067 6574 3a20 6765 * *note get: ge │ │ │ │ -00268550: 742c 202d 2d20 6765 7420 7468 6520 636f t, -- get the co │ │ │ │ -00268560: 6e74 656e 7473 206f 6620 6120 6669 6c65 ntents of a file │ │ │ │ -00268570: 0a0a 5761 7973 2074 6f20 7573 6520 7468 ..Ways to use th │ │ │ │ -00268580: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d is method:.===== │ │ │ │ -00268590: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -002685a0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2050 ===.. * *note P │ │ │ │ -002685b0: 726f 6772 616d 203c 3c20 5468 696e 673a rogram << Thing: │ │ │ │ -002685c0: 2050 726f 6772 616d 203c 3c20 5468 696e Program << Thin │ │ │ │ -002685d0: 672c 202d 2d20 7275 6e20 7072 6f67 7261 g, -- run progra │ │ │ │ -002685e0: 6d20 7769 7468 2069 6e70 7574 0a20 2020 m with input. │ │ │ │ -002685f0: 2072 6564 6972 6563 7469 6f6e 0a2d 2d2d redirection.--- │ │ │ │ -00268600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00268610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00268620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268450: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00268460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268470: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00268480: 0a7c 2020 2020 206f 3120 3d20 3420 2020 .| o1 = 4 │ │ │ │ +00268490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002684a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +002684b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002684c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002684d0: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ +002684e0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +002684f0: 7465 2066 696e 6450 726f 6772 616d 3a20 te findProgram: │ │ │ │ +00268500: 6669 6e64 5072 6f67 7261 6d2c 202d 2d20 findProgram, -- │ │ │ │ +00268510: 6c6f 6164 2065 7874 6572 6e61 6c20 7072 load external pr │ │ │ │ +00268520: 6f67 7261 6d0a 2020 2a20 2a6e 6f74 6520 ogram. * *note │ │ │ │ +00268530: 7275 6e50 726f 6772 616d 3a20 7275 6e50 runProgram: runP │ │ │ │ +00268540: 726f 6772 616d 2c20 2d2d 2072 756e 2061 rogram, -- run a │ │ │ │ +00268550: 6e20 6578 7465 726e 616c 2070 726f 6772 n external progr │ │ │ │ +00268560: 616d 0a20 202a 202a 6e6f 7465 2067 6574 am. * *note get │ │ │ │ +00268570: 3a20 6765 742c 202d 2d20 6765 7420 7468 : get, -- get th │ │ │ │ +00268580: 6520 636f 6e74 656e 7473 206f 6620 6120 e contents of a │ │ │ │ +00268590: 6669 6c65 0a0a 5761 7973 2074 6f20 7573 file..Ways to us │ │ │ │ +002685a0: 6520 7468 6973 206d 6574 686f 643a 0a3d e this method:.= │ │ │ │ +002685b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +002685c0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +002685d0: 7465 2050 726f 6772 616d 203c 3c20 5468 te Program << Th │ │ │ │ +002685e0: 696e 673a 2050 726f 6772 616d 203c 3c20 ing: Program << │ │ │ │ +002685f0: 5468 696e 672c 202d 2d20 7275 6e20 7072 Thing, -- run pr │ │ │ │ +00268600: 6f67 7261 6d20 7769 7468 2069 6e70 7574 ogram with input │ │ │ │ +00268610: 0a20 2020 2072 6564 6972 6563 7469 6f6e . redirection │ │ │ │ +00268620: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 00268630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00268640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -00268650: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -00268660: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -00268670: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -00268680: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -00268690: 7932 2d31 2e32 362e 3035 2b64 732f 4d32 y2-1.26.05+ds/M2 │ │ │ │ -002686a0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -002686b0: 6765 732f 0a4d 6163 6175 6c61 7932 446f ges/.Macaulay2Do │ │ │ │ -002686c0: 632f 6675 6e63 7469 6f6e 732f 7275 6e50 c/functions/runP │ │ │ │ -002686d0: 726f 6772 616d 2d64 6f63 2e6d 323a 3135 rogram-doc.m2:15 │ │ │ │ -002686e0: 373a 302e 0a1f 0a46 696c 653a 204d 6163 7:0....File: Mac │ │ │ │ -002686f0: 6175 6c61 7932 446f 632e 696e 666f 2c20 aulay2Doc.info, │ │ │ │ -00268700: 4e6f 6465 3a20 7573 696e 6720 736f 636b Node: using sock │ │ │ │ -00268710: 6574 732c 204e 6578 743a 206e 6f74 6966 ets, Next: notif │ │ │ │ -00268720: 792c 2050 7265 763a 2063 6f6d 6d75 6e69 y, Prev: communi │ │ │ │ -00268730: 6361 7469 6e67 2077 6974 6820 7072 6f67 cating with prog │ │ │ │ -00268740: 7261 6d73 2c20 5570 3a20 7379 7374 656d rams, Up: system │ │ │ │ -00268750: 2066 6163 696c 6974 6965 730a 0a75 7369 facilities..usi │ │ │ │ -00268760: 6e67 2073 6f63 6b65 7473 0a2a 2a2a 2a2a ng sockets.***** │ │ │ │ -00268770: 2a2a 2a2a 2a2a 2a2a 0a0a 4974 2773 2065 ********..It's e │ │ │ │ -00268780: 6173 7920 746f 2075 7365 2073 6f63 6b65 asy to use socke │ │ │ │ -00268790: 7473 2061 7320 7468 6f75 6768 2074 6865 ts as though the │ │ │ │ -002687a0: 7920 7765 7265 2066 696c 6573 2e20 2053 y were files. S │ │ │ │ -002687b0: 696d 706c 7920 7265 706c 6163 6520 7468 imply replace th │ │ │ │ -002687c0: 6520 6669 6c65 0a6e 616d 6520 6279 2061 e file.name by a │ │ │ │ -002687d0: 2073 7472 696e 6720 6f66 2074 6865 2066 string of the f │ │ │ │ -002687e0: 6f72 6d20 2468 6f73 743a 7365 7276 6963 orm $host:servic │ │ │ │ -002687f0: 6520 7768 6572 6520 686f 7374 2069 7320 e where host is │ │ │ │ -00268800: 7468 6520 6e61 6d65 206f 6620 4950 206e the name of IP n │ │ │ │ -00268810: 756d 6265 720a 6f66 2068 6f73 7420 746f umber.of host to │ │ │ │ -00268820: 2063 6f6e 7461 6374 2c20 616e 6420 7365 contact, and se │ │ │ │ -00268830: 7276 6963 6520 6973 2074 6865 2070 6f72 rvice is the por │ │ │ │ -00268840: 7420 6e75 6d62 6572 206f 7220 6e61 6d65 t number or name │ │ │ │ -00268850: 2074 6f20 7573 652e 2020 4966 2073 6572 to use. 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If hos │ │ │ │ -00268890: 7420 6973 206f 6d69 7474 6564 2c20 7468 t is omitted, th │ │ │ │ -002688a0: 656e 2061 6e20 696e 636f 6d69 6e67 0a63 en an incoming.c │ │ │ │ -002688b0: 6f6e 6e65 6374 696f 6e20 7769 6c6c 2062 onnection will b │ │ │ │ -002688c0: 6520 6c69 7374 656e 6564 2066 6f72 2e0a e listened for.. │ │ │ │ -002688d0: 0a54 6865 2066 6f6c 6c6f 7769 6e67 2063 .The following c │ │ │ │ -002688e0: 6f64 6520 7769 6c6c 2069 6c6c 7573 7472 ode will illustr │ │ │ │ -002688f0: 6174 6520 7477 6f2d 7761 7920 636f 6d6d ate two-way comm │ │ │ │ -00268900: 756e 6963 6174 696f 6e20 7573 696e 6720 unication using │ │ │ │ -00268910: 736f 636b 6574 7320 7369 6d69 6c61 720a sockets similar. │ │ │ │ -00268920: 746f 2074 6865 2069 6e74 6572 6163 7469 to the interacti │ │ │ │ -00268930: 6f6e 2075 7365 6420 6279 2077 6562 2073 on used by web s │ │ │ │ -00268940: 6572 7665 7273 2c20 616e 6420 796f 7520 ervers, and you │ │ │ │ -00268950: 6d61 7920 7472 7920 6974 206f 7574 206f may try it out o │ │ │ │ -00268960: 6e20 796f 7572 206d 6163 6869 6e65 2c0a n your machine,. │ │ │ │ -00268970: 756e 6c65 7373 2061 2066 6972 6577 616c unless a firewal │ │ │ │ -00268980: 6c20 7072 6576 656e 7473 2069 742e 0a0a l prevents it... │ │ │ │ -00268990: 6966 2028 7069 6420 3d20 666f 726b 2829 if (pid = fork() │ │ │ │ -002689a0: 2920 3d3d 2030 2074 6865 6e20 280a 2020 ) == 0 then (. │ │ │ │ -002689b0: 2020 2074 7279 2022 243a 3735 3030 2220 try "$:7500" │ │ │ │ -002689c0: 3c3c 2022 6869 2074 6865 7265 2220 3c3c << "hi there" << │ │ │ │ -002689d0: 2063 6c6f 7365 3b0a 2020 2020 2065 7869 close;. exi │ │ │ │ -002689e0: 7420 303b 0a20 2020 2020 290a 736c 6565 t 0;. ).slee │ │ │ │ -002689f0: 7020 320a 6765 7420 2224 6c6f 6361 6c68 p 2.get "$localh │ │ │ │ -00268a00: 6f73 743a 3735 3030 220a 7761 6974 2070 ost:7500".wait p │ │ │ │ -00268a10: 6964 0a0a 5468 6520 636f 6465 2075 7365 id..The code use │ │ │ │ -00268a20: 7320 2a6e 6f74 6520 666f 726b 3a20 666f s *note fork: fo │ │ │ │ -00268a30: 726b 2c20 746f 2063 7265 6174 6520 6120 rk, to create a │ │ │ │ -00268a40: 7365 7061 7261 7465 2070 726f 6365 7373 separate process │ │ │ │ -00268a50: 2074 6861 7420 7769 6c6c 206c 6973 7465 that will liste │ │ │ │ -00268a60: 6e0a 666f 7220 6120 636f 6e6e 6563 7469 n.for a connecti │ │ │ │ -00268a70: 6f6e 206f 6e20 706f 7274 2037 3530 3020 on on port 7500 │ │ │ │ -00268a80: 616e 6420 7468 656e 2073 656e 6420 7573 and then send us │ │ │ │ -00268a90: 2061 206d 6573 7361 6765 2e20 2054 6865 a message. The │ │ │ │ -00268aa0: 202a 6e6f 7465 2073 6c65 6570 3a0a 736c *note sleep:.sl │ │ │ │ -00268ab0: 6565 702c 2063 6f6d 6d61 6e64 2070 6175 eep, command pau │ │ │ │ -00268ac0: 7365 7320 666f 7220 6120 7768 696c 6520 ses for a while │ │ │ │ -00268ad0: 746f 206d 616b 6520 7375 7265 2074 6865 to make sure the │ │ │ │ -00268ae0: 2063 6869 6c64 2070 726f 6365 7373 2068 child process h │ │ │ │ -00268af0: 6173 2068 6164 2074 696d 650a 746f 2073 as had time.to s │ │ │ │ -00268b00: 7461 7274 206c 6973 7465 6e69 6e67 2e20 tart listening. │ │ │ │ -00268b10: 5468 656e 2077 6520 7573 6520 616e 206f Then we use an o │ │ │ │ -00268b20: 7264 696e 6172 7920 696e 7075 7420 636f rdinary input co │ │ │ │ -00268b30: 6d6d 616e 642c 206e 616d 656c 7920 2a6e mmand, namely *n │ │ │ │ -00268b40: 6f74 6520 6765 743a 0a67 6574 2c2c 2074 ote get:.get,, t │ │ │ │ -00268b50: 6f20 6f62 7461 696e 2074 6865 206d 6573 o obtain the mes │ │ │ │ -00268b60: 7361 6765 2e20 4669 6e61 6c6c 792c 2077 sage. Finally, w │ │ │ │ -00268b70: 6520 2a6e 6f74 6520 7761 6974 3a20 7761 e *note wait: wa │ │ │ │ -00268b80: 6974 2c20 666f 7220 7468 6520 6368 696c it, for the chil │ │ │ │ -00268b90: 640a 7072 6f63 6573 7320 746f 2066 696e d.process to fin │ │ │ │ -00268ba0: 6973 682c 2061 7320 7765 2073 686f 756c ish, as we shoul │ │ │ │ -00268bb0: 642e 0a0a 5365 6520 616c 736f 0a3d 3d3d d...See also.=== │ │ │ │ -00268bc0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -00268bd0: 2067 6574 5757 573a 2067 6574 5757 572c getWWW: getWWW, │ │ │ │ -00268be0: 202d 2d20 6765 7420 6120 7765 6220 7061 -- get a web pa │ │ │ │ -00268bf0: 6765 0a2a 204d 656e 753a 0a0a 0a43 7265 ge.* Menu:...Cre │ │ │ │ -00268c00: 6174 696e 6720 6120 736f 636b 6574 0a2a ating a socket.* │ │ │ │ -00268c10: 206f 7065 6e4c 6973 7465 6e65 723a 206f openListener: o │ │ │ │ -00268c20: 7065 6e4c 6973 7465 6e65 725f 6c70 5374 penListener_lpSt │ │ │ │ -00268c30: 7269 6e67 5f72 702e 2020 6f70 656e 2061 ring_rp. open a │ │ │ │ -00268c40: 2070 6f72 7420 666f 7220 6c69 7374 656e port for listen │ │ │ │ -00268c50: 696e 670a 2a20 6f70 656e 496e 3a20 6f70 ing.* openIn: op │ │ │ │ -00268c60: 656e 496e 5f6c 7053 7472 696e 675f 7270 enIn_lpString_rp │ │ │ │ -00268c70: 2e20 2020 6f70 656e 2061 6e20 696e 7075 . open an inpu │ │ │ │ -00268c80: 7420 6669 6c65 0a2a 206f 7065 6e49 6e4f t file.* openInO │ │ │ │ -00268c90: 7574 3a3a 2020 2020 2020 2020 2020 2020 ut:: │ │ │ │ -00268ca0: 2020 2020 2020 206f 7065 6e20 616e 2069 open an i │ │ │ │ -00268cb0: 6e70 7574 206f 7574 7075 7420 6669 6c65 nput output file │ │ │ │ -00268cc0: 0a2a 206f 7065 6e4f 7574 3a20 6f70 656e .* openOut: open │ │ │ │ -00268cd0: 4f75 745f 6c70 5374 7269 6e67 5f72 702e Out_lpString_rp. │ │ │ │ -00268ce0: 2020 6f70 656e 2061 6e20 6f75 7470 7574 open an output │ │ │ │ -00268cf0: 2066 696c 650a 2a20 6f70 656e 4f75 7441 file.* openOutA │ │ │ │ -00268d00: 7070 656e 643a 206f 7065 6e4f 7574 4170 ppend: openOutAp │ │ │ │ -00268d10: 7065 6e64 5f6c 7053 7472 696e 675f 7270 pend_lpString_rp │ │ │ │ -00268d20: 2e20 206f 7065 6e20 616e 206f 7574 7075 . open an outpu │ │ │ │ -00268d30: 7420 6669 6c65 2066 6f72 2061 7070 656e t file for appen │ │ │ │ -00268d40: 6469 6e67 0a0a 4d61 6e69 7075 6c61 7469 ding..Manipulati │ │ │ │ -00268d50: 6e67 2061 2073 6f63 6b65 740a 2a20 656e ng a socket.* en │ │ │ │ -00268d60: 646c 3a3a 2020 2020 2020 2020 2020 2020 dl:: │ │ │ │ -00268d70: 2020 2020 2020 2020 2020 2020 656e 6420 end │ │ │ │ -00268d80: 616e 206f 7574 7075 7420 6c69 6e65 0a2a an output line.* │ │ │ │ -00268d90: 2066 6c75 7368 3a3a 2020 2020 2020 2020 flush:: │ │ │ │ -00268da0: 2020 2020 2020 2020 2020 2020 2020 2066 f │ │ │ │ -00268db0: 6c75 7368 206f 7574 7075 7420 746f 2066 lush output to f │ │ │ │ -00268dc0: 696c 650a 2a20 636c 6f73 653a 3a20 2020 ile.* close:: │ │ │ │ -00268dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268de0: 2020 2020 636c 6f73 6520 6120 6669 6c65 close a file │ │ │ │ -00268df0: 0a2a 2063 6c6f 7365 496e 3a3a 2020 2020 .* closeIn:: │ │ │ │ -00268e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268e10: 2063 6c6f 7365 2061 6e20 696e 7075 7420 close an input │ │ │ │ -00268e20: 6669 6c65 0a2a 2063 6c6f 7365 4f75 743a file.* closeOut: │ │ │ │ -00268e30: 3a20 2020 2020 2020 2020 2020 2020 2020 : │ │ │ │ -00268e40: 2020 2020 2063 6c6f 7365 2061 6e20 6f75 close an ou │ │ │ │ -00268e50: 7470 7574 2066 696c 650a 0a49 6e74 6572 tput file..Inter │ │ │ │ -00268e60: 6163 7469 6e67 2061 2073 6f63 6b65 740a acting a socket. │ │ │ │ -00268e70: 2a20 6765 743a 3a20 2020 2020 2020 2020 * get:: │ │ │ │ -00268e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268e90: 6765 7420 7468 6520 636f 6e74 656e 7473 get the contents │ │ │ │ -00268ea0: 206f 6620 6120 6669 6c65 0a2a 2067 6574 of a file.* get │ │ │ │ -00268eb0: 633a 3a20 2020 2020 2020 2020 2020 2020 c:: │ │ │ │ -00268ec0: 2020 2020 2020 2020 2020 2067 6574 2061 get a │ │ │ │ -00268ed0: 2062 7974 650a 2a20 7265 6164 3a3a 2020 byte.* read:: │ │ │ │ -00268ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268ef0: 2020 2020 2020 7265 6164 2066 726f 6d20 read from │ │ │ │ -00268f00: 6120 6669 6c65 0a2a 2073 6361 6e4c 696e a file.* scanLin │ │ │ │ -00268f10: 6573 3a3a 2020 2020 2020 2020 2020 2020 es:: │ │ │ │ -00268f20: 2020 2020 2020 2061 7070 6c79 2061 2066 apply a f │ │ │ │ -00268f30: 756e 6374 696f 6e20 746f 2065 6163 6820 unction to each │ │ │ │ -00268f40: 6c69 6e65 206f 6620 6120 6669 6c65 0a2a line of a file.* │ │ │ │ -00268f50: 2066 696c 654c 656e 6774 683a 3a20 2020 fileLength:: │ │ │ │ -00268f60: 2020 2020 2020 2020 2020 2020 2020 2074 t │ │ │ │ -00268f70: 6865 206c 656e 6774 6820 6f66 2061 2066 he length of a f │ │ │ │ -00268f80: 696c 650a 2a20 6865 6967 6874 2846 696c ile.* height(Fil │ │ │ │ -00268f90: 6529 3a20 6865 6967 6874 5f6c 7046 696c e): height_lpFil │ │ │ │ -00268fa0: 655f 7270 2e20 2067 6574 2077 696e 646f e_rp. get windo │ │ │ │ -00268fb0: 7720 6865 6967 6874 0a2a 2077 6964 7468 w height.* width │ │ │ │ -00268fc0: 2846 696c 6529 3a20 7769 6474 685f 6c70 (File): width_lp │ │ │ │ -00268fd0: 4669 6c65 5f72 702e 2020 6765 7420 7769 File_rp. get wi │ │ │ │ -00268fe0: 6e64 6f77 2077 6964 7468 0a2a 2061 7445 ndow width.* atE │ │ │ │ -00268ff0: 6e64 4f66 4669 6c65 3a20 6174 456e 644f ndOfFile: atEndO │ │ │ │ -00269000: 6646 696c 655f 6c70 4669 6c65 5f72 702e fFile_lpFile_rp. │ │ │ │ -00269010: 2020 7465 7374 2066 6f72 2065 6e64 206f test for end o │ │ │ │ -00269020: 6620 6669 6c65 0a2a 2065 6368 6f4f 6e3a f file.* echoOn: │ │ │ │ -00269030: 3a20 2020 2020 2020 2020 2020 2020 2020 : │ │ │ │ -00269040: 2020 2020 2020 2074 7572 6e20 6f6e 2065 turn on e │ │ │ │ -00269050: 6368 6f69 6e67 0a2a 2065 6368 6f4f 6666 choing.* echoOff │ │ │ │ -00269060: 3a3a 2020 2020 2020 2020 2020 2020 2020 :: │ │ │ │ -00269070: 2020 2020 2020 2074 7572 6e20 6f66 6620 turn off │ │ │ │ -00269080: 6563 686f 696e 670a 2a20 6973 4f70 656e echoing.* isOpen │ │ │ │ -00269090: 3a3a 2020 2020 2020 2020 2020 2020 2020 :: │ │ │ │ -002690a0: 2020 2020 2020 2020 7768 6574 6865 7220 whether │ │ │ │ -002690b0: 6120 6669 6c65 206f 7220 6461 7461 6261 a file or databa │ │ │ │ -002690c0: 7365 2069 7320 6f70 656e 0a2a 2069 7352 se is open.* isR │ │ │ │ -002690d0: 6561 6479 3a20 6973 5265 6164 795f 6c70 eady: isReady_lp │ │ │ │ -002690e0: 4669 6c65 5f72 702e 2020 2077 6865 7468 File_rp. wheth │ │ │ │ -002690f0: 6572 2061 2066 696c 6520 6861 7320 6461 er a file has da │ │ │ │ -00269100: 7461 2061 7661 696c 6162 6c65 2066 6f72 ta available for │ │ │ │ -00269110: 2072 6561 6469 6e67 0a2a 2069 734f 7574 reading.* isOut │ │ │ │ -00269120: 7075 7446 696c 653a 2069 734f 7574 7075 putFile: isOutpu │ │ │ │ -00269130: 7446 696c 655f 6c70 4669 6c65 5f72 702e tFile_lpFile_rp. │ │ │ │ -00269140: 2020 7768 6574 6865 7220 6120 6669 6c65 whether a file │ │ │ │ -00269150: 2069 7320 6f70 656e 2066 6f72 206f 7574 is open for out │ │ │ │ -00269160: 7075 740a 2a20 6973 496e 7075 7446 696c put.* isInputFil │ │ │ │ -00269170: 653a 2069 7349 6e70 7574 4669 6c65 5f6c e: isInputFile_l │ │ │ │ -00269180: 7046 696c 655f 7270 2e20 2077 6865 7468 pFile_rp. wheth │ │ │ │ -00269190: 6572 2061 2066 696c 6520 6973 206f 7065 er a file is ope │ │ │ │ -002691a0: 6e20 666f 7220 696e 7075 740a 2a20 6973 n for input.* is │ │ │ │ -002691b0: 4c69 7374 656e 6572 3a20 6973 4c69 7374 Listener: isList │ │ │ │ -002691c0: 656e 6572 5f6c 7046 696c 655f 7270 2e20 ener_lpFile_rp. │ │ │ │ -002691d0: 2077 6865 7468 6572 2061 2066 696c 6520 whether a file │ │ │ │ -002691e0: 6973 206f 7065 6e20 666f 7220 6c69 7374 is open for list │ │ │ │ -002691f0: 656e 696e 670a 2a20 6f70 656e 4669 6c65 ening.* openFile │ │ │ │ -00269200: 733a 3a20 2020 2020 2020 2020 2020 2020 s:: │ │ │ │ -00269210: 2020 2020 2020 6c69 7374 2074 6865 206f list the o │ │ │ │ -00269220: 7065 6e20 6669 6c65 730a 2a20 636f 6e6e pen files.* conn │ │ │ │ -00269230: 6563 7469 6f6e 436f 756e 743a 3a20 2020 ectionCount:: │ │ │ │ -00269240: 2020 2020 2020 2020 2020 7468 6520 6e75 the nu │ │ │ │ -00269250: 6d62 6572 206f 6620 636f 6e6e 6563 7469 mber of connecti │ │ │ │ -00269260: 6f6e 730a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ons.------------ │ │ │ │ -00269270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268670: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +00268680: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +00268690: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +002686a0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +002686b0: 6175 6c61 7932 2d31 2e32 362e 3035 2b64 aulay2-1.26.05+d │ │ │ │ +002686c0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +002686d0: 6163 6b61 6765 732f 0a4d 6163 6175 6c61 ackages/.Macaula │ │ │ │ +002686e0: 7932 446f 632f 6675 6e63 7469 6f6e 732f y2Doc/functions/ │ │ │ │ +002686f0: 7275 6e50 726f 6772 616d 2d64 6f63 2e6d runProgram-doc.m │ │ │ │ +00268700: 323a 3135 373a 302e 0a1f 0a46 696c 653a 2:157:0....File: │ │ │ │ +00268710: 204d 6163 6175 6c61 7932 446f 632e 696e Macaulay2Doc.in │ │ │ │ +00268720: 666f 2c20 4e6f 6465 3a20 7573 696e 6720 fo, Node: using │ │ │ │ +00268730: 736f 636b 6574 732c 204e 6578 743a 206e sockets, Next: n │ │ │ │ +00268740: 6f74 6966 792c 2050 7265 763a 2063 6f6d otify, Prev: com │ │ │ │ +00268750: 6d75 6e69 6361 7469 6e67 2077 6974 6820 municating with │ │ │ │ +00268760: 7072 6f67 7261 6d73 2c20 5570 3a20 7379 programs, Up: sy │ │ │ │ +00268770: 7374 656d 2066 6163 696c 6974 6965 730a stem facilities. │ │ │ │ +00268780: 0a75 7369 6e67 2073 6f63 6b65 7473 0a2a .using sockets.* │ │ │ │ +00268790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4974 ************..It │ │ │ │ +002687a0: 2773 2065 6173 7920 746f 2075 7365 2073 's easy to use s │ │ │ │ +002687b0: 6f63 6b65 7473 2061 7320 7468 6f75 6768 ockets as though │ │ │ │ +002687c0: 2074 6865 7920 7765 7265 2066 696c 6573 they were files │ │ │ │ +002687d0: 2e20 2053 696d 706c 7920 7265 706c 6163 . Simply replac │ │ │ │ +002687e0: 6520 7468 6520 6669 6c65 0a6e 616d 6520 e the file.name │ │ │ │ +002687f0: 6279 2061 2073 7472 696e 6720 6f66 2074 by a string of t │ │ │ │ +00268800: 6865 2066 6f72 6d20 2468 6f73 743a 7365 he form $host:se │ │ │ │ +00268810: 7276 6963 6520 7768 6572 6520 686f 7374 rvice where host │ │ │ │ +00268820: 2069 7320 7468 6520 6e61 6d65 206f 6620 is the name of │ │ │ │ +00268830: 4950 206e 756d 6265 720a 6f66 2068 6f73 IP number.of hos │ │ │ │ +00268840: 7420 746f 2063 6f6e 7461 6374 2c20 616e t to contact, an │ │ │ │ +00268850: 6420 7365 7276 6963 6520 6973 2074 6865 d service is the │ │ │ │ +00268860: 2070 6f72 7420 6e75 6d62 6572 206f 7220 port number or │ │ │ │ +00268870: 6e61 6d65 2074 6f20 7573 652e 2020 4966 name to use. If │ │ │ │ +00268880: 2073 6572 7669 6365 0a69 7320 6f6d 6974 service.is omit │ │ │ │ +00268890: 7465 642c 2074 6865 6e20 706f 7274 2032 ted, then port 2 │ │ │ │ +002688a0: 3530 3020 6973 2075 7365 642e 2020 4966 500 is used. If │ │ │ │ +002688b0: 2068 6f73 7420 6973 206f 6d69 7474 6564 host is omitted │ │ │ │ +002688c0: 2c20 7468 656e 2061 6e20 696e 636f 6d69 , then an incomi │ │ │ │ +002688d0: 6e67 0a63 6f6e 6e65 6374 696f 6e20 7769 ng.connection wi │ │ │ │ +002688e0: 6c6c 2062 6520 6c69 7374 656e 6564 2066 ll be listened f │ │ │ │ +002688f0: 6f72 2e0a 0a54 6865 2066 6f6c 6c6f 7769 or...The followi │ │ │ │ +00268900: 6e67 2063 6f64 6520 7769 6c6c 2069 6c6c ng code will ill │ │ │ │ +00268910: 7573 7472 6174 6520 7477 6f2d 7761 7920 ustrate two-way │ │ │ │ +00268920: 636f 6d6d 756e 6963 6174 696f 6e20 7573 communication us │ │ │ │ +00268930: 696e 6720 736f 636b 6574 7320 7369 6d69 ing sockets simi │ │ │ │ +00268940: 6c61 720a 746f 2074 6865 2069 6e74 6572 lar.to the inter │ │ │ │ +00268950: 6163 7469 6f6e 2075 7365 6420 6279 2077 action used by w │ │ │ │ +00268960: 6562 2073 6572 7665 7273 2c20 616e 6420 eb servers, and │ │ │ │ +00268970: 796f 7520 6d61 7920 7472 7920 6974 206f you may try it o │ │ │ │ +00268980: 7574 206f 6e20 796f 7572 206d 6163 6869 ut on your machi │ │ │ │ +00268990: 6e65 2c0a 756e 6c65 7373 2061 2066 6972 ne,.unless a fir │ │ │ │ +002689a0: 6577 616c 6c20 7072 6576 656e 7473 2069 ewall prevents i │ │ │ │ +002689b0: 742e 0a0a 6966 2028 7069 6420 3d20 666f t...if (pid = fo │ │ │ │ +002689c0: 726b 2829 2920 3d3d 2030 2074 6865 6e20 rk()) == 0 then │ │ │ │ +002689d0: 280a 2020 2020 2074 7279 2022 243a 3735 (. try "$:75 │ │ │ │ +002689e0: 3030 2220 3c3c 2022 6869 2074 6865 7265 00" << "hi there │ │ │ │ +002689f0: 2220 3c3c 2063 6c6f 7365 3b0a 2020 2020 " << close;. │ │ │ │ +00268a00: 2065 7869 7420 303b 0a20 2020 2020 290a exit 0;. ). │ │ │ │ +00268a10: 736c 6565 7020 320a 6765 7420 2224 6c6f sleep 2.get "$lo │ │ │ │ +00268a20: 6361 6c68 6f73 743a 3735 3030 220a 7761 calhost:7500".wa │ │ │ │ +00268a30: 6974 2070 6964 0a0a 5468 6520 636f 6465 it pid..The code │ │ │ │ +00268a40: 2075 7365 7320 2a6e 6f74 6520 666f 726b uses *note fork │ │ │ │ +00268a50: 3a20 666f 726b 2c20 746f 2063 7265 6174 : fork, to creat │ │ │ │ +00268a60: 6520 6120 7365 7061 7261 7465 2070 726f e a separate pro │ │ │ │ +00268a70: 6365 7373 2074 6861 7420 7769 6c6c 206c cess that will l │ │ │ │ +00268a80: 6973 7465 6e0a 666f 7220 6120 636f 6e6e isten.for a conn │ │ │ │ +00268a90: 6563 7469 6f6e 206f 6e20 706f 7274 2037 ection on port 7 │ │ │ │ +00268aa0: 3530 3020 616e 6420 7468 656e 2073 656e 500 and then sen │ │ │ │ +00268ab0: 6420 7573 2061 206d 6573 7361 6765 2e20 d us a message. │ │ │ │ +00268ac0: 2054 6865 202a 6e6f 7465 2073 6c65 6570 The *note sleep │ │ │ │ +00268ad0: 3a0a 736c 6565 702c 2063 6f6d 6d61 6e64 :.sleep, command │ │ │ │ +00268ae0: 2070 6175 7365 7320 666f 7220 6120 7768 pauses for a wh │ │ │ │ +00268af0: 696c 6520 746f 206d 616b 6520 7375 7265 ile to make sure │ │ │ │ +00268b00: 2074 6865 2063 6869 6c64 2070 726f 6365 the child proce │ │ │ │ +00268b10: 7373 2068 6173 2068 6164 2074 696d 650a ss has had time. │ │ │ │ +00268b20: 746f 2073 7461 7274 206c 6973 7465 6e69 to start listeni │ │ │ │ +00268b30: 6e67 2e20 5468 656e 2077 6520 7573 6520 ng. Then we use │ │ │ │ +00268b40: 616e 206f 7264 696e 6172 7920 696e 7075 an ordinary inpu │ │ │ │ +00268b50: 7420 636f 6d6d 616e 642c 206e 616d 656c t command, namel │ │ │ │ +00268b60: 7920 2a6e 6f74 6520 6765 743a 0a67 6574 y *note get:.get │ │ │ │ +00268b70: 2c2c 2074 6f20 6f62 7461 696e 2074 6865 ,, to obtain the │ │ │ │ +00268b80: 206d 6573 7361 6765 2e20 4669 6e61 6c6c message. Finall │ │ │ │ +00268b90: 792c 2077 6520 2a6e 6f74 6520 7761 6974 y, we *note wait │ │ │ │ +00268ba0: 3a20 7761 6974 2c20 666f 7220 7468 6520 : wait, for the │ │ │ │ +00268bb0: 6368 696c 640a 7072 6f63 6573 7320 746f child.process to │ │ │ │ +00268bc0: 2066 696e 6973 682c 2061 7320 7765 2073 finish, as we s │ │ │ │ +00268bd0: 686f 756c 642e 0a0a 5365 6520 616c 736f hould...See also │ │ │ │ +00268be0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +00268bf0: 6e6f 7465 2067 6574 5757 573a 2067 6574 note getWWW: get │ │ │ │ +00268c00: 5757 572c 202d 2d20 6765 7420 6120 7765 WWW, -- get a we │ │ │ │ +00268c10: 6220 7061 6765 0a2a 204d 656e 753a 0a0a b page.* Menu:.. │ │ │ │ +00268c20: 0a43 7265 6174 696e 6720 6120 736f 636b .Creating a sock │ │ │ │ +00268c30: 6574 0a2a 206f 7065 6e4c 6973 7465 6e65 et.* openListene │ │ │ │ +00268c40: 723a 206f 7065 6e4c 6973 7465 6e65 725f r: openListener_ │ │ │ │ +00268c50: 6c70 5374 7269 6e67 5f72 702e 2020 6f70 lpString_rp. op │ │ │ │ +00268c60: 656e 2061 2070 6f72 7420 666f 7220 6c69 en a port for li │ │ │ │ +00268c70: 7374 656e 696e 670a 2a20 6f70 656e 496e stening.* openIn │ │ │ │ +00268c80: 3a20 6f70 656e 496e 5f6c 7053 7472 696e : openIn_lpStrin │ │ │ │ +00268c90: 675f 7270 2e20 2020 6f70 656e 2061 6e20 g_rp. open an │ │ │ │ +00268ca0: 696e 7075 7420 6669 6c65 0a2a 206f 7065 input file.* ope │ │ │ │ +00268cb0: 6e49 6e4f 7574 3a3a 2020 2020 2020 2020 nInOut:: │ │ │ │ +00268cc0: 2020 2020 2020 2020 2020 206f 7065 6e20 open │ │ │ │ +00268cd0: 616e 2069 6e70 7574 206f 7574 7075 7420 an input output │ │ │ │ +00268ce0: 6669 6c65 0a2a 206f 7065 6e4f 7574 3a20 file.* openOut: │ │ │ │ +00268cf0: 6f70 656e 4f75 745f 6c70 5374 7269 6e67 openOut_lpString │ │ │ │ +00268d00: 5f72 702e 2020 6f70 656e 2061 6e20 6f75 _rp. open an ou │ │ │ │ +00268d10: 7470 7574 2066 696c 650a 2a20 6f70 656e tput file.* open │ │ │ │ +00268d20: 4f75 7441 7070 656e 643a 206f 7065 6e4f OutAppend: openO │ │ │ │ +00268d30: 7574 4170 7065 6e64 5f6c 7053 7472 696e utAppend_lpStrin │ │ │ │ +00268d40: 675f 7270 2e20 206f 7065 6e20 616e 206f g_rp. open an o │ │ │ │ +00268d50: 7574 7075 7420 6669 6c65 2066 6f72 2061 utput file for a │ │ │ │ +00268d60: 7070 656e 6469 6e67 0a0a 4d61 6e69 7075 ppending..Manipu │ │ │ │ +00268d70: 6c61 7469 6e67 2061 2073 6f63 6b65 740a lating a socket. │ │ │ │ +00268d80: 2a20 656e 646c 3a3a 2020 2020 2020 2020 * endl:: │ │ │ │ +00268d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268da0: 656e 6420 616e 206f 7574 7075 7420 6c69 end an output li │ │ │ │ +00268db0: 6e65 0a2a 2066 6c75 7368 3a3a 2020 2020 ne.* flush:: │ │ │ │ +00268dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268dd0: 2020 2066 6c75 7368 206f 7574 7075 7420 flush output │ │ │ │ +00268de0: 746f 2066 696c 650a 2a20 636c 6f73 653a to file.* close: │ │ │ │ +00268df0: 3a20 2020 2020 2020 2020 2020 2020 2020 : │ │ │ │ +00268e00: 2020 2020 2020 2020 636c 6f73 6520 6120 close a │ │ │ │ +00268e10: 6669 6c65 0a2a 2063 6c6f 7365 496e 3a3a file.* closeIn:: │ │ │ │ +00268e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268e30: 2020 2020 2063 6c6f 7365 2061 6e20 696e close an in │ │ │ │ +00268e40: 7075 7420 6669 6c65 0a2a 2063 6c6f 7365 put file.* close │ │ │ │ +00268e50: 4f75 743a 3a20 2020 2020 2020 2020 2020 Out:: │ │ │ │ +00268e60: 2020 2020 2020 2020 2063 6c6f 7365 2061 close a │ │ │ │ +00268e70: 6e20 6f75 7470 7574 2066 696c 650a 0a49 n output file..I │ │ │ │ +00268e80: 6e74 6572 6163 7469 6e67 2061 2073 6f63 nteracting a soc │ │ │ │ +00268e90: 6b65 740a 2a20 6765 743a 3a20 2020 2020 ket.* get:: │ │ │ │ +00268ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268eb0: 2020 2020 6765 7420 7468 6520 636f 6e74 get the cont │ │ │ │ +00268ec0: 656e 7473 206f 6620 6120 6669 6c65 0a2a ents of a file.* │ │ │ │ +00268ed0: 2067 6574 633a 3a20 2020 2020 2020 2020 getc:: │ │ │ │ +00268ee0: 2020 2020 2020 2020 2020 2020 2020 2067 g │ │ │ │ +00268ef0: 6574 2061 2062 7974 650a 2a20 7265 6164 et a byte.* read │ │ │ │ +00268f00: 3a3a 2020 2020 2020 2020 2020 2020 2020 :: │ │ │ │ +00268f10: 2020 2020 2020 2020 2020 7265 6164 2066 read f │ │ │ │ +00268f20: 726f 6d20 6120 6669 6c65 0a2a 2073 6361 rom a file.* sca │ │ │ │ +00268f30: 6e4c 696e 6573 3a3a 2020 2020 2020 2020 nLines:: │ │ │ │ +00268f40: 2020 2020 2020 2020 2020 2061 7070 6c79 apply │ │ │ │ +00268f50: 2061 2066 756e 6374 696f 6e20 746f 2065 a function to e │ │ │ │ +00268f60: 6163 6820 6c69 6e65 206f 6620 6120 6669 ach line of a fi │ │ │ │ +00268f70: 6c65 0a2a 2066 696c 654c 656e 6774 683a le.* fileLength: │ │ │ │ +00268f80: 3a20 2020 2020 2020 2020 2020 2020 2020 : │ │ │ │ +00268f90: 2020 2074 6865 206c 656e 6774 6820 6f66 the length of │ │ │ │ +00268fa0: 2061 2066 696c 650a 2a20 6865 6967 6874 a file.* height │ │ │ │ +00268fb0: 2846 696c 6529 3a20 6865 6967 6874 5f6c (File): height_l │ │ │ │ +00268fc0: 7046 696c 655f 7270 2e20 2067 6574 2077 pFile_rp. get w │ │ │ │ +00268fd0: 696e 646f 7720 6865 6967 6874 0a2a 2077 indow height.* w │ │ │ │ +00268fe0: 6964 7468 2846 696c 6529 3a20 7769 6474 idth(File): widt │ │ │ │ +00268ff0: 685f 6c70 4669 6c65 5f72 702e 2020 6765 h_lpFile_rp. ge │ │ │ │ +00269000: 7420 7769 6e64 6f77 2077 6964 7468 0a2a t window width.* │ │ │ │ +00269010: 2061 7445 6e64 4f66 4669 6c65 3a20 6174 atEndOfFile: at │ │ │ │ +00269020: 456e 644f 6646 696c 655f 6c70 4669 6c65 EndOfFile_lpFile │ │ │ │ +00269030: 5f72 702e 2020 7465 7374 2066 6f72 2065 _rp. test for e │ │ │ │ +00269040: 6e64 206f 6620 6669 6c65 0a2a 2065 6368 nd of file.* ech │ │ │ │ +00269050: 6f4f 6e3a 3a20 2020 2020 2020 2020 2020 oOn:: │ │ │ │ +00269060: 2020 2020 2020 2020 2020 2074 7572 6e20 turn │ │ │ │ +00269070: 6f6e 2065 6368 6f69 6e67 0a2a 2065 6368 on echoing.* ech │ │ │ │ +00269080: 6f4f 6666 3a3a 2020 2020 2020 2020 2020 oOff:: │ │ │ │ +00269090: 2020 2020 2020 2020 2020 2074 7572 6e20 turn │ │ │ │ +002690a0: 6f66 6620 6563 686f 696e 670a 2a20 6973 off echoing.* is │ │ │ │ +002690b0: 4f70 656e 3a3a 2020 2020 2020 2020 2020 Open:: │ │ │ │ +002690c0: 2020 2020 2020 2020 2020 2020 7768 6574 whet │ │ │ │ +002690d0: 6865 7220 6120 6669 6c65 206f 7220 6461 her a file or da │ │ │ │ +002690e0: 7461 6261 7365 2069 7320 6f70 656e 0a2a tabase is open.* │ │ │ │ +002690f0: 2069 7352 6561 6479 3a20 6973 5265 6164 isReady: isRead │ │ │ │ +00269100: 795f 6c70 4669 6c65 5f72 702e 2020 2077 y_lpFile_rp. w │ │ │ │ +00269110: 6865 7468 6572 2061 2066 696c 6520 6861 hether a file ha │ │ │ │ +00269120: 7320 6461 7461 2061 7661 696c 6162 6c65 s data available │ │ │ │ +00269130: 2066 6f72 2072 6561 6469 6e67 0a2a 2069 for reading.* i │ │ │ │ +00269140: 734f 7574 7075 7446 696c 653a 2069 734f sOutputFile: isO │ │ │ │ +00269150: 7574 7075 7446 696c 655f 6c70 4669 6c65 utputFile_lpFile │ │ │ │ +00269160: 5f72 702e 2020 7768 6574 6865 7220 6120 _rp. whether a │ │ │ │ +00269170: 6669 6c65 2069 7320 6f70 656e 2066 6f72 file is open for │ │ │ │ +00269180: 206f 7574 7075 740a 2a20 6973 496e 7075 output.* isInpu │ │ │ │ +00269190: 7446 696c 653a 2069 7349 6e70 7574 4669 tFile: isInputFi │ │ │ │ +002691a0: 6c65 5f6c 7046 696c 655f 7270 2e20 2077 le_lpFile_rp. w │ │ │ │ +002691b0: 6865 7468 6572 2061 2066 696c 6520 6973 hether a file is │ │ │ │ +002691c0: 206f 7065 6e20 666f 7220 696e 7075 740a open for input. │ │ │ │ +002691d0: 2a20 6973 4c69 7374 656e 6572 3a20 6973 * isListener: is │ │ │ │ +002691e0: 4c69 7374 656e 6572 5f6c 7046 696c 655f Listener_lpFile_ │ │ │ │ +002691f0: 7270 2e20 2077 6865 7468 6572 2061 2066 rp. whether a f │ │ │ │ +00269200: 696c 6520 6973 206f 7065 6e20 666f 7220 ile is open for │ │ │ │ +00269210: 6c69 7374 656e 696e 670a 2a20 6f70 656e listening.* open │ │ │ │ +00269220: 4669 6c65 733a 3a20 2020 2020 2020 2020 Files:: │ │ │ │ +00269230: 2020 2020 2020 2020 2020 6c69 7374 2074 list t │ │ │ │ +00269240: 6865 206f 7065 6e20 6669 6c65 730a 2a20 he open files.* │ │ │ │ +00269250: 636f 6e6e 6563 7469 6f6e 436f 756e 743a connectionCount: │ │ │ │ +00269260: 3a20 2020 2020 2020 2020 2020 2020 7468 : th │ │ │ │ +00269270: 6520 6e75 6d62 6572 206f 6620 636f 6e6e e number of conn │ │ │ │ +00269280: 6563 7469 6f6e 730a 2d2d 2d2d 2d2d 2d2d ections.-------- │ │ │ │ 00269290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002692a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002692b0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -002692c0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -002692d0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -002692e0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -002692f0: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -00269300: 352b 6473 2f4d 322f 4d61 6361 756c 6179 5+ds/M2/Macaulay │ │ │ │ -00269310: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ -00269320: 756c 6179 3244 6f63 2f6f 765f 6669 6c65 ulay2Doc/ov_file │ │ │ │ -00269330: 732e 6d32 3a35 3039 3a30 2e0a 1f0a 4669 s.m2:509:0....Fi │ │ │ │ -00269340: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ -00269350: 2e69 6e66 6f2c 204e 6f64 653a 206f 7065 .info, Node: ope │ │ │ │ -00269360: 6e4c 6973 7465 6e65 725f 6c70 5374 7269 nListener_lpStri │ │ │ │ -00269370: 6e67 5f72 702c 204e 6578 743a 206f 7065 ng_rp, Next: ope │ │ │ │ -00269380: 6e49 6e5f 6c70 5374 7269 6e67 5f72 702c nIn_lpString_rp, │ │ │ │ -00269390: 2055 703a 2075 7369 6e67 2073 6f63 6b65 Up: using socke │ │ │ │ -002693a0: 7473 0a0a 6f70 656e 4c69 7374 656e 6572 ts..openListener │ │ │ │ -002693b0: 2853 7472 696e 6729 202d 2d20 6f70 656e (String) -- open │ │ │ │ -002693c0: 2061 2070 6f72 7420 666f 7220 6c69 7374 a port for list │ │ │ │ -002693d0: 656e 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a ening.********** │ │ │ │ -002693e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002693f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00269400: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 756e *******.. * Fun │ │ │ │ -00269410: 6374 696f 6e3a 202a 6e6f 7465 206f 7065 ction: *note ope │ │ │ │ -00269420: 6e4c 6973 7465 6e65 723a 206f 7065 6e4c nListener: openL │ │ │ │ -00269430: 6973 7465 6e65 725f 6c70 5374 7269 6e67 istener_lpString │ │ │ │ -00269440: 5f72 702c 0a20 202a 2055 7361 6765 3a20 _rp,. * Usage: │ │ │ │ -00269450: 0a20 2020 2020 2020 2066 203d 206f 7065 . f = ope │ │ │ │ -00269460: 6e4c 6973 7465 6e65 7220 730a 2020 2a20 nListener s. * │ │ │ │ -00269470: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -00269480: 732c 2061 202a 6e6f 7465 2073 7472 696e s, a *note strin │ │ │ │ -00269490: 673a 2053 7472 696e 672c 2c20 6f66 2074 g: String,, of t │ │ │ │ -002694a0: 6865 2066 6f72 6d20 2224 696e 7465 7266 he form "$interf │ │ │ │ -002694b0: 6163 653a 706f 7274 222e 2020 426f 7468 ace:port". Both │ │ │ │ -002694c0: 2070 6172 7473 0a20 2020 2020 2020 2061 parts. a │ │ │ │ -002694d0: 7265 206f 7074 696f 6e61 6c2e 2020 4966 re optional. If │ │ │ │ -002694e0: 2074 6865 2070 6f72 7420 6973 206f 6d69 the port is omi │ │ │ │ -002694f0: 7474 6564 2c20 7468 6520 636f 6c6f 6e20 tted, the colon │ │ │ │ -00269500: 6973 206f 7074 696f 6e61 6c2e 0a20 202a is optional.. * │ │ │ │ -00269510: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -00269520: 2a20 662c 2061 202a 6e6f 7465 2066 696c * f, a *note fil │ │ │ │ -00269530: 653a 2046 696c 652c 2c20 616e 206f 7065 e: File,, an ope │ │ │ │ -00269540: 6e20 6c69 7374 656e 6572 206f 6e20 7468 n listener on th │ │ │ │ -00269550: 6520 7370 6563 6966 6965 6420 696e 7465 e specified inte │ │ │ │ -00269560: 7266 6163 6520 6f66 0a20 2020 2020 2020 rface of. │ │ │ │ -00269570: 2074 6865 206c 6f63 616c 2068 6f73 7420 the local host │ │ │ │ -00269580: 6174 2074 6865 2073 7065 6369 6669 6564 at the specified │ │ │ │ -00269590: 2073 6572 7669 6365 2070 6f72 742e 2049 service port. I │ │ │ │ -002695a0: 6620 7468 6520 706f 7274 2069 7320 6f6d f the port is om │ │ │ │ -002695b0: 6974 7465 642c 0a20 2020 2020 2020 2069 itted,. i │ │ │ │ -002695c0: 7420 6973 2074 616b 656e 2074 6f20 6265 t is taken to be │ │ │ │ -002695d0: 2070 6f72 7420 3235 3030 2e20 2049 6620 port 2500. If │ │ │ │ -002695e0: 7468 6520 696e 7465 7266 6163 6520 6973 the interface is │ │ │ │ -002695f0: 206f 6d69 7474 6564 2c20 7468 6520 6c69 omitted, the li │ │ │ │ -00269600: 7374 656e 6572 0a20 2020 2020 2020 2061 stener. a │ │ │ │ -00269610: 6363 6570 7473 2063 6f6e 6e65 6374 696f ccepts connectio │ │ │ │ -00269620: 6e73 206f 6e20 616c 6c20 696e 7465 7266 ns on all interf │ │ │ │ -00269630: 6163 6573 2e0a 0a44 6573 6372 6970 7469 aces...Descripti │ │ │ │ -00269640: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00269650: 5573 6520 6f70 656e 496e 4f75 7420 6620 Use openInOut f │ │ │ │ -00269660: 746f 2061 6363 6570 7420 616e 2069 6e63 to accept an inc │ │ │ │ -00269670: 6f6d 696e 6720 636f 6e6e 6563 7469 6f6e oming connection │ │ │ │ -00269680: 206f 6e20 7468 6520 6c69 7374 656e 6572 on the listener │ │ │ │ -00269690: 2c20 7265 7475 726e 696e 6720 610a 6e65 , returning a.ne │ │ │ │ -002696a0: 7720 696e 7075 7420 6f75 7470 7574 2066 w input output f │ │ │ │ -002696b0: 696c 6520 7468 6174 2073 6572 7665 7320 ile that serves │ │ │ │ -002696c0: 6173 2074 6865 2063 6f6e 6e65 6374 696f as the connectio │ │ │ │ -002696d0: 6e2e 2020 5468 6520 6675 6e63 7469 6f6e n. The function │ │ │ │ -002696e0: 202a 6e6f 7465 0a69 7352 6561 6479 3a20 *note.isReady: │ │ │ │ -002696f0: 6973 5265 6164 795f 6c70 4669 6c65 5f72 isReady_lpFile_r │ │ │ │ -00269700: 702c 2063 616e 2062 6520 7573 6564 2074 p, can be used t │ │ │ │ -00269710: 6f20 6465 7465 726d 696e 6520 7768 6574 o determine whet │ │ │ │ -00269720: 6865 7220 616e 2069 6e63 6f6d 696e 670a her an incoming. │ │ │ │ -00269730: 636f 6e6e 6563 7469 6f6e 2068 6173 2061 connection has a │ │ │ │ -00269740: 7272 6976 6564 2c20 7769 7468 6f75 7420 rrived, without │ │ │ │ -00269750: 626c 6f63 6b69 6e67 2e0a 0a53 6565 2061 blocking...See a │ │ │ │ -00269760: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -00269770: 2a20 2a6e 6f74 6520 6f70 656e 496e 4f75 * *note openInOu │ │ │ │ -00269780: 743a 206f 7065 6e49 6e4f 7574 2c20 2d2d t: openInOut, -- │ │ │ │ -00269790: 206f 7065 6e20 616e 2069 6e70 7574 206f open an input o │ │ │ │ -002697a0: 7574 7075 7420 6669 6c65 0a20 202a 202a utput file. * * │ │ │ │ -002697b0: 6e6f 7465 2069 7352 6561 6479 3a20 6973 note isReady: is │ │ │ │ -002697c0: 5265 6164 795f 6c70 4669 6c65 5f72 702c Ready_lpFile_rp, │ │ │ │ -002697d0: 202d 2d20 7768 6574 6865 7220 6120 6669 -- whether a fi │ │ │ │ -002697e0: 6c65 2068 6173 2064 6174 6120 6176 6169 le has data avai │ │ │ │ -002697f0: 6c61 626c 6520 666f 720a 2020 2020 7265 lable for. re │ │ │ │ -00269800: 6164 696e 670a 0a57 6179 7320 746f 2075 ading..Ways to u │ │ │ │ -00269810: 7365 2074 6869 7320 6d65 7468 6f64 3a0a se this method:. │ │ │ │ -00269820: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00269830: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ -00269840: 6f74 6520 6f70 656e 4c69 7374 656e 6572 ote openListener │ │ │ │ -00269850: 2853 7472 696e 6729 3a20 6f70 656e 4c69 (String): openLi │ │ │ │ -00269860: 7374 656e 6572 5f6c 7053 7472 696e 675f stener_lpString_ │ │ │ │ -00269870: 7270 2c20 2d2d 206f 7065 6e20 6120 706f rp, -- open a po │ │ │ │ -00269880: 7274 2066 6f72 0a20 2020 206c 6973 7465 rt for. liste │ │ │ │ -00269890: 6e69 6e67 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d ning.----------- │ │ │ │ -002698a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002698b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002692b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002692c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002692d0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +002692e0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +002692f0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +00269300: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +00269310: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00269320: 3236 2e30 352b 6473 2f4d 322f 4d61 6361 26.05+ds/M2/Maca │ │ │ │ +00269330: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +00269340: 4d61 6361 756c 6179 3244 6f63 2f6f 765f Macaulay2Doc/ov_ │ │ │ │ +00269350: 6669 6c65 732e 6d32 3a35 3039 3a30 2e0a files.m2:509:0.. │ │ │ │ +00269360: 1f0a 4669 6c65 3a20 4d61 6361 756c 6179 ..File: Macaulay │ │ │ │ +00269370: 3244 6f63 2e69 6e66 6f2c 204e 6f64 653a 2Doc.info, Node: │ │ │ │ +00269380: 206f 7065 6e4c 6973 7465 6e65 725f 6c70 openListener_lp │ │ │ │ +00269390: 5374 7269 6e67 5f72 702c 204e 6578 743a String_rp, Next: │ │ │ │ +002693a0: 206f 7065 6e49 6e5f 6c70 5374 7269 6e67 openIn_lpString │ │ │ │ +002693b0: 5f72 702c 2055 703a 2075 7369 6e67 2073 _rp, Up: using s │ │ │ │ +002693c0: 6f63 6b65 7473 0a0a 6f70 656e 4c69 7374 ockets..openList │ │ │ │ +002693d0: 656e 6572 2853 7472 696e 6729 202d 2d20 ener(String) -- │ │ │ │ +002693e0: 6f70 656e 2061 2070 6f72 7420 666f 7220 open a port for │ │ │ │ +002693f0: 6c69 7374 656e 696e 670a 2a2a 2a2a 2a2a listening.****** │ │ │ │ +00269400: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00269410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00269420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00269430: 2046 756e 6374 696f 6e3a 202a 6e6f 7465 Function: *note │ │ │ │ +00269440: 206f 7065 6e4c 6973 7465 6e65 723a 206f openListener: o │ │ │ │ +00269450: 7065 6e4c 6973 7465 6e65 725f 6c70 5374 penListener_lpSt │ │ │ │ +00269460: 7269 6e67 5f72 702c 0a20 202a 2055 7361 ring_rp,. * Usa │ │ │ │ +00269470: 6765 3a20 0a20 2020 2020 2020 2066 203d ge: . f = │ │ │ │ +00269480: 206f 7065 6e4c 6973 7465 6e65 7220 730a openListener s. │ │ │ │ +00269490: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +002694a0: 2020 2a20 732c 2061 202a 6e6f 7465 2073 * s, a *note s │ │ │ │ +002694b0: 7472 696e 673a 2053 7472 696e 672c 2c20 tring: String,, │ │ │ │ +002694c0: 6f66 2074 6865 2066 6f72 6d20 2224 696e of the form "$in │ │ │ │ +002694d0: 7465 7266 6163 653a 706f 7274 222e 2020 terface:port". │ │ │ │ +002694e0: 426f 7468 2070 6172 7473 0a20 2020 2020 Both parts. │ │ │ │ +002694f0: 2020 2061 7265 206f 7074 696f 6e61 6c2e are optional. │ │ │ │ +00269500: 2020 4966 2074 6865 2070 6f72 7420 6973 If the port is │ │ │ │ +00269510: 206f 6d69 7474 6564 2c20 7468 6520 636f omitted, the co │ │ │ │ +00269520: 6c6f 6e20 6973 206f 7074 696f 6e61 6c2e lon is optional. │ │ │ │ +00269530: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +00269540: 2020 2020 2a20 662c 2061 202a 6e6f 7465 * f, a *note │ │ │ │ +00269550: 2066 696c 653a 2046 696c 652c 2c20 616e file: File,, an │ │ │ │ +00269560: 206f 7065 6e20 6c69 7374 656e 6572 206f open listener o │ │ │ │ +00269570: 6e20 7468 6520 7370 6563 6966 6965 6420 n the specified │ │ │ │ +00269580: 696e 7465 7266 6163 6520 6f66 0a20 2020 interface of. │ │ │ │ +00269590: 2020 2020 2074 6865 206c 6f63 616c 2068 the local h │ │ │ │ +002695a0: 6f73 7420 6174 2074 6865 2073 7065 6369 ost at the speci │ │ │ │ +002695b0: 6669 6564 2073 6572 7669 6365 2070 6f72 fied service por │ │ │ │ +002695c0: 742e 2049 6620 7468 6520 706f 7274 2069 t. If the port i │ │ │ │ +002695d0: 7320 6f6d 6974 7465 642c 0a20 2020 2020 s omitted,. │ │ │ │ +002695e0: 2020 2069 7420 6973 2074 616b 656e 2074 it is taken t │ │ │ │ +002695f0: 6f20 6265 2070 6f72 7420 3235 3030 2e20 o be port 2500. │ │ │ │ +00269600: 2049 6620 7468 6520 696e 7465 7266 6163 If the interfac │ │ │ │ +00269610: 6520 6973 206f 6d69 7474 6564 2c20 7468 e is omitted, th │ │ │ │ +00269620: 6520 6c69 7374 656e 6572 0a20 2020 2020 e listener. │ │ │ │ +00269630: 2020 2061 6363 6570 7473 2063 6f6e 6e65 accepts conne │ │ │ │ +00269640: 6374 696f 6e73 206f 6e20 616c 6c20 696e ctions on all in │ │ │ │ +00269650: 7465 7266 6163 6573 2e0a 0a44 6573 6372 terfaces...Descr │ │ │ │ +00269660: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +00269670: 3d3d 0a0a 5573 6520 6f70 656e 496e 4f75 ==..Use openInOu │ │ │ │ +00269680: 7420 6620 746f 2061 6363 6570 7420 616e t f to accept an │ │ │ │ +00269690: 2069 6e63 6f6d 696e 6720 636f 6e6e 6563 incoming connec │ │ │ │ +002696a0: 7469 6f6e 206f 6e20 7468 6520 6c69 7374 tion on the list │ │ │ │ +002696b0: 656e 6572 2c20 7265 7475 726e 696e 6720 ener, returning │ │ │ │ +002696c0: 610a 6e65 7720 696e 7075 7420 6f75 7470 a.new input outp │ │ │ │ +002696d0: 7574 2066 696c 6520 7468 6174 2073 6572 ut file that ser │ │ │ │ +002696e0: 7665 7320 6173 2074 6865 2063 6f6e 6e65 ves as the conne │ │ │ │ +002696f0: 6374 696f 6e2e 2020 5468 6520 6675 6e63 ction. The func │ │ │ │ +00269700: 7469 6f6e 202a 6e6f 7465 0a69 7352 6561 tion *note.isRea │ │ │ │ +00269710: 6479 3a20 6973 5265 6164 795f 6c70 4669 dy: isReady_lpFi │ │ │ │ +00269720: 6c65 5f72 702c 2063 616e 2062 6520 7573 le_rp, can be us │ │ │ │ +00269730: 6564 2074 6f20 6465 7465 726d 696e 6520 ed to determine │ │ │ │ +00269740: 7768 6574 6865 7220 616e 2069 6e63 6f6d whether an incom │ │ │ │ +00269750: 696e 670a 636f 6e6e 6563 7469 6f6e 2068 ing.connection h │ │ │ │ +00269760: 6173 2061 7272 6976 6564 2c20 7769 7468 as arrived, with │ │ │ │ +00269770: 6f75 7420 626c 6f63 6b69 6e67 2e0a 0a53 out blocking...S │ │ │ │ +00269780: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +00269790: 0a0a 2020 2a20 2a6e 6f74 6520 6f70 656e .. * *note open │ │ │ │ +002697a0: 496e 4f75 743a 206f 7065 6e49 6e4f 7574 InOut: openInOut │ │ │ │ +002697b0: 2c20 2d2d 206f 7065 6e20 616e 2069 6e70 , -- open an inp │ │ │ │ +002697c0: 7574 206f 7574 7075 7420 6669 6c65 0a20 ut output file. │ │ │ │ +002697d0: 202a 202a 6e6f 7465 2069 7352 6561 6479 * *note isReady │ │ │ │ +002697e0: 3a20 6973 5265 6164 795f 6c70 4669 6c65 : isReady_lpFile │ │ │ │ +002697f0: 5f72 702c 202d 2d20 7768 6574 6865 7220 _rp, -- whether │ │ │ │ +00269800: 6120 6669 6c65 2068 6173 2064 6174 6120 a file has data │ │ │ │ +00269810: 6176 6169 6c61 626c 6520 666f 720a 2020 available for. │ │ │ │ +00269820: 2020 7265 6164 696e 670a 0a57 6179 7320 reading..Ways │ │ │ │ +00269830: 746f 2075 7365 2074 6869 7320 6d65 7468 to use this meth │ │ │ │ +00269840: 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d od:.============ │ │ │ │ +00269850: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +00269860: 2a20 2a6e 6f74 6520 6f70 656e 4c69 7374 * *note openList │ │ │ │ +00269870: 656e 6572 2853 7472 696e 6729 3a20 6f70 ener(String): op │ │ │ │ +00269880: 656e 4c69 7374 656e 6572 5f6c 7053 7472 enListener_lpStr │ │ │ │ +00269890: 696e 675f 7270 2c20 2d2d 206f 7065 6e20 ing_rp, -- open │ │ │ │ +002698a0: 6120 706f 7274 2066 6f72 0a20 2020 206c a port for. l │ │ │ │ +002698b0: 6973 7465 6e69 6e67 0a2d 2d2d 2d2d 2d2d istening.------- │ │ │ │ 002698c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002698d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002698e0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -002698f0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00269900: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00269910: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00269920: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -00269930: 3035 2b64 732f 4d32 2f4d 6163 6175 6c61 05+ds/M2/Macaula │ │ │ │ -00269940: 7932 2f70 6163 6b61 6765 732f 0a4d 6163 y2/packages/.Mac │ │ │ │ -00269950: 6175 6c61 7932 446f 632f 6f76 5f73 7973 aulay2Doc/ov_sys │ │ │ │ -00269960: 7465 6d2e 6d32 3a34 3232 3a30 2e0a 1f0a tem.m2:422:0.... │ │ │ │ -00269970: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ -00269980: 6f63 2e69 6e66 6f2c 204e 6f64 653a 206f oc.info, Node: o │ │ │ │ -00269990: 7065 6e49 6e5f 6c70 5374 7269 6e67 5f72 penIn_lpString_r │ │ │ │ -002699a0: 702c 204e 6578 743a 206f 7065 6e49 6e4f p, Next: openInO │ │ │ │ -002699b0: 7574 2c20 5072 6576 3a20 6f70 656e 4c69 ut, Prev: openLi │ │ │ │ -002699c0: 7374 656e 6572 5f6c 7053 7472 696e 675f stener_lpString_ │ │ │ │ -002699d0: 7270 2c20 5570 3a20 7573 696e 6720 736f rp, Up: using so │ │ │ │ -002699e0: 636b 6574 730a 0a6f 7065 6e49 6e28 5374 ckets..openIn(St │ │ │ │ -002699f0: 7269 6e67 2920 2d2d 206f 7065 6e20 616e ring) -- open an │ │ │ │ -00269a00: 2069 6e70 7574 2066 696c 650a 2a2a 2a2a input file.**** │ │ │ │ -00269a10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00269a20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00269a30: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20 .. * Function: │ │ │ │ -00269a40: 2a6e 6f74 6520 6f70 656e 496e 3a20 6f70 *note openIn: op │ │ │ │ -00269a50: 656e 496e 5f6c 7053 7472 696e 675f 7270 enIn_lpString_rp │ │ │ │ -00269a60: 2c0a 2020 2a20 5573 6167 653a 200a 2020 ,. * Usage: . │ │ │ │ -00269a70: 2020 2020 2020 6f70 656e 496e 2066 6e0a openIn fn. │ │ │ │ -00269a80: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -00269a90: 2020 2a20 666e 2c20 6120 2a6e 6f74 6520 * fn, a *note │ │ │ │ -00269aa0: 7374 7269 6e67 3a20 5374 7269 6e67 2c0a string: String,. │ │ │ │ -00269ab0: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -00269ac0: 2020 202a 2061 202a 6e6f 7465 2066 696c * a *note fil │ │ │ │ -00269ad0: 653a 2046 696c 652c 2c20 616e 206f 7065 e: File,, an ope │ │ │ │ -00269ae0: 6e20 696e 7075 7420 6669 6c65 2077 686f n input file who │ │ │ │ -00269af0: 7365 2066 696c 656e 616d 6520 6973 2066 se filename is f │ │ │ │ -00269b00: 6e2e 2046 696c 656e 616d 6573 0a20 2020 n. Filenames. │ │ │ │ -00269b10: 2020 2020 2073 7461 7274 696e 6720 7769 starting wi │ │ │ │ -00269b20: 7468 2021 206f 7220 7769 7468 2024 2061 th ! or with $ a │ │ │ │ -00269b30: 7265 2074 7265 6174 6564 2073 7065 6369 re treated speci │ │ │ │ -00269b40: 616c 6c79 2c20 7365 6520 2a6e 6f74 6520 ally, see *note │ │ │ │ -00269b50: 6f70 656e 496e 4f75 743a 0a20 2020 2020 openInOut:. │ │ │ │ -00269b60: 2020 206f 7065 6e49 6e4f 7574 2c2e 0a0a openInOut,... │ │ │ │ -00269b70: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -00269b80: 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d =======..+------ │ │ │ │ -00269b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269bb0: 2d2d 2b0a 7c69 3120 3a20 2274 6573 742d --+.|i1 : "test- │ │ │ │ -00269bc0: 6669 6c65 2220 3c3c 2022 6869 2074 6865 file" << "hi the │ │ │ │ -00269bd0: 7265 2220 3c3c 2063 6c6f 7365 3b7c 0a2b re" << close;|.+ │ │ │ │ -00269be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269c00: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -00269c10: 6720 3d20 6f70 656e 496e 2022 7465 7374 g = openIn "test │ │ │ │ -00269c20: 2d66 696c 6522 2020 2020 2020 2020 2020 -file" │ │ │ │ -00269c30: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00269c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269c50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00269c60: 7c6f 3220 3d20 7465 7374 2d66 696c 6520 |o2 = test-file │ │ │ │ +002698e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002698f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269900: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +00269910: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +00269920: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +00269930: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00269940: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00269950: 2e32 362e 3035 2b64 732f 4d32 2f4d 6163 .26.05+ds/M2/Mac │ │ │ │ +00269960: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00269970: 0a4d 6163 6175 6c61 7932 446f 632f 6f76 .Macaulay2Doc/ov │ │ │ │ +00269980: 5f73 7973 7465 6d2e 6d32 3a34 3232 3a30 _system.m2:422:0 │ │ │ │ +00269990: 2e0a 1f0a 4669 6c65 3a20 4d61 6361 756c ....File: Macaul │ │ │ │ +002699a0: 6179 3244 6f63 2e69 6e66 6f2c 204e 6f64 ay2Doc.info, Nod │ │ │ │ +002699b0: 653a 206f 7065 6e49 6e5f 6c70 5374 7269 e: openIn_lpStri │ │ │ │ +002699c0: 6e67 5f72 702c 204e 6578 743a 206f 7065 ng_rp, Next: ope │ │ │ │ +002699d0: 6e49 6e4f 7574 2c20 5072 6576 3a20 6f70 nInOut, Prev: op │ │ │ │ +002699e0: 656e 4c69 7374 656e 6572 5f6c 7053 7472 enListener_lpStr │ │ │ │ +002699f0: 696e 675f 7270 2c20 5570 3a20 7573 696e ing_rp, Up: usin │ │ │ │ +00269a00: 6720 736f 636b 6574 730a 0a6f 7065 6e49 g sockets..openI │ │ │ │ +00269a10: 6e28 5374 7269 6e67 2920 2d2d 206f 7065 n(String) -- ope │ │ │ │ +00269a20: 6e20 616e 2069 6e70 7574 2066 696c 650a n an input file. │ │ │ │ +00269a30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00269a40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00269a50: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469 ****.. * Functi │ │ │ │ +00269a60: 6f6e 3a20 2a6e 6f74 6520 6f70 656e 496e on: *note openIn │ │ │ │ +00269a70: 3a20 6f70 656e 496e 5f6c 7053 7472 696e : openIn_lpStrin │ │ │ │ +00269a80: 675f 7270 2c0a 2020 2a20 5573 6167 653a g_rp,. * Usage: │ │ │ │ +00269a90: 200a 2020 2020 2020 2020 6f70 656e 496e . openIn │ │ │ │ +00269aa0: 2066 6e0a 2020 2a20 496e 7075 7473 3a0a fn. * Inputs:. │ │ │ │ +00269ab0: 2020 2020 2020 2a20 666e 2c20 6120 2a6e * fn, a *n │ │ │ │ +00269ac0: 6f74 6520 7374 7269 6e67 3a20 5374 7269 ote string: Stri │ │ │ │ +00269ad0: 6e67 2c0a 2020 2a20 4f75 7470 7574 733a ng,. * Outputs: │ │ │ │ +00269ae0: 0a20 2020 2020 202a 2061 202a 6e6f 7465 . * a *note │ │ │ │ +00269af0: 2066 696c 653a 2046 696c 652c 2c20 616e file: File,, an │ │ │ │ +00269b00: 206f 7065 6e20 696e 7075 7420 6669 6c65 open input file │ │ │ │ +00269b10: 2077 686f 7365 2066 696c 656e 616d 6520 whose filename │ │ │ │ +00269b20: 6973 2066 6e2e 2046 696c 656e 616d 6573 is fn. Filenames │ │ │ │ +00269b30: 0a20 2020 2020 2020 2073 7461 7274 696e . startin │ │ │ │ +00269b40: 6720 7769 7468 2021 206f 7220 7769 7468 g with ! or with │ │ │ │ +00269b50: 2024 2061 7265 2074 7265 6174 6564 2073 $ are treated s │ │ │ │ +00269b60: 7065 6369 616c 6c79 2c20 7365 6520 2a6e pecially, see *n │ │ │ │ +00269b70: 6f74 6520 6f70 656e 496e 4f75 743a 0a20 ote openInOut:. │ │ │ │ +00269b80: 2020 2020 2020 206f 7065 6e49 6e4f 7574 openInOut │ │ │ │ +00269b90: 2c2e 0a0a 4465 7363 7269 7074 696f 6e0a ,...Description. │ │ │ │ +00269ba0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d ===========..+-- │ │ │ │ +00269bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269bd0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 2274 ------+.|i1 : "t │ │ │ │ +00269be0: 6573 742d 6669 6c65 2220 3c3c 2022 6869 est-file" << "hi │ │ │ │ +00269bf0: 2074 6865 7265 2220 3c3c 2063 6c6f 7365 there" << close │ │ │ │ +00269c00: 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ;|.+------------ │ │ │ │ +00269c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00269c30: 3220 3a20 6720 3d20 6f70 656e 496e 2022 2 : g = openIn " │ │ │ │ +00269c40: 7465 7374 2d66 696c 6522 2020 2020 2020 test-file" │ │ │ │ +00269c50: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00269c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00269c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269c80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00269c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269cb0: 2020 2020 7c0a 7c6f 3220 3a20 4669 6c65 |.|o2 : File │ │ │ │ +00269c80: 2020 7c0a 7c6f 3220 3d20 7465 7374 2d66 |.|o2 = test-f │ │ │ │ +00269c90: 696c 6520 2020 2020 2020 2020 2020 2020 ile │ │ │ │ +00269ca0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00269cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00269cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269cd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00269ce0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00269cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ -00269d10: 3a20 6669 6c65 4c65 6e67 7468 2067 2020 : fileLength g │ │ │ │ -00269d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269d30: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00269d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269d60: 7c0a 7c6f 3320 3d20 3820 2020 2020 2020 |.|o3 = 8 │ │ │ │ +00269cd0: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ +00269ce0: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ +00269cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00269d00: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00269d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00269d30: 7c69 3320 3a20 6669 6c65 4c65 6e67 7468 |i3 : fileLength │ │ │ │ +00269d40: 2067 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ +00269d50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00269d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00269d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269d80: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00269d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269db0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6174 ------+.|i4 : at │ │ │ │ -00269dc0: 456e 644f 6646 696c 6520 6720 2020 2020 EndOfFile g │ │ │ │ -00269dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269de0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00269d80: 2020 2020 7c0a 7c6f 3320 3d20 3820 2020 |.|o3 = 8 │ │ │ │ +00269d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00269da0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00269db0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00269dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ +00269de0: 3a20 6174 456e 644f 6646 696c 6520 6720 : atEndOfFile g │ │ │ │ 00269df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269e00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00269e10: 3420 3d20 6661 6c73 6520 2020 2020 2020 4 = false │ │ │ │ +00269e00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00269e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00269e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269e30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00269e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269e60: 2d2d 2b0a 7c69 3520 3a20 7265 6164 2067 --+.|i5 : read g │ │ │ │ -00269e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269e80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00269e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00269e30: 7c0a 7c6f 3420 3d20 6661 6c73 6520 2020 |.|o4 = false │ │ │ │ +00269e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00269e50: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00269e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269e80: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7265 ------+.|i5 : re │ │ │ │ +00269e90: 6164 2067 2020 2020 2020 2020 2020 2020 ad g │ │ │ │ 00269ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269eb0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ -00269ec0: 6869 2074 6865 7265 2020 2020 2020 2020 hi there │ │ │ │ -00269ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269ee0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00269ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00269f10: 7c69 3620 3a20 6174 456e 644f 6646 696c |i6 : atEndOfFil │ │ │ │ -00269f20: 6520 6720 2020 2020 2020 2020 2020 2020 e g │ │ │ │ -00269f30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00269f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269f60: 2020 2020 7c0a 7c6f 3620 3d20 7472 7565 |.|o6 = true │ │ │ │ +00269eb0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00269ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00269ed0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00269ee0: 3520 3d20 6869 2074 6865 7265 2020 2020 5 = hi there │ │ │ │ +00269ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00269f00: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00269f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269f30: 2d2d 2b0a 7c69 3620 3a20 6174 456e 644f --+.|i6 : atEndO │ │ │ │ +00269f40: 6646 696c 6520 6720 2020 2020 2020 2020 fFile g │ │ │ │ +00269f50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00269f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00269f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269f80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00269f90: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00269fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 ----------+.|i7 │ │ │ │ -00269fc0: 3a20 636c 6f73 6520 6720 2020 2020 2020 : close g │ │ │ │ -00269fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00269fe0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00269f80: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ +00269f90: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ +00269fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00269fb0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00269fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00269fe0: 7c69 3720 3a20 636c 6f73 6520 6720 2020 |i7 : close g │ │ │ │ 00269ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a010: 7c0a 7c6f 3720 3d20 7465 7374 2d66 696c |.|o7 = test-fil │ │ │ │ -0026a020: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ -0026a030: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0026a040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a060: 2020 2020 2020 7c0a 7c6f 3720 3a20 4669 |.|o7 : Fi │ │ │ │ -0026a070: 6c65 2020 2020 2020 2020 2020 2020 2020 le │ │ │ │ -0026a080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a090: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0026a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026a0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0026a0c0: 3820 3a20 7265 6d6f 7665 4669 6c65 2022 8 : removeFile " │ │ │ │ -0026a0d0: 7465 7374 2d66 696c 6522 2020 2020 2020 test-file" │ │ │ │ -0026a0e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0026a0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026a100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026a110: 2d2d 2b0a 0a41 2066 696c 656e 616d 6520 --+..A filename │ │ │ │ -0026a120: 7374 6172 7469 6e67 2077 6974 6820 7e2f starting with ~/ │ │ │ │ -0026a130: 2077 696c 6c20 6861 7665 2074 6865 2074 will have the t │ │ │ │ -0026a140: 696c 6465 2072 6570 6c61 6365 6420 6279 ilde replaced by │ │ │ │ -0026a150: 2074 6865 2075 7365 7227 7320 686f 6d65 the user's home │ │ │ │ -0026a160: 0a64 6972 6563 746f 7279 2e0a 0a53 6565 .directory...See │ │ │ │ -0026a170: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -0026a180: 2020 2a20 2a6e 6f74 6520 6f70 656e 4f75 * *note openOu │ │ │ │ -0026a190: 743a 206f 7065 6e4f 7574 5f6c 7053 7472 t: openOut_lpStr │ │ │ │ -0026a1a0: 696e 675f 7270 2c20 2d2d 206f 7065 6e20 ing_rp, -- open │ │ │ │ -0026a1b0: 616e 206f 7574 7075 7420 6669 6c65 0a20 an output file. │ │ │ │ -0026a1c0: 202a 202a 6e6f 7465 206f 7065 6e4f 7574 * *note openOut │ │ │ │ -0026a1d0: 4170 7065 6e64 3a20 6f70 656e 4f75 7441 Append: openOutA │ │ │ │ -0026a1e0: 7070 656e 645f 6c70 5374 7269 6e67 5f72 ppend_lpString_r │ │ │ │ -0026a1f0: 702c 202d 2d20 6f70 656e 2061 6e20 6f75 p, -- open an ou │ │ │ │ -0026a200: 7470 7574 2066 696c 6520 666f 720a 2020 tput file for. │ │ │ │ -0026a210: 2020 6170 7065 6e64 696e 670a 2020 2a20 appending. * │ │ │ │ -0026a220: 2a6e 6f74 6520 6f70 656e 496e 4f75 743a *note openInOut: │ │ │ │ -0026a230: 206f 7065 6e49 6e4f 7574 2c20 2d2d 206f openInOut, -- o │ │ │ │ -0026a240: 7065 6e20 616e 2069 6e70 7574 206f 7574 pen an input out │ │ │ │ -0026a250: 7075 7420 6669 6c65 0a20 202a 202a 6e6f put file. * *no │ │ │ │ -0026a260: 7465 2066 696c 654c 656e 6774 683a 2066 te fileLength: f │ │ │ │ -0026a270: 696c 654c 656e 6774 682c 202d 2d20 7468 ileLength, -- th │ │ │ │ -0026a280: 6520 6c65 6e67 7468 206f 6620 6120 6669 e length of a fi │ │ │ │ -0026a290: 6c65 0a20 202a 202a 6e6f 7465 2072 6561 le. * *note rea │ │ │ │ -0026a2a0: 643a 2072 6561 642c 202d 2d20 7265 6164 d: read, -- read │ │ │ │ -0026a2b0: 2066 726f 6d20 6120 6669 6c65 0a20 202a from a file. * │ │ │ │ -0026a2c0: 202a 6e6f 7465 2063 6c6f 7365 3a20 636c *note close: cl │ │ │ │ -0026a2d0: 6f73 652c 202d 2d20 636c 6f73 6520 6120 ose, -- close a │ │ │ │ -0026a2e0: 6669 6c65 0a20 202a 202a 6e6f 7465 2061 file. * *note a │ │ │ │ -0026a2f0: 7445 6e64 4f66 4669 6c65 3a20 6174 456e tEndOfFile: atEn │ │ │ │ -0026a300: 644f 6646 696c 655f 6c70 4669 6c65 5f72 dOfFile_lpFile_r │ │ │ │ -0026a310: 702c 202d 2d20 7465 7374 2066 6f72 2065 p, -- test for e │ │ │ │ -0026a320: 6e64 206f 6620 6669 6c65 0a20 202a 202a nd of file. * * │ │ │ │ -0026a330: 6e6f 7465 2046 696c 6520 3c3c 2054 6869 note File << Thi │ │ │ │ -0026a340: 6e67 3a20 7072 696e 7469 6e67 2074 6f20 ng: printing to │ │ │ │ -0026a350: 6120 6669 6c65 2c20 2d2d 2070 7269 6e74 a file, -- print │ │ │ │ -0026a360: 2074 6f20 6120 6669 6c65 0a0a 5761 7973 to a file..Ways │ │ │ │ -0026a370: 2074 6f20 7573 6520 7468 6973 206d 6574 to use this met │ │ │ │ -0026a380: 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d hod:.=========== │ │ │ │ -0026a390: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -0026a3a0: 202a 202a 6e6f 7465 206f 7065 6e49 6e28 * *note openIn( │ │ │ │ -0026a3b0: 5374 7269 6e67 293a 206f 7065 6e49 6e5f String): openIn_ │ │ │ │ -0026a3c0: 6c70 5374 7269 6e67 5f72 702c 202d 2d20 lpString_rp, -- │ │ │ │ -0026a3d0: 6f70 656e 2061 6e20 696e 7075 7420 6669 open an input fi │ │ │ │ -0026a3e0: 6c65 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d le.------------- │ │ │ │ -0026a3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026a400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a000: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0026a010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a030: 2020 2020 7c0a 7c6f 3720 3d20 7465 7374 |.|o7 = test │ │ │ │ +0026a040: 2d66 696c 6520 2020 2020 2020 2020 2020 -file │ │ │ │ +0026a050: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026a060: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0026a070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a080: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ +0026a090: 3a20 4669 6c65 2020 2020 2020 2020 2020 : File │ │ │ │ +0026a0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a0b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0026a0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a0e0: 2b0a 7c69 3820 3a20 7265 6d6f 7665 4669 +.|i8 : removeFi │ │ │ │ +0026a0f0: 6c65 2022 7465 7374 2d66 696c 6522 2020 le "test-file" │ │ │ │ +0026a100: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0026a110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a130: 2d2d 2d2d 2d2d 2b0a 0a41 2066 696c 656e ------+..A filen │ │ │ │ +0026a140: 616d 6520 7374 6172 7469 6e67 2077 6974 ame starting wit │ │ │ │ +0026a150: 6820 7e2f 2077 696c 6c20 6861 7665 2074 h ~/ will have t │ │ │ │ +0026a160: 6865 2074 696c 6465 2072 6570 6c61 6365 he tilde replace │ │ │ │ +0026a170: 6420 6279 2074 6865 2075 7365 7227 7320 d by the user's │ │ │ │ +0026a180: 686f 6d65 0a64 6972 6563 746f 7279 2e0a home.directory.. │ │ │ │ +0026a190: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ +0026a1a0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6f70 ==.. * *note op │ │ │ │ +0026a1b0: 656e 4f75 743a 206f 7065 6e4f 7574 5f6c enOut: openOut_l │ │ │ │ +0026a1c0: 7053 7472 696e 675f 7270 2c20 2d2d 206f pString_rp, -- o │ │ │ │ +0026a1d0: 7065 6e20 616e 206f 7574 7075 7420 6669 pen an output fi │ │ │ │ +0026a1e0: 6c65 0a20 202a 202a 6e6f 7465 206f 7065 le. * *note ope │ │ │ │ +0026a1f0: 6e4f 7574 4170 7065 6e64 3a20 6f70 656e nOutAppend: open │ │ │ │ +0026a200: 4f75 7441 7070 656e 645f 6c70 5374 7269 OutAppend_lpStri │ │ │ │ +0026a210: 6e67 5f72 702c 202d 2d20 6f70 656e 2061 ng_rp, -- open a │ │ │ │ +0026a220: 6e20 6f75 7470 7574 2066 696c 6520 666f n output file fo │ │ │ │ +0026a230: 720a 2020 2020 6170 7065 6e64 696e 670a r. appending. │ │ │ │ +0026a240: 2020 2a20 2a6e 6f74 6520 6f70 656e 496e * *note openIn │ │ │ │ +0026a250: 4f75 743a 206f 7065 6e49 6e4f 7574 2c20 Out: openInOut, │ │ │ │ +0026a260: 2d2d 206f 7065 6e20 616e 2069 6e70 7574 -- open an input │ │ │ │ +0026a270: 206f 7574 7075 7420 6669 6c65 0a20 202a output file. * │ │ │ │ +0026a280: 202a 6e6f 7465 2066 696c 654c 656e 6774 *note fileLengt │ │ │ │ +0026a290: 683a 2066 696c 654c 656e 6774 682c 202d h: fileLength, - │ │ │ │ +0026a2a0: 2d20 7468 6520 6c65 6e67 7468 206f 6620 - the length of │ │ │ │ +0026a2b0: 6120 6669 6c65 0a20 202a 202a 6e6f 7465 a file. * *note │ │ │ │ +0026a2c0: 2072 6561 643a 2072 6561 642c 202d 2d20 read: read, -- │ │ │ │ +0026a2d0: 7265 6164 2066 726f 6d20 6120 6669 6c65 read from a file │ │ │ │ +0026a2e0: 0a20 202a 202a 6e6f 7465 2063 6c6f 7365 . * *note close │ │ │ │ +0026a2f0: 3a20 636c 6f73 652c 202d 2d20 636c 6f73 : close, -- clos │ │ │ │ +0026a300: 6520 6120 6669 6c65 0a20 202a 202a 6e6f e a file. * *no │ │ │ │ +0026a310: 7465 2061 7445 6e64 4f66 4669 6c65 3a20 te atEndOfFile: │ │ │ │ +0026a320: 6174 456e 644f 6646 696c 655f 6c70 4669 atEndOfFile_lpFi │ │ │ │ +0026a330: 6c65 5f72 702c 202d 2d20 7465 7374 2066 le_rp, -- test f │ │ │ │ +0026a340: 6f72 2065 6e64 206f 6620 6669 6c65 0a20 or end of file. │ │ │ │ +0026a350: 202a 202a 6e6f 7465 2046 696c 6520 3c3c * *note File << │ │ │ │ +0026a360: 2054 6869 6e67 3a20 7072 696e 7469 6e67 Thing: printing │ │ │ │ +0026a370: 2074 6f20 6120 6669 6c65 2c20 2d2d 2070 to a file, -- p │ │ │ │ +0026a380: 7269 6e74 2074 6f20 6120 6669 6c65 0a0a rint to a file.. │ │ │ │ +0026a390: 5761 7973 2074 6f20 7573 6520 7468 6973 Ways to use this │ │ │ │ +0026a3a0: 206d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d method:.======= │ │ │ │ +0026a3b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0026a3c0: 3d0a 0a20 202a 202a 6e6f 7465 206f 7065 =.. * *note ope │ │ │ │ +0026a3d0: 6e49 6e28 5374 7269 6e67 293a 206f 7065 nIn(String): ope │ │ │ │ +0026a3e0: 6e49 6e5f 6c70 5374 7269 6e67 5f72 702c nIn_lpString_rp, │ │ │ │ +0026a3f0: 202d 2d20 6f70 656e 2061 6e20 696e 7075 -- open an inpu │ │ │ │ +0026a400: 7420 6669 6c65 0a2d 2d2d 2d2d 2d2d 2d2d t file.--------- │ │ │ │ 0026a410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0026a420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026a430: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -0026a440: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -0026a450: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -0026a460: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -0026a470: 6163 6175 6c61 7932 2d31 2e32 362e 3035 acaulay2-1.26.05 │ │ │ │ -0026a480: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0026a490: 2f70 6163 6b61 6765 732f 0a4d 6163 6175 /packages/.Macau │ │ │ │ -0026a4a0: 6c61 7932 446f 632f 6f76 5f73 7973 7465 lay2Doc/ov_syste │ │ │ │ -0026a4b0: 6d2e 6d32 3a34 3434 3a30 2e0a 1f0a 4669 m.m2:444:0....Fi │ │ │ │ -0026a4c0: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ -0026a4d0: 2e69 6e66 6f2c 204e 6f64 653a 206f 7065 .info, Node: ope │ │ │ │ -0026a4e0: 6e49 6e4f 7574 2c20 4e65 7874 3a20 6f70 nInOut, Next: op │ │ │ │ -0026a4f0: 656e 4f75 745f 6c70 5374 7269 6e67 5f72 enOut_lpString_r │ │ │ │ -0026a500: 702c 2050 7265 763a 206f 7065 6e49 6e5f p, Prev: openIn_ │ │ │ │ -0026a510: 6c70 5374 7269 6e67 5f72 702c 2055 703a lpString_rp, Up: │ │ │ │ -0026a520: 2075 7369 6e67 2073 6f63 6b65 7473 0a0a using sockets.. │ │ │ │ -0026a530: 6f70 656e 496e 4f75 7420 2d2d 206f 7065 openInOut -- ope │ │ │ │ -0026a540: 6e20 616e 2069 6e70 7574 206f 7574 7075 n an input outpu │ │ │ │ -0026a550: 7420 6669 6c65 0a2a 2a2a 2a2a 2a2a 2a2a t file.********* │ │ │ │ -0026a560: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026a570: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ -0026a580: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -0026a590: 2020 206f 7065 6e49 6e4f 7574 2066 0a20 openInOut f. │ │ │ │ -0026a5a0: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ -0026a5b0: 202a 2066 2c20 6120 2a6e 6f74 6520 7374 * f, a *note st │ │ │ │ -0026a5c0: 7269 6e67 3a20 5374 7269 6e67 2c20 6f72 ring: String, or │ │ │ │ -0026a5d0: 2061 202a 6e6f 7465 2066 696c 653a 2046 a *note file: F │ │ │ │ -0026a5e0: 696c 652c 0a20 202a 204f 7574 7075 7473 ile,. * Outputs │ │ │ │ -0026a5f0: 3a0a 2020 2020 2020 2a20 616e 206f 7065 :. * an ope │ │ │ │ -0026a600: 6e20 696e 7075 7420 6f75 7470 7574 2066 n input output f │ │ │ │ -0026a610: 696c 650a 0a44 6573 6372 6970 7469 6f6e ile..Description │ │ │ │ -0026a620: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 .===========..Th │ │ │ │ -0026a630: 6572 6520 6172 6520 7661 7269 6f75 7320 ere are various │ │ │ │ -0026a640: 6f70 7469 6f6e 7320 666f 7220 7468 6520 options for the │ │ │ │ -0026a650: 6172 6775 6d65 6e74 2066 2e0a 0a20 202a argument f... * │ │ │ │ -0026a660: 2061 2073 7472 696e 6720 6e6f 7420 7374 a string not st │ │ │ │ -0026a670: 6172 7469 6e67 2077 6974 6820 2120 6f72 arting with ! or │ │ │ │ -0026a680: 2024 3a20 7468 6520 7374 7269 6e67 2069 $: the string i │ │ │ │ -0026a690: 7320 7461 6b65 6e20 6173 2074 6865 206e s taken as the n │ │ │ │ -0026a6a0: 616d 6520 6f66 2061 6e0a 2020 2020 696e ame of an. in │ │ │ │ -0026a6b0: 7075 7420 6f75 7470 7574 2066 696c 6520 put output file │ │ │ │ -0026a6c0: 746f 206f 7065 6e2e 2020 466f 7220 6578 to open. For ex │ │ │ │ -0026a6d0: 616d 706c 652c 2069 6e20 556e 6978 2c20 ample, in Unix, │ │ │ │ -0026a6e0: 6974 206d 6967 6874 2062 6520 6120 6e61 it might be a na │ │ │ │ -0026a6f0: 6d65 6420 7069 7065 2e0a 2020 2020 4120 med pipe.. A │ │ │ │ -0026a700: 6669 6c65 6e61 6d65 2073 7461 7274 696e filename startin │ │ │ │ -0026a710: 6720 7769 7468 207e 2f20 7769 6c6c 2068 g with ~/ will h │ │ │ │ -0026a720: 6176 6520 7468 6520 7469 6c64 6520 7265 ave the tilde re │ │ │ │ -0026a730: 706c 6163 6564 2062 7920 7468 6520 7573 placed by the us │ │ │ │ -0026a740: 6572 2773 2068 6f6d 650a 2020 2020 6469 er's home. di │ │ │ │ -0026a750: 7265 6374 6f72 792e 0a20 202a 2061 2073 rectory.. * a s │ │ │ │ -0026a760: 7472 696e 6720 6f66 2074 6865 2066 6f72 tring of the for │ │ │ │ -0026a770: 6d20 2221 636d 6422 3a20 7468 6520 636f m "!cmd": the co │ │ │ │ -0026a780: 6d6d 616e 6420 636d 6420 7769 6c6c 2062 mmand cmd will b │ │ │ │ -0026a790: 6520 7374 6172 7465 642c 2061 6e64 2074 e started, and t │ │ │ │ -0026a7a0: 776f 2070 6970 6573 0a20 2020 2077 696c wo pipes. wil │ │ │ │ -0026a7b0: 6c20 6265 206f 7065 6e65 642c 2063 6f6e l be opened, con │ │ │ │ -0026a7c0: 6e65 6374 6564 2074 6f20 6974 7320 7374 nected to its st │ │ │ │ -0026a7d0: 616e 6461 7264 2069 6e70 7574 2061 6e64 andard input and │ │ │ │ -0026a7e0: 2073 7461 6e64 6172 6420 6f75 7470 7574 standard output │ │ │ │ -0026a7f0: 2066 696c 650a 2020 2020 6465 7363 7269 file. descri │ │ │ │ -0026a800: 7074 6f72 732e 2057 6172 6e69 6e67 3a20 ptors. Warning: │ │ │ │ -0026a810: 7069 7065 7320 686f 6c64 206f 6e6c 7920 pipes hold only │ │ │ │ -0026a820: 3430 3936 2062 6974 6573 2c20 736f 2069 4096 bites, so i │ │ │ │ -0026a830: 6620 796f 7520 7772 6974 6520 6d6f 7265 f you write more │ │ │ │ -0026a840: 2074 6861 6e0a 2020 2020 7468 6174 2074 than. that t │ │ │ │ -0026a850: 6f20 7468 6520 7265 7375 6c74 696e 6720 o the resulting │ │ │ │ -0026a860: 696e 7075 7420 6f75 7470 7574 2066 696c input output fil │ │ │ │ -0026a870: 6520 2861 7320 696e 7075 7420 666f 7220 e (as input for │ │ │ │ -0026a880: 7468 6520 636f 6d6d 616e 6429 2077 6974 the command) wit │ │ │ │ -0026a890: 686f 7574 0a20 2020 2072 6561 6469 6e67 hout. reading │ │ │ │ -0026a8a0: 2061 6e79 2064 6174 612c 2079 6f75 206d any data, you m │ │ │ │ -0026a8b0: 6179 2062 6c6f 636b 2077 6869 6c65 2074 ay block while t │ │ │ │ -0026a8c0: 6865 2063 6f6d 6d61 6e64 2069 7320 626c he command is bl │ │ │ │ -0026a8d0: 6f63 6b65 6420 7761 6974 696e 6720 746f ocked waiting to │ │ │ │ -0026a8e0: 0a20 2020 2077 7269 7465 206d 6f72 6520 . write more │ │ │ │ -0026a8f0: 6f75 7470 7574 3b20 696e 2074 6869 7320 output; in this │ │ │ │ -0026a900: 6361 7365 2c20 4d61 6361 756c 6179 3220 case, Macaulay2 │ │ │ │ -0026a910: 7769 6c6c 2061 7070 6561 7220 746f 2068 will appear to h │ │ │ │ -0026a920: 616e 672e 0a20 202a 2061 2073 7472 696e ang.. * a strin │ │ │ │ -0026a930: 6720 6f66 2074 6865 2066 6f72 6d20 2224 g of the form "$ │ │ │ │ -0026a940: 686f 7374 6e61 6d65 3a73 6572 7669 6365 hostname:service │ │ │ │ -0026a950: 223a 2061 2063 6f6e 6e65 6374 696f 6e20 ": a connection │ │ │ │ -0026a960: 7769 6c6c 2062 6520 6d61 6465 2074 6f20 will be made to │ │ │ │ -0026a970: 7468 650a 2020 2020 7370 6563 6966 6965 the. specifie │ │ │ │ -0026a980: 6420 7365 7276 6963 6520 6174 2074 6865 d service at the │ │ │ │ -0026a990: 2073 7065 6369 6669 6564 2068 6f73 742e specified host. │ │ │ │ -0026a9a0: 2020 4966 2074 6865 2073 6572 7669 6365 If the service │ │ │ │ -0026a9b0: 2070 6f72 7420 6973 206f 6d69 7474 6564 port is omitted │ │ │ │ -0026a9c0: 2c0a 2020 2020 616c 6f6e 6720 7769 7468 ,. along with │ │ │ │ -0026a9d0: 2074 6865 2063 6f6c 6f6e 2c20 7468 656e the colon, then │ │ │ │ -0026a9e0: 2070 6f72 7420 3235 3030 2069 7320 7573 port 2500 is us │ │ │ │ -0026a9f0: 6564 2e20 2049 6620 7468 6520 686f 7374 ed. If the host │ │ │ │ -0026aa00: 6e61 6d65 2069 7320 6f6d 6974 7465 642c name is omitted, │ │ │ │ -0026aa10: 0a20 2020 2061 6e20 696e 636f 6d69 6e67 . an incoming │ │ │ │ -0026aa20: 2063 6f6e 6e65 6374 696f 6e20 7769 6c6c connection will │ │ │ │ -0026aa30: 2062 6520 7761 6974 6564 2066 6f72 2e0a be waited for.. │ │ │ │ -0026aa40: 2020 2a20 6120 6c69 7374 656e 6572 2063 * a listener c │ │ │ │ -0026aa50: 7265 6174 6564 2070 7265 7669 6f75 736c reated previousl │ │ │ │ -0026aa60: 7920 6279 202a 6e6f 7465 206f 7065 6e4c y by *note openL │ │ │ │ -0026aa70: 6973 7465 6e65 723a 0a20 2020 206f 7065 istener:. ope │ │ │ │ -0026aa80: 6e4c 6973 7465 6e65 725f 6c70 5374 7269 nListener_lpStri │ │ │ │ -0026aa90: 6e67 5f72 702c 3a20 6120 6e65 7720 636f ng_rp,: a new co │ │ │ │ -0026aaa0: 6e6e 6563 7469 6f6e 2077 696c 6c20 6265 nnection will be │ │ │ │ -0026aab0: 2063 7265 6174 6564 2e20 2054 6f20 6176 created. To av │ │ │ │ -0026aac0: 6f69 640a 2020 2020 626c 6f63 6b69 6e67 oid. blocking │ │ │ │ -0026aad0: 2074 6865 204d 6163 6175 6c61 7932 2070 the Macaulay2 p │ │ │ │ -0026aae0: 726f 6365 7373 2077 6869 6c65 2077 6169 rocess while wai │ │ │ │ -0026aaf0: 7469 6e67 2066 6f72 2074 6865 2069 6e63 ting for the inc │ │ │ │ -0026ab00: 6f6d 696e 6720 636f 6e6e 6563 7469 6f6e oming connection │ │ │ │ -0026ab10: 2c0a 2020 2020 7573 6520 2a6e 6f74 6520 ,. use *note │ │ │ │ -0026ab20: 6973 5265 6164 793a 2069 7352 6561 6479 isReady: isReady │ │ │ │ -0026ab30: 5f6c 7046 696c 655f 7270 2c2e 0a0a 496e _lpFile_rp,...In │ │ │ │ -0026ab40: 206f 7264 6572 2074 6f20 6f70 656e 2061 order to open a │ │ │ │ -0026ab50: 2073 6f63 6b65 7420 7375 6363 6573 7366 socket successf │ │ │ │ -0026ab60: 756c 6c79 2c20 7468 6572 6520 6d75 7374 ully, there must │ │ │ │ -0026ab70: 2062 6520 6120 7072 6f63 6573 7320 6163 be a process ac │ │ │ │ -0026ab80: 6365 7074 696e 670a 636f 6e6e 6563 7469 cepting.connecti │ │ │ │ -0026ab90: 6f6e 7320 666f 7220 7468 6520 6465 7369 ons for the desi │ │ │ │ -0026aba0: 7265 6420 7365 7276 6963 6520 6f6e 2074 red service on t │ │ │ │ -0026abb0: 6865 2073 7065 6369 6669 6564 2068 6f73 he specified hos │ │ │ │ -0026abc0: 742e 0a0a 5468 6520 7661 7269 6f75 7320 t...The various │ │ │ │ -0026abd0: 666f 726d 7320 6c69 7374 6564 2061 626f forms listed abo │ │ │ │ -0026abe0: 7665 2063 616e 2062 6520 7573 6564 2061 ve can be used a │ │ │ │ -0026abf0: 6c73 6f20 7769 7468 2061 6c6c 206f 7468 lso with all oth │ │ │ │ -0026ac00: 6572 2069 6e70 7574 206f 7574 7075 740a er input output. │ │ │ │ -0026ac10: 6f70 6572 6174 696f 6e73 2074 6861 7420 operations that │ │ │ │ -0026ac20: 6f70 656e 2066 696c 6573 2c20 7375 6368 open files, such │ │ │ │ -0026ac30: 2061 7320 2a6e 6f74 6520 6f70 656e 496e as *note openIn │ │ │ │ -0026ac40: 3a20 6f70 656e 496e 5f6c 7053 7472 696e : openIn_lpStrin │ │ │ │ -0026ac50: 675f 7270 2c2c 202a 6e6f 7465 0a6f 7065 g_rp,, *note.ope │ │ │ │ -0026ac60: 6e4f 7574 3a20 6f70 656e 4f75 745f 6c70 nOut: openOut_lp │ │ │ │ -0026ac70: 5374 7269 6e67 5f72 702c 2c20 2a6e 6f74 String_rp,, *not │ │ │ │ -0026ac80: 6520 6765 743a 2067 6574 2c2c 2061 6e64 e get: get,, and │ │ │ │ -0026ac90: 202a 6e6f 7465 203c 3c3a 203c 3c2c 2c20 *note <<: <<,, │ │ │ │ -0026aca0: 7769 7468 2064 6174 610a 7472 616e 7366 with data.transf │ │ │ │ -0026acb0: 6572 2070 6f73 7369 626c 6520 6f6e 6c79 er possible only │ │ │ │ -0026acc0: 2069 6e20 7468 6520 6469 7265 6374 696f in the directio │ │ │ │ -0026acd0: 6e20 7370 6563 6966 6965 642e 2020 4120 n specified. A │ │ │ │ -0026ace0: 706f 7373 6962 6c79 2063 6f6e 6675 7369 possibly confusi │ │ │ │ -0026acf0: 6e67 0a61 7379 6d6d 6574 7279 2069 7320 ng.asymmetry is │ │ │ │ -0026ad00: 7468 6174 2077 6974 6820 6f70 656e 496e that with openIn │ │ │ │ -0026ad10: 2022 2166 6f6f 2220 6f72 2077 6974 6820 "!foo" or with │ │ │ │ -0026ad20: 6765 7420 2221 666f 6f22 2074 6865 2073 get "!foo" the s │ │ │ │ -0026ad30: 7461 6e64 6172 6420 696e 7075 7420 6f66 tandard input of │ │ │ │ -0026ad40: 0a74 6865 2063 6f6d 6d61 6e64 2066 6f6f .the command foo │ │ │ │ -0026ad50: 2069 7320 636c 6f73 6564 2c20 6275 7420 is closed, but │ │ │ │ -0026ad60: 7769 7468 206f 7065 6e4f 7574 2022 2166 with openOut "!f │ │ │ │ -0026ad70: 6f6f 2220 7468 6520 7374 616e 6461 7264 oo" the standard │ │ │ │ -0026ad80: 206f 7574 7075 7420 6f66 2074 6865 0a63 output of the.c │ │ │ │ -0026ad90: 6f6d 6d61 6e64 2066 6f6f 2069 7320 636f ommand foo is co │ │ │ │ -0026ada0: 6e6e 6563 7465 6420 746f 2074 6865 2073 nnected to the s │ │ │ │ -0026adb0: 7461 6e64 6172 6420 6f75 7470 7574 206f tandard output o │ │ │ │ -0026adc0: 6620 7468 6520 7061 7265 6e74 204d 6163 f the parent Mac │ │ │ │ -0026add0: 6175 6c61 7932 0a70 726f 6365 7373 2e0a aulay2.process.. │ │ │ │ -0026ade0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ -0026adf0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6f70 ==.. * *note op │ │ │ │ -0026ae00: 656e 496e 3a20 6f70 656e 496e 5f6c 7053 enIn: openIn_lpS │ │ │ │ -0026ae10: 7472 696e 675f 7270 2c20 2d2d 206f 7065 tring_rp, -- ope │ │ │ │ -0026ae20: 6e20 616e 2069 6e70 7574 2066 696c 650a n an input file. │ │ │ │ -0026ae30: 2020 2a20 2a6e 6f74 6520 6f70 656e 4f75 * *note openOu │ │ │ │ -0026ae40: 743a 206f 7065 6e4f 7574 5f6c 7053 7472 t: openOut_lpStr │ │ │ │ -0026ae50: 696e 675f 7270 2c20 2d2d 206f 7065 6e20 ing_rp, -- open │ │ │ │ -0026ae60: 616e 206f 7574 7075 7420 6669 6c65 0a20 an output file. │ │ │ │ -0026ae70: 202a 202a 6e6f 7465 206f 7065 6e4c 6973 * *note openLis │ │ │ │ -0026ae80: 7465 6e65 723a 206f 7065 6e4c 6973 7465 tener: openListe │ │ │ │ -0026ae90: 6e65 725f 6c70 5374 7269 6e67 5f72 702c ner_lpString_rp, │ │ │ │ -0026aea0: 202d 2d20 6f70 656e 2061 2070 6f72 7420 -- open a port │ │ │ │ -0026aeb0: 666f 7220 6c69 7374 656e 696e 670a 0a57 for listening..W │ │ │ │ -0026aec0: 6179 7320 746f 2075 7365 206f 7065 6e49 ays to use openI │ │ │ │ -0026aed0: 6e4f 7574 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d nOut:.========== │ │ │ │ -0026aee0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ -0026aef0: 2a20 226f 7065 6e49 6e4f 7574 2846 696c * "openInOut(Fil │ │ │ │ -0026af00: 6529 220a 2020 2a20 226f 7065 6e49 6e4f e)". * "openInO │ │ │ │ -0026af10: 7574 2853 7472 696e 6729 220a 0a46 6f72 ut(String)"..For │ │ │ │ -0026af20: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -0026af30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0026af40: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -0026af50: 6e6f 7465 206f 7065 6e49 6e4f 7574 3a20 note openInOut: │ │ │ │ -0026af60: 6f70 656e 496e 4f75 742c 2069 7320 6120 openInOut, is a │ │ │ │ -0026af70: 2a6e 6f74 6520 636f 6d70 696c 6564 2066 *note compiled f │ │ │ │ -0026af80: 756e 6374 696f 6e3a 0a43 6f6d 7069 6c65 unction:.Compile │ │ │ │ -0026af90: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d dFunction,...--- │ │ │ │ -0026afa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026afb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026afc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a450: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +0026a460: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +0026a470: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +0026a480: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +0026a490: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +0026a4a0: 362e 3035 2b64 732f 4d32 2f4d 6163 6175 6.05+ds/M2/Macau │ │ │ │ +0026a4b0: 6c61 7932 2f70 6163 6b61 6765 732f 0a4d lay2/packages/.M │ │ │ │ +0026a4c0: 6163 6175 6c61 7932 446f 632f 6f76 5f73 acaulay2Doc/ov_s │ │ │ │ +0026a4d0: 7973 7465 6d2e 6d32 3a34 3434 3a30 2e0a ystem.m2:444:0.. │ │ │ │ +0026a4e0: 1f0a 4669 6c65 3a20 4d61 6361 756c 6179 ..File: Macaulay │ │ │ │ +0026a4f0: 3244 6f63 2e69 6e66 6f2c 204e 6f64 653a 2Doc.info, Node: │ │ │ │ +0026a500: 206f 7065 6e49 6e4f 7574 2c20 4e65 7874 openInOut, Next │ │ │ │ +0026a510: 3a20 6f70 656e 4f75 745f 6c70 5374 7269 : openOut_lpStri │ │ │ │ +0026a520: 6e67 5f72 702c 2050 7265 763a 206f 7065 ng_rp, Prev: ope │ │ │ │ +0026a530: 6e49 6e5f 6c70 5374 7269 6e67 5f72 702c nIn_lpString_rp, │ │ │ │ +0026a540: 2055 703a 2075 7369 6e67 2073 6f63 6b65 Up: using socke │ │ │ │ +0026a550: 7473 0a0a 6f70 656e 496e 4f75 7420 2d2d ts..openInOut -- │ │ │ │ +0026a560: 206f 7065 6e20 616e 2069 6e70 7574 206f open an input o │ │ │ │ +0026a570: 7574 7075 7420 6669 6c65 0a2a 2a2a 2a2a utput file.***** │ │ │ │ +0026a580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026a590: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026a5a0: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +0026a5b0: 2020 2020 2020 206f 7065 6e49 6e4f 7574 openInOut │ │ │ │ +0026a5c0: 2066 0a20 202a 2049 6e70 7574 733a 0a20 f. * Inputs:. │ │ │ │ +0026a5d0: 2020 2020 202a 2066 2c20 6120 2a6e 6f74 * f, a *not │ │ │ │ +0026a5e0: 6520 7374 7269 6e67 3a20 5374 7269 6e67 e string: String │ │ │ │ +0026a5f0: 2c20 6f72 2061 202a 6e6f 7465 2066 696c , or a *note fil │ │ │ │ +0026a600: 653a 2046 696c 652c 0a20 202a 204f 7574 e: File,. * Out │ │ │ │ +0026a610: 7075 7473 3a0a 2020 2020 2020 2a20 616e puts:. * an │ │ │ │ +0026a620: 206f 7065 6e20 696e 7075 7420 6f75 7470 open input outp │ │ │ │ +0026a630: 7574 2066 696c 650a 0a44 6573 6372 6970 ut file..Descrip │ │ │ │ +0026a640: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +0026a650: 0a0a 5468 6572 6520 6172 6520 7661 7269 ..There are vari │ │ │ │ +0026a660: 6f75 7320 6f70 7469 6f6e 7320 666f 7220 ous options for │ │ │ │ +0026a670: 7468 6520 6172 6775 6d65 6e74 2066 2e0a the argument f.. │ │ │ │ +0026a680: 0a20 202a 2061 2073 7472 696e 6720 6e6f . * a string no │ │ │ │ +0026a690: 7420 7374 6172 7469 6e67 2077 6974 6820 t starting with │ │ │ │ +0026a6a0: 2120 6f72 2024 3a20 7468 6520 7374 7269 ! or $: the stri │ │ │ │ +0026a6b0: 6e67 2069 7320 7461 6b65 6e20 6173 2074 ng is taken as t │ │ │ │ +0026a6c0: 6865 206e 616d 6520 6f66 2061 6e0a 2020 he name of an. │ │ │ │ +0026a6d0: 2020 696e 7075 7420 6f75 7470 7574 2066 input output f │ │ │ │ +0026a6e0: 696c 6520 746f 206f 7065 6e2e 2020 466f ile to open. Fo │ │ │ │ +0026a6f0: 7220 6578 616d 706c 652c 2069 6e20 556e r example, in Un │ │ │ │ +0026a700: 6978 2c20 6974 206d 6967 6874 2062 6520 ix, it might be │ │ │ │ +0026a710: 6120 6e61 6d65 6420 7069 7065 2e0a 2020 a named pipe.. │ │ │ │ +0026a720: 2020 4120 6669 6c65 6e61 6d65 2073 7461 A filename sta │ │ │ │ +0026a730: 7274 696e 6720 7769 7468 207e 2f20 7769 rting with ~/ wi │ │ │ │ +0026a740: 6c6c 2068 6176 6520 7468 6520 7469 6c64 ll have the tild │ │ │ │ +0026a750: 6520 7265 706c 6163 6564 2062 7920 7468 e replaced by th │ │ │ │ +0026a760: 6520 7573 6572 2773 2068 6f6d 650a 2020 e user's home. │ │ │ │ +0026a770: 2020 6469 7265 6374 6f72 792e 0a20 202a directory.. * │ │ │ │ +0026a780: 2061 2073 7472 696e 6720 6f66 2074 6865 a string of the │ │ │ │ +0026a790: 2066 6f72 6d20 2221 636d 6422 3a20 7468 form "!cmd": th │ │ │ │ +0026a7a0: 6520 636f 6d6d 616e 6420 636d 6420 7769 e command cmd wi │ │ │ │ +0026a7b0: 6c6c 2062 6520 7374 6172 7465 642c 2061 ll be started, a │ │ │ │ +0026a7c0: 6e64 2074 776f 2070 6970 6573 0a20 2020 nd two pipes. │ │ │ │ +0026a7d0: 2077 696c 6c20 6265 206f 7065 6e65 642c will be opened, │ │ │ │ +0026a7e0: 2063 6f6e 6e65 6374 6564 2074 6f20 6974 connected to it │ │ │ │ +0026a7f0: 7320 7374 616e 6461 7264 2069 6e70 7574 s standard input │ │ │ │ +0026a800: 2061 6e64 2073 7461 6e64 6172 6420 6f75 and standard ou │ │ │ │ +0026a810: 7470 7574 2066 696c 650a 2020 2020 6465 tput file. de │ │ │ │ +0026a820: 7363 7269 7074 6f72 732e 2057 6172 6e69 scriptors. Warni │ │ │ │ +0026a830: 6e67 3a20 7069 7065 7320 686f 6c64 206f ng: pipes hold o │ │ │ │ +0026a840: 6e6c 7920 3430 3936 2062 6974 6573 2c20 nly 4096 bites, │ │ │ │ +0026a850: 736f 2069 6620 796f 7520 7772 6974 6520 so if you write │ │ │ │ +0026a860: 6d6f 7265 2074 6861 6e0a 2020 2020 7468 more than. th │ │ │ │ +0026a870: 6174 2074 6f20 7468 6520 7265 7375 6c74 at to the result │ │ │ │ +0026a880: 696e 6720 696e 7075 7420 6f75 7470 7574 ing input output │ │ │ │ +0026a890: 2066 696c 6520 2861 7320 696e 7075 7420 file (as input │ │ │ │ +0026a8a0: 666f 7220 7468 6520 636f 6d6d 616e 6429 for the command) │ │ │ │ +0026a8b0: 2077 6974 686f 7574 0a20 2020 2072 6561 without. rea │ │ │ │ +0026a8c0: 6469 6e67 2061 6e79 2064 6174 612c 2079 ding any data, y │ │ │ │ +0026a8d0: 6f75 206d 6179 2062 6c6f 636b 2077 6869 ou may block whi │ │ │ │ +0026a8e0: 6c65 2074 6865 2063 6f6d 6d61 6e64 2069 le the command i │ │ │ │ +0026a8f0: 7320 626c 6f63 6b65 6420 7761 6974 696e s blocked waitin │ │ │ │ +0026a900: 6720 746f 0a20 2020 2077 7269 7465 206d g to. write m │ │ │ │ +0026a910: 6f72 6520 6f75 7470 7574 3b20 696e 2074 ore output; in t │ │ │ │ +0026a920: 6869 7320 6361 7365 2c20 4d61 6361 756c his case, Macaul │ │ │ │ +0026a930: 6179 3220 7769 6c6c 2061 7070 6561 7220 ay2 will appear │ │ │ │ +0026a940: 746f 2068 616e 672e 0a20 202a 2061 2073 to hang.. * a s │ │ │ │ +0026a950: 7472 696e 6720 6f66 2074 6865 2066 6f72 tring of the for │ │ │ │ +0026a960: 6d20 2224 686f 7374 6e61 6d65 3a73 6572 m "$hostname:ser │ │ │ │ +0026a970: 7669 6365 223a 2061 2063 6f6e 6e65 6374 vice": a connect │ │ │ │ +0026a980: 696f 6e20 7769 6c6c 2062 6520 6d61 6465 ion will be made │ │ │ │ +0026a990: 2074 6f20 7468 650a 2020 2020 7370 6563 to the. spec │ │ │ │ +0026a9a0: 6966 6965 6420 7365 7276 6963 6520 6174 ified service at │ │ │ │ +0026a9b0: 2074 6865 2073 7065 6369 6669 6564 2068 the specified h │ │ │ │ +0026a9c0: 6f73 742e 2020 4966 2074 6865 2073 6572 ost. If the ser │ │ │ │ +0026a9d0: 7669 6365 2070 6f72 7420 6973 206f 6d69 vice port is omi │ │ │ │ +0026a9e0: 7474 6564 2c0a 2020 2020 616c 6f6e 6720 tted,. along │ │ │ │ +0026a9f0: 7769 7468 2074 6865 2063 6f6c 6f6e 2c20 with the colon, │ │ │ │ +0026aa00: 7468 656e 2070 6f72 7420 3235 3030 2069 then port 2500 i │ │ │ │ +0026aa10: 7320 7573 6564 2e20 2049 6620 7468 6520 s used. If the │ │ │ │ +0026aa20: 686f 7374 6e61 6d65 2069 7320 6f6d 6974 hostname is omit │ │ │ │ +0026aa30: 7465 642c 0a20 2020 2061 6e20 696e 636f ted,. an inco │ │ │ │ +0026aa40: 6d69 6e67 2063 6f6e 6e65 6374 696f 6e20 ming connection │ │ │ │ +0026aa50: 7769 6c6c 2062 6520 7761 6974 6564 2066 will be waited f │ │ │ │ +0026aa60: 6f72 2e0a 2020 2a20 6120 6c69 7374 656e or.. * a listen │ │ │ │ +0026aa70: 6572 2063 7265 6174 6564 2070 7265 7669 er created previ │ │ │ │ +0026aa80: 6f75 736c 7920 6279 202a 6e6f 7465 206f ously by *note o │ │ │ │ +0026aa90: 7065 6e4c 6973 7465 6e65 723a 0a20 2020 penListener:. │ │ │ │ +0026aaa0: 206f 7065 6e4c 6973 7465 6e65 725f 6c70 openListener_lp │ │ │ │ +0026aab0: 5374 7269 6e67 5f72 702c 3a20 6120 6e65 String_rp,: a ne │ │ │ │ +0026aac0: 7720 636f 6e6e 6563 7469 6f6e 2077 696c w connection wil │ │ │ │ +0026aad0: 6c20 6265 2063 7265 6174 6564 2e20 2054 l be created. T │ │ │ │ +0026aae0: 6f20 6176 6f69 640a 2020 2020 626c 6f63 o avoid. bloc │ │ │ │ +0026aaf0: 6b69 6e67 2074 6865 204d 6163 6175 6c61 king the Macaula │ │ │ │ +0026ab00: 7932 2070 726f 6365 7373 2077 6869 6c65 y2 process while │ │ │ │ +0026ab10: 2077 6169 7469 6e67 2066 6f72 2074 6865 waiting for the │ │ │ │ +0026ab20: 2069 6e63 6f6d 696e 6720 636f 6e6e 6563 incoming connec │ │ │ │ +0026ab30: 7469 6f6e 2c0a 2020 2020 7573 6520 2a6e tion,. use *n │ │ │ │ +0026ab40: 6f74 6520 6973 5265 6164 793a 2069 7352 ote isReady: isR │ │ │ │ +0026ab50: 6561 6479 5f6c 7046 696c 655f 7270 2c2e eady_lpFile_rp,. │ │ │ │ +0026ab60: 0a0a 496e 206f 7264 6572 2074 6f20 6f70 ..In order to op │ │ │ │ +0026ab70: 656e 2061 2073 6f63 6b65 7420 7375 6363 en a socket succ │ │ │ │ +0026ab80: 6573 7366 756c 6c79 2c20 7468 6572 6520 essfully, there │ │ │ │ +0026ab90: 6d75 7374 2062 6520 6120 7072 6f63 6573 must be a proces │ │ │ │ +0026aba0: 7320 6163 6365 7074 696e 670a 636f 6e6e s accepting.conn │ │ │ │ +0026abb0: 6563 7469 6f6e 7320 666f 7220 7468 6520 ections for the │ │ │ │ +0026abc0: 6465 7369 7265 6420 7365 7276 6963 6520 desired service │ │ │ │ +0026abd0: 6f6e 2074 6865 2073 7065 6369 6669 6564 on the specified │ │ │ │ +0026abe0: 2068 6f73 742e 0a0a 5468 6520 7661 7269 host...The vari │ │ │ │ +0026abf0: 6f75 7320 666f 726d 7320 6c69 7374 6564 ous forms listed │ │ │ │ +0026ac00: 2061 626f 7665 2063 616e 2062 6520 7573 above can be us │ │ │ │ +0026ac10: 6564 2061 6c73 6f20 7769 7468 2061 6c6c ed also with all │ │ │ │ +0026ac20: 206f 7468 6572 2069 6e70 7574 206f 7574 other input out │ │ │ │ +0026ac30: 7075 740a 6f70 6572 6174 696f 6e73 2074 put.operations t │ │ │ │ +0026ac40: 6861 7420 6f70 656e 2066 696c 6573 2c20 hat open files, │ │ │ │ +0026ac50: 7375 6368 2061 7320 2a6e 6f74 6520 6f70 such as *note op │ │ │ │ +0026ac60: 656e 496e 3a20 6f70 656e 496e 5f6c 7053 enIn: openIn_lpS │ │ │ │ +0026ac70: 7472 696e 675f 7270 2c2c 202a 6e6f 7465 tring_rp,, *note │ │ │ │ +0026ac80: 0a6f 7065 6e4f 7574 3a20 6f70 656e 4f75 .openOut: openOu │ │ │ │ +0026ac90: 745f 6c70 5374 7269 6e67 5f72 702c 2c20 t_lpString_rp,, │ │ │ │ +0026aca0: 2a6e 6f74 6520 6765 743a 2067 6574 2c2c *note get: get,, │ │ │ │ +0026acb0: 2061 6e64 202a 6e6f 7465 203c 3c3a 203c and *note <<: < │ │ │ │ +0026acc0: 3c2c 2c20 7769 7468 2064 6174 610a 7472 <,, with data.tr │ │ │ │ +0026acd0: 616e 7366 6572 2070 6f73 7369 626c 6520 ansfer possible │ │ │ │ +0026ace0: 6f6e 6c79 2069 6e20 7468 6520 6469 7265 only in the dire │ │ │ │ +0026acf0: 6374 696f 6e20 7370 6563 6966 6965 642e ction specified. │ │ │ │ +0026ad00: 2020 4120 706f 7373 6962 6c79 2063 6f6e A possibly con │ │ │ │ +0026ad10: 6675 7369 6e67 0a61 7379 6d6d 6574 7279 fusing.asymmetry │ │ │ │ +0026ad20: 2069 7320 7468 6174 2077 6974 6820 6f70 is that with op │ │ │ │ +0026ad30: 656e 496e 2022 2166 6f6f 2220 6f72 2077 enIn "!foo" or w │ │ │ │ +0026ad40: 6974 6820 6765 7420 2221 666f 6f22 2074 ith get "!foo" t │ │ │ │ +0026ad50: 6865 2073 7461 6e64 6172 6420 696e 7075 he standard inpu │ │ │ │ +0026ad60: 7420 6f66 0a74 6865 2063 6f6d 6d61 6e64 t of.the command │ │ │ │ +0026ad70: 2066 6f6f 2069 7320 636c 6f73 6564 2c20 foo is closed, │ │ │ │ +0026ad80: 6275 7420 7769 7468 206f 7065 6e4f 7574 but with openOut │ │ │ │ +0026ad90: 2022 2166 6f6f 2220 7468 6520 7374 616e "!foo" the stan │ │ │ │ +0026ada0: 6461 7264 206f 7574 7075 7420 6f66 2074 dard output of t │ │ │ │ +0026adb0: 6865 0a63 6f6d 6d61 6e64 2066 6f6f 2069 he.command foo i │ │ │ │ +0026adc0: 7320 636f 6e6e 6563 7465 6420 746f 2074 s connected to t │ │ │ │ +0026add0: 6865 2073 7461 6e64 6172 6420 6f75 7470 he standard outp │ │ │ │ +0026ade0: 7574 206f 6620 7468 6520 7061 7265 6e74 ut of the parent │ │ │ │ +0026adf0: 204d 6163 6175 6c61 7932 0a70 726f 6365 Macaulay2.proce │ │ │ │ +0026ae00: 7373 2e0a 0a53 6565 2061 6c73 6f0a 3d3d ss...See also.== │ │ │ │ +0026ae10: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ +0026ae20: 6520 6f70 656e 496e 3a20 6f70 656e 496e e openIn: openIn │ │ │ │ +0026ae30: 5f6c 7053 7472 696e 675f 7270 2c20 2d2d _lpString_rp, -- │ │ │ │ +0026ae40: 206f 7065 6e20 616e 2069 6e70 7574 2066 open an input f │ │ │ │ +0026ae50: 696c 650a 2020 2a20 2a6e 6f74 6520 6f70 ile. * *note op │ │ │ │ +0026ae60: 656e 4f75 743a 206f 7065 6e4f 7574 5f6c enOut: openOut_l │ │ │ │ +0026ae70: 7053 7472 696e 675f 7270 2c20 2d2d 206f pString_rp, -- o │ │ │ │ +0026ae80: 7065 6e20 616e 206f 7574 7075 7420 6669 pen an output fi │ │ │ │ +0026ae90: 6c65 0a20 202a 202a 6e6f 7465 206f 7065 le. * *note ope │ │ │ │ +0026aea0: 6e4c 6973 7465 6e65 723a 206f 7065 6e4c nListener: openL │ │ │ │ +0026aeb0: 6973 7465 6e65 725f 6c70 5374 7269 6e67 istener_lpString │ │ │ │ +0026aec0: 5f72 702c 202d 2d20 6f70 656e 2061 2070 _rp, -- open a p │ │ │ │ +0026aed0: 6f72 7420 666f 7220 6c69 7374 656e 696e ort for listenin │ │ │ │ +0026aee0: 670a 0a57 6179 7320 746f 2075 7365 206f g..Ways to use o │ │ │ │ +0026aef0: 7065 6e49 6e4f 7574 3a0a 3d3d 3d3d 3d3d penInOut:.====== │ │ │ │ +0026af00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0026af10: 0a0a 2020 2a20 226f 7065 6e49 6e4f 7574 .. * "openInOut │ │ │ │ +0026af20: 2846 696c 6529 220a 2020 2a20 226f 7065 (File)". * "ope │ │ │ │ +0026af30: 6e49 6e4f 7574 2853 7472 696e 6729 220a nInOut(String)". │ │ │ │ +0026af40: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0026af50: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0026af60: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0026af70: 6374 202a 6e6f 7465 206f 7065 6e49 6e4f ct *note openInO │ │ │ │ +0026af80: 7574 3a20 6f70 656e 496e 4f75 742c 2069 ut: openInOut, i │ │ │ │ +0026af90: 7320 6120 2a6e 6f74 6520 636f 6d70 696c s a *note compil │ │ │ │ +0026afa0: 6564 2066 756e 6374 696f 6e3a 0a43 6f6d ed function:.Com │ │ │ │ +0026afb0: 7069 6c65 6446 756e 6374 696f 6e2c 2e0a piledFunction,.. │ │ │ │ +0026afc0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 0026afd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026afe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -0026aff0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -0026b000: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -0026b010: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -0026b020: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -0026b030: 7932 2d31 2e32 362e 3035 2b64 732f 4d32 y2-1.26.05+ds/M2 │ │ │ │ -0026b040: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -0026b050: 6765 732f 0a4d 6163 6175 6c61 7932 446f ges/.Macaulay2Do │ │ │ │ -0026b060: 632f 6f76 5f73 7973 7465 6d2e 6d32 3a35 c/ov_system.m2:5 │ │ │ │ -0026b070: 3337 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 37:0....File: Ma │ │ │ │ -0026b080: 6361 756c 6179 3244 6f63 2e69 6e66 6f2c caulay2Doc.info, │ │ │ │ -0026b090: 204e 6f64 653a 206f 7065 6e4f 7574 5f6c Node: openOut_l │ │ │ │ -0026b0a0: 7053 7472 696e 675f 7270 2c20 4e65 7874 pString_rp, Next │ │ │ │ -0026b0b0: 3a20 6f70 656e 4f75 7441 7070 656e 645f : openOutAppend_ │ │ │ │ -0026b0c0: 6c70 5374 7269 6e67 5f72 702c 2050 7265 lpString_rp, Pre │ │ │ │ -0026b0d0: 763a 206f 7065 6e49 6e4f 7574 2c20 5570 v: openInOut, Up │ │ │ │ -0026b0e0: 3a20 7573 696e 6720 736f 636b 6574 730a : using sockets. │ │ │ │ -0026b0f0: 0a6f 7065 6e4f 7574 2853 7472 696e 6729 .openOut(String) │ │ │ │ -0026b100: 202d 2d20 6f70 656e 2061 6e20 6f75 7470 -- open an outp │ │ │ │ -0026b110: 7574 2066 696c 650a 2a2a 2a2a 2a2a 2a2a ut file.******** │ │ │ │ -0026b120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026b130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -0026b140: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e * Function: *n │ │ │ │ -0026b150: 6f74 6520 6f70 656e 4f75 743a 206f 7065 ote openOut: ope │ │ │ │ -0026b160: 6e4f 7574 5f6c 7053 7472 696e 675f 7270 nOut_lpString_rp │ │ │ │ -0026b170: 2c0a 2020 2a20 5573 6167 653a 200a 2020 ,. * Usage: . │ │ │ │ -0026b180: 2020 2020 2020 6f70 656e 4f75 7420 666e openOut fn │ │ │ │ -0026b190: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ -0026b1a0: 2020 202a 2066 6e2c 2061 202a 6e6f 7465 * fn, a *note │ │ │ │ -0026b1b0: 2073 7472 696e 673a 2053 7472 696e 672c string: String, │ │ │ │ -0026b1c0: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -0026b1d0: 2020 2020 2a20 6120 2a6e 6f74 6520 6669 * a *note fi │ │ │ │ -0026b1e0: 6c65 3a20 4669 6c65 2c2c 2061 6e20 6f70 le: File,, an op │ │ │ │ -0026b1f0: 656e 206f 7574 7075 7420 6669 6c65 2077 en output file w │ │ │ │ -0026b200: 686f 7365 2066 696c 656e 616d 6520 6973 hose filename is │ │ │ │ -0026b210: 2066 6e2e 0a20 2020 2020 2020 2046 696c fn.. Fil │ │ │ │ -0026b220: 656e 616d 6573 2073 7461 7274 696e 6720 enames starting │ │ │ │ -0026b230: 7769 7468 2021 206f 7220 7769 7468 2024 with ! or with $ │ │ │ │ -0026b240: 2061 7265 2074 7265 6174 6564 2073 7065 are treated spe │ │ │ │ -0026b250: 6369 616c 6c79 2c20 7365 6520 2a6e 6f74 cially, see *not │ │ │ │ -0026b260: 650a 2020 2020 2020 2020 6f70 656e 496e e. openIn │ │ │ │ -0026b270: 4f75 743a 206f 7065 6e49 6e4f 7574 2c2e Out: openInOut,. │ │ │ │ -0026b280: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -0026b290: 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d =========..+---- │ │ │ │ -0026b2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026b2b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -0026b2c0: 6720 3d20 6f70 656e 4f75 7420 2274 6573 g = openOut "tes │ │ │ │ -0026b2d0: 742d 6669 6c65 227c 0a7c 2020 2020 2020 t-file"|.| │ │ │ │ -0026b2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b2f0: 2020 2020 2020 7c0a 7c6f 3120 3d20 7465 |.|o1 = te │ │ │ │ -0026b300: 7374 2d66 696c 6520 2020 2020 2020 2020 st-file │ │ │ │ -0026b310: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0026b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b330: 2020 2020 7c0a 7c6f 3120 3a20 4669 6c65 |.|o1 : File │ │ │ │ +0026afe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026aff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026b000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026b010: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +0026b020: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +0026b030: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +0026b040: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +0026b050: 6175 6c61 7932 2d31 2e32 362e 3035 2b64 aulay2-1.26.05+d │ │ │ │ +0026b060: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +0026b070: 6163 6b61 6765 732f 0a4d 6163 6175 6c61 ackages/.Macaula │ │ │ │ +0026b080: 7932 446f 632f 6f76 5f73 7973 7465 6d2e y2Doc/ov_system. │ │ │ │ +0026b090: 6d32 3a35 3337 3a30 2e0a 1f0a 4669 6c65 m2:537:0....File │ │ │ │ +0026b0a0: 3a20 4d61 6361 756c 6179 3244 6f63 2e69 : Macaulay2Doc.i │ │ │ │ +0026b0b0: 6e66 6f2c 204e 6f64 653a 206f 7065 6e4f nfo, Node: openO │ │ │ │ +0026b0c0: 7574 5f6c 7053 7472 696e 675f 7270 2c20 ut_lpString_rp, │ │ │ │ +0026b0d0: 4e65 7874 3a20 6f70 656e 4f75 7441 7070 Next: openOutApp │ │ │ │ +0026b0e0: 656e 645f 6c70 5374 7269 6e67 5f72 702c end_lpString_rp, │ │ │ │ +0026b0f0: 2050 7265 763a 206f 7065 6e49 6e4f 7574 Prev: openInOut │ │ │ │ +0026b100: 2c20 5570 3a20 7573 696e 6720 736f 636b , Up: using sock │ │ │ │ +0026b110: 6574 730a 0a6f 7065 6e4f 7574 2853 7472 ets..openOut(Str │ │ │ │ +0026b120: 696e 6729 202d 2d20 6f70 656e 2061 6e20 ing) -- open an │ │ │ │ +0026b130: 6f75 7470 7574 2066 696c 650a 2a2a 2a2a output file.**** │ │ │ │ +0026b140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026b150: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026b160: 2a2a 0a0a 2020 2a20 4675 6e63 7469 6f6e **.. * Function │ │ │ │ +0026b170: 3a20 2a6e 6f74 6520 6f70 656e 4f75 743a : *note openOut: │ │ │ │ +0026b180: 206f 7065 6e4f 7574 5f6c 7053 7472 696e openOut_lpStrin │ │ │ │ +0026b190: 675f 7270 2c0a 2020 2a20 5573 6167 653a g_rp,. * Usage: │ │ │ │ +0026b1a0: 200a 2020 2020 2020 2020 6f70 656e 4f75 . openOu │ │ │ │ +0026b1b0: 7420 666e 0a20 202a 2049 6e70 7574 733a t fn. * Inputs: │ │ │ │ +0026b1c0: 0a20 2020 2020 202a 2066 6e2c 2061 202a . * fn, a * │ │ │ │ +0026b1d0: 6e6f 7465 2073 7472 696e 673a 2053 7472 note string: Str │ │ │ │ +0026b1e0: 696e 672c 0a20 202a 204f 7574 7075 7473 ing,. * Outputs │ │ │ │ +0026b1f0: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74 :. * a *not │ │ │ │ +0026b200: 6520 6669 6c65 3a20 4669 6c65 2c2c 2061 e file: File,, a │ │ │ │ +0026b210: 6e20 6f70 656e 206f 7574 7075 7420 6669 n open output fi │ │ │ │ +0026b220: 6c65 2077 686f 7365 2066 696c 656e 616d le whose filenam │ │ │ │ +0026b230: 6520 6973 2066 6e2e 0a20 2020 2020 TRUNCATED DUE TO SIZE LIMIT: 10485760 bytes